Classical and mixed multilayered plate/shell models for ... · This dissertation is the...

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POLITECNICO DI TORINO SCUOLA DI DOTTORATO Dottorato di Ricerca in Ingegneria Aerospaziale - XXI ciclo Doctorat en mècanique de l’Université Paris Ouest - Nanterre La Défense Ph.D. Dissertation Classical and mixed multilayered plate/shell models for multifield problems analysis S ALVATORE B RISCHETTO Tutors prof. Erasmo Carrera prof. Olivier Polit April 2009

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POLITECNICO DI TORINO

SCUOLA DI DOTTORATODottorato di Ricerca in Ingegneria Aerospaziale - XXI ciclo

Doctorat en mècanique de l’Université Paris Ouest - Nanterre LaDéfense

Ph.D. Dissertation

Classical and mixed multilayered plate/shellmodels for multifield problems analysis

SALVATORE BRISCHETTO

Tutorsprof. Erasmo Carrera

prof. Olivier Polit

April 2009

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a mia moglie

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Acknowledgements

This dissertation is the accomplishment of my research activity as a Ph.D. student in co-tutoring between the Dipartimento di Ingegneria Aeronautica e Spaziale of Politecnico di Torinoand the Laboratoire Energétique Mécanique Electromagnétisme of Université Paris Ouest -Nanterre La Défense. The present work benefited from the insights, guidance and support ofseveral people, who I would like to acknowledge here.

Above all, I am grateful to Prof. Erasmo Carrera, my supervisor at Politecnico di Torino, forthis great opportunity. In these years, the possibility of working at his side has been a privilegeand an honor for me. His suggestions and advise will be a precious teaching for my work andmy life.

I would like to express my gratitude to Prof. Olivier Polit, my supervisor at UniversitéParis Ouest - Nanterre La Défense, for his kind welcome in LEME and for his help and supportin the development of the new shell finite element implemented in this thesis. My period inLEME was very stimulating, and I wish to thank my colleagues in the laboratory for this, inparticular Dr. Michele D’Ottavio who made my period spent in Paris more lighthearted.

Last but not least, my deep gratitude also goes to former and current colleagues in my office,the famous "room 30", at Politecnico di Torino. From each one I have got a distinctive featureto improve myself. In particular, a warm thanks goes to Dr. Gaetano Giunta, a brotherly friend,with whom I have shared the most part of this experience, Dr. Alessandro Robaldo, who was anexample of efficiency and work method, and Simone for his help for the typographical design ofthis thesis.

Torino, April 2009 Salvatore Brischetto

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Summary

The main improvements in future aircraft and spacecraft could depend on an increasing use ofconventional and unconventional multilayered structures. Some of these structures have beenused for over three decades: carbon fiber reinforced laminates; sandwich structures with honey-comb or metallic foams as core-layers; layered ceramic-metallic structures employed as thermalprotection. New unconventional materials could be used in the near future: e.g. piezoelectricones, which are commonly used in the so-called smart structures and functionally graded mate-rials, which have a continuous variation of physical properties in a particular direction. Layersmade of such materials can be combined in different ways to obtain structures which are able tofulfill several structural requirements.

Aerospace vehicles are often exposed to high sun irradiation and thermal cycling. The re-lated structures are simultaneously loaded by high thermal and mechanical loads. If a networkof piezoelectric actuators and sensors are embedded in multilayered structures, a self-controllingand self-monitoring smart system is created. This new engineered class of materials has resultedin significant improvements in the performance of integrated systems, actuation technologies,shape control, vibration and acoustic control and condition monitoring. The described examplesclearly explain that most multilayered structures are subjected to different loadings: mechani-cal, thermal and/or electric loads. This fact leads to the definition of multifield problems.

In particular applications, the aforementioned structures appear as two-dimensional andare known as plates and shells. The advent of new materials in aerospace structures and theuse of multilayered configurations has led to a significant increase in the development of refinedtheories for the modelling of plates and shells. Classical two-dimensional models, which werefrequently used in the past, are inappropriate for the analysis of these new structures: theirmodelling involves complicated effects that are not considered in the hypotheses used in classicalmodels. To overcome these limitations, a new set of two-dimensional models, which employCarrera’s Unified Formulation, are presented. These models can have higher orders of expansionin the thickness direction for each considered multifield variable component, and permit twodifferent multilayered descriptions to be considered: - equivalent single layer models, where thelayers are seen as one equivalent plate; - layer wise, where each layer is considered separately.

Analytical and numerical models have been implemented in this work to study multifieldproblems for multilayered structures. The dissertation is organized in three main parts: - anextension of the geometrical relations and constitutive equations to multifield problems andmultilayered plates and shells; - the introduction of Carrera’s Unified Formualtion (CUF), theuse of variational statements, and their extension to multifield problems; - the results of severalmultifield couplings.

Constitutive equations for multifield problems are given in the first part and they are ex-

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tended to functionally graded materials by employing opportune thickness functions to describethe physical properties that continuously change in the thickness direction. The multifield con-stitutive equations, obtained in a generalized way by employing thermodynamic considerations,are discussed for several multifield couplings and rewritten opportunely for the case of mixedmodels. Geometrical relations for plates and shells are discussed, with particular attention totheir extension to multifield problems.

The second part is devoted to CUF, with the introduction of a general and unified mannerof describing the variables related to mechanical, thermal and electrical fields in multilayeredplates and shells. The proposed two-dimensional models are defined as refined or advanced,according to the considered variational statement: refined models are based on the principle ofvirtual displacements (PVD), advanced models employ Reissner’s mixed variational theorem(RMVT). Refined models for multifield problems are obtained simply by adding the thermaland electrical contributions to the well-known PVD for the pure mechanical case; in this case,the discussion on the several possible sub-cases is very intuitive. In some cases, it is necessaryto a priori model some variables which cannot be obtained correctly via post-processing (e.gtransverse shear/normal stresses and normal electric displacement). These variational state-ments, which are based on RMVT, are obtained by adding opportune Lagrange multipliers tothe principle of virtual displacements, and coherently rewriting the constitutive equations. Anexhaustive discussion on several sub-cases for RMVT applications is made: they are not sointuitive as in the PVD case.

The third part is about the results of the pure mechanical analysis, the thermo-mechanicalanalysis and the electro-mechanical coupling. For the pure mechanical analysis, both analyticaland finite element solutions are compared in order to note the importance of refined and ad-vanced models; a new finite element shell has been proposed in the case of homogeneous shells.The thermo-mechanical analysis is devoted in particular to the case of multilayered compositestructures and/or functionally graded layers: a calculated temperature profile or a full-couplingbetween mechanical and thermal field is fundamental in these for a correct structural investi-gation. The electro-mechanical analysis, in particular those for smart structures, recognizes theimportance of advanced models in order to a priori satisfy the interlaminar continuity of somemultifield transverse variables.

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Résumé

L’amélioration des futurs avions et vaisseaux spatiaux pourrait dépendre principalement del’utilisation de structures multicouches conventionnelles et non conventionnelles. Certainesde ces structures ont été utilisées depuis une trentaine d’années: multicouches renforcées pardes fibres de carbone; structures sandwich à coeur nid d’abeille ou mousse métallique; struc-tures multicouches céramique métallique utilisées comme protection thermique. De nouveauxmatériaux non conventionnels pourraient etre utilisés dans un futur proche: les piézoélec-triques utilisés dans les structures intelligentes et les matériaux à gradient fonctionnel qui sontcaractérisés par une variation continue de leurs propriétés physiques dans une direction. Lescouches obtenues à partir de ces matériaux peuvent etre combinées afin d’obtenir la structurequi répondra à un cahier des charges précis.

Les véhicules aérospatiaux sont exposés aux radiations solaires et à des cycles thermiques.Les structures qui les composent sont donc soumises à des sollicitations mécaniques et ther-miques. Si un réseau de capteurs et d’actionneurs piézoélectriques est intégré dans une struc-ture multicouche, un système intelligent de controle et de surveillance est ainsi obtenu. Cettenouvelle classe de matériau a permis une amélioration significative dans les performances de cessystèmes pour actionner des structures, controler les vibrations, le bruit et la santé structurale.Ces exemples permettent donc de montrer que les structures multicouches sont soumises à deschargements de différents types: mécanique, thermique et donc électrique. C’est ainsi que l’ondéfinit les problèmes multiphysiques ou multichamps.

Les structures identifiées ci-dessus sont principalement bidimensionnelles et appelées plaqueset coques. L’arrivée de ces nouveaux matériaux dans les structures aéronautiques et l’utilisationde différentes couches a amené au développement de théories raffinées pour la modélisation desplaques et des coques. Les modèles bidimensionnels classiques, fréquemment utilisés par lepassé, sont inadaptés pour l’analyse de ces nouvelles structures: leur modélisation nécessite laprise en compte d’un certain niveau de complexité, négligé dans ces modèles. Afin de surmonterces limitations, un nouvel ensemble de modèles, basé sur la Formulation Unifiée de Carrera (enanglais CUF) est présenté. Ces modèles peuvent etre d’un haut degré polynomial dans la direc-tion de l’épaisseur pour chaque fonction inconnue du champ multiphysique, permettant ainsi dedécrire le multicouche de deux façons: - modèle couche équivalente, où les couches sont traitéscomme une unique structure; - layerwise, où chaque couche est considérée individuellement.

Des approches analytiques et numériques ont été développées et implémentées dans ce tra-vail afin d’étudier les structures multicouches sous sollicitations multiphysiques. Le docu-ment est organisé en trois principales parties: - une extension des relations géométriques et deslois de comportement aux problèmes multiphysiques et aux plaques et coques multicouches; -l’introduction de la CUF, des principes variationnels, et leur extension aux problèmes multi-

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physiques; - des résultats pour différents problèmes multiphysiques couplés.Les lois de comportement pour les problèmes multiphysiques sont présentées dans la pre-

mière partie et étendues aux matériaux à gradient fonctionnel en utilisant des fonctions adap-tées pour décrire les variations dans l’épaisseur. Ces lois sont obtenues en utilisant les lois dela thermodynamique, sont ensuite étudiées pour différents couplages et réécrites dans le casparticulier des approches mixtes. Les relations géométriques des plaques et coques sont aussidiscutées dans le cas des problèmes multiphysiques.

La seconde partie est dédiée à la CUF, en utilisant une présentation générale et unifiée per-mettant de décrire les variables mécaniques, thermiques et électriques pour les plaques et lescoques. Les modèles proposés sont raffinés ou avancés, en fonction de la formulation variation-nelle: raffinés dans le cas du principe des puissances virtuelles (PVD, en anglais), avancés sion utilise le théorème variationnel mixte de Reissner (RMVT, en anglais). Les modèles raf-finés sont déduits directement du PVD mécanique en ajoutant les contributions thermique etélectrique, et on discutera assez intuitivement des configurations possible . Par contre, les mod-èles avancés utilisant donc RMVT, sont obtenus dans le cas multiphysique en introduisant desmutiplicateurs de Lagrange et en réécrivant correctement les lois de comportement. Une étudeassez précise de différents cas particuliers sera ensuite effectuée.

La troisième partie concerne les résultats dans le cas d’analyse mécanique, thermo-mécaniqueet électro-mécanique. Dans le cas mécanique, des solutions analytique et élément fini (EF) sontcomparées afin d’évaluer les modèles raffinés et avancés; un nouvel EF de coque homogèneest proposé. L’analyse thermo-mécanique est dédié aux structures multicouches et à gradientfonctionnel: un profil de température calculé et un couplage fort entre champ mécanique et ther-mique est nécessaire pour une analyse correcte. Enfin, l’étude du couplage électro-mécaniqueen considérant des structures intelligentes permet d’identifier l’importance des modèles avancésafin de satisfaire a priori les continuités interlaminaires de certaines grandeurs.

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Contents

1 Introduction 151.1 Multifield problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Multilayered structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 Aluminium and Titanium alloys . . . . . . . . . . . . . . . . . . . 191.2.2 Composite materials . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.3 Sandwiches: foam and honeycomb cores . . . . . . . . . . . . . . 281.2.4 Piezoelectric materials . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.5 Functionally graded materials . . . . . . . . . . . . . . . . . . . . 36

2 Constitutive and geometrical equations 412.1 Generalized Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.1 Transformation of material coefficients . . . . . . . . . . . . . . . 452.1.2 Poisson’s locking phenomena: plane stress constitutive relations 48

2.2 Constitutive equations for multifield problems . . . . . . . . . . . . . . . 502.2.1 Mechanical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.2 Electro-mechanical case . . . . . . . . . . . . . . . . . . . . . . . . 552.2.3 Thermo-mechanical case . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 Constitutive equations for mixed models . . . . . . . . . . . . . . . . . . 562.4 Constitutive equations for FGMs . . . . . . . . . . . . . . . . . . . . . . . 572.5 Geometrical relations for shells and plates . . . . . . . . . . . . . . . . . . 59

2.5.1 Shells: multifield geometrical relations . . . . . . . . . . . . . . . 642.5.2 Plates: multifield geometrical relations . . . . . . . . . . . . . . . 65

3 Two-dimensional plate/shell theories 673.1 Plate/shell theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.1.1 Three-dimensional problems . . . . . . . . . . . . . . . . . . . . . 683.1.2 Two-dimensional approaches . . . . . . . . . . . . . . . . . . . . . 68

3.2 Complicating effects of layered structures . . . . . . . . . . . . . . . . . . 713.3 Classical theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1 Classical lamination theory, CLT . . . . . . . . . . . . . . . . . . . 743.3.2 First order shear deformation theory, FSDT . . . . . . . . . . . . . 753.3.3 Vlasov-Reddy theory, VRT . . . . . . . . . . . . . . . . . . . . . . 76

3.4 Carrera’s unified formulation: refined models . . . . . . . . . . . . . . . 783.4.1 Equivalent single layer theories, ESL . . . . . . . . . . . . . . . . . 78

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3.4.2 Murakami’s zigzag function, MZZF . . . . . . . . . . . . . . . . . 803.4.3 Layer wise theories, LW . . . . . . . . . . . . . . . . . . . . . . . . 823.4.4 Refined models for the thermo-mechanical case . . . . . . . . . . 843.4.5 Refined models for the electro-mechanical case . . . . . . . . . . . 84

3.5 Carrera’s unified formulation: advanced models . . . . . . . . . . . . . . 843.5.1 Transverse shear/normal stresses modelling . . . . . . . . . . . . 853.5.2 Advanced models for the thermo-mechanical case . . . . . . . . . 863.5.3 Advanced models for the electro-mechanical case . . . . . . . . . 87

4 Variational statements for multifield problems 894.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Principle of virtual displacements, PVD . . . . . . . . . . . . . . . . . . . 90

4.2.1 PVD for the mechanical case . . . . . . . . . . . . . . . . . . . . . 924.2.2 PVD for the mechanical case with an external thermal load . . . . 934.2.3 PVD for the electro-mechanical case . . . . . . . . . . . . . . . . . 944.2.4 PVD for the thermo-mechanical case . . . . . . . . . . . . . . . . . 944.2.5 PVD for the thermo-electrical case . . . . . . . . . . . . . . . . . . 96

4.3 Reissner’s mixed variational theorem, RMVT . . . . . . . . . . . . . . . . 964.3.1 RMVT for the mechanical case . . . . . . . . . . . . . . . . . . . . 994.3.2 RMVT for the electro-mechanical case . . . . . . . . . . . . . . . . 1004.3.3 RMVT for the thermo-mechanical case . . . . . . . . . . . . . . . . 100

4.4 A general extension of RMVT . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4.1 RMVT1 for the thermo-electro-mechanical case . . . . . . . . . . 1024.4.2 RMVT2 for the thermo-electro-mechanical case . . . . . . . . . . 105

5 Differential equations and FE matrices for multifield problems 1115.1 PVD for the mechanical case, PVD-M . . . . . . . . . . . . . . . . . . . . 111

5.1.1 PVD-M: differential equations for plates and shells . . . . . . . . 1125.1.2 PVD-M: plate finite element . . . . . . . . . . . . . . . . . . . . . . 1165.1.3 PVD-M: shell finite element . . . . . . . . . . . . . . . . . . . . . . 119

5.2 PVD for the mechanical case with an external temperature load, PVD-M(T)1345.2.1 Assumed temperature profile, Ta . . . . . . . . . . . . . . . . . . . 1375.2.2 Calculated temperature profile, Tc . . . . . . . . . . . . . . . . . . 138

5.3 PVD for the electro-mechanical case, PVD-EM . . . . . . . . . . . . . . . 1425.4 PVD for the thermo-mechanical case, PVD-TM . . . . . . . . . . . . . . . 145

5.4.1 Imposed temperature on surfaces . . . . . . . . . . . . . . . . . . 1475.4.2 Mechanical load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.5 RMVT for the electro-mechanical case, RMVT-EM and RMVT-M . . . . . 1515.6 RMVT2 for the electro-mechanical case, RMVT2-EM . . . . . . . . . . . . 158

6 Mechanical analysis 1676.1 Preliminary assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.1.1 Quasi-3D analysis of composite and sandwich plates . . . . . . . 1676.1.2 Quasi-3D analysis of composite shells . . . . . . . . . . . . . . . . 170

6.2 Static and free-vibrations analysis of sandwich plates with soft core . . . 173

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6.3 Static analysis of sandwich shells with soft core . . . . . . . . . . . . . . . 1756.4 Functionally graded material plates . . . . . . . . . . . . . . . . . . . . . 1776.5 Sandwich plates with core in FGM . . . . . . . . . . . . . . . . . . . . . . 1806.6 Functionally graded material shells . . . . . . . . . . . . . . . . . . . . . . 1846.7 Sandwich shells with core in FGM . . . . . . . . . . . . . . . . . . . . . . 1866.8 Finite element analysis of shells . . . . . . . . . . . . . . . . . . . . . . . . 1876.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7 Thermo-mechanical analysis 1937.1 Thermal analysis of multilayered plates . . . . . . . . . . . . . . . . . . . 1937.2 Thermal analysis of multilayered shells . . . . . . . . . . . . . . . . . . . 1957.3 Thermal analysis of functionally graded material plates . . . . . . . . . . 1997.4 Thermal analysis of functionally graded material shells . . . . . . . . . . 2097.5 Thermo-mechanical coupling in homogeneous plates . . . . . . . . . . . 2127.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8 Electro-mechanical analysis 2258.1 Static analysis of piezoelectric plates . . . . . . . . . . . . . . . . . . . . . 2258.2 Static analysis of piezoelectric shells . . . . . . . . . . . . . . . . . . . . . 2288.3 Vibrations analysis of piezoelectric plates and shells . . . . . . . . . . . . 2408.4 Static analysis of plates including functionally graded piezoelectric layers 2458.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9 Conclusions and outlook 259

10 Conclusions et perspectives 263

Bibliography 266

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Chapter 1

Introduction

Multilayered plates and shells are two-dimensional structures obtained stacking layers untilthe desired thickness and stiffness are reached. Generally, the material differs from one layer toanother: homogeneous isotropic and orthotropic, fiber reinforced composites, piezoelectric andfunctionally graded materials can be taken into account. In aeronautics and space field suchstructures are subjected to several loadings: mechanical, thermal and electrical ones, this factleads to the definition of multifield problems. In the proposed analytical and numerical models,three physical fields are considered: mechanical, electrical and thermal field with the possibilityof interactions between them.

1.1 Multifield problems

The next generation of aircraft and spacecraft will be manufactured as multilayeredstructures under the action of a combination of two or more physical fields. In or-der to make a modelling of structures possible, without reference to subatomic di-mensions, four fields have been defined: mechanical, thermal, electrical and magneticfield. These all are based on measurable material properties (e.g., the Young’s mod-ulus for the mechanical case). Such properties describe the behavior of the materialin a suitable scale for engineering purposes [1]. Examples of multifield problems arethe sun irradiation over the wings of modern aircraft (thermo-mechanical problem);smart structures involving distributed actuators and sensors, and one or more micro-processors which analyze the responses from the sensors and use an integrated controltheory to command the actuators to apply localized strains/displacements to alter thesystem response. A smart structure has the capacity for respond to a changing externalenvironment (such as loads or shape change) as well as to a changing internal environ-ment (such as damage or failure) [2], [3]. Piezoelectric materials are the most popularsmart materials: they undergo deformation (strain) when an electric field is appliedacross them, and conversely produce voltage when a strain is applied, and thus can beused both as actuators and sensors [4]. Problems related to smart structures embed-ding piezoelectric materials are defined as electro-mechanical problems. The thermalstress analysis of smart structures represents an other interesting problem, an appli-cation is the use of piezoelectric layers embedded in multilayered shells and plates

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16 CHAPTER 1

Figure 1.1: Coupling between the three considered physical fields.

in order to control the thermal deformations [5], in this case a three field problem isinvestigated (thermo-electro-mechanical coupling). In smart structures considered inthis work, only piezoelectric layers are employed, an other possibility could be the useof piezomagnetic materials as done in [6] and [7], in these two papers refined [6] andadvanced [7] models are extended to the coupled magneto-electro-elastic analysis ofplates, for details about the magnetic field readers can refer to them. In the present dis-sertation three physical fields are considered: mechanical, electrical and thermal fieldand their possible interactions, see Figure 1.1. The possible interactions are:

• three fields problem: electro-thermo-mechanical coupling;

• two fields problem: thermo-mechanical coupling, electro-mechanical coupling,thermo-electrical coupling;

• one field problem: mechanical problem, thermal problem, electrical problem.

It is obvious that some problems result more interesting than others, for this reasononly some interactions are investigated, in analytical and numerical way, in the presentwork. In particular, results about the mechanical analysis, the thermo-mechanical anal-ysis and the electro-mechanical analysis are discussed in apposite chapters.

Multified problems grow out of different loading types acting on multilayered struc-tures and/or different physical properties considered in the layers. For example, apiezoelectric material exhibits an electric response even if only a mechanical load isapplied. For these reasons, an overview about multilayered structures and propertiesof their materials is given in the next section.

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INTRODUCTION 17

Figure 1.2: Example of a multilayered plate.

Figure 1.3: Examples of multilayered shells: cylindrical panel and cylindrical shell.

1.2 Multilayered structures

Multilayered structures considered in this work are two-dimensional elements embed-ding several layers with different mechanical, thermal and electrical properties. Astwo-dimensional structures we consider those with a dimension, usually the thick-ness, negligible with respect to the other two in the in-plane directions. Typical two-dimensional structures are plates and shells. Plates do not have any curvature alongthe two in-plane directions, they are flat panels as indicated in Figure 1.2. Shells aretwo-dimensional structures with curvature along the two in-plane directions. In thecase of plates, a rectilinear Cartesian reference system is employed as indicated in Fig-ure 1.2. In the case of shells, the introduction of a curvilinear Cartesian reference sys-tem is necessary as indicated in Figures 1.3 and 1.4. In both plate and shell cases, thethird axis in the thickness direction is always rectilinear. Several shell geometries canbe considered depending on the curvatures. When one of the two radii of curvatureare infinite, cylindrical panels or shells are considered. In the first case the structureis open, in the second one the dimension concerning the radius of curvature different

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18 CHAPTER 1

Figure 1.4: Example of a multilayered shell: spherical panel.

from infinite has value a = 2πRα or b = 2πRβ . When both radii of curvature are differ-ent from infinite, the considered shell or panel is defined as spherical. An example ofspherical panel is given in Figure 1.4.

Several materials are considered for layers embedded in multilayered structures. Afirst possibility are the homogeneous materials, typical homogeneous materials usedin aeronautics and space field are the aluminium and titanium alloys [8], they presenthigh strength-to-weight ratio and excellent mechanical properties. A natural develop-ment are composite materials, where two or more materials are combined on a macro-scopic scale in order to obtain better engineering properties than the conventional ma-terials (for example metals). Composite materials are commonly formed in three dif-ferent types [9]: (1) fibrous composites, which consist of fibers of one material in a matrixmaterial of another; (2) particulate composites, which are composed of macro size parti-cles of one material in a matrix of another; (3) laminated composites, which are made oflayers of different materials, including composites of the first two types. The particlesand matrix in particulate composites can be either metallic or non metallic. Other typi-cal aeronautics multilayered structures are the so-called sandwich structures. They areused to provide a stronger and stiffer structure for the same weight, or conversely alighter structure to carry the same load as a homogenous or compact-laminate flexu-ral member. These structures are constituted by two stiff skins (faces) and a soft core,and they are widely used to build large parts of aircraft, spacecraft, ship and automo-tive vehicle structures. Most of the recent applications have used skins constituted bylayered structures made of anisotropic composite materials. Several important issuesshould be considered in the design, analysis and construction of sandwich structuresand these have been fully discussed in the well-known books by Plantema [10], Allen[11], Zenkert [12], Bitzer [13] and Vinson [14] as well as in the handbook sections byMarshall [15] and Corden [16]. In the case of smart structures, some layers are in piezo-electric materials, they use the so-called piezoelectric effect which connects the electricaland mechanical fields [17]. The electroelastic state is defined as a linear problem. We

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INTRODUCTION 19

assume that: (a) displacements are small compared to the body thickness; (b) the de-formations, the mechanical stresses, and the electric field are directly proportional. Weneglect temperature and magnetic effects [18]. One of the most important innovationsof the present work is the possibility to consider the so-called Functionally Graded Ma-terials (FGMs) embedded in multilayered structures. These materials can be used toprovide the desired thermo-mechanical and piezoelectric properties, via the spatialvariation in their composition. FGMs vary the elastic, electric and thermal propertiesin the thickness direction via a gradually changing of the volume fraction of the con-stituents [19], [20]. One of the advantages of a monotonous variation of the volumefraction of constituent phases is the elimination of stress discontinuity, which is of-ten encountered in laminated composites and accordingly leads to the avoidance ofdelamination-related problems [21], [22].

The above proposed materials are discussed in depth in next sections.

1.2.1 Aluminium and Titanium alloys

Alloy is defined as a solid solution or homogeneous mixture of two or more elements,usually one of which is a metal. It usually has different properties from those of itscomponent elements. Alloying one metal with others often enhances its properties.The physical properties, such as Young’s modulus, density, reactivity, electrical andthermal conductivity, of an alloy are not so different from those of its elements, butengineering properties, such as tensile strength and shear strength may be substan-tially different from those of the constituent materials [23]. Alloys may exhibit markeddifferences in behavior even if small amounts of one element occur [24]. Some alloyscan be obtained by melting and mixing two or more metals, typical examples are brassand bronze. A possible classification of alloys can be made by the number of theirconstituents: binary alloy has two components, ternary alloy has three components,and so on. An other possible classification is made depending on their method of for-mation: substitution alloys where the atoms of the components have approximately thesame size and the various atoms are simply substituted for one another in the crystalstructure, interstitial alloys where the atoms of one component are substantially smallerthan the other and the smaller atoms fit into the spaces between the larger atoms. Thereare a large number of alloys, depending by their base element, for example aluminium,iron, nickel, titanium, zirconium alloys are very commonly. In this work, the most usedalloys are the aluminium ones and the titanium alloys, they are largely employed inaeronautics field.

Aluminium alloys

Aluminium and its alloys have been the prime material of construction for the aircraftindustry throughout most of its history. Today, even if titanium and composites aregrowing in use, 60% of commercial civil aircraft airframes are made from aluminiumalloys. Without aluminium, civil aviation would not be economically viable, becauseof acceptable cost, low component mass (derived from its low density), appropriate

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Property ValueAtomic number 13Atomic weight (g/mol) 26.98Valency 3Crystal Structure Face centered cubicMelting Point (C) 660.2Boiling Point (C) 2480Mean Specific Heat (0− 100C)(cal/g.C) 0.219Thermal Conductivity (0− 100C)(cal/cms.C) 0.57Coefficient of Linear Expansion (0− 100C)(10−6/C) 23.5Electrical Resistivity at 20C (µΩcm) 2.69Density (g/cm3) 2.6898Modulus of Elasticity (GPa) 68.3Poisson ratio 0.34

Table 1.1: Typical properties of aluminium [25].

mechanical properties, structural integrity. Thanks these properties, aluminium alloysare attractive in other areas of transport too. There are many examples of its use incommercial vehicles, rail cars (both passenger and freight), marine hulls, superstruc-tures and military vehicles. Table 1.1 indicates the properties for pure aluminium [25]:it is soft, ductile, corrosion resistant and has a high electrical conductivity. For thesereasons it is used for foil and conductor cables, but alloying with other elements pro-vides the higher strengths needed for other applications. The main alloying elementsare copper, zinc, magnesium, silicon, manganese and lithium. Other small additionscould be made: chromium, titanium, zirconium, lead, bismuth and nickel; iron is in-variably present in small quantities. There are over 300 wrought alloys. They arenormally identified by a four figure system which originated in USA and is now uni-versally accepted. Table 1.2 describes the system for wrought alloys. Cast alloys havesimilar designations but use a five digit system (see Table 1.2). In Table 1.3 a list ofthe designations, characteristics and common uses of some widely employed alloys isreported.

In multilayered structures, aluminium alloys are largely employed and stackedwith other materials.

Titanium alloys

Titanium and titanium alloys have been introduced in the early 1950s, and in a rel-atively short time they became very important in the aerospace, energy and chemicalindustries. This quick spreading is due to the combination of high strength-to-weightratio, excellent mechanical properties, and corrosion resistance. Titanium is consideredone of the best material choice for many critical applications: titanium alloys are usedfor demanding applications such as static and rotating gas turbine engine components;

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Major Alloying Element Wrought CastNone (99% + Aluminium) 1XXX 1XXX0Copper 2XXX 2XXX0Manganese 3XXXSilicon 4XXX 4XXX0Magnesium 5XXX 5XXX0Magnesium +Silicon 6XXX 6XXX0Zinc 7XXX 7XXX0Lithium 8XXXUnused 9XXX0

Table 1.2: Designations for alloyed wrought and cast aluminium alloys [25].

some of the most critical and highly-stressed civilian and military airframe parts aremade of these alloys [26], [27]. In recent years further applications are extended to nu-clear power plants, food processing plants, oil refinery heat exchangers, marine com-ponents and medical protheses. The high cost of titanium alloy components may limittheir use to applications for which lower-cost alloys, such as aluminium and stainlesssteels.

The physical properties of titanium and some titanium alloys are summarised inTable 1.4. The density of an alloy is dependent upon the amount and density of thealloying constituents. Where the weight is important, it may be worthwhile to comparespecific properties of alloys, e.g. the specific strength [28]. In order to remark thisaspect, in Table 1.5 the specific strengths of some titanium alloys are compared withthose of other structural elements: the specific properties of titanium alloys are betterthan those of classical structural metals. This peculiarity highly suggests the use oftitanium alloys in aeronautics and space fields. It is difficult to give a reliable value forPoisson’s ratio for titanium alloys since anisotropy leads to small differences in bothelastic and shear moduli which, when taken together to calculate Poisson’s ratio, canlead to values varying from 0.287 to 0.391. However, the generally accepted value forcommercially pure titanium is 0.36.

1.2.2 Composite materials

Composite materials consist of two or more combined materials which have desirableproperties that cannot be obtained with any of the constituents alone [9]. Typical exam-ples are fiber-reinforced composite materials which have high strength and high mod-ulus fibers in a matrix material. In such composites, fibers are the main load-carryingmembers and the matrix material keeps the fibers together, acts as a load-transfermedium between fibers, and protects them from being exposed to the environment.Fibers are stiffer and stronger than the same material in bulk form, matrix materialshave their usual bulk-form properties. Fibers have a very high length-to-diameter ra-tio, paradoxically short fibers (whiskers) exhibit better structural properties than long

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Alloy Characteristics Common Uses1050/1200 Good formability, Food and

weldability and chemical industry.corrosion resistance.

2014A Heat treatable. High strength. Airframes.Non-weldable.Poor corrosion resistance.

3103/3003 Non-heat treatable. Vehicle panelling,Medium strength work hardening structures exposedalloy. Good weldability, to marine atmospheres,formability and mine cages.corrosion resistance.

5251/5052 Non-heat treatable. Vehicle panelling,Medium strength work structures exposedhardening alloy. Good to marine atmospheres,weldability, formability mine cages.and corrosion resistance.

5454 Non-heat treatable. Pressure vessels andUsed at temperatures road tankers. Transport(65− 200)C. Good of ammonium nitrate,weldability and petroleum.corrosion resistance. Chemical plants.

5083/5182 Non-heat treatable. Pressure vessels andGood weldability road transport applicationsand corrosion resistance. below 65C.Very resistant to sea water, Ship building,industrial atmospheres. A superior structure in general.alloy for cryogenic use(in annealed condition).

6063 Heat treatable. Architectural extrusionsMedium strength alloy. (internal and external),Good weldability window frame.and corrosion resistance. Irrigation pipes.Used for intricate profiles.

6061/6082 Heat treatable. Stressed structural members,Medium strength alloy. bridges, cranes,Good weldability and roof trusses,corrosion resistance. beer barrels.

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INTRODUCTION 23

6005A Heat treatable. Thin walled wideProperties very similar extrusions.to 6082. Preferableas air quenchable, thereforehas less distortion problems.Not notch sensitive.

7020 Heat treatable. Armoured vehicles,Age hardens naturally therefore military bridges,will recover properties in motor cycle andheat affected zone after welding. bicycle frames.Susceptible to stress corrosion.Good ballistic deterrent properties.

7075 Heat treatable. Airframes.Very high strength.Non-weldable.Poor corrosion resistance.

Table 1.3: Some common aluminium alloys: characteristics and common uses [25].

Alloy Density Spec. Heat Therm. Cond. Therm Exp. Coeff(gcm−3) (Jg−1K−1) (Wm−1K−1) 0− 300C(10−6K−1)

Commercially Pure 4.51 0.54 16.3 9.2Ti-3%Al-2.5%V 4.48 - 7.6 7.9Ti-2.5%Cu 4.56 - 16.0 9.1Ti-6%Al-6%V-2%Sn 4.54 0.65 7.2 9.4Ti-8%Al-1%Mo-1%V 4.37 - 6.5 9.0Ti-6%Al-4%V 4.42 0.56 7.2 9.2

Table 1.4: Physical properties of titanium and titanium alloys [28].

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Alloy Yield Tensile 107 Cycle FatigueStr/Density Str/Density Str/Density

(106NmKg−1) (106NmKg−1) (106NmKg−1)Commercially Pure 78 107 54Ti-6%Al-4%V 206 226 135Ti-6%Al-2%Sn-4%Zr-2%Mo 202 223 123Ti-4%Al-4%Mo-2%Sn-0.5%Si 225 247 136Ti-10%V-2%Fe-3%Al 264 282 155Maraging Steel 170 202 121Steel 95 105 6818/8 Stainless Steel 68 75 40

Table 1.5: Strength of some titanium alloys at room temperature, normalised by den-sity, compared with other structural metals [28].

fibers. Materials are studied at various levels: atomic level, nano-level, single-crystallevel, a group of crystals. In this work we consider a basic unit of materials that haveproperties such as the modulus, strength, thermal coefficient of expansion, electricalresistance and so on, whose magnitudes depend on the direction. Fibers are materialswhere the desired properties are maximized in a given direction. Where materials areprocessed such that the basic units are randomly oriented, the resulting material tendsto have the same value of the property, in an average statistical sense, in all directions.Such materials are called isotropic materials, a typical example is the matrix material.The fibers and matrix materials usually employed in composites can be metallic ornon-metallic, the fiber materials can be common metals like aluminum, copper, iron,nickel, steel, titanium, or organic material like glass, boron and graphite [9].

In the case of structural applications, for example in aeronautics field, fiber-reinforcedcomposite materials are often a thin layer called lamina. A lamina is a macro unit of ma-terial whose material properties are determined through appropriate laboratory tests.Typical structural elements, such as bars, beams, plates or shells are formed by stackingthe layers to obtain desired strength and stiffness. Fiber orientation in each lamina andstacking sequence of the layers can be chosen to achieve desired strength and stiffnessfor a specific application.

In composite materials, fibers are the reinforcement material, and matrix is the basematerial. Three different types of composite materials are possible: - fibrous com-posites, where fibers of one material are in a matrix material of another; - particulatecomposites, where macro size particles of one material are in a matrix of another; -laminated composites, which are made of layers of different materials, including com-posites of the first two types. In composites, the particles and matrix can be metallicor nonmetallic, this permits four possible combinations: metallic in nonmetallic, non-metallic in metallic, nonmetallic in nonmetallic, and metallic in metallic. The stiffnessand strength of fibrous composites come from fibers which are stiffer and stronger thanthe same material in bulk form. Shorter fibers (whiskers) have better strength and stiff-

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INTRODUCTION 25

Figure 1.5: Various types of fiber-reinforced composite laminae.

ness properties than long fibers. Whiskers are about 1 to 10 microns in diameter and 10to 100 times as long. Fibers may be 5 microns to 0.005 inches. Some forms of graphitefibers are 5 to 10 microns in diameter [29].

A lamina or ply represents a fundamental building block. A fiber-reinforced lam-ina consists of many fibers embedded in a matrix material, which can be a metal likealuminum, or a nonmetal like thermoset or thermoplastic polymer. As indicated inFigure 1.5, the fibers can be continuous or discontinuous, woven, unidirectional, bidi-rectional, or randomly distributed. Unidirectional fiber-reinforced laminae exhibit thehighest strength and modulus in the fiber direction, but they have very low strengthand modulus in the direction transverse to the fibers. Discontinuous fiber-reinforcedcomposites have lower strength and modulus than continuous fiber-reinforced com-posites. A poor bonding between a fiber and matrix results in poor transverse proper-ties and failures such as fiber pull out, fiber breakage and fiber buckling [30].

A collection of laminae stacked to obtain the desired stiffness and thickness is calledlaminate. In Figure 1.6 some unidirectional fiber-reinforced laminae are stacked in thesame or different directions. The sequence of various orientations of a fiber-reinforcedcomposite layer in a laminated is called lamination scheme or stacking sequence. Thelayers are usually bounded together with the same matrix material as that in a lamina.The lamination scheme and material properties of individual lamina provide an addedflexibility to designers to tailor the stiffness and strength of the laminate to match thestructural stiffness and strength requirements.

The main disadvantages of laminates made of fiber-reinforced composite materialsare the delamination and the fiber debonding. Delamination is caused by the mismatch

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Figure 1.6: A laminate made up of laminae with different fiber orientations.

of material properties between layers, which produces shear stresses between the lay-ers, especially at the edges of a laminate. Fiber debonding is caused by the mismatchof material properties between matrix and fiber. Also, during manufacturing of lami-nates, material defects such as interlaminar voids, delamination, incorrect orientation,damaged fibers and variation in thickness may be introduced [31].

In formulating the constitutive equations of a lamina we assume that: (a) a lamina isa continuum: no gaps or empty spaces exist; (b) a lamina behaves as a linear elastic ma-terial. The assumption (a) permits to consider the macromechanical behavior of a lam-ina. The assumption (b) implies that the generalized Hooke’s law is valid. Compositematerials are heterogeneous from the microscopic point of view. They are assumedto be homogeneous from the macroscopic point of view. In contracted notation, thegeneralized Hooke’s law for an anisotropic material under isothermal conditions is:

σi = Cij εj , (1.1)

where σi are the stress components, εj are the strain components, and Cij are the mate-rial coefficients, all referred to an orthogonal Cartesian coordinate system (x1, x2, x3).The material coordinate system (x1, x2, x3) is indicated in Figure 1.7. The material co-ordinate axis x1 is parallel to the fiber, the x2-axis is transverse to the fibers direction inthe plane of the lamina, and the x3-axis is perpendicular to the plane of the lamina.

The orthotropic material properties of a lamina are obtained either by the theoret-ical approach or through suitable laboratory tests. In the case of theoretical approach(micromechanics approach), the assumptions to determine the engineering constantsof a continuous fiber-reinforced composite material are:

• perfect bonding exists between fibers and matrix;

• fibers are parallel and uniformly distributed throughout;

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INTRODUCTION 27

Figure 1.7: Unidirectional fiber-reinforced composite layer with the material coordi-nate system (x1, x2, x3) (the x1-axis is oriented along the fiber direction).

• the matrix is free of voids or microcracks and initially in a stress-free state;

• both fibers and matrix are isotropic and obey Hooke’s law;

• the applied loads are either parallel or perpendicular to the fiber direction.

For a fiber reinforced material we can define:

Ef = modulus of the fiber, Em = modulus of the matrix,

νf = Poisson’s ratio of the fiber, νm = Poisson’s ratio of the matrix,

vf = fiber volume fraction, vm = matrix volume fraction,

in this way the lamina engineering constants are given by:

E1 = Efvf + Emvm , ν12 = νfvf + νmvm , (1.2)

E2 =EfEm

Efvm + Emvf

, G12 =GfGm

Gfvm + Gmvf

,

where E1 is the longitudinal modulus, E2 is the transverse one, ν12 is the major Pois-son’s ratio, and G12 is the shear modulus. It is important to remember that:

Gf =Ef

2(1 + νf ), Gm =

Em

2(1 + νm). (1.3)

The engineering parameters E1, E2, E3, G12, G13, G23, ν12, ν13 and ν23 of an or-thotropic material can be determined experimentally using an appropriate test speci-men made up of the material.

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Material E1 E2 E3 G12 G13 G23 ν12 ν13 ν23

Aluminum 10.6 10.6 10.6 3.38 3.38 3.38 0.33 0.33 0.33Copper 18.0 18.0 18.0 6.39 6.39 6.39 0.33 0.33 0.33Steel 30.0 30.0 30.0 11.24 11.24 11.24 0.29 0.29 0.29Gr.-Ep (AS) 20.0 1.30 1.30 1.03 1.03 0.90 0.30 0.30 0.49Gr.-Ep (T) 19.0 1.50 1.50 1.00 0.90 0.90 0.22 0.22 0.49Gl.-Ep (1) 7.80 2.60 2.60 1.30 1.30 0.50 0.25 0.25 0.34Gl.-Ep (2) 5.60 1.20 1.30 0.60 0.60 0.50 0.26 0.26 0.34Br.-Ep 30.0 3.00 3.00 1.00 1.00 0.60 0.30 0.25 0.25

Table 1.6: Values of the engineering constants for several materials. The Young’s andshear moduli are in msi (million psi) where 1 psi=6894.76 Pa.

The values of the engineering constants are presented in Table 1.6 for several mate-rials [9], the following abbreviations are used: Gr.-Ep(AS)=graphite-epoxy (AS/3501),Gr.-Ep(T)=graphite-epoxy (T300/934), Gl.-Ep=glass-epoxy, Br.-Ep=boron-epoxy.

1.2.3 Sandwiches: foam and honeycomb cores

Sandwich structures are widely used in the aerospace, aircraft, marine, and automotiveindustries because they are lightweight with high bending stiffness. In general, the facesheets of sandwich panels consist of metals or laminated composites while the coreis made of corrugated sheet, foam, or honeycomb. Recently, fibrous core sandwichpanels have been developed by replacing the conventional core material with fibersaligned at small angles of inclination to the faceplates [32]. The concept of sandwichconstruction has been traced back to the mid 19th century, while the broad introductionof the sandwich concept in aircraft structures started at the beginning of World War II.The commonly used core materials include aluminum, alloys, titanium, stainless steel,and polymer composites. The core supports the skin, increases bending and torsionalstiffness, and carries most of the shear load [33].

Structural sandwiches most often have two faces, identical in material and thick-ness, which primarily resist the in-plane and lateral (bending) loads. However, in spe-cial cases the faces may differ in either thickness or material or both, because one faceis the primary load-carrying and low-temperature portion, while the other face mustwithstand an elevated temperature, corrosive environment, etc. In the case of uniformcore, the sandwich with identical faces is called symmetric sandwich, the latter with dif-ferent faces is the so-called asymmetric sandwich [14].

The core of a sandwich structure can be any material or architecture, but in general,as indicated in Figure 1.8, there are four main types: - foam or solid core; - honeycombcore; - web core; - a corrugated or truss core. Foam or solid cores are relatively inex-pensive and can consist of balsa wood, and an almost infinite selection of foam/plasticmaterials with a wide variety of densities and shear moduli, see Figure 1.9. In Figure1.10, a typical honeycomb-core architecture is given. The two most common types are

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INTRODUCTION 29

Figure 1.8: Types of sandwich construction.

the hexagonally-shaped cell structure (hexcell) and the square cell. Web core construc-tion is like a group of I-beams with their flanges welded together. Truss core sandwichis a triangulated core construction. In the web core and truss core constructions, thespace in the core could be used for liquid storage or as heat exchanger [14].

In the proposed construction the primary loading, both in-plane and bending, arecarried by the faces, while the core resists transverse shear loads (analogous to the webof an I-beam), and keep the faces in place. In foam-core and honeycomb-core sand-wiches all of the in-plane and bending loads are carried by the faces only. However,in web-core and truss-core sandwiches a portion of the in-plane and bending loads arealso carried by the core elements.

The most common foam cores are:(1) Polyurethane (PUR), a thermosetting material, widely used;(2) Polyisocyanurate (PIR), a thermosetting material;(3) Phenolic foam (PF), a thermosetting material, not yet widely used;(4) Polystyrene (expanded, EPS and extruded, XPS), a thermoplastic material.

Sandwich construction has been used primarily in the aircraft industry since the1940s, with the development of the British Mosquito bomber, and later logically ex-tended to missile and spacecraft structures. An excellent overview of the uses of corematerials and applications is given by Bitzer in [34]. He lists the quantity of hon-eycomb sandwich being used in various Boeing aircraft (see Table 1.7). In Table 1.7,wetted surfaces are defined as the airplane’s surfaces that would be wet if the aircraftwere submerged in water. In the Boeing 747, the fuselage cylindrical shell is primarilyNomex-honeycomb sandwich, and the floors, side panels, overhead bins, and ceilingare also of sandwich construction. The Beech Starship, the first all sandwich aircraft,uses Nomex honeycomb with graphite or Kevlar faces for the entire structure. A majorportion of the space shuttle is a composite-faced honeycomb-core sandwich. Europeleads the way in the use of sandwich constructions for lightweight railcars, while inthe U.S. some of the rapid transit trains use honeycomb sandwich. The U.S. Navy isusing honeycomb-sandwich bulkheads to reduce the ship weight above the waterline.

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Figure 1.9: Typical foam core construction.

Figure 1.10: Example of an honeycomb core.

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INTRODUCTION 31

Boeing Aircraft Percent of Wetted Surface707 8727 18737 26747 36

757-767 46

Table 1.7: Use of sandwich construction in Boeing aircraft.

Core Density(Kg/m3)H45 48H60 60LD7 90

Al5052 91.2Al3000 43÷59

Table 1.8: Density for different types of core [35].

Sailboats, racing boats, and auto racing cars are all employing sandwich construction.Sandwich construction is also used in snow skis, water skis, kayaks, canoes, pool ta-bles, and platform tennis paddles. Honeycomb-sandwich construction is also excellentfor absorbing mechanical and sound energy. It has a high-crush strength-to-weight ra-tio. It can also be used to transmit heat or to be an insulative barrier. In the former, ametallic honeycomb is used plus natural convection; for the latter, a nonmetallic core isused with the cells filled with a foam. For a sound barrier, the honeycomb core is filledwith a fiberglass batting, and a thin porous Tedlar skin can be used for the interiorface. Also, honeycomb core has been used in direct fans, wind tunnel, air conditioners,heaters, grills and registers [14].

The weight for such structures is an important parameter, in Table 1.8 the densityfor some foam and honeycomb cores are compared. H45 and H60 are H series of foamcores, LD7 is a Balsa lightweight core, Al5052 and Al3000 are Aluminum alloy honey-comb cores.

1.2.4 Piezoelectric materials

The phenomena of piezoelectricity is a peculiarity of certain class of crystalline materi-als. The piezoelectric effect is a linear energy conversion between mechanical and elec-trical fields. The linear conversion between the two fields is in both directions defininga direct or converse piezoelectric effect. The direct piezoelectric effect generates an electricpolarization by applying mechanical stresses. On the contrary, the converse piezoelectriceffect induces mechanical stresses or strains by applying an electric field. These twoeffects represent the coupling between the mechanical and electrical field. The first

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application was in the field of submarine sensory in World War I. A growing interestevolved after the introduction of the piezoceramics PZT (Lead-Zirconate-Titanate) in thelate first half of the 20th century. These ceramic materials show much higher perfor-mance and thus lead to a broadening of the possible applications. Still applicationswere limited to sound and ultrasound devices. These early piezoelectric materials aredescribed in [36]. Kawai [37] in the late seventies discovered an other class of piezo-electric materials, the so-called polyvinylidenfluorid PVDF, a semi crystalline polymerwith high sensor capability. In recent years piezoelectricity has found renewed inter-est, as active intelligent structures with self-monitoring and self-adaptive capabilities.Interesting reviews on these topics can be found in Chopra [2], Tani et alii [38], Raoand Sumar [39].

First applications for piezoelectric materials were sound, ultrasound sensors and

Figure 1.11: Example of a smart structure: sensor-actuator network for a plate.

sources. These are still actual, but in recent years a range of new applications evolved.The use of piezoelectric materials in the so-called adaptive structures or smart structuresopened a new interesting field in the last 20 years. A typical and very simple exampleof smart structure is the plate indicated in Figure 1.11, a network of sensors and actua-tors is embedded in it to control the deformations and to apply the corrections. Of therange of possible applications of adaptive structures, an overview is given in the next[40], with a focus on aerospace engineering.

Vibration damping. Nearly every structure in aerospace engineering is subjected tovibrations. In some cases such dynamic loads can be more dangerous than theapplied static loads. By implementing sensors and actuators in such structures,the dynamic vibrations can be measured and then actively damped. Typical ex-amples are vibration problems for the rotor wings in helicopters, sound dampingin the cockpit or cabin of civil planes.

Shape adaption of aerodynamics surfaces. In modern airplane the aerodynamic sur-faces can be optimized only for a certain airspeed and flight altitude. Wings thatare able to change their geometry according to the actual demands could lead toan increase in efficiency.

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INTRODUCTION 33

Active aeroelastic control. Typical problems of aeroelasticity like flutter or buffetingcan be reduced by the use of adaptive materials.

Shape control of optical and electromagnetic devices. Structures in aerospace field aresubjected to rapid and high temperature variations due to changing exposure tothe sunlight. Optical surfaces like mirrors and lenses, electromagnetic antennasand reflectors are highly sensitive to thermal deformations. A remedy to theseproblems could be the use of adaptive materials.

Health monitoring. In aerospace structures microscopic cracks are tolerable up to acertain limit. Smart structures could monitor these stresses and then apply anadditional control mechanism to maintain the safety.

However, different applications require different properties, like high or low fre-quency actuation, high deformation, high sensory capabilities and so on. For this aim,different materials can show advantages in some fields. Alternative smart materials tothose treated in this dissertation could be Shape Memory Alloys (SMA), Polymer Gels(PG) or electro-magnetostrictive materials as described in [6] and [7].

In this work the adaptive materials considered for smart structures are the crystallinematerials which show piezoelectric properties, and the piezoelectric polymers andsemi-crystalline polymers with ferroelectric properties. The group of crystalline ma-terials considers the natural crystals (e.g. Quartz (SiO2), Rochelle Salt (KNa(C4H4O6) ·4H2O), Tourmaline (SiO2+B, Al)) and the manufactured ceramics (e.g Barium Titanate(BaTiO3), Lead Zirconate Titanate (PZT )). The second group considers piezoelectricpolymers and semi-crystalline polymers such as the polyvinylidenfluoride (PV DF ).Table 1.9 gives the basic parameters of typical piezoceramics (PZT) and typical piezo-electric polymer (PVDF) (see also [40] and [4]). E11 and E33 are the Young’s moduli,ν is the Poisson’s ratio, ρ is the density. Tc is the Curie temperature, the relative per-mittivities ε11/ε0 and ε33/ε0 are expressed with respect to the reference permittivityε0 = 8.85 × 1012As/V m. The meaning of piezoelectric coefficients d33, d31 and d15 isclarified later.

Crystalline material must be polarized to express a piezoelectric effect. On the mi-croscopic level polarized domains exist, but their directions are randomly distributed.In order to activate the material, an external polarization is necessary. If a sufficientlyhigh electric field, expressed by the potential ΦP , is applied on the crystalline material,the domains reorder more or less in the same direction and the macroscopic polar-ization is produced. After the poling the material has a remanent polarization anda remanent elongation, as can be seen from the hysteresis curves in the Figure 1.12.In this activated state, any applied potential lower than the polarization potential ΦP

leads to a temporary deformation and viceversa. The chosen coordinate system forthe polarization is indicated in Figure 1.13. The definition of an appropriate referencesystem is fundamental: by depending the chosen polarization, piezoelectric materialshave a different coupling between the electric field and the mechanical deformationsor stresses. The materials considered in this work are polarized in direction 3 as indi-cated in Figure 1.13. For further details about this topic, readers can refer to Ikeda [4],Rogacheva [17] and Yang and Yu [41].

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34 CHAPTER 1

material: PZT-5H PZT-5A PIC 151 PVDFproducer: Morgan Morgan PI Ceramic Kynar

E11 [GPa] 71 69 79 2E33 [GPa] 111 106 77 2ν [−] 0.31 - - -ρ [kg/m3] 7450 7700 7800 1800

ε11/ε0 [−] - 1700 1980 12ε33/ε0 [−] 3400 1730 2100 12d33 [m/V ]× 10−12 593 374 450 -33d31 [m/V ]× 10−12 -274 -171 -210 23d15 [m/V ]× 10−12 741 585 580 -TC [C] 195 365 250 -

Table 1.9: Material properties of some piezoelectric ceramics (PZT) and polymers(PVDF).

Figure 1.12: Poling of piezoelectric material. Hysteresis of Polarization P (left), hys-teresis of Strain S (right).

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INTRODUCTION 35

E

Figure 1.13: Reference system for polarization in transverse direction of a piezoelectricmaterial.

The effect of the electric field on the elastic field (converse effect), and of the elas-tic field on the electric field (direct effect) is assumed to be linear. So the coupling isrepresented by linear factors: the piezoelectric coefficients. The mechanical system isrepresented by the stresses σ and the strains ε, the electrical system by the dielectricdisplacement D and the electric field E . In [4] four possible definitions of the coeffi-cients for the coupling of the two systems are given. In Table 1.10 these definitionsare clearly summarized. For each type of coefficient exists different components which

piezoelectric converse directcoefficient effect effect

a σ = aT D E = a εd ε = dT E D = d σb ε = bT D E = b σe σ = eT E D = e ε

Table 1.10: Piezoelectric coefficients. Coupling between mechanical and electricalfields.

relate one electric field component to one component of the mechanical field.The typical notation is exemplarily explained for the converse effect in the case of

d. The components of the electric field E are named as E1, E2 and E3 in 1, 2 and 3 di-rection, respectively. The components of the strain tensor in engineering notation arereferred to as 1 to 6, representing the components 11, 22, 33, 23, 13 and 12. The differentcomponents of d are thus named dij with i referring to the electric field direction andj to the stress or strain components. In the case of polarized polycrystalline ceramicmaterials with the crystal symmetry associated to the crytallographic class, five differ-ent coefficients dij exist: d15, d24, d31, d32 and d33, of which the first and the second pairhave the same value. The array-form of the piezoelectric coefficients states:

d =

0 0 0 0 d15 00 0 0 d24 0 0

d31 d32 d33 0 0 0

. (1.4)

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36 CHAPTER 1

The meaning of coefficients d33, d31 and d15 is clearly explained in Figure 1.14.

E EE133

e e e3 1 5

Figure 1.14: Meaning and effects of some piezoelectric coefficients.

1.2.5 Functionally graded materials

The severe temperature loads involved in many engineering applications, such as ther-mal barrier coatings, engine components or rocket nozzles, require high temperatureresistant materials. In Japan in the late 1980s the concept of Functionally Graded Ma-terials (FGMs) has been proposed as a thermal barrier material. FGMs are advancedcomposite materials wherein the composition of each material constituent varies grad-ually with respect to spatial coordinates [20]. Therefore, in FGMs the macroscopicmaterial properties vary continuously, distinguishing them from laminated compos-ite materials in which the abrupt change of material properties across layer interfacesleads to large interlaminar stresses allowing for damage development. As in the caseof laminated composite materials, FGMs combine the desirable properties of the con-stituent phases to obtain a superior performance, but avoid the problem of interfacialstresses [21], [22].

Functionally Graded Materials (FGMs) have a large variety of applications , duetheir properties, not only to provide the desired thermomechanical properties, but alsoto obtain appropriate piezoelectric, and magnetic properties, via the spatial variationin their composition. So, FGMs can be applied in several fields such as tribology, elec-tronics, biomechanics, aeronautics, and space research. The special feature of gradedspatial compositions associated to FGMs provides freedom in the design and manu-facturing of novel structures; on the other hand, it also poses great challenges in nu-merical modeling and simulation of the FGM structures [19]. Embedding a networkof piezoceramic actuators and sensors in FGM structure creates a self-controlling andself-monitoring smart system. This newly engineered class of materials has resultedin significant improvements in the performance of integrated systems, actuation tech-nologies, shape control, vibration and acoustic control and condition monitoring. Analternative solution could be the use of piezoelectric materials, functionally graded inthe thickness direction (FGPM), in order to build smart structures which are exten-sively used as sensors and actuators. The development of piezoelectric materials andstructures with functionally graded properties along the layer-thickness direction to

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INTRODUCTION 37

improve the mechanical and electrical properties at layer interfaces, has received in-creasing attention in recent years [42]. A typical example of a material functionallygraded in the thickness direction and employed as thermal barrier coating is given inFigure 1.15. Other interesting possibilities are the multilayered FGM, and the function-ally graded W/Cu obtained by sintering processing (see Figure 1.16).

In the field of FGMs we face substantially three problems, namely: (1) develop-

Figure 1.15: Typical microstructure of a thermal barrier coating functionally graded ina desired direction.

ment of processing routes for functionally graded materials, (2) determination of thespatially varying material properties (material modeling), and (3) modeling of struc-tures comprising FGMs and FGPMs. Even though the attention of the present work isfocused on the latter topic, a short discussion including a brief literature overview ofthe first two topics [43] is given in this section.

Figure 1.16: Six layers FGM structure with a linear gradient in the thickness direction(left). Sintered functionally graded W/Cu structure (right).

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38 CHAPTER 1

Processing routes

In a functionally graded material (FGM) the properties change gradually with posi-tion. The property gradient in the material is caused by a position-dependent chemicalcomposition, microstructure or atomic order. The manufacturing process of a FGM canusually be divided in building the spatially inhomogeneous structure (gradation) andtransformation of this structure into a bulk material (consolidation) [44].

Production of a NiTi-TiCx functionally graded material composite is possible throughuse of a combustion synthesis (CS) reaction employing the propagating mode (SHS). Dis-tinct interfaces with good material interaction and bonding can be observed betweeneach layer of the FGM. The TiCx particle size decreases with increasing NiTi contentin the final product as a result of minimized Ostwald ripening. Microindentation per-formed across the length of the FGM reveals a decrease in hardness as NiTi content isincreased [45].

Many fabrication methods were proposed to obtain glass-alumina FGMs, one ofthese is the production via percolation of molten glass into a sintered polycrystallinealumina substrate and via plasma spraying, the glass composition is designed in or-der to minimize the difference between the coefficients of thermal expansion of theconstituent phases, which may induce thermal residual stresses in service or duringfabrication [46]. The plasma spray method is very common in the case of functionallygraded Al2O3/ZrO2 thermal barrier coating [47].

In order to prepare Ni-Al2O3 graded composite coatings by an electroplating prepa-ration, a rotating cathode can be used. A regular octagonal cathode is employed bychanging the relative position between anode and cathode. The simplicity to control,the low equipment cost, and the potential for the economic mass production of com-posite coatings, permit to consider this technique as a new interesting way to fabricategraded composite coatings [48].

An other interesting method is the electrophoretic deposition (EPD) combined with apressureless sintering, in this way an experimental alumina/zirconia planar FGM canbe prepared, this material exhibits excellent hardness in the exterior layers, comparableto that of pure alumina [49].

Powder metallurgy is a suitable approach for the preparation of FGMs, but its effectson the electronic properties have to be carefully checked. Powder metallurgical pro-cessing may introduce atomic defects and local strains into the material and, thereby,alter the carrier concentration. Such material may be in non-equilibrium conditions atthe operating temperature with unstable thermoelectric properties. This effect can bereduced and eliminated by appropriate annealing procedures [50].

A consistent compositionally gradient protective layer can be prepared by chemicalvapor deposition by changing the reactant mixture composition gradually from propaneto dimethyldichlorosilane. Thus an FGM with a continuous composition distributionis obtained. No thermal cracks are observed and the compositionally gradient layerremains adhesive to the base composite following repeated rapid cooling tests from1000C to 0C [51].

The centrifugal method is applied to obtain a graded distribution in manufacturedFGMs. In this case a controlling composition method is required to monitoring the

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INTRODUCTION 39

movement of solid particles. The graded distribution in FGMs manufactured by thecentrifugal method is significantly influenced by many processing parameters, whichinclude the difference in density between particles and molten metal, the applied Gnumber, the particle size, the viscosity of the molten metal, the mean volume fractionof particles, the ring thickness and the solidification time [52].

Multi-layer Mg2Si/Al functionally graded composites can be produced by a direc-tional remelting and quenching process. The structures of functionally graded materialscontain three regions (unmelting, partial remelting and remelting) and form five layers(unmelting layer, transition layer, semisolid layer, partial remelting layer and remelt-ing layer) [53].

A novel one-step method, is the resistance sintering under ultra-high pressure, it hasbeen developed to fabricate W/Cu functionally graded materials without the additionof any sintering additive. A five-layered W/Cu FGM had been successfully fabricatedby resistance sintering under ultra-high pressure of 8GPa and 20kW power input for50 s. The relative density of the FGM is more than 97%. The relative density of the pureW layer is more than 96% [54].

For further information about the above processing methods, readers can refer tothe cited literature and to the overview work by Kieback et alii [44].

Material modelling

We are concerned with graded composite materials, consisting of one or more dis-persed phases of spatially variable volume fractions embedded in a matrix of anotherphase, that are subdivided by internal percolation thresholds or wider transition zonesbetween the different matrix phases. A detailed description of the geometry of actualgraded composite microstructures is usually not available, except perhaps for infor-mation on volume fraction distribution and approximate shape of the dispersed phaseor phases. Therefore, evaluation of thermomechanical response and local stresses ingraded materials must rely on analysis of micromechanical models with idealized ge-ometries. While such idealizations may have much in common with those that havebeen developed for analysis of macroscopically homogeneous composites, there aresignificant differences between the analytical models for the two classes of materials.It is well known that the response of macroscopically homogeneous systems can bedescribed in terms of certain thermoelastic moduli that are evaluated for a selectedrepresentative volume element, subjected to uniform overall thermomechanical fields.However, such representative volumes are not easily defined for systems with variablephase volume fractions, subjected to nonuniform overall fields [55]. The characteriza-tion of an FGM is not easy and it changes depending the considered material. The mostcommon methods based on micromechanical models are the rule of mixtures [55], the3-D phases distribution micromechanical models [56], the Voronoi Cell Finite ElementMethod (VCFEM) [57], the stress waves methods [58], and the stochastic microme-chanical models [59].

The rule of mixtures is an extension of the classical mixture rule for the compositematerials. For example in case of a Glass-Alumina FGM, the glass, the Alumina and the

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40 CHAPTER 1

residual elements are considered as three different phases with three different volumefractions Vg, Va and Vp.

The 3-D phases micromechanical models consider a three-dimensional, arbitrary andnon-linear distribution for the phases. For this aim is necessary to define an appropri-ate Local Representative Volume Element (LRVE) in order to define the local strainsand stresses.

An other method to define the elastic properties of a FGM is the definition of theVoronoi Cell Finite Element Method (VCFEM). In order to apply this method, the struc-ture of the FGM must be discrete and some empty zones must be introduced to con-sider the porosity of the FGM.

The characterization of FGMs can be done by using elastic waves to exciting the ma-terial. In order to apply this method a Linearly Inhomogeneous Element (LIE) must bedefined. By solving the equations of motion is possible to obtain the relation betweenthe displacements and the mechanical properties of the functionally graded material.

The elastic properties of FGM can be also obtained by using a stochastic microme-chanical model, in this case a stochastic approach is introduced to define the volumefraction of elements and the material properties of the constituents. A Mori-Tanakascheme [60] must be employed for the homogenization of the FGM.

For further information about the above material modellings for FGMs, readers canrefer to the previous cited literature.

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Chapter 2

Constitutive and geometrical equations

Constitutive equations characterize the individual material and its reaction to applied loads.First, generalized Hooke’s law is considered for mechanical case by employing a linear constitu-tive model for infinitesimal deformations. These equations are obtained in material coordinatesand then modified in a general reference system depending by the problem. The plane stressconditions are shortly discussed in order to avoid the Poisson’s locking phenomena. In the sec-ond part, constitutive equations are obtained for the thermo-electro-mechanical case by usingas thermodynamic function, the Gibbs free energy-per-unit of volume. Gibbs free energy canbe written as a quadratic form in case of linear interactions. Constitutive equations for me-chanical case, thermo-mechanical case and electro-mechanical case are obtained as particularcases of the most general thermo-electro-mechanical case. The discussed constitutive equationsare extended to functionally graded materials by considering the coefficients involved in suchequations depending by the thickness coordinate z. Such coefficients are approximated by usingopportune weights given by particular thickness functions which are a combination of Legendrepolynomials.

2.1 Generalized Hooke’s law

Constitutive equations characterize the individual material and its reaction to appliedloads. Elastic materials are considered, for which the constitutive behavior is only afunction of the current state of deformation. The material is called hyperelastic if thework done by the stresses during the deformation depends only on the initial stateand the current configuration [61], [62]. A material body is homogeneous if its proper-ties are the same throughout the body, it is called heterogeneous if the material proper-ties are a function of position. A body is defined anisotropic if it has different materialproperties in different directions at a given point: the material properties are direction-dependent. An isotropic body has the same material properties in all directions at apoint. An isotropic or anisotropic material can be nonhomogeneous or homogeneous[63]. A material body is defined as ideally elastic if, under isothermal conditions, it re-covers its original form completely upon removal of the forces causing deformations,so a one-to-one relationship between the state of stress and the state of strain in the cur-rent configuration exists. This fact implies that the material coefficients in constitutive

41

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42 CHAPTER 2

relations between stress and strain components are assumed constant during the defor-mation [64], [9]. Here, we consider the constitutive equations of linear elasticity for thecase of infinitesimal deformations, these are known as generalized Hooke’s law. Further,we can suppose that the reference configuration has a residual stress state indicatedwith σ0. The most general form of the linear constitutive equations for infinitesimaldeformations is:

σij = Cijklεkl + σ0ij , (2.1)

where Cijkl is called stiffness tensor and it is the fourth-order tensor of material parame-ters. In general, it has (3)2 × (3)2 = 81 scalar components. The number of independentcomponents of Cijkl can be reduced considering the symmetry of σij , symmetry of εkl

and symmetry of Cijkl as detailed described in [9].The number of independent material stiffness components is reduced to 6× (3)2 =

54 in the absence of body couples, so for the principle of angular momentum the stresstensor is symmetric (σij = σji) and the tensor Cijkl is symmetric in the first two sub-scripts. By its definition, the strain tensor is symmetric too, εkl = εlk, then Cijkl issymmetric in the last two subscripts; in this way the number of independent materialstiffness components is reduced to 6× 6 = 36.

If we also assume the material hyperelastic, exist a strain energy density functionU0(εij) [9] which permits to say that Cijkl = Cklij . This reduces the number of inde-pendent material stiffness components from 36 to 21. This fact permits to express theEq.(2.1) in an alternative form using single subscript notation for stresses and strains,and double subscript notation for the material stiffness coefficients:

σi = Cijεj + σ0i . (2.2)

The notation employed in Eq.(2.2) renders σi, εj and Cij non-tensor components. Thesingle subscript notation for stresses and strains is called engineering notation or Voigt-Kelvin notation. The notation for stresses and strains is:

σ1 = σ11, σ2 = σ22, σ3 = σ33, σ4 = σ23, σ5 = σ13, σ6 = σ12 ,

ε1 = ε11, ε2 = ε22, ε3 = ε33, ε4 = 2ε23, ε5 = 2ε13, ε6 = 2ε12 , (2.3)11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6 ,

the meaning of the 6 components of the considered vectors is clearly explained in Fig-ure 2.1 in the case of the 6 stress components in material reference system. In matrixnotation, Eq.(2.2) can be written as:

σ1

σ2

σ3

σ4

σ5

σ6

=

C11 C12 C13 C14 C15 C16

C21 C22 C23 C24 C25 C26

C31 C32 C33 C34 C35 C36

C41 C42 C43 C44 C45 C46

C51 C52 C53 C54 C55 C56

C61 C62 C63 C64 C65 C66

ε1

ε2

ε3

ε4

ε5

ε6

+

σ01

σ02

σ03

σ04

σ05

σ06

. (2.4)

The coefficients of matrix Cij in Eq.(2.4) are symmetric (Cij = Cji) because the ma-terial is supposed hyperelastic. So, for the most general elastic material, we have 21independent stiffness coefficients.

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 43

Figure 2.1: Components of the stress vector in a material body.

The Eqs.(2.2) and (2.4) are invertible and the strain components are related to stressones by:

εi = Sijσj + ε0i , ε0

i = −Sijσ0j , (2.5)

where Sij are the material compliance parameters with [S] = [C]−1. The compliancetensor is the inverse of the stiffness tensor (S = C−1), so in analogy with Eq.(2.4) ispossible to write:

ε1

ε2

ε3

ε4

ε5

ε6

=

S11 S12 S13 S14 S15 S16

S21 S22 S23 S24 S25 S26

S31 S32 S33 S34 S35 S36

S41 S42 S43 S44 S45 S46

S51 S52 S53 S54 S55 S56

S61 S62 S63 S64 S65 S66

σ1

σ2

σ3

σ4

σ5

σ6

+

ε01

ε02

ε03

ε04

ε05

ε06

. (2.6)

In the next part of this chapter we consider as reference configuration the stress andstrain free configuration: σ0

i = 0 and ε0i = 0.

Further reduction in the number of independent stiffness parameters comes fromthe so-called material symmetry. When elastic material parameters at a point have thesame values for every pair of coordinate systems that are mirror images of each otherin a certain plane, that plane is called a material plane of symmetry, this symmetry is adirectional property and not a positional property. We discuss about various plane ofsymmetry and forms of associated stress-strain relations, in particular monoclinic mate-rials, orthotropic materials and isotropic materials. For further details about the materialsymmetry, readers can refer to [9].

Monoclinic materials

When the elastic coefficients at a point have the same value for every pair of coordi-nate systems which are the mirror images of each other with respect to a plane, the

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44 CHAPTER 2

material is called a monoclinic material. The consequence of this considerations is thatthe following pairs of elastic coefficients are equal to zero: -) C14 = 0 and C15 = 0; -)C24 = 0 and C25 = 0; -) C34 = 0 and C35 = 0; -) C46 = 0 and C56 = 0. So the independentparameters are 21− 8 = 13 and the Eq.(2.4) changes in:

σ1

σ2

σ3

σ4

σ5

σ6

=

C11 C12 C13 0 0 C16

C12 C22 C23 0 0 C26

C13 C23 C33 0 0 C36

0 0 0 C44 C45 00 0 0 C45 C55 0

C16 C26 C36 0 0 C66

ε1

ε2

ε3

ε4

ε5

ε6

. (2.7)

Orthotropic materials

In orthotropic materials three mutually orthogonal planes of symmetry exist, so the num-ber of independent elastic coefficients is reduced from 13 to 9. Considering the Eq.(2.7),the stress-strain relations for an orthotropic material change in:

σ1

σ2

σ3

σ4

σ5

σ6

=

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

ε1

ε2

ε3

ε4

ε5

ε6

. (2.8)

Most often, the material properties are determined in a laboratory in terms of the engi-neering constants such as Young’s modulus, shear modulus and Poisson’s ratios. The9 independent material coefficients in Eq.(2.8) can be expressed by 9 independent ma-terial engineering constants:

E1, E2, E3, G23, G13, G12, ν12, ν13, ν23 , (2.9)

the relations between material coefficients and engineering constants are:

C11 =1− ν23ν23

E2E3∆, C12 =

ν21 + ν31ν23

E2E3∆=

ν12 + ν32ν13

E1E3∆,

C13 =ν31 + ν21ν32

E2E3∆=

ν13 + ν12ν23

E1E2∆,

C22 =1− ν13ν31

E1E3∆, C23 =

ν32 + ν12ν31

E1E3∆=

ν23 + ν21ν13

E1E3∆, (2.10)

C33 =1− ν12ν21

E1E2∆, C44 = G23 , C55 = G31 , C66 = G12 ,

∆ =1− ν12ν21 − ν23ν32 − ν31ν13 − 2ν21ν32ν13

E1E2E3

.

For the Poisson’s ratio is valid the following relation:νij

Ei

=νji

Ej

(no sum on i, j) . (2.11)

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 45

Applications of a normal stress to a rectangular block of isotropic or orthotropic mate-rial lead to only extension in the direction of the applied stress and contraction perpen-dicular to it, whereas an anisotropic material experiences extension in the direction ofthe applied normal stress, contraction perpendicular to it, as well as shear strain. Theapplication of a shearing stress to an anisotropic material causes shearing strain as wellas normal strains [9].

Isotropic materials

When there exist no preferred directions in the material, infinite number of planes ofmaterial symmetry are considered. Such materials are called isotropic and the numberof independent elastic coefficients are reduced from 9 to 2:

E1 = E2 = E3 = E , G12 = G13 = G23 = G , ν12 = ν23 = ν13 = ν (2.12)

withG =

E

2(1 + ν). (2.13)

Consequently, Eq.(2.8) takes the form:

σ1

σ2

σ3

σ4

σ5

σ6

= Λ

1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 1

2(1− 2ν) 0 0

0 0 0 0 12(1− 2ν) 0

0 0 0 0 0 12(1− 2ν)

ε1

ε2

ε3

ε4

ε5

ε6

(2.14)

whereΛ =

E

(1 + ν)(1− 2ν). (2.15)

Alternative, the stress-strain relations can be written in a compact form by using2 different independent constants known as Lamè constants λ and µ. The relations be-tween the Lamè constants λ and µ and engineering constants E, G and ν are obtainedin [9] and [62] by writing the fourth-order isotropic tensor Cijkl in a different way:

E =µ(3λ + 2µ)

(λ + µ), ν =

λ

2(µ + λ), G = µ . (2.16)

2.1.1 Transformation of material coefficients

The constitutive relations in Section 2.1 for an orthotropic material were written interms of stress and strain components that are referred to a coordinate system whichcoincides with the principal material coordinate system (x1, x2, x3). But in the problemformulation the employed coordinate system does not coincide with the principal ma-terial one, for example in composite laminates each layer has its material coordinate

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46 CHAPTER 2

system, and these have different orientations with respect to the most general lami-nate coordinates called problem coordinate system (x, y, z). In Figure 2.2 the materialcoordinate system and the problem coordinate system are clearly indicated for a layerembedded in a laminate. The angle φ between the in-plane material coordinates x1, x2

Figure 2.2: Material coordinate system (x1, x2, x3) and problem coordinate system (x,y, z) in a lamina.

and the problem coordinates x, y is considered counterclockwise, the third coordinatescoincide (x3 = z). The relations between the two reference systems are [9], [29]:

x1

x2

x3

=

cos φ sin φ 0− sin φ cos φ 0

0 0 1

xyz

= [L]

xyz

. (2.17)

The inverse of Eq.(2.17) is:

xyz

=

cos φ − sin φ 0sin φ cos φ 0

0 0 1

x1

x2

y3

= [L]T

x1

x2

x3

. (2.18)

The inverse of [L] is equal to its transpose: [L]−1 = [L]T . Omitted details about thematrix [L] of direction cosines lij are given in [9], [29], [65].

Each quantity related to material reference system is indicated by the subscript m,those related to problem reference system by the subscript p. The direct cosines lij aredefined as:

lij = (ei)m · (ej)p (2.19)

where (ei)m and (ej)p are the orthonormal basis vectors in the material and problemcoordinate systems, respectively. So the stress tensors can be transformed accordingto:

(σkq)m = lkilqj(σij)p , (σkq)p = likljq(σij)m . (2.20)

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 47

The Eqs.(2.20) can be rewritten in matrix form as:

[σ]m = [L][σ]p[L]T , [σ]p = [L]T [σ]m[L] , (2.21)

where

[σ]p =

σxx σxy σxz

σxy σyy σyz

σxz σyz σzz

, [σ]m =

σ11 σ12 σ13

σ12 σ22 σ23

σ13 σ23 σ33

. (2.22)

Carrying out the matrix multiplications in Eq.(2.21) and rearranging in terms of thesingle-subscript stress components, we obtain:

σxx

σyy

σzz

σyz

σxz

σxy

=

cos2φ sin2φ 0 0 0 − sin 2φsin2φ cos2φ 0 0 0 sin 2φ

0 0 1 0 0 00 0 0 cos φ sin φ 00 0 0 − sin φ cos φ 0

sin φ cos φ − sin φ cos φ 0 0 0 cos2φ− sin2φ

σ1

σ2

σ3

σ4

σ5

σ6

(2.23)and

σ1

σ2

σ3

σ4

σ5

σ6

=

cos2φ sin2φ 0 0 0 sin 2φsin2φ cos2φ 0 0 0 − sin 2φ

0 0 1 0 0 00 0 0 cos φ − sin φ 00 0 0 sin φ cos φ 0

− sin φ cos φ sin φ cos φ 0 0 0 cos2φ− sin2φ

σxx

σyy

σzz

σyz

σxz

σxy

.

(2.24)The Eqs.(2.23) and (2.24) in compact form can be expressed as:

σp = [T ]σm , (2.25)

σm = [R]σp . (2.26)

The same procedure can be applied for the transformation of strain components, theomitted steps are detailed in [9] and [29]. Since strains are also second-order tensorquantities, transformation equations obtained for stresses in Eqs.(2.21), are also validfor tensor components of strains:

[ε]m = [L][ε]p[L]T , [ε]p = [L]T [ε]m[L] . (2.27)

The Eqs.(2.23) and (2.24) are valid for strains when the stress components are replacedwith tensor components of strains for the two coordinate systems. However, the single-column formats in Eqs.(2.23) and (2.24) for stresses are not valid for single-columnformats of strain because of the definition:

2ε12 = ε6 , 2ε13 = ε5 , 2ε23 = ε4 . (2.28)

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48 CHAPTER 2

For this reason Eqs.(2.25) and (2.26) are modified in case of strains in:

εp = [R]Tεm , (2.29)

εm = [T ]Tεp . (2.30)

The only remaining quantities that need to be transformed from material coordinatesystem to the problem coordinates are the material stiffnesses Cij . These can be easilyobtained considering the Eqs.(2.25), (2.26) and Eqs.(2.29), (2.30):

σp = [T ]σm = [T ][C]mεm = [T ][C]m[T ]Tεp = [C]pεp (2.31)

where [C]m is the 6 × 6 material stiffness matrix in the material coordinates and [T ] isthe transformation matrix in Eq.(2.23). We can define [Q] = [C]p and [C] = [C]m, so:

[Q] = [T ][C][T ]T . (2.32)

In the case of orthotropic material the stress-strain relations in problem coordinates is:

σxx

σyy

σzz

σyz

σxz

σxy

=

Q11 Q12 Q13 0 0 Q16

Q12 Q22 Q23 0 0 Q26

Q13 Q23 Q33 0 0 Q36

0 0 0 Q44 Q45 00 0 0 Q45 Q55 0

Q16 Q26 Q36 0 0 Q66

εxx

εyy

εzz

γyz

γxz

γxy

(2.33)

withγyz = 2εyz , γxz = 2εxz , γxy = 2εxy . (2.34)

The meaning of coefficients Qij in explicit form and their relations with Cij and φ aregiven in [9]. The Eq.(2.33) can be written for each kth layer of a laminate.

2.1.2 Poisson’s locking phenomena: plane stress constitutive rela-tions

The thickness locking (TL) mechanism, also known as Poisson’s locking phenomena,is caused by the use of simplified kinematic assumptions in the plate/shell analy-sis [66], [67]. Two-dimensional plate/shell structures can be analyzed as particularcase of three-dimensional (3D) continuum by eliminating, via ’a priori’ integration,the thickness coordinate z. Such integration can be made by following two differentmethods: the method of asymptotic expansions or the axiomatic methods. The in-troduction of axiomatic and/or asymptotic approximations could introduce some notdesired mechanisms which are not in the three-dimensional solution. One of these isthe Poisson’s locking which is related to the use of plane-strain/plane-stress hypoth-esis in thin plate/shell theory. The analysis of thin plate/shell problems is, in fact,often associated to plane-stress assumptions (thin surface problem) while plane-strain

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 49

hypothesis is usually referred to beam theory. Discussion on plane strain, plane stressand/or plane elasto-static problems can be also found in [66], [67] and [68]. However,in most of equivalent-single layer theories assumptions are in a ’contradictory manner’made on strain fields. Plane strain assumptions are:

γyz = γxz = εzz = 0 , (2.35)

and they are used in-place of the more natural plane stress assumptions:

σyz = σxz = σzz = 0 . (2.36)

That contradiction introduces a ’locking mechanism’ that make the plate/shell modelnot applicable in some cases. Thickness locking (TL, also known as Poisson’s lock-ing) is the name assigned to that mechanism: TL does not permit to equivalent singlelayer analysis with transverse displacement w constant or linear through the thickness(that means transverse strain εzz zero or constant) to lead to 3D solution in thin plateproblems. A known technique to contrast TL consists of modifying the elastic stiffnesscoefficients by forcing the ’contradictory’ condition known as transverse normal stresszero condition:

σzz = 0 . (2.37)

Poisson’s locking appears if and only if a plate theory shows a constant distribution oftransverse normal strain εzz; that is to avoid TL the plate/shell theories would requireat least a parabolic distribution of transverse displacement component w, for such the-ories the Hooke’s law as presented in Section 2.1 is suitable. For further details see thecomplete discussion reported in the books by Washizu [69], Librescu [70] and Reddy[9] as well as the discussion quoted in papers [66] and [67].

By imposing the condition σ3 = 0 in Eq.(2.8), the modified stiffness coefficients inmaterial reference system (reduced stiffness coefficients) can be obtained. These last areindicated as Cij , and they are used in equivalent single layers theories with transversedisplacement w constant or linear in thickness direction, in order to avoid the Poisson’slocking phenomena:

C11 =E1

1− ν12ν21

, C12 =ν12E2

1− ν12ν21

, C13 = C13 ,

C22 =E2

1− ν12ν21

, C23 = C23 , C33 = C33 , (2.38)

C44 = C44 = G23 , C55 = C55 = G13 , C66 = C66 = G12 .

Eq.(2.38) means that to avoid Poisson’s locking phenomena, Eq.(2.8) must be used,but by considering the reduced stiffness C11, C12 and C22. The new obtained stiffnesscoefficients matrix Cij can be then rotated in according to Section 2.1.1.

Layer wise theories, with a linear expansion in the thickness direction for w, do notshow TL because the transverse normal strain has a piece-wise constant distributionalong the thickness direction z.

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50 CHAPTER 2

2.2 Constitutive equations for multifield problems

Constitutive equations for the electro-thermo-mechanical problem are obtained in thissection in according with that reported in [1] and [71]; then three particular casesare discussed: - pure mechanical problem; - thermo-mechanical problem; - electro-mechanical problem.

The coupling between the mechanical, thermal and electrical fields can be deter-mined by using thermodynamical principles and Maxwell’s relations [72]-[75]. Forthis aim, it is necessary to define a Gibbs free-energy function G and a thermopiezoelectricenthalpy density H [64], [4]:

G(εij, Ei, θ) = σijεij − EiDi − ηθ , (2.39)H(εij, Ei, θ, ϑi) = G(εij, Ei, θ)− F (ϑi) , (2.40)

where σij and εij are the stress and strain components, Ei is the electric field vector,Di isthe electric displacement vector. η is the variation of entropy per unit of volume and θthe temperature considered with respect to the reference temperature T0. The functionF (ϑi) is the dissipation function, it depends by the spatial temperature gradient ϑi:

F (ϑi) =1

2κijϑiϑj − τ0hi , (2.41)

where κij is the symmetric, positive semidefinite conductivity tensor. In the secondterm, τ0 is a thermal relaxation parameter and hi is the temporal derivative of the heatflux hi. The thermal relaxation parameter is omitted in the present work. Further de-tails about the dissipation function F (ϑi) can be found in [76], [77] and [78], whereinteresting considerations are made about the inclusion or not of the dissipation func-tion F (ϑi) (e.g., it must be considered in the thermo-mechanical analysis of a structurewith imposed temperature on its surfaces).

The thermopiezoelectric enthalpy density H can be expanded in order to obtain aquadratic form for a linear interaction:

H(εij, Ei, θ, ϑi) =1

2Qijklεijεkl − eijkεijEk − λijεijθ − 1

2εklEkEl − pkEkθ (2.42)

− 1

2χθ2 − 1

2κijϑiϑj ,

where Qijkl is the elastic coefficients tensor considered for an orthotropic material inthe problem reference system [9]. eijk are the piezoelectric coefficients and εkl are thepermittivity coefficients [17]. λij are thermo-mechanical coupling coefficients, pk arethe pyroelectric coefficients and χ = ρCv

T0where ρ is the material density, Cv is the

specific heat per unit mass and T0 is the reference temperature [4].The constitutive equations are obtained by considering the following relations:

σij =∂H

∂εij

, Dk = −∂H

∂Ek

, η = −∂H

∂θ, hi = −∂H

∂ϑi

. (2.43)

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 51

By considering Eqs.(2.42) and (2.43), we can obtain the constitutive equations for thethermo-electro-mechanical problem:

σij = Qijklεkl − eijkEk − λijθ , (2.44)Dk = eijkεij + εklEl + pkθ , (2.45)

η = λijεij + pkEk + χθ , (2.46)hi = κijϑj . (2.47)

Above equations can be written in single-subscript notation by using the indexes m =q = 1, 2, 3, 4, 5, 6 and i = j = 1, 2, 3 (see also Eqs.(2.3)):

σm = Qmqεq − emiEi − λmθ , (2.48)Di = eiqεq + εijEj + piθ , (2.49)η = λqεq + pjEj + χθ , (2.50)hi = κijϑj . (2.51)

From the equations written in single-subscript notations, it is very easy to write theirmatrix form; the matrices and vectors are indicated in bold scripture. Considering ageneric multilayered structure, Eqs.(2.48)-(2.51) are written for a generic layer k in theproblem reference system (x, y, z) as:

σk = Qkεk − ekT Ek − λkθk , (2.52)

Dk = ekεk + εkEk + pkθk , (2.53)

ηk = λkT εk + pkT Ek + χkθk , (2.54)

hk = κkϑk , (2.55)

where the temperature θk, the term χk and the entropy for unite volume ηk are scalarvariables in each layer k. The 6× 1 vectors of stress and strain components are:

σk =

σxx

σyy

σzz

σyz

σxz

σxy

k

, εk =

εxx

εyy

εzz

γyz

γxz

γxy

k

. (2.56)

The 3 × 1 vectors of electrical field Ek, electrical displacement Dk, heat flux hk andspatial gradient of temperature ϑk are:

Ek =

Ex

Ey

Ez

k

, Dk =

Dx

Dy

Dz

k

, hk =

hx

hy

hz

k

, ϑk =

ϑx

ϑy

ϑz

k

. (2.57)

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52 CHAPTER 2

The 3× 1 array of pyroelectric coefficients pk and the 6× 1 array of thermo-mechanicalcoupling coefficients λk are:

pk =

p1

p2

p3

k

, λk = Qk αk =

λ1

λ2

λ3

00λ6

k

, (2.58)

where the elastic coefficients matrix Qk of Hooke’s law in problem reference systemfor an orthotropic material is:

Qk =

Q11 Q12 Q13 0 0 Q16

Q12 Q22 Q23 0 0 Q26

Q13 Q23 Q33 0 0 Q36

0 0 0 Q44 Q45 00 0 0 Q45 Q55 0

Q16 Q26 Q36 0 0 Q66

k

, (2.59)

the vector αk has dimension 6× 1 and contains the thermal expansion coefficients:

αk =

α1

α2

α3

000

k

. (2.60)

The matrices εk of permittivity coefficients and κk of conductivity coefficients havedimension 3× 3:

εk =

ε11 ε12 0ε12 ε22 00 0 ε33

k

, κk =

κ11 κ12 0κ12 κ22 00 0 κ33

k

. (2.61)

The matrix of piezoelectric coefficients ek has dimension 3× 6:

ek =

0 0 0 e14 e15 00 0 0 e24 e25 0

e31 e32 e33 0 0 e36

k

. (2.62)

In order to use the relations proposed in Eqs.(2.52)-(2.55) in the variational state-ments, that will be presented in the next chapters, it is convenient to split them inin-plane components (subscript p) and out-of-plane components (subscript n). Other

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 53

two new subscripts are introduced: the subscript C for those variables in the varia-tional statements which need the substitution of constitutive equations; the subscriptG for those variables in constitutive equations which need the substitution of geometri-cal relations (these last are introduced at the end of the present chapter). The proposedconstitutive equations must be valid for both plate and shell geometries, for this reasona curvilinear cartesian reference system is introduced (α, β, z) in place of the less gen-eral rectilinear ones (x, y, z). In Section 2.5 geometrical relations for shells are obtainedand their degeneration in geometrical relations for plates is discussed. The split stressand strain components vectors are:

σkpC =

σαα

σββ

σαβ

k

, σknC =

σαz

σβz

σzz

k

, εkpG =

εαα

εββ

γαβ

k

, εknG =

γαz

γβz

εzz

k

. (2.63)

In Eqs.(2.63), it is important to notice the change of position between the αz and βzcomponents. The vectors 3× 1 of electrical field, electrical displacement, heat flux andspatial gradient of the temperature in curvilinear coordinates, split in in-plane andout-plane components, are:

EkpG =

k

, EknG =

Ez

k, Dk

pC =

k

, DknC =

Dz

k, (2.64)

hkpC =

k

, hknC =

hz

k, ϑk

pG =

ϑα

ϑβ

k

, ϑknG =

ϑz

k.

By considering Eqs.(2.63) and (2.64), the split form of Eqs.(2.52)-(2.55) is:

σkpC = Qk

ppεkpG + Qk

pnεknG − ekT

pp EkpG − ekT

np EknG − λk

pθk , (2.65)

σknC = Qk

npεkpG + Qk

nnεknG − ekT

pn EkpG − ekT

nnEknG − λk

nθk , (2.66)

DkpC = ek

ppεkpG + ek

pnεknG + εk

ppEkpG + εk

pnEknG + pk

pθk , (2.67)

DknC = ek

npεkpG + ek

nnεknG + εk

npEkpG + εk

nnEknG + pk

nθk , (2.68)

ηkC = λkT

p εkpG + λkT

n εknG + pkT

p EkpG + pkT

n EknG + χkθk , (2.69)

hkp = κk

ppϑkpG + κk

pnϑknG , (2.70)

hkn = κk

npϑkpG + κk

nnϑknG . (2.71)

The explicit forms of the new matrices in Eqs.(2.65)-(2.71) are:

• Stiffness matrices:

Qkpp =

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

k

, Qkpn =

0 0 Q13

0 0 Q23

0 0 Q36

k

, (2.72)

Qknp =

0 0 00 0 0

Q13 Q23 Q36

k

, Qknn =

Q55 Q45 0Q45 Q44 00 0 Q33

k

.

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54 CHAPTER 2

• Piezoelectric coefficients:

ekpp =

[0 0 00 0 0

]k

, ekpn =

[e15 e14 0e25 e24 0

]k

, (2.73)

eknp =

[e31 e32 e36

]k, ek

nn =[

0 0 e33

]k.

• Permittivity coefficients:

εkpp =

[ε11 ε12

ε12 ε22

]k

, εkpn =

[00

]k

, (2.74)

εknp =

[0 0

]k, εk

nn =[

ε33

]k.

• Thermo-mechanical coupling coefficients:

λkp =

λ1

λ2

λ6

k

, λkn =

00λ3

k

. (2.75)

• Pyroelectric coefficients:

pkp =

[p1

p2

]k

, pkn =

[p3

]k. (2.76)

• Conductivity coefficients:

κkpp =

[κ11 κ12

κ12 κ22

]k

, κkpn =

[00

]k

, (2.77)

κknp =

[0 0

]k, κk

nn =[

κ33

]k.

2.2.1 Mechanical case

In case of pure mechanical problems, electrical and temperature loads are not appliedon the structure and the spatial gradient of temperature does not exist. The couplingsbetween the mechanical field and the electrical field, and between the mechanical fieldand the thermal field are not considered. For these reasons, Eqs.(2.65)-(2.71) degeneratein:

σkpC = Qk

ppεkpG + Qk

pnεknG , (2.78)

σknC = Qk

npεkpG + Qk

nnεknG . (2.79)

Eqs.(2.78)-(2.79) are the Hooke’s law written in the problem reference system and splitin in-plane and out-of-plane components.

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 55

2.2.2 Electro-mechanical case

In case of electro-mechanical problem, two physical fields interact, no thermal loadsand spatial temperature gradients are applied on the structure. The couplings betweenmechanical and thermal fields, and between electrical and thermal fields are not con-sidered. Eqs.(2.65)-(2.71) degenerate in:

σkpC = Qk

ppεkpG + Qk

pnεknG − ekT

pp EkpG − ekT

np EknG , (2.80)

σknC = Qk

npεkpG + Qk

nnεknG − ekT

pn EkpG − ekT

nnEknG , (2.81)

DkpC = ek

ppεkpG + ek

pnεknG + εk

ppEkpG + εk

pnEknG , (2.82)

DknC = ek

npεkpG + ek

nnεknG + εk

npEkpG + εk

nnEknG . (2.83)

Typical examples of electro-mechanical analysis of multilayered structures, where Eqs.(2.80)-(2.83) are employed, can be found in [79], [80] and [42].

2.2.3 Thermo-mechanical case

In case of thermo-mechanical problems, electrical loads are not applied on the struc-ture. The coupling between the mechanical and electrical fields, and between the ther-mal and electrical fields are not considered. Eqs.(2.65)-(2.71) degenerate in:

σkpC = Qk

ppεkpG + Qk

pnεknG − λk

pθk , (2.84)

σknC = Qk

npεkpG + Qk

nnεknG − λk

nθk , (2.85)

ηkC = λkT

p εkpG + λkT

n εknG + χkθk , (2.86)

hkp = κk

ppϑkpG + κk

pnϑknG , (2.87)

hkn = κk

npϑkpG + κk

nnϑknG . (2.88)

Eqs.(2.84)-(2.88) must be particularized depending the considered thermo-mechanicalcase.

If a partial coupling is considered in the structure and the temperature is only seenas an external load, as in [43] and [81], the considered constitutive equations are:

σkpC = Qk

ppεkpG + Qk

pnεknG − λk

pθk , (2.89)

σknC = Qk

npεkpG + Qk

nnεknG − λk

nθk . (2.90)

In case of a full coupling between the thermal and mechanical field (the tempera-ture is seen as a primary variable in analogy with the displacements), if a temperatureis imosed on the structure surface, the internal virtual thermal work δθkηc cannot beconsidered, so the relative constitutive equations are:

σkpC = Qk

ppεkpG + Qk

pnεknG − λk

pθk , (2.91)

σknC = Qk

npεkpG + Qk

nnεknG − λk

nθk , (2.92)

hkp = κk

ppϑkpG + κk

pnϑknG , (2.93)

hkn = κk

npϑkpG + κk

nnϑknG . (2.94)

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56 CHAPTER 2

The inclusion of the dissipative function F (ϑi) in Eq.(2.40) and information about theheat flux h are fundamental to understand how the temperature profile evolves alongthe thickness of the structure [76]-[78].

The second case of full coupling considers the structure subjected to a mechanicalload, the dissipative function F (ϑi) is not included and constitutive equations are:

σkpC = Qk

ppεkpG + Qk

pnεknG − λk

pθk , (2.95)

σknC = Qk

npεkpG + Qk

nnεknG − λk

nθk , (2.96)

ηkC = λkT

p εkpG + λkT

n εknG + χkθk . (2.97)

The meaning of these constitutive equations are clarified in next chapters, wherethe opportune variational statements and governing equations are discussed.

2.3 Constitutive equations for mixed models

Two different variational statements are here employed in order to obtain the govern-ing equations for the multifield analysis of multilayered plates and shells. The first oneis the Principle of Virtual Displacements (PVD) [82] and its extension to electro-thermo-mechanical problems, these models are defined as refined models and classical constitu-tive equations, as obtained in Section 2.2, are used. The second employed variationalstatement is the Reissner’s Mixed Variational Theorem (RMVT) [83] and its extensions tomultifield problems. The models obtained from RMVT are called mixed models: in thepure mechanical case, they permit to consider as primary variables the three displace-ment components and the transverse shear/normal stresses [84]. Mixed models areextended to multifield problems in order to consider other a priori variables, such asthe normal electrical displacement and/or the normal magnetic inductance. In orderto write the RMVT for mechanical and multifield problems, the constitutive equationsin Section 2.2 must be rearranged to make them consistent with the employed vari-ational statement. In Chapter 4 and 5, where the variational statements are proposedand developed, the constitutive equations are rewritten in according to those proposedin the previous section. However, to clarify better this point, we consider an example ofconstitutive equations, in case of pure mechanical problem, rearranged for the mixedmodels.

In mixed models for the mechanical case, the primary variables are the displace-ments and the out-plane stresses σnM , these last are not obtained by Eq.(2.79), but aremodelized (subscript M). From Eqs.(2.78, 2.79) the new ones for RMVT are:

σkpC = Q

k

pp εkpG + Q

k

pn σknM , (2.98)

εknC = Q

k

np εkpG + Q

k

nn σknM , (2.99)

where the new coefficients are:

Qk

pp = Qkpp −Qk

pnQk−1

nn Qknp , Q

k

pn = QkpnQ

k−1

nn , (2.100)

Qk

np = −Qk−1

nn Qnp , Qk

nn = Qk−1

nn .

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 57

In Eqs.(2.100), only Qk

pp is a stiffness coefficients matrix.

2.4 Constitutive equations for FGMs

In the case of Functionally Graded Materials (FGMs), the properties change with conti-nuity along a particular direction of plates and shells. In the present work, mechanical,electrical and thermal properties can vary with continuity along the thickness direc-tion z. The matrices of elastic coefficients Q, piezoelectric coefficients e, dielectriccoefficients ε, thermo-mechanical coupling coefficients λ, pyroelectric coefficients p,conductivity coefficients κ and the scalar χ are given for a kth FGM layer as:

Q(z) = Q0 ∗ f(z) ,

e(z) = e0 ∗ g(z) ,

ε(z) = ε0 ∗ h(z) ,

λ(z) = λ0 ∗ i(z) , (2.101)p(z) = p0 ∗ l(z) ,

κ(z) = κ0 ∗m(z) ,

χ(z) = χ0 ∗ n(z) .

The functions f(z), g(z), h(z), i(z), l(z), m(z) and n(z) are general continuous func-tions of the thickness coordinate z. The variation in z of these material properties aredescribed via particular thickness functions, that are a combination of Legendre poly-nomials (see [85] and [86]):

Q(z) = Fb(z)Qb + Fγ(z)Qγ + Ft(z)Qt = FrQr ,

e(z) = Fb(z)eb + Fγ(z)eγ + Ft(z)et = Frer ,

ε(z) = Fb(z)εb + Fγ(z)εγ + Ft(z)εt = Frεr ,

λ(z) = Fb(z)λb + Fγ(z)λγ + Ft(z)λt = Frλr , (2.102)p(z) = Fb(z)pb + Fγ(z)pγ + Ft(z)pt = Frpr ,

κ(z) = Fb(z)κb + Fγ(z)κγ + Ft(z)κt = Frκr ,

χ(z) = Fb(z)χb + Fγ(z)χγ + Ft(z)χt = Frχr ,

t and b are the top and bottom values, and γ terms denote the higher order terms ofexpansion. The thickness functions Fr(ζk) have been defined at the k-layer level, theyare a linear combination of Legendre polynomials Pj = Pj(ζk) of the jth-order definedin ζk-domain (ζk = 2zk

hkwith zk local coordinate and hk thickness, both referred to kth

layer, so −1 ≤ ζk ≤ 1). For example, the first five Legendre polynomials are:

P0 = 1, P1 = ζk, P2 =(3ζk

2 − 1)

2, P3 =

5ζk3

2− 3ζk

2, P4 =

35ζk4

8− 15ζk

2

4+

3

8, (2.103)

their combinations for the thickness functions are:

F10 = Ft =P0 + P1

2, F1 = Fb =

P0 − P1

2, Fγ = Pγ − Pγ−2 with γ = 2, . . . , 9 .

(2.104)

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58 CHAPTER 2

In the present work a value of r equal to 10 has been chosen, which always guaranteesa good approximation of the FGMs properties. To obtain the Qr, er εr, λr, pr, κr andχr values, it is sufficient to solve a simple algebraic system as shown in Eq.(2.105) forthe stiffness coefficients matrix:

Qij(z1)

...

Qij(zNr)

=

Fb(z1) · · · Fγ(z1) · · · Ft(z1)

......

...

Fb(zNr) · · · Fγ(zNr) · · · Ft(zNr)

Qijb...

Qijγ...

Qijt

. (2.105)

The values of Q(z), e(z), ε(z), λ(z), p(z), κ(z) and χ(z), and the thickness functionsFr(z) are known in ten different locations along the thickness of the plate. By solvingthe system in Eq.(2.105) the values Qr are obtained, and the same is for the other mate-rial properties (see [42] and [43]). The material properties, varying with continuity inthe thickness direction z, can be recovered as illustrated in Eq.(2.106) and in Figure 2.3:

(Qpp(z), Qpn(z),Qnp(z),Qnn(z)) = Fr(Qppr,Qpnr,Qnpr,Qnnr) ,

(epp(z), epn(z), enp(z), enn(z)) = Fr(eppr, epnr, enpr, ennr) ,

(εpp(z), εpn(z), εnp(z), εnn(z)) = Fr(εppr, εpnr, εnpr, εnnr) ,

(λp(z),λn(z)) = Fr(λpr,λnr) , (2.106)(pp(z), pn(z)) = Fr(ppr,pnr) ,

(κpp(z),κpn(z),κnp(z),κnn(z)) = Fr(κppr, κpnr,κnpr,κnnr) ,

χ(z) = Frχr .

Constitutive relations considered in Eqs.(2.65)-(2.71), have the material coefficientsdepending by the thickness coordinates z in the case of FGMs:

σkpC = Qk

pp(z)εkpG + Qk

pn(z)εknG − ekT

pp (z)EkpG − ekT

np (z)EknG − λk

p(z)θk , (2.107)

σknC = Qk

np(z)εkpG + Qk

nn(z)εknG − ekT

pn (z)EkpG − ekT

nn(z)EknG − λk

n(z)θk , (2.108)

DkpC = ek

pp(z)εkpG + ek

pn(z)εknG + εk

pp(z)EkpG + εk

pn(z)EknG + pk

p(z)θk , (2.109)

DknC = ek

np(z)εkpG + ek

nn(z)εknG + εk

np(z)EkpG + εk

nn(z)EknG + pk

n(z)θk , (2.110)

ηkC = λkT

p (z)εkpG + λkT

n (z)εknG + pkT

p (z)EkpG + pkT

n (z)EknG + χk(z)θk , (2.111)

hkp = κk

pp(z)ϑkpG + κk

pn(z)ϑknG , (2.112)

hkn = κk

np(z)ϑkpG + κk

nn(z)ϑknG . (2.113)

By considering the approximation given in Eqs.(2.106), it is possible to obtain a generalform of constitutive relations for the Eqs.(2.107)-(2.113), they are valid for both casesof functionally graded materials and materials with constant properties through the

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 59

Q

Q Q Q Q

Q Q Q Q

QQQ Q

Q

QQ

Q Q Q Q

Q Q

Q

Q Q Q

QQ

Q

Q Q Q Q

QQ

Q Q

Q

Q

Q

Q Q Q Q

Q

Q

Q

Q Q

Q

QQQQ

Q Q Q Q

Q

Q

Q Q Q

Q

Q

Q

Q QQ Q

Q

Q

Q

Q

Q

Q (z)=F (z)Q

e (z)=F (z)e

(z)=F (z)

(z)=F (z)

p (z)=F (z)p

(z)=F (z)

ij r ijr

ij r ijr

i r ir

ij r ijr

i r ir

r r

l l

e e

c c

ij r ijr(z)=F (z)kk

Figure 2.3: Example of assembling on index r for the FGM properties.

thickness direction z:

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Fre

kTpprEk

pG − FrekTnprEk

nG − Frλkprθ

k , (2.114)

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Fre

kTpnrEk

pG − FrekTnnrEk

nG − Frλknrθ

k , (2.115)

DkpC = Fre

kpprε

kpG + Fre

kpnrε

knG + Frε

kpprEk

pG + FrεkpnrEk

nG + Frpkprθ

k , (2.116)

DknC = Fre

knprε

kpG + Fre

knnrε

knG + Frε

knprEk

pG + FrεknnrEk

nG + Frpknrθ

k , (2.117)

ηkC = Frλ

kTpr εk

pG + FrλkTnr εk

nG + FrpkTpr Ek

pG + FrpkTnr Ek

nG + Frχkrθ

k , (2.118)

hkp = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG , (2.119)

hkn = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG , (2.120)

where k = 1, . . . , Nl indicates the considered layers, and r = 1, . . . , 10 is the loop toapproximate the FGM properties varying with the z coordinate. In the case of materialswith constant properties in z, the loop on r index is not necessary and the materialcoefficients are constant.

For the mixed constitutive equations in Section 2.3, the approximation in Eqs.(2.106)is directly applied to modified coefficients in Eqs.(2.100).

2.5 Geometrical relations for shells and plates

We define a thin shell as a three-dimensional body bounded by two closely spacedcurved surfaces, the distance between the two surfaces must be small in comparisonwith the other dimensions. The middle surface of the shell is the locus of points whichlie midway between these surfaces. The distance between the surfaces measured alongthe normal to the middle surface is the thickness of the shell at that point [87]. Shellsmay be seen as generalizations of a flat plate [88]; conversely, a flat plate is a special

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60 CHAPTER 2

case of a shell having no curvature. In this section the fundamental equations of thinshell theory are presented in order to obtain the geometrical relations for multifieldproblems. Geometrical relations for plates are seen as particular case of those for shells.

The material is assumed to be linearly elastic and homogeneous, displacements areassumed to be small, thereby yielding linear equations; shear deformation and rotaryinertia effects are neglected, and the thickness is taken to be small.

The deformation of a thin shell is completely determined by the displacements ofits middle surface [87]. The equation of the undeformed middle surface is given, interms of two independent parameters α and β, by the radius vector:

~r = ~r(α, β) . (2.121)

Eq.(2.121) determines a space curve on the surface. Such curves are called β curvesand α curves, see Figure 2.4. We can assume that the parameters α and β alwaysvary within a definite region, and that a one-to-one correspondence exists between thepoints on this region and points on the portion of the surface of interest:

~r,α =∂~r

∂α, ~r,β =

∂~r

∂β. (2.122)

The vectors ~r,α and ~r,β are tangent to the α and β curves, respectively. Their length is:

| ~r,α |= A , | ~r,β |= B . (2.123)

Consequently ~r,α/A and ~r,β/B are unit vectors tangent to the coordinates curves. Theangle between the coordinate curves is χ:

~r,α

A· ~r,β

B= cos χ , (2.124)

where~r,α

A= iα ,

~r,β

B= iβ , in =

iα × iβsin χ

, (2.125)

in is the unit vector of the normal to the surface and is orthogonal to the vectors iα andiβ . The unit vectors iα, iβ and in are usually called the basic vectors of the surface [87].

First quadratic form

If we consider two points (α, β) and (α + dα, β + dβ) arbitrarily near to each other andboth lying on the surface, the increment of the vector ~r in moving from the first pointto the second one is:

d~r = ~r,αdα + ~r,βdβ . (2.126)By considering Eqs.(2.123), (2.124), (2.125) and (2.126), we can obtain the square of thedifferential of the arc length on the surface:

d~r · d~r = ds2 = A2dα2 + 2AB cos χ dα dβ + B2dβ2 . (2.127)

The right-hand side of Eq.(2.127) is the first quadratic form of the surface. This formdetermines the infinitesimal lengths, the angle between the curves, and the area on thesurface: the intrinsic geometry of the surface. However, it does not determine a surfaceby itself. The terms A2, AB cos χ, and B2 are called first fundamental quantities.

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 61

Figure 2.4: Middle surface coordinates.

Second quadratic form

The problem of finding the curvature of a curve which lies on the surface, can be solvedby considering the second quadratic form of the surface. ~r = ~r(s) is the vectorial equationof a curve on the surface (s is the arc length from a certain origin). τ is the unit vectoralong the tangent to the curve:

τ =d~r

ds= ~r,α

ds+ ~r,β

ds. (2.128)

According to Frenet’s formula [89], the derivative of this vector is:

ds=

N

ρ, (2.129)

where 1/ρ is the curvature of the curve, and N is the unit vector of the principal normalto the curve.

By omitting the middle passages, detailed described in [87], is possible to obtainthe expression for the second quadratic form: Ldα2 + 2Mdαdβ + Ndβ2.

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62 CHAPTER 2

L, M and N are the coefficients of the form. The second quadratic form is thusrelated to the curvatures of the curves on the surface. The curvatures of the α curvesand the β curves take β = constant and α = constant, respectively:

1

= − L

A2,

1

= − N

B2. (2.130)

When A, B, Rα and Rβ are given, they determine a surface uniquely, except to positionand orientation in space [87]. Rα and Rβ are the radii of curvature.

Strain-displacement equations

To describe the location of an arbitrary point in the space occupied by a thin shell, theposition vector is defined as:

~R(α, β, z) = ~r(α, β) + zin , (2.131)

where z measures the distance of the point from the corresponding point on the middlesurface along in and varies over the thickness (−h/2 ≤ z ≤ h/2). The magnitude of anarbitrary infinitesimal change in the vector ~R(α, β, z) is determined by:

(ds)2 = d~R · d~R = ( ~dr + zdin + indz)( ~dr + zdin + indz) . (2.132)

Remembering the orthogonality of the coordinate system and the chain rule:

din =∂in∂α

dα +∂in∂β

dβ , (2.133)

one obtains:(ds)2 = g1dα2 + g2dβ2 + g3dz2 , (2.134)

whereg1 = [A(1 +

z

)]2 , g2 = [B(1 +z

)]2 , g3 = 1 . (2.135)

The quantities g1, g2, g3, A, B, Rα, Rβ are connected by the equations of Lamb [90],since the three-dimensional space (the space in which the three independent variablesα, β, z vary) is an Euclidean space.

The fundamental shell element is the differential element bounded by two surfacesdz apart at a distance z from the middle surface, and four ruled surfaces whose gen-erators are the normals to the middle surface along the parametric curves α = α0,α = α0 + dα, β = β0 and β = β0 + dβ [87]. The lengths of the edges of this fundamentalelement are (see Figure 2.5):

ds(z)α = A(1 + z/Rα)dα , (2.136)

ds(z)β = B(1 + z/Rβ)dβ ,

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 63

Figure 2.5: Notation and positive directions in shell coordinates.

the differential areas of the edge faces of the fundamental element are (see Figure 2.5):

dA(z)α = A(1 + z/Rα)dαdz , (2.137)

dA(z)β = B(1 + z/Rβ)dβdz ,

while the volume of the fundamental element is:

dV (z) = [A(1 + z/Rα)][B(1 + z/Rβ)]dαdβdz . (2.138)

The well-known strain-displacement equations of three-dimensional theory of elas-ticity in orthogonal curvilinear coordinates have been obtained in [68]:

ei =∂

∂αi

( Ui√gi

)+

1

2gi

3∑

k=1

∂gi

∂αk

Uk√gk

, i = 1, 2, 3 , (2.139)

γij =1√gigj

[gi

∂αj

( Ui√gi

)+ gj

∂αi

( Uj√gj

)], i, j = 1, 2, 3 i 6= j , (2.140)

where the ei, γij and Ui are normal strains, shear strains, and displacement compo-nents, respectively, at an arbitrary point. In the shell coordinates the indices 1, 2 and3 are replaced by α, β and z, respectively, except for the displacements U1, U2 and U3,which are replaced by u, v, and w, respectively. Coefficients of the metric tensor are

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64 CHAPTER 2

given by Eq.(2.135), thus yielding:

eα =1

(1 + z/Rα)

( 1

A

∂u

∂α+

v

AB

∂A

∂β+

w

), (2.141)

eβ =1

(1 + z/Rβ)

( u

AB

∂B

∂α+

1

B

∂v

∂β+

w

), (2.142)

ez =∂w

∂z, (2.143)

γαβ =A(1 + z/Rα)

B(1 + z/Rβ)

∂β

[ u

A(1 + z/Rα)

]+

B(1 + z/Rβ)

A(1 + z/Rα)

∂α

[ v

B(1 + z/Rβ)

], (2.144)

γαz =1

A(1 + z/Rα)

∂w

∂α+ A(1 + z/Rα)

∂z

[ u

A(1 + z/Rα)

], (2.145)

γβz =1

B(1 + z/Rβ)

∂w

∂β+ B(1 + z/Rβ)

∂z

[ v

B(1 + z/Rβ)

]. (2.146)

2.5.1 Shells: multifield geometrical relations

By considering Eqs.(2.141)-(2.146), in case of shells with constant radii of curvature,the coefficients A and B are equal to 1. In this section geometrical relations are writtenin matrix form. The strains eα, eβ and ez are replaced by the notation εαα, εββ and εzz,respectively. The separation in in-plane (p) and out-plane (n) components is consideredtoo:

εkpG = [εαα, εββ, γαβ]kT = (Dk

p + Akp) uk , (2.147)

εknG = [γαz, γβz, εzz]

kT = (Dknp + Dk

nz −Akn) uk ,

where for each layer k the vector of displacement components is u = (u, v, w). Theexplicit form of the introduced arrays follows:

Dkp =

∂α

Hkα

0 0

0∂β

Hkβ

0∂β

Hkβ

∂α

Hkα

0

, Dk

np =

0 0 ∂α

Hkα

0 0∂β

Hkβ

0 0 0

, Dk

nz =

∂z 0 00 ∂z 00 0 ∂z

, (2.148)

Akp =

0 0 1

HkαRk

α

0 0 1Hk

βRkβ

0 0 0

, Ak

n =

1Hk

αRkα

0 0

0 1Hk

βRkβ

0

0 0 0

. (2.149)

Details on Eqs.(2.147)-(2.149) are given in [80]. In [80] geometrical relations, which linkthe electrical field with the electric potential Φk, are also given:

EkpG = [Eα, Eβ]kT = −Dk

ep Φk , (2.150)

EknG = [Ez]

k = −Dken Φk , (2.151)

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CONSTITUTIVE AND GEOMETRICAL EQUATIONS 65

where the meaning of arrays is:

Dkep =

[∂α

Hkα

∂β

Hkβ

], Dk

en =[∂z

]. (2.152)

In analogy with Eqs.(2.150), (2.151) and (2.152), it is possible to define geometrical re-lations between the temperature θk and its spatial gradient ϑk:

ϑkpG = [ϑα, ϑβ]kT = −Dk

tp θk , (2.153)

ϑknG = [ϑz]

k = −Dktn θk , (2.154)

where the meaning of arrays is:

Dktp =

[∂α

Hkα

∂β

Hkβ

], Dk

tn =[∂z

]. (2.155)

In the proposed differential arrays the symbols ∂α, ∂β and ∂z indicate the partial deriva-tives ∂

∂α, ∂

∂β, and ∂

∂z, respectively. The parameters Hk

α and Hkβ are

√g1 and

√g2, respec-

tively, where g1 and g2 are clearly indicated in Eq.(2.135).

2.5.2 Plates: multifield geometrical relations

Geometrical relations for shell in Section 2.5.1, degenerate in geometrical relations forplates when the radii of curvature Rk

α and Rkβ are infinite. So the parameters Hk

α andHk

β are equal to 1, and the orthogonal curvilinear coordinates (α, β, z) degenerate inthe rectilinear ones (x, y, z). So:

εkpG = [εxx, εyy, γxy]

kT = Dp uk , (2.156)

εknG = [γxz, γyz, εzz]

kT = (Dnp + Dnz) uk , (2.157)

EkpG = [Ex, Ey]

kT = −Dep Φk , (2.158)

EknG = [Ez]

k = −Den Φk , (2.159)

ϑkpG = [ϑx, ϑy]

kT = −Dtp θk , (2.160)

ϑknG = [ϑz]

k = −Dtn θk . (2.161)

The new differential operators does not depend on the layer k:

Dp =

∂x 0 00 ∂y 0∂y ∂x 0

, Dnp =

0 0 ∂x

0 0 ∂y

0 0 0

, Dnz =

∂z 0 00 ∂z 00 0 ∂z

, (2.162)

Dep = Dtp =

[∂x

∂y

], Den = Dtn =

[∂z

].

The symbols in differential operators matrices are: ∂x = ∂∂x

, ∂y = ∂∂y

and ∂z = ∂∂z

.Further details for geometrical relations of plates, in case of thermo-electro-mechanicalproblems, can be found in [71].

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Chapter 3

Two-dimensional plate/shell theories

Two-dimensional plate/shell models are introduced in the axiomatic framework. First, classicaltheories for shells/plates, such as classical lamination theory and first order shear deformationtheory, are discussed. Refined and advanced models are obtained by means of Carrera’s UnifiedFormulation (CUF). CUF permits to obtain, in a general and unified manner, several modelsthat can differ by the chosen order of expansion in the thickness direction and by the equiva-lent single layer or layer wise approach. We define as refined theories those which have higherorders of expansion in the thickness direction for the three displacement components. Thesetheories can be extended to multifield problems by considering the modelling of temperatureand electric potential. Refined models are based on the principle of virtual displacements andits extensions to multifield problems. Advanced theories are those based upon the Reissner’smixed variational theorem and its extensions to multifield problems; in this case transverseshear/normal quantities such as the transverse shear/normal stresses and the transverse normalelectric displacement are "a priori" modelled. A complete system of acronyms is introducedto characterize these two-dimensional theories. Proposed two-dimensional models are obtaineddirectly for shells: plates are particular cases when the curvilinear orthogonal coordinates de-generate in rectilinear orthogonal ones.

3.1 Plate/shell theories

The analysis, design and construction of layered structures is a cumbersome subject.New, different and complicated effects arise to add to those that are already known fortraditional one-layered, isotropic structures [91]. Of all the possible topics, the presentchapter covers the aspects related to two-dimensional modeling of layered plate andshell structures [9]. The refined and advanced Equivalent Single Layer (ESL) and LayerWise (LW) two-dimensional models considered in this dissertation, are obtained in aunified and general manner by using Carrera’s Unified Formulation (CUF) [92], [82].

Several approaches can be used for the analysis of plates and shells: - three di-mensional approaches; - continuum based methods; - axiomatic and asymptotic two-dimensional theories. The theories largely employed in this work are axiomatic two-dimensional models, and they are extensively discussed in this chapter.

67

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68 CHAPTER 3

3.1.1 Three-dimensional problems

A first obvious approach to multilayered plates and shells can be the three-dimensional(3D) analysis. The 3D analysis can be implemented by solving, in a strong or in a weaksense, the fundamental differential equations of three-dimensional elasticity: equilib-rium equations, compatibility equations and physical constitutive relations [91]. Ar-rays of differential operators in above equations are defined in a 3D continuum bodywith domain Σ and boundary Γ. The unknown quantities, such as the displacements,stresses, strains, are defined in each point P (α, β, z) of a given reference system (α, β,z). When a plate/shell problem is approached by the direct solution of equilibrium,compatibility and constitutive equations, a 3D analysis has been acquired. In the caseof pure mechanical problems, typical examples of 3D solutions for layered structuresare given in [93]-[96].

However, 3D solutions are difficult to obtain, and often cannot be given in strongform for each case of geometry, laminate lay-out, boundary and loading conditions.Moreover, if we consider a finite element implementation of the 3D approaches, itscomputational cost results often prohibitive for practical problems. For all these rea-sons, two-dimensional (2D) approaches had a great diffusion instead of 3D ones.

3.1.2 Two-dimensional approaches

Plates and shells are, by definition, two-dimensional structures, because in them a di-mension, in general the thickness h, is at least of one order of magnitude lower than torepresentative in-plane dimensions a and b measured on the reference plate/shell sur-face Ω. This fact permits to reduce the 3D problem to a 2D one. Such a reduction can be

Figure 3.1: Geometrical notations for a multilayered plate.

seen as a transformation of the problem defined in each point PΣ(x, y, z) of the 3D con-tinuum occupied by the considered plate into a problem defined in each point PΩ(x, y)of a reference plate surface Ω. A typical multilayered plate is given in Figure 3.1, (x,y,z)is an orthogonal rectilinear coordinate system, Ω is the middle reference surface of themultilayered structure, Ωk is the reference surface for each k-layer of thickness hk. Alocal orthogonal rectilinear coordinate system (xk,yk,zk) can be defined for each layer.

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 69

The reduction from a 3D problem to a 2D one can be made for shell structures too,in this case the considered structures have curvatures in the two in-plane curvilineardirections α and β. A typical multilayered shell is given in Figure 3.2. The reference

,

K=1

K=2

K

K=Nl

h

Figure 3.2: Geometrical notations for a multilayered shell.

surface Ω is a curvilinear surface and the 2D problem is obtained considering the pointsPΩ(α, β) in stead of the points PΣ(α, β, z). In Figure 3.2, (α, β, z) is the curvilinear or-thogonal reference system. Plates are seen as particular case of shell geometries; forthis reason, the 2D refined and advanced theories presented in this chapter are directlywritten for the shell geometry.

2D modeling of plates and shells is a classical problem of the theory of structures.The elimination of the thickness coordinate z is usually performed upon integrationof equilibrium equations, compatibility equations and physical constitutive relations.The elimination of the z-coordinate can be made according to several methodologies,these methodologies lead to a significant number of approaches and techniques [97]-[99]. A possible classification of 2D approaches, even though in literature there is acertain degree of arbitrage, can be made in the following:

• continuum based or stress resultants based models;

• asymptotic type approaches;

• axiomatic type approaches.

Continuum based or stress resultants based models

According to [99], plate and shell theories can be obtained from a generalized continuaby using of Cosserat surface concept [100]. In this kind of approach, a 3D continuum,i.e. a shell, is seen as a surface on which stress resultants are defined. Then, the 2Dapproximations are introduced at a certain level and integration in the thickness direc-tion is performed. The most important advantage of such models is the easy way ofconsidering both geometric and physical nonlinear behavior in plate/shell theories.

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70 CHAPTER 3

Asymptotic type approaches

In case of asymptotic approaches a perturbation parameter δ is defined, it is usuallythe ratio between the shell thickness and a characteristic length (δ = h/a). The 3Dgoverning equations are expanded in terms of δ, for instance the equilibrium equationsEΣ can be written as:

EΣ ≈ E1Σδp1 + E2

Σδp2 + . . . + ENΣ δpN , (3.1)

where the exponents pi of the perturbation parameter δ are in general real numbers.The obtained 2D theories are related to the same order in δ. In order to obtain the ex-pansion in Eq.(3.1) several variational statements can be used. Interesting asymptoticapproaches for shell structures are given in [101] and [102]. The main advantage of anasymptotic approach is that it gives a consistent approximation: in it all the terms havethe same order of magnitude as the introduced perturbation parameter δ. 3D solutionsare approached when δ → 0. The extension of asymptotic approaches to multilayeredstructures introduces further difficulties, for example in order to take into account theanisotropy of composite layers a further mechanical parameter must be introduced (forexample the orthotropic ratio of the lamina EL/ET ).

Figure 3.3: Comparison between some 2D approaches and 3D exact solution in case ofa generic function f .

Axiomatic type approach

The 2D models which are detailed in this work are axiomatic approaches. In this casethe displacement field and/or stress field are postulated in the thickness direction z:

f(α, β, z) = f1(α, β)F1(z) + f2(α, β)F2(z) + . . . + fN(α, β)FN(z) , (3.2)

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 71

where the generic function f can be the vector of displacements u = (u, v, w) in caseof a displacement formulation, the vector of strain components ε in case of a strain for-mulation and the vector of stress components σ in case of a stress formulation. Mixedformulations can be accounted for too, for example by considering as functions f bothdisplacement components u and transverse shear/normal stress components σn =(σαz, σβz, σzz). The functions fi are the introduced unknowns which are defined on Ω,Fi are the polynomials which have been introduced as the base function of the expan-sion in z. N is the order of expansion in z-direction. Different variational statementsmust be applied depending on the formulation: a displacement formulation is based onthe Principle of Virtual Displacements (PVD) [9], while a mixed formulation could usethe Reissner’s Mixed Variational Theorem (RMVT) [83]. This work considers multi-field problems, so further unknowns can be chosen as functions f : the temperature,the electric potential and the normal electric displacement.

Axiomatic type approaches offer the advantage of introducing intuitive approxima-tions into the plate/shell behavior [103]. In Figure 3.3, two cases of axiomatic andasymptotic approaches are compared with respect to a 3D solution for a generic func-tion f .

3.2 Complicating effects of layered structures

In the case of multilayered structures new complicating effects can arise with respectto the isotropic one-layered plates and shells. These effects have a fundamental role inthe developments of any plate/shell theory [9]. For this reasons, classical 2D theoriesresult often inadequate for the analysis of such structures [29]. The main complicatingeffects introduced by the multilayered structures are:

• in-plane anisotropy;

• transverse anisotropy: zigzag effects and interlaminar continuity (C0z require-

ments).

In-plane anisotropy

In the case of laminates made by anisotropic layers, an high in-plane anisotropy canbe exhibited. This means that the structure has different mechanical-physical prop-erties in different in-plane directions [9]. If compared to the traditional isotropic one-layered structures, multilayered composite plates/shells could show higher transverseshear/normal flexibility with respect to in-plane deformability. A consequence of thisin-plane anisotropy is the coupling between shear and axial strains [29]. Such a cou-pling leads to many complications in the solution procedure of an anisotropic struc-ture. The 2D models must consider these effects, an example is given by the Higherorder Shear Deformation Theory (HSDT); but depending the magnitude of the in-planeanisotropy, such theories could be not sufficient.

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72 CHAPTER 3

Transverse anisotropy, zigzag effects and interlaminar continuity

A further complicating effect of multilayered structures can be the transverse anisotropy:they exhibit different mechanical-physical properties in the thickness direction z [9].Transverse discontinuous mechanical properties cause displacement field u in the thick-ness direction which can exhibit a rapid change of their slopes in correspondence toeach layer interface, this effect is known as the zigzag (ZZ) form of the displacementfield in the thickness direction z [104], and it is clearly shown for a sandwich struc-ture (two stiffer faces and a soft core) in Figure 3.4. In order to consider the ZZ form

Figure 3.4: Typical zigzag effect for a sandwich structure in bending response.

of displacements in deformed multilayered structures, a layer wise approach can berequested as illustrated in [105] or an opportune zigzag function can be added to dis-placements field as done in [106] and [107], however these topics are elaborated onthe next sections. In-plane stresses σp = (σαα, σββ, σαβ) can in general be discontinu-ous at each layer interface, on the contrary the transverse stresses σn = (σαz, σβz, σzz)must be continuous at each layer interface for equilibrium reasons, as clearly illus-trated in Figure 3.5 for a multilayered plate. This conditions are called in literatureas Interlaminar Continuity (IC) [108], [9]. From a qualitative point of view, in Figure3.6 the observable behavior of in-plane stresses, displacements and transverse stressesthrough the thickness z of a multilayered plate is clearly indicated. In-plane compo-nents of stres can be discontinuous or continuous and they are given in the figureonly for a comparison purpose. Displacements must be continuous in z direction forcompatibility reasons, transverse shear/normal stresses must be continuous in thick-ness z-direction for equilibrium reasons, so u and σn are C0-continuous functions inthe z direction. Moreover, displacements and transverse stresses have discontinuousfirst derivatives with correspondence to each interface because the mechanical prop-erties change in each layer (ZZ effect). In [108] and [109] ZZ and IC conditions arereferred as C0

z -requirements. The fulfillment of C0z -requirements is a crucial point in the

development of appropriate 2D models for multilayered structures. Displacements formu-lations must fulfill the C0

z -requirements for the displacement components; mixed for-mulations must fulfill C0

z -requirements for both displacements and transverse stresses.In the case of multilayered anisotropic structures, classical theories, such those basedon Cauchy-Poisson-Kirchhoff-Love [110]-[113] hypothesis or Reissner-Mindlin [114],

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 73

Figure 3.5: Interlaminar Continuity in a multilayered plate: continuity and discontinu-ity of stress components at layer interfaces.

u v w

K=1

K=2

K=3

In-plane stresses Displacements Transverse stresses

Figure 3.6: C0z -requirements for displacements and stresses in a three-layered compos-

ite plate.

[115] hypothesis, which will be discussed in the next section, fulfill the IC conditionsfor displacements, but not the ZZ form of u. For these reasons they can often resultinappropriate for the study of multilayered composite plates and shells.

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74 CHAPTER 3

3.3 Classical theories

Classical theories have been originally developed for one-layered isotropic structures.These can be divided in two main groups: Love First Approximation Theories (LFAT)and Love Second Approximation Theories (LSAT). LFAT are based on the well-knownCauchy-Poisson-Kirchhoff-Love thin shell assumptions [110]-[113]: normals to the ref-erence surface Ω remain normal in the deformed states and do not change in length,this means that transverse shear and transverse normal strains are negligible with re-spect to the other strains. When one or more of these LFAT postulates are removed, weobtain the so-called LSAT [97], for example the effects of transverse shear and/or trans-verse normal stresses can be taken into account. Consequently to the introduction ofmultilayered structures, several LFAT and LSAT have been extended to multilayeredplates and shells. However, these extensions remains in the framework of EquivalentSingle Layer (ESL) theories: the layers in the multilayered structure are seen as onlyone equivalent plate or shell, and the 2D approximation does not consider the depen-dency by the index layer k.

3.3.1 Classical lamination theory, CLT

A possible application of LFAT to multilayered structures is the Classical LaminationTheory (CLT), see books by Reddy [9] and Jones [29]. CLT is based on Kirchhoff hy-pothesis [112]:

• straight lines perpendicular to the midsurface (i.e., transverse normals) beforedeformation remain straight after deformation;

• the transverse normals do not experience elongation (i.e, they are inextensible);

• the transverse normals rotate such that they remain perpendicular to the midsur-face after deformation.

These hypothesis are clearly summarized in Figure 3.7. The first two assumptionsimply that transverse displacement is independent of the transverse (or thickness) co-ordinate and the transverse normal strain εzz is zero. The third assumption results inzero transverse shear strains: γxz = γyz=0. The displacement field for a plate is:

u(x, y, z) = u0(x, y)− z∂w0

∂x,

v(x, y, z) = v0(x, y)− z∂w0

∂y, (3.3)

w(x, y, z) = w0(x, y) .

Only three degrees of freedom are used for these 2D theory: the displacements in thethree directions referred to the midsurface Ω. In CLT, in order to avoid the Poisson’slocking phenomena, the σzz = 0 condition must be enforced as seen in Section 2.1.2.For further details about this topic, readers can refer to [66], [67]. In case of a three-layered plate, typical displacements trough the thickness direction z are given in Figure

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 75

Figure 3.7: Undeformed and deformed geometry of a plate in according to Kirchhoffhypothesis.

3.8. Figure 3.9 indicates the typical behavior of in-plane displacement componentsu, v (linear and equivalent single layer) and transverse shear stresses (zero for all themultilayer) in the thickness direction z.

3.3.2 First order shear deformation theory, FSDT

A typical LSAT in case of multilayered structures is the First order Shear DeformationTheory (FSDT). The third part of Kirchhoff hypothesis is removed, so the transversenormals do not remain perpendicular to the midsurface after deformation. In this way,transverse shear strains γxz and γyz are included in the theory. However, the inexten-sibility of transverse normal remains, so displacement w is constant in the thicknessdirection z. FSDT is the extension of the so-called Reissner-Mindlin model [114], [115]to multilayered structures. The displacement model in the case of FSDT for a plate is:

u(x, y, z) = u0(x, y) + zΦx(x, y) ,

v(x, y, z) = v0(x, y) + zΦy(x, y) , (3.4)w(x, y, z) = w0(x, y) .

The hypothesis of FSDT are clearly shown in Figure 3.10. The displacement fieldof FSDT has five unknowns (for CLT they were three): they are the displacements ofmidsurface (u0, v0, w0) and the rotations of a transverse normal about the x- and y-axes(Φy, Φx). In the case of CLT the rotations coincide with the derivatives: Φx = −∂w0

∂xand

Φy = −∂w0

∂y. Only strain εzz is zero, so stresses σxz and σyz are different from zero. Figure

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76 CHAPTER 3

Figure 3.8: Displacement components through the thickness direction z in case of CLTand FSDT.

Figure 3.9: CLT: displacements u and v, and transverse shear stresses through the thick-ness direction z.

3.11 indicates the typical behavior of in-plane displacement components u, v (linearand equivalent single layer) and transverse shear stresses (constant in each layer) inthe thickness direction z. Poisson’s locking phenomena exists because the transversenormal strain εzz remains zero: it can be avoided by the enforced σzz = 0 condition asseen in Section 2.1.2 and in [66], [67].

3.3.3 Vlasov-Reddy theory, VRT

A refinement of Reissner-Mindlin theory was made by Vlasov in the case of one-layered isotropic structures [116]. This theory permits the fulfillment of the homo-geneous conditions for the transverse shear stresses in correspondence to the top andbottom of the plate/shell. Reddy [117] and Reddy and Phan [118] have shown thatsuch an inclusion leads to significant improvements with respect to FSDT in case oflayered structures (static and dynamic analysis). The resulting model is called Vlasov-

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 77

Figure 3.10: Undeformed and deformed geometry of a plate in according to Reissner-Mindlin hypothesis.

Figure 3.11: FSDT: displacements u and v, and transverse shear stresses through thethickness direction z.

Reddy Theory (RVT), and its displacement model for a plate is:

u(x, y, z) = u0(x, y) + zΦx(x, y) + z3(− 4

3h2)(Φx +

∂w0

∂x) ,

v(x, y, z) = v0(x, y) + zΦy(x, y) + z3(− 4

3h2)(Φy +

∂w0

∂y) , (3.5)

w(x, y, z) = w0(x, y) .

The model in Eq.(3.5), as each ESL theory with transverse displacement w constant orlinear in z, needs to the correction of Poisson’s locking phenomena [66], [67].

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78 CHAPTER 3

3.4 Carrera’s unified formulation: refined models

We define as refined models those displacements models where higher orders of expan-sion in the thickness direction z are assumed for all the three displacement compo-nents. These axiomatic 2D models can be seen in Equivalent Single Layer (ESL) formwhen the layers included in the multilayered are considered as one equivalent struc-ture, and in Layer Wise (LW) form when each layer embedded in the multilayered isseparately considered in order to write the expansions in z for each layer k. In the caseof multifield problems, refined models are those where the extension is made by con-sidering as primary variables the temperature and the electric potential in addition tothe displacement vector. These models are obtained by using the Principle of VirtualDisplacements (PVD) [91] and its extensions to multifield problems [71], [4].

Carrera’s Unified Formulation (CUF) is a technique which handles a large varietyof plate/shell models in a unified manner [92]. According to CUF, the governing equa-tions are written in terms of a few fundamental nuclei which do not formally dependon the order of expansion N used in the z direction and on the description of variables(LW or ESL) [109], [119]. The application of a two-dimensional method for plates andshells permits to express the unknown variables as a set of thickness functions depend-ing only on the thickness coordinate z and the correspondent variable depending onthe in-plane coordinates α and β. So that the generic variable f(α, β, z), for instance adisplacement, and its variation δf(α, β, z) are written according to the following gen-eral expansion:

f(α, β, z) = Fτ (z)f τ (α, β) , δf(α, β, z) = Fs(z)δf s(α, β) , (3.6)with τ, s = 1, . . . , N .

Bold letters denote arrays, (α,β) are the in-plane coordinates and z the thickness one.The summing convention with repeated indexes τ and s is assumed. The order of ex-pansion N goes from first to higher order values, and depending on the used thicknessfunctions, a model can be: ESL when the variable is assumed for the whole multilayerand a Taylor expansion is employed as thickness functions F (z); LW when the variableis considered independent in each layer and a combination of Legendre polynomialsare used as thickness functions F (z). In CUF the maximum order of expansion N inz direction is fourth, in the present dissertation higher values have been employed,until to N = 14, in order to analyze complicated cases such as the thermal analysis offunctionally graded material structures.

3.4.1 Equivalent single layer theories, ESL

The displacement u = (u, v, w) is described according to Equivalent Single Layer de-scription if the unknowns are the same for the whole plate [120], [121]. The z expansion

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 79

is obtained via Taylor polynomials, that is:

u = F0 u0 + F1 u1 + . . . + FN uN = Fτ uτ ,

v = F0 v0 + F1 v1 + . . . + FN vN = Fτ vτ , (3.7)w = F0 w0 + F1 w1 + . . . + FN wN = Fτ wτ ,

with τ = 0, 1, . . . , N ; N is the order of expansion that ranges from 1 (linear) to 14:

F0 = z0 = 1, F1 = z1 = z, . . . , FN = zN . (3.8)

Eq.(3.7) can be written in a vectorial form:

u(α, β, z) = Fτ (z)uτ (α, β) , δu(α, β, z) = Fs(z)δus(α, β) , (3.9)with τ, s = 1, . . . , N .

The 2D models obtained from Eqs.(3.7)-(3.9) are denoted by the acronym EDN whereE indicates that an Equivalent Single Layer approach have been employed, D indi-cates that the theory is a displacement formulation, N indicates the order of expansionin the thickness direction. For example, a ED2 model has a quadratic expansion inz, a ED4 has a fourth order of expansion in z, and so on. In Figure 3.12 a typicaldisplacement field is indicated for a three-layered structure in case of a ED4 model.Figure 3.13 considers the displacement and the transverse stresses along z directionfor a ED2 model: displacements are quadratic in z, so transverse stresses are linear(no more constant as classical theories) but discontinuous at each interface. Simpler

Figure 3.12: ED4: displacements u, v and w through the thickness direction z.

theories can be obtained from EDN models, such those which discarding the εzz ef-fect; in this case is sufficient to impose that the transverse displacement w is constantin z, such theories are denoted as EDNd. ED1d model coincides with the FSDT. CLTis obtained from FSDT via an opportune penalty technique which imposes an infiniteshear correction factor. It is important to remember that all the theories EDNd whichhave transverse displacement constant and transverse normal strain εzz zero and theED1 model, shows Poisson’s locking phenomena; this can be overcame via plane stressconditions in constitutive equations [66], [67].

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80 CHAPTER 3

Figure 3.13: ED2: displacements and transverse shear stresses through the thicknessdirection z.

3.4.2 Murakami’s zigzag function, MZZF

The proposed ESL models in Section 3.4.1 do not consider the typical ZZ form of dis-placements in z direction, which is typical of multilayered structures with transverseanisotropy [104]. A remedy to this limitation can be the introduction of an opportunezigzag function in the ESL displacement model, in order to recover the ZZ form of dis-placements without the use of LW models. These last have an intrinsic ZZ behavior,but are more computational expansive with respect to the ESL models [105]. A possi-ble choice for the zigzag function is the so-called Murakami’s Zig-Zag Function (MZZF)[122], [123]. MZZF can be simply added to displacement model and give remarkableimprovements in the solution by satisfying the typical ZZ form of displacements inmultilayered structures.

The MZZF Z(z) is defined according to:

Z(z) = (−1)kζk , (3.10)

with the not dimensioned layer coordinate ζk = (2zk)/hk, where zk is the transversethickness coordinate and hk is the thickness of the layer, so −1 ≤ ζk ≤ 1. Z(z) has thefollowing properties: - it is piece-wise linear function of layer coordinates zk; - Z(z) hasunit amplitude for the whole layers; - the slope Z ′(z) = dZ/dz assumes opposite signbetween two-adjacent layers. Its amplitude is layer thickness independent [123]. Thedisplacement model including MZZF is:

u = F0 u0 + F1 u1 + . . . + FN uN + FZ uZ = Fτ uτ ,

v = F0 v0 + F1 v1 + . . . + FN vN + FZ vZ = Fτ vτ , (3.11)w = F0 w0 + F1 w1 + . . . + FN wN + FZ wZ = Fτ wτ ,

with τ = 0, 1, . . . , (N + 1), N is the order of expansion which ranges from 1 (linear) to14:

F0 = z0 = 1, F1 = z1 = z, . . . , FN = zN , FN+1 = FZ = (−1)kζk. (3.12)

The acronym to indicate such models is EDZN , where E states for ESL approach,D for displacements formulation, N is the order of expansion in z direction. Z indi-cates that the MZZF has been added [124]. The following remarks can be made: - the

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 81

additional degree of freedom uZ has a meaning of displacement; - the amplitude uZ

is layer independent: uZ has an intrinsic equivalent single layer description; - MZZFcan be used for both in-plane and out-of-plane displacement components [125], [126].Figure 3.14 clearly explains the meaning of MZZF and how to add it to displacementcomponents. The vectorial form of Eq.(3.11) can be written by considering as (N + 1)th

Figure 3.14: Displacements model in EDZ1 and EDZ3 theories. Inclusion of MZZF inan ESL model.

thickness function the MZZF FZ = (−1)kζk:

u(α, β, z) = Fτ (z)uτ (α, β) , δu(α, β, z) = Fs(z)δus(α, β) , (3.13)with τ, s = 1, . . . , (N + 1) .

In Figure 3.15 typical displacements and transverse shear stresses along the thick-ness z are shown for a EDZ1 model: the inclusion of MZZF permits to recover thetypical ZZ form of displacement vector in case of multilayered transverse-anisotropystructures. In analogy with the EDN models, there is the possibility to impose con-

Figure 3.15: EDZ1: displacements and transverse shear stresses through the thicknessdirection z.

stant transverse displacements w, such models are denoted as EDZNd models. BothEDZNd and EDZ1 models have the necessity to correct the Poisson’s locking phe-nomena as indicated in [66], [67].

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82 CHAPTER 3

3.4.3 Layer wise theories, LW

When each layer of a multilayered structure is described as independent plates/shells,a Layer Wise (LW) approach is accounted for [9]. The displacement uk = (u, v, w)k isdescribed for each layer k, in this way the ZZ form of displacement in multilayeredtransverse-anisotropy structures is easily obtained [127]-[130]. The recovering of ZZeffect via Layer Wise models is detailed in [131] and in Figure 3.16. The z expansion

Figure 3.16: Linear expansion in z direction for displacement components: LW ap-proach vs. ESL approach.

for displacement components is made for each layer k:

uk = F0 uk0 + F1 uk

1 + . . . + FN ukN = Fτ uk

τ ,

vk = F0 vk0 + F1 vk

1 + . . . + FN vkN = Fτ vk

τ , (3.14)

wk = F0 wk0 + F1 wk

1 + . . . + FN wkN = Fτ wk

τ ,

with τ = 0, 1, . . . , N , N is the order of expansion that ranges from 1 (linear) to 14.k = 1, . . . , Nl where Nl indicates the number of layers. The Eq.(3.14) written in vectorialform is:

uk(α, β, z) = Fτ (z)ukτ (α, β) , δuk(α, β, z) = Fs(z)δuk

s(α, β) , (3.15)with τ, s = t, b, r and k = 1, . . . , Nl ,

where t and b indicate the top and bottom of each layer k, respectively; r indicatesthe higher orders of expansion in the thickness direction: r = 2, . . . , N . The thick-ness functions Fτ (ζk) and Fs(ζk) have now been defined at the k-layer level, they area linear combination of Legendre polynomials Pj = Pj(ζk) of the jth-order defined inζk-domain (ζk = 2zk

hkwith zk local coordinate and hk thickness, both referred to kth layer,

so −1 ≤ ζk ≤ 1). The first five Legendre polynomials are:

P0 = 1, P1 = ζk, P2 =(3ζk

2 − 1)

2, P3 =

5ζk3

2− 3ζk

2, P4 =

35ζk4

8− 15ζk

2

4+

3

8, (3.16)

their combinations for the thickness functions are:

Ft = F0 =P0 + P1

2, Fb = F1 =

P0 − P1

2, Fr = Pr − Pr−2 with r = 2, . . . , N . (3.17)

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 83

The chosen functions have the following interesting properties:

ζk = 1 : Ft = 1; Fb = 0; Fr = 0 at top , (3.18)ζk = −1 : Ft = 0; Fb = 1; Fr = 0 at bottom . (3.19)

That is interface values of the variables are considered as variable unknowns. This factpermits to easily imposing the compatibility conditions for displacements at each layerinterface. The acronym to indicate such theories is LDN where L states for Layer Wiseapproach, D indicates displacements formulation and N is the order of expansion ineach layer k. A typical displacement behavior for a three layered structure is indicatedin Figure 3.17 for a LD2 model. Figure 3.18 indicates displacements and transverse

Figure 3.17: LD2: displacements u, v and w through the thickness direction z.

shear stresses for a LD3 model. The transverse shear/normal stresses are obtainedvia constitutive equations and this fact does not ensure the Interlaminar Continuity(IC). IC is enforced by a priori modelling transverse shear/normal stresses. In LayerWise models, even if a linear expansion in z is considered for transverse displacementw, Poisson’s locking phenomena does not appear: the transverse normal strain εzz ispiece-wise constant in the thickness direction [66], [67].

Figure 3.18: LD3: displacements and transverse shear stresses through the thicknessdirection z.

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84 CHAPTER 3

3.4.4 Refined models for the thermo-mechanical case

In the case of thermo-mechanical problems the primary variables are the displacementvector u = (u, v, w) and the scalar temperature θ. By considering the higher spatialgradient of temperature field, the variable θk is always modelled as Layer Wise [71]:

θk(α, β, z) = Fτ (z)θkτ (α, β) , δθk(α, β, z) = Fs(z)δθk

s (α, β) , (3.20)with τ, s = t, b, r and k = 1, . . . , Nl ,

where t and b indicate the top and bottom of each layer k, respectively. r indicatesthe higher orders of expansion in the thickness direction: r = 2, . . . , N . The thicknessfunctions are a combination of Legendre polynomials as indicated in Section 3.4.3. Thetemperature θ can be considered as an external load or as a primary variable [71], [43].A 2D model for thermo-mechanical problems is defined as ESL, ESL+MZZF or LWdepending on the choice made for the displacement vector: the temperature is alwaysconsidered Layer Wise [43], [81].

3.4.5 Refined models for the electro-mechanical case

In the case of electro-mechanical problems the primary variables are the displacementvector u = (u, v, w) and the scalar electric potential Φ. By considering the higher spatialgradient of electric potential, the variable Φk is always modelled as Layer Wise [79],[42]:

Φk(α, β, z) = Fτ (z)Φkτ (α, β) , δΦk(α, β, z) = Fs(z)δΦk

s(α, β) , (3.21)with τ, s = t, b, r and k = 1, . . . , Nl ,

where t and b indicate the top and bottom of each layer k, respectively. r indicates thehigher orders of expansion in the thickness direction: r = 2, . . . , N . The thickness func-tions are a combination of Legendre polynomials as indicated in Section 3.4.3. A 2Dmodel for electro-mechanical problems is defined as ESL, ESL+MZZF or LW depend-ing on the choice made for the displacement vector: the electric potential is alwaysconsidered Layer Wise [132], [133].

3.5 Carrera’s unified formulation: advanced models

In the case of multifield problems, we define as advanced models those 2D models ob-tained by employing Reissner’s mixed variational theorem (RMVT) [83] and its ex-tensions to thermo-electro-mechanical coupling [1]. These extensions permit to model"a priori" some transverse quantities that in PVD applications are obtained via post-processing. Transverse shear/normal stresses σn = (σαz, σβz, σzz) and/or transversenormal electric displacement Dz are modelled "a priori" and considered in Layer Wiseform. The main advantage of obtaining these variables directly from the governingequations is the fulfillment of the Interlaminar Continuity (IC) [86], [134]. These ad-vanced models are obtained by means of Carrera’s Unified Formulation (CUF) [91]which has been detailed in Section 3.4.

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 85

3.5.1 Transverse shear/normal stresses modelling

An advanced model for a pure mechanical problem considers as primary variables bothdisplacements u = (u, v, w) and transverse shear/normal stresses σn = (σαz, σβz, σzz)[86]. The displacements can be modelled as ESL (Section 3.4.1), ESL+MZZF (Section3.4.2) and LW (Section 3.4.3), and this choice permits to define the considered advancedmodel as ESL, ESL+MZZF or LW, respectively: the transverse shear/normal stressesσnM are always LW (the subscript M means that the stresses are modelled and notobtained from the constitutive equations). The LW model for stresses is:

σkαz = F0 σk

αz0 + F1 σkαz1 + . . . + FN σk

αzN = Fτ σkαzτ ,

σkβz = F0 σk

βz0 + F1 σkβz1 + . . . + FN σk

βzN = Fτ σkβzτ , (3.22)

σkzz = F0 σk

zz0 + F1 σkzz1 + . . . + FN σk

zzN = Fτ σkzzτ ,

with τ = 0, 1, . . . , N , N is the order of expansion that ranges from 1 (linear) to 14.k = 1, . . . , Nl where Nl indicates the number of layers. The Eq.(3.22) written in vectorialform is:

σknM(α, β, z) = Fτ (z)σk

nMτ (α, β) , δσknM(α, β, z) = Fs(z)δσk

nMs(α, β) , (3.23)with τ, s = t, b, r and k = 1, . . . , Nl ,

where t and b indicate the top and bottom of each layer k, respectively. r indicatesthe higher orders of expansion in the thickness direction: r = 2, . . . , N . The thicknessfunctions Fτ (ζk) and Fs(ζk) have now been defined at the k-layer level, they are a linearcombination of Legendre polynomials. The use of such thickness functions, thanks theproperty remarked in Eqs.(3.18) and (3.19), permits to easily write the InterlaminarContinuity for the transverse stresses:

σknt = σk+1

nb with k = 1, . . . , (Nl − 1) , (3.24)

that means: in each interface the top value of the layer k is equal to the bottom valueof the layer (k + 1). The same property can be used for displacements in Layer Wiseform, in order to impose the compatibility conditions:

ukt = uk+1

b with k = 1, . . . , (Nl − 1) . (3.25)

We define as EMN models those with displacements in ESL form (E) and trans-verse stresses in Layer Wise form, M means mixed formulation (use of RMVT), N isthe order of expansion that is the same for both variables. EMZN models considerthe displacements modelled in ESL form with the inclusion of MZZF. LMN modelsconsider both displacements and transverse stresses in Layer Wise form. Figure 3.19gives the displacements and transverse stresses for a EM2 model. The displacementsare considered ESL, the transverse stresses are a priori modelled and directly obtainedfrom the governing equations: they are considered in LW form, and this permits tosatisfy both ZZ form and IC. If transverse stresses are obtained from constitutive equa-tions via post-processing, the IC could be not ensured. Figure 3.20 gives displacements

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86 CHAPTER 3

Figure 3.19: EM2: displacements and transverse shear stresses through the thicknessdirection z.

and stresses for a LM2 model, in this case displacements are LW too, and ZZ form andIC are ensured for both displacement and transverse stress components. The trans-verse stresses obtained from constitutive equations could not satisfy the InterlaminarContinuity [134].

Figure 3.20: LM2: displacements and transverse shear stresses through the thicknessdirection z.

3.5.2 Advanced models for the thermo-mechanical case

Advanced models for thermo-mechanical cases are a natural extension of those dis-cussed in Section 3.5.1: the internal thermal work is included and the temperatureθk is modelled via Layer Wise approach as done in Section 3.4.4 [71]. So, in EMN ,EMZN and LMN models, transverse stresses σnM and temperature θ are in LW form;displacements u are in ESL, ESL+MZZF and LW form, respectively [1].

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TWO-DIMENSIONAL PLATE/SHELL THEORIES 87

3.5.3 Advanced models for the electro-mechanical case

In the case of electro-mechanical problems, several extensions of RMVT can be ac-counted for [83], [1]. In such models displacements u and electrical potential Φ arealways considered in the governing equations, the electric potential Φ is modelled al-ways in LW form as discussed in Section 3.4.5, the displacement components u aremodelled as ESL, ESL+MZZF or LW, and this choice defines the considered advancedmodel as ESL, ESL+MZZF or LW.

Three different extensions of RMVT to electro-mechanical problems can be con-sidered and in addition to displacements and electric potential Φ, the other modelledvariables are:

1. by using only one Lagrange multiplier [83] the transverse stresses σnM are mod-elled a priori (LW form as described in Section 3.5.1) [79];

2. by using only one Lagrange multiplier the transverse electric displacement Dz isobtained a priori in LW form;

3. by considering two Lagrange multipliers both transverse stresses and transversenormal electric displacement are modelled a priori in LW form.

The Layer Wise expansion for the normal electric displacement Dz is:

Dkz (α, β, z) = Fτ (z)Dk

zτ (α, β) , δDkz (α, β, z) = Fs(z)δDk

zs(α, β) , (3.26)with τ, s = t, b, r and k = 1, . . . , Nl ,

where t and b indicate the top and bottom of each layer k, respectively. r indicates thehigher orders of expansion in the thickness direction: r = 2, . . . , N .

The "a priori" variables for these three advanced models are:

1. displacements u, transverse stresses σnM and electric potential Φ for case 1;

2. displacements u, electric potential Φ and transverse normal electric displacementDz for the case 2;

3. displacements u, electric potential Φ, transverse stresses σnM and transverse nor-mal electric displacement Dz for the case 3.

The acronyms for such advanced models are detailed in the next Chapter, where varia-tional statements are discussed.

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Chapter 4

Variational statements for multifieldproblems

Variational statements, considered for multifield problems, are the Principle of Virtual Dis-placements (PVD) and the Reissner’s Mixed Variational Theorem (RMVT). PVD extended tothermo-electro-mechanical problems is easily obtained by considering the thermal, electrical andmechanical internal works and the opportune constitutive equations. By using Carrera’s Uni-fied Formulation (CUF), several refined two-dimensional models are given. Particular casesof PVD are simply obtained by discarding thermal, electrical or mechanical internal works.The extension of RMVT to multifield problems is obtained by rearranging the constitutiveequations, and using opportune Lagrange multipliers in the variational statement. The two-dimensional models for multifield problems obtained via RMVT are called advanced models.In this chapter several extensions of PVD and RMVT to multifield problems are discussed,the governing equations are only introduced in a general form, their closed and finite elementapplications are detailed in the next chapter.

4.1 Introduction

Thermo-electro-mechanical problems for multilayered structures had a growing inter-est in recent years [135]. For this reason several three-dimensional solutions are avail-able in literature [136]-[138] for problems considering three different involved physicalfields. Other interesting reference solutions are those related to the electro-mechanicalproblem [139]-[143], and those for thermo-mechanical coupling [144]-[147]. However,by considering the introduction of new materials in aerospace field, i.e. composites,sandwiches and functionally graded materials, the pure mechanical problem remainsa fundamental topic, as illustrated in several works about the three-dimensional anal-ysis of plates and shells [148]-[151]. The possibility of treating in a unified manner therefined and advanced two-dimensional models for multifield problems could repre-sent a fundamental topic in the structural analysis. In this chapter the refined mod-els are obtained by means of the Principle of Virtual Displacement (PVD) and its ex-tensions to multifield problems. The most general case is the PVD written for thethermo-electro-mechanical case, the other PVD applications (thermo-mechanical case,

89

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90 CHAPTER 4

electro-mechanical case, and so on) can be considered as particular cases of the mostgeneral thermo-electro-mechanical application [71]. Reissner’s Mixed Variational The-orem (RMVT) [83] extended to thermo-electro-mechanical problems gives several ad-vanced two-dimensional models, where some transverse variables such as the trans-verse shear/normal stresses [84] and/or transverse normal electrical displacement area priori modelled: in the PVD case these variables are obtained with an opportunepost-processing via constitutive equations. The variables obtained a priori satisfy theInterlaminar Continuity (IC). The extension of RMVT to thermo-electro-mechanicalproblems is not so intuitive as the PVD case; problems such as the thermo-mechanicalor the electro-mechanical ones, are not always particular cases of the most generalthermo-electro-mechanical RMVT extension. In order to clarify these aspects, PVD andRMVT extensions for multifield problems are discussed in this chapter for each possi-ble combination (one, two or three involved physical fields). Governing equations arewritten in a general form in order to simplify the relationships between general andparticular cases.

4.2 Principle of virtual displacements, PVD

In the recent past, several two-dimensional approaches have been successfully ex-tended to multilfield problems [152]-[158]. Refined models, obtained via the exten-sion of the Principle of Virtual Displacements (PVD) to thermo-electro-mechanical case(TEM) and employing Carrera’s Unified Formulation (CUF), give the possibility tochoose the order of expansion in the thickness direction and the multilayer description(Equivalent Single Layer (ESL) or Layer Wise (LW)).

The PVD for a thermo-electro-elastic medium can be derived from the Hamilton’sprinciple as indicated in [71] and [1]:

δ

∫ t

t0

(Ec − Ep)dt = 0 ⇒ δ

∫ t

t0

Ecdt− δ

∫ t

t0

Epdt = 0 , (4.1)

where Ec and Ep are the kinetic and potential energy, respectively. δ is the variationalsymbol, t0 the initial time and t a generic instant [1]. The total potential energy Ep

includes the thermopiezoelectric enthalpy density H as described in Eq.(2.40) and thework done by surface tractions tj , electric charge Q and thermal forces P on the dis-placements uj , electric potential Φ and temperature θ, respectively:

Ep =

V

HdV −∫

Γ

(tjuj − QΦ− P θ)dΓ , (4.2)

where V is the volume and Γ the boundary of the reference surface Ω.The variation of the kinetic energy Ec is the well-known relation [1], [71]:

δ

∫ t

t0

Ecdt = δ

∫ t

t0

dt

V

(1

2ρuiui)dV =

∫ t

t0

V

ρuiδuidV dt (4.3)

=

V

ρuiδuidV |t1t0 −∫ t

t0

V

ρuiδuidV dt ,

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 91

since δui vanishes at t0 and t1, the following expression can be obtained:

δ

∫ t

t0

Ecdt = −∫ t

t0

V

ρuiδuidV dt = −∫ t

t0

δLindt , (4.4)

where ρ is the mass density, ui and ui are the first and second temporal derivative ofdisplacement ui, respectively. δLin is the virtual variation of the work done by theinertial loads.

The variation of the potential energy Ep can be rewritten in according to Eqs.(2.39)-(2.42):

δ

∫ t

t0

Epdt = δ

∫ t

t0

[ ∫

V

(G(εij, Ei, θ)− F (ϑi)

)dV −

Γ

(tjuj − QΦ− P θ)dΓ]dt , (4.5)

where G is the Gibbs free-energy function and F is the dissipation function [76], [77],[78]. The variables of the problem are the strains vector εij , the electric field Ei, thetemperature θ and the vector containing the spatial derivatives of the temperature ϑi.The contribution given by the external loads is the virtual variation of the externalwork:

δLe = δ

Γ

(tjuj − QΦ− P θ)dΓ . (4.6)

By using Eq.(4.6), Eq.(4.5) can be rewritten as:

δ

∫ t

t0

Epdt = δ

∫ t

t0

V

(G(εij, Ei, θ)− F (ϑi)

)dV dt−

∫ t

t0

δLedt . (4.7)

Differentiating each term in Eq.(4.7):

δ

∫ t

t0

Epdt =

∫ t

t0

V

( ∂G

∂εij

δεij +∂G

∂Ei

δEi +∂G

∂θδθ − ∂F

∂ϑi

δϑi

)dV dt−

∫ t

t0

δLedt . (4.8)

By considering the relations given in Eq.(2.43), the Eq.(4.8) can be rewritten as:

δ

∫ t

t0

Epdt =

∫ t

t0

V

(σijδεij −DiδEi − ηδθ − hiδϑi

)dV dt−

∫ t

t0

δLedt , (4.9)

where σij are the stress components, Di is the vector containing the electric displace-ment components, η is the variation of entropy per unit of volume and θ is the tempera-ture. By putting together Eq.(4.9) and Eq.(4.4) in according to Eq.(4.1), the final versionof the Principle of Virtual Displacements (PVD) in case of thermo-electro-mechanical(TEM) problems is:

∫ t

t0

V

(σijδεij −DiδEi − ηδθ − hiδϑi

)dV dt =

∫ t

t0

δLedt−∫ t

t0

δLindt . (4.10)

By discarding the dependency by the time t and introducing the vectorial form ofEqs.(2.65)-(2.71) with the split in-plane (p) and out-plane (n) components, Eq.(4.10)

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92 CHAPTER 4

can be rewritten as:∫

V

(δεT

pGσpC + δεTnGσnC − δET

pGDpC − δETnGDnC − δθηC − δϑT

pGhpC − δϑTnGhnC

)dV

= δLe − δLin . (4.11)

In Eq.(4.11) the bold letters denote vectors, T means the transpose of a vector. Sub-script C and G suggest the substitution of constitutive and geometrical relations, re-spectively (see Chapter 2). The constitutive equations related to the variational state-ment in Eq.(4.11) are:

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Fre

kTpprEk

pG − FrekTnprEk

nG − Frλkprθ

k ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Fre

kTpnrEk

pG − FrekTnnrEk

nG − Frλknrθ

k ,

DkpC = Fre

kpprε

kpG + Fre

kpnrε

knG + Frε

kpprEk

pG + FrεkpnrEk

nG + Frpkprθ

k ,

DknC = Fre

knprε

kpG + Fre

knnrε

knG + Frε

knprEk

pG + FrεknnrEk

nG + Frpknrθ

k , (4.12)

ηkC = Frλ

kTpr εk

pG + FrλkTnr εk

nG + FrpkTpr Ek

pG + FrpkTnr Ek

nG + Frχkrθ

k ,

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

These constitutive equations have been obtained in Chapter 2: k indicates the con-sidered layer and Fr are the thickness functions introduced for functionally gradedmaterials (FGMs).

This extension of PVD to thermo-electro-mechanical case is here called PVD-TEM.The general form of governing equations is:

Kuuu + KuΦΦ + Kuθθ = pu −Muuu

KΦuu + KΦΦΦ + KΦθθ = pΦ (4.13)Kθuu + KθΦΦ + Kθθθ = pθ .

The matrices K are just considered assembled at multilayer level and expanded for thechosen order of expansion in the thickness direction. The vectors contain the degrees offreedom for the displacement u, the electric potential Φ and the temperature θ. Muu isthe inertial matrix and u is the second temporal derivative of the displacement vector.Mechanical, electrical and thermal loads are indicated with pu, pΦ and pθ, respectively.The full procedure to obtain the governing equations is illustrated in Chapter 5: - cho-sen of the appropriate variational statement; - substitution of constitutive equations(C) and geometrical ones (G); - introduction of CUF for the two-dimensional approx-imation; - integration by parts to obtain closed form (CF) governing equations; - useof shape functions in the plane to obtain finite element (FE) governing equations. Theextension of PVD, as illustrated in Eqs.(4.11)-(4.13) (PVD-TEM), has several particularcases which are discussed in the following sections.

4.2.1 PVD for the mechanical case

In the case of pure mechanical problems the PVD, here stated as PVD-M, has onlythe displacement u as primary variable [9]. The variational statement, the constitutive

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 93

equations and the governing equations can be considered as particular cases of themost general case illustrated in Section 4.2. From Eq.(4.11), the variational statement issimplified by discarding the internal thermal and electrical works:

V

(δεT

pGσpC + δεTnGσnC

)dV = δLe − δLin . (4.14)

The relative constitutive equations are the well-known Hooke’s law [9], [85], which canbe considered as particular case of the constitutive equations given in Eqs.(4.12):

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG , (4.15)

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG .

By substituting the Eqs.(4.15) in the variational statement of Eq.(4.14), the governingequation for the pure mechanical case is obtained [43]:

Kuuu = pu −Muuu . (4.16)

It is important to notice that Eq.(4.16) can be obtained in a simpler way by delating inEq.(4.13) the second and third line and the second and third column, in fact the matrixKuu is the same in the cases of PVD-M and PVD-TEM.

4.2.2 PVD for the mechanical case with an external thermal load

A possibility to study the thermal loads applied to elastic structures [159]-[173] is toconsider the stress components as an algebraic summation of mechanical stresses (d)and thermal ones (t) as given in Eqs.(2.89) and (2.90):

σkpC = σk

pd − σkpt = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Frλ

kprθ

k , (4.17)

σknC = σk

nd − σknt = FrQ

knprε

kpG + FrQ

knnrε

knG − Frλ

knrθ

k .

The variational statement is the same of Eq.(4.14) with both mechanical and thermalstresses included in it:∫

V

(δεT

pG(σpd − σpt) + δεTnG(σnd − σnt)

)dV = δLe − δLin . (4.18)

The case whit the temperature seen as an external load is called PVD-M(T), the gov-erning equation obtained from Eq.(4.18) is:

Kuuu = −Kuθθ + pu −Muuu . (4.19)

In Eq.(4.19) the vector pu is the external mechanical load and pθ = −Kuθθ is the exter-nal thermal load. As discussed in [43], the structure can be loaded by both mechanicaland thermal loads, or by one of these separately. For this reason, governing equationin Eq.(4.16) can be considered as a particular case of Eq.(4.19). The thermal load canbe determined by considering a temperature profile θ through the thickness directionimposed a priori [81] or calculated by solving Fourier’s heat conduction equation [43],[172]. The two cases of assumed temperature profile (Ta) and calculated temperatureprofile (Tc) are discussed in Chapter 5.

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94 CHAPTER 4

4.2.3 PVD for the electro-mechanical case

In the case of electro-mechanical coupling, for example the use of piezoelectric mate-rials and/or the application of an electrical load [174]-[189], the relative PVD can besimply obtained by discarding the internal thermal work in Eq.(4.11):

V

(δεT

pGσpC + δεTnGσnC − δET

pGDpC − δETnGDnC

)dV = δLe − δLin . (4.20)

The relative constitutive equations for the variational statement in Eq.(4.20) can beobtained as particular case of constitutive relations in Eq.(4.12) for the case of thermo-electro-mechanical case:

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Fre

kTpprEk

pG − FrekTnprEk

nG ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Fre

kTpnrEk

pG − FrekTnnrEk

nG , (4.21)

DkpC = Fre

kpprε

kpG + Fre

kpnrε

knG + Frε

kpprEk

pG + FrεkpnrEk

nG ,

DknC = Fre

knprε

kpG + Fre

knnrε

knG + Frε

knprEk

pG + FrεknnrEk

nG .

In this work, the PVD extended to electro-mechanical case is called as PVD-EM. Typ-ical applications of such an extension can be found in [190]-[193]. By substitutingEqs.(4.21) in the variational statement of Eq.(4.20), governing equations for the casePVD-EM are obtained. As in each PVD application, the governing equation for thePVD-EM can be considered as particular case of the most general PVD-TEM simplydelating the third line and the third column in Eq.(4.13):

Kuuu + KuΦΦ = pu −Muuu (4.22)KΦuu + KΦΦΦ = pΦ .

The matrices Kuu, KuΦ, KΦu and KΦΦ are the same for each proposed extension ofPVD [132], [133].

4.2.4 PVD for the thermo-mechanical case

The PVD proposed in Section 4.2.2 is defined as a partial coupling between the thermaland mechanical fields [81], [43]. In order to obtain a fully-coupling between the thermaland mechanical field, the temperature θ must be considered as a primary variable of theproblem in analogy with the displacement u. By considering the PVD-TEM in Eq.(4.11)and by discarding the internal electric work, the principle of virtual displacements inthe case of thermo-mechanical coupling (PVD-TM) is given:

V

(δεT

pGσpC + δεTnGσnC − δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.23)

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 95

For the PVD-TM, the constitutive equations are a particular case of the most generalones in Eq.(4.12):

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Frλ

kprθ

k ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Frλ

knrθ

k ,

ηkC = Frλ

kTpr εk

pG + FrλkTnr εk

nG + Frχkrθ

k , (4.24)

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

The governing equations have the following form:

Kuuu + Kuθθ = pu −Muuu (4.25)Kθuu + Kθθθ = pθ .

The relations given in Eq.(4.23), (4.24) and (4.25) for the case of thermo-mechanical cou-pling are very general. As discussed in [72], [73], [76] and [78], in the case of thermo-mechanical coupling two different investigations can be made: - temperature imposedat the top and bottom of the structure (PVD-TM1); - external applied mechanical load(PVD-TM2).

PVD-TM1

In the case of a temperature value imposed at the top and bottom of the structure, it isnot possible in the Eq.(4.23) to consider the virtual variation of temperature δθ, so:∫

V

(δεT

pGσpC + δεTnGσnC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.26)

The relative constitutive equations are the same indicated in Eq.(4.24): it’s obvious thatthe entropy η is not used. The governing equations have the same form indicated inEq.(4.25), but the matrices Kθu and Kθθ are different for the PVD-TM1 and the PVD-TM2. It can be noticed that the results obtained by the PVD-TM1 are very similarto the results obtained via the PVD-M(T) when the temperature profile through thethickness direction is calculated by means of Fourier’s heat conduction equation.

PVD-TM2

In the case of a mechanical load applied at the top or bottom of the structure, thegradient of temperature ϑ is not considered. So in case of thermo-mechanical coupling,the PVD in Eq.(4.23) gets:∫

V

(δεT

pGσpC + δεTnGσnC − δθηC

)dV = δLe − δLin . (4.27)

The PVD-TM2 permits to investigate how the temperature profile evolves in the thick-ness direction when a mechanical load is applied, or the free vibrations problem. Theemployed constitutive equations are the same of Eq.(4.24), but the heat flux hC is notconsidered. Formally, the governing equations are the same of Eq.(4.25), but the mean-ing of the matrices K is different from the PVD-TM1 case.

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96 CHAPTER 4

4.2.5 PVD for the thermo-electrical case

An other possible combination is the coupling between the thermal and electrical field,without considering the mechanical field. This extension could be very interesting forparticular applications different from those related to the structural analysis as justmade in [194], [195] and [196]. This extension of PVD is defined as PVD-TE, the varia-tional statement is obtained from Eq.(4.11) by discarding the internal mechanical work:

V

(− δET

pGDpC − δETnGDnC − δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.28)

The constitutive relations for the PVD-TE are a particular case of those proposed inEq.(4.12):

DkpC = Frε

kpprEk

pG + FrεkpnrEk

nG + Frpkprθ

k ,

DknC = Frε

knprEk

pG + FrεknnrEk

nG + Frpknrθ

k ,

ηkC = Frp

kTpr Ek

pG + FrpkTnr Ek

nG + Frχkrθ

k , (4.29)

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

In a symbolic form the system of governing equations is:

Kθθθ + KθΦΦ = pθ (4.30)KΦθθ + KΦΦΦ = pΦ .

In the open literature there are not any works about this coupling, but using the anal-ogy with the PVD-TM, similar conclusions could be obtained in the case of PVD-TE.It seems correct to give two different extensions of the PVD-TE: - PVD-TE1 when atemperature is imposed at the top and bottom of the structure, in this case the virtualvariation of θ cannot be considered in Eq.(4.28); - PVD-TE2 when an electric load isconsidered and in this case no temperature gradient is present in Eq.(4.28). The anal-ogy between the cases PVD-TE1, PVD-TE2 and PVD-TM1, PVD-TM2 seems verystrong.

4.3 Reissner’s mixed variational theorem, RMVT

The Reissner’s Mixed Variational Theorem (RMVT) [83] permits to assume two inde-pendent sets of variables: a set of primary unknowns as the PVD case, and a set of ex-tensive variables which are modelled a "priori" in the thickness direction. The main ad-vantage of using the RMVT is a priori and complete fulfillment of the C0

z -requirementsfor the modelled extensive variables [84]. In the case of multifield problems [197],[198] different extensions of RMVT are given by starting from the PVD-TEM: - trans-verse shear/normal stresses σn as extensive variables (RMVT-TEM); - transverse nor-mal electric displacement Dz as extensive variable (RMVT1-TEM); - both σn and Dz

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 97

as extensive variables (RMVT2-TEM). The ways to obtain these three cases are: a La-grange’s multiplier for σn is added in the former case, a different Lagrange’s multiplierforDz is considered in the second case, two Lagrange’s multipliers are added in the lastcase. When a new Lagrange’s multiplier is added [83], the constitutive equations mustbe rearranged in order to explicit the modelled variables. For this reason, each pro-posed extension of RMVT cannot be seen as a particular case of the other two. How-ever, for each extension of RMVT to thermo-electro-mechanical problems, the othermultifield couplings (M, EM, TM, TE) can be seen as particular cases when no newLagrange’s multipliers have been added.

In this section, by starting from the PVD-TEM developed in Section 4.2, the so-called RMVT-TEM is obtained in order to model a priori the transverse shear/normalstresses σn to satisfy the C0

z -requirements [84]. The other RMVT cases proposed inthis section, are the RMVT-M, RMVT-EM, RMVT-TM, these are particular cases ofthe most general RMVT-TEM because the Lagrange’s multiplier does not change and,consequently, matrices K do not change in the governing equations.

By considering the variational statement in Eq.(4.11) for the PVD-TEM, the RMVT-TEM is obtained modelling a priori the transverse shear/normal stresses σnM (thenew subscript M is introduced to remark that the transverse stresses are now mod-elled and not obtained via constitutive equations). The added Lagrange’s multiplier isδσT

nM(εnG − εnC). The condition to add this multiplier is that the transverse strains εn

calculated by means of geometrical relations (G) and by using the constitutive equa-tions (C) must be the same or almost the same. In this way the balance of the internalwork does not change or remains almost the same:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)− δETpGDpC

− δETnGDnC − δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.31)

The relative constitutive equations are obtained from Eqs.(2.107)-(2.113) consideringin them the transverse stresses σn as modelled (M) and the transverse strains εn asobtained from constitutive equations (C). The coefficients in the proposed constitutiveequations are rearranged, so the approximation for FGMs (introduction of thicknessfunctions Fr for the dependency by thickness coordinate z) must be introduced afterthe modification of constitutive equations. The dependency by the z coordinate inEqs.(4.32) is considered, but not indicated in order to do not make heavy the notation:

σkpC = Qk

ppεkpG + Qk

pnεknC − ekT

pp EkpG − ekT

np EknG − λk

pθk ,

σknM = Qk

npεkpG + Qk

nnεknC − ekT

pn EkpG − ekT

nnEknG − λk

nθk ,

DkpC = ek

ppεkpG + ek

pnεknC + εk

ppEkpG + εk

pnEknG + pk

pθk ,

DknC = ek

npεkpG + ek

nnεknC + εk

npEkpG + εk

nnEknG + pk

nθk , (4.32)

ηkC = λkT

p εkpG + λkT

n εknC + pkT

p EkpG + pkT

n EknG + χkθk ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

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98 CHAPTER 4

The rearranged constitutive equations for RMVT-TEM are:

σkpC = C

k

σpεpεk

pG + Ck

σpσnσk

nM + Ck

σpEpEk

pG + Ck

σpEnEk

nG + Ck

σpθθk ,

εknC = C

k

εnεpεk

pG + Ck

εnσnσk

nM + Ck

εnEpEk

pG + Ck

εnEnEk

nG + Ck

εnθθk ,

DkpC = C

k

Dpεpεk

pG + Ck

Dpσnσk

nM + Ck

DpEpEk

pG + Ck

DpEnEk

nG + Ck

Dpθθk ,

DknC = C

k

Dnεpεk

pG + Ck

Dnσnσk

nM + Ck

DnEpEk

pG + Ck

DnEnEk

nG + Ck

Dnθθk , (4.33)

ηkC = C

k

ηεpεk

pG + Ck

ησnσk

nM + Ck

ηEpEk

pG + Ck

ηEnEk

nG + Ck

ηθθk ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

In according to [1] and [71] the meaning of coefficients C in Eqs.(4.33) is:

Ck

σpεp= Qk

pp −QkpnQ

knn

−1Qk

np , Ck

σpσn= Qk

pnQknn

−1, C

k

σpEp= Qk

pnQknn

−1ekT

pn − ekTpp ,

Ck

σpEn= Qk

pnQknn

−1ekT

nn − ekTnp , C

k

σpθ = QkpnQ

knn

−1λk

n − λkp , C

k

εnεp= −Qk

nn

−1Qk

np ,

Ck

εnσn= Qk

nn

−1, C

k

εnEp= Qk

nn

−1ekT

pn , Ck

εnEn= Qk

nn

−1ekT

nn , Ck

εnθ = Qknn

−1λk

n ,

Ck

Dpεp= ek

pp − ekpnQ

knn

−1Qk

np , Ck

Dpσn= ek

pnQknn

−1, C

k

DpEp= ek

pnQknn

−1ekT

pn + εkpp ,

Ck

DpEn= ek

pnQknn

−1ekT

nn + εkpn , C

k

Dpθ = ekpnQ

knn

−1λk

n + pkp , (4.34)

Ck

Dnεp= ek

np − eknnQ

knn

−1Qk

np , Ck

Dnσn= ek

nnQknn

−1, C

k

DnEp= ek

nnQknn

−1ekT

pn + εknp ,

Ck

DnEn= ek

nnQknn

−1ekT

nn + εknn , C

k

Dnθ = eknnQ

knn

−1λk

n + pkn ,

Ck

ηεp= λkT

p − λkTn Qk

nn

−1Qk

np , Ck

ησn= λkT

n Qknn

−1, C

k

ηEp= λkT

n Qknn

−1ekT

pn + pkTp ,

Ck

ηEn= λkT

n Qknn

−1εkT

nn + pkTn , C

k

ηθ = λkTn Qk

nn

−1λk

n + χk .

In Eq.(4.33) the coefficients C depend by the thickness coordinate z for the case offunctionally graded materials. The approximation by means of thickness functions Fr

[85] can be introduced (details can be found in Chapter 2):

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpEnrEknG + FrC

k

σpθrθk ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnEnrEknG + FrC

k

εnθrθk ,

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpEnrEknG + FrC

k

Dpθrθk ,

DknC = FrC

k

DnεprεkpG + FrC

k

DnσnrσknM + FrC

k

DnEprEkpG + FrC

k

DnEnrEknG + FrC

k

Dnθrθk ,

ηkC = FrC

k

ηεprεkpG + FrC

k

ησnrσknM + FrC

k

ηEprEkpG + FrC

k

ηEnrEknG + FrC

k

ηθrθk ,

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG , (4.35)

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 99

By using the variational statement in Eq.(4.31), the constitutive relations in Eqs.(4.35),and the CUF (see Chapter 3), the governing equations are obtained. In a symbolicform, these are:

Kuuu + Kuσσn + KuΦΦ + Kuθθ = pu −Muuu

Kσuu + Kσσσn + KσΦΦ + Kσθθ = 0 (4.36)KΦuu + KΦσσn + KΦΦΦ + KΦθθ = pΦ

Kθuu + Kθσσn + KθΦΦ + Kθθθ = pθ .

In Eqs.(4.36), the transverse shear/normal stresses σn are primary variables of theproblem, and they are directly obtained from the governing equations, this fact permitsto have transverse stresses that fulfill a priori and completely the C0

z -requirements. TheRMVT-TEM has four variables (u, σn, Φ and θ), the PVD-TEM has three variables (u,Φ and θ). It is important to notice that the matrices K for PVD-TEM in Eq.(4.13) arecompletely different from those for RMVT-TEM in Eq.(4.36): this happens because aLagrange’s multiplier has been added and constitutive equations have been rewritten.

4.3.1 RMVT for the mechanical case

Reissner’s mixed variational theorem, with the transverse shear/nomal stresses mod-elled a priori, in case of pure mechanical problems is called RMVT-M. As variables ofthe problem we have the displacements u and the transverse stresses σn [86]. RMVT-M can be obtained as a particular case of RMVT-TEM simply discarding the internalthermal and electrical works, so from Eq.(4.31), it is possible to obtain [199], [200]:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC))dV = δLe − δLin . (4.37)

With respect to the RMVT-TEM, no new Lagrange’s multipliers are added, so the con-stitutive equations can be considered as particular case of those in Eqs.(4.35):

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM , (4.38)

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM .

The governing equations can be obtained by using the Eq.(4.37) and the Eqs.(4.38)[201], [202]:

Kuuu + Kuσσn = pu −Muuu

Kσuu + Kσσσn = 0 . (4.39)

The governing equations in Eq.(4.39) can be simply obtained by eliminating the thirdand fourth column and the third and fourth line in Eq.(4.36). The remaining matricesK are the same for the RMVT-TEM and RMVT-M. The matrix Kuu is completely dif-ferent from that in Eqs.(4.16) for the PVD-M because of the introduction of a Lagrangemultiplier and the consequently rearrangement of constitutive equations.

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100 CHAPTER 4

4.3.2 RMVT for the electro-mechanical case

In the case of electro-mechanical problems the variational statement can be obtainedfrom that for the RMVT-TEM discarding the internal thermal work, so from Eq.(4.31):

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)

− δETpGDpC − δET

nGDnC

)dV = δLe − δLin . (4.40)

This RMVT is here called as RMVT-EM because it is the extension to the electro-mechanical case [198]. The constitutive equations are a particular case of those inEqs.(4.35):

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpEnrEknG ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnEnrEknG , (4.41)

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpEnrEknG ,

DknC = FrC

k

DnεprεkpG + FrC

k

DnσnrσknM + FrC

k

DnEprEkpG + FrC

k

DnEnrEknG .

The governing equations are obtained by using the variational statement in Eq.(4.40)and the constitutive relations in Eq.(4.41) [79], [205]:

Kuuu + Kuσσn + KuΦΦ = pu −Muuu

Kσuu + Kσσσn + KσΦΦ = 0 (4.42)KΦuu + KΦσσn + KΦΦΦ = pΦ .

These governing equations can be also seen as particular case of those in Eq.(4.36) sim-ply eliminating the fourth column and the fourth line. The matrices K are the same forthe cases RMVT-TEM and RMVT-EM because no further Lagrange multipliers havebeen added. For the same reason, the matrices in Eq.(4.42) are completely differentfrom those in the case of PVD: a Lagrange’s multiplier for the transverse stresses hasbeen added [79].

4.3.3 RMVT for the thermo-mechanical case

The last possible extension is the thermo-mechanical case, such an extension is herecalled as RMVT-TM. The extension to the thermo-electrical case is not possible be-cause the chosen Lagrange’s multiplier is a mechanical quantity which includes thetransverse stresses σn. The variational statement for RMVT-TM is a particular case ofRMVT-TEM, see Eq.(4.31), when the electrical internal work is discarding:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)

− δθηC − δϑTpGhpC − δϑT

nGhnC

)dV = δLe − δLin . (4.43)

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 101

From Eq.(4.35), simply eliminating the electrical contributes, it is possible to obtain theconstitutive equations for the RMVT-TM:

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpθrθk ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnθrθk ,

ηkC = FrC

k

ηεprεkpG + FrC

k

ησnrσknM + FrC

k

ηθrθk , (4.44)

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

By using the Eqs.(4.43) and (4.44), or by eliminating the third column and third line inEq.(4.36), it is possible to obtain the governing equations in case of RMVT-TM:

Kuuu + Kuσσn + Kuθθ = pu −Muuu

Kσuu + Kσσσn + Kσθθ = 0 (4.45)Kθuu + Kθσσn + Kθθθ = pθ .

The matrices K are in common with the other RMVT extensions presented in thissection, but are completely different from those presented in Section 4.2 in the case ofPVD applications. In analogy with the variational statements in Eqs.(4.26) and (4.27),it is possible to consider two cases depending on if temperature values are imposed(RMVT-TM1 and term δθ is discarded in Eq.(4.43)), or mechanical loads are applied(RMVT-TM2 and terms δϑG is discarded in Eq.(4.43)).

The partial thermo-mechanical coupling can be accounted for in analogy to thePVD-M(T) of Section 4.2.2; in this case the temperature is only seen as an externalload. The case of RMVT for mechanical case and external thermal load is here calledas RMVT-M(T), exhaustive details can be found in [172] and [204].

4.4 A general extension of RMVT

The main idea of Reissner’s mixed variational theorem [83] can be extended to othercases in a more general way. In the case of multifield problems other transverse vari-ables can be modelled a priori, such as the transverse normal electric displacementDz in the case of an electrical field and/or the transverse normal magnetic inductionin the case of a magnetic field [1], [80]. In this work thermo-electro-mechanical prob-lems are considered, so other two extensions of RMVT are possible: - introduction of anew Lagrange’s multipier for the modelling of the transverse normal electric displace-ment (RMVT1-TEM); - introduction of two Lagrange’s multipliers for both transversestresses and transverse normal electric displacement (RMVT2-TEM).

RMVT1-TEM cannot be seen as a particular case of RMVT-TEM or PVD-TEM be-cause a different Lagrange’s multiplier is introduced (which is connected to the trans-verse normal electric displacement), however it has several particular cases dependingon the discarded physical fields.

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102 CHAPTER 4

RMVT2-TEM is not a particular case of RMVT-TEM, RMVT1-TEM and PVD-TEM,because it has different Lagrange’s multipliers. The subcase of RMVT2-TEM can be ob-tained only discarding the thermal field, because there are two Lagrange’s multipliers:one for a mechanical quantity and an other one for an electrical quantity [1].

4.4.1 RMVT1 for the thermo-electro-mechanical case

By considering the variational statement in Eq.(4.11) for the PVD-TEM, the RMVT1-TEM is obtained by modelling a priori the transverse normal electric displacementDnM (the new subscript M is introduced to remark that the transverse normal elec-tric displacement is now modelled and not obtained via constitutive equations). Theadded Lagrange’s multiplier is δDT

nM(EnG−EnC). The condition to add this multiplieris that the transverse normal electric field En calculated by means of geometrical rela-tions (G) and by using the constitutive equations (C) must be the same or almost thesame. In this way the internal work does not change or remains almost the same:

V

(δεT

pGσpC + δεTnGσnC − δET

pGDpC − δETnGDnM

− δDTnM(EnG − EnC)− δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.46)

The relative constitutive equations are obtained from Eqs.(4.12) considering in themthe transverse normal electric displacement Dn as modelled (M) and the transversenormal electric field En as obtained from constitutive equations (C). The coefficientsin the proposed constitutive equations are rearranged, so the approximation for FGMs(introduction of thickness functions Fr for the dependency by the thickness coordinatez) must be introduced after the modification of constitutive equations. The depen-dency by z coordinate in Eqs.(4.47) is considered but not indicated in order to do notmake heavy the notation:

σkpC = Qk

ppεkpG + Qk

pnεknG − ekT

pp EkpG − ekT

np EknC − λk

pθk ,

σknC = Qk

npεkpG + Qk

nnεknG − ekT

pn EkpG − ekT

nnEknC − λk

nθk ,

DkpC = ek

ppεkpG + ek

pnεknG + εk

ppEkpG + εk

pnEknC + pk

pθk , (4.47)

DknM = ek

npεkpG + ek

nnεknG + εk

npEkpG + εk

nnEknC + pk

nθk ,

ηkC = λkT

p εkpG + λkT

n εknG + pkT

p EkpG + pkT

n EknC + χkθk ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 103

The rearranged constitutive equations for RMVT1-TEM are:

σkpC = C

kσpεp

εkpG + C

kσpεn

εknG + C

kσpEp

EkpG + C

kσpDn

DknM + C

kσpθθ

k ,

σknC = C

kσnεp

εkpG + C

kσnεn

εknG + C

kσnEp

EkpG + C

kσnDn

DknM + C

kσnθθ

k ,

DkpC = C

kDpεp

εkpG + C

kDpεn

εknG + C

kDpEp

EkpG + C

kDpDn

DknM + C

kDpθθ

k ,

EknC = C

kEnεp

εkpG + C

kEnεn

εknG + C

kEnEp

EkpG + C

kEnDn

DknM + C

kEnθθ

k , (4.48)

ηkC = C

kηεp

εkpG + C

kηεn

εknG + C

kηEp

EkpG + C

kηDn

DknM + C

kηθθ

k ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

In according to [1], the meaning of coefficients C in Eqs.(4.48) is:

Ckσpεp

= Qkpp + ekT

np εknn

−1ek

np , Ckσpεn

= Qkpn + ekT

np εknn

−1ek

nn , CkσpEp

= ekTnp εk

nn

−1εk

np − ekTpp ,

CkσpDn

= −ekTnp εk

nn

−1, C

kσpθ = ekT

np εknn

−1pk

n − λkp , C

kσnεp

= Qknp + ekT

nnεknn

−1ek

np ,

Ckσnεn

= Qknn + ekT

nnεknn

−1ek

nn , CkσnEp

= ekTnnεk

nn

−1εk

np − ekTpn , C

kσnDn

= −ekTnnεk

nn

−1,

Ckσnθ = ekT

nnεknn

−1pk

n − λkn , C

kDpεp

= ekpp − εk

pnεknn

−1ek

np ,

CkDpεn

= ekpn − εk

pnεknn

−1ek

nn , CkDpEp

= εkpp − εk

pnεknn

−1εk

np ,

CkDpDn

= εkpnε

knn

−1, C

kDpθ = pk

p − εkpnε

knn

−1pk

n , (4.49)

CkEnεp

= −εknn

−1ek

np , CkEnεn

= −εknn

−1ek

nn , CkEnEp

= −εknn

−1εk

np ,

CkEnDn

= εknn

−1, C

kEnθ = −εk

nn

−1pk

n ,

Ckηεp

= λkTp − pkT

n εknn

−1ek

np , Ckηεn

= λkTn − pkT

n εknn

−1ek

nn , CkηEp

= pkTp − pkT

n εknn

−1εk

np ,

CkηDn

= pkTn εk

nn

−1, C

kηθ = χk − pkT

n εknn

−1pk

n .

In Eqs.(4.48) the coefficients C depend by the thickness coordinate z in case of func-tionally graded materials. The approximation by means of thickness functions Fr [85]can be introduced (details can be found in Chapter 2):

σkpC = FrC

kσpεprε

kpG + FrC

kσpεnrε

knG + FrC

kσpEprEk

pG + FrCkσpDnrDk

nM + FrCkσpθrθ

k ,

σknC = FrC

kσnεprε

kpG + FrC

kσnεnrε

knG + FrC

kσnEprEk

pG + FrCkσnDnrDk

nM + FrCkσnθrθ

k ,

DkpC = FrC

kDpεprε

kpG + FrC

kDpεnrε

knG + FrC

kDpEprEk

pG + FrCkDpDnrDk

nM + FrCkDpθrθ

k ,

EknC = FrC

kEnεprε

kpG + FrC

kEnεnrε

knG + FrC

kEnEprEk

pG + FrCkEnDnrDk

nM + FrCkEnθrθ

k ,

ηkC = FrC

kηεprε

kpG + FrC

kηεnrε

knG + FrC

kηEprEk

pG + FrCkηDnrDk

nM + FrCkηθrθ

k ,

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG , (4.50)

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

By using the variational statement in Eq.(4.46), the constitutive relations in Eqs.(4.50),and the CUF (see Chapter 3), it is possible to obtain the governing equations. In a

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104 CHAPTER 4

symbolic form, these are:

Kuuu + KuΦΦ + KuDDn + Kuθθ = pu −Muuu

KΦuu + KΦΦΦ + KΦDDn + KΦθθ = pΦ (4.51)KDuu + KDΦΦ + KDDDn + KDθθ = 0

Kθuu + KθΦΦ + KθDDn + Kθθθ = pθ .

In Eqs.(4.51), the transverse normal electric displacement Dn is a primary variable ofthe problem, and it is directly obtained from the governing equations, this fact permitsto have transverse normal electric displacement that fulfills a priori and completely theC0

z -requirements. The RMVT1-TEM has four variables (u, Dn, Φ and θ), the PVD-TEMin Eq.(4.13) has three variables (u, Φ and θ). It is important to notice that the matricesK in Eq.(4.13) are completely different from those for RMVT1-TEM in Eq.(4.51).

Two particular cases of the RMVT1-TEM can be considered: - RMVT1-TE whenthe internal mechanical work is not considered; - RMVT1-EM when the internal ther-mal work is not considered.

RMVT1-TE

The variational statement is obtained from that in Eq.(4.46) simply discarding the in-ternal mechanical work:∫

V

(− δET

pGDpC − δETnGDnM − δDT

nM(EnG − EnC)− δθηC − δϑTpGhpC − δϑT

nGhnC

)dV

= δLe − δLin . (4.52)

Constitutive equations are also a particular case of those for the RMVT1-TEM, in factfrom Eqs.(4.50) simply discarding the mechanical contribution, it is possible to obtain:

DkpC = FrC

kDpEprEk

pG + FrCkDpDnrDk

nM + FrCkDpθrθ

k ,

EknC = FrC

kEnEprEk

pG + FrCkEnDnrDk

nM + FrCkEnθrθ

k ,

ηkC = FrC

kηEprEk

pG + FrCkηDnrDk

nM + FrCkηθrθ

k , (4.53)

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

By using Eq.(4.52) and Eqs.(4.53), it is possible to obtain the governing equations, thesecan be considered as a particular case of those for RMVT1-TEM in Eq.(4.51) simplyeliminating the first column and the first line, the matrices K remain the same becauseno new Lagrange’s multipliers have been added:

KΦΦΦ + KΦDDn + KΦθθ = pΦ

KDΦΦ + KDDDn + KDθθ = 0 (4.54)KθΦΦ + KθDDn + Kθθθ = pθ .

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 105

RMVT1-EM

The variational statement is obtained from that in Eq.(4.46) simply discarding the in-ternal thermal work:

V

(δεT

pGσpC + δεTnGσnC − δET

pGDpC − δETnGDnM − δDT

nM(EnG − EnC))dV

= δLe − δLin . (4.55)

Constitutive equations are also a particular case of those for the RMVT1-TEM, in factfrom Eqs.(4.50) simply discarding the thermal contributions, it is possible to obtain:

σkpC = FrC

kσpεprε

kpG + FrC

kσpεnrε

knG + FrC

kσpEprEk

pG + FrCkσpDnrDk

nM ,

σknC = FrC

kσnεprε

kpG + FrC

kσnεnrε

knG + FrC

kσnEprEk

pG + FrCkσnDnrDk

nM ,

DkpC = FrC

kDpεprε

kpG + FrC

kDpεnrε

knG + FrC

kDpEprEk

pG + FrCkDpDnrDk

nM , (4.56)

EknC = FrC

kEnεprε

kpG + FrC

kEnεnrε

knG + FrC

kEnEprEk

pG + FrCkEnDnrDk

nM .

By using Eq.(4.55) and Eqs.(4.56), it is possible to obtain the governing equations, thesecan be considered as particular case of those for RMVT1-TEM in Eq.(4.51) simply elim-inating the fourth column and the fourth line:

Kuuu + KuΦΦ + KuDDn = pu −Muuu

KΦuu + KΦΦΦ + KΦDDn = pΦ (4.57)KDuu + KDΦΦ + KDDDn = 0 .

The matrices K in governing equations for RMVT1-TEM, RMVT1-TE, RMVT1-EMare the same because the same Lagrange’s multiplier is employed.

4.4.2 RMVT2 for the thermo-electro-mechanical case

The starting point is the variational statement in Eq.(4.11) for the PVD-TEM, the RMVT2-TEM is obtained by modelling a priori both transverse shear/normal stresses σnM andtransverse normal electric displacement DnM (the new subscript M is introduced toremark that the transverse stresses and normal electric displacements are now mod-elled and not obtained via constitutive equations). The added Lagrange’s multipliersare δσT

nM(εnG− εnC) and δDTnM(EnG−EnC). The condition to include these multipliers

is that the transverse strains εn and the normal electric field En calculated by means ofgeometrical relations (G) and by using constitutive equations (C) must be the same oralmost the same. In this way the internal work does not change or remains almost thesame [79], [80]:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)− δETpGDpC − δET

nGDnM

− δDTnM(EnG − EnC)− δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (4.58)

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106 CHAPTER 4

The relative constitutive equations are obtained from Eqs.(4.12) by considering thetransverse stresses σn and the normal electric displacement Dn as modelled (M) andthe transverse strains εn and the normal electric field En as obtained from constitutiveequations (C). The coefficients in the proposed constitutive equations are rearranged,so the approximation for FGMs (introduction of thickness functions Fr for the depen-dency by the thickness coordinate z) must be introduced after the modification of con-stitutive equations. The dependency by the z coordinate in Eqs.(4.59) is considered butnot introduced in order to do not make heavy the notation:

σkpC = Qk

ppεkpG + Qk

pnεknC − ekT

pp EkpG − ekT

np EknC − λk

pθk ,

σknM = Qk

npεkpG + Qk

nnεknC − ekT

pn EkpG − ekT

nnEknC − λk

nθk ,

DkpC = ek

ppεkpG + ek

pnεknC + εk

ppEkpG + εk

pnEknC + pk

pθk ,

DknM = ek

npεkpG + ek

nnεknC + εk

npEkpG + εk

nnEknC + pk

nθk , (4.59)

ηkC = λkT

p εkpG + λkT

n εknC + pkT

p EkpG + pkT

n EknC + χkθk ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

The rearranged constitutive equations for RMVT2-TEM are:

σkpC = C

k

σpεpεk

pG + Ck

σpσnσk

nM + Ck

σpEpEk

pG + Ck

σpDnDk

nM + Ck

σpθθk ,

εknC = C

k

εnεpεk

pG + Ck

εnσnσk

nM + Ck

εnEpEk

pG + Ck

εnDnDk

nM + Ck

εnθθk ,

DkpC = C

k

Dpεpεk

pG + Ck

Dpσnσk

nM + Ck

DpEpEk

pG + Ck

DpDnDk

nM + Ck

Dpθθk ,

EknC = C

k

Enεpεk

pG + Ck

Enσnσk

nM + Ck

EnEpEk

pG + Ck

EnDnDk

nM + Ck

Enθθk , (4.60)

ηkC = C

k

ηεpεk

pG + Ck

ησnσk

nM + Ck

ηEpEk

pG + Ck

ηDnDk

nM + Ck

ηθθk ,

hkpC = κk

ppϑkpG + κk

pnϑknG ,

hknC = κk

npϑkpG + κk

nnϑknG .

In according to [205], [79] and [80], the meaning of coefficients C in Eqs.(4.60) is:

Ck

σpεp= Qk

pp −QkpnQ

knn

−1Qk

np − (QkpnQ

knn

−1ekT

nn − ekTnp )(ek

nnQknn

−1ekT

nn + εknn)−1

(eknp − ek

nnQknn

−1Qk

np) ,

Ck

σpσn= Qk

pnQknn

−1 − (QkpnQ

knn

−1ekT

nn − ekTnp )(ek

nnQknn

−1ekT

nn + εknn)−1(ek

nnQknn

−1) ,

Ck

σpEp= Qk

pnQknn

−1ekT

pn − ekTpp − (Qk

pnQknn

−1ekT

nn − ekTnp )(ek

nnQknn

−1ekT

nn + εknn)−1

(eknnQ

knn

−1ekT

pn + εknp) ,

Ck

σpDn= (Qk

pnQknn

−1ekT

nn − ekTnp )(ek

nnQknn

−1ekT

nn + εknn)−1 ,

Ck

σpθ = QkpnQ

knn

−1λk

n − λkp − (Qk

pnQknn

−1ekT

nn − ekTnp )(ek

nnQknn

−1ekT

nn + εknn)−1

(eknnQ

knn

−1λk

n + pkn) ,

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 107

Ck

εnεp= −Qk

nn

−1Qk

np −Qknn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1Qk

np − eknp) ,

Ck

εnσn= Qk

nn

−1 −Qknn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1ek

nnQknn

−1,

Ck

εnEp= Qk

nn

−1ekT

pn −Qknn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ekT

pn + εknp) ,

Ck

εnDn= Qk

nn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1 ,

Ck

εnθ = Qknn

−1λk

n −Qknn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λk

n + pkn) ,

Ck

Dpεp= ek

pp − ekpnQ

knn

−1Qk

np − ekpnQ

knn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

np − eknnQ

knn

−1Qk

np)

− εkpn(ek

nnQknn

−1ekT

nn + εknn)−1(ek

np − eknnQ

knn

−1Qk

np) , (4.61)

Ck

Dpσn= ek

pnQknn

−1 − ekpnQ

knn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1ek

nnQknn

−1

− εkpn(ek

nnQknn

−1ekT

nn + εknn)−1ek

nnQknn

−1,

Ck

DpEp= εk

pp + ekpnQ

knn

−1ekT

pn − ekpnQ

knn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ekT

pn + εknp)

− εknp(e

knnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ekT

pn + εknp) ,

Ck

DpDn= ek

pnQknn

−1ekT

nn(eknnQk

nn

−1ekT

nn + εknn)−1 + εk

pn(eknnQk

nn

−1ekT

nn + εknn)−1 ,

Ck

Dpθ = ekpnQ

knn

−1λk

n − ekpnQ

knn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λkT

n + pkn)

− (eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λkT

n + pkn) + pk

p ,

Ck

Enεp= −(ek

nnQknn

−1ekT

nn + εknn)−1(ek

np − eknnQ

knn

−1Qk

np) ,

Ck

Enσn= −(ek

nnQknn

−1ekT

nn + εknn)−1ek

nnQknn

−1,

Ck

EnEp= −(ek

nnQknn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ekT

pn + εknp) ,

Ck

EnDn= (ek

nnQknn

−1ekT

nn + εknn)−1 ,

Ck

Enθ = −(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λk

n + pkn) ,

Ck

ηεp= λkT

p − λkTn Qk

nn

−1Qk

np − λkTn Qk

nn

−1ekT

nn(eknnQk

nn

−1ekT

nn + εknn)−1(ek

np − eknnQ

knn

−1Qk

np)

− pkTn (ek

nnQknn

−1ekT

nn + εknn)−1(ek

np − eknnQk

nn

−1Qk

np) ,

Ck

ησn= λkT

n Qknn

−1 − λkTn Qk

nn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1ek

nnQknn

−1

− pkTn (ek

nnQknn

−1ekT

nn + εknn)−1ek

nnQknn

−1,

Ck

ηEp= λkT

n Qknn

−1ekT

pn − λkTn Qk

nn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ek

np + εknp)

− pkTn (ek

nnQknn

−1ekT

nn + εknn)−1(ek

nnQknn

−1ekT

pn + εknp) ,

Ck

ηDn= λkT

n Qknn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1 + pkT

n (eknnQ

knn

−1ekT

nn + εknn)−1 ,

Ck

ηθ = λkTn Qk

nn

−1λk

n − λkTn Qk

nn

−1ekT

nn(eknnQ

knn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λk

n + pkn)

− pkTn (ek

nnQknn

−1ekT

nn + εknn)−1(ek

nnQknn

−1λk

n + pkn) + χk .

In Eqs.(4.60) the coefficients C depend by the thickness coordinate z in case of func-tionally graded materials. The approximation by means of thickness functions Fr [85]

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108 CHAPTER 4

can be introduced (details can be found in Chapter 2):

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpDnrDknM + FrC

k

σpθrθk ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnDnrDknM + FrC

k

εnθrθk ,

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpDnrDknM + FrC

k

Dpθrθk ,

EknC = FrC

k

EnεprεkpG + FrC

k

EnσnrσknM + FrC

k

EnEprEkpG + FrC

k

EnDnrDknM + FrC

k

Enθrθk ,

ηkC = FrC

k

ηεprεkpG + FrC

k

ησnrσknM + FrC

k

ηEprEkpG + FrC

k

ηDnrDknM + FrC

k

ηθrθk , (4.62)

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG ,

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

By using the variational statement in Eq.(4.58), the constitutive relations in Eqs.(4.62),and the CUF (see Chapter 3), the governing equations are obtained. In a symbolicform, these are:

Kuuu + Kuσσn + KuΦΦ + KuDDn + Kuθθ = pu −Muuu

Kσuu + Kσσσn + KσΦΦ + KσDDn + Kσθθ = 0

KΦuu + KΦσσn + KΦΦΦ + KΦDDn + KΦθθ = pΦ (4.63)KDuu + KDσσn + KDΦΦ + KDDDn + KDθθ = 0

Kθuu + Kθσσn + KθΦΦ + KθDDn + Kθθθ = pθ .

In Eqs.(4.63), the transverse shear/normal stresses σn and the normal electric displace-ment Dn are primary variables of the problem, and they are directly obtained from thegoverning equations, this fact permits to fulfill a priori and completely the C0

z require-ments. The RMVT2-TEM has five variables (u, σn, Φ, Dn and θ). It is important tonotice that the matrices K in Eq.(4.63) are completely different from those for RMVT-TEM, RMVT1-TEM and PVD-TEM: two different Lagrange’s multipliers have beenadded.

As particular case of RMVT2-TEM in Eq.(4.63) we can consider the so called RMVT2-EM obtained discarding the thermal internal work.

RMVT2-EM

The variational statement is obtained from that in Eq.(4.58) simply discarding the in-ternal thermal work:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)− δETpGDpC − δET

nGDnM

− δDTnM(EnG − EnC)

)dV = δLe − δLin . (4.64)

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VARIATIONAL STATEMENTS FOR MULTIFIELD PROBLEMS 109

Constitutive equations are a particular case of those for the RMVT2-TEM, in fact fromEqs.(4.62) simply discarding the thermal contributions, it is possible to obtain:

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpDnrDknM ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnDnrDknM ,

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpDnrDknM , (4.65)

EknC = FrC

k

EnεprεkpG + FrC

k

EnσnrσknM + FrC

k

EnEprEkpG + FrC

k

EnDnrDknM .

By using Eq.(4.64) and Eqs.(4.65), it is possible to obtain the governing equations,these can be considered as a particular case of those for RMVT2-TEM in Eq.(4.63) sim-ply eliminating the fifth column and the fifth line:

Kuuu + Kuσσn + KuΦΦ + KuDDn = pu −Muuu

Kσuu + Kσσσn + KσΦΦ + KσDDn = 0

KΦuu + KΦσσn + KΦΦΦ + KΦDDn = pΦ (4.66)KDuu + KDσσn + KDΦΦ + KDDDn = 0 .

The matrices K in governing equations for RMVT2-TEM and RMVT2-EM are thesame because the same Lagrange’s multipliers are considered.

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Chapter 5

Differential equations and FE matricesfor multifield problems

Governing equations are here obtained in terms of some few basic elements called fundamentalnuclei. Expanding them by means of opportune indexes and loops, it is possible to obtain thestiffness matrices of the considered multilayered structures. The use of such nuclei permitsto obtain in a unified manner several refined and advanced models which differ for the chosenorder of expansion in the thickness direction, for the choice of the modelled multifield variables,and for the multilayer description: equivalent single layer (ESL) or layer wise (LW). Some ofthe proposed variational statements in the previous chapter are here developed to obtain thegoverning equations for the analysis which will be proposed in the next three chapters. Thesegoverning equations can be obtained in closed-form solution or by finite element matrices. Bothplate and shell geometries are considered.

5.1 PVD for the mechanical case, PVD-M

In the case of pure mechanical problems, the Principle of Virtual Displacements (PVD)states as indicated in Eq.(4.14):

V

(δεT

pGσpC + δεTnGσnC

)dV = δLe − δLin . (5.1)

By considering a laminate of Nl layers, and the integral on the volume Vk of each layeras an integral on the in plane domain Ωk plus the integral in the thickness-directiondomain Ak, it is possible to write:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nC

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin , (5.2)

where δLke and δLk

in are the external and inertial virtual works at the k-layer level,respectively. The relative constitutive equations, written in order to consider the case

111

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112 CHAPTER 5

of some functionally graded material (FGM) layers embedded in the structure [85], arethose obtained in Eq.(4.15):

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG , (5.3)

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG .

By considering a generic layer k and substituting the Eqs.(5.3) in the variational state-ment of Eq.(5.2):

Ωk

Ak

δεk

pG

T(FrQ

kpprε

kpG + FrQ

kpnrε

knG) + δεk

nG

T(FrQ

knprε

kpG + FrQ

knnrε

knG)

dΩkdz

= δLke − δLk

in . (5.4)

Further steps to obtain the fundamental nuclei are: the substitution of geometricalrelations, the introduction of Carrera’s Unified Formulation (CUF) [92]. Governingequations can be written in differential form or by the Finite Element (FE) method.

5.1.1 PVD-M: differential equations for plates and shells

In Eq.(5.4), we can substitute directly the geometrical relations for shells, in fact thosefor the plate geometry are considered as particular cases. By using the geometricalrelations in Eqs.(2.147), the variational statement is:∫

Ωk

Ak

[((Dk

p + Akp) δuk

)T (FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)uk (5.5)

+((Dk

np + Dknz −Ak

n) δuk)T (

FrQknpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)uk

]

dΩkdz = δLke − δLk

in .

In Eq.(5.5), by introducing the CUF [92] as proposed in Eq.(3.6):∫

Ωk

Ak

[((Dk

p + Akp) Fsδu

ks

)T (FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)Fτu

+((Dk

np + Dknz −Ak

n) Fsδuks

)T (FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)

Fτukτ

]dΩkdz = δLk

e − δLkin . (5.6)

In Eq.(5.6), in order to obtain a strong form of differential equations on the domainΩk and the relative boundary conditions on edge Γk, the integration by parts is used,which permits to move the differential operator from the infinitesimal variation of thegeneric variable δak to the finite quantity ak [92]. For a generic variable ak, the inte-gration by parts states:

Ωk

(Dk

Ωδak)T

akdΩk = −∫

Ωk

δakT (DkT

Ω ak)dΩk +

Γk

δakT (IkT

Ω ak)dΓk , (5.7)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 113

where Ω = p, np. The matrices to perform the integration by parts have the followingform, in analogy with matrices for the geometrical relations in Eqs.(2.148):

Ikp =

1Hk

α0 0

0 1Hk

β0

1Hk

β

1Hk

α0

, Ik

np =

0 0 1

Hkα

0 0 1Hk

β

0 0 0

. (5.8)

By considering the integration by parts, and the governing equation in the followingform:

δuks : Kkτsr

uu ukτ = pk

us − M kτsruu uk

τ , (5.9)

with related boundary conditions on edge Γk:

Πkτsruu uk

τ = Πkτsruu uk

τ , (5.10)

where pkus is the mechanical load, M kτsr

uu is the inertial contribution in form of funda-mental nucleus, uk

τ is the vector of the degrees of freedom for the displacements, ukτ is

the second temporal derivative of ukτ , Kkτsr

uu is the fundamental nucleus for the stiffnessmatrix, Πkτsr

uu is the fundamental nucleus for the boundary conditions:

Kkτsruu =

Ak

[(−Dk

p + Akp

)T (FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+

(5.11)(−Dk

np + Dknz −Ak

n

)T (FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)]

FsFτHkαHk

βdz ,

Πkτsruu =

Ak

[IkT

p

(FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+ (5.12)

IkTnp

(FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)]FsFτH

kαHk

βdz ,

M kτsruu =

Ak

(FrρkrI)FsFτH

kαHk

βdz . (5.13)

ρkr is the mass density of the kth layer and I is the (3× 3) identity matrix.

In order to write the explicit form of the nucleus in Eq.(5.11), the following integralsin the z thickness-direction can be defined:

(Jkτsr, Jkτsrα , Jkτsr

β , Jkτsrαβ

, Jkτsrβα

, Jkτsrαβ ) =

Ak

FτFsFr

(1, Hk

α, Hkβ ,

Hkα

Hkβ

,Hk

β

Hkα

, HkαHk

β

)dz ,

(5.14)

(Jkτzsr, Jkτzsrα , Jkτzsr

β , Jkτzsrαβ

, Jkτzsrβα

, Jkτzsrαβ ) =

Ak

∂Fτ

∂zFsFr

(1, Hk

α, Hkβ ,

Hkα

Hkβ

,Hk

β

Hkα

, HkαHk

β

)dz ,

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114 CHAPTER 5

(Jkτszr, Jkτszrα , Jkτszr

β , Jkτszrαβ

, Jkτszrβα

, Jkτszrαβ ) =

Ak

Fτ∂Fs

∂zFr

(1, Hk

α, Hkβ ,

Hkα

Hkβ

,Hk

β

Hkα

, HkαHk

β

)dz ,

(Jkτzszr, Jkτzszrα , Jkτzszr

β , Jkτzszrαβ

, Jkτzszrβα

, Jkτzszrαβ ) =

Ak

∂Fτ

∂z

∂Fs

∂zFr

(1, Hk

α, Hkβ ,

Hkα

Hkβ

,Hk

β

Hkα

, HkαHk

β

)dz .

By using the Eqs.(5.14), by developing the matrix products in Eq.(5.11) and employinga Navier-type closed form solution [85], the algebraic explicit form of the nucleus Kkτsr

uu

of dimension (3× 3) is:

Kuu11 = Qk55rJ

kτzszrαβ +

1

Rkα

Qk55r(−Jkτzsr

β − Jkτszrβ +

1

Rkα

Jkτsrβ/α ) + Qk

11rJkτsrβ/α α2 + Qk

66rJkτsrα/β β2 ,

Kuu12 = Jkτsrαβ(Qk12r + Qk

66r) = Kuu21 , (5.15)

Kuu13 = Qk55r(J

kτzsrβ α− 1

Rkα

Jkτsrβ/α α)−Qk

13rJkτszrβ α− 1

Rkα

Qk11rJ

kτsrβ/α α−Qk

12rJkτsrα

1

Rkβ

,

Kuu22 = Qk44rJ

kτzszrαβ +

1

Rkβ

Qk44r(−Jkτzsr

α − Jkτszrα +

1

Rkβ

Jkτsrα/β ) + Qk

22rJkτsrα/β β2 + Qk

66rJkτsrβ/α α2 ,

Kuu23 = Qk44r(J

kτzsrα β − 1

Rkβ

Jkτsrα/β β)−Qk

23rJkτszrα β − 1

Rkβ

Qk22rJ

kτsrα/β β − 1

Rkα

Qk12rJ

kτsrβ ,

Kuu31 = Qk55rJ

kτszrβ α−Qk

55r

1

Rkα

Jkτsrβ/α α−Qk

13rJkτzsrβ α− 1

Rkα

Qk11rJ

kτsrβ/α α− 1

Rkβ

Qk12rJ

kτsrα ,

Kuu32 = Qk44r(J

kτszrα β − 1

Rkβ

Jkτsrα/β β)−Qk

23rJkτzsrα β − 1

Rkβ

Qk22rJ

kτsrα/β β − 1

Rkα

Qk12rJ

kτsrβ ,

Kuu33 = Qk55rJ

kτsrβ/α α2 + Qk

44rJkτsrα/β β2 + Qk

33rJkτzszrαβ +

1

Rkα

(1

Rkα

Qk11rJ

kτsrβ/α + Qk

13rJkτszrβ

+ Qk13rJ

kτzsrβ ) +

2

RkαRk

β

JkτsrQk12r +

1

Rkβ

(1

Rkβ

Qk22rJ

kτsrα/β + Qk

23rJkτzsrα + Qk

23rJkτszrα ) .

α = mπ/a and β = nπ/b, with m and n as the wave numbers in in-plane directions,and a and b as the shell dimensions in α and β directions, respectively.

Nuclei for shells considered in Eqs.(5.15) degenerate in those for plates simply con-sidering 1

Rα= 1

Rβ= 0, and Hα = Hβ = 1. All the integrals in the z-direction, in the case

of plate, are of type Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.The explicit form of matrices Πkτsr

uu in Eq.(5.12) and M kτsruu in Eq.(5.13) can be found

in [193].Navier-type closed form solution is obtained via substitution of harmonic expres-

sions for the displacements as well as considering the following material coefficientsequal to zero: Q16r = Q26r = Q36r = Q45r = 0. The following harmonic assumptionscan be made for the variables, which correspond to simply supported boundary con-

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 115

ditions:

ukτ =

∑m,n

(Ukτ ) cos

(mπαk

ak

)sin

(nπβk

bk

), k = 1, Nl ,

vkτ =

∑m,n

(V kτ ) sin

(mπαk

ak

)cos

(nπβk

bk

), τ = t, b, r ,

wkτ =

∑m,n

(W kτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), r = 2, N ,

(5.16)

where Ukτ , V k

τ , W kτ are the amplitudes.

By starting from the (3×3) fundamental nucleus in Eqs.(5.15), the stiffness matrix ofthe considered multilayer is obtained by expanding via the index r in case of FGM lay-ers (see Section 2.4), via the indexes τ and s for the order of expansion in the thicknessdirection (see Section 3.4) and via the index k for the multilayer assembling (equiva-lent single layer (ESL) or Layer Wise (LW)). An example of assembling procedure for athree layered structure with the internal core in FGM is given in Figure 5.1. If there areno FGM layers, the assembling on index r is not considered [85], [43]. The acronyms

t t t

s s s

r=1,

.....

, 10

r

Layer 1 Layer 2 Layer 3

k kk

Multilayer

Constant properties Constant properties

FGM

t, s: order of

expansion assembling

r: FGM assembling

k: multilayer

assembling

: fundamental nuclei

t t t

s s s

r=1,

.....

, 10

r

Layer 1 Layer 2 Layer 3

k kk

Multilayer

Constant properties Constant properties

FGM

t, s: order of

expansion assembling

r: FGM assembling

k: multilayer

assembling : fundamental nuclei

Figure 5.1: Three layered structure with the internal layer in FGM. ESL (left) and LW(right) assembling procedures.

to indicate the two-dimensional models based on the PVD-M are: EDN, EDZN andLDN, where D means displacements formulation based on the PVD variational state-ment, N is the order of expansion in the thickness direction (until to N=14), E means

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116 CHAPTER 5

equivalent single layer approach, Z is added for an ESL approach with zig-zag func-tion, L indicates layer wise approach. Further details about these acronyms and theirmeaning, as well as its degeneration in classical theories are given in Chapter 3.

5.1.2 PVD-M: plate finite element

The Eq.(5.5) can be also solved by means of the Finite Element Method (FEM) [135],which introduces in the plate several generic elements of surface Ω (mesh of the plate).Each generic element can be transformed in a master element Ω with a given numberof nodes, see Figure 5.2. Each considered master element can have a certain number

Figure 5.2: Mesh for a generic plate by means of several master elements.

of nodes where the considered variables can be expressed, these elements can have 4nodes (Q4), 8 nodes (Q8) and 9 nodes (Q9) [206]. In the finite element method, theunknowns are expressed in terms of their nodal values, via the shape functions Nj .These last assume unit value in the nodes, and permit to express the unknowns inthe points different from the nodes as linear combinations of the 4, 8 or 9 values inthe nodes [206]. In the Figure 5.3 the elements Q4, Q8 and Q9 are clearly indicated,a natural coordinate system (ξ, η) is defined which goes always from −1 to +1. Theapproach employed here for the plate is an isoparametric approach [135]. The shapefunctions Nj for the Q4, Q8 and Q9 elements are the well-known formulas given in[206]. The variable uk

τ and its virtual variation δuks can be expressed in terms of nodal

values qkτj and δqk

si via the shape functions Nj and Ni:

ukτ (x, y) = Nj qk

τj , δuks(x, y) = Ni δq

ksi j, i = 1, 2, . . . , Nn , (5.17)

where Nn denotes the number of nodes of the element, and

qkτj =

qkuτj

qkvτj

qkwτj

, δqk

si =

δqkusi

δqkvsi

δqkwsi

. (5.18)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 117

Figure 5.3: Finite element method: 4, 8 and 9 nodes master elements with naturalcoordinate system and nodes numeration.

By considering the CUF as indicated in Section 3.4, by means of thickness functions Fτ

and Fs, the final expression of the displacement field can be obtained:

uk(x, y, z) = Fτ Nj qkτj , δuk(x, y, z) = Fs Ni δq

ksi . (5.19)

By introducing the Eqs.(5.19) in Eq.(5.5), and considering the geometrical relations forplates (see Section 2.5.2), the virtual internal work for a generic layer k is:

δLki =

Ωk

(δqk

si)T[Dp Ni

]T [(JkτsrQk

ppr (DpNj) + JkτsrQkpnr (DnpNj) + JkτzsrQk

pnr NjI)qk

τj

]

+ (δqksi)

T[Dnp Ni

]T [(JkτsrQk

npr (DpNj) + JkτsrQknnr (DnpNj) + JkτzsrQk

nnr NjI)qk

τj

]

+ (δqksi)

T[NiI

]T [(JkτszrQk

npr (DpNj) + JkτszrQknnr (DnpNj) + JkτzszrQk

nnr

NjI)qk

τj

]dΩk = δqk

si

TKkτsrij

uu qkτj = δLk

e − δLkin ,

(5.20)

where I is the [3× 3] unit array.By conveniently regrouping the terms in Eq.(5.20), the following fundamental nu-

cleus can be easily obtained:

Kkτsrijuu =

Ωk

(DT

p Ni)[Qk

ppr Jkτsr (Dp Nj) + Qkpnr Jkτsr (Dnp Nj) + Qk

pnr Jkτzsr (Nj I)]

+ (DTnp Ni)

[Qk

npr Jkτsr (Dp Nj) + Qknnr Jkτsr (Dnp Nj) + Qk

nnr Jkτzsr (Nj I)]

+ (Ni I)[Qk

npr Jkτszr (Dp Nj) + Qknnr Jkτszr (Dnp Nj) +

Qknnr Jkτ,zs,zr (Nj I)

]dΩk ,

(5.21)

The same procedure can be applied to the external work done by forces. Since theprocedure does not present any differences from the classical material cases, it is not

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118 CHAPTER 5

reported here. The complete derivation can be found in [199]. Anyway the final ex-pression for the external work reads:

δLke = δqk

si

TKkτsrij

up pτj = δqksi

TP k

us , (5.22)

where pτj contains the load nodal amplitudes. The PVD leads to the following gov-erning equation in case of FEM:

δqksi : Kkτsrij

uu qkτj = P k

us −M kτsrijuu qk

τj , (5.23)

the nucleus Kkτsrijuu has two additional indexes with respect to the nucleus for closed

form solution in Section 5.1.1: i and j are related to shape functions and to nodes of theconsidered finite element. The matrix M kτsrij

uu for the inertial forces is obtained writingthe virtual variation of the inertial work δLk

in as done for the external one in Eq.(5.22).By developing the arrays products, the explicit form of the (3 × 3) fundamental

nucleus is:

Kuu11 = Q55r / NiNj .ΩkJkτzszr + Q11r / Ni,xNj,x .Ωk

Jkτsr +

Q16r / Ni,xNj,y .ΩkJkτsr + Q16r / Ni,yNj,x .Ωk

Jkτsr + Q66r / Ni,yNj,y .ΩkJkτsr ,

Kuu12 = Q45r / NiNj .ΩkJkτzszr + Q16r / Ni,xNj,x .Ωk

Jkτsr +

Q12r / Ni,xNj,y .ΩkJkτsr + Q66r / Ni,yNj,x .Ωk

Jkτsr + Q26r / Ni,yNj,y .ΩkJkτsr ,

Kuu13 = Q13r / Ni,xNj .ΩkJkτszr + Q36r / Ni,yNj .Ωk

Jkτszr +

Q55r / NiNj,x .ΩkJkτzsr + Q45r / NiNj,y .Ωk

Jkτzsr ,

Kuu21 = Q45r / NiNj .ΩkJkτzszr + Q12r / Ni,yNj,x .Ωk

Jkτsr +

Q26r / Ni,yNj,y .ΩkJkτsr + Q16r / Ni,xNj,x .Ωk

Jkτsr + Q66r / Ni,xNj,y .ΩkJkτsr ,

Kuu22 = Q44r / NiNj .ΩkJkτzszr + Q26r / Ni,yNj,x .Ωk

Jkτsr +

Q22r / Ni,yNj,y .ΩkJkτsr + Q66r / Ni,xNj,x .Ωk

Jkτsr + Q26r / Ni,xNj,y .ΩkJkτsr ,

Kuu23 = Q36r / Ni,xNj .ΩkJkτszr + Q23r / Ni,yNj .Ωk

Jkτszr + (5.24)

Q45r / NiNj,x .ΩkJkτzsr + Q44r / NiNj,y .Ωk

Jkτzsr ,

Kuu31 = Q55r / Ni,xNj .ΩkJkτszr + Q45r / Ni,yNj .Ωk

Jkτszr +

Q13r / NiNj,x .ΩkJkτzsr + Q36r / NiNj,y .Ωk

Jkτzsr ,

Kuu32 = Q45r / Ni,xNj .ΩkJkτszr + Q44r / Ni,yNj .Ωk

Jkτszr +

Q36r / NiNj,x .ΩkJkτzsr + Q23r / NiNj,y .Ωk

Jkτzsr ,

Kuu33 = Q33r / NiNj .ΩkJkτzszr + Q45r / Ni,yNj,x .Ωk

Jkτsr +

Q44r / Ni,yNj,y .ΩkJkτsr + Q55r / Ni,xNj,x .Ωk

Jkτsr + Q45r / Ni,xNj,y .ΩkJkτsr .

where / . . . .Ωkindicates the integral on the surface Ωk of the products of the shape

functions and their derivatives with respect to the two in plane-coordinates (x,y).The finite element fundamental nucleus Kkτsrij

uu has two new indexes with respectto the same nucleus in closed form Kkτsr

uu [85]. The assembling procedures for the

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 119

number of layers, the functionally graded layers and the orders of expansion in thethickness direction are the same of those indicated in Figure 5.1 for the ESL and LWapproach, an additional loop on indexes i and j is considered for the assembling proce-dure on the number of nodes (FE assembling), as indicated in Figure 5.4. The acronyms

Figure 5.4: FE assembling procedure, additional loop on nodes (i,j).

used for the kinematic models are the same of the closed-form case: EDN, EDZN andLDN.

5.1.3 PVD-M: shell finite element

The shell finite element method is much more complicated than the plate finite elementmethod, this last does not introduce further complications with respect to the closed-form solution. In this work, a new finite shell model has been developed. It is anextension of the well-known degenerative approaches. By fixing the kinematic modelfor the displacements approximation in the thickness direction, only materials withconstant properties through the thickness z are considered. A ED2 model is employed,which means Equivalent Single Layer approach in displacements formulation, with aquadratic z-expansion for each displacement component u, v and w. The chosen mas-ter element is an eight-node (Q8). This finite element shell has been called 9P-8N-FEbecause nine variables are considered in the thickness expansion for the displacementmodel (three for each displacement component) and an eight node FE has been used.In [207] a nine node degenerated shell element is presented, in this case transverse nor-mal strain is assumed equal to zero and the element displacement field is expressed interms of three displacements of the mid-surface and two rotations of the mid-surface

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120 CHAPTER 5

normal. In each nodal point of this degenerated shell element, there are three dis-placements in the global directions x, y and z and two normal local rotations β1 andβ2. The 9P-8N-FE considers a 8 nodes finite element with 9 parameters for each node.The considered approach is an extension of the classical degenerated approaches, inthis case three terms of expansion are considered for each displacement component,so the transverse displacement is quadratically varying in the thickness direction, thismeans that the transverse normal strain εzz is not zero, so it is not a pure degeneratedapproach. A great advantage of the 9P-8N-FE model is that it is free from Poisson’slocking [66], [67]: by considering a quadratic expansion for the displacement w in thethickness direction, the transverse normal starin εzz is linear and the 3D Hooke’s lawcan be used.

The considered reference systems useful to develop the new shell element are: - theglobal cartesian (x, y, z); - the natural curvilinear (ξ, η, ζ); - the orthonormal in the node(xk, yk, zk); - the orthonormal in a generic point (x′ , y

′ , z′) (see Figure 5.5). The main

x,u

y,vz,w

e1

e3

e2

x

z h

ax

ahaz

x

hz

x’,u’

y’,v’

z’,w’

e’1

e’2e’3

x

yz

k

kk

a1

a2a3

Figure 5.5: Employed reference systems in the nine components kinematic model for aeight-node shell element (9P-8N-FE).

idea is to consider the different terms referred to a middle surface in the thickness ofthe shell [208], [209], [210]. This assumption can represent a limit with respect to the so-called geometrically exact shell models [211], [212], in particular in case of moderatelythick shells and for non-linear problems [213].

The global reference system is a rectilinear orthogonal system considered in thespace where the shell is situated. The three coordinates are x, y and z and the relative

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 121

displacement components in these three directions are indicated with u, v and w, re-spectively. The base of this reference system is an orthonormal base indicated with thevectors ~e1, ~e2 and ~e3.

The orthogonal curvilinear system for the natural coordinates is indicated with(ξ, η, ζ), the base of this system is given by the three tangent vectors to the three co-ordinates, these vectors are ~aξ, ~aη and ~aζ , respectively.

In each node is possible to consider a local tern of orthonormal vectors indicatedwith ~a1, ~a2 and ~a3.

Finally, for a generic point in the middle surface, a local cartesian reference systemcan be indicated: the three coordinates are x′, y′ and z′ and the relative displacementcomponents are u′, v′ and w′, respectively. The orthonormal base of this reference sys-tem is given by the vectors ~e′1, ~e′2 and ~e′3. The used reference systems are indicated inFigure 5.5.

By considering the ED2 model in Section 3.4.1 (ESL approach in displacements for-mulation and quadratic expansion in the thickness direction for each component), thiscan be rewritten in the local reference system as:

u′(x′, y′, z′) = u′1(x′, y′) + z′ϕ′1(x′, y′) + z

′2ψ′1(x′, y′) ,

v′(x′, y′, z′) = u′2(x′, y′) + z′ϕ′2(x′, y′) + z

′2ψ′2(x′, y′) , (5.25)

w′(x′, y′, z′) = u′3(x′, y′) + z′ϕ′3(x′, y′) + z

′2ψ′3(x′, y′) .

The three displacement components and the nine degrees of freedom are written in thelocal reference system (x′ , y

′ , z′).

By introducing the shape functions for the eight nodes element, the three displace-ment components are expressed with respect to the degrees of freedom of each node k.It is important to remark that the natural coordinate ζ and the local z′ are coincident.The shape functions in the node are indicated with Nk and they are functions of thenatural coordinates ξ and η:

u′(ξ, η, ζ) = Nk(ξ, η)u1k + ζNk(ξ, η)ϕ1k + ζ2Nk(ξ, η)ψ1k ,

v′(ξ, η, ζ) = Nk(ξ, η)u2k + ζNk(ξ, η)ϕ2k + ζ2Nk(ξ, η)ψ2k , (5.26)w′(ξ, η, ζ) = Nk(ξ, η)u3k + ζNk(ξ, η)ϕ3k + ζ2Nk(ξ, η)ψ3k ,

where the shape functions for the considered eight nodes element are those indicatedin [206] and [214]:

N1 =−(1− ξ)(1− η)(1 + ξ + η)

4, N2 =

−(1 + ξ)(1− η)(1− ξ + η)

4,

N3 =−(1 + ξ)(1 + η)(1− ξ − η)

4, N4 =

−(1− ξ)(1 + η)(1 + ξ − η)

4,

N5 =(1− ξ2)(1− η)

2, N6 =

(1− η2)(1 + ξ)

2, (5.27)

N7 =(1− ξ2)(1 + η)

2, N8 =

(1− η2)(1− ξ)

2.

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122 CHAPTER 5

By considering the derivatives of the shape functions with respect to the natural co-ordinates (ξ, η) and indicating them with the symbols Nk,ξ and Nk,η, respectively; it ispossible to obtain the derivatives of the three displacement components with respectto the natural coordinates (ξ, η, ζ):

u′,ξ = Nk,ξu1k + ζNk,ξϕ1k + ζ2Nk,ξψ1k

u′,η = Nk,ηu1k + ζNk,ηϕ1k + ζ2Nk,ηψ1k ,

u′,ζ = Nkϕ1k + 2ζNkψ1k ,

v′,ξ = Nk,ξu2k + ζNk,ξϕ2k + ζ2Nk,ξψ2k ,

v′,η = Nk,ηu2k + ζNk,ηϕ2k + ζ2Nk,ηψ2k , (5.28)

v′,ζ = Nkϕ2k + 2ζNkψ2k ,

w′,ξ = Nk,ξu3k + ζNk,ξϕ3k + ζ2Nk,ξψ3k ,

w′,η = Nk,ηu3k + ζNk,ηϕ3k + ζ2Nk,ηψ3k ,

w′,ζ = Nkϕ3k + 2ζNkψ3k .

The displacement components have been written with respect to the local referencesystem (x′, y′, z′), and derived with respect to the natural coordinates (ξ, η, ζ). The localdisplacements u′, v′ and w′ derived with respect to coordinates (ξ, η, ζ) are transformedin local displacements derived with respect to the global coordinates (x, y, z). For thisaim, it is important to remark the meaning of Jacobian matrix [J ], it is a [3 × 3] matrixcontaining the derivatives of global coordinates with respect to natural ones:

[J ] =

∂x∂ξ

∂y∂ξ

∂z∂ξ

∂x∂η

∂y∂η

∂z∂η

∂x∂ζ

∂y∂ζ

∂z∂ζ

. (5.29)

In order to pass from local displacements derived with respect to natural coordinates,to local displacements derived with respect to global coordinates, the inverse of Jaco-bian matrix, indicated with [J ]−1, is employed:

[J ]−1 =

J∗11 J∗12 J∗13

J∗21 J∗22 J∗23

J∗31 J∗32 J∗33

. (5.30)

By using the matrix form, it is possible to write:

u′,x v′,x w′,x

u′,y v′,y w′,y

u′,z v′,z w′,z

=

J∗11 J∗12 J∗13

J∗21 J∗22 J∗23

J∗31 J∗32 J∗33

u′,ξ v′,ξ w′,ξ

u′,η v′,η w′,η

u′,ζ v′,ζ w′,ζ

. (5.31)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 123

In explicit form, the nine derivatives with respect to the global coordinates are:

u′,x = J∗11u′,ξ + J∗12u

′,η + J∗13u

′,ζ ,

v′,x = J∗11v′,ξ + J∗12v

′,η + J∗13v

′,ζ ,

w′,x = J∗11w

′,ξ + J∗12w

′,η + J∗13w

′,ζ ,

u′,y = J∗21u′,ξ + J∗22u

′,η + J∗23u

′,ζ ,

v′,y = J∗21v′,ξ + J∗22v

′,η + J∗23v

′,ζ , (5.32)

w′,y = J∗21w

′,ξ + J∗22w

′,η + J∗23w

′,ζ ,

u′,z = J∗31u′,ξ + J∗32u

′,η + J∗33u

′,ζ ,

v′,z = J∗31v′,ξ + J∗32v

′,η + J∗33v

′,ζ ,

w′,z = J∗31w

′,ξ + J∗32w

′,η + J∗33w

′,ζ .

By substituting Eqs.(5.28) in Eqs.(5.32) and collecting with respect to the nine degreesof freedom in each node, we obtain the explicit form of local displacements derivedwith respect to the global coordinates:

u′,x =(J∗11Nk,ξ + J∗12Nk,η)u1k + (J∗11ζNk,ξ + J∗12ζNk,η + J∗13Nk)ϕ1k+

(J∗11ζ2Nk,ξ + J∗12ζ

2Nk,η + J∗132ζNk)ψ1k ,

v′,x =(J∗11Nk,ξ + J∗12Nk,η)u2k + (J∗11ζNk,ξ + J∗12ζNk,η + J∗13Nk)ϕ2k+

(J∗11ζ2Nk,ξ + J∗12ζ

2Nk,η + J∗132ζNk)ψ2k ,

w′,x =(J∗11Nk,ξ + J∗12Nk,η)u3k + (J∗11ζNk,ξ + J∗12ζNk,η + J∗13Nk)ϕ3k+

(J∗11ζ2Nk,ξ + J∗12ζ

2Nk,η + J∗132ζNk)ψ3k ,

u′,y =(J∗21Nk,ξ + J∗22Nk,η)u1k + (J∗21ζNk,ξ + J∗22ζNk,η + J∗23Nk)ϕ1k+

(J∗21ζ2Nk,ξ + J∗22ζ

2Nk,η + J∗232ζNk)ψ1k ,

v′,y =(J∗21Nk,ξ + J∗22Nk,η)u2k + (J∗21ζNk,ξ + J∗22ζNk,η + J∗23Nk)ϕ2k+ (5.33)

(J∗21ζ2Nk,ξ + J∗22ζ

2Nk,η + J∗232ζNk)ψ2k ,

w′,y =(J∗21Nk,ξ + J∗22Nk,η)u3k + (J∗21ζNk,ξ + J∗22ζNk,η + J∗23Nk)ϕ3k+

(J∗21ζ2Nk,ξ + J∗22ζ

2Nk,η + J∗232ζNk)ψ3k ,

u′,z =(J∗31Nk,ξ + J∗32Nk,η)u1k + (J∗31ζNk,ξ + J∗32ζNk,η + J∗33Nk)ϕ1k+

(J∗31ζ2Nk,ξ + J∗32ζ

2Nk,η + J∗332ζNk)ψ1k ,

v′,z =(J∗31Nk,ξ + J∗32Nk,η)u2k + (J∗31ζNk,ξ + J∗32ζNk,η + J∗33Nk)ϕ2k+

(J∗31ζ2Nk,ξ + J∗32ζ

2Nk,η + J∗332ζNk)ψ2k ,

w′,z =(J∗31Nk,ξ + J∗32Nk,η)u3k + (J∗31ζNk,ξ + J∗32ζNk,η + J∗33Nk)ϕ3k+

(J∗31ζ2Nk,ξ + J∗32ζ

2Nk,η + J∗332ζNk)ψ3k .

The final step is: the displacements written with respect to the local cartesian refer-ence system and derived with respect to coordinates (x′, y′, z′). For this aim, the matrix[Θ] containing the scalar products between the base vectors of global reference system

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124 CHAPTER 5

(~e1, ~e2, ~e3) and the base vectors of local reference system (~e′1, ~e′2, ~e′3) is introduced:

[Θ] =

~e′1 · ~e1

~e′2 · ~e1~e′3 · ~e1

~e′1 · ~e2~e′2 · ~e2

~e′3 · ~e2

~e′1 · ~e3~e′2 · ~e3

~e′3 · ~e3

=

t11 t21 t31

t12 t22 t32

t13 t23 t33

. (5.34)

By using the matrix [Θ], the local displacements derived with respect to the local coor-dinates (x′, y′, z′) are:

u′,x′ v′,x′ w′,x′

u′,y′ v′,y′ w′,y′

u′,z′ v′,z′ w′,z′

= [Θ]T

u′,x v′,x w′,x

u′,y v′,y w′,y

u′,z v′,z w′,z

. (5.35)

The local strains can be obtained using the results from Eq.(5.35). In the local referencesystem the relations for the six strain components are:

ε′x′x′ = u′,x′ ,

ε′y′y′ = v′,y′ ,

ε′z′z′ = w′,z′ ,

γ′y′z′ = v′,z′ + w′,y′ , (5.36)

γ′x′z′ = u′,z′ + w′,x′ ,

γ′x′y′ = u′,y′ + v′,x′ .

By considering Eqs.(5.33)-(5.35), the Eqs.(5.36) can be written in explicit forms whichdepend by the nine degrees of freedom of each node:

ε′x′x′ =1 · 0 ·Nk · u1k + 1 · A1 ·Nk,ξ · u1k + 1 ·B1 ·Nk,η · u1k+

1 ·G1 ·Nk · ϕ1k + ζ · A1 ·Nk,ξ · ϕ1k + ζ ·B1 ·Nk,η · ϕ1k+ (5.37)ζ · 2G1 ·Nk · ψ1k + ζ2 · A1 ·Nk,ξ · ψ1k + ζ2 ·B1 ·Nk,η · ψ1k ,

ε′y′y′ =1 · 0 ·Nk · u2k + 1 · A2 ·Nk,ξ · u2k + 1 ·B2 ·Nk,η · u2k+

1 ·G2 ·Nk · ϕ2k + ζ · A2 ·Nk,ξ · ϕ2k + ζ ·B2 ·Nk,η · ϕ2k+ (5.38)ζ · 2G2 ·Nk · ψ2k + ζ2 · A2 ·Nk,ξ · ψ2k + ζ2 ·B2 ·Nk,η · ψ2k ,

ε′z′z′ =1 · 0 ·Nk · u3k + 1 · A3 ·Nk,ξ · u3k + 1 ·B3 ·Nk,η · u3k+

1 ·G3 ·Nk · ϕ3k + ζ · A3 ·Nk,ξ · ϕ3k + ζ ·B3 ·Nk,η · ϕ3k+ (5.39)ζ · 2G3 ·Nk · ψ3k + ζ2 · A3 ·Nk,ξ · ψ3k + ζ2 ·B3 ·Nk,η · ψ3k ,

γ′y′z′ =1 · 0 ·Nk · u2k + 1 · A3 ·Nk,ξ · u2k + 1 ·B3 ·Nk,η · u2k+

1 · 0 ·Nk · u3k + 1 · A2 ·Nk,ξ · u3k + 1 ·B2 ·Nk,η · u3k+

1 ·G3 ·Nk · ϕ2k + ζ · A3 ·Nk,ξ · ϕ2k + ζ ·B3 ·Nk,η · ϕ2k+ (5.40)1 ·G2 ·Nk · ϕ3k + ζ · A2 ·Nk,ξ · ϕ3k + ζ ·B2 ·Nk,η · ϕ3k+

ζ · 2G3 ·Nk · ψ2k + ζ2 · A3 ·Nk,ξ · ψ2k + ζ2 ·B3 ·Nk,η · ψ2k

ζ · 2G2 ·Nk · ψ3k + ζ2 · A2 ·Nk,ξ · ψ3k + ζ2 ·B2 ·Nk,η · ψ3k ,

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 125

γ′x′z′ =1 · 0 ·Nk · u1k + 1 · A3 ·Nk,ξ · u1k + 1 ·B3 ·Nk,η · u1k+

1 · 0 ·Nk · u3k + 1 · A1 ·Nk,ξ · u3k + 1 ·B1 ·Nk,η · u3k+

1 ·G3 ·Nk · ϕ1k + ζ · A3 ·Nk,ξ · ϕ1k + ζ ·B3 ·Nk,η · ϕ1k+ (5.41)1 ·G1 ·Nk · ϕ3k + ζ · A1 ·Nk,ξ · ϕ3k + ζ ·B1 ·Nk,η · ϕ3k+

ζ · 2G3 ·Nk · ψ1k + ζ2 · A3 ·Nk,ξ · ψ1k + ζ2 ·B3 ·Nk,η · ψ1k

ζ · 2G1 ·Nk · ψ3k + ζ2 · A1 ·Nk,ξ · ψ3k + ζ2 ·B1 ·Nk,η · ψ3k ,

γ′x′y′ =1 · 0 ·Nk · u1k + 1 · A2 ·Nk,ξ · u1k + 1 ·B2 ·Nk,η · u1k+

1 · 0 ·Nk · u2k + 1 · A1 ·Nk,ξ · u2k + 1 ·B1 ·Nk,η · u2k+

1 ·G2 ·Nk · ϕ1k + ζ · A2 ·Nk,ξ · ϕ1k + ζ ·B2 ·Nk,η · ϕ1k+ (5.42)1 ·G1 ·Nk · ϕ2k + ζ · A1 ·Nk,ξ · ϕ2k + ζ ·B1 ·Nk,η · ϕ2k+

ζ · 2G2 ·Nk · ψ1k + ζ2 · A2 ·Nk,ξ · ψ1k + ζ2 ·B2 ·Nk,η · ψ1k

ζ · 2G1 ·Nk · ψ2k + ζ2 · A1 ·Nk,ξ · ψ2k + ζ2 ·B1 ·Nk,η · ψ2k .

The six components are listed in the following way:

[ε′]T = [ε′x′x′ ε′y′y′ ε′z′z′ γ′y′z′ γ′x′z′ γ′x′y′ ] = [ε′1 ε′2 ε′3 ε′4 ε′5 ε′6] .

The coefficients Aj , Bj and Gj contain, for each strain component, the geometrical in-formation of shell, such as the Jacobian and the scalar products. The explicit form ofthe geometrical coefficients for each strain component are:

A1 = t11J∗11 + t21J

∗21 + t31J

∗31 ,

B1 = t11J∗12 + t21J

∗22 + t31J

∗32 ,

G1 = t11J∗13 + t21J

∗23 + t31J

∗33 ,

A2 = t12J∗11 + t22J

∗21 + t32J

∗31 , (5.43)

B2 = t12J∗12 + t22J

∗22 + t32J

∗32 ,

G2 = t12J∗13 + t22J

∗23 + t32J

∗33 ,

A3 = t13J∗11 + t23J

∗21 + t33J

∗31 ,

B3 = t13J∗12 + t23J

∗22 + t33J

∗32 ,

G3 = t13J∗13 + t23J

∗23 + t33J

∗33 .

The strains can be split in in-plane and out-plane components, and we can considerthem as matrix products, where the involved matrices contains separately the thick-ness coordinates, the geometrical parameters and the degrees of freedom. The startingpoint is the Eqs.(5.37)-(5.42) and the vector [BDOF ] of dimension (27 × 1), where thedegrees of freedom and their derivatives are ordinate in the following way:

[BDOF ]T =[[NU ] [NΦ] [NΨ]] = (5.44)=[Nku1k Nk,ξu1k Nk,ηu1k Nku2k Nk,ξu2k Nk,ηu2k Nku3k Nk,ξu3k Nk,ηu3k

Nkϕ1k Nk,ξϕ1k Nk,ηϕ1k Nkϕ2k Nk,ξϕ2k Nk,ηϕ2k Nkϕ3k Nk,ξϕ3k Nk,ηϕ3k

Nkψ1k Nk,ξψ1k Nk,ηψ1k Nkψ2k Nk,ξψ2k Nk,ηψ2k Nkψ3k Nk,ξψ3k Nk,ηψ3k].

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126 CHAPTER 5

In a compact form the in-plane and out-plane strains are expressed as:

[ε′p] = [Fζ ][Geop][BDOF ] , (5.45)[ε′n] = [Fζ ][Geon][BDOF ] , (5.46)

where the in-plane and out-plane strains are:

[ε′p]T = [ε′x′x′ ε′y′y′ γ′x′y′ ] , [ε′n]T = [γ′y′z′ γ′x′z′ ε′z′z′ ] , (5.47)

the matrix [Fζ ], containing the thickness parameters, has dimension (3× 9):

[Fζ ] =

1 ζ ζ2 0 0 0 0 0 00 0 0 1 ζ ζ2 0 0 00 0 0 0 0 0 1 ζ ζ2

. (5.48)

The two matrices [Geop] and [Geon] contains the geometrical parameters, they havedimension (9×27), each geometrical matrix can be split in three submatrices of dimen-sion (9× 9):

[Geop] =[[Gup] [Gϕp] [Gψp]

], (5.49)

[Geon] =[[Gun] [Gϕn] [Gψn]

], (5.50)

where

[Gup] =

0 A1 B1 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 A2 B2 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 A2 B2 0 A1 B1 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

, (5.51)

[Gϕp] =

G1 0 0 0 0 0 0 0 00 A1 B1 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 G2 0 0 0 0 00 0 0 0 A2 B2 0 0 00 0 0 0 0 0 0 0 0

G2 0 0 G1 0 0 0 0 00 A2 B2 0 A1 B1 0 0 00 0 0 0 0 0 0 0 0

, (5.52)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 127

[Gψp] =

0 0 0 0 0 0 0 0 02G1 0 0 0 0 0 0 0 00 A1 B1 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 2G2 0 0 0 0 00 0 0 0 A2 B2 0 0 00 0 0 0 0 0 0 0 0

2G2 0 0 2G1 0 0 0 0 00 A2 B2 0 A1 B1 0 0 0

, (5.53)

[Gun] =

0 0 0 0 A3 B3 0 A2 B2

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 A3 B3 0 0 0 0 A1 B1

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 A3 B3

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

, (5.54)

[Gϕn] =

0 0 0 G3 0 0 G2 0 00 0 0 0 A3 B3 0 A2 B2

0 0 0 0 0 0 0 0 0G3 0 0 0 0 0 G1 0 00 A3 B3 0 0 0 0 A1 B1

0 0 0 0 0 0 0 0 00 0 0 0 0 0 G3 0 00 0 0 0 0 0 0 A3 B3

0 0 0 0 0 0 0 0 0

, (5.55)

[Gψn] =

0 0 0 0 0 0 0 0 00 0 0 2G3 0 0 2G2 0 00 0 0 0 A3 B3 0 A2 B2

0 0 0 0 0 0 0 0 02G3 0 0 0 0 0 2G1 0 00 A3 B3 0 0 0 0 A1 B1

0 0 0 0 0 0 0 0 00 0 0 0 0 0 2G3 0 00 0 0 0 0 0 0 A3 B3

. (5.56)

The vector (27 × 1) indicated in Eq.(5.44) can be written in matrix form split in thematrix [B] containing the shape functions and their derivatives (dimension (27 × 72))

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128 CHAPTER 5

and the array containing the degrees of freedom [DOF ] (dimension (72× 1)):

[BDOF ] = [B][DOF ] . (5.57)

The vector containing the degrees of freedom is split in three subarrays [DOFu], [DOFϕ]and [DOFψ], each one of dimension (24× 1):

[DOF ]T = [[DOFu]T [DOFϕ]T [DOFψ]T ] = (5.58)= [u1k=1,8 u2k=1,8 u3k=1,8 ϕ1k=1,8 ϕ2k=1,8 ϕ3k=1,8

ψ1k=1,8 ψ2k=1,8 ψ3k=1,8] .

The matrix of shape functions and their derivatives [B] of dimension (27 × 72), hasthree blocks of dimension (9 × 24) along its diagonal, these three blocks are the sameand they are indicated with [Bf ] = [Bu] = [Bϕ] = [Bψ]:

[B] =

[Bu] [0] [0]

[0] [Bϕ] [0]

[0] [0] [Bψ]

, (5.59)

the explicit form of each block is:

[Bf ] =

[N ] [0] [0]

[0] [N ] [0]

[0] [0] [N ]

, (5.60)

where the matrix [N ] contains the shape functions and their derivatives for each node(k = 1, 8):

[N ] =

N1 N2 N3 N4 N5 N6 N7 N8

N1,ξ N2,ξ N3,ξ N4,ξ N5,ξ N6,ξ N7,ξ N8,ξ

N1,η N2,η N3,η N4,η N5,η N6,η N7,η N8,η

. (5.61)

By using Eqs.(5.57),(5.58), the Eqs.(5.45),(5.46) can be written as:

[ε′p] = [Fζ ][Geop][B][DOF ] , (5.62)[ε′n] = [Fζ ][Geon][B][DOF ] . (5.63)

By using the procedure illustrated in Eqs. (5.49),(5.50),(5.58) and (5.59), the Eqs.(5.62)and (5.63) are written in the following way:

[ε′p] = [Fζ ]([Gup][Bu][DOFu] + [Gϕp][Bϕ][DOFϕ] + [Gψp][Bψ][DOFψ]

), (5.64)

[ε′n] = [Fζ ]([Gun][Bu][DOFu] + [Gϕn][Bϕ][DOFϕ] + [Gψn][Bψ][DOFψ]

). (5.65)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 129

These are the geometrical relations written in matrix form which are used in the varia-tional statement in order to obtain the local stiffness matrix.

By considering the Hooke’s law, as described in Section 2.1, a relation between stresscomponents and strain ones holds. The vector of dimension (6×1) containing the stresscomponents in local reference system is in relation with the vector of dimension (6×1)containing the strain components via the matrix [Q] of dimension [6×6] containing theelastic coefficients. In general the relation is:

[σ′] = [Q][ε′] . (5.66)

The vectors of stress and strain components in the local reference system are:

[σ′]T = [σ′x′x′ σ′y′y′ σ′z′z′ σ′y′z′ σ′x′z′ σ′x′y′ ] , (5.67)

[ε′]T = [ε′x′x′ ε′y′y′ ε′z′z′ γ′y′z′ γ′x′z′ γ′x′y′ ] . (5.68)

In the case of orthotropic material, the matrix containing the elastic coefficients has thefollowing structure:

[Q] =

Q11 Q12 Q13 0 0 Q16

Q12 Q22 Q23 0 0 Q26

Q13 Q23 Q33 0 0 Q36

0 0 0 Q44 Q45 00 0 0 Q45 Q55 0

Q16 Q26 Q36 0 0 Q66

. (5.69)

In order to use the constitutive equations in the Principle of Virtual Displacements, itis very useful to split the stress and strain components in in-plane (p) and out-plane(n) contributes:

σ′x′x′σ′y′y′σ′x′y′σ′y′z′σ′x′z′σ′z′z′

=

Q11 Q12 Q16 0 0 Q13

Q12 Q22 Q26 0 0 Q23

Q16 Q26 Q66 0 0 Q36

0 0 0 Q44 Q45 00 0 0 Q45 Q55 0

Q13 Q23 Q36 0 0 Q33

ε′x′x′ε′y′y′ε′x′y′ε′y′z′ε′x′z′ε′z′z′

. (5.70)

The vectors of in-plane and out-plane strain and stress components are:

[σ′p]T = [σ′x′x′ σ′y′y′ σ′x′y′ ] , [σ′n]T = [σ′y′z′ σ′x′z′ σ′z′z′ ] ,

[ε′p]T = [ε′x′x′ ε′y′y′ ε′x′y′ ] , [ε′n]T = [ε′y′z′ ε′x′z′ ε′z′z′ ] . (5.71)

In this way the constitutive equations are split in the following form:

[σ′p] = [Qpp][ε′p] + [Qpn][ε′n] , (5.72)

[σ′n] = [Qnp][ε′p] + [Qnn][ε′n] , (5.73)

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130 CHAPTER 5

where:

[Qpp] =

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

, [Qpn] =

0 0 Q13

0 0 Q23

0 0 Q36

, (5.74)

[Qnp] =

0 0 00 0 0

Q13 Q23 Q36

, [Qnn] =

Q44 Q45 0Q45 Q55 00 0 Q33

. (5.75)

The Principle of Virtual Displacements (PVD) in Eq.(5.2) can be written for one layer,and the inertial work can be discarded:

δLi =

V

δε′T σ′dV =

Ω

A

(δε′TpGσ′pC + δε′TnGσ′

nC)dΩdζ , (5.76)

where δLi is the virtual variation of the internal work, V is the volume of the consid-ered layer, Ω is the integration domain in the plane, A is the integration domain inthe thickness direction, T means transpose of a vector or matrix. Subscript G meanssubstitution of geometrical relations, subscript C means substitution of constitutiveequations written in split form. Bold letters before the substitution of geometrical andconstitutive equations mean arrays or matrices, in this section the arrays and matricesare indicate also with the symbol [ ].

By substituting the constitutive equations in the subscript C, the PVD is:

δLi =

Ω

A

δε′TpG

([Qpp]ε

′pG+[Qpn]ε′nG]

)+δε′TnG

([Qnp]ε

′pG+[Qnn]ε′nG

)dΩdζ . (5.77)

After substitution of geometrical relations, given in Eqs.(5.64) and(5.65), in the sub-script G, the PVD gets:

δLi =

Ω

A

(δ[DOFu]T ([Fζ ][Gup][Bu])T + δ[DOFϕ]T ([Fζ ][Gϕp][Bϕ])T (5.78)

+ δ[DOFψ]T ([Fζ ][Gψp][Bψ])T

)(([Qpp][Fζ ][Gup][Bu][DOFu] + [Qpp][Fζ ][Gϕp][Bϕ][DOFϕ]

+ [Qpp][Fζ ][Gψp][Bψ][DOFψ])

+([Qpn][Fζ ][Gun][Bu][DOFu] + [Qpn][Fζ ][Gϕn][Bϕ][DOFϕ]

+ [Qpn][Fζ ][Gψn][Bψ][DOFψ]))

+

(δ[DOFu]T ([Fζ ][Gun][Bu])T + δ[DOFϕ]T ([Fζ ][Gϕn][Bϕ])T

+ δ[DOFψ]T ([Fζ ][Gψn][Bψ])T

)(([Qnp][Fζ ][Gup][Bu][DOFu] + [Qnp][Fζ ][Gϕp][Bϕ][DOFϕ]

+ [Qnp][Fζ ][Gψp][Bψ][DOFψ])

+([Qnn][Fζ ][Gun][Bu][DOFu] + [Qnn][Fζ ][Gϕn][Bϕ][DOFϕ]

+ [Qnn][Fζ ][Gψn][Bψ][DOFψ]))

dΩdζ .

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 131

For a eight node shell element the system of governing equations is:

δ[DOF ] : [K ′] [DOF ] = [P ] . (5.79)

In Eq.(5.79) the vector [P] has dimension (72 × 1) and it is the mechanical load. Thevector of degrees of freedom [DOF ] of dimension (72 × 1) is that in Eq.(5.58). Thestiffness matrix of the shell element has dimension (72 × 72) and it is considered asa block of 9 matrices of dimension (24 × 24). It is important to remark that [DOFu]represents the first order of expansion terms for the considered eight nodes, [DOFϕ]contains the second order terms and [DOFψ] is for the higher order terms in ζ . Thematrix [K ′] is:

[K ′] =

[K ′uu]

[K ′

] [K ′

][K ′

ϕu

] [K ′

ϕϕ

] [K ′

ϕψ

][K ′

ψu

] [K ′

ψϕ

] [K ′

ψψ

]

. (5.80)

The nine subarrays of the stiffness matrix [K ′] have dimension (24× 24), they are con-sidered for each layer at the element level. The considered element is a shell elementwith eight nodes, the assembly procedure to obtain the stiffness matrix at multilayerlevel is the Equivalent Single Layer (ESL) one. The nine sub-arrays are:

[K ′uu] =

Ω

A

[Bu]T [Gup]

T [Fζ ]T [Qpp][Fζ ][Gup][Bu] + [Bu]T [Gup]

T [Fζ ]T [Qpn][Fζ ][Gun][Bu]

(5.81)

+ [Bu]T [Gun]T [Fζ ]T [Qnp][Fζ ][Gup][Bu] + [Bu]T [Gun]T [Fζ ]

T [Qnn][Fζ ][Gun][Bu]

dΩdζ ,

[K ′uϕ] =

Ω

A

[Bu]T [Gup]

T [Fζ ]T [Qpp][Fζ ][Gϕp][Bϕ] + [Bu]T [Gup]

T [Fζ ]T [Qpn][Fζ ][Gϕn][Bϕ]

(5.82)

+ [Bu]T [Gun]T [Fζ ]T [Qnp][Fζ ][Gϕp][Bϕ] + [Bu]T [Gun]T [Fζ ]

T [Qnn][Fζ ][Gϕn][Bϕ]

dΩdζ ,

[K ′uψ] =

Ω

A

[Bu]T [Gup]

T [Fζ ]T [Qpp][Fζ ][Gψp][Bψ] + [Bu]T [Gup]

T [Fζ ]T [Qpn][Fζ ][Gψn][Bψ]

(5.83)

+ [Bu]T [Gun]T [Fζ ]T [Qnp][Fζ ][Gψp][Bψ] + [Bu]T [Gun]T [Fζ ]

T [Qnn][Fζ ][Gψn][Bψ]

dΩdζ ,

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132 CHAPTER 5

[K ′ϕu] =

Ω

A

[Bϕ]T [Gϕp]

T [Fζ ]T [Qpp][Fζ ][Gup][Bu] + [Bϕ]T [Gϕp]

T [Fζ ]T [Qpn][Fζ ][Gun][Bu]

(5.84)

+ [Bϕ]T [Gϕn]T [Fζ ]T [Qnp][Fζ ][Gup][Bu] + [Bϕ]T [Gϕn]T [Fζ ]

T [Qnn][Fζ ][Gun][Bu]

dΩdζ ,

[K ′ϕϕ] =

Ω

A

[Bϕ]T [Gϕp]

T [Fζ ]T [Qpp][Fζ ][Gϕp][Bϕ] + [Bϕ]T [Gϕp]

T [Fζ ]T [Qpn][Fζ ][Gϕn][Bϕ]

(5.85)

+ [Bϕ]T [Gϕn]T [Fζ ]T [Qnp][Fζ ][Gϕp][Bϕ] + [Bϕ]T [Gϕn]T [Fζ ]

T [Qnn][Fζ ][Gϕn][Bϕ]

dΩdζ ,

[K ′ϕψ] =

Ω

A

[Bϕ]T [Gϕp]

T [Fζ ]T [Qpp][Fζ ][Gψp][Bψ] + [Bϕ]T [Gϕp]

T [Fζ ]T [Qpn][Fζ ][Gψn][Bψ]

(5.86)

+ [Bϕ]T [Gϕn]T [Fζ ]T [Qnp][Fζ ][Gψp][Bψ] + [Bϕ]T [Gϕn]T [Fζ ]

T [Qnn][Fζ ][Gψn][Bψ]

dΩdζ ,

[K ′ψu] =

Ω

A

[Bψ]T [Gψp]

T [Fζ ]T [Qpp][Fζ ][Gup][Bu] + [Bψ]T [Gψp]

T [Fζ ]T [Qpn][Fζ ][Gun][Bu]

(5.87)

+ [Bψ]T [Gψn]T [Fζ ]T [Qnp][Fζ ][Gup][Bu] + [Bψ]T [Gψn]T [Fζ ]

T [Qnn][Fζ ][Gun][Bu]

dΩdζ ,

[K ′ψϕ] =

Ω

A

[Bψ]T [Gψp]

T [Fζ ]T [Qpp][Fζ ][Gϕp][Bϕ] + [Bψ]T [Gψp]

T [Fζ ]T [Qpn][Fζ ][Gϕn][Bϕ]

(5.88)

+ [Bψ]T [Gψn]T [Fζ ]T [Qnp][Fζ ][Gϕp][Bϕ] + [Bψ]T [Gψn]T [Fζ ]

T [Qnn][Fζ ][Gϕn][Bϕ]

dΩdζ ,

[K ′ψψ] =

Ω

A

[Bψ]T [Gψp]

T [Fζ ]T [Qpp][Fζ ][Gψp][Bψ] + [Bψ]T [Gψp]

T [Fζ ]T [Qpn][Fζ ][Gψn][Bψ]

(5.89)

+ [Bψ]T [Gψn]T [Fζ ]T [Qnp][Fζ ][Gψp][Bψ] + [Bψ]T [Gψn]T [Fζ ]

T [Qnn][Fζ ][Gψn][Bψ]

dΩdζ .

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 133

The integrals in ζ direction are analytically solved, they are four matrices of dimension(9×9), which are integrated in each layer with bounds ζt and ζb (coordinates of the topand bottom of each layer). These four matrices are called [App], [Apn], [Anp] and [Ann].All stiffness matrices are written for a generic layer. In case of multilayer structures thematrices [App], [Apn], [Anp] and [Ann] are calculated for each given layer and in an ESLapproach they are simply summed. The matrices which depend on the thickness ζ are:

[App] =

∫ ζt

ζb

([Fζ ]

T [Qpp][Fζ ])dζ , [Apn] =

∫ ζt

ζb

([Fζ ]

T [Qpn][Fζ ])dζ , (5.90)

[Anp] =

∫ ζt

ζb

([Fζ ]

T [Qnp][Fζ ])dζ , [Ann] =

∫ ζt

ζb

([Fζ ]

T [Qnn][Fζ ])dζ . (5.91)

By employing Eqs.(5.90) and (5.91), the stiffness matrices can be rewritten in a differentway. |J | means determinant of the Jacobian matrix:

[K ′uu] =

Ω

[Bu]T [Gup]

T [App][Gup][Bu] + [Bu]T [Gup]T [Apn][Gun][Bu] (5.92)

+ [Bu]T [Gun]T [Anp][Gup][Bu] + [Bu]T [Gun]T [Ann][Gun][Bu]

|J |dξdη ,

[K ′uϕ] =

Ω

[Bu]T [Gup]

T [App][Gϕp][Bϕ] + [Bu]T [Gup]T [Apn][Gϕn][Bϕ] (5.93)

+ [Bu]T [Gun]T [Anp][Gϕp][Bϕ] + [Bu]T [Gun]T [Ann][Gϕn][Bϕ]

|J |dξdη ,

[K ′uψ] =

Ω

[Bu]T [Gup]

T [App][Gψp][Bψ] + [Bu]T [Gup]T [Apn][Gψn][Bψ] (5.94)

+ [Bu]T [Gun]T [Anp][Gψp][Bψ] + [Bu]T [Gun]T [Ann][Gψn][Bψ]

|J |dξdη ,

[K ′ϕu] =

Ω

[Bϕ]T [Gϕp]

T [App][Gup][Bu] + [Bϕ]T [Gϕp]T [Apn][Gun][Bu] (5.95)

+ [Bϕ]T [Gϕn]T [Anp][Gup][Bu] + [Bϕ]T [Gϕn]T [Ann][Gun][Bu]

|J |dξdη ,

[K ′ϕϕ] =

Ω

[Bϕ]T [Gϕp]

T [App][Gϕp][Bϕ] + [Bϕ]T [Gϕp]T [Apn][Gϕn][Bϕ] (5.96)

+ [Bϕ]T [Gϕn]T [Anp][Gϕp][Bϕ] + [Bϕ]T [Gϕn]T [Ann][Gϕn][Bϕ]

|J |dξdη ,

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134 CHAPTER 5

[K ′ϕψ] =

Ω

[Bϕ]T [Gϕp]

T [App][Gψp][Bψ] + [Bϕ]T [Gϕp]T [Apn][Gψn][Bψ] (5.97)

+ [Bϕ]T [Gϕn]T [Anp][Gψp][Bψ] + [Bϕ]T [Gϕn]T [Ann][Gψn][Bψ]

|J |dξdη ,

[K ′ψu] =

Ω

[Bψ]T [Gψp]

T [App][Gup][Bu] + [Bψ]T [Gψp]T [Apn][Gun][Bu] (5.98)

+ [Bψ]T [Gψn]T [Anp][Gup][Bu] + [Bψ]T [Gψn]T [Ann][Gun][Bu]

|J |dξdη ,

[K ′ψϕ] =

Ω

[Bψ]T [Gψp]

T [App][Gϕp][Bϕ] + [Bψ]T [Gψp]T [Apn][Gϕn][Bϕ] (5.99)

+ [Bψ]T [Gψn]T [Anp][Gϕp][Bϕ] + [Bψ]T [Gψn]T [Ann][Gϕn][Bϕ]

|J |dξdη ,

[K ′ψψ] =

Ω

[Bψ]T [Gψp]

T [App][Gψp][Bψ] + [Bψ]T [Gψp]T [Apn][Gψn][Bψ] (5.100)

+ [Bψ]T [Gψn]T [Anp][Gψp][Bψ] + [Bψ]T [Gψn]T [Ann][Gψn][Bψ]

|J |dξdη .

The integrals in the domain Ω are numerically computed via points and weights ofGauss. The chosen number of Gauss points is nine. The split forms obtained inEqs.(5.64), (5.65) and in Eqs.(5.90), (5.91), have been introduced in order to correct theshear and membrane lockings, further details about the correction of the numericallockings will be given in the near future. The great advantage of the 9P-8N-FE modelis that it is free from Poisson’s locking and no plane-stress conditions must be imposedin Hooke’s law.

The transformation matrix in Eq.(5.34) is rearranged by nodes, it is called ΘTR andhas dimension [72× 72]. The global stiffness matrix is given by:

[K] = [ΘTR] [K ′] [ΘTR]T .

5.2 PVD for the mechanical case with an external temper-ature load, PVD-M(T)

In case of a thermal stress analysis of plates and shells, a possible extension of thePVD is the so called PVD-M(T) as indicated in Section 4.2.2, where the temperature is

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 135

seen as an external load without any coupling with the mechanical field [172]. In thevariational statement obtained in Eq.(4.18) the stresses are seen as an algebraic additionof mechanical (d) and thermal (t) contributions:

Nl∑

k=1

Ωk

Ak

δεk

pG

T(σk

pd − σkpt) + δεk

nG

T(σk

nd − σknt)

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin . (5.101)

The constitutive equations written in order to consider the inclusion of FGM layershave been obtained in Eq.(4.17):

σkpC = σk

pd − σkpt = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Frλ

kprθ

k , (5.102)

σknC = σk

nd − σknt = FrQ

knprε

kpG + FrQ

knnrε

knG − Frλ

knrθ

k ,

where the arrays λp and λn permit the partial coupling between the mechanical fieldand the temperature as detailed in Chapter 2.

The steps to obtain the inertial matrix M kτsruu , with the inclusion of inertial forces, do

not introduce new concepts with respect to the mechanical case, so it is not discussedanymore in this chapter.

By substituting the Eqs.(5.102) in Eq.(5.101) and considering the geometrical rela-tions for shell of Section 2.5.1 and the CUF of Section 3.4 , for a generic layer k it ispossible to write:∫

Ωk

Ak

[((Dk

p + Akp) Fsδu

ks

)T (FrQ

kppr(D

kp + Ak

p) Fτukτ + FrQ

kpnr(D

knp + Dk

nz −Akn)Fτu

− FrλkprFτθ

)+

((Dk

np + Dknz −Ak

n)Fsδuks

)T

(5.103)(FrQ

knpr(D

kp + Ak

p) Fτukτ + FrQ

knnr(D

knp + Dk

nz −Akn)Fτu

kτ − Frλ

knrFτθ

)]dΩkdz = δLk

e .

The Eq.(5.6) can be seen as particular case of Eq.(5.103) when the temperature θ is notconsidered. The omitted steps to obtain the Eq.(5.103) are detailed in [43] and [81]. Byusing the integration by parts as reported in Eq.(5.7), the governing equations are:

δuks : Kkτsr

uu ukτ = −Kkτsr

uθ θkτ + pk

us , (5.104)

with related boundary conditions on edge Γk:

Πkτsruu uk

τ − Πkτsrθθ θk

τ = Πkτsruu uk

τ − Πkτsrθθ θk

τ , (5.105)

where (-Kkτsruθ θk

τ ) is the thermal load pkθs and pk

us is the external mechanical one. FromEqs.(5.104) and (5.105), simply discarding the thermal contribution, it is possible toobtain the governing equations and the boundary conditions for the pure mechanicalcase (see Eqs.(5.9) and (5.10)).

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136 CHAPTER 5

Fundamental nuclei Kkτsruu and Πkτsr

uu are the same of Eqs.(5.11) and (5.12), the newfundamental nuclei are:

Kkτsruθ =

Ak

[(−Dk

p + Akp

)T (− Frλ

kpr

)+

(−Dk

np + Dknz −Ak

n

)T (− Frλ

knr

)]

(5.106)

FsFτHkαHk

βdz ,

Πkτsrθθ =

Ak

[IkT

p

(− Frλ

kpr

)+ IkT

np

(− Frλ

knr

)]FsFτH

kαHk

βdz . (5.107)

Considering the integrals in Eqs.(5.14), developing the matrix products in Eq.(5.106)and employing a Navier-type closed form solution [85], the algebraic explicit form ofthe nucleus Kkτsr

uθ of dimension (3× 1) is:

Kuθ1 = αJkτsrβ λk

1r , Kuθ2 = βJkτsrα λk

2r ,

Kuθ3 = − Jkτsrβ

1

Rkα

λk1r − Jkτsr

α

1

Rkβ

λk2r − Jkτzsr

αβ λk3r .

(5.108)

α = mπ/a and β = nπ/b, with m and n as the wave numbers in in-plane directions αand β, respectively. a and b as the shell dimensions.

Nucleus for shell, considered in Eqs.(5.108), degenerate in that for plates simplyconsidering 1

Rα= 1

Rβ= 0 and Hα = Hβ = 1; in this case the integrals in z-direction are

Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.Navier-type closed form solution is obtained via substitution of harmonic expres-

sions for the temperature as well as considering the following material coefficient equalto zero: λ6r = 0. The following harmonic assumptions can be made for the temperaturevariable in addition to those seen for the displacements in Eq.(5.16), they correspondto simply supported boundary conditions:

θkτ =

∑m,n

(θkτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), (5.109)

where θkτ is the temperature amplitude.

By starting from the (3 × 1) fundamental nucleus in Eqs.(5.108), the matrix of theconsidered multilayer structures is obtained by expanding via the index r in case ofFGM layers (see Section 2.4), via the indexes τ and s for the order of expansion inthe thickness direction (see Section 3.4) and via the index k for the multilayer assem-bling. Independently by the chosen kinematics for the displacements (EDN, EDZNand LDN), the temperature is always seen in layer wise approach. This means thatthe LW assembling procedure is the same of nucleus Kkτsr

uu as indicated in Figure 5.1,on the contrary the ESL assembling is that indicated in Figure 5.6: rows are assembledESL and columns are assembled LW.

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 137

t t t

s s s

r=1,

.....

, 10

r

Layer 1 Layer 2 Layer 3

k kk

Multilayer

Constant properties Constant properties

FGM

t, s: order of

expansion assembling

r: FGM assembling

k: multilayer

assembling

: fundamental nuclei

Figure 5.6: Three layered structure with the internal layer in FGM. ESL assemblingprocedure for nucleus Kuθ.

If the values of temperature are given at upper and lower surfaces of the shell,in order to determine the temperature profile through the thickness z for the thermalload pk

θs = −Kkτsuθ θk

τ , two different methods can be considered: - imposed the temper-ature bi-sinusoidal in the plane, a linear temperature profile through the thickness z isassumed from the top to the bottom of the shell (Ta) [81], [172]; - imposed the temper-ature bi-sinusoidal in the plane, Fourier’s conductivity equation is solved in order toobtain the calculated temperature profile in the thickness direction (Tc) [43], [172].

5.2.1 Assumed temperature profile, Ta

If the values of the temperature are known at the top and bottom surface of the shell,an assumed profile T (z) which varies linearly from the top to the bottom is given. Thetemperature is assumed bi-sinusoidal in the plane (α, β) at the top and bottom shellsurfaces:

T (α, β, z) = T (z) sin(mπ

aα) sin(

bβ) , (5.110)

with amplitudes T (+h/2) = Ttop and T (−h/2) = Tbot. a and b are the shell dimensions.m and n are the waves number. In the case of assumed temperature profile (Ta) a linearthrough the thickness distribution is considered from Ttop to Tbot. Independently by thenumber of considered layers the linear profile is always the same as indicated in Figure5.7: here examples of one-layered, three-layered and ten-layered shells are reported for

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138 CHAPTER 5

Figure 5.7: Linear assumed profile of temperature through the thickness of the one-layered, two-layered and ten-layered cylindrical shell.

a temperature profile which goes from +0.5 at the top to −0.5 at the bottom. The tem-perature profile is approximated as displacements in case of the Layer Wise approach:

T k(z) = Fτ θkτ with τ = t, b, r and r = 2, . . . , 14 , (5.111)

t and b indicate the top and bottom of the considered kth layer. The thickness functionsFτ are a combination of Legendre polynomials (see Section 3.4.3).

If the temperature is assumed linear through the thickness, the values at the top andbottom surfaces, and therefore Ft and Fb, would be sufficient to describe the assumedprofile via CUF [81]. The acronyms used for these cases are EDN(Ta), EDZN(Ta) andLDN(Ta): the kinematics for the displacements is the same of Section 5.1.1, Ta meansthat a linear temperature profile through the thickness is assumed.

5.2.2 Calculated temperature profile, Tc

The calculation procedure for the actual temperature profile in case of one or morelayers is reported in the following, in order to obtain the values of θk

τ for the Eq.(5.111).If the considered shell is subjected to a bi-sinusoidal temperature at the top and at

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 139

the bottom, the thermal boundary conditions are:

T = 0 at α = 0, a and β = 0, b ,

T = Tb sin(mπα

a

)sin

(nπβ

b

)at z = −h

2with b : bottom , (5.112)

T = Tt sin(mπα

a

)sin

(nπβ

b

)at z = +

h

2with t : top ,

where m and n are the waves number along the two in-plane shell directions (α,β). aand b are the shell dimensions, h is the shell thickness, and Tb and Tt are the amplitudesof the temperature at the bottom and top, respectively.

In case of multi-layered structures, continuity conditions for the temperature T andthe transverse normal heat flux qz hold in the thickness direction at each kth layer in-terface, reading:

T kt = T k+1

b , qkz t = qk+1

z b , for k = 1, . . . , Nl − 1 , (5.113)

where Nl is the number of layers in the considered structure.The relationship between transverse normal heat flux and temperature is given as:

qkz = Kk

3

∂T k

∂z. (5.114)

In general for the kth homogeneous orthotropic layer, the differential Fourier’s equa-tion of heat conduction reads [172], [135]:

(Kk1

H2α

) ∂2T

∂α2+

(Kk2

H2β

) ∂2T

∂β2+ Kk

3

∂2T

∂z2= 0 , (5.115)

Kk1 , Kk

2 and Kk3 are the thermal conductivities along the three shell directions α, β and

z; they are constant in each layer in case of classical materials, but they depend bythe thickness coordinate in case of FGMs. ∂ indicates the partial derivative. Hα =(1 + zk/Rk

α) and Hβ = (1 + zk/Rkβ) are the metric coefficients. In case of plates the

Eq.(5.115) has Hα = Hβ = 1 and the coefficients Kk1 , Kk

2 and Kk3 depend on z because

some layers k can be in FGM [43]. In case of shell we can define three new coefficientsK∗k

1 =Kk

1 (z)

H2α

, K∗k2 =

Kk2 (z)

H2β

and K∗k3 = Kk

3 (z), which in a generic layer k depend on the

thickness coordinate of the shell. K∗k1 and K∗k

2 can depend by the thickness coordinatez for two reasons: possible use of FGM layers, presence of curvature in case of shells;K∗k

3 can depend by z coordinate only in case of FGM layers. So:

K∗k1

∂2T

∂α2+ K∗k

2

∂2T

∂β2+ K∗k

3

∂2T

∂z2= 0 . (5.116)

The Eq.(5.116) has not constant coefficients in the layer k. It can be solved by introduc-ing, for each layer k, a given number of mathematical layers (Nml). Considering with

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140 CHAPTER 5

Nl the number of physical layers, a new index j can be defined which goes from 1 to(Nl ×Nml). So the continuity of temperature and transverse heat flux can be written ineach jth mathematical interface:

T jt = T j+1

b , qjz t = qj+1

z b , for j = 1, . . . , (Nl ×Nml − 1) , (5.117)

where

qjz = Kj

3

∂T j

∂z. (5.118)

Eq.(5.116) can be rewritten for each mathematical layer j:

K∗j1

∂2T

∂α2+ K∗j

2

∂2T

∂β2+ K∗j

3

∂2T

∂z2= 0 . (5.119)

In these mathematical layers we can suppose K∗j1 , K∗j

2 and K∗j3 constant because in

each mathematical layer, we can calculate the values of Hα and Hβ and the materialproperties of the FGM at a given value of the coordinate z.

For a mathematical layer, both governing equations and boundary conditions aresatisfied by assuming the following temperature field:

T (α, β, z) = f(z) sin(mπα

a

)sin

(nπβ

b

), (5.120)

withf(z) = T0 exp

(sj z

), (5.121)

T0 is a constant and sj a parameter. Substituting Eq.(5.120) in Eq.(5.119) and solvingfor sj :

sj1,2 = ±

√K∗j

1 (mπa

)2 + K∗j2 (nπ

b)2

K∗j3

. (5.122)

Therefore:

f(z) = T j01 exp

(sj1 z

)+ T j

02 exp(sj1z

)or f(z) = Cj

1 cosh(sj1 z

)+ Cj

2 sinh(sj1 z

).

(5.123)The solution for a mathematical layer j can be written as:

Tc(α, β, z) = T j =[Cj

1 cosh(sj1z

)+ Cj

2 sinh(sj1z

)]sin

(mπα

a

)sin

(nπβ

b

). (5.124)

wherein the coefficients Cj1 and Cj

2 are constant for each mathematical layer j.In Eq.(5.123) for each mathematical layer j two unknowns (Cj

1 and Cj2) remain.

Therefore, if the number of layers is Nl, the number of total mathematical layers isNl ×Nml, the number of unknowns is 2Nl ×Nml, and we need 2Nl ×Nml equations todetermine the unknowns.

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 141

The temperature at the top and the bottom surface are known, so we have alreadytwo conditions:

Tbot = C11 cosh

(s11 zbot

)+ C1

2 sinh(s11 zbot

),

Ttop = CNl×Nml1 cosh

(sNl×Nml1 ztop

)+ CNl×Nml

2 sinh(sNl×Nml1 ztop

).

(5.125)

Another (Nl ×Nml − 1) equations can be obtained from the continuity of temperatureat each mathematical interface, and finally (Nl × Nml − 1) equations result from thecontinuity of the normal heat flux through the interfaces, see Eq.(5.117). Thus, wehave:

Cj1 cosh

(sj1 zj

t

)+ Cj

2 sinh(sj1 zj

t

)− Cj+11 cosh

(sj+11 zj+1

b

)− Cj+12 sinh

(sj+11 zj+1

b

)= 0 ,

sj1 K∗j

3

[Cj

1 cosh(sj1 zj

t ) + Cj2 sinh(sj

1 zjt )

]

− sj+11 K∗j+1

3

[Cj+1

1 cosh(sj+11 zj+1

b

)+ Cj+1

2 sinh(sj+11 zj+1

b

)]= 0 .

(5.126)

In Eqs.(5.125) and (5.126), ztop and zbot indicate the coordinates of top and bottom of thewhole shell. zj

t and zj+1b represent the top of the jth mathematical layer and the bottom

of the (j + 1)th mathematical layer, respectively.By solving the system given by Eqs.(5.125) and (5.126), we gain the 2Nl ×Nml coef-

ficients Cj1 and Cj

2 . The actual temperature amplitude in the thickness shell direction isgiven by:

Tc(z) = T j = Cj1 cosh

(sj1 z

)+ Cj

2 sinh(sj1 z

)with j = 1, . . . , (Nl ×Nml) . (5.127)

We compute the temperature at different values zN in the thickness of multilayeredshell. By solving the system in Eq.(5.128), we obtain the N values of θk

τ for the CUF:

Tc(z1)Tc(z2)

...

Tc(zN)

=

F0(z1) F1(z1) · · · FN(z1)F0(z2) F1(z2) · · · FN(z2)

......

......

F0(zN) F1(zN) · · · FN(zN)

θk0

θk1......

θkN

. (5.128)

So, if we consider a generic multilayered shell, the temperature profile is approximatedby Eq.(5.111) and the N values of θk

τ are given by Eq.(5.128). In Eq.(5.128) Tc is calcu-lated by means of the mathematical layers j in certain points that permit the analogywith the physical layers k.

The acronyms used for these cases are EDN(Tc), EDZN(Tc) and LDN(Tc): the kine-matics for the displacements is the same of Section 5.1.1, Tc means that the temperatureprofile has been calculated by means of Fourier’s conductivity equation and then ap-proximated in LW form via CUF [43].

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142 CHAPTER 5

5.3 PVD for the electro-mechanical case, PVD-EM

In case of electro-mechanical problems, the Principle of Virtual Displacements is thePVD-EM as proposed in Eq.(4.20):

V

(δεT

pGσpC + δεTnGσnC − δET

GDC

)dV = δLe − δLin . (5.129)

By considering a laminate of Nl layers, and the integral on the volume Vk of each layeras an integral on the in plane domain Ωk plus the integral in the thickness-directiondomain Ak, it is possible to write:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nC − δEkG

T DkC

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin , (5.130)

where δLke and δLk

in are the external and inertial virtual work at k-layer level, respec-tively. The relative constitutive equations written to consider the case of some function-ally graded material (FGM) layers embedded in the structure [42], are those obtainedin Eq.(4.21); if the splitting of electric displacement and electric field in in-plane andout-plane components is not considered, the relations are:

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Fre

kT

pr EkG ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Fre

kT

nr EkG , (5.131)

DkC = Fre

kprε

kpG + Fre

knrε

knG + Frε

krEk

G .

By substituting the Eqs.(5.131), the geometrical relations of Section 2.5.1 in the varia-tional statement of Eq.(5.130), and considering a generic layer k [135], [71]:

Ωk

Ak

[((Dk

p + Akp)δu

k)T ((

FrQkppr(D

kp + Ak

p) + FrQkpnr(D

knp+Dk

nz −Akn)

)uk+

FrekT

pr (Dkep+Dk

en)Φk)

+((Dk

np+Dknz −Ak

n)δuk)T ((

FrQknpr(D

kp + Ak

p)

+ FrQknnr(D

knp+Dk

nz −Akn)

)uk + Fre

kT

nr (Dkep+Dk

en)Φk)

+((Dk

ep+Dken)δΦk

)T

((Fre

kpr(D

kp + Ak

p) + Freknr(D

knp+Dk

nz −Akn)

)uk + Frε

kr(D

kep + Dk

en)Φk) ]

dΩk dz = δLke . (5.132)

In Eq.(5.132), CUF [92], as presented in Section 3.4.5, can be introduced:∫

Ωk

Ak

[((Dk

p + Akp)Fsδu

ks

)T ((FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp+Dk

nz −Akn)

)Fτu

kτ+

FrekT

pr (Dkep+Dk

en)FτΦkτ

)+

((Dk

np+Dknz −Ak

n)Fsδuks

)T ((FrQ

knpr(D

kp + Ak

p)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 143

+ FrQknnr(D

knp+Dk

nz −Akn)

)Fτu

kτ + Fre

kT

nr (Dkep+Dk

en)FτΦkτ

)+

((Dk

ep+Dken)FsδΦ

ks

)T

((Fre

kpr(D

kp + Ak

p) + Freknr(D

knp+Dk

nz −Akn)

)Fτu

kτ + Frε

kr(D

kep + Dk

en)FτΦkτ

) ]

dΩk dz = δLke . (5.133)

In Eq.(5.133), in order to obtain a strong form of differential equations on the domainΩk and the relative boundary conditions on edge Γk, the integration by parts is used,which permits to move the differential operator from the infinitesimal variation of thegeneric variable δak to the finite quantity ak [92]. For a generic variable ak, the inte-gration by parts states as illustrated in Eq.(5.7) with matrices given in Eqs.(5.8). Thegoverning equations have the following form [42]:

δuks : Kkτsr

uu ukτ + Kkτsr

uΦ Φkτ = pk

us −M kτsruu uk

τ (5.134)

δΦks : Kkτsr

Φu ukτ + Kkτsr

ΦΦ Φkτ = pk

Φs .

The arrays pkus and pk

Φs indicate the variationally consistent mechanical and electricloadings, respectively. Along with these governing equations, the following boundaryconditions on the edge Γk of the in-plane integration domain Ωk hold:

Πkτsruu uk

τ + ΠkτsruΦ Φk

τ = Πkτsruu uk

τ + ΠkτsruΦ Φk

τ

ΠkτsrΦu uk

τ + ΠkτsrΦΦ Φk

τ =ΠkτsrΦu uk

τ + ΠkτsrΦΦ Φk

τ . (5.135)

By comparing the Eq.(5.133), after the integration by parts, with the Eqs.(5.134) and(5.135), the fundamental nuclei can be obtained:

Kkτsruu =

Ak

[(−Dk

p + Akp

)T (FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+

(−Dk

np + Dknz −Ak

n

)T (FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)]

FsFτHkαHk

βdz , (5.136)

KkτsruΦ =

Ak

[(−Dk

p + Akp

)T (Fre

kTpr (Dk

ep + Dken)

)+

(−Dk

np + Dknz −Ak

n

)T

(Fre

kTnr (Dk

ep + Dken)

)]FsFτH

kαHk

βdz , (5.137)

KkτsrΦu =

Ak

[(−Dk

ep + Dken

)T (Fre

kTpr (Dk

p + Akp) + Fre

kTnr (Dk

np + Dknz −Ak

n))]

FsFτHkαHk

βdz , (5.138)

KkτsrΦΦ =

Ak

[(−Dk

ep + Dken

)T (Frε

kr(D

kep + Dk

en)) ]

FsFτHkαHk

βdz . (5.139)

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144 CHAPTER 5

The nuclei for boundary conditions on edge Γk are [42]:

Πkτsruu =

Ak

[IkT

p

(FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+ IkT

np

(FrQ

knpr(D

kp + Ak

p)

+ FrQknnr(D

knp + Dk

nz −Akn)

)]FsFτH

kαHk

βdz , (5.140)

ΠkτsruΦ =

Ak

[IkT

p

(Fre

kTpr (Dk

ep + Dken)

)+ IkT

np

(Fre

kTnr (Dk

ep + Dken)

)]FsFτH

kαHk

βdz ,

(5.141)

ΠkτsrΦu =

Ak

[IkT

ep

(Fre

kTpr (Dk

p + Akp) + Fre

kTnr (Dk

np + Dknz −Ak

n))]

FsFτHkαHk

βdz , (5.142)

ΠkτsrΦΦ =

Ak

[IkT

ep (Frεkr(D

kep + Dk

en)) ]

FsFτHkαHk

βdz . (5.143)

In order to perform the integration by parts, the matrices Ikp and Ik

np are those presentedin Eq.(5.8), while Ik

ep is:

Ikep =

[1

Hkα

1Hk

β

]. (5.144)

In order to write the explicit form of the nuclei in Eq.(5.136)-(5.139), the integrals inthe z thickness-direction are defined as in Eq.(5.14), by developing the matrix prod-ucts in Eqs.(5.136)-(5.139) and employing a Navier-type closed form solution [42], thealgebraic explicit form of the nuclei can be obtained. The nucleus Kkτsr

uu of dimension(3× 3) is the same of Eq.(5.15), the other three are:

KuΦ11 = α(−Jkτszrβ ek

31r + Jkτzsrβ ek

15r −1

Rkα

Jkτsrβα

ek15r) ,

KuΦ21 = β(Jkτzsrα ek

24r −1

Rkβ

Jkτsrαβ

ek24r − ek

32rJkτszrα ) ,

KuΦ31 = α2ek15rJ

kτsrβα

+ β2ek24rJ

kτsrαβ

+ ek33rJ

kτzszrαβ +

1

Rkα

ek31rJ

kτszrβ +

1

Rkβ

ek32rJ

kτszrα ,

(5.145)

Φk is a scalar so KkτsruΦ has dimension (3× 1).

KΦu11 = − αek15r(J

kτszrβ − Jkτsr

βα

1

Rkα

) + αek31rJ

kτzsrβ ,

KΦu12 = − βek24r(J

kτszrα − Jkτsr

αβ

1

Rkβ

) + βek32rJ

kτzsrα ,

KΦu13 = − α2ek15rJ

kτsrβα

− β2ek24rJ

kτsrαβ

− ek33rJ

kτzszrαβ − ek

31rJkτzsrβ

1

Rkα

− ek32rJ

kτzsrα

1

Rkβ

,

(5.146)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 145

KkτsrΦu has dimension (1× 3).

KΦΦ11 = Jkτsrβα

α2ε11r + Jkτsrαβ

β2ε22r + ε33rJkτzszrαβ , (5.147)

KkτsrΦΦ has dimension (1× 1).α = mπ/a and β = nπ/b, with m and n as the wave numbers in in-plane directions

and a and b as the shell dimensions in α and β directions, respectively.Nuclei for shell considered in Eqs.(5.145)-(5.147) degenerate in those for plates sim-

ply considering 1Rα

= 1Rβ

= 0, and Hα = Hβ = 1; in this case integrals in z-direction areJkτsr, Jkτzsr, Jkτszr and Jkτzszr.

Navier-type closed form solution is obtained via substitution of harmonic expres-sions for the displacements and electric potential as well as considering the followingmaterial coefficients equal to zero: Q16r = Q26r = Q36r = Q45r = 0 and e25r = e14r =e36r = ε12r = 0. The harmonic assumptions for the displacements are reported inEqs.(5.16), the electric potential is:

Φkτ =

∑m,n

(Φkτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), (5.148)

where Φkτ is the electric potential amplitude.

The electric potential Φ is always considered in layer wise form, while the displace-ment u can be considered both LW or ESL. For this reason the nucleus Kkτsr

uu is as-sembled as indicated in Figure 5.1 (ESL or LW depending the choice made for thedisplacements). Kkτsr

ΦΦ is always assembled LW (see Figure 5.1). KkτsruΦ and Kkτsr

Φu ifassembled in LW form, follow the procedure indicated in Figure 5.1. In case of ESLapproach, Kkτsr

uΦ is assembled as in Figure 5.6, KkτsrΦu is assembled as in Figure 5.8.

A PVD-EM model can be ESL or LW depending the choice made for the displace-ments, in fact the electric potential is always in LW form. The acronyms used are thefollowing: EDN(EM) when the displacement is in ESL form and the electric potentialis in LW form; EDZN(EM) when the displacement uses the zig-zag function and theelectric potential is in layer wise form; LDN(EM) when both displacements and electricpotential are in LW form. The order of expansion N in the thickness direction has beenextended until to 14, N is the same for the displacements and the electric potential evenif a different multilayer approach is used for them [42].

5.4 PVD for the thermo-mechanical case, PVD-TM

In case of the fully coupling between the mechanical and thermal field the variationalstatement is the PVD-TM as indicated in Eq.(4.23):

V

(δεT

pGσpC + δεTnGσnC − δθηC − δϑT

pGhpC − δϑTnGhnC

)dV = δLe − δLin . (5.149)

By considering a laminate of Nl layers and the volume Vk for each layer as an integralon the in-plane surface Ωk and an integral in the thickness direction domain Ak, the

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146 CHAPTER 5

t t t

ss s

r=1,

.....

, 10

r

Layer 1 Layer 2 Layer 3

k kk

Multilayer

Constant properties Constant properties

FGM

t, s: order of

expansion assembling

r: FGM assembling

k: multilayer

assembling

: fundamental nuclei

Figure 5.8: Three layered structure with the internal layer in FGM. ESL assemblingprocedure for nucleus Kkτsr

Φu .

Eq.(5.149) can be rewritten as:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nC − δθkηkC − δϑk

pG

Thk

pC − δϑknG

Thk

nC

dΩkdz

=

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin , (5.150)

where δLke and δLk

in are the external and inertial virtual work at k-layer level, respec-tively.

The governing equations have the following form:

δuks : Kkτsr

uu ukτ + Kkτsr

uθ θkτ = pk

us −M kτsruu uk

τ (5.151)

δθks : Kkτsr

θu ukτ + Kkτsr

θθ θkτ = pk

θs .

The arrays pkus and pk

θs indicate the variationally consistent mechanical and thermalloadings, respectively. Along with these governing equations the following boundaryconditions on the edge Γk of the in-plane integration domain Ωk hold:

Πkτsruu uk

τ + Πkτsruθ θk

τ = Πkτsruu uk

τ + Πkτsruθ θk

τ

Πkτsrθu uk

τ + Πkτsrθθ θk

τ =Πkτsrθu uk

τ + Πkτsrθθ θk

τ . (5.152)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 147

As indicated in [71], the temperature θk is a variable of the problem. The displace-ments uk can be seen in ESL, ESL+ZZF and LW form giving the theories EDN(TM),EDZN(TM) and LW(TM), respectively. Independently by the choice made for the dis-placements, the temperature is always seen in LW form. For this reason the LW as-sembling for all four nuclei is that indicated in Figure 5.1, while the ESL assemblingchanges depending the considered nucleus: Kkτsr

uu is assembled as in Figure 5.1, Kkτsrθθ

is always assembled in LW form as in Figure 5.1, Kkτsruθ is assembled as in Figure 5.6

and Kkτsrθu is assembled as in Figure 5.8.

As discussed in [78], [72], the PVD-TM include only the internal thermal work madeby the gradient of temperature in the case of temperature applied at the top and bottomof the structure (PVD-TM1, see the first part of the Section 4.2.4); it includes only theinternal thermal work made by the temperature in the case of mechanical load appliedon the structure (PVD-TM2, see the second part of the Section 4.2.4).

5.4.1 Imposed temperature on surfaces

In case of temperature imposed at the top and bottom of the structure, in the Eq.(5.150)the term δθkηk

C cannot be considered because it does not exist a virtual variation oftemperature. So for the PVD-TM1 the variational statement is:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nC − δϑkpG

Thk

pC − δϑknG

Thk

nC

dΩkdz

=

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin . (5.153)

By considering the constitutive equations as obtained in Eqs.(4.24), simply discardingthe entropy ηk

c :

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Frλ

kprθ

k ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Frλ

knrθ

k ,

hkpC = Frκ

kpprϑ

kpG + Frκ

kpnrϑ

knG , (5.154)

hknC = Frκ

knprϑ

kpG + Frκ

knnrϑ

knG .

The geometrical relations for shells have been obtained in Section 2.5.1, and Carrera’sUnified Formulation is described in Section 3.4. The Eq.(5.153) is rewritten in the fol-lowing form in the case of a generic layer k:

Ωk

Ak

[((Dk

p + Akp)Fsδu

ks

)T ((FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp+Dk

nz −Akn)

)Fτu

kτ−

FrλkprFτθ

)+

((Dk

np+Dknz −Ak

n)Fsδuks

)T ((FrQ

knpr(D

kp + Ak

p) + FrQknnr

(Dknp+Dk

nz −Akn)

)Fτu

kτ − Frλ

knrFτθ

)+

(Dk

tpFsδθks

)T ((Frκ

kppr(−Dk

tp) + Frκknpr(−Dk

tn))

Fτθkτ

)+

(Dk

tnFsδθks

)T ((Frκ

knpr(−Dk

tp) + Frκkppr(−Dk

tn))Fτθ

)]dΩk dz = δLk

e . (5.155)

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148 CHAPTER 5

Integrating by parts the Eq.(5.155) by means of the Eq.(5.7), the fundamental nuclei are:

Kkτsruu =

Ak

[(−Dk

p + Akp

)T (FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+

(−Dk

np + Dknz −Ak

n

)T (FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)]

FsFτHkαHk

βdz , (5.156)

Kkτsruθ =

Ak

[(−Dk

p + Akp

)T (− Frλ

kpr

)+

(−Dk

np + Dknz −Ak

n

)T (− Frλ

knr

)]

FsFτHkαHk

βdz , (5.157)

Kkτsrθu = 0 (5.158)

Kkτsrθθ =

Ak

[DkT

tp FrκkpprD

ktp + DkT

tp FrκkpnrD

ktn −DkT

tn FrκknprD

ktp −DkT

tn FrκknnrD

ktn

]

FsFτHkαHk

βdz . (5.159)

The nuclei for boundary conditions on the edge Γk are:

Πkτsruu =

Ak

[IkT

p

(FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp + Dk

nz −Akn)

)+

IkTnp

(FrQ

knpr(D

kp + Ak

p) + FrQknnr(D

knp + Dk

nz −Akn)

)]FsFτH

kαHk

βdz , (5.160)

Πkτsruθ =

Ak

[IkT

p

(− Frλ

kpr

)+ IkT

np

(− Frλ

knr

)]FsFτH

kαHk

βdz , (5.161)

Πkτsrθu = 0 (5.162)

Πkτsrθθ =

Ak

[IkT

tp Frκkppr(−Dk

tp) + IkTtp Frκ

kpnr(−Dk

tn)]FsFτH

kαHk

βdz . (5.163)

In order to perform the integration by parts, the matrices Ikp and Ik

np are those presentedin Eq.(5.8), while Ik

tp is:

Iktp =

[1

Hkα

1Hk

β

]. (5.164)

In order to write the explicit form of the nuclei in Eq.(5.156)-(5.159), the integrals inthe z thickness-direction are defined as in Eq.(5.14). By developing the matrix prod-ucts in Eqs.(5.156)-(5.159) and employing a Navier-type closed form solution [42], thealgebraic explicit form of the nuclei can be obtained. The nucleus Kkτsr

uu of dimension(3× 3) is the same of Eq.(5.15), the other three are:

Kuθ11 = αJkτsrβ λk

1r , Kuθ21 = βJkτsrα λk

2r ,

Kuθ31 = − 1

Rkα

Jkτsrβ λ1r − 1

Rkβ

Jkτsrα λ2r − Jkτszr

αβ λ3r ,(5.165)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 149

θk is a scalar, so Kkτsruθ has dimension (3× 1).

Kθu11 = Kθu12 = Kθu13 = 0 , (5.166)

Kkτsrθu has dimension (1× 3).

Kθθ11 = − Jkτsrβα

α2κ11r − Jkτsrαβ

β2κ22r − κ33rJkτzszrαβ , (5.167)

Kkτsrθθ has dimension (1× 1).α = mπ/a and β = nπ/b, with m and n as the waves numbers in in-plane directions,

and a and b as the shell dimensions.The nuclei for shells considered in Eqs.(5.165)-(5.167) degenerate in those for plates

simply considering 1Rα

= 1Rβ

= 0, and Hα = Hβ = 1 in order to obtain the integrals inz-direction Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.

Navier-type closed form solution is obtained via substitution of harmonic expres-sions for the displacements and temperature as well as considering the following ma-terial coefficients equal to zero: Q16r = Q26r = Q36r = Q45r = 0 and λ6r = κ12r = 0. Theharmonic assumptions for the displacements are reported in Eqs.(5.16), while for thetemperature is:

θkτ =

∑m,n

(θkτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), (5.168)

where θkτ is the temperature amplitude.

5.4.2 Mechanical load

In case of mechanical load applied on the structure, in the Eq.(5.150) the terms δϑkpG

Thk

pC

and δϑknG

Thk

nC are not considered because it does not exist a gradient of temperaturevariation. So for the PVD-TM2 the variational statement is:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nC − δθkηkC

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin . (5.169)

By considering the constitutive equations as obtained in Eqs.(4.24), simply discardingthe heat fluxes hk

pC and hknC :

σkpC = FrQ

kpprε

kpG + FrQ

kpnrε

knG − Frλ

kprθ

k ,

σknC = FrQ

knprε

kpG + FrQ

knnrε

knG − Frλ

knrθ

k , (5.170)

ηkC = Frλ

kTpr εk

pG + FrλkTnr εk

nG + Frχkrθ

k .

The geometrical relations for shells have been obtained in Section 2.5.1, Carrera’s Uni-fied Formulation is described in Section 3.4. The Eq.(5.169) is rewritten in the following

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150 CHAPTER 5

form for a generic layer k:∫

Ωk

Ak

[((Dk

p + Akp)Fsδu

ks

)T ((FrQ

kppr(D

kp + Ak

p) + FrQkpnr(D

knp+Dk

nz −Akn)

)Fτu

kτ−

FrλkprFτθ

)+

((Dk

np+Dknz −Ak

n)Fsδuks

)T ((FrQ

knpr(D

kp + Ak

p) + FrQknnr

(Dknp+Dk

nz −Akn)

)Fτu

kτ − Frλ

knrFτθ

)− Fsδθ

kTs

((Frλ

kTpr (Dk

p + Akp) + Frλ

kTnr

(Dknp + Dk

nz −Akn)

)Fτu

kτ + Frχ

krFτθ

)]dΩk dz = δLk

e . (5.171)

Integrating by parts the Eq.(5.171) by means of the Eq.(5.7), the fundamental nucleiKkτsr

uu and Kkτsruθ are the same of PVD-TM1 in Section 5.4.1, while nuclei Kkτsr

θu andKkτsr

θθ are:

Kkτsrθu =

Ak

[− Frλ

kTpr (Dk

p + Akp)− Frλ

kTnr (Dk

np + Dknz −Ak

n)]FsFτH

kαHk

βdz , (5.172)

Kkτsrθθ =

Ak

[− Frχ

kr

]FsFτH

kαHk

βdz . (5.173)

Nuclei for boundary conditions on the edge Γk Πkτsruu and Πkτsr

uθ are the same of PVD-TM1 in Section 5.4.1, while the other two state:

Πkτsrθu = Πkτsr

θθ = 0 . (5.174)

In order to write the explicit form of the nuclei in Eq.(5.172)-(5.173), the integrals in thez thickness-direction are defined as in Eq.(5.14). By developing the matrix productsin Eqs.(5.172)-(5.173) and employing a Navier-type closed form solution [42], the alge-braic explicit form of the nuclei can be obtained. The nuclei Kkτsr

uu of dimension (3× 3)and Kkτsr

uθ (3× 1) are the same of Section 5.4.1, the other two are:

Kθu11 = αJkτsrβ λk

1r , Kθu12 = βJkτsrα λk

2r ,

Kθu13 = − 1

Rkα

Jkτsrβ λ1r − 1

Rkβ

Jkτsrα λ2r − Jkτzsr

αβ λ3r ,(5.175)

θk is a scalar, so Kkτsrθu has dimension (1× 3).

Kθθ11 = − Jkτsrαβ χr , (5.176)

Kkτsrθθ has dimension (1× 1).α = mπ/a and β = nπ/b, with m and n as the waves numbers in in-plane directions

and a and b as the shell dimensions.Nuclei for shells considered in Eqs.(5.175)-(5.176) degenerate in those for plates

simply considering 1Rα

= 1Rβ

= 0 and Hα = Hβ = 1, in order to obtain the integralsin z-direction Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 151

Navier-type closed form solution is obtained via substitution of harmonic expres-sions for the displacements and temperature as well as considering the following ma-terial coefficients equal to zero: Q16r = Q26r = Q36r = Q45r = 0 and λ6r = κ12r = 0. Theharmonic assumptions of the displacements are reported in Eqs.(5.16), for the temper-ature see in Eq.(5.168).

The PVD-TM2 is also employed for the dynamic analysis, simply considering theinertial forces in the same governing equations.

5.5 RMVT for the electro-mechanical case, RMVT-EM andRMVT-M

In case of electro-mechanical coupling a possible extension of Reissner’s Mixed Vari-ational Theorem (RMVT) [83] is that indicated in Eq.(4.40) where the internal electricwork is simply added [79], [203]:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)− δETpGDpC − δET

nGDnC

)dV

= δLe − δLin . (5.177)

By considering a multilayered structure constituted by Nl layers, the Eq.(5.177) can berewritten as:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nM + δσknM

T(εk

nG − εknC)− δEk

pG

T DkpC − δEk

nG

T DknC

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin . (5.178)

The relative constitutive equations, written for the FGMs have been obtained fromEq.(4.41) where the thermal contributes have been simply discarded [135], [198]:

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpEnrEknG ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnEnrEknG , (5.179)

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpEnrEknG ,

DknC = FrC

k

DnεprεkpG + FrC

k

DnσnrσknM + FrC

k

DnEprEkpG + FrC

k

DnEnrEknG .

By substituting the Eqs.(5.179) in the variational statement of Eq.(5.178), by consideringthe geometrical relations of Section 2.5.1, and substituting CUF [92] as presented inSection 3.4.5, for a generic layer k:

Ωk

Ak

[((Dk

p + Akp)Fsδu

ks

)T (FrC

k

σpεpr(Dkp + Ak

p)Fτukτ + FrC

k

σpσnrFτσknMτ−

FrCk

σpEprDkepFτΦ

kτ − FrC

k

σpEnrDkenFτΦ

)+

((Dk

np + Dknz −Ak

n)Fsδuks

)T (Fτσ

knMτ

)

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152 CHAPTER 5

+(Fsδσ

knMs

)T ((Dk

np + Dknz −Ak

n)Fτukτ − FrC

k

εnεpr(Dkp + Ak

p)Fτukτ−

FrCk

εnσnrFτσknMτ + FrC

k

εnEprDkepFτΦ

kτ + FrC

k

εnEnrDkenFτΦ

)+

(Dk

epFsδΦks

)T

(FrC

k

Dpεpr(Dkp + Ak

p)Fτukτ + FrC

k

DpσnrFτσknMτ − FrC

k

DpEprDkepFτΦ

kτ − FrC

k

DpEnrDkenFτΦ

)

+(Dk

enFsδΦks

)T (FrC

k

Dnεpr(Dkp + Ak

p)Fτukτ + FrC

k

DnσnrFτσknMτ − FrC

k

DnEprDkepFτΦ

kτ−

FrCk

DnEnrDkenFτΦ

)]dΩk dz = δLk

e − δLkin . (5.180)

In Eq.(5.180), in order to obtain a strong form of differential equations on the domainΩk and the relative boundary conditions on edge Γk, the integration by parts is used,which permits to move the differential operator from the infinitesimal variation of thegeneric variable δak to the finite quantity ak [92]. For a generic variable ak, the integra-tion by parts states as illustrated in Eq.(5.7) with matrices of Eqs.(5.8). The governingequations have the following form [79]:

δuks : Kkτsr

uu ukτ + Kkτsr

uσ σknMτ + Kkτsr

uΦ Φkτ = pk

us −M kτsruu uk

τ

δσkns : Kkτsr

σu ukτ + Kkτsr

σσ σknMτ + Kkτsr

σΦ Φkτ = 0 (5.181)

δΦks : Kkτsr

Φu ukτ + Kkτsr

Φσ σknMτ + Kkτsr

ΦΦ Φkτ = pk

Φs .

The arrays pkus and pk

Φs indicate the variationally consistent loadings. Along with thesegoverning equations the following boundary conditions on the edge Γk of the in-planeintegration domain Ωk hold:

Πkτsruu uk

τ + Πkτsruσ σk

nMτ + ΠkτsruΦ Φk

τ =Πkτsruu uk

τ + Πkτsruσ σk

nMτ + ΠkτsruΦ Φk

τ

ΠkτsrΦu uk

τ + ΠkτsrΦσ σk

nMτ + ΠkτsrΦΦ Φk

τ =ΠkτsrΦu uk

τ + ΠkτsrΦσ σk

nMτ + ΠkτsrΦΦ Φk

τ (5.182)

Comparing the Eq.(5.180), after the integration by parts, with the Eqs.(5.181) and (5.182),the fundamental nuclei can be obtained:

Kkτsruu =

Ak

[(−Dk

p + Akp)

T (FrCk

σpεpr(Dkp + Ak

p))]FsFτH

kαHk

βdz , (5.183)

Kkτsruσ =

Ak

[(−Dk

p + Akp)

T (FrCk

σpσnr) + (−Dknp + Dk

nz −Akn)T

]FsFτH

kαHk

βdz , (5.184)

KkτsruΦ =

Ak

[(−Dk

p + Akp)

T (−FrCk

σpEprDkep − FrC

k

σpEnrDken)

]FsFτH

kαHk

βdz , (5.185)

Kkτsrσu =

Ak

[(Dk

np + Dknz −Ak

n)− (FrCk

εnεpr)(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.186)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 153

Kkτsrσσ =

Ak

[− FrC

k

εnσnr

]FsFτH

kαHk

βdz , (5.187)

KkτsrσΦ =

Ak

[FrC

k

εnEprDkep + FrC

k

εnEnrDken

]FsFτH

kαHk

βdz , (5.188)

KkτsrΦu =

Ak

[(−Dk

ep

TFrC

k

Dpεpr + Dken

TFrC

k

Dnεpr)(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.189)

KkτsrΦσ =

Ak

[−Dk

ep

TFrC

k

Dpσnr + Dken

TFrC

k

Dnσnr

]FsFτH

kαHk

βdz , (5.190)

KkτsrΦΦ =

Ak

[−Dk

ep

T(−FrC

k

DpEprDkep − FrC

k

DpEnrDken) + Dk

en

T

(−FrCk

DnEprDkep − FrC

k

DnEnrDken)

]FsFτH

kαHk

βdz . (5.191)

The nuclei for boundary conditions on edge Γk are [79], [198]:

Πkτsruu =

Ak

[Ik

p

TFrC

k

σpεpr(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.192)

Πkτsruσ =

Ak

[Ik

p

TFrC

k

σpσnr + Iknp

T]FsFτH

kαHk

βdz , (5.193)

ΠkτsruΦ =

Ak

[Ik

p

T(−FrC

k

σpEprDkep − FrC

k

σpEnrDken)

]FsFτH

kαHk

βdz , (5.194)

ΠkτsrΦu =

Ak

[Ik

ep

T(FrC

k

Dpεpr(Dkp + Ak

p))]FsFτH

kαHk

βdz , (5.195)

ΠkτsrΦσ =

Ak

[Ik

ep

TFrC

k

Dpσnr

]FsFτH

kαHk

βdz , (5.196)

ΠkτsrΦΦ =

Ak

[Ik

ep

T(−FrC

k

DpEprDkep − FrC

k

DpEnrDken)

]FsFτH

kαHk

βdz . (5.197)

In order to perform the integration by parts, the matrices Ikp and Ik

np are those presentedin Eq.(5.8), while Ik

ep is that presented in Eq.(5.144). In order to write the explicit formof the nuclei in Eqs.(5.183)-(5.191), the integrals in the z thickness-direction are definedas in Eq.(5.14). By developing the matrix products in Eqs.(5.183)-(5.191) and employinga Navier-type closed form solution [79], the algebraic explicit form of the nuclei can beobtained.

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154 CHAPTER 5

Nucleus Kkτsruu of dimension (3× 3) is:

Kuu11 = α2Jkτsrβα

Ckσpεpr11 + β2Jkτsr

αβ

Ckσpεpr33 , Kuu12 = Jkτsr(Ck

σpεpr12 + Ckσpεpr33)αβ ,

Kuu13 = − 1

Rkα

Jkτsrβα

αCkσpεpr11 −

1

Rkβ

JkτsrαCkσpεpr12 , Kuu21 = Jkτsr(Ck

σpεpr21 + Ckσpεpr33)αβ ,

Kuu22 = β2Jkτsrαβ

Ckσpεpr22 + α2Jkτsr

βα

Ckσpεpr33 , Kuu23 = − 1

Rkα

JkτsrβCkσpεpr21 −

1

Rkβ

Jkτsrαβ

αCkσpεpr22 ,

Kuu31 = − 1

Rkα

Jkτsrβα

αCkσpεpr11 −

1

Rkβ

JkτsrαCkσpεpr21 ,

Kuu32 = − 1

Rkα

JkτsrβCkσpεpr12 −

1

Rkβ

Jkτsrαβ

βCkσpεpr22 ,

Kuu33 =1

Rkα

2Jkτsrβα

Ckσpεpr11 +

1

RkαRk

β

Jkτsr(Ckσpεpr12 + Ck

σpεpr21) +1

Rkβ

2Jkτsrαβ

Ckσpεpr22 .

(5.198)

Nucleus Kkτsruσ of dimension (3× 3) is:

Kuσ11 = − 1

Rkα

Jkτsβ + Jkτsz

αβ , Kuσ12 = 0 , Kuσ13 = −αJkτsrβ Ck

σpσnr13 ,

Kuσ21 = 0 , Kuσ22 = − 1

Rkβ

Jkτsα + Jkτsz

αβ , Kuσ23 = −βJkτsrα Ck

σpσnr23 ,

Kuσ31 = αJkτsβ , Kuσ32 = βJkτs

α , Kuσ33 = Jkτszαβ +

1

Rkα

Jkτsrβ Ck

σpσnr13 +1

Rkβ

Jkτsrα Ck

σpσnr23 .

(5.199)

Nucleus KkτsruΦ of dimension (3× 1) is:

KuΦ11 = αJkτzsrβ Ck

σpEnr11 , KuΦ21 = βJkτzsrα Ck

σpEnr21 ,

KuΦ31 = − 1

Rkα

Jkτzsrβ Ck

σpEnr11 −1

Rkβ

Jkτzsrα Ck

σpEnr21 .(5.200)

Nucleus Kkτsrσu of dimension (3× 3) is:

Kσu11 = − 1

Rkα

Jkτsβ + Jkτzs

αβ , Kσu12 = 0 ; Kσu13 = αJkτsβ ,

Kσu21 = 0 , Kσu22 = − 1

Rkβ

Jkτsα + Jkτzs

αβ , Kuσ23 = βJkτsα , Kσu31 = αJkτsr

β Ckεnεpr31 ,

Kσu32 = βJkτsrα Ck

εnεpr32 , Kσu33 = Jkτzsαβ − 1

Rkα

Jkτsrβ Ck

εnεpr31 −1

Rkβ

Jkτsrα Ck

εnεpr32 .

(5.201)

Nucleus Kkτsrσσ of dimension (3× 3) is:

Kσσ11 = − Jkτsrαβ Ck

εnσnr11 , Kσσ12 = 0 , Kσσ13 = 0 ,

Kσσ21 = 0 , Kσσ22 = −Jkτsrαβ Ck

εnσnr22 , Kσσ23 = 0 ,

Kσσ31 = 0 , Kσσ32 = 0 , Kσσ33 = −Jkτsrαβ Ck

εnσnr33 .

(5.202)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 155

Nucleus KkτsrσΦ of dimension (3× 1) is:

KσΦ11 = αJkτsrβ Ck

εnEpr11 , KσΦ21 = βJkτsrα Ck

εnEpr22 , KσΦ31 = Jkτzsrαβ Ck

εnEnr31 . (5.203)

Nucleus KkτsrΦu of dimension (1× 3) is:

KΦu11 = − αJkτszrβ Ck

Dnεpr11 , KΦu12 = −βJkτszrα Ck

Dnεpr12 ,

KΦu13 =1

Rkα

Jkτszrβ Ck

Dnεpr11 +1

Rkβ

Jkτszrα Ck

Dnεpr12 .(5.204)

Nucleus KkτsrΦσ of dimension (1× 3) is:

KΦσ11 = αJkτsrβ Ck

Dnσnr11 , KΦσ12 = βJkτsrα Ck

Dnσnr22 , KΦσ13 = Jkτszrαβ Ck

Dnσnr13 . (5.205)

Nucleus KkτsrΦΦ of dimension (1× 1) is:

KΦΦ11 = − Jkτzszrαβ Ck

DnEnr11 − α2Jkτsrβα

CkDpEpr11 − β2Jkτsr

αβ

CkDpEpr22 . (5.206)

α = mπ/a and β = nπ/b, with m and n as the waves numbers in in-plane directionsand a and b as the shell dimensions.

Nuclei for shell considered in Eqs.(5.198)-(5.206) degenerate in those for plates sim-ply considering 1

Rα= 1

Rβ= 0, and Hα = Hβ = 1, in order to obtain the integrals in

z-direction Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.Navier-type closed form solution is obtained via substitution of harmonic expres-

sions for the displacements, electric potential and transverse stresses as well as consid-ering the following material coefficients equal to zero: Q16r = Q26r = Q36r = Q45r = 0and e25r = e14r = e36r = ε12r = 0. The harmonic assumptions for the displacements,transverse shear/normal stresses and electric potential are:

(ukτ , σ

kαzτ ) =

∑m,n

(Ukτ , σk

αzτ ) cos

(mπαk

ak

)sin

(nπβk

bk

), k = 1, Nl (5.207)

(vkτ , σ

kβzτ ) =

∑m,n

(V kτ , σk

βzτ ) sin

(mπαk

ak

)cos

(nπβk

bk

), τ = t, b, r (5.208)

(wkτ , σ

kzzτ , Φ

kτ ) =

∑m,n

(W kτ , σk

zzτ , Φkτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), r = 2, N (5.209)

where Ukτ , V k

τ , W kτ , Φk

τ , σkαzτ , σk

βzτ and σkzzτ are the amplitudes of the considered vari-

ables.The electric potential Φ and transverse shear/normal stresses σn are always consid-

ered in layer wise form, while the displacement u can be considered both LW or ESL. ARMVT-EM model can be ESL or LW depending the chose made for the displacementsbecause the electric potential and the stresses are always in LW form. The acronymsused are the following: EMN(EM) when the displacement is in ESL form and the othervariables are in LW approach; EMZN(EM) when the displacement use the zig-zag

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156 CHAPTER 5

function and the electric potential and stresses are in layer wise form; LMN(EM) whenall the variables are in LW form. The order of expansion N in the thickness directionhas been extended until to 14, N is the same for the displacements, the electric poten-tial and the transverse shear/normal stresses even if a different multilayer approach isused for them [79]. The LW assembling procedure is the same for all the nine nuclei asindicated in Figure 5.1. Nucleus Kkτsr

uu is assembled in ESL form as described in Figure5.1, nuclei Kkτsr

σσ , KkτsrσΦ , Kkτsr

Φσ and KkτsrΦΦ are always assembled in LW form as indi-

cated in Figure 5.1. The ESL assembling procedure for nuclei Kkτsruσ and Kkτsr

uΦ is thatindicated in Figure 5.6, for nuclei Kkτsr

σu and KkτsrΦu is that indicated in Figure 5.8. Some

terms in fundamental nuclei in case of RMVT applications present integrals J withoutthe index r, this happens because these terms do not multiply quantities which dependby the thickness coordinate z, so the loop on index r is not necessary.

The coefficients C have a form given by the Eqs.(4.34). In case of FunctionallyGraded Materials (FGMs), the coefficients C(z) are first calculated and then approxi-mated by means of thickness functions Fr by the formula C(z) = Fr(z)Cr [86].

The array Cσpεp(z) has dimension (3× 3):

Cσpεp11 = Q11 − Q213

Q33

, Cσpεp12 = Q12 − Q13Q23

Q33

, Cσpεp13 = 0 , Cσpεp21 = Q12 − Q13Q23

Q33

,

Cσpεp22 = Q22 − Q223

Q33

, Cσpεp23 = 0 , Cσpεp31 = 0 , Cσpεp32 = 0 , Cσpεp33 = Q66 .

(5.210)

The array Cσpσn(z) has dimension (3× 3):

Cσpσn11 = 0 , Cσpσn12 = 0 , Cσpσn13 =Q13

Q33

, Cσpσn21 = 0 ,

Cσpσn22 = 0 , Cσpσn23 =Q23

Q33

, Cσpσn31 = 0 , Cσpσn32 = 0 , Cσpσn33 = 0 .

(5.211)

The array CσpEp(z) has dimension (3× 2):

CσpEp11 = CσpEp12 = CσpEp21 = CσpEp22 = CσpEp31 = CσpEp32 = 0 . (5.212)

The array CσpEn(z) has dimension (3× 1):

CσpEn11 = − e31 +e33Q13

Q33

, CσpEn21 = −e32 +e33Q23

Q33

, CσpEn31 = 0 . (5.213)

The array Cεnεp(z) has dimension (3× 3):

Cεnεp11 = Cεnεp12 = Cεnεp13 = Cεnεp21 = Cεnεp22 = Cεnεp23 = 0 ,

Cεnεp31 = − Q13

Q33

, Cεnεp32 = −Q23

Q33

, Cεnεp33 = 0 .(5.214)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 157

The array Cεnσn(z) has dimension (3× 3):

Cεnσn11 =1

Q55

, Cεnσn12 = Cεnσn13 = Cεnσn21 = 0 , Cεnσn22 =1

Q44

,

Cεnσn23 = Cεnσn31 = Cεnσn32 = 0 , Cεnσn33 =1

Q33

.(5.215)

The array CεnEp(z) has dimension (3× 2):

CεnEp11 =e15

Q55

, CεnEp12 = CεnEp21 = 0 , CεnEp22 =e24

Q44

, CεnEp31 = CεnEp32 = 0 .

(5.216)

The array CεnEn(z) has dimension (3× 1):

CεnEn11 = CεnEn21 = 0 , CεnEn31 =e33

Q33

. (5.217)

The array CDpεp(z) has dimension (2× 3):

CDpεp11 = CDpεp12 = CDpεp13 = CDpεp21 = CDpεp22 = CDpεp23 = 0 . (5.218)

The array CDpσn(z) has dimension (2× 3):

CDpσn11 =e15

Q55

, CDpσn12 = CDpσn13 = CDpσn21 = 0 , CDpσn22 =e24

Q44

, CDpσn23 = 0 .

(5.219)

The array CDpEp(z) has dimension (2× 2):

CDpEp11 =e215

Q55

+ ε11 , CDpEp12 = CDpEp21 = 0 , CDpEp22 =e224

Q44

+ ε22 . (5.220)

The array CDpEn(z) has dimension (2× 1):

CDpEn11 = CDpEn21 = 0 . (5.221)

The array CDnεp(z) has dimension (1× 3):

CDnεp11 = e31 − e33Q13

Q33

, CDnεp12 = e32 − e33Q23

Q33

, CDnεp13 = 0 . (5.222)

The array CDnσn(z) has dimension (1× 3):

CDnσn11 = CDnσn12 = 0 , CDnσn13 =e33

Q33

. (5.223)

The array CDnEp(z) has dimension (1× 2):

CDnEp11 = CDnEp12 = 0 . (5.224)

The array CDnEn(z) has dimension (1× 1):

CDnEn11 =e233

Q33

+ ε33 . (5.225)

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158 CHAPTER 5

RMVT-M

In case of pure mechanical problem Reissner’s Mixed Variational Theorem (RMVT)[83], here stated as RMVT-M, considers as variables the displacement u and the trans-verse shear/normal stresses σn [199], [200]. No further Lagrange’s multipliers havebeen added with respect to the RMVT-EM discussed in the Eq.(5.177), so governingequations can be considered as a particular case of those presented in Eq.(5.181) sim-ply discarding the elements for the electro-mechanical coupling. The fundamental nu-clei are the same of Eqs.(5.183), (5.184), (5.186), (5.187) and Eqs.(5.198), (5.199), (5.201),(5.202).

The acronyms used for the RMVT-M are EMN, EMZN and LMN. Further detailsabout the RMVT for the mechanical case can be found in [201] and [202].

5.6 RMVT2 for the electro-mechanical case, RMVT2-EM

A further extension of the RMVT to the electro-mechanical analysis considers as vari-ables of the problem the displacements u, the electric potential Φ, the transverse shear/normal stresses σn and the normal electric displacement Dn. This last is modelled, in-troducing a further Lagrange’s multiplier in the variational statement of the Eq.(5.177).For this reason the variational statement RMVT-EM cannot be considered as a par-ticular case of this new RMVT2-EM one where the normal electric displacement isobtained a priori: the obtained fundamental nuclei are completely different from thosein Eqs.(5.183)-(5.191) [1], [80]. The variational statement is that indicated in Eq.(4.64)in according to [80], [205]:

V

(δεT

pGσpC + δεTnGσnM + δσT

nM(εnG − εnC)− δETpGDpC − δET

nGDnM

− δDTnM(EnG − EnC)

)dV = δLe − δLin . (5.226)

By considering the multilayered structure made of Nl layers, the integral on the volumeV in Eq.(5.226) can be rewritten as:

Nl∑

k=1

Ωk

Ak

δεk

pG

Tσk

pC + δεknG

Tσk

nM + δσknM

T(εk

nG − εknC)− δEk

pG

T DkpC − δEk

nG

T DknM

− δDknM

T(Ek

nG − EknC)

dΩkdz =

Nl∑

k=1

δLke −

Nl∑

k=1

δLkin , (5.227)

where the subscript M indicates that the considered variables are now modelled a pri-ori and not obtained by the constitutive equations. The related constitutive equations,written for the FGMs, have been obtained from Eq.(4.65) where the thermal contributes

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 159

have been simply discarded [80]:

σkpC = FrC

k

σpεprεkpG + FrC

k

σpσnrσknM + FrC

k

σpEprEkpG + FrC

k

σpDnrDknM ,

εknC = FrC

k

εnεprεkpG + FrC

k

εnσnrσknM + FrC

k

εnEprEkpG + FrC

k

εnDnrDknM ,

DkpC = FrC

k

DpεprεkpG + FrC

k

DpσnrσknM + FrC

k

DpEprEkpG + FrC

k

DpDnrDknM , (5.228)

EknC = FrC

k

EnεprεkpG + FrC

k

EnσnrσknM + FrC

k

EnEprEkpG + FrC

k

EnDnrDknM .

As it will be detailed at the end of this Section, the coefficients Ck

are completely dif-ferent from the C

kused in Eq.(5.179) for the RMVT-EM.

By substituting the Eqs.(5.228) in the variational statement of Eq.(5.227), the CUF[92] as presented in Section 3.5.3 and the geometrical relations of Section 2.5.1, for ageneric layer k [80]:∫

Ωk

Ak

[((Dk

p + Akp)Fsδu

ks

)T (FrC

k

σpεpr(Dkp + Ak

p)Fτukτ + FrC

k

σpσnrFτσknMτ−

FrCk

σpEprDkepFτΦ

kτ + FrC

k

σpDnrFτDknMτ

)+

((Dk

np + Dknz −Ak

n)Fsδuks

)T (Fτσ

knMτ

)

+(Fsδσ

knMs

)T ((Dk

np + Dknz −Ak

n)Fτukτ − FrC

k

εnεpr(Dkp + Ak

p)Fτukτ− (5.229)

FrCk

εnσnrFτσknMτ + FrC

k

εnEprDkepFτΦ

kτ − FrC

k

εnDnrFτDknMτ

)+

(Dk

epFsδΦks

)T

(FrC

k

Dpεpr(Dkp + Ak

p)Fτukτ + FrC

k

DpσnrFτσknMτ − FrC

k

DpEprDkepFτΦ

kτ + FrC

k

DpDnrFτDknMτ

)

+(Dk

enFsδΦks

)T (FτDk

nMτ

)−

(FsDk

nMs

)T (−Dk

enFτΦkτ − FrC

k

Enεpr(Dkp + Ak

p)Fτukτ−

FrCk

EnσnrFτσknMτ + FrC

k

EnEprDkepFτΦ

kτ − FrC

k

EnDnrFτDknMτ

)]dΩk dz = δLk

e .

In Eq.(5.229), in order to obtain a strong form of differential equations on the domainΩk and the relative boundary conditions on edge Γk, the integration by parts is used,which permits to move the differential operator from the infinitesimal variation of thegeneric variable δak to the finite quantity ak [92]. For a generic variable ak, the integra-tion by parts states as illustrated in Eq.(5.7) with matrices of Eqs.(5.8).The governingequations have the following form [80]:

δuks : Kkτsr

uu ukτ + Kkτsr

uσ σknMτ + Kkτsr

uΦ Φkτ + Kkτsr

uD DknMτ = pk

us −M kτsruu uk

τ

δσkns : Kkτsr

σu ukτ + Kkτsr

σσ σknMτ + Kkτsr

σΦ Φkτ + Kkτsr

σD DknMτ = 0 (5.230)

δΦks : Kkτsr

Φu ukτ + Kkτsr

Φσ σknMτ + Kkτsr

ΦΦ Φkτ + Kkτsr

ΦD DknMτ = pk

Φs

δDkns : Kkτsr

Du ukτ + Kkτsr

Dσ σknMτ + Kkτsr

DΦ Φkτ + Kkτsr

DD DknMτ = 0 .

The arrays pk indicate the variationally consistent loadings. Along with these gov-erning equations the following boundary conditions on the edge Γk of the in-plane

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160 CHAPTER 5

integration domain Ωk hold:

Πkτsruu uk

τ + Πkτsruσ σk

nMτ + ΠkτsruΦ Φk

τ + ΠkτsruD Dk

nMτ = Πkτsruu uk

τ + Πkτsruσ σk

nMτ

+ ΠkτsruΦ Φk

τ + ΠkτsruD Dk

nMτ (5.231)

ΠkτsrΦu uk

τ + ΠkτsrΦσ σk

nMτ + ΠkτsrΦΦ Φk

τ + ΠkτsrΦD Dk

nMτ = ΠkτsrΦu uk

τ + ΠkτsrΦσ σk

nMτ

+ ΠkτsrΦΦ Φk

τ + ΠkτsrΦD Dk

nMτ .

By comparing the Eq.(5.229), after the integration by parts, with the Eqs.(5.230) and(5.231), the fundamental nuclei are obtained:

Kkτsruu =

Ak

[(−Dk

p + Akp)

T (FrCk

σpεpr(Dkp + Ak

p))]FsFτH

kαHk

βdz , (5.232)

Kkτsruσ =

Ak

[(−Dk

p + Akp)

T (FrCk

σpσnr) + (−Dknp + Dk

nz −Akn)T

]FsFτH

kαHk

βdz , (5.233)

KkτsruΦ =

Ak

[(−Dk

p + Akp)

T (−FrCk

σpEprDkep)

]FsFτH

kαHk

βdz , (5.234)

KkτsruD =

Ak

[(−Dk

p + Akp)

T (FrCk

σpDnr)]FsFτH

kαHk

βdz , (5.235)

Kkτsrσu =

Ak

[(Dk

np + Dknz −Ak

n)− (FrCk

εnεpr)(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.236)

Kkτsrσσ =

Ak

[− FrC

k

εnσnr

]FsFτH

kαHk

βdz , (5.237)

KkτsrσΦ =

Ak

[FrC

k

εnEprDkep

]FsFτH

kαHk

βdz , (5.238)

KkτsrσD =

Ak

[− FrC

k

εnDnr

]FsFτH

kαHk

βdz , (5.239)

KkτsrΦu =

Ak

[−Dk

ep

TFrC

k

Dpεpr(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.240)

KkτsrΦσ =

Ak

[−Dk

ep

TFrC

k

Dpσnr

]FsFτH

kαHk

βdz , (5.241)

KkτsrΦΦ =

Ak

[Dk

ep

TFrC

k

DpEprDkep

]FsFτH

kαHk

βdz , (5.242)

KkτsrΦD =

Ak

[−Dk

ep

TFrC

k

DpDnr + Dken

T]FsFτH

kαHk

βdz , (5.243)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 161

KkτsrDu =

Ak

[FrC

k

Enεpr(Dkp

T+ Ak

p

T)]FsFτH

kαHk

βdz , (5.244)

KkτsrDσ =

Ak

[FrC

k

Enσnr

]FsFτH

kαHk

βdz , (5.245)

KkτsrDΦ =

Ak

[Dk

en − FrCk

EnEprDkep

]FsFτH

kαHk

βdz , (5.246)

KkτsrDD =

Ak

[FrC

k

EnDnr

]FsFτH

kαHk

βdz . (5.247)

The nuclei for boundary condition on edge Γk are [80]:

Πkτsruu =

Ak

[Ik

p

TFrC

k

σpεpr(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.248)

Πkτsruσ =

Ak

[Ik

p

TFrC

k

σpσnr + Iknp

T]FsFτH

kαHk

βdz , (5.249)

ΠkτsruΦ =

Ak

[−Ik

p

TFrC

k

σpEprDkep

]FsFτH

kαHk

βdz , (5.250)

ΠkτsruD =

Ak

[Ik

p

TFrC

k

σpDnr

]FsFτH

kαHk

βdz , (5.251)

ΠkτsrΦu =

Ak

[Ik

ep

TFrC

k

Dpεpr(Dkp + Ak

p)]FsFτH

kαHk

βdz , (5.252)

ΠkτsrΦσ =

Ak

[Ik

ep

TFrC

k

Dpσnr

]FsFτH

kαHk

βdz , (5.253)

ΠkτsrΦΦ =

Ak

[− Ik

ep

TFrC

k

DpEprDkep

]FsFτH

kαHk

βdz , (5.254)

ΠkτsrΦD =

Ak

[Ik

ep

TFrC

k

DpDnr

]FsFτH

kαHk

βdz . (5.255)

In order to perform the integration by parts, the matrices Ikp and Ik

np are those presentedin Eq.(5.8), while Ik

ep is that presented in Eq.(5.144). In order to write the explicit form ofthe nuclei in Eq.(5.232)-(5.247), the integrals in the z thickness-direction are defined asin Eq.(5.14). By developing the matrix products in Eqs.(5.232)-(5.247) and employinga Navier-type closed form solution [80], the algebraic explicit form of the nuclei isobtained.

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162 CHAPTER 5

Nucleus Kkτsruu of dimension (3× 3) is:

Kuu11 = α2Jkτsrβα

Ckσpεpr11 + β2Jkτsr

αβ

Ckσpεpr33 , Kuu12 = Jkτsr(Ck

σpεpr12 + Ckσpεpr33)αβ ,

Kuu13 = − 1

Rkα

Jkτsrβα

αCkσpεpr11 −

1

Rkβ

JkτsrαCkσpεpr12 , Kuu21 = Jkτsr(Ck

σpεpr21 + Ckσpεpr33)αβ ,

Kuu22 = β2Jkτsrαβ

Ckσpεpr22 + α2Jkτsr

βα

Ckσpεpr33 , Kuu23 = − 1

Rkα

JkτsrβCkσpεpr21 −

1

Rkβ

Jkτsrαβ

αCkσpεpr22 ,

Kuu31 = − 1

Rkα

Jkτsrβα

αCkσpεpr11 −

1

Rkβ

JkτsrαCkσpεpr21 ,

Kuu32 = − 1

Rkα

JkτsrβCkσpεpr12 −

1

Rkβ

Jkτsrαβ

βCkσpεpr22 ,

Kuu33 =1

Rkα

2Jkτsrβα

Ckσpεpr11 +

1

RkαRk

β

Jkτsr(Ckσpεpr12 + Ck

σpεpr21) +1

Rkβ

2Jkτsrαβ

Ckσpεpr22 .

(5.256)

Nucleus Kkτsruσ of dimension (3× 3) is:

Kuσ11 = − 1

Rkα

Jkτsβ + Jkτsz

αβ , Kuσ12 = 0 , Kuσ13 = −αJkτsrβ Ck

σpσnr13 ,

Kuσ21 = 0 , Kuσ22 = − 1

Rkβ

Jkτsα + Jkτsz

αβ , Kuσ23 = −βJkτsrα Ck

σpσnr23 ,

Kuσ31 = αJkτsβ , Kuσ32 = βJkτs

α , Kuσ33 = Jkτszαβ +

1

Rkα

Jkτsrβ Ck

σpσnr13 +1

Rkβ

Jkτsrα Ck

σpσnr23 .

(5.257)

Nucleus KkτsruΦ of dimension (3× 1) is:

KuΦ11 = KuΦ21 = KuΦ31 = 0 . (5.258)

Nucleus KkτsruD of dimension (3× 1) is:

KuD11 = − αJkτsrβ Ck

σpDnr11 , KuD21 = −βJkτsrα Ck

σpDnr21 ,

KuD31 =1

Rkα

Jkτsrβ Ck

σpDnr11 +1

Rkβ

Jkτsrα Ck

σpDnr21 .(5.259)

Nucleus Kkτsrσu of dimension (3× 3) is:

Kσu11 = − 1

Rkα

Jkτsβ + Jkτzs

αβ , Kσu12 = 0 , Kσu13 = αJkτsβ ,

Kσu21 = 0 , Kσu22 = − 1

Rkβ

Jkτsα + Jkτzs

αβ , Kuσ23 = βJkτsα , Kσu31 = αJkτsr

β Ckεnεpr31 ,

Kσu32 = βJkτsrα Ck

εnεpr32 , Kσu33 = Jkτzsαβ − 1

Rkα

Jkτsrβ Ck

εnεpr31 −1

Rkβ

Jkτsrα Ck

εnεpr32 .

(5.260)

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 163

Nucleus Kkτsrσσ of dimension (3× 3) is:

Kσσ11 = − Jkτsrαβ Ck

εnσnr11 , Kσσ12 = 0 , Kσσ13 = 0 ,

Kσσ21 = 0 , Kσσ22 = −Jkτsrαβ Ck

εnσnr22 , Kσσ23 = 0 ,

Kσσ31 = 0 , Kσσ32 = 0 , Kσσ33 = −Jkτsrαβ Ck

εnσnr33 .

(5.261)

Nucleus KkτsrσΦ of dimension (3× 1) is:

KσΦ11 = αJkτsrβ Ck

εnEpr11 , KσΦ21 = βJkτsrα Ck

εnEpr22 , KσΦ31 = 0 . (5.262)

Nucleus KkτsrσD of dimension (3× 1) is:

KσD11 = KσD21 = 0 , KσD31 = −Jkτsrαβ Ck

εnDnr31 . (5.263)

Nucleus KkτsrΦu of dimension (1× 3) is:

KΦu11 = KΦu12 = KΦu13 = 0. (5.264)

Nucleus KkτsrΦσ of dimension (1× 3) is:

KΦσ11 = αJkτsrβ Ck

Dpσnr11 , KΦσ12 = βJkτsrα Ck

Dpσnr22 , KΦσ13 = 0 . (5.265)

Nucleus KkτsrΦΦ of dimension (1× 1) is:

KΦΦ11 = − α2Jkτsrβα

CkDpEpr11 − β2Jkτsr

αβ

CkDpEpr22 . (5.266)

Nucleus KkτsrΦD of dimension (1× 1) is:

KΦΦ11 = Jkτzsαβ . (5.267)

Nucleus KkτsrDu of dimension (1× 3) is:

KDu11 = − αJkτsrβ Ck

Enεpr11 , KDu12 = −βJkτsrα Ck

Enεpr12 ,

KDu13 =1

Rkα

Jkτsrβ Ck

Enεpr11 +1

Rkβ

Jkτsrα Ck

Enεpr12 .(5.268)

Nucleus KkτsrDσ of dimension (1× 3) is:

KDσ11 = KDσ12 = 0 , KDσ13 = Jkτsrαβ Ck

Enσnr13 . (5.269)

Nucleus KkτsrDΦ of dimension (1× 1) is:

KDΦ11 = Jkτzsαβ . (5.270)

Nucleus KkτsrDD of dimension (1× 1) is:

KDD11 = Jkτsαβ Ck

EnDnr11 . (5.271)

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164 CHAPTER 5

α = mπ/a and β = nπ/b, with m and n as the waves numbers in in-plane directionsand a and b as the shell dimensions.

Nuclei for shells considered in Eqs.(5.256)-(5.271) degenerate in those for plates sim-ply considering 1

Rα= 1

Rβ= 0, and Hα = Hβ = 1, in order to obtain the integrals in

z-direction Jkτsr, Jkτzsr, Jkτszr and Jkτzszr.Navier-type closed form solution is obtained via substitution of harmonic expres-

sions for the displacements and electric potential as well as considering the follow-ing material coefficients equal to zero: Q16r = Q26r = Q36r = Q45r = 0 and e25r =e14r = e36r = ε12r = 0. The harmonic assumptions for the displacements, transverseshear/normal stresses and electric potential are those discussed in Eqs.(5.207)-(5.209),the transverse normal electric displacement Dn:

(Dknτ ) =

∑m,n

(Dk

nτ ) sin

(mπαk

ak

)sin

(nπβk

bk

), (5.272)

where Dk

nτ is the amplitude.The electric potential Φ, transverse shear/normal stresses σn and the transverse

normal electric displacement Dkn are always considered in layer wise form, while the

displacement u can be considered both LW or ESL. A RMVT2-EM model is ESL orLW depending the choice made for the displacements because the other variables arealways in LW form. The acronyms used are the following: EMN(EM2) when the dis-placement is in ESL form and the other variables are LW; EMZN(EM2) when the dis-placement uses the zig-zag function and the other variables are in layer wise form;LMN(EM2) when all the variables are in LW form. The order of expansion N in thethickness direction has been extended until to 14, N is the same for the displacements,the electric potential, the transverse shear/normal stresses and the transverse normalelectric displacement even if a different multilayer approach is used for them [80]. TheLW assembling procedure is the same for all the sixteen nuclei as indicated in Figure5.1. Nucleus Kkτsr

uu is assembled in ESL form as described in Figure 5.1, nuclei Kkτsrσσ ,

KkτsrσΦ , Kkτsr

Φσ , KkτsrΦΦ , Kkτsr

σD , KkτsrΦD , Kkτsr

Dσ , KkτsrDΦ and Kkτsr

DD are always assembled in LWform as indicated in Figure 5.1. The ESL assembling procedure for nuclei Kkτsr

uσ , KkτsruΦ

and KkτsruD is that indicated in Figure 5.6, for nuclei Kkτsr

σu , KkτsrΦu and Kkτsr

Du is that indi-cated in Figure 5.8. Some terms in fundamental nuclei, in case of RMVT applications,present integrals J without the index r, this happens because these terms do not mul-tiply quantities which depend by the thickness coordinate z, so the loop on index r isnot necessary.

The coefficients C have a form given by the Eqs.(4.61). In case of FunctionallyGraded Materials (FGMs), the coefficients C(z) are first calculated and then approxi-mated by means of thickness functions Fr by the formula C(z) = Fr(z)Cr [86].

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DIFFERENTIAL EQUATIONS AND FE MATRICES FOR MULTIFIELD PROBLEMS 165

The array Cσpεp(z) has dimension (3× 3):

Cσpεp11 =e233Q11 − 2e31e33Q13 −Q2

13ε33 + Q33(e231 + Q11ε33)

e233 + Q33ε33

,

Cσpεp12 = Q12 +e32(−e33Q13 + e31Q33)−Q23(e31e33 + Q13ε33)

e233 + Q33ε33

, Cσpεp13 = 0 ,

Cσpεp21 = Q12 +e32(−e33Q13 + e31Q33)−Q23(e31e33 + Q13ε33)

e233 + Q33ε33

,

Cσpεp22 =e233Q22 − 2e32e33Q23 −Q2

23ε33 + Q33(e232 + Q22ε33)

e233 + Q33ε33

, Cσpεp23 = 0 ,

Cσpεp31 = Cσpεp32 = 0 , Cσpεp33 = Q66 .

(5.273)

The array Cσpσn(z) has dimension (3× 3):

Cσpσn11 = Cσpσn12 = 0 , Cσpσn13 =e31e33 + Q13ε33

e233 + Q33ε33

,

Cσpσn21 = Cσpσn22 = 0 , Cσpσn23 =e32e33 + Q23ε33

e233 + Q33ε33

,

Cσpσn31 = Cσpσn32 = Cσpσn33 = 0 .

(5.274)

The array CσpEp(z) has dimension (3× 2):

CσpEp11 = CσpEp12 = CσpEp21 = 0 ,

CσpEp22 = CσpEp31 = CσpEp32 = 0 .(5.275)

The array CσpDn(z) has dimension (3× 1):

CσpDn11 =e33Q13 − e31Q33

e233 + Q33ε33

, CσpDn21 =e33Q23 − e32Q33

e233 + Q33ε33

, CσpDn31 = 0 . (5.276)

The array Cεnεp(z) has dimension (3× 3):

Cεnεp11 = Cεnεp12 = Cεnεp13 = Cεnεp21 = Cεnεp22 = Cεnεp23 = 0 ,

Cεnεp31 =−2e2

33Q13 + e31e33Q33 −Q13Q33ε33

Q33(e233 + Q33ε33)

,

Cεnεp32 =−2e2

33Q23 + e32e33Q33 −Q23Q33ε33

Q33(e233 + Q33ε33)

, Cεnεp33 = 0 .

(5.277)

The array Cεnσn(z) has dimension (3× 3):

Cεnσn11 =1

Q55

, Cεnσn12 = Cεnσn13 = Cεnσn21 = 0 , Cεnσn22 =1

Q44

,

Cεnσn23 = Cεnσn31 = Cεnσn32 = 0 , Cεnσn33 =ε33

e233 + Q33ε33

.(5.278)

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166 CHAPTER 5

The array CεnEp(z) has dimension (3× 2):

CεnEp11 =e15

Q55

, CεnEp12 = CεnEp21 = 0 ,

CεnEp22 =e24

Q44

, CεnEp31 = CεnEp32 = 0 .(5.279)

The array CεnDn(z) has dimension (3× 1):

CεnDn11 = CεnDn21 = 0 , CεnDn31 =e33

e233 + Q33ε33

. (5.280)

The array CDpεp(z) has dimension (2× 3):

CDpεp11 = CDpεp12 = CDpεp13 = CDpεp21 = CDpεp22 = CDpεp23 = 0 . (5.281)

The array CDpσn(z) has dimension (2× 3):

CDpσn11 =e15

Q55

, CDpσn12 = CDpσn13 = 0 ,

CDpσn21 = 0 , CDpσn22 =e24

Q44

, CDpσn23 = 0 .(5.282)

The array CDpEp(z) has dimension (2× 2):

CDpEp11 =e215

Q55

+ ε11 , CDpEp12 = CDpEp21 = 0 , CDpEp22 =e224

Q44

+ ε22 . (5.283)

The array CDpDn(z) has dimension (2× 1):

CDpDn11 = CDpDn21 = 0 . (5.284)

The array CEnεp(z) has dimension (1× 3):

CEnεp11 =e33Q13 − e31Q33

e233 + Q33ε33

, CEnεp12 =e33Q23 − e32Q33

e233 + Q33ε33

, CEnεp13 = 0 . (5.285)

The array CEnσn(z) has dimension (1× 3):

CEnσn11 = CEnσn12 = 0 , CEnσn13 = − e33

e233 + Q33ε33

. (5.286)

The array CEnEp(z) has dimension (1× 2):

CEnEp11 = CEnEp12 = 0 . (5.287)

The array CEnDn(z) has dimension (1× 1):

CEnDn11 =1

e233

Q33+ ε33

. (5.288)

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Chapter 6

Mechanical analysis

This chapter discusses the results for the pure mechanical analysis of multilayered compositeand sandwich plates/shells with the possible inclusion of layers in functionally graded mate-rial. First, some preliminary assessments are given in order to demonstrate the capability ofthe refined and advanced models to obtain the quasi-3D results. Then, some new benchmarksare proposed: - sandwiches with soft core (where the ZZ effect is very evident); - structuresincluding FGM layers (where the classical models are not efficient); - sandwiches with core inFGM (where the problems related to classical sandwiches are overcame). Finally, some resultsabout the new proposed finite element shell are given in order to demonstrate the capability ofthis new element for the analysis of shells.

6.1 Preliminary assessments

By using some 3D results given in the open literature, the refined and advanced modelsbased on CUF [92] are here validated: LW models with higher orders of expansion inthe thickness direction give a quasi-3D evaluation in case of multilayered plates andshells. So, they can be used as reference solutions in the benchmarks where no 3Dsolutions are given in the open literature.

In case of multilayered plates (composite and sandwiche structures), some compar-isons are made with the exact solutions given by Pagano [94]. In case of multilayeredshells, the refined and advanced models are compared with the exact solutions of Ren[148] and those of Varadan and Bhaskar [149]. Only few results are given in the tables:a complete overview about the capability of the 2D models based on CUF [92] is givenin [215] and [216], where further results about bi-sinusoidal, distributed and concen-trated mechanical loadings applied on multilayered plates and shells are discussed.

6.1.1 Quasi-3D analysis of composite and sandwich plates

Two assessments are given in the case of plates: the 3D exact solutions are those in-dicated in Pagano [94], simply supported plates with a bi-sinusoidal mechanical load

167

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168 CHAPTER 6

applied at the top surface:

pz(x, y) = pz sinmπx

asin

nπy

b, (6.1)

with the amplitude pz = 1 psi, waves number m = n = 1 and a and b are the dimen-sions of the plate.

The first case is a symmetric three-ply laminate with geometrical and material datagiven in Table 6.1. The considered plate and the employed reference system are given

Properties Layer 1− 2− 3EL [106psi] 25ET [106psi] 1GLT [106psi] 0.5GTT [106psi] 0.2νLT = νTT 0.25h1 = h2 = h3 [m] h/3a [m] 1b = 3a [m] 3lamination sequence 0/90/0

Table 6.1: Case 1. Elastic and geometrical properties of the rectangular composite plate.

in Figure 6.1. In Tables 6.2 and 6.3, the results are given in terms of normalized maxi-

Figure 6.1: Employed reference system and geometrical properties for the case 1: threelayered composite plate.

mum amplitudes:

(σxx, σyy, σxy) =(σxx, σyy, σxy)

pz(ah)2

, (σyz) =(σyz)

pz(ah), w =

100ET w

pzh( ah)4

. (6.2)

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MECHANICAL ANALYSIS 169

σxx(±h/2) σyy(±h/6) σyz(0) σxy(±h/2) w(0)1.14 0.109 -0.0269

3D[94] -1.10 -0.119 0.0334 0.0281 2.820.623 0.0251 -0.0083

CLT -0.623 -0.0251 0.0106 0.0083 0.5030.614 0.0833 -0.0187

FSDT -0.614 -0.0833 0.0234 0.0187 2.051.11 0.100 -0.0254

ED4 -1.06 -0.111 0.0346 0.0266 2.621.14 0.112 -0.0267

EDZ3 -1.09 -0.112 0.0317 0.0279 2.811.14 0.109 -0.0269

LD4 -1.10 -0.119 0.0334 0.0281 2.821.14 0.109 -0.0269

LM4 -1.10 -0.119 0.0334 0.0281 2.82

Table 6.2: Case 1. Thickness ratio a/h = 4. Three layered rectangular composite plate, normal-ized maximum stresses and deflections.

Two thickness ratios are investigated: thick plate (a/h = 4) and thin plate (a/h =100). The difficulty of classical theories (CLT and FSDT) are clearly remarked, in partic-ular for the thick plate. The refined models, in particular the LW ones with higher or-ders of expansion (N = 4) are completely in according with the 3D solution by Pagano[94].

The second case is a square sandwich plate with two external faces of thicknessh1 = h3 = 0.1h and material as indicated in Table 6.1 (the fibers orientation for thefaces is θ = 0). The core has thickness h2 = 0.8h, and material data are those givenin Table 6.4. The normalized quantities are the same of Eq.(6.2) using the Young’smodulus of the faces, the geometrical properties are indicated in Figure 6.2. The

Figure 6.2: Employed reference system and geometrical properties for the case 2:square sandwich plate.

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170 CHAPTER 6

σxx(±h/2) σyy(±h/6) σyz(0) σxy(±h/2) w(0)0.624 0.0253 -0.0083

3D[94] -0.624 -0.0253 0.0108 0.0083 0.5080.623 0.0251 -0.0083

CLT -0.623 -0.0251 0.0106 0.0083 0.5030.623 0.0252 -0.0083

FSDT -0.623 -0.0252 0.0106 0.0083 0.5060.624 0.0252 -0.0083

ED4 -0.624 -0.0252 0.0121 0.0083 0.5070.624 0.0252 -0.0083

EDZ3 -0.624 -0.0252 0.0107 0.0083 0.5080.624 0.0253 -0.0083

LD4 -0.624 -0.0253 0.0108 0.0083 0.5080.624 0.0253 -0.0083

LM4 -0.624 -0.0253 0.0108 0.0083 0.508

Table 6.3: Case 1. Thickness ratio a/h = 100. Three layered rectangular composite plate,normalized maximum stresses and deflections.

Properties CoreE11 = E22 [106psi] 0.04E33 [106psi] 0.5G13 = G23 [106psi] 0.06G12 [106psi] 0.016ν31 = ν32 = ν12 0.25h2 [m] 0.8ha = b [m] 1

Table 6.4: Case 2. Elastic and geometrical properties for the core of the sandwich plate.

results given in Tables 6.5 and 6.6, remark the importance of refined and advancedmodels and the problems related to classical theories, in particular for thick plates.In the case of sandwich plates, the ZZ effect can be captured by using the layer wisemodels. The results for the cases 1 and 2, permit to say that a LM4 model gives a quasi-3D evaluation of stresses and displacements in case of multilayered structures (thickand thin plates).

6.1.2 Quasi-3D analysis of composite shells

In order to assess the refined and advanced models based on CUF for the multilayeredshells, two assessments are proposed: a cylindrical panel with 3D exact solutions pro-posed by Ren [148], and a cylindrical shell with 3D exact solutions given by Varadanand Bhaskar [149].

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MECHANICAL ANALYSIS 171

σxx(±h/2) σyy(±h/2) σyz(0) σxy(±h/2)1.099 0.0569 -0.0446

3D[94] -1.099 -0.0569 0.0306 0.04461.097 0.0543 -0.0433

CLT -1.097 -0.0543 0.0177 0.04331.095 0.0552 -0.0438

FSDT -1.095 -0.0552 0.0179 0.04381.099 0.0572 -0.0444

ED4 -1.099 -0.0572 0.0352 0.04461.100 0.0579 -0.0445

EDZ3 -1.101 -0.0583 0.0311 0.04471.099 0.0568 -0.0445

LD4 -1.099 -0.0570 0.0306 0.04471.099 0.0568 -0.0445

LM4 -1.099 -0.0570 0.0306 0.0445

Table 6.5: Case 2. Thickness ratio a/h = 50. Sandwich square plate, normalized maximumstresses.

The first case is a simply supported two-layered curved panel, in cylindrical bend-ing, with a mechanical load applied at its top surface:

pz(x, y) = pz sinmπα

a, (6.3)

with the amplitude pz = 1 psi and waves number in α-direction m = 1. It is made oftwo layers of equal thickness h1 = h2 = 0.5h, material properties are those indicatedin Table 6.1, the lamination sequence is 90/0; the geometry is that indicated in Figure6.3. The radius of curvature in β direction is Rβ = ∞, the radius of curvature in αdirection is Rα = 10 with angle Φ equal to π

3. The two dimensions of the panel are:

a = π3Rα and b = 1.

The maximum stresses and deflections investigated in Tables 6.7 and 6.8 are givenin normalized form:

(σαα, σββ) =(σαα, σββ)

pz(Rα

h)2

, (σαz) =(σαz)

pz(Rα

h), w =

10ELw

pzh(Rα

h)4

. (6.4)

The second case is a three-layered cylindrical shell with a mechanical load appliedas an internal pressure:

pz(x, y) = pz sinmπα

asin

nπβ

b(6.5)

with the amplitude pz = 1 psi at the bottom, waves number in α-direction m = 8 andin β-direction n = 1. It is made of three layers of equal thickness h1 = h2 = h3 = h/3and material properties as seen in Table 6.1, the lamination sequence is 0/90/0; thegeometry is that indicated in Figure 6.4. The radius of curvature in β direction is Rβ =

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172 CHAPTER 6

σxx(±h/2) σyy(±h/2) σyz(0) σxy(±h/2)1.098 0.0550 -0.0437

3D[94] -1.098 -0.0550 0.0297 0.04371.097 0.0543 -0.0433

CLT -1.097 -0.0543 0.0177 0.04331.096 0.0545 -0.0434

FSDT -1.096 -0.0545 0.0177 0.04341.098 0.0554 -0.0436

ED4 -1.098 -0.0554 0.0343 0.04361.099 0.0559 -0.0436

EDZ3 -1.099 -0.0560 0.0303 0.04371.097 0.0549 -0.0436

LD4 -1.097 -0.0550 0.0297 0.04371.097 0.0549 -0.0436

LM4 -1.097 -0.0550 0.0297 0.0437

Table 6.6: Case 2. Thickness ratio a/h = 100. Sandwich square plate, normalized maximumstresses.

∞, the radius of curvature in α direction is Rα = 10 with angle Φ equal to 2π. The twodimensions of the panel are: a = 2πRα and b = 40.

The maximum stresses and deflections investigated in Tables 6.9 and 6.10 are givenin normalized form:

(σαα, σββ, σαβ) =10(σαα, σββ, σαβ)

pz(Rα

h)2

, (σαz, σβz) =10(σαz, σβz)

pz(Rα

h)

, w =10ELw

pzh(Rα

h)4

, σzz = σzz .

(6.6)By considering the results in Tables 6.7-6.10, two main considerations can be made:

w(0) σαα(h/2) σββ(h/2) σαz(h/4)3D[148] 0.493 2.245 0.0250 0.879CLT 0.445 2.246 0.0225 0.560FSDT 0.488 2.246 0.0225 0.560ED4 0.496 2.266 0.0245 0.797LD4 0.493 2.245 0.0249 0.881LM4 0.493 2.245 0.0249 0.879

Table 6.7: Case 1. Thickness ratio Rα/h = 10. Two-layered cylindrical panel, normalizedmaximum stresses and deflections.

the classical models such as CLT and FSDT are not able to furnish correct values ofdisplacements and stresses in case of multilayered shells, refined and advanced modelsmust be considered in these cases. The LM4 model gives a quasi-3D evaluation of thestresses and displacements in multilayered shells.

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MECHANICAL ANALYSIS 173

Figure 6.3: Employed reference system and geometrical properties for the case 1: two-layered composite Ren shell.

Figure 6.4: Employed reference system and geometrical properties for the case 2: three-layered composite Varadan & Bhaskar shell.

6.2 Static and free-vibrations analysis of sandwich plateswith soft core

In this section some new results are presented for multilayered structures, with par-ticular attention to two problems: - the importance of refined and advanced models(in particular the layer wise theories) in the static analysis of sandwich plates with softcore; - the use of refined theories to determine the fundamental frequencies for the freevibrations problem of sandwiches with very soft core. These benchmarks have beenintroduced in order to remark the problems connected to the ZZ effect due to the trans-verse anisotropy, for this aim a sandwich with two external faces in isotropic Al2024and an ideally soft core has been considered: - Benchmark 1, core with properties ofNomex divided by a factor 100; - Benchmark 2, core with properties of Nomex diviedby a factor 1012. The proposed plate is indicated in Figure 6.5. The two proposedbenchmarks are not real cases, but are very useful to investigate such problems, how-

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174 CHAPTER 6

w(0) σαα(h/2) σββ(h/2) σαz(h/4)3D[148] 0.399 2.153 0.0215 0.865CLT 0.399 2.153 0.0215 0.536FSDT 0.399 2.153 0.0215 0.536ED4 0.399 2.153 0.0211 0.769LD4 0.399 2.153 0.0215 0.865LM4 0.399 2.153 0.0215 0.865

Table 6.8: Case 1. Thickness ratio Rα/h = 500. Two-layered cylindrical panel, normalizedmaximum stresses and deflections.

w(0) σαα(h/2) σββ(h/2) σαβ(h/2) σβz(−h/6) σαz(0) σzz(0)3D[149] 1.223 4.683 0.0739 0.0374 0.0826 -3.264 -1.27LD1 1.190 4.442 0.0624 0.0360 0.0451 -3.148 -1.23LD2 1.221 4.671 0.0722 0.0373 0.0881 -3.129 -1.23LD4 1.223 4.683 0.0739 0.0374 0.0826 -3.137 -1.23LM2 1.222 4.675 0.0721 0.0374 0.0891 -3.152 -1.23LM4 1.223 4.683 0.0739 0.0374 0.0826 -3.264 -1.27

Table 6.9: Case 2. Thickness ratio Rα/h = 10. Three-layered cylindrical shell, normalizedmaximum stresses and deflections.

ever a complete overview is given in [105] where preliminary investigations are madefor classical sandwiches, and then sandwiches with very soft core are introduced topoint up the results and conclusions.

The plate is considered simply supported with a mechanical load applied at thetop as indicated in Eq.(6.1). The material and geometrical properties are indicated inTable 6.11. The static analysis made for the Benchmark 1 is given in Table 6.12 in termsof normalized transverse displacements w, the LM4 model is considered as referencesolution, in fact in Section 6.1 has been demonstrated that it gives a quasi-3D evaluationfor multilayered structures. In Table 6.12 is clearly indicated the importance of LWtheories to investigate sandwich structures with a soft core, in this case two parametersmust be considered: a geometrical one that is the length-to-thickness-ratio (LTR) anda mechanical parameter that is the face-to-core-stiffness-ratio (FCSR), for details see[105]. The error due to LTR can be contrasted by the use of higher order of expansions,for the error due to FCSR, the use of LW models becomes mandatory, in alternativethe inclusion of MZZF in ESL models (for example the EDZ3) gives good results if theplate is moderately thin.

An other interesting analysis can be made for the benchmark 2, which considers afree vibrations problem: given the waves number in the plane direction (m = n = 1)a certain number of frequencies are obtained depending the degrees of freedom of theemployed theory through the thickness direction. The benchmark 2 is an extreme case,because the core is very soft and the two faces can be considered as independent, so in

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MECHANICAL ANALYSIS 175

w(0) σαα(h/2) σββ(h/2) σαβ(h/2) σβz(−h/6) σαz(0) σzz(0)3D[149] 0.1027 0.7895 0.0559 -0.0766 0.1051 -0.691 -9.12LD1 0.1027 0.7906 0.0567 -0.0766 0.0530 -0.142 -0.170LD2 0.1027 0.7896 0.0559 -0.0766 0.1057 -0.142 -0.182LD4 0.1027 0.7896 0.0559 -0.0766 0.1051 -0.143 -0.159LM2 0.1027 0.7896 0.0559 -0.0766 0.1057 -0.677 -8.86LM4 0.1027 0.7895 0.0559 -0.0766 0.1051 -0.691 -9.12

Table 6.10: Case 2. Thickness ratio Rα/h = 500. Three-layered cylindrical shell, normalizedmaximum stresses and deflections.

Figure 6.5: Benchmark 1 and 2. Employed reference system and geometrical propertiesfor the case of the considered sandwich plate.

the free vibrations problem the two faces vibrate as two independent plates with thesame frequencies. Obviously, this effect can only be captured by means of a LW modelwhich see the three layers as independent. This fact is clearly indicated in Table 6.13:the LD2 model gives 21 frequencies through the thickness z, these are double valuesfor each frequency, which means that the two skins vibrate separately, this conditionof independent faces is confirmed by the Figure 6.6 where transverse shear stress σxz

is given along the thickness direction z (case of thick and thin plate).The results presented in this section permit to understand the importance of higher

order theories (problems connected to LTR) and layer wise models (problems con-nected to FCSR) in case of sandwich plates. Benchmarks 1 and 2 are two extremecases, but they are very useful to remark these problems. Other results about sand-wich plates can be found in [105]. The use of Murakami’s zig-zag function (MZZF) inthe ESL models to study sandwich structures has been delved in [124]-[126]: in Table6.12, the efficiency of the EDZ3 model has clearly described.

6.3 Static analysis of sandwich shells with soft core

In this section, it is shortly discussed the case of sandwich shell panels with soft core,the introduction of the curvature in the plane directions, does not modify the conclu-

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176 CHAPTER 6

Properties Layers 1-3 (Al2024)E [GPa] 73ν 0.34G [GPa] 27.239ρ[Kg/m3] 2800h1 = h3 [m] 0.1a = b [m] 4,10,100,1000

Layer 2 (B1)EL = ET [GPa] 0.0001Ez [GPa] 0.7585ν 0.01G [GPa] 0.225ρ[Kg/m3] 0.32h2 [m] 0.8a = b [m] 4,10,100,1000

Layer 2 (B2)EL = ET [µPa] 0.01Ez [µPa] 75.85ν 0.01G [µPa] 22.5ρ[Kg/m3] 3.2× 10−11

h2 [m] 0.8a = b [m] 4,10,100,1000

Table 6.11: Benchmarks 1 and 2. Elastic and geometrical properties for the sandwich plateswith soft core.

sions already obtained for the plate in Section 6.2. For further details about this topic,readers can refer to [131].

The chosen geometry is those of Ren shell as indicated in Figure 6.3 of Section 6.1.2,loading conditions are those of Eq.(6.3). In this case we consider a sandwich structurewith two external skins in Al2024 and a core in Nomex as indicated in Table 6.14. Thetransverse displacement w is evaluated in z = 0 for different 2D models, the compar-isons are made with respect to a reference solution that is the LM4 model which givesa quasi-3D evaluation, see Table 6.15. In Table 6.16, it is remarked the importance ofmixed models based on RMVT to obtain correct values of transverse shear/normalstresses. Figure 6.7 shows the validity of mixed models in layer wise form to obtain acorrect evaluation of the transverse normal stresses through the thickness z.

The main conclusions of this section are: - for displacements and in-plane stressesthe use of layer wise models in PVD form is sufficient; - the use of mixed ones becomesmandatory to obtain correct values of transverse shear/normal stresses. The resultsand considerations here omitted can be found in [131].

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MECHANICAL ANALYSIS 177

a/h 4 10 100 1000Ref 1370.6 Err.(%) 1260.3 Err.(%) 149.51 Err.(%) 7.1901 Err.(%)CLT 5.5815 (99.6) 5.5815 (99.6) 5.5815 (96.3) 5.5815 (22.4)FSDT 9.8241 (99.3) 6.2603 (99.5) 5.5883 (96.3) 5.5815 (22.4)ED4 112.53 (91.8) 24.083 (98.1) 5.7693 (96.1) 5.5834 (22.3)EDZ3 996.59 (27.3) 944.52 (25.0) 144.19 (3.56) 7.1890 (0.01)LD4 1370.6 (0.00) 1260.3 (0.00) 149.51 (0.00) 7.1901 (0.00)LM1 1349.8 (1.52) 1257.3 (0.24) 149.50 (0.01) 7.1901 (0.00)

Table 6.12: Benchmark 1. Comparison between the quasi-3D solution versus 2D models. Staticanalysis. Transverse displacement w = w(100(ET )skinh3)/(pza

4) at the mid surface z = 0.

-2

0

2

4

6

8

10

-0.4 -0.2 0 0.2 0.4

3D

LD4

ED4

0

50

100

150

200

250

-0.4 -0.2 0 0.2 0.4

3D

LD4

ED4

Figure 6.6: Benchmark 2. Sandwich plate with high-reduced core stiffness. Static analysis. σxz

vs z. Comparison between 3D solution, higher-order and LW theories. a/h = 4 on the left.a/h = 100 on the right.

6.4 Functionally graded material plates

Simply supported plates made of one layer in functionally graded material (FGM) areanalyzed in this section in case of static analysis. Two different cases are considered: -case 1: an FGM layer with the shear modulus continuously changing in the thicknessdirection z according to an exponential law; - case 2: an FGM layer with Young’s mod-ulus continuously changing in z according to a polynomial law. The plate is square asindicated in Figure 6.8 with a bi-sinusoidal loading applied at its top surface accordingto Eq.(6.1): m = n = 1 and pz = 1Pa.

For the case 1 the shear modulus G, according to Kashtalyan [151], changes as:

G(z) = G1eγ(z/h−1) , G1 =

E1

2(1 + ν), 0 ≤ z ≤ h . (6.7)

In Eq.(6.7) the Poisson’s ratio is constant and the exponential γ can assume severalvalues. For a very thick plate (a/h = 3), the 3D exact solution is given by Kashtalyan

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178 CHAPTER 6

Frequency CLT ED2 EDZ2 LD2ω1 271.39 271.39 17.388 0.6059ω2 472.43 472.41 271.39 0.6059ω3 9.4628 27514 472.43 271.39ω4 211953 0.7064 271.39ω5 9.4571 472.78 472.43ω6 211952 429769 472.43ω7 13557 211953 6491.9ω8 13549 271.39 6491.5ω9 430475 430475 16336ω10 211952 960992ω11 211605 960992ω12 211604 211605ω13 211605ω14 211605ω15 211604ω16 429772ω17 429763ω18 473162ω19 473162ω20 473161ω21 473161

Table 6.13: Benchmark 2. Sandwich plate with high reduced core stiffness. Free vibrations

problem. ω = ω√

a4(ρ)skin

(ET )skinh2 . m,n = 1. a/h = 100.

[151] for the transverse displacement w = (G1w/pzh) considered in the middle throughthe thickness.

For the case 2, in a moderately thick plate (a/h = 10), the Young’s modulus changesaccording to Zenkour [217] with a polynomial law:

E(z) = Em + (Ec − Em)(2z + h

2h)κ , −h

2≤ z ≤ h

2. (6.8)

The FGM is completely metallic at the bottom where Em = 70GPa and completelyceramic at the top where Ec = 380GPa; the Poisson’s ratio is considered constant andequal to 0.3, the exponent κ goes from 1 to 10. In this case Zenkour gives a generalizedshear deformation theory as reference solution [217]; so in order to test our refinedand advanced models, in [85] and [86] a quasi-3D reference solution for the FGMs hasbeen developed employing a layer wise model with fourth order of expansion in thethickness direction and by dividing the FGM layer in 100 mathematical layers withconstant properties, in [85] this reference solution has been called Nml = 100 (it is verycomputational expensive but it is incisive). The normalized given quantities in the

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MECHANICAL ANALYSIS 179

Properties Layers 1-3 (Al2024)E [GPa] 73ν 0.34G [GPa] 27.239h1 = h3 [m] 0.1

Layer 2 (Nomex)EL = ET [MPa] 0.01Ez [MPa] 75.85ν 0.01G [MPa] 22.5h2 [m] 0.8

Table 6.14: Ren shell geometry. Material properties for the two skins and the internal core inNomex.

Rα/h 4 10 50 100Ref 234.88 Err.(%) 47.587 Err.(%) 5.2040 Err.(%) 3.8562 Err.(%)CLT 3.7775 (98.4) 3.5612 (92.5) 3.4318 (34.0) 3.4150 (11.4)FSDT 5.0668 (97.8) 3.7577 (92.1) 3.4393 (33.9) 3.4168 (11.3)ED4 34.654 (85.5) 8.5193 (82.1) 3.6278 (30.3) 3.4650 (10.1)EDZ1 833.55 (71.8) 27.321 (42.6) 5.1794 (0.47) 3.8548 (0.04)LD2 234.87 (0.00) 47.587 (0.00) 5.2040 (0.00) 3.8562 (0.00)LM1 233.41 (0.62) 47.587 (0.00) 5.2040 (0.00) 3.8562 (0.00)

Table 6.15: Ren shell geometry with EskinETcore

= 73 × 105. Comparison between the quasi-3Dsolution versus 2D models. Static analysis. Transverse displacement w = w(10Eskinh3)/(pzR

4α)

at the mid surface z = 0.

tables are:

σyy =h

apz

σyy(h

3) , σxy =

h

apz

σxy(−h

3) , σyz =

h

apz

σyz(h

6) . (6.9)

For cases 1 and 2, closed form solutions and numerical finite element solutions havebeen given in [85] and [86] for both displacements models (PVD) and mixed models(RMVT). Here, only few results are discussed in order to verify the capability of CUF tostudy the FGM plates, and to remark the importance of refined and advanced modelswith respect to classical ones. The employed FE solution uses a 6 × 6 mesh with a Q8element. Closed form solutions are indicated with the acronym CF.

In Table 6.17, the transverse displacement w obtained by Kashtalyan (3D exact so-lution) [151] is compared with refined and advanced models obtained by CUF. Forboth Closed Form (CF) and Finite Element (FE) solutions, if higher orders of expan-sion are considered, the 3D values are obtained using both PVD and RMVT models. Itis important to remember that the order of expansion for the approximation of elastic

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180 CHAPTER 6

Rα/h 10 100 10 100σβz σzz

Ref 0.4125 Err.(%) 0.4168 Err.(%) 1.0000 Err.(%) 1.0000 Err.(%)EM1 0.0023 (99.4) 0.0022 (99.5) 138.41 (>100) 13201 (>100)EM4 0.0679 (83.5) 0.0654 (84.3) -60.874 (>100) -5516.8 (>100)EMZ3 0.4069 (1.36) 0.4169 (0.00) 106.93 (>100) 10355 (>100)LM1 0.4131 (0.14) 0.4168 (0.00) 0.8394 (16.1) 0.8258 (17.4)LM2 0.4130 (0.12) 0.4168 (0.00) 1.0342 (3.42) 1.0043 (0.43)LM3 0.4125 (0.00) 0.4168 (0.00) 1.0008 (0.08) 1.0000 (0.00)

Table 6.16: Ren shell geometry with EskinETcore

= 73 × 105. Comparison between the quasi-3Dsolution versus 2D models. Static analysis. Transverse stresses σβz = σβz(h/pzRα) in z = 0 andσzz = σzz in z = h/2.

Figure 6.7: Ren shell geometry with EskinETcore

= 73 × 105. σzz vs z for Rα/h = 4 on the left andRα/h = 100 on the right.

coefficients in case of FGMs is Nr = 10.Table 6.18 considers in-plane and transverse shear stresses for the case 2 proposed

by Zenkour [217], the solution given in [217] is a generalized shear deformation the-ory, so it cannot be used as reference solution for the refined and advanced models,the used reference solution is the so-called Nml = 100 detailed in [85]. It is evidentthe advantage of using mixed models (LM4) to obtain the transverse shear stresses.This fact is confirmed by the Figure 6.9, the LM4 model gives a 3D evaluation of theσzz through the thickness direction. The use of mixed models is mandatory for thetransverse shear/normal stresses in case of FGM plates. All the results omitted in thissection, can be found in [85] and [86].

6.5 Sandwich plates with core in FGM

The use of FGM face sheets and/or core in sandwich structures can be very usefulto contrast failure mechanisms and crack propagations, to increase critical and buck-

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MECHANICAL ANALYSIS 181

Figure 6.8: Employed reference system and geometrical properties for the cases 1 and2 of the one-layered FGM plate.

(CF)(CF)

(CF)(CF)

Figure 6.9: Case2. Transverse normal stress σzz through the thickness z for the FGM plate.Quasi-3D solution vs refined and advanced models. Exponential law κ = 7 on the left andκ = 10 on the right.

ling loads, to decrease the wear rate, the stress concentration effects and the interfacialshear stresses, and to improve thermal shock resistance [134]. For these reasons, in thissection sandwich plates with two external skins in "classical" material and an internalFGM core are discussed. First an assessment, for which the 3D exact solution is pro-vided by Kashtalyan and Menshykova [150], is given to demonstrate that refined andadvanced models are able to investigate the sandwich plates with FGM core. Then wepropose a new benchmark where the core is an FGM as proposed by Zenkour [217](see Section 6.4).

A validation for the employed ESL and LW models, based on PVD and RMVT, ismade by using the 3D solution proposed by Kashtalyan and Menshykova [150] for thecase of a three-layered plate with a core in FGM, see Figure 6.10. The global thicknessof the plate is h0 = 2h where h and −h represent the top and bottom coordinates of theplate, respectively. hc and −hc are the coordinates of the bottom 4th layer and the top1st layer, respectively. The global thickness of the core is 2hc. The plate has a global

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182 CHAPTER 6

Closed Form (CF) Finite Element (FE)γ 3D[151] LD4 LD2 LM4 FSDT LD4 LD310−1 -1.4146 -1.4145 -1.3480 -1.4145 -1.2285 -1.4148 -1.416710−2 -1.3496 -1.3495 -1.2856 -1.3495 -1.1746 -1.3497 -1.351610−3 -1.3433 -1.3431 -1.2796 -1.3431 -1.1693 -1.3434 -1.345310−4 -1.3426 -1.3425 -1.2789 -1.3425 -1.1688 -1.3427 -1.344610−5 -1.3426 -1.3424 -1.2789 -1.3424 -1.1687 -1.3427 -1.344610−6 -1.3426 -1.3424 -1.2789 -1.3424 -1.1687 -1.3427 -1.3446−10−1 -1.2740 -1.2739 -1.2132 -1.2739 -1.1116 -1.2741 -1.2759−10−2 -1.3355 -1.3354 -1.2722 -1.3354 -1.1629 -1.3357 -1.3375−10−3 -1.3419 -1.3417 -1.2782 -1.3417 -1.1681 -1.3420 -1.3439−10−4 -1.3425 -1.3424 -1.2788 -1.3424 -1.1686 -1.3426 -1.3445−10−5 -1.3425 -1.3424 -1.2789 -1.3424 -1.1687 -1.3427 -1.3445−10−6 -1.3425 -1.3424 -1.2789 -1.3424 -1.1687 -1.3427 -1.34460 -1.3430 -1.3424 -1.2789 -1.3424 -1.1687 -1.3427 -1.3445

Table 6.17: Case 1. Normalized out-of-plane displacement w in functionally gradedplate and a homogeneous plate. Comparison between exact solution, closed form (CF)and finite element method (FE).

thickness h0 = 1.0m, the thickness of the two faces is hf = 0.1h0, and the thicknessof the core is 2hc = 0.8h0. The considered thickness ratio is a/h0 = 3. ReferenceYoung’s modulus is E0 = 73GPa with Poisson’s ratio ν = 0.3, and the two faces haveconsequently a constant shear modulus Gf = G0 = 28.08GPa (layers 1 and 4). Thecore can be divided in parts 2 and 3. The value of the shear modulus in the middlereference surface is indicated with Gc. In layer 2, the exponential law for the shearmodulus is G(z) = G0exp[−γ(z/hc + 1)] with hc = 0.4m and −0.4 ≤ z ≤ 0. For layer3, the exponential law is G(z) = G0exp[γ(z/hc − 1)] with hc = 0.4m and 0 ≤ z ≤ 0.4.Four different cases are considered, which correspond to four different shear modulusratios Gc/Gf (0.9, 0.99, 0.999, 1.0), different Gc/Gf means the exponential γ equal to0.105360, 0.010050, 0.001000 and 0.0, respectively. The ratio Gc/Gf = 1.0 means a threelayered plate with the same "classical" material. The examples in Figure 6.10 are givenfor Gc/Gf = 0.9 and Gc/Gf = 0.999. The considered square plate is simply supportedwith a bi-sinusoidal loading applied at its top surface (pz = 1.0 and m = n = 1). Byconsidering the results in Table 6.19, it is clear that classical theories such as the FSDTand the ESL models with low order of expansions are inappropriate for such types ofproblems. The use of LW models is mandatory: in case of transverse normal stresses,mixed models in layer wise form, with at least a quadratic expansion in the z direction,is sufficient.

The new proposed benchmark is a three layered plate with the bottom layer inmetallic material with Young’s modulus Em = 70GPa and the top layer in ceramicwith Ec = 380GPa. The core consists of an FGM with Young’s modulus varying inz, according to Zenkour’s formula [217] (see Eq.(6.8)). The proposed plate is given in

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MECHANICAL ANALYSIS 183

κ σyy σyz σxy κ σyy σyz σxy

[217] 1 1.4894 0.2622 0.6110 8 0.9466 0.2121 0.5856Nml = 100 1.5062 0.2510 0.6081 0.9687 0.2262 0.5879LD4(CF ) 1.5064 0.2509 0.6112 0.9660 0.2118 0.5880LD4(FE) 1.5411 0.3631 0.6244 0.9879 0.2670 0.6007LM4(CF ) 1.5062 0.2510 0.6110 0.9661 0.2309 0.5882[217] 2 1.3954 0.2763 0.5441 9 0.9092 0.2072 0.5875Nml = 100 1.4147 0.2496 0.5421 0.9320 0.2249 0.5902LD4(CF ) 1.4139 0.2516 0.5437 0.9286 0.2107 0.5903LD4(FE) 1.4463 0.3601 0.5555 0.9496 0.2640 0.6030LM4(CF ) 1.4140 0.2497 0.5422 0.9285 0.2298 0.5905

Table 6.18: Case 2. Results for stresses in functionally graded plate. Comparison be-tween quasi-3D solution Nml = 100, GSDT by Zenkour [217], closed form (CF) andfinite element method (FE) solutions based on CUF.

Figure 6.10: Assessment. Sandwich plate with core in FGM. Case proposed by Kashtalyanand Menshykova [150].

Figure 6.11 where the thickness of each face is hf = 0.1h and the core has hc = 0.8h.In the core, Young’s modulus E(z) changes exponentially in z according to an expo-nential κ that can assume values of 1, 5 or 10, see Figure 6.11. A fourth case has beenadded: a core with a constant Young’s modulus, that is, an average between Ec andEm. Poisson’s ratio is constant for the three layers (ν = 0.3). Two thickness ratios areinvestigated, a/h = 4 and a/h = 100, that is, h = 0.25 and 0.01, respectively. Displace-ments, in-plane stresses and transverse shear/normal stresses are given in [134]. Here,the results given by advanced models, such as LD4 and LM4, could be considered asreference values for models that will be proposed in the future by other scientists. Inthe case of a thick plate (a/h = 4), the use of LW models is mandatory, if we considerthin plates (a/h = 100), ESL models could be used by employing higher orders of ex-pansion. In Figure 6.12 displacements and stresses through the thickness direction zare given for the proposed benchmark. Several FGM cores are compared by varyingthe law through z (κ = 1, 5, 10) and by considering a core with constant Young’s modu-lus. The employed theory is always a LD4 model. The displacements in the z directionobtained using a core with a constant Young’s modulus are more conservative than the

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184 CHAPTER 6

Gc/Gf 0.9 0.99 0.999 1.0 0.9 0.99 0.999 1.0w σzz

3D[150] 1.4227 1.3500 1.3433 1.3430 1.0000 1.0000 1.0000 1.0000FSDT 1.1924 1.1714 1.1690 1.1687 1.1536 1.1709 1.1723 1.1724ED3 1.4223 1.3518 1.3453 1.3446 1.1766 1.2226 1.2226 1.2271ED4 1.4199 1.3496 1.3432 1.3424 0.9060 0.9940 1.0026 1.0035LD2 1.4213 1.3476 1.3413 1.3406 1.0119 1.0119 1.0119 1.0119LD4 1.4227 1.3496 1.3432 1.3426 1.0000 1.0000 1.0000 1.0000LM2 1.4190 1.3464 1.3402 1.3395 1.0046 1.0071 1.0067 1.0066LM4 1.4227 1.3495 1.3432 1.3425 1.0002 1.0001 1.0001 1.0000

Table 6.19: Assessment. Sandwich plate with core in FGM. Case proposed by Kashtalyan andMenshykova. Transverse displacement w = w G0

pzh0in z = 0 and transverse normal stress σzz at

the top of the plate. Thickness ratio a/h = 3.

Figure 6.11: Benchmark. Sandwich plate with core in FGM according to Zenkour’s law[217].

FGM core cases, but this depends on the chosen law in z for the material properties.The use of an FGM core permits in-plane stresses continuous in the z direction to beobtained: this is not possible with a "classical" core where the typical discontinuity ofin-plane stresses for the sandwich structures is clearly shown.

6.6 Functionally graded material shells

In case of shells, we consider a Ren shell geometry as indicated in Section 6.1.2. Thegeometry data are detailed in [148], in case of FGM a bi-sinusoidal load with m = n = 1is applied at the top, the material is that proposed by Zenkour [217] with the Young’smodulus changing in according to Eq.(6.8). No exact solutions are provided in liter-ature, so a reference solution is proposed by dividing the shell in 100 mathematicallayers with constant properties and considering for each layer a LM4 theory, this solu-tion in the proposed tables is indicated as Nml = 100. In [218] it is demonstrated that

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MECHANICAL ANALYSIS 185

-0.04

-0.02

0

0.02

0.04

-0.1 -0.05 0 0.05 0.1 0.15 0.2

z

u

K=1

K=5

K=10

Ec=cost

-0.04

-0.02

0

0.02

0.04

0.4 0.5 0.6 0.7 0.8 0.9z

w

K=1

K=5

K=10

Ec=cost

-0.04

-0.02

0

0.02

0.04

-20 -10 0 10 20 30 40

z

σxx

K=1

K=5

K=10

Ec=cost

-0.04

-0.02

0

0.02

0.04

-20 -15 -10 -5 0 5 10

z

σxy

K=1K=5

K=10Ec=cost

-0.04

-0.02

0

0.02

0.04

0 0.5 1 1.5 2 2.5 3

z

σxz

K=1

K=5

K=10

Ec=cost

-0.04

-0.02

0

0.02

0.04

0 0.2 0.4 0.6 0.8 1

z

σzz

K=1

K=5

K=10

Ec=cost

Figure 6.12: Benchmark. Sandwich plate with core in FGM according to Zenkour’slaw [217]. Displacements (u, w) = (ux, uz)10Ech3

pza4 , stress σxx, σxy, σxz and σzz vs z forLD4 theory. Comparison between different parameter values κ. The thickness ratio isa/h = 10.

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186 CHAPTER 6

w(0)Rα/h 4 10 100 1000

k=1 Nml = 100 0.0018 0.0170 52.781 4201.3LM4 0.0013 0.0170 52.783 4201.4LD4 0.0013 0.0170 52.783 4201.4LM2 0.0013 0.0162 52.693 4201.4LD2 0.0014 0.0162 52.692 4201.4

FSDT 0.0054 0.0170 43.735 3792.2k=4 Nml = 100 0.0032 0.0314 79.739 7081.1

LM4 0.0022 0.0315 79.734 7081.6LD4 0.0021 0.0315 79.734 7081.6LM2 0.0028 0.0287 79.345 7081.6LD2 0.0032 0.0288 79.344 7081.6

FSDT 0.0090 0.0277 65.603 6384.2k=10 Nml = 100 0.0042 0.0404 92.018 9370.8

LM4 0.0022 0.0405 92.033 9373.1LD4 0.0021 0.0404 92.014 9373.1LM2 0.0044 0.0359 91.356 9373.0LD2 0.0049 0.0355 91.304 9373.0

FSDT 0.0121 0.0358 75.561 8436.5

Table 6.20: One-layered FGM Ren shell geometry. Transverse displacement w = w1010 inz = 0.

the Nml = 100 solution is a quasi-3D solution for functionally graded shells. The resultsabout the FGM shells have been added in order to investigate the effect of the curva-ture: no further considerations have been added with respect to the case of plate inSection 6.4. The Table 6.20 considers the transverse displacement w in z = 0, the com-parison is made between several 2D theories and the quasi-3D solution for differentthickness ratios Rα/h and several exponentials k for the material law. The importanceof higher orders of expansion in the thickness direction z and the inefficiency of theclassical models, such as the FSDT, are clearly demonstrated. For the displacements,there are no differences between the PVD and RMVT models, these differences areclear in the case of transverse shear/normal stresses as detailed in [218] and [85]. Fig-ure 6.13 considers the in-plane stress σxy for k = 10: in case of thick shells, it is clear theimportance of refined models.

6.7 Sandwich shells with core in FGM

In analogy to Section 6.5, a sandwich shell with a core in functionally graded materialhas been proposed in this section. The materials data and the geometry for the consid-ered layers are the same of Section 6.5 with the core in FGM as proposed by Zenkour

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MECHANICAL ANALYSIS 187

-0.1

-0.05

0

0.05

0.1

-1 -0.5 0 0.5 1

σ xy

z

LD4LM4LD2

FSDT

-4

-2

0

2

4

-0.04 -0.02 0 0.02 0.04

σ xy

z

LD4LM4LD2

FSDT

Figure 6.13: One-layered FGM Ren shell geometry. In-plane stress σxy vs z for k = 10. Thickshell Rα/h = 4 on the left and thin shell Rα/h = 100 on the right.

[217] (Young’s modulus changing in according to Eq.(6.8)). The Nml = 100 model hasbeen proposed as reference solution. Several values of k are investigated, further, acore in classical material is considered where the Young’s modulus is an average be-tween the Em at the bottom and the Ec at the top. The introduction of the curvature,does not change the conclusions already obtained for the plate in Section 6.5. The re-fined models proposed in this thesis permit to obtain a quasi-3D investigation of suchproblems, as detailed for the transverse normal stress in Table 6.21. The value of theσzz must be equal to 1 at the top, for the homogeneous loading conditions and as con-firmed by the Nml = 100 model. The case proposed is a three-layered sandwich shell,so there are substantial differences between the ESL and LW models. In Table 6.21,even if higher orders of expansion are employed, the ESL theories do not give satisfac-tory values of σzz. The use of LW models is mandatory for such problems as indicatedby the LM4, LD4 and LM2 results. The introduction of the MZZF in the ESL mod-els gives some improvements in case of moderately thin shells. The advantage of LM4model with respect to the LD4 one is the possibility of fulfil the continuity of transverseshear/normal stresses at the interfaces, as detailed in [134].

6.8 Finite element analysis of shells

In this section, the new implemented finite element shell 9P-8N-FE, proposed in theSection 5.1.3, is validated by means of several assessments. This finite element is anine parameter kinematic model (9P), quadratic order of expansion in the thicknessdirection for each displacement component, and the element is an eight node Q8 (8N).The proposed approach is an extension of the degenerated approaches: displacementsare written by referring to a generic mid-surface. The shear locking phenomena hasbeen corrected as described in Polit [208]. This method, applied to 9P-8N-FE, givescorrect results for plate assessments, but it needs some improvements for the shellgeometry. The membrane locking phenomena will be corrected in the near future byusing the same idea applied for the shear locking phenomena. The Poisson’s locking,

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188 CHAPTER 6

σzz(z = h/2)a/h 4 10 100 1000

k=1 Nml = 100 1.0000 1.0000 1.0000 1.0000LM4 1.0000 1.0000 1.0000 1.0000LD4 1.0009 1.0000 1.0000 1.0000LD2 1.0765 1.0257 1.0124 1.0000EM4 1.4163 1.0952 0.9675 0.9756ED4 1.4204 1.0959 0.9674 0.9756ED2 1.0733 1.2953 1.1173 0.8956

EMZ3 2.0956 1.4603 1.2973 1.1088FSDT 0.0797 0.4464 42.278 776.77

k=5 Nml = 100 1.0000 1.0000 1.0000 1.0000LM4 1.0000 1.0000 1.0000 1.0000LD4 1.0010 1.0000 1.0000 1.0000LD2 1.0845 1.0307 1.0138 0.9998EM4 1.2988 0.8036 0.6404 0.8164ED4 1.3011 0.7977 0.6332 0.8159ED2 1.2457 1.5951 1.3997 0.9338

EMZ3 2.7617 1.5077 1.3505 1.0326FSDT 0.0934 0.5155 49.864 1149.6

k=10 Nml = 100 1.0000 1.0000 1.0000 1.0000LM4 1.0000 1.0000 1.0000 1.0000LD4 1.0010 1.0000 1.0000 1.0000LD2 1.0856 1.0298 1.0147 0.9998EM4 1.2027 0.5142 0.3908 0.5989ED4 1.2179 0.5140 0.3883 0.6029ED2 1.3376 1.7534 1.5234 1.0486

EMZ3 3.1867 1.6443 1.4872 1.0481FSDT 0.0987 0.5417 52.575 1320.4

E=225 GPa Nml = 100 1.0000 1.0000 1.0000 1.0000LM4 1.0000 1.0000 1.0000 1.0000LD4 1.0008 1.0000 1.0000 1.0000LD2 1.0757 1.0219 1.0130 1.0003EM4 1.3804 0.9608 0.8565 0.8178ED4 1.4589 1.0541 0.9422 0.8964ED2 1.2160 1.5163 1.2584 1.1895

EMZ3 2.4376 1.7073 1.5219 1.2415FSDT 0.0786 0.4383 41.286 824.18

Table 6.21: Sandwich Ren shell geometry with core in FGM. Transverse normal stress σzz atthe top.

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MECHANICAL ANALYSIS 189

which is not a numerical phenomena, has just been corrected because the 9P-8N-FEmodel is an ED2 model as already seen for the analytical and FE plate solutions. So,as indicated in the Section 2.1.2 and in works [66], [67], the use of a linear transversenormal strain εzz is sufficient to avoid the Poisson’s locking phenomena.

The proposed assessments are three: - a clamped square plate with a concentratedload in its middle; - a simply supported plate with a concentrated load in its middle; -the well-known pinched cylindrical shell.

The clamped plate is indicated in Figure 6.14, the dimensions are a = b = 10m with

Figure 6.14: Clamped or simply supported homogeneous isotropic plate with a me-chanical concentrated load.

thickness h = 0.01m. The material is isotropic with Young’s modulus E = 10.92Paand Poisson’s ratio ν = 0.3. The load in z direction is Pz = 4N . The results in Table6.22 compare the maximum transverse displacement in the center of the plate. Thereference result is given by Batoz and Dhatt in [214]: wmax = 0.0056PL2

D= 2.24 × 106.

The CL8 theory is the shell finite element by Polit [208] with the correction of the shearlocking. The 9P-8N-FE(iso) is the proposed model without the shear locking correction,

Ref [214] 9P-8N-FE(iso) 9P-8N-FE CL82.24× 106

2× 2 2.20× 102 2.13× 106 2.23× 106

4× 4 7.20× 103 2.18× 106 2.24× 106

8× 8 7.60× 105 2.22× 106 2.24× 106

16× 16 2.10× 106 2.23× 106 2.24× 106

32× 32 2.22× 106 2.24× 106 2.24× 106

Table 6.22: Clamped isotropic plate with concentrated load at its top surface. Thickness ratioa/h = 1000. Maximum transverse displacement w in the middle of the plate.

even if a 32× 32 mesh is applied the reference solution is not obtained. The 9P-8N-FEis the same model including the shear locking correction, now the convergence speedis higher and the reference solution is obtained.

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190 CHAPTER 6

The second assessment considers the plate already illustrated in Figure 6.14 withthe same material and geometrical properties, but in this case it is simply supported.For a/h = 1000, the reference solution is given by Batoz and Dhatt in [214]: wmax =

0.0116PL2

D= 4.64 × 106. The maximum transverse displacement w is given in Table

Ref [214] 9P-8N-FE(iso) 9P-8N-FE4.64× 106

2× 2 3.42× 103 4.63× 106

4× 4 4.30× 105 4.63× 106

8× 8 3.93× 106 4.64× 106

16× 16 4.54× 106 4.64× 106

32× 32 4.62× 106 4.64× 106

Table 6.23: Simply supported isotropic plate with concentrated load at its top surface. Thick-ness ratio a/h = 1000. Maximum transverse displacement w in the middle of the plate.

6.23. The correction of shear locking is very efficient, in fact the reference solution isobtained by means of a 8× 8 mesh. In conclusion, for both plates, the model is correctand it is free from shear locking phenomena thanks the procedure given in Polit [208].

The third case is the well-known pinched cylindrical shell as illustrated in [208] and[219]. The geometrical properties and loading conditions are given in Figure 6.15. The

Figure 6.15: Pinched cylindrical shell configuration in homogeneous and isotropic ma-terial.

two surfaces in the α-z plane are considered as two diaphragms, the dimension in β

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MECHANICAL ANALYSIS 191

direction is L = 6m, the radius of curvature is R = 3m and the thickness is h = 0.03m.The value of the loading is Pz = 1N . The shell is made of an isotropic material withYoung’s modulus E = 3 × 1010Pa and Poisson’s ratio ν = 0.3. In case of a finite el-ement investigation, by using the symmetry, only a quarter of the proposed shell canbe considered as illustrated in Figure 6.15 (area A-B-C-D), in this case the consideredmechanical loading is Pz/4. For the 9P-8N-FE model, a 16 × 16 mesh is applied onthis quarter. In Table 6.24 the maximum displacement w in the point C of the shell isconsidered. The reference solution is a three-dimensional one as given in [219]. The 9P-

Ref [219] 9P-8N-FE Err.(%) CL8 Err.(%) Q9*F Err.(%) Q9RF Err.(%)164.240 145.800 (11.2) 164.700 (0.28) 139.250 (15.2) 140.338 (14.5)

Table 6.24: Pinched cylindrical shell. Maximum displacement w = EhwPz

in the point C. Thick-ness ratio R/h = 100.

8N-FE model gives an error of 11.2%, because both shear and membrane lockings havenot yet corrected in the shell case. These lockings will be corrected as done by Polit[208] in case of the CL8 theory (shell finite element Q8 with five degrees of freedom foreach node). The two added models in the Table 6.24 are given in [207] by Huang andHinton. The Q9*F and Q9RF are the so-called QUAD9*F and QUAD9RF, respectively.These are 9 node elements with 5 degrees of freedom for each node, in them no correc-tions for the shear and membrane lockings have been applied, so we can compare themodel 9P-8N-FE with these last and understand if the model is correctly implementedand if the error is only due to locking phenomena: the error in percentage are almostthe same, a 8× 8 mesh is applied in [207].

6.9 Conclusions

This chapter has considered the static and dynamic mechanical analysis of multilay-ered plates and shells. Composite and sandwich structures have been investigated,with the possibility of the inclusion of some FGM layers. The importance of higherorders of expansion for the 2D models has been remarked, and the importance of thelayer wise theories has been outlined for particular benchmarks such as the sandwichshells and plates with soft core. The mixed models, where the transverse shear/normalstresses are a priori modelled, are mandatory in order to obtain the continuity of suchvariables in the thickness direction. Both analytical and numerical examples have beenprovided: for the case of shells a new finite element (8 nodes element with 9 degrees offreedom for each node) has been introduced. By using some well-known benchmarksin open literature, this FE element has been validated, even if an affective correction forthe shear and membrane locking phenomena must be introduced in the near future.

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Chapter 7

Thermo-mechanical analysis

This chapter considers the thermal stress analysis of multilayered plates and shells, with thepossible inclusion of some functionally graded material layers. In order to determine the thermalload two different cases are considered: - the temperature distribution in the thickness directionis assumed linear; - the temperature distribution in the thickness direction is calculated viaFourier’s heat conduction equation. An other possibility is the full coupling between the thermaland mechanical field, in this case both the temperature and displacements are primary variablesof the problem. The full coupled problem permits to consider three different cases: - staticanalysis considering a temperature imposed at the top and bottom of the structure (the resultsare coincident with the case where the thermal load is obtained calculating the temperatureprofile via Fourier’s equation); - static analysis by considering a mechanical load applied tothe structure (the effects of the thermal coupling are evaluated and the profile of temperaturein relation with the considered displacement field is obtained); - free vibration problem wherethe effect of the thermal coupling on the frequencies is investigated and the temperature field,related to a particular vibration mode through the thickness, is obtained.

7.1 Thermal analysis of multilayered plates

This section permits to asses the refined models for the thermal stress investigation ofmultilayered structures. The uncoupled problem is considered, which means tempera-ture included as an external load. A satisfactory thermal stress analysis is only possibleif: - advanced and refined computational models are developed; - correct thermal loadsare recognized. For the latter issue, an appropriate temperature profile must be defined(for example by solving Fourier’s heat conduction equation): an assumed linear tem-perature profile is advisable only for thin and homogeneous structures [220].

In case of plates a comparison is made between the refined 2D models based on theCUF and the 3D exact solution by Bhaskar et alii [160], this last is obtained consider-ing a linear assumed temperature profile. So two aspects are remarked: - the refinedtheories with an assumed temperature profile (Ta), such as the layer wise ones, givethe 3D solution; - if the same 2D models consider a calculated temperature profile,the results are quite different, for the cases of thick and moderately thick plates. Theplate is a three-layered composite with lamination sequence as indicated in Figure 7.1

193

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194 CHAPTER 7

(0/90/0), the three layers have a thickness h1 = h2 = h3 = h/3. The temperatureis Tt = +1C at the top and Tb = −1C at the bottom and varies linearly throughthe thickness z (the temperature is seen as a sovratemperature referred to the exter-nal room temperature). The properties of the carbon fiber reinforced layers are those

0

90

0h/3

h/3

h/3

T =+1t

T =-1b

z

x,y

O

O

O

Figure 7.1: Multilayered composite plate with an assumed temperature profile throughthe thickness z.

indicated in [160]: the ratio between longitudinal and transverse Young’s modulus isEL/ET = 25, the shear modulus are GLT /ET = 0.5 and GTT /ET = 0.2, the Poisson’sratio is νLT = νTT = 0.25, the ratio between the transverse and longitudinal thermalexpansion coefficients is αT /αL = 1125, and finally the conduction coefficients in lon-gitudinal and transverse directions are KL = 36.42W/mC and KT = 0.96W/mC,respectively. The plate is square with a = b and the total thickness is h = 1. The tem-perature distribution in the xy-plane is bi-sinusoidal with waves number m = n = 1.

In Tables 7.1, 7.2 and 7.3 the transverse displacement and the in-plane stresses arecompared with the exact 3D solutions by Bhaskar et alii [160]. This reference solutionis calculated by considering an assumed liner temperature profile (Ta), the 2D modelsare given by using both assumed linear temperature profile (Ta) and calculated tem-perature profile (Tc). The importance of the kinematic model is clearly reported inthe Tables 7.1-7.3 by comparing the 3D solutions with the 2D models (both employingan assumed temperature profile Ta): the plate is three-layered made, so a layer wisemodel with higher orders of expansion in z (for example a LD4) gives the 3D solutionfor both thick and thin plates. The inadequacy of classical models such as FSDT andCLT is clearly remarked. The ESL models give a quasi-3D evaluation if thin plates areconsidered and higher orders of expansion are applied. If the considered 2D modelsuse a calculated temperature profile Tc, the results are quite different: in particular forthick plates, where as demonstrated in Figure 7.2, the temperature profile is never lin-ear. Figure 7.2 compares the assumed and calculated temperature profiles for severalthickness ratios a/h, the calculated temperature profile Tc coincides with the assumedone Ta only if the plate is sufficiently thin, this fact explains the differences given inTables 7.1-7.3. Further results about one-layered and multilayered plates subjected tothermal loads can be found in [172] and [204]. The main conclusions are: - the impor-

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THERMO-MECHANICAL ANALYSIS 195

a/h 2 4 10 20 50 100w = w/(a/h)2 at z = h/2

3D(Ta)[160] 96.79 42.69 17.39 12.12 10.50 10.263D(Tc) - - - - - -LD4(Ta) 96.78 42.69 17.39 12.12 10.50 10.26LD4(Tc) 75.52 37.88 16.91 12.03 10.48 10.26LD3(Ta) 96.73 42.68 17.39 12.12 10.50 10.26LD3(Tc) 70.16 37.50 16.91 12.03 10.48 10.26ED4(Ta) 98.21 42.05 16.90 11.96 10.47 10.25ED4(Tc) 76.41 37.17 16.42 11.87 10.46 10.25ED2(Ta) 83.47 34.74 14.96 11.42 10.38 10.23ED2(Tc) 43.18 26.11 14.10 11.25 10.35 10.22FSDT(Ta) 41.28 29.08 18.33 15.91 15.17 15.06FSDT(Tc) 33.55 25.82 17.81 15.79 15.15 15.05CLT(Ta) 15.02 15.02 15.02 15.02 15.02 15.02CLT(Tc) 12.05 13.27 14.59 14.91 15.00 15.02

Table 7.1: Multilayered composite plate. Transverse displacement w at the top. 3D solution vsrefined and classical models. Linear assumed temperature profile (Ta) vs calculated tempera-ture profile (Tc).

tance of refined models for a correct evaluation of displacement and stress componentsin multilayered structures; - it has been shown that results obtained assuming a lineartemperature profile in the thickness direction can be sometimes meaningless: the im-portance of calculating the actual temperature profile to obtain a correct thermal load-ing has been shown by comparing the 2D models with the 3D one which uses a lineartemperature profile [160]. This preliminary assessment permits to use with confidencethe refined models in the next section, for the thermal analysis of multilayered shells.

7.2 Thermal analysis of multilayered shells

The effects of the through-the-thickness temperature distribution on the response oflayered composite shells are investigated in this section. Thanks to the conclusionsobtained in the Section 7.1, the layer wise theory LD4 with a calculated temperatureprofile can be considered as reference solution. The stress analysis must be precededby an accurate thermal analysis, which provides the required temperature input data[147]. The effectiveness of CUF to treat such problems is clearly reported in [81] whereseveral comparisons have been made with theories proposed by Khare et alii [166] andKhdeir [169]. In case of shells, three benchmarks are proposed to remark the impor-tance of both refined models and calculated temperature profile.

The first case is an isotropic one-layered cylindrical shell, the dimensions are a = 1and b = π

3Rβ = 10.47197551. The radii of curvature in the α and β directions are 1

Rα= 0

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196 CHAPTER 7

a/h 2 4 10 20 50 100σxx at z = h/2

3D(Ta)[160] 1390 1183 1026 982.0 967.5 965.43D(Tc) - - - - - -LD4(Ta) 1390 1183 1026 982.0 967.5 965.4LD4(Tc) 893.4 953.2 976.9 969.1 965.4 964.8LD3(Ta) 1382 1183 1026 982.0 967.5 965.4LD3(Tc) 873.5 986.3 986.3 971.6 965.9 964.9ED4(Ta) 1319 1171 1004 963.4 950.8 949.0ED4(Tc) 885.5 985.9 966.8 953.9 949.3 948.6ED2(Ta) 123.3 486.4 802.8 873.7 895.4 898.6ED2(Tc) -382.6 143.7 707.0 847.1 891.0 897.5FSDT(Ta) 46.22 525.6 948.0 1043 1072 1076FSDT(Tc) -36.22 425.7 912.5 1033 1070 1076CLT(Ta) 1078 1078 1078 1078 1078 1078CLT(Tc) 808.2 918.8 1039 1067 1076 1077

Table 7.2: Multilayered composite plate. In-plane stress σxx at the top. 3D solution vs re-fined and classical models. Linear assumed temperature profile (Ta) vs calculated temperatureprofile (Tc).

a/h 2 4 10 20 50 100σxy at z = −h/2

3D(Ta)[160] 269.3 157.0 76.29 57.35 51.41 50.533D(Tc) - - - - - -LD4(Ta) 269.3 157.0 76.29 57.35 51.41 50.53LD4(Tc) 215.1 140.3 74.21 56.91 51.34 50.51LD3(Ta) 269.1 157.0 76.29 57.35 51.41 50.53LD3(Tc) 244.6 144.1 74.46 56.96 51.35 50.52ED4(Ta) 256.1 148.0 73.44 56.51 51.26 50.49ED4(Tc) 203.4 131.8 71.40 56.07 51.20 50.48ED2(Ta) 172.1 114.6 65.28 54.26 50.89 50.40ED2(Tc) 90.06 86.38 61.54 53.43 50.77 50.37FSDT(Ta) 232.8 159.1 94.10 79.48 75.00 74.35FSDT(Tc) 189.3 141.3 91.46 78.87 74.91 74.33CLT(Ta) 74.14 74.13 74.13 74.13 74.13 74.13CLT(Tc) 59.47 65.48 72.01 73.56 74.04 74.11

Table 7.3: Multilayered composite plate. In-plane stress σxy at the bottom. 3D solution vs re-fined and classical models. Linear assumed temperature profile (Ta) vs calculated temperatureprofile (Tc).

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THERMO-MECHANICAL ANALYSIS 197

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

Figure 7.2: Multilayered composite plate: comparison between assumed temperatureprofile Ta and calculated temperature profile Tc with order of expansion N = 1 andN = 4 for several thickness ratios (a/h = 2, 4, 20, 100).

and 1Rβ

= 0.1, respectively. The considered thicknesses are h = 2.5, 1.0, 0.1, 0.01, whichmean thickness ratios Rβ/h = 4, 10, 100, 1000. The considered material is AluminumAl5086 with Young’s modulus E = 70.3GPa, Poisson’s ratio ν = 0.33, thermal expan-sion coefficient α = 24 × 10−6C−1 and conductivity coefficient K = 130W/mC. Thetransverse displacement w and the in-plane stress σxy are given, in the middle of theshell, in Table 7.4. Different thickness ratios Rβ/h are investigated. Only ESL theoriesare reported in the table because the shell is one-layered. Ta means a linear assumedtemperature profile in the thickness direction, Tc means a calculated temperature ap-proximated in the thickness direction with the same order of the employed kinematicmodel.

The second case is an isotropic two-layered cylindrical shell with the same di-mensions and geometrical properties as the shell indicated in case 1. The consid-ered total thicknesses are h = 2.5, 1.0, 0.1, 0.01 which mean thickness ratios Rβ/h =4, 10, 100, 1000. The bottom layer is in Al5086 (the same as case 1) while the top layer isin Titanium Ti22 with Young’s modulus E = 110GPa, Poisson’s ratio ν = 0.32, thermalexpansion coefficient α = 8.6×10−6C−1 and conductivity coefficient K = 21.9W/mC.The two considered layers have the same thickness h/2. The non dimensioned quan-

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198 CHAPTER 7

tities in the tables and figures are dimensioned with the data of aluminum Al5086.Non-dimensioned transverse displacement w is considered at z = h/4 in Table 7.5. Inthis analysis, both LW and ESL models are considered because the shell is two-layered.The difference between LW and ESL models, and between the assumed and calculatedtemperature profile are clearly shown in Table 7.5.

The third case is a cylindrical shell with ten carbon fiber reinforced layers with lam-ination sequence 0/90. The proposed shell has dimensions a = b = 1. The radii ofcurvature in the α and β directions are 1

Rα= 0 and 1

Rβ= 0.2, 0.1, 0.02. The considered

total thicknesses is h = 0.1. The ratio between Young’s modulus in the longitudinaland transverse direction is EL/ET = 25.0. The shear modulus ratio is GLT /GTT = 2.5,Poisson’s ratio is νLT = νTT = 0.25. The ratio between the thermal expansion coefficientin the transverse and longitudinal direction is αT /αL = 3.0. The conductivity coeffi-cients are KL = 36.42W/mC in the longitudinal direction and KT = 0.96W/mC inthe transverse direction. Each layer has a thickness h/10. The results for the transversedisplacement w are reported in Table 7.6. The assumed and calculated temperatureprofile are never the same because of the 10 layers.

For all the cases proposed in this section, the temperature amplitude imposed onthe top surface is +0.5, and −0.5 on the bottom. In the case of assumed temperatureprofile (Ta), it is always considered linear from the top to the bottom as indicated inFigure 7.3. In the case of calculated temperature profile (Tc), it is obtained by solvingFourier’s heat conduction equation, as described in Section 5.2.2. The thus-calculatedtemperature profile is always approximated in Layer Wise form, even though the em-ployed theory for the displacements is in Equivalent Single Layer form. The orderof expansion used to approximate the temperature profile is the same as that of thedisplacements. The temperature profile in the plane (α, β) is always considered bi-sinusoidal with wave numbers m = n = 1.

In the case of a one-layered shell, when the structure is very thin, the assumed pro-file Ta coincides with the calculated Tc one, therefore a thin shell means a linear tem-perature profile through the thickness. For thick shells, the difference between profilesTa and Tc is much more evident, as illustrated in Figure 7.4 which clearly explains theresults of Table 7.4; there are two sources of error for the thermo-mechanical analysisof one-layered homogeneous shells: - the employed kinematic model for the displace-ment; - the considered temperature profile. For thick shells, the temperature profileis not linear and it must be calculated by Fourier’s equation and then approximatedwith a higher order of expansion (N=4). The transverse displacement w through thethickness z is given in Figure 7.7 for the case of a thick and thin shell, respectively. Fig-ure 7.7 clearly shows two main aspects for the thermo-mechanical analysis of shells:- assumed vs calculated temperature profile; - the importance of a higher-order of ex-pansion for the unknowns.

In the case of a two-layered shell, even though a thin shell is considered, there is adifference between the results obtained with Ta and those obtained with Tc. This factis clearly explained in Figure 7.5: the calculated temperature profile for thin shells islinear in each layer, but its slope changes because the layer in Al5086 has a differentconductivity coefficient K from the K of the layer in Ti22. A different K means differ-

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THERMO-MECHANICAL ANALYSIS 199

Figure 7.3: Linear assumed profile of temperature through the thickness (Ta) of theone-layered, two-layered and ten-layered cylindrical shell.

ent slopes because of the continuity of transverse normal heat flux qz: the Section 5.2.2clearly explains why the temperature profile is linear for each layer when the shell isthin but with different slopes. The in-plane stress σxy through the thickness z, is givenin Figure 7.8 for both thick and thin shells. Because of what is explained in Figure 7.5,the differences between an assumed and a calculated temperature profile, even thoughthe shell is thin, are confirmed.

In the case of the ten-layered composite shell, the difference between the Ta andTc profiles is clearly reported in Figure 7.6 which explains Table 7.6: the importanceof LW theories, higher orders of expansion and calculated temperature profile Tc forthe analysis of multilayered composite shells. The solution added in the Table 7.6,called HOST12, is an ESL model with cubic expansion in z direction for the three dis-placement components (similar to the ED3 model obtained by CUF). In [166] for theHOST12 only an assumed temperature profile has been employed. The in-plane stressσxy through z is given in Figure 7.9 for thin and very thin shells. The same conclusionsobtained for displacements are confirmed in case of stresses.

Further results and considerations about such problems can be found in [220].

7.3 Thermal analysis of functionally graded material plates

In this section the deformations of a simply supported, functionally graded, rectan-gular plate subjected to thermo-mechanical loadings are analysed. Displacements ac-

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200 CHAPTER 7

w σxy

Rβ/h 10 100 1000 10 100 1000ED4(Ta) 0.9468 1.2007 0.1151 0.3209 ∗ 104 0.1916 ∗ 105 0.1822 ∗ 105

ED4(Tc) 0.9062 1.1987 0.1151 0.2847 ∗ 104 0.1913 ∗ 105 0.1822 ∗ 105

ED3(Ta) 0.9602 1.2007 0.1151 0.3270 ∗ 104 0.1916 ∗ 105 0.1822 ∗ 105

ED3(Tc) 0.9415 1.9999 0.1151 0.1379 ∗ 105 0.1928 ∗ 105 0.1822 ∗ 105

ED2(Ta) 0.8398 1.1994 0.1151 0.2976 ∗ 104 0.1914 ∗ 105 0.1822 ∗ 105

ED2(Tc) 0.8398 1.1994 0.1151 0.2972 ∗ 104 0.1914 ∗ 105 0.1822 ∗ 105

ED1(Ta) 1.9784 1.8359 0.2189 0.4977 ∗ 104 0.2244 ∗ 105 0.2642 ∗ 105

ED1(Tc) 1.9784 1.8359 0.2189 0.4977 ∗ 104 0.2244 ∗ 105 0.2642 ∗ 105

FSDT (Ta) 1.9818 1.7943 0.1715 0.4086 ∗ 104 0.2846 ∗ 105 0.2712 ∗ 105

FSDT (Tc) 1.9818 1.7943 0.1715 0.4086 ∗ 104 0.2846 ∗ 105 0.2712 ∗ 105

CLT (Ta) 1.9869 1.7985 0.1716 0.4087 ∗ 104 0.2852 ∗ 105 0.2713 ∗ 105

CLT (Tc) 1.9869 1.7985 0.1716 0.4087 ∗ 104 0.2852 ∗ 105 0.2713 ∗ 105

Table 7.4: Case 1. Isotropic one-layered shell. Non-dimensional transverse displace-ment w = 10uzh

a2αT1and in-plane stress σxy in z = 0. T1 = 1.0 is the gradient of the linear

temperature profile.

wRβ/h 4 10 100 1000LD4(Ta) 0.4002 0.7472 0.7468 0.0325LD4(Tc) 0.3977 0.6385 0.7952 -0.0530LD3(Ta) 0.4242 0.7487 0.7468 0.0325LD3(Tc) 0.2982 0.6354 0.7951 -0.0530ED4(Ta) 0.4068 0.7416 0.7468 0.0325ED4(Tc) 0.3075 0.6127 0.7947 −0.0533ED2(Ta) -0.3183 0.5089 0.7465 0.0326ED2(Tc) -0.2408 0.4300 0.7930 −0.0535FSDT (Ta) 1.2351 1.2694 1.1054 0.0463FSDT (Tc) 1.2418 1.3566 1.1884 −0.0797CLT (Ta) 1.2908 1.2914 1.1096 0.0463CLT (Tc) 1.3017 1.3956 1.1966 −0.0797

Table 7.5: Case 2. Two-layered isotropic shell in aluminium and titanium. Non-dimensional transverse displacement w = 10uzh

a2αAlT1in z = h/4. Equivalent Single Layer

vs Layer Wise Theories. T1 = 1.0 is the gradient of the linear temperature profile.

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THERMO-MECHANICAL ANALYSIS 201

wRβ/h 50 100 500HOST12(Ta)[166] 1.0224 1.0299 1.0325LD4(Ta) 1.0207 1.0283 1.0306LD4(Tc) 0.9643 0.9715 0.9737LD2(Ta) 1.0207 1.0283 1.0306LD2(Tc) 0.9613 0.9684 0.9706ED4(Ta) 1.0208 1.0279 1.0301ED4(Tc) 0.9642 0.9709 0.9730ED3(Ta) 1.0208 1.0279 1.0301ED3(Tc) 0.9640 0.9707 0.9728FSDT (Ta) 1.0468 1.0533 1.0551FSDT (Tc) 0.9872 0.9933 0.9951CLT (Ta) 1.0496 1.0540 1.0552CLT (Tc) 0.9898 0.9940 0.9951

Table 7.6: Case 3. Ten-layered carbon fiber reinforced cylindrical shell (laminationsequence: 0/90). Non-dimensional transverse displacement w = uz

b2αLT1in z = 0.

Equivalent Single Layer vs Layer Wise theories. T1 = 1.0 is the gradient of the lineartemperature profile.

-1

-0.5

0

0.5

1

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

Figure 7.4: Case 1. Assumed linear temperature (Ta) vs calculated temperature profile(Tc) for one-layered isotropic cylindrical shell in case of Rβ/h = 4, 100. N is the orderof expansion employed to approximate the temperature profile.

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202 CHAPTER 7

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.004

-0.002

0

0.002

0.004

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

Figure 7.5: Case 2. Assumed linear temperature (Ta) vs calculated temperature profile(Tc) for two-layered isotropic cylindrical shell in case of Rβ/h = 10, 1000. N is the orderof expansion employed to approximate the temperature profile.

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

z

T

Ta(N=1)Tc(N=1)Tc(N=4)

Figure 7.6: Case 3. Assumed linear temperature (Ta) vs calculated temperature pro-file (Tc) for ten-layered composite fiber reinforced cylindrical shell in case of Rβ/h =50, 500. N is the order of expansion employed to approximate the temperature profile.

Figure 7.7: Case 1. Isotropic one-layered cylindrical shell. Non-dimensional transversedisplacement w through z for Rβ/h = 10 (left) and Rβ/h = 1000 (right). Assumedtemperature profile (Ta) vs calculated temperature profile (Tc) for refined models.

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THERMO-MECHANICAL ANALYSIS 203

-0.4

-0.2

0

0.2

0.4

-100000 -50000 0 50000 100000

z

σxy

ED4(Ta)ED4(Tc)LD4(Ta)LD4(Tc)

-0.04

-0.02

0

0.02

0.04

-60000 -40000 -20000 0 20000 40000 60000 80000

z

σxy

ED4(Ta)ED4(Tc)LD4(Ta)LD4(Tc)

Figure 7.8: Case 2. Isotropic two-layered cylindrical shell. In-plane stress σxy throughz for Rβ/h = 10 (left) and Rβ/h = 100 (right). Assumed temperature profile (Ta) vscalculated temperature profile (Tc) for refined models.

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4 0.6

z

σxy

ED4(Ta)ED4(Tc)LD4(Ta)LD4(Tc)

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4 0.6

z

σxy

ED4(Ta)ED4(Tc)LD4(Ta)LD4(Tc)

Figure 7.9: Case 3. Ten-layered carbon fiber reinforced cylindrical shell. In-plane stressσxy through z for Rβ/h = 50 (left) and Rβ/h = 500 (right). Assumed temperatureprofile (Ta) vs calculated temperature profile (Tc) for refined models.

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204 CHAPTER 7

counting for the temperature as an external loading only. Because of the functionallygraded mechanical and thermal properties in the thickness direction, the required tem-perature field is not assumed a priori because it is never linear even if the plate is thin.The temperature profile is determined separately by solving Fourier’s equation as in-dicated in Section 5.2.2. Numerical results for temperature, displacement and stressdistributions are provided for different volume fractions of the metallic and ceramicconstituent as well as for different plate thickness ratios. They correlate very well withthree-dimensional solutions given in the literature [146]. For further results, omittedin this section, readers can refer to [43] and [221].

The proposed assessment considers only one FGM layer, with two different load-ing conditions: a pure mechanical or thermal load, both applied at the top of the plate.In case of only one layer, there is no difference between ESL or LW models. The pro-posed kinematics used for the description of the displacement u and the temperatureT along the thickness direction have the same order of expansion (N = 1, . . . , 14). Theproposed model is able to describe multilayered FGM/classical structures too. Otherresults about the thermo-mechanical analysis of sandwich plates with an FGM core canbe found in [221].A rectangular plate comprising a single functionally graded layer, as shown in Figure7.10, is analysed. As a typical example for high-temperature applications, the con-stituent materials of the functionally graded plate are taken to be Monel (70Ni-30Cu),a nickel-based alloy, and the ceramic zirconia (ZrO2). The required material propertiesare those reported in [146] where the 3D exact solutions can be also found. We havefor Monel and Zirconia, respectively:

Bm = 227.24 GPa, Gm = 65.55 GPa, αm = 15× 10−6 / K, Km = 25 W/mK,

Bc = 125.83 GPa, Gc = 58.08 GPa, αc = 10× 10−6 / K, Kc = 2.09 W/mK.(7.1)

For this two-phase composite material, different micromechanical models can be ap-plied for the computation of the effective local material properties. According to [146],we use the following formulas:

• The effective bulk modulus B and shear modulus G are given by the mean fieldestimate of Mori and Tanaka [60]:

B −Bm

Bc −Bm

=V2

1 + (1− V2)Bc−Bm

Bm+ 43µm

, (7.2)

G−Gm

Gc −Gm

=V2

1 + (1− V2)Gc−Gm

Gm+f1

with f1 =Gm(9Bm + 8Gm)

6(Bm + 2Gm). (7.3)

• The effective heat conduction coefficient K is given by the model of Hatta andTaya [222]:

K −Km

Kc −Km

=V2

1 + (1− V2)Kc−Km

3Km

. (7.4)

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THERMO-MECHANICAL ANALYSIS 205

• For the coefficient of thermal expansion α a correspondence relation holds [223],[224], reading:

α− αm

αc − αm

=1B− 1

Bm

1Bc− 1

Bm

. (7.5)

In Eqs.(7.2)-(7.5), the indices m and c refer to the metallic and ceramic phase, respec-tively. V2 is the volume fraction of the ceramic phase that is assumed for the computa-tions as:

V2 = Vc = (z/h)ng , (7.6)

where by changing the exponent ng different material gradients can be accomplished.Figure 7.11 shows the through-thickness distribution of the volume fraction Vc of theceramic phase and the resulting evolution of the bulk modulus. At the top surface, the

Figure 7.10: Considered plate for the numerical assessment: a single-layered, function-ally graded plate subjected to pure thermal or pure mechanical bi-sinusoidal loads.

plate is subjected to pure mechanical or pure thermal, transverse bi-sinusoidal loads,see Figure 7.10, reading:

p+z = p+

z sin(mπ x

a

)sin

(nπ y

b

), T+ = T+ sin

(m π x

a

)sin

(nπ y

b

). (7.7)

Here m, n are the wave numbers and a, b the plate dimensions, respectively. A quan-tity with a superimposed hat denotes the amplitude of the respective load. Since wepropose a linear theory, more complicated load cases can be accomplished by super-imposing the pure thermal and mechanical contributions.

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206 CHAPTER 7

-0.5

-0.25

0

0.25

0.5

0 0.25 0.5 0.75 1

z / h

volume fraction

ng = 5

ng = 2

ng = 1

ng = 0.5

ng = 0.2

-0.5

-0.25

0

0.25

0.5

125 150 175 200 225

z / h

bulk modulus

ng = 5

ng = 2

ng = 1

ng = 0.5

ng = 0.2

Figure 7.11: Through-thickness distribution of the volume fraction Vc of the ceramicphase (left) and of the bulk modulus (right).

As we consider an analytical Navier-type solution, the plate is assumed to be sim-ply supported, i.e. the boundary conditions read:

v = w = 0 at x = 0, a ,

u = w = 0 at y = 0, b ,

T = 0 at x = 0, a and y = 0, b ,

(7.8)

which is fulfilled by the assumed harmonic in-plane displacement and temperaturefields. In addition, we assume m = n = 1 for the wave numbers.

As done in [146], non-dimensionalized quantities are introduced:

(u, v, w) =(u, v, w)

P a, σij =

σij

P B∗ , T =α∗ T

P, (7.9)

where either P = p+z /B∗ or P = α∗ T+ is taken for the applied load p+

z or for theapplied temperature T+ at the top, respectively. The scale factors are B∗ = 1 GPa andα∗ = 1 × 10−6. The indexes i and j can be x, y and z. The 3D solution calculated in[146], in case of thermal load is obtained using a calculated temperature profile (Tc).

Since in the models proposed in this section, the temperature is taken into accountas an external loading only, its through-thickness distribution must be given a priorias an input to the formulation. In the literature the temperature field is often assumedto vary linearly from top to bottom surface, see e.g. Bhaskar et al. [160]. In order tocompute the actual through-thickness distribution of the temperature for any givenmaterial gradient ng and plate dimensions, the method indicated in Section 5.2.2 hasbeen used. The boundary conditions at the top and bottom surface of the plate read,respectively:

T+ = 1 at z = +h

2, T− = 0 at z = −h

2. (7.10)

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THERMO-MECHANICAL ANALYSIS 207

Figure 7.12 depicts the through-thickness distribution of the non-dimensionalized tem-perature T for a plate thickness ratio of a / h = 10, a / b = 1 and different material gra-dients ng. For comparison reasons the temperature variation for a "classical" layer withconstant material properties is also shown. It can be clearly seen that the temperaturedistribution of a functionally graded layer differs significantly from that of a "classical"layer. Furthermore, it is obvious that the temperature distribution is strongly nonlin-ear which contradicts the assumption of a linear temperature variation often found inthe literature. Concerning the alteration of the temperature distribution due to differ-ent material gradients ng the influence of the material composition is considerable. In

-0.5

-0.25

0

0.25

0.5

0 0.25 0.5 0.75 1

z / h

T

ng = 0.5ng = 1ng = 2

const MAT

Figure 7.12: Through-thickness distribution of the temperature T.

Figures 7.13 and 7.14 the through-thickness distribution of the transverse deflection wis shown for different plate thickness ratios (a / h = 4, 10, 50) and material gradients(ng = 0.5, 1, 2), respectively, taking a / b = 1. It can be clearly seen from Figure 7.13 thatin the case of a thermal loading the transverse deflection varies considerably throughthe thickness. Therefore, the usual assumption of a constant through-thickness dis-tribution of w made by most lower order plate theories is not justified in the thermalcase. As it is seen from Figure 7.13, the influence of different material gradients ng issubstantial for a thermal loading. This is due to the combined effects of the varyingthermal field (and therefore the loading) as well as the altering mechanical properties.However, in the case of a pure mechanical loading, the influence of different materialgradients ng is less pronounced, see Figure 7.14. Furthermore, as can be seen fromthis figure, the variation of the transverse deflection w through the thickness is small.Therefore, in the mechanical case the assumption of a constant through-thickness dis-tribution can be valid in case of thin structures.

The results obtained by CUF for a pure thermal or a pure mechanical loading arecompared with three-dimensional solutions given by Reddy and Cheng in [146]. Ta-bles 7.7-7.10 provide results for the in-plane displacement u, transverse deflection wand stresses σij in non-dimensionalized form for different plate thickness ratios, tak-ing a / b = 1 and the exponential index ng = 2. It can be concluded that CUF yieldsvery accurate results (compared to 3D solutions) for both the mechanical and ther-

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208 CHAPTER 7

-0.5

-0.25

0

0.25

0.5

0 5 10 15 20 25 30

z / h

–uz

a / h = 4a / h = 10a / h = 50

-0.5

-0.25

0

0.25

0.5

5.5 5.75 6 6.25 6.5

z / h

–uz

ng = 0.5

ng = 1

ng = 2

Figure 7.13: Through-thickness distribution of the non-dimensionalized transverse de-flection w due to a thermal load for different plate thickness ratios (ng = 2) (left) andfor different material gradients (a/h = 10) (right).

-0.5

-0.25

0

0.25

0.5

-25 -20 -15 -10 -5 0 5

z / h

–uz

a / h = 4a / h = 10a / h = 50

-0.5

-0.25

0

0.25

0.5

-0.2 -0.175 -0.15

z / h

–uz

ng = 0.5

ng = 1

ng = 2

Figure 7.14: Through-thickness distribution of the non-dimensionalized transverse de-flection w due to a mechanical load for different plate thickness ratios (ng = 2) (left)and for different material gradients (a/h = 10) (right).

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THERMO-MECHANICAL ANALYSIS 209

mal case, and for very thick plates too. However, in order to capture all effects ofthe displacement and stress distributions, higher orders plate theories are necessary.By comparing the cases of thermal and mechanical loading, it is demonstrated that athermal load requires higher order thickness assumptions. This happens because theprofile of temperature is never linear in case of FGMs, so higher order of expansionsare required to correctly approximate such profiles.

a/h = 4 a/h = 103D LD1 LD3 LD5 3D LD1 LD3 LD5

w(t) -13.46 -11.23 -13.46 -13.46 -168.9 -126.2 -168.9 -168.9w(m) -13.70 -10.83 -13.71 -13.70 -170.7 -125.9 -170.7 -170.7w(b) -12.73 -10.43 -12.73 -12.73 -168.5 -125.5 -168.5 -168.5u(t) 4.021 2.932 4.020 4.022 26.17 19.29 26.17 26.17u(m) -0.08998 -0.05419 -0.09095 -0.08991 0.7108 0.7807 0.7109 0.7112u(b) -4.069 -3.041 -4.067 -4.069 -24.72 -17.73 -24.72 -24.72

Table 7.7: Mechanical load. Non-dimensional transverse displacement w × 103 andin-plane displacement u × 103 at top (t), middle (m) and bottom (b) of the consideredplate (ng = 2). 3D solution is reported in [146].

a/h = 4 a/h = 103D LD1 LD3 LD5 3D LD1 LD3 LD5

σxx(t) -3.154 -2.8473 -3.152 -3.154 -18.17 -16.96 -18.17 -18.17σxx(m) -0.2037 -0.3079 -0.2027 -0.2038 -0.8722 -1.498 -0.8738 -0.8726σxx(b) 3.631 3.633 3.633 3.632 22.06 22.88 22.06 22.06σxz(m) -0.9500 -0.6854 -0.9535 -0.9500 -2.396 -1.739 -2.398 -2.396σzz(m) -0.5130 -0.7165 -0.5110 -0.5131 -0.5142 -1.691 -0.5166 -0.5143

Table 7.8: Mechanical load. Non-dimensional stresses σij at top (t), middle (m) andbottom (b) of the considered plate (ng = 2). 3D solution is reported in [146].

7.4 Thermal analysis of functionally graded material shells

By considering the same material and thermal/mechanical loadings of the the Section7.3, now the shell geometry is investigated in order to analyze the effects of the cur-vature Rβ . The shell geometry is those presented by Ren [148]: a = 1m, b = π

3Rβ =

10.47197m, Rα = ∞, Rβ = 10m. No 3D exact solutions are available in the open lit-erature for this case, so a quasi-3D solution is given by dividing the considered one-layered FGM shell in 100 mathematical layers with constant properties and consideringfor each layer a LD4 model, this reference solution is here called Nml = 100. In [85] and

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210 CHAPTER 7

a/h = 4 a/h = 503D LD1(Tc) LD6(Tc) LD14(Tc) 3D LD1(Tc) LD6(Tc) LD14(Tc)

w(t) 3.043 5.452 3.074 3.043 28.53 55.61 28.81 28.54w(m) 2.143 4.453 2.170 2.144 28.45 55.52 28.74 28.46w(b) 1.901 3.456 1.928 1.901 28.43 55.44 28.71 28.44u(t) -1.681 -3.111 -1.694 -1.681 -1.703 -3.092 -1.714 -1.703u(m) -0.6822 -1.370 -0.6841 -0.6822 -0.8081 -1.348 -0.8101 -0.8080u(b) 0.08240 0.3710 0.08999 0.08266 0.08528 0.3695 0.09209 0.08553

Table 7.9: Thermal load. Non-dimensional transverse displacement w and in-planedisplacement u at top (t), middle (m) and bottom (b) of the considered plate (ng = 2).3D solution is reported in [146].

a/h = 4 a/h = 503D LD1(Tc) LD6(Tc) LD14(Tc) 3D LD1(Tc) LD6(Tc) LD14(Tc)

σxx(t) -1018 -241.2 -1015 -1018 -1003 -235.7 -1001 -1003σxx(m) -204.8 -989.9 -197.2 -204.7 -251.2 -980.1 -244.4 -251.2σxx(b) -73.53 884.6 -108.8 -74.03 -76.10 893.1 -107.4 -76.59σxz(m) 4.186 3.9833 5.036 4.203 0.3122 0.3292 0.3742 0.3135σzz(m) 6.217 -521.8 17.10 6.300 0.04067 -469.61 9.3567 0.1178

Table 7.10: Thermal load. Non-dimensional stresses σij at top (t), middle (m) andbottom (b) of the considered plate (ng = 2). 3D solution is reported in [146].

[86], it has been demonstrated that such discrete model gives a quasi-3D evaluation forthe static analysis of FGM structures. The displacements and stresses are normalizedas in the plate case of Section 7.3.

Tables 7.11 and 7.12 consider the displacements and stresses in case of mechanicalload applied at the top of the shell. The shell is one-layered, so there are no differencesbetween the ESL and LW models. A LD3 model gives a quasi-3D evaluation in case ofdisplacements for both thick and thin shells; in case of stresses a LD6 model is required.

In case of thermal load, the results in terms of displacements and stresses are givenin Tables 7.13 and 7.14. In case of FGMs, the temperature profile is never linear, soit must be calculated and then approximated with higher orders of expansion; thisfact explains why for thick and thin shells refined theories as the LD14 are requestedto obtain a quasi-3D evaluation of displacements and stresses through the thicknessdirection z.

In conclusion, the presence of the curvature does not introduce further considera-tions with respect to the plate case considered in Section 7.3.

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THERMO-MECHANICAL ANALYSIS 211

Rβ/h = 10 Rβ/h = 500Ref LD1 LD3 LD6 Ref LD1 LD3 LD6

w(t) 0.0039 0.0032 0.0040 0.0039 2.2199 2.0431 2.2199 2.2199w(m) 0.0021 0.0021 0.0022 0.0021 2.2212 2.0437 2.2211 2.2212w(b) 0.0013 0.0011 0.0013 0.0013 2.2214 2.0443 2.2214 2.2214u(t) -0.0012 -0.0004 -0.0011 -0.0012 -0.4870 -0.4580 -0.4870 -0.4870u(m) 0.0006 0.0003 0.0006 0.0006 0.2099 0.1832 0.2100 0.2100u(b) 0.0010 0.0011 0.0010 0.0010 0.9072 0.8244 0.9071 0.9072

Table 7.11: Mechanical load. Non-dimensional transverse displacement w and in-planedisplacement u at top (t), middle (m) and bottom (b) of the considered shell (ng = 2).Ref is the Nml = 100 solution.

Rβ/h = 10 Rβ/h = 500Ref LD1 LD3 LD6 Ref LD1 LD3 LD6

σββ(t) 0.6770 0.3703 0.7392 0.6758 441.03 479.99 437.16 440.23σββ(m) 0.1571 0.2135 0.1413 0.1566 381.70 374.26 381.94 381.69σββ(b) -0.2194 -0.2399 -0.3737 -0.2196 229.93 32.776 235.44 229.89σαz(m) 0.4440 0.3303 0.4513 0.4435 6.4016 5.3208 6.4139 6.4017σzz(m) 0.4486 0.4602 0.4404 0.4482 0.4494 38.780 0.9361 0.4526

Table 7.12: Mechanical load. Non-dimensional stresses σij at top (t), middle (m) andbottom (b) of the considered shell (ng = 2). Ref is the Nml = 100 solution.

Rβ/h = 50 Rβ/h = 1000Ref LD2(Tc) LD8(Tc) LD14(Tc) Ref LD2(Tc) LD8(Tc) LD14(Tc)

w(t) 7.1337 8.8684 7.1548 7.1361 43.590 48.034 43.653 43.600w(m) 6.4131 8.0312 6.4331 6.4153 43.553 47.990 43.617 43.563w(b) 6.1942 7.8766 6.2143 6.1964 43.554 47.997 43.618 43.564u(t) -3.5466 -4.1620 -3.5545 -3.5477 -1.7868 -1.8872 -1.7886 -1.7871u(m) -1.4532 -1.5217 -1.4547 -1.4535 -1.1021 -1.1326 -1.1029 -1.1023u(b) 0.4833 0.9074 0.4880 0.4837 -0.4178 -0.3785 -0.4176 -0.4178

Table 7.13: Thermal load. Non-dimensional transverse displacement w and in-planedisplacement u at top (t), middle (m) and bottom (b) of the considered shell (ng = 2).Ref is the Nml = 100 solution.

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212 CHAPTER 7

Rβ/h = 50 Rβ/h = 1000Ref LD2(Tc) LD8(Tc) LD14(Tc) Ref LD2(Tc) LD8(Tc) LD14(Tc)

σββ(t) -1481.4 -1409.8 -1470.8 -1470.4 -1170.2 -1098.6 -1159.3 -1159.2σββ(m) -422.78 -205.82 -419.74 -422.76 159.55 392.59 159.63 159.60σββ(b) 8.4708 -628.63 20.795 7.9878 991.05 554.54 986.16 990.98σαz(m) 26.448 -7.3846 26.664 26.459 -5.2242 -6.6837 -5.2415 -5.2262σzz(m) 5.0735 319.18 7.5271 5.1982 0.2428 259.60 -1.7681 0.3165

Table 7.14: Thermal load. Non-dimensional stresses σij at top (t), middle (m) andbottom (b) of the considered shell (ng = 2). Ref is the Nml = 100 solution.

7.5 Thermo-mechanical coupling in homogeneous plates

The thermal stress analysis of plates can be also investigated by considering a full cou-pling between the mechanical and thermal field. The PVD is extended to such prob-lems simply adding the internal thermal work; both displacements and temperatureare primary variables of this model which is called PVD-TM. In this section in order tovalidate this extension, three different benchmarks are proposed: - a simply supportedhomogeneous plate with a mechanical load applied on its top surface; - a simply sup-ported homogeneous plate with temperature values imposed at the top and bottomsurfaces; - a free vibrations problem for a simply supported homogeneous plate. Forthe three proposed cases the plate is homogeneous with only one isotropic layer, it issquare (a = b) and several thickness ratios a/h are considered. The material propertiesare reported in the Table 7.15.

a/h 5,10,50,100h[m] 1m 1n 1E[GPa] 73ν 0.3ρ[Kg/m3] 2800CH [J/KgK] 897α[1/K] 25× 10−6

K[W/mC] 130

Table 7.15: One-layered isotropic square plate for the full coupled thermo-mechanicalanalysis.

The first case considers the isotropic plate subjected to a mechanical load on itstop surface, it is bi-sinusoidal with waves number m = n = 1 and amplitude pz =

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THERMO-MECHANICAL ANALYSIS 213

−200000Pa. The displacements can be obtained considering only the mechanical field,and then they are compared with those obtained by considering the effects of the ther-mal field. Several two-dimensional models are given in the Table 7.16, if 2D modelsinclude the thermo-mechanical coupling, they are defined as (TM). In the Table 7.16

a/h 5 10 50 100

LD4(TM) w(0) -0.0570 -0.7988 -476.35 -7610.0LD4 w(0) -0.0574 -0.8061 -480.86 -7682.2LD4(TM) u(h/2) 0.0145 0.1185 14.930 119.47LD4 u(h/2) 0.0147 0.1197 15.072 120.60LD2(TM) w(0) -0.0557 -0.7936 -476.22 -7609.5LD2 w(0) -0.0561 -0.8009 -480.73 -7681.7LD2(TM) u(h/2) 0.0141 0.1177 14.926 119.46LD2 u(h/2) 0.0143 0.1189 15.068 120.59FSDT(TM) w(0) -0.0561 -0.7895 -471.78 -7537.7FSDT w(0) -0.0570 -0.8039 -480.80 -7682.0FSDT(TM) u(h/2) 0.0142 0.1172 14.788 118.34FSDT u(h/2) 0.0151 0.1206 15.076 120.61CLT(TM) w(0) -0.0471 -0.7534 -470.87 -7534.1CLT w(0) -0.0479 -0.7678 -479.90 -7678.4CLT(TM) u(h/2) 0.0148 0.1183 14.793 118.34CLT u(h/2) 0.0151 0.1206 15.076 120.61

Table 7.16: Isotropic plate with a mechanical load applied on its top surface. Displace-ments in [mm] for several 2D theories and thickness ratios.

the transverse displacement w in the middle of the plate and the in-plane displacementu at the top are investigated for several thickness ratios. The plate is one-layered, so nodifferences are exhibited between the ESL and LW models. Higher orders of expansionare mandatory only for thick plates, in case of thin plates lower orders of expansion canbe considered. The results without thermo-mechanical coupling give displacementsbigger than those which consider the effect of the thermal field. These differences areabout 0.5%-1.3% independently by the considered displacement component and by theinvestigated thickness ratio. These differences between the pure mechanical case andthe thermo-mechanical coupling are confirmed by the considerations given in the bookby Nowinski [64]. The effects of the thermal field on this bending problem are clarifiedin the Figures 7.15-7.18, where the thickness ratios a/h = 5, 10, 50, 100 are investigated.For each thickness ratio the displacement w through the thickness z is given for a LD4model for both cases of pure mechanical problem and thermo-mechanical coupling.It is confirmed that the thermal coupling (TM) gives a displacement smaller than thepure mechanical analysis, in fact as demonstrated by the figures on the right side, atemperature profile through the thickness is pointed out. These profiles have an in-crease of temperature for the part of plate that is in compression, and a decrease of

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214 CHAPTER 7

temperature for the portion of plate in extension. In the figures on the right side, it isremarked how an higher order of expansion is required to obtain a correct temperatureprofile (LD1(TM) vs. LD4(TM)).

-0.4

-0.2

0

0.2

0.4

-0.058 -0.0575 -0.057 -0.0565 -0.056 -0.0555 -0.055 -0.0545 -0.054

z

w

LD4(TM)LD4

-0.4

-0.2

0

0.2

0.4

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

z

T

LD4(TM)LD1(TM)

Figure 7.15: Thickness ratio a/h = 5. Isotropic square plate with a mechanical loadapplied on its top surface. Displacement w for the pure mechanical case and for thethermo-mechanical coupling (left). Temperature profile along z (right).

-0.4

-0.2

0

0.2

0.4

-0.81 -0.805 -0.8 -0.795 -0.79

z

w

LD4(TM)LD4

-0.4

-0.2

0

0.2

0.4

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

z

T

LD4(TM)LD1(TM)

Figure 7.16: Thickness ratio a/h = 10. Isotropic square plate with a mechanical loadapplied on its top surface. Displacement w for the pure mechanical case and for thethermo-mechanical coupling (left). Temperature profile along z (right).

The second case considers the same plate with a bi-sinusoidal temperature in theplane (m = n = 1) applied at the top and bottom surfaces. In the conducted anal-ysis a temperature on the top θt = 1.0 and one on the bottom θb = 0.0 are applied.If these values are imposed, in order to calculate the thermal load two methods canbe followed as seen in the previous sections: - by assuming a linear temperature pro-file through the thickness (Ta); - by calculating the temperature profile by means ofFourier’s conduction equation (Tc). The originality of this section is the application ofthe full thermo-mechanical analysis to this case, in order to consider the temperature

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THERMO-MECHANICAL ANALYSIS 215

-0.4

-0.2

0

0.2

0.4

-484 -482 -480 -478 -476 -474 -472 -470

z

w

LD4(TM)LD4

-0.4

-0.2

0

0.2

0.4

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

z

T

LD4(TM)LD1(TM)

Figure 7.17: Thickness ratio a/h = 50. Isotropic square plate with a mechanical loadapplied on its top surface. Displacement w for the pure mechanical case and for thethermo-mechanical coupling (left). Temperature profile along z (right).

-0.4

-0.2

0

0.2

0.4

-7700 -7680 -7660 -7640 -7620 -7600

z

w

LD4(TM)LD4

-0.4

-0.2

0

0.2

0.4

-3 -2 -1 0 1 2 3

z

T

LD4(TM)LD1(TM)

Figure 7.18: Thickness ratio a/h = 100. Isotropic square plate with a mechanical loadapplied on its top surface. Displacement w for the pure mechanical case and for thethermo-mechanical coupling (left). Temperature profile along z (right).

an independent variable as the displacement, in this way both displacements and tem-perature are given directly from the model without the necessity to determine a prioria temperature profile through the thickness. In the Table 7.17, for several thicknessratios and for the transverse and in-plane displacements, the results are compared forthe three cases of assumed temperature profile (Ta), calculated temperature profile (Tc)and temperature considered as a priori variable (TM ). For both thick and thin plates,the results obtained with LD4(Tc) and LD4(TM) are in agreement, when the plate ismoderately thin the results for the LD4(Ta) model are correct too. Obviously, the dif-ferences are very small because the plate is homogeneous and isotropic, but the resultsin the Table 7.17 are very useful to validate the full-coupled models. If lower ordersof expansion are considered, the three models coincide because the evaluation of thetemperature depends on the chosen order of expansion; for example it is always linearin the case of FSDT and CLT analysis. The coupling effect is much more evident in the

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216 CHAPTER 7

displacement u, as seen in particular for thick plates. The values compared in the Table7.17, are considered only in particular positions through the thickness h, further inter-esting information can be obtained from the Figures 7.19-7.23. For very thick plates(a/h = 2 and a/h = 5), the results for the assumed temperature profile are not correct,and the best evaluations for the displacement and the temperature are given by thecoupled case (TM): the temperature through the thickness z for a/h = 2 is correctlycalculated by using a LD4(TM) model. In case of moderately thick plates, the resultsfor the calculated temperature profile and the full-coupling are in agreement for bothdisplacements and temperature. It’s obvious that the three methods are completelycoincident for thin plates. However the advantage of using a (TM) model remains:the temperature profile is easily obtained without assuming or calculating a priori thetemperature. The full-coupled models will be very useful in future investigations, incase of multilayered structures and FGM layers.

a/h 2 5 10 50 100

LD4(Ta) w(0) 0.0057 0.0403 0.1638 4.1154 16.464LD4(Tc) w(0) 0.0054 0.0398 0.1633 4.1149 16.463LD4(TM) w(0) 0.0054 0.0398 0.1633 4.1134 16.463LD4(Ta) u(h/2) -0.0125 -0.0268 -0.0522 -0.2587 -0.5173LD4(Tc) u(h/2) -0.0119 -0.0266 -0.0521 -0.2587 -0.5173LD4(TM) u(h/2) -0.0103 -0.0258 -0.0517 -0.2586 -0.5172LD2(Ta) w(0) 0.0053 0.0398 0.1633 4.1149 16.463LD2(Tc) w(0) 0.0053 0.0398 0.1633 4.1149 16.463LD2(TM) w(0) 0.0053 0.0398 0.1633 4.1149 16.463LD2(Ta) u(h/2) -0.0120 -0.0267 -0.0521 -0.2587 -0.5173LD2(Tc) u(h/2) -0.0106 -0.0259 -0.0517 -0.2586 -0.5172LD2(TM) u(h/2) -0.0106 -0.0259 -0.0517 -0.2586 -0.5172FSDT(Ta) w(0) 0.0092 0.0577 0.2311 5.7785 23.114FSDT(Tc) w(0) 0.0092 0.0577 0.2311 5.7785 23.114FSDT(TM) w(0) 0.0092 0.0577 0.2311 5.7785 23.114FSDT(Ta) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261FSDT(Tc) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261FSDT(TM) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261CLT(Ta) w(0) 0.0092 0.0577 0.2311 5.7785 23.114CLT(Tc) w(0) 0.0092 0.0577 0.2311 5.7785 23.114CLT(TM) w(0) 0.0092 0.0577 0.2311 5.7785 23.114CLT(Ta) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261CLT(Tc) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261CLT(TM) u(h/2) -0.0145 -0.0363 -0.0726 -0.3630 -0.7261

Table 7.17: Isotropic plate with a temperature applied on its top and bottom surfaces.Displacements in [mm] for several 2D theories and thickness ratios.

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THERMO-MECHANICAL ANALYSIS 217

-0.4

-0.2

0

0.2

0.4

0 0.005 0.01 0.015 0.02

z

w

LD4(TM)LD4(Tc)LD4(Ta)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

T

LD4(TM)LD4(Tc)LD4(Ta)

Figure 7.19: Thickness ratio a/h = 2. Isotropic square plate with an imposed tempera-ture. Displacement w for assumed temperature profile, calculated temperature profileand for the thermo-mechanical coupling (left). The temperature profiles are reportedon the right side.

The third case is the free vibrations problem of the same considered plate (simplysupported and isotropic one-layered). By imposing the vibration mode in the in-planedirections (m = n = 1), the fundamental frequency is calculated by means of severaltwo-dimensional models and by considering or not the thermo-mechanical coupling.When the thermo-mechanical coupling is considered, two different configurations areinvestigated: a free vibrations problem where the temperature at the top and bottomsurfaces are imposed equal to the external temperature (imposed conditions), and afree vibrations problem where the temperature at the top and bottom surfaces of theplate is not imposed (free conditions). In the imposed conditions the values of θ arereported to the external room temperature, so they mean θt = θb = 0. In the free con-ditions the temperatures θt (at the top) and θb (at the bottom) can assume the valuesgiven by the model. The Table 7.18 considers the fundamental frequency in Hz form = n = 1 for several thickness ratios. If the model considers the thermo-mechanicalcoupling, higher values of frequency are obtained, in analogy with the static case wherelower values of displacements were calculated. The error in percentage is very closeto that exhibited for the static analysis of the first case (mechanical load applied at thetop) and to the considerations in the book by Nowinski [64]. A small difference canbe outlined for the cases of free and imposed conditions (models with *) in the caseof thermo-mechanical coupling (TM). The classical theories such as the FSDT and CLTdo not give the frequency for the imposed case because in such theories the degreesof freedom are not sufficient to impose the condition θt = θb = 0. The importance ofhigher orders of expansion is remarked for the thick plate. The obtained frequenciesare the eigenvalues of the considered system, for each obtained eigenvalue (in our casethe fundamental frequency), it is possible to obtain the relative eigenvectors which rep-resent the vibration mode through the thickness. The eigenvector goes from -1 to +1 be-cause they are normalized with respect to the maximum value, this is done because theeigenvectors represent only the qualitative behavior of the thickness vibration modes.

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218 CHAPTER 7

-0.4

-0.2

0

0.2

0.4

0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054

z

w

LD4(TM)LD4(Tc)LD4(Ta)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

T

LD4(TM)LD4(Tc)LD4(Ta)

Figure 7.20: Thickness ratio a/h = 5. Isotropic square plate with an imposed tempera-ture. Displacement w for assumed temperature profile, calculated temperature profileand for the thermo-mechanical coupling (left). The temperature profiles are reportedon the right side.

-0.4

-0.2

0

0.2

0.4

0.16 0.165 0.17 0.175 0.18

z

w

LD4(TM)LD4(Tc)LD4(Ta)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

T

LD4(TM)LD4(Tc)LD4(Ta)

Figure 7.21: Thickness ratio a/h = 10. Isotropic square plate with an imposed tempera-ture. Displacement w for assumed temperature profile, calculated temperature profileand for the thermo-mechanical coupling (left). The temperature profiles are reportedon the right side.

In Figures 7.24 and 7.25 the thickness vibration modes are plotted in terms of displace-ments and temperature for the thick (a/h = 5) and thin (a/h = 50) plate in case offree and imposed conditions, respectively. The thickness mode in term of temperatureis in according with the thickness mode given in terms of displacement componentsu, v, w: the thickness mode associated with the fundamental frequency is a bendingmode, so the mode in term of temperature gives a temperature greater than zero forthe compressed part and less than zero for the extended part. The employed modelis the LD4 theory which gives a quasi-3D evaluation. In the Figure 7.26 the thicknessvibration modes for the moderately thin plate (a/h=50) are given for the case of notimposed temperature at the bottom and top surfaces. By considering m = n = 1, theLD1(TM) model has six degrees of freedom: six frequencies and six vibration modes

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THERMO-MECHANICAL ANALYSIS 219

-0.4

-0.2

0

0.2

0.4

4.1 4.11 4.12 4.13 4.14 4.15

z

w

LD4(TM)LD4(Tc)LD4(Ta)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

T

LD4(TM)LD4(Tc)LD4(Ta)

Figure 7.22: Thickness ratio a/h = 50. Isotropic square plate with an imposed tempera-ture. Displacement w for assumed temperature profile, calculated temperature profileand for the thermo-mechanical coupling (left). The temperature profiles are reportedon the right side.

-0.4

-0.2

0

0.2

0.4

16.4 16.45 16.5 16.55 16.6

z

w

LD4(TM)LD4(Tc)LD4(Ta)

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

T

LD4(TM)LD4(Tc)LD4(Ta)

Figure 7.23: Thickness ratio a/h = 100. Isotropic square plate with an imposed temper-ature. Displacement w for assumed temperature profile, calculated temperature profileand for the thermo-mechanical coupling (left). The temperature profiles are reportedon the right side.

are reported in terms of displacements (right side) and temperature (left side).

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220 CHAPTER 7

a/h 5 10 50 100

LD4(TM) 173.10 47.158 1.9481 0.4875LD4(TM)* 172.88 47.094 1.9454 0.4869LD4 172.40 46.946 1.9390 0.4852LD2(TM) 174.86 47.306 1.9484 0.4876LD2(TM)* 174.15 47.093 1.9392 0.4853LD2 174.15 47.093 1.9392 0.4853FSDT(TM) 175.52 47.517 1.9577 0.4899FSDT 174.10 47.088 1.9392 0.4853CLT(TM) 189.87 48.607 1.9596 0.4900CLT 188.08 48.148 1.9411 0.4854

Table 7.18: Free vibrations problem of the isotropic plate. Fundamental frequency f inHz for several 2D theories and thickness ratios. Waves number m = n = 1. The fullcoupling problem is indicated with (TM)* in case of imposed conditions and (TM) incase of free conditions.

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

u

v

w

Figure 7.24: Free configuration of the isotropic plate. Fundamental frequency, thick-ness vibration modes in terms of displacements and temperature for a/h=5 on the topand a/h=50 on the bottom. The employed model is the LD4(TM).

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THERMO-MECHANICAL ANALYSIS 221

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

u

v

w

Figure 7.25: Imposed configuration of the isotropic plate. Fundamental frequency,thickness vibration modes in terms of displacements and temperature for a/h=5 onthe top and a/h=50 on the bottom. The employed model is the LD4(TM)*.

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222 CHAPTER 7

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f1=1.9577 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f1=1.9577 Hz

θ

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f2=44.783 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f2=44.783 Hz

θ

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f3=66.811 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f3=66.811 Hz

θ

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THERMO-MECHANICAL ANALYSIS 223

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f4=1748.09 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f4=1748.01 Hz

θ

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f5=3306.14 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f5=3306.14 Hz

θ

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f6=1746.42 Hz

u

v

w

-0.4

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

z

f6=1746.42 Hz

θ

Figure 7.26: Moderately thin a/h = 50 plate, free conditions. The employed model is aLD1(TM). Thickness vibration modes for the six frequencies in terms of displacementsand temperature.

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224 CHAPTER 7

7.6 Conclusions

The thermo-mechanical analysis of multilayered plates and shells has been investi-gated. The importance of refined models, in particular the layer wise ones, has beenremarked in the case of multilayered plates and shells. In order to define the thermalload, an appropriate temperature profile through the thickness must be recognized,this can be done by solving Fourier’s conduction equation. The possibility of assum-ing a linear temperature profile, is correct only if the structure is homogeneous andthin; for multilayered and/or FGM structures this simplification can introduce verylarge errors. An other possibility is to consider the temperature as a primary variableas the displacement, this fully-coupled problem gives satisfactory analysis as the casewhere Fourier’s equation is solved. The fully-coupled problem is also used to investi-gate the effects of the thermal field on the static (applied mechanical load) and dynamic(free vibrations problem) analysis of multilayered plates and shells.

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Chapter 8

Electro-mechanical analysis

In case of smart structures, obtained by the inclusion of some piezoelectric layers, the electro-mechanical analysis is a fundamental aspect in their design. In this chapter refined and ad-vanced models, obtained by CUF, are validated in case of multilayered smart structures, andapplied to investigate their physical behavior. The inclusion of some functionally graded piezo-electric layers is also discussed. The free vibrations problem is a fundamental topic because ofthe health monitoring and the vibration control, which are two typical applications of smartstructures. In the proposed cases both sensor and actuator configurations are analyzed. Theuse of advanced mixed models permits to obtain a priori the transverse normal electric displace-ment and the transverse shear/normal stresses, with the possibility of imposing their continuityat interfaces.

8.1 Static analysis of piezoelectric plates

A simply supported square plate is considered, it has four layers: two external in piezo-electric material (PZT-4) and two internal in carbon fiber reinforced material (Gr/EP).The two cases of sensor (a bi-sinusoidal distribution of mechanical pressure is applied)and actuator (a bi-sinusoidal distribution of electric voltage is applied) are consideredin Figure 8.1. The material properties are those in Table 8.1. The two external lay-ers in piezoelectric material have a thickness he = 0.1htot. The two internal layers areunidirectional composite with fiber orientation 0/90 and thickness hi = 0.4htot. Thethree-dimensional solutions are given in [142]. The mechanical load in the sensor casehas amplitude pz = 1.0, waves number m = n = 1, and it is applied at the top. The ac-tuator configuration has an electric potential applied at the top with amplitude Φ = 1.0and waves number m = n = 1.

For the sensor case, the results are given in Tables 8.2-8.5 and in Figure 8.2. In Ta-ble 8.2 the three-dimensional solution for the in-plane stress σxx is given by Heyligher[142] along the thickness coordinate z. At the interfaces (where the coordinate z/his given twice) the discontinuity of the in-plane stress is remarked. The use of layerwise models and higher orders of expansion permits to obtain the 3D solution. Nosignificative differences are shown between the PVD-EM, RMVT-EM and RMVT2-EMvariational statement; in-plane stresses are always obtained via post-processing. Ta-

225

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226 CHAPTER 8

Figure 8.1: Considered multilayered piezoelectric plates: sensor configuration (left)and actuator configuration (right).

PZT − 4 Gr/EPE1[GPa] 81.3 132.38E2[GPa] 81.3 10.756E3[GPa] 64.5 10.756ν12[−] 0.329 0.24ν13[−] 0.432 0.24ν23[−] 0.432 0.49G23[GPa] 25.6 3.606G13[GPa] 25.6 5.6537G12[GPa] 30.6 5.6537e15[C/m2] 12.72 0e24[C/m2] 12.72 0e31[C/m2] −5.20 0e32[C/m2] −5.20 0e33[C/m2] 15.08 0ε11/ε0[−] 1475 3.5ε22/ε0[−] 1475 3.0ε33/ε0[−] 1300 3.0ε11[pC/V m] 1.306× 104 −ε22[pC/V m] 1.306× 104 −ε33[pC/V m] 1.151× 104 −

Table 8.1: Electro-mechanical properties of the considered materials for the plates insensor and actuator configuration.

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ELECTRO-MECHANICAL ANALYSIS 227

ble 8.3 compares the 3D solution for the electric potential with several higher ordermodels. In each variational statement the electric potential Φ is a primary variable, soeach model (layer wise approach and higher orders of expansion) gives the 3D solu-tion and permits to obtain the electric potential continuous at the interfaces. In caseof the transverse normal stress σzz in Table 8.4, the 3D evaluation is obtained via layerwise approaches and higher orders of expansion, but the interlaminar continuity iscaptured only by using the RMVT-EM and RMVT2-EM variational statements. In caseof PVD models, the transverse shear/normal stresses are obtained via post-processingand they are discontinuous at the interfaces. Table 8.5 gives the three-dimensionalevaluation of the normal electric displacement Dz through the thickness z. The pos-sibility of obtaining the interlaminar continuity of this variable is only given by theRMVT2-EM variational statement where the Dz is calculated a priori. The other mod-els such as LD4(EM) and LM4(EM) give satisfactory values of the normal electric dis-placement but they consider it as discontinuous at the interfaces: it is calculated viapost-processing. In Figure 8.2, a thick plate (a/h = 4) in sensor configuration is consid-ered: the discontinuity of the in-plane stress at the interfaces is clearly indicated.

For the actuator case, the results are given in Tables 8.6, 8.7 and in Figure 8.3. Thesame conclusions given for the sensor plate are confirmed in case of actuator config-uration: in order to obtain the continuity in z direction at the interfaces, the relativequantity must be a primary variable and a priori modelled. Tables 8.6 and 8.7 considerthe electric potential and the in-plane displacement along z. By employing higher or-ders of expansion and layer wise approaches the three-dimensional solution by Hey-ligher [142] is obtained, the potential Φ and the displacement u are primary variablesin each considered variational statement. Figure 8.3 gives the electric potential for athick and a thin plate, respectively. In case of thin plates lower orders of expansion canbe employed, the ESL models give satisfactory results because in them the electric po-tential is given in LW form: they are considered ESL models because of the approachused for the displacement. Further results about this topic can be found in [79] and[80]. Further interesting considerations are given in [133] and [198].

Figure 8.2: Sensor case, a/h = 4. In-plane stress σxx vs. z on the left. Transverse normalstress σzz vs. z on the right.

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228 CHAPTER 8

z/h 3D LD4(EM) LM4(EM) LD3(EM) LM3(EM)0.500 6.5643 6.5642 6.5642 6.5649 6.56440.400 3.6408 3.6408 3.6407 3.6400 3.36810.400 2.8855 2.8858 2.8857 2.8890 2.88750.300 1.4499 1.4551 1.4551 1.4541 1.45450.200 0.2879 0.2880 0.2882 0.2862 0.28720.100 −0.7817 −0.7829 −0.7830 −0.7813 −0.78190.000 −1.9266 −1.9266 −1.9257 −1.9302 −1.93530.000 0.0991 0.0991 0.0990 0.1033 0.1089−0.100 −0.0149 −0.0150 −0.0150 −0.0166 −0.0154−0.200 −0.1280 −0.1281 −0.1280 −0.1280 −0.1290−0.300 −0.2426 −0.2427 −0.2427 −0.2411 −0.2418−0.400 −0.3616 −0.3617 −0.3617 −0.3658 −0.3639−0.400 −4.2348 −4.2349 −4.2348 −4.2342 −4.2335−0.500 −6.8658 −6.8658 −6.8658 −6.8664 −6.8663

Table 8.2: Through the thickness distribution of in-plane stress σxx. Comparison ofvarious approaches vs 3D solution by Heyligher [142]. a/h = 4; sensor case.

8.2 Static analysis of piezoelectric shells

In this section several shell geometries including piezoelectric layers are investigatedfor both sensor and actuator configurations. First, some assessments are proposedin order to validate the refined and advanced models, than a new benchmark is dis-cussed.

The first assessment is a Ren shell geometry [148] with three internal cross-ply lay-ers (90/0/90) and two external piezoelectric layers, the data for the considered ma-terials are given in Table 8.8. The thickness of each piezoelectric layer is 1/100 of thethickness of the composite layer. The shell is in sensor configuration: mechanical loadapplied at the top with amplitude pz = 1.0 and waves number m = n = 1, the elec-tric potential applied at the top and bottom of the shell is zero (Φt = Φb = 0). Thiscase was proposed by Chen et alii [140] which provide also a three-dimensional solu-tion. In Table 8.9 the refined and advanced models are compared with the 3D solution[140]: transverse and in-plane displacements valuated in the middle of the shell. Theresults in Table 8.9 demonstrate that the refined and advanced models (LW approachand higher orders) work properly in case of multilayered piezoelectric shells: they givethe 3D solution for each considered thickness ratio (thick and thin shells). In order toinvestigate the displacements, both PVD and the two extensions of the RMVT are ableto give the correct values.

The second assessment permits to understand better the advantages given by theRMVT models with respect to the PVD one. The considered shell is a Ren geom-etry [148] with only a piezoelectric layer. The considered material is a PVDF, its

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ELECTRO-MECHANICAL ANALYSIS 229

z/h 3D LD4(EM) LM4(EM) LD3(EM) LM3(EM)0.500 0.0000 0.0000 0.0000 0.0000 0.00000.400 0.0598 0.0598 0.0598 0.0598 0.05980.400 0.0598 0.0598 0.0598 0.0598 0.05980.300 0.0589 0.0590 0.0590 0.0590 0.05900.200 0.0589 0.0589 0.0589 0.0589 0.05890.100 0.0596 0.0596 0.0596 0.0596 0.05960.000 0.0611 0.0611 0.0611 0.0611 0.06110.000 0.0611 0.0611 0.0611 0.0611 0.0611−0.100 0.0634 0.0634 0.0634 0.0634 0.0634−0.200 0.0665 0.0666 0.0666 0.0666 0.0666−0.300 0.0706 0.0706 0.0706 0.0706 0.0706−0.400 0.0756 0.0756 0.0756 0.0756 0.0756−0.400 0.0756 0.0756 0.0756 0.0756 0.0756−0.500 0.0000 0.0000 0.0000 0.0000 0.0000

Table 8.3: Through the thickness distribution of electric potential Φ× 103. Comparisonof various approaches vs 3D solution by Heyligher [142]. a/h = 4; sensor case.

properties are given in the third column of Table 8.8. The loading conditions arethe same of the first assessment: the shell is in sensor configuration (pz = 1.0 withwaves number m = n = 1 at the top) and closed circuit conditions (Φt = Φb = 0).

In Table 8.10 the displacements at different locations through the thickness z areinvestigated. Comparisons between the 3D solution in [141] and the refined and ad-vanced 2D models are made. The displacements are primary variables in the PVD-EM,RMVT-EM and RMVT2-EM variational statements, so each model with higher ordersof expansion give the three-dimensional solution, even if the considered shell is thick(Rβ/h = 2, 4, 6, 10). The results for the PVD-EM statement are not given in Table 8.10for sake of brevity, but they are perfectly coincident with those for the RMVT-EM andRMVT2-EM. Thanks the results obtained in Table 8.10, in the next benchmark we canuse the LW models with higher orders of expansion as reference solutions. In Table8.11 other variables such as the stresses, the electric potential and the electric displace-ments are investigated. By considering higher orders of expansion, the 3D solutions byDumir et alii [141] are obtained. The importance of mixed models is clearly remarked,in fact the results obtained directly from the model are better than those obtained viapost-processing: the electric potential is a primary variable in PVD-EM, RMVT-EMand RMVT2-EM models; the transverse shear/normal stresses are primary variablesin RMVT-EM and RMVT2-EM models; the normal electric displacement is a primaryvariable only in the RMVT2-EM variational statement.

The two presented assessments have been very useful to demonstrate that refinedmodels give the three-dimensional solution for the electro-mechanical analysis of shells.In the new proposed benchmark we can use the LW models with fourth order of ex-

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230 CHAPTER 8

z/h 3D LD4(EM) LM4(EM) LM4(EM2) LD3(EM) LM3(EM)0.500 1.0000 1.0000 1.0000 1.0000 1.0013 1.00000.400 0.9515 0.9515 0.9516 0.9516 0.9502 0.95220.400 0.9515 0.9518 0.9516 0.9516 0.9611 0.95220.300 0.8520 0.8517 0.8517 0.8517 0.8479 0.85150.200 0.7375 0.7376 0.7375 0.7375 0.7376 0.73760.100 0.6169 0.6168 0.6169 0.6169 0.6207 0.61710.000 0.4983 0.4986 0.4984 0.4984 0.4888 0.49770.000 0.4983 0.4982 0.4984 0.4984 0.5067 0.4977−0.100 0.3805 0.3805 0.3804 0.3804 0.3771 0.3807−0.200 0.2614 0.2613 0.2614 0.2614 0.2613 0.2613−0.300 0.1482 0.1485 0.1484 0.1485 0.1518 0.1482−0.400 0.0487 0.0485 0.0487 0.0487 0.0403 0.0492−0.400 0.0487 0.0487 0.0487 0.0487 0.0499 0.0492−0.500 0.0000 0.0000 0.0000 0.0000 −0.0012 0.0000

Table 8.4: Through the thickness distribution of transverse normal stress σzz. Compar-ison of various approaches vs 3D solution by Heyligher [142]. a/h = 4; sensor case.

pansion in the thickness direction as reference solutions. We consider a cylindricalshell with the geometry given by Varadan and Bhaskar [149]: dimensions a = 40 andb = 2πRβ = 62.831853, radii of curvature Rα = ∞ and Rβ = 10, waves number m = 1and n = 8. A four-layers configuration is investigated: two external piezoelectric lay-ers with material properties indicated in the first column of the Table 8.1, and twointernal carbon fiber reinforced layers with sequence lamination 0/90 and materialproperties indicated in the second column of the Table 8.1. The total thickness of theshell is h, each piezoelectric layer has thickness he = 0.1h and each composite layerhas thickness hi = 0.4h. This benchmark has been added to the investigation becausethe first assessment considers a multilayer configuration but the piezoelectric layersare very thin, the second benchmark is a one-layered piezoelectric shell. Both actuatorand sensor configurations are treated. For the actuator case the distribution of electricpotential Φ applied at top surface is,

Φ(α, β) = Φsin(πα

a) sin(

8πβ

b) , (8.1)

with Φtop = 1, Φb = 0 and P z = 0. A mechanical transverse pressure is applied withcorrespondence to the bottom surface in the sensor case:

Pz(α, β) = P zsin(πα

a) sin(

8πβ

b) , (8.2)

with Φtop = Φb = 0, and P z = 1.Table 8.12 considers the results for the actuator configuration: for the electric poten-

tial and the transverse displacements there are no differences between the employed

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ELECTRO-MECHANICAL ANALYSIS 231

z/h 3D LD4(EM) LM4(EM) LM4(EM2) LM3(EM) LFM3(EM2)0.500 160.58 160.58 160.61 160.58 160.14 160.600.400 −0.3382 −0.3348 −0.4734 −0.3383 −4.1211 −0.33800.400 −0.3382 −0.3384 −0.3384 −0.3383 −0.3369 −0.33800.300 −0.1276 −0.1277 −0.1277 −0.1277 −0.1283 −0.12780.200 0.0813 0.0813 0.0813 0.0813 0.0813 0.08130.100 0.2913 0.2914 0.2914 0.2914 0.2920 0.29160.000 0.5052 0.5053 0.5053 0.5053 0.5038 0.50500.000 0.5052 0.5053 0.5053 0.5053 0.5070 0.5050−0.100 0.7259 0.7262 0.7262 0.7262 0.7256 0.7264−0.200 0.9563 0.9565 0.9565 0.9565 0.9565 0.9565−0.300 1.1995 1.2000 1.2000 1.2000 1.2007 1.1999−0.400 1.4587 1.4590 1.4590 1.4590 1.4573 1.4593−0.400 1.4587 1.4559 1.3599 1.4590 2.9753 1.4593−0.500 −142.46 −142.46 −142.43 −142.46 −142.69 −142.47

Table 8.5: Through the thickness distribution of normal electric displacementDz×1013.Comparison of various approaches vs 3D solution by Heyligher [142]. a/h = 4; sensorcase.

variational statements because in each model the electric potential and the displace-ments are primary variables. The use of higher order of expansion is mandatory forthe thick shells. For the transverse normal stress the correct results are given by bothmixed models (RMVT-EM and RMVT2-EM), the PVD-EM models do not give a satis-factory analysis because the transverse stresses are obtained via post-processing andthere is not the possibility to impose the interlaminar continuity. The transverse normalelectric displacement is obtained a priori only by the RMVT2-EM models, so these re-sults can be considered as reference solutions and the continuity of the normal electricdisplacement at interfaces can be easily imposed by means of the LMN(EM2) models.The same considerations obtained for the actuator configuration, can be confirmed forthe sensor case detailed investigated in the Table 8.13: importance of LW approaches,higher orders of expansion and mixed models to obtain a priori the transverse mechan-ical and/or electrical variables. Figure 8.4 gives the transverse normal stress throughthe thickness z in case of sensor (left) and actuator (right) configuration. The use ofmixed models (RMVT-EM and RMVT2-EM) permits to obtain a quasi-3D evaluation ofsuch variables along the thickness (the homogeneous loading conditions for the sensorand actuator cases are fully satisfied) and the interlaminar continuity at the interfaces.

For further results and considerations about these topics, omitted in this work forsake of brevity, readers can refer to [79] and [80].

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232 CHAPTER 8

z/h 3D LD4(EM) LM4(EM) LM4(EM2) LD3(EM) LM3(EM2)

0.500 1.0000 1.0000 1.0000 1.0000 1.0000 1.00000.400 0.9929 0.9929 0.9929 0.9929 0.9929 0.99290.400 0.9929 0.9929 0.9929 0.9929 0.9929 0.99290.300 0.8415 0.8418 0.8418 0.8418 0.8418 0.84180.200 0.7014 0.7015 0.7015 0.7015 0.7014 0.70140.100 0.5707 0.5709 0.5709 0.5709 0.5709 0.57090.000 0.4476 0.4477 0.4477 0.4477 0.4477 0.44760.000 0.4476 0.4477 0.4477 0.4477 0.4477 0.4476−0.100 0.3305 0.3307 0.3307 0.3307 0.3307 0.3307−0.200 0.2179 0.2179 0.2179 0.2179 0.2179 0.2179−0.300 0.1081 0.1082 0.1082 0.1082 0.1082 0.1082−0.400 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010−0.400 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010−0.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table 8.6: Through the thickness distribution of electric potential Φ. Comparison ofvarious approaches vs 3D solution by Heyligher [142]. a/h = 4; actuator case.

z/h 3D LD4(EM) LM4(EM) LD3(EM) LM3(EM)

0.500 −32.764 −32.765 −32.765 −32.766 −32.7660.400 4.7356 4.7352 4.7351 4.7350 4.73460.400 4.7356 4.7352 4.7351 4.7350 4.73460.300 2.9808 2.9902 2.9905 2.9915 2.99050.200 1.7346 1.7346 1.7348 1.7286 1.73040.100 0.8008 0.8045 0.8042 0.8056 0.80730.000 0.0295 0.0297 0.0306 0.0296 0.02240.000 0.0295 0.0297 0.0306 0.0296 0.0224−0.100 −0.4404 −0.4395 −0.4398 −0.4395 −0.4378−0.200 −0.8815 −0.8811 −0.8810 −0.8813 −0.8795−0.300 −1.3206 −1.3207 −1.3206 −1.3208 −1.3217−0.400 −1.7839 −1.7834 −1.7834 −1.7835 −1.7832−0.400 −1.7839 −1.7834 −1.7834 −1.7835 −1.7832−0.500 −2.8625 −2.8618 −2.8618 −2.8618 −2.8618

Table 8.7: Through the thickness distribution of displacement u× 1012. Comparison ofvarious approaches vs 3D solution by Heyligher [142]. a/h = 4; actuator case.

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ELECTRO-MECHANICAL ANALYSIS 233

Figure 8.3: Actuator case. Electric potential Φ vs. z for thickness ratio a/h = 4 (left) anda/h = 100 (right).

Piezo Composite PV DFE1[GPa] 2 172 2E2 = E3[GPa] 2 6.9 2ν12[−] 0.29 0.25 1/3ν13[−] 0.29 0.25 1/3ν23[−] 0.29 0.25 1/3G23[GPa] 0.7752 1.4 0.75G13[GPa] 0.7752 3.4 0.75G12[GPa] 0.7752 3.4 0.75e15[C/m2] 0 0 0e24[C/m2] 0 0 0e31[C/m2] 0.046 0 −0.0015e32[C/m2] 0 0 0.0285e33[C/m2] 0 0 −0.051ε11[pC/V m] 106.0 13060 106.2ε22[pC/V m] 106.0 13060 106.2ε33[pC/V m] 106.0 13060 106.2

Table 8.8: Electro-mechanical properties for the five-layered Ren shell proposed byChen et alii [140] and the one-layered PVDF Ren shell proposed by Dumir et alii [141].

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234 CHAPTER 8

Rβ/hc 2 4 10 50 100w(z = 0)

3D 1.440 0.459 0.144 0.0808 0.0785LM4(EM) 1.443 0.458 0.144 0.0810 0.0787LD4(EM) 1.443 0.458 0.144 0.0810 0.0787LM4(EM2) 1.443 0.458 0.144 0.0810 0.0787LM3(EM) 1.442 0.458 0.144 0.0810 0.0787LD3(EM) 1.442 0.458 0.144 0.0810 0.0787LM3(EM2) 1.442 0.458 0.144 0.0810 0.0787

v(z = 0)3D 5.294 1.549 0.480 0.269 0.262LM4(EM) 5.308 1.547 0.479 0.270 0.262LD4(EM) 5.305 1.547 0.479 0.270 0.262LM4(EM2) 5.308 1.549 0.479 0.270 0.262LM3(EM) 5.306 1.547 0.479 0.270 0.262LD3(EM) 5.299 1.547 0.479 0.270 0.262LM3(EM2) 5.306 1.547 0.479 0.270 0.262

Table 8.9: Mechanical displacements for a five layered piezo-mechanic Ren shell insensor configuration. Comparison with 3D exact solution [140]. w = 10 E3 w hc

3

Pz Rβ4 and

v = 100 E3 v hc3

Pz Rβ4 . hc is the thickness of the composite.

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ELECTRO-MECHANICAL ANALYSIS 235

Rβ/h 2 4 6 10 20 100 500w(z = −h/2)

3D −28.65 −20.55 −18.72 −17.60 −16.96 −16.55 −16.48LM4(EM) −28.63 −20.55 −18.72 −17.60 −16.95 −16.55 −16.48LM3(EM) −28.46 −20.54 −18.72 −17.60 −16.95 −16.55 −16.48LM4(EM2) −28.63 −20.55 −18.72 −17.60 −16.95 −16.55 −16.48LM3(EM2) −28.46 −20.54 −18.72 −17.60 −16.95 −16.55 −16.48

w(z = 0)3D −31.47 −21.10 −18.96 −17.68 −16.98 −16.55 −16.48LM4(EM) −31.45 −21.10 −18.96 −17.68 −16.98 −16.55 −16.48LM3(EM) −31.41 −21.10 −18.96 −17.68 −16.98 −16.55 −16.48LM4(EM2) −31.45 −21.10 −18.96 −17.68 −16.98 −16.55 −16.48LM3(EM2) −31.41 −21.10 −18.96 −17.68 −16.98 −16.55 −16.48

w(z = h/2)3D −31.31 −20.65 −18.73 −17.60 −16.95 −16.55 −16.48LM4(EM) −31.30 −20.65 −18.73 −17.60 −16.95 −16.55 −16.48LM3(EM) −31.14 −20.64 −18.73 −17.60 −16.95 −16.55 −16.48LM4(EM2) −31.30 −20.65 −18.73 −17.60 −16.95 −16.55 −16.48LM3(EM2) −31.14 −20.64 −18.73 −17.60 −16.95 −16.55 −16.48

v(z = −h/2)3D −23.61 −13.09 −10.23 −8.177 −6.778 −5.738 −5.539LM4(EM) −23.59 −13.09 −10.23 −8.177 −6.778 −5.738 −5.539LM3(EM) −23.38 −13.08 −10.22 −8.177 −6.778 −5.738 −5.539LM4(EM2) −23.59 −13.09 −10.22 −8.177 −6.778 −5.738 −5.539LM3(EM2) −23.38 −13.08 −10.22 −8.177 −6.778 −5.738 −5.539

v(z = h/2)3D 2.046 −0.8806 −2.331 −3.572 −4.528 −5.297 −5.451LM4(EM) 2.045 −0.8806 −2.331 −3.572 −4.528 −5.297 −5.451LM3(EM) 2.029 −0.8799 −2.330 −3.572 −4.527 −5.297 −5.451LM4(EM2) 2.045 −0.8806 −2.331 −3.572 −4.527 −5.297 −5.451LM3(EM2) 2.029 −0.8799 −2.330 −3.572 −4.527 −5.297 −5.451

Table 8.10: One layered piezoelectric Ren shell in sensor configuration. Comparisonwith 3D exact solutions [141] for the displacements.

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236 CHAPTER 8

Rβ/h 2 4 6 10 20 100 500σzz(z = 0)

3D −0.2906 0.2170 0.6356 1.420 3.319 18.34 93.34LM4(EM) −0.2359 0.2453 0.6541 1.430 3.325 18.34 93.34LM3(EM) −0.2922 0.2181 0.6365 1.420 3.320 18.34 93.34LM4(EM2) −0.2359 0.2453 0.6541 1.430 3.325 18.34 93.34LM3(EM2) −0.2922 0.2181 0.6365 1.420 3.320 18.34 93.34

σβz(z = 0)3D −0.6653 −0.6238 −0.6055 −0.5893 −0.5762 −0.5653 −0.5631LM4(EM) −0.6742 −0.6261 −0.6065 −0.5896 −0.5763 −0.5653 −0.5631LM3(EM) −0.6706 −0.6252 −0.6061 −0.5895 −0.5763 −0.5653 −0.5631LM4(EM2) −0.6742 −0.6261 −0.6065 −0.5896 −0.5763 −0.5653 −0.5631LM3(EM2) −0.6706 −0.6252 −0.6061 −0.5895 −0.5763 −0.5653 −0.5631

103Φ(z = 0)3D 1.734 2.443 2.541 2.560 2.540 2.504 2.494LM4(EM) 1.729 2.442 2.540 2.560 2.540 2.504 2.494LM3(EM) 1.847 2.477 2.556 2.565 2.541 2.504 2.494LM4(EM2) 1.729 2.442 2.540 2.560 2.540 2.504 2.494LM3(EM2) 1.847 2.477 2.556 2.565 2.541 2.504 2.494

10Dz(z = −h/2)3D 5.908 2.550 1.532 0.7830 0.2659 −0.1171 −0.1902LM4(EM2) 6.381 2.670 1.584 0.8008 0.2702 −0.1169 −0.1902LM3(EM2) 7.326 3.299 2.017 1.0622 0.4003 −0.0911 −0.1851

10Dz(z = h/2)3D 0.8480 0.0213 −0.0996 −0.1575 −0.1860 −0.2041 −0.2075LM4(EM2) 1.1484 0.1161 −0.0557 −0.1413 −0.1820 −0.2039 −0.2075LM3(EM2) −0.1072 −0.5774 −0.5155 −0.4121 −0.3143 −0.2298 −0.2126

10Dβ(z = 0)3D −0.3070 −0.4324 −0.4497 −0.4531 −0.4496 −0.4431 −0.4415LM4(EM2) −0.3060 −0.4323 −0.4497 −0.4531 −0.4496 −0.4431 −0.4414LM3(EM2) −0.3270 −0.4384 −0.4524 −0.4541 −0.4498 −0.4431 −0.4414

Table 8.11: One layered piezoelectric Ren shell in sensor configuration. Comparisonwith 3D exact solutions [141] for the stresses and electric variables.

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ELECTRO-MECHANICAL ANALYSIS 237

Figure 8.4: Proposed benchmark: multilayered piezoelectric shell. Sensor case (topleft) and actuator case (top right). σzz vs z. Rβ

h= 4. Zooms for sensor case (left) and

actuator case (right) are reported at bottom.

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238 CHAPTER 8

Rβ/h 2 4 10 100Φ (z = 0)

LD4(EM) 0.4064 0.4829 0.5029 0.5009LM1(EM) 0.4014 0.4824 0.5029 0.5009LM4(EM) 0.4064 0.4829 0.5029 0.5009LM1(EM2) 0.3805 0.4784 0.5041 0.5012LM4(EM2) 0.4064 0.4829 0.5029 0.5009

Dz109 (z = h/2)LD4(EM) −1.0754 −0.6666 −0.3322 −0.3494LM1(EM) −0.6686 −0.4157 −0.2172 −0.3373LM4(EM) −1.0754 −0.6666 −0.3322 −0.3494LM1(EM2) −1.0844 −0.6674 −0.3285 −0.3623LM4(EM2) −1.0654 −0.6603 −0.3269 −0.3622

w1011 (z = 0)LD4(EM) −1.1542 −1.0208 −1.0048 2.4869LM1(EM) −1.2671 −1.0582 −1.0290 2.4730LM4(EM) −1.1542 −1.0208 −1.0048 2.4869LM1(EM2) −1.2320 −1.0342 −1.0112 2.4838LM4(EM2) −1.1542 −1.0208 −1.0048 2.4869

σzz (z = h/2)LD4(EM) 0.1416 0.0902 0.0757 −0.1835LM1(EM) −0.0062 0.0021 0.0023 0.0006LM4(EM) 0.0000 0.0000 0.0000 0.0000LM1(EM2) 0.0055 0.0074 0.0043 0.0008LM4(EM2) 0.0000 0.0000 0.0000 0.0000

Table 8.12: Proposed benchmark: multilayered piezoelectric Varadan and Bhaskarcylindrical shell. Comparison of various approaches. Actuator case.

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ELECTRO-MECHANICAL ANALYSIS 239

Rβ/h 2 4 10 100

Φ (z = 0)LD4(EM) 0.0039 0.0157 0.0485 0.3414LM1(EM) 0.0036 0.0154 0.0484 0.3415LM4(EM) 0.0039 0.0157 0.0485 0.3414LM1(EM2) 0.0013 0.0133 0.0458 0.3287LM4(EM2) 0.0039 0.0156 0.0485 0.3414

w109 (z = 0)LD4(EM) 0.2633 0.9437 7.9334 4403.2LM1(EM) 0.2548 0.9386 7.9314 4404.3LM4(EM) 0.2633 0.9437 7.9334 4403.2LM1(EM2) 0.2539 0.9377 7.9271 4403.0LM4(EM2) 0.2633 0.9437 7.9334 4403.2

σzz (z = −h/2)LD4(EM) −2.2444 −6.0302 −52.912 −32549LM1(EM) −1.1772 −1.0658 −0.8449 11.453LM4(EM) −1.0000 −1.0000 −1.0000 −1.0000LM1(EM2) −1.1791 −1.0667 −0.8440 11.512LM4(EM2) −1.0000 −1.0000 −1.0000 −1.0000

Table 8.13: Proposed benchmark: multilayered piezoelectric Varadan and Bhaskarcylindrical shell. Comparison of various approaches. Sensor case.

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240 CHAPTER 8

8.3 Vibrations analysis of piezoelectric plates and shells

The free vibrations problem of multilayered plates and shells including some piezo-electric layers are here investigated. By imposing the waves number in the plane, acertain number of vibration modes through the thickness direction are obtained. Thenumber of frequencies is equal to the number of degrees of freedom through the thick-ness of the employed 2D model. This discussion is valid in case of closed form so-lution and 2D models, actually in the real case the number of vibration modes is ∞3.Two cases are investigated in this section: a plate geometry and a shell geometry; foreach case, by imposing the waves number the so-called fundamental frequencies (thatmeans the lowest frequencies for the given waves number m,n) are given.

The first case is a five-layered plate, the two external layers are in piezoelectric withthickness h1 = h5 = h

10, the three internal ones are carbon fiber reinforced layers with

sequence lamination 0/90/0 and thickness h2 = h3 = h4 = 415

h. The propertiesfor these two materials are given in the Table 8.1, the three-dimensional solution wasproposed by Heyliger and Saravanos [178]. The considered plate is square (a = b) inclosed circuit configuration (electric potential applied at the top and bottom equal tozero) as indicated in Figure 8.5. In order to obtain the reference solution, Heyliger andSaravanos [178] have employed for both materials a mass density ρ = 1Kg/m3, thisdoes not have a physic sense but it is however acceptable for the mathematical pointof view. The results given in the Table 8.14 are given as the first three fundamental

Figure 8.5: Considered multilayered piezoelectric plate for the free vibrations problem:closed circuit configuration.

circular frequencies ω = 2πf for waves number m = n = 1. Two thickness ratios areinvestigated: thick plate (a/h = 4) and moderately thin plate (a/h = 50). For the thickplate case the 3D solution [178] is obtained by using LW models and higher orders ofexpansion. The use of ESL models, even if higher orders are considered, gives an errorlarger than 2%. Classical theories such as the FSDT and CLT are completely inappro-priate for such cases. In order to obtain the frequencies (which are the eigenvalues

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ELECTRO-MECHANICAL ANALYSIS 241

a/h = 4 a/h = 50Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3

3D 57074.5 191301 250769 618.118 15681.6 21492.8

LM1(EM2) 57094.0 194697 253958 618.143 15683.3 21493.9Err(%) (−0.03) (−1.77) (−1.27) (0.00) (−0.01) (−0.01)LM4(EM2) 57074.0 191301 250768 618.106 15681.6 21492.6Err(%) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)LM1(EM) 57056.6 194696 253955 617.996 15683.3 21493.9Err(%) (0.03) (−1.77) (−1.27) (0.02) (−0.01) (−0.01)LM4(EM) 57074.0 191301 250768 618.105 15681.6 21492.6Err(%) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)EM2(EM2) 68710.5 195835 262094 620.214 15693.3 21504.6Err(%) (−20.39) (−2.37) (−4.52) (−0.34) (−0.07) (−0.05)EM3(EM2) 58576.0 195807 259498 618.504 15693.0 21499.7Err(%) (−2.63) (−2.35) (−3.48) (−0.06) (−0.07) (−0.03)LD1(EM) 57252.5 194840 255646 619.023 15683.4 21494.4Err(%) (−0.31) (−1.85) (−1.94) (−0.15) (−0.01) (−0.01)LD4(EM) 57074.0 191301 250768 618.106 15681.6 21492.6Err(%) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)ED3(EM) 58818.6 195825 259586 618.551 15694.2 21500.1Err(%) (−3.06) (−2.36) (−3.52) (−0.07) (−0.08) (−0.03)ED4(EM) 58713.8 194592 254740 618.465 15693.5 21497.8Err(%) (−2.87) (−1.72) (−1.58) (−0.06) (−0.08) (−0.02)FSDT (EM) 67878.5 195086 262887 590.917 15607.0 21031.4Err(%) (−18.93) (−1.98) (−4.83) (4.40) (0.48) (2.15)CLT (EM) 88173.4 195086 262887 592.439 15607.0 21031.4Err(%) (−54.49) (−1.98) (−4.83) (4.15) (0.48) (2.15)

Table 8.14: Closed circuit vibration problem for the multilayered piezoelectric plate,first three modes ω = ω/100. m = n = 1. 3D results [178] vs classical, refined andmixed models.

of the governing equations), the use of mixed models is not mandatory, the PVD-EMmodels are sufficient to obtain a quasi-3D evaluation of the first three frequencies ofthe free vibrations problem: the use of mixed models is mandatory to obtain the cor-rect thickness modes in terms of displacements, stresses, electric potential and electricdisplacement. In Figures 8.6-8.9 the thickness modes through the thickness are givenin terms of the above mentioned variables. In Figure 8.6 the three displacement com-ponents through the thickness are given, in this case there are no differences betweenLD4(EM), LM4(EM) and LM4(EM2) because the displacement is a primary variable.In order to give the mode through the thickness in term of stress σzz, the use of mixedmodels (LM4(EM) and LM4(EM2)) is mandatory, because in them it is a primary vari-

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242 CHAPTER 8

Figure 8.6: Multilayered piezoelectric plate, a/h = 50, mode 1. Mechanical displace-ments (u, v, w) vs z. Closed circuit. Refined and advanced 2D models.

Figure 8.7: Multilayered piezoelectric plate, a/h = 50, mode 1. Normal stress σzz vs z.Closed circuit. Refined and advanced 2D models.

able and this permits to obtain the interlaminar continuity and the correct boundaryhomogeneous conditions, see Figure 8.7. The through-the-thickness mode in term ofelectric potential is given in Figure 8.8, the potential Φ is a primary variable in eachconsidered model, so the same considerations made for the displacements are here

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ELECTRO-MECHANICAL ANALYSIS 243

Figure 8.8: Multilayered piezoelectric plate, a/h = 50, mode 1. Electric potential Φ vsz. Closed circuit. Refined and advanced 2D models.

Figure 8.9: Multilayered piezoelectric plate, a/h = 50, mode 1. Normal electric dis-placement Dz vs z. Closed circuit. Refined and advanced 2D models.

confirmed, the electric potential is continuous and satisfy the closed-circuit boundaryconditions. Finally, the Figure 8.9 gives the thickness mode in term of transverse nor-mal electric displacement; by using a LM4(EM2) model (where the Dz is a primaryvariable) there is the possibility to obtain the interlaminar continuity at the interfaces.By considering the Table 8.14, in case of thin plate (a/h = 50) the ESL models (if higherorders of expansion are applied) give correct values of the first three fundamental fre-quencies if compared with the 3D solution by Heyliger and Saravanos [178]: the erroris less than 1% for both refined and mixed models.

The second investigated case is a cylindrical ring shell made of two layers, an inter-nal layer in Titanium and an external one in piezoelectric PZT-4. The geometry of thisshell is given in the Figure 8.10, the free vibrations problem is investigated by consider-ing the closed circuit configuration (Φt = Φb = 0). The data for the employed materials

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244 CHAPTER 8

and the geometrical parameters are given in Table 8.15. The three-dimensional solu-

Properties PZT − 4 TitaniumE1[GPa] 81.3 114E2[GPa] 81.3 114E3[GPa] 64.5 114ν12[−] 0.329 0.3ν13[−] 0.432 0.3ν23[−] 0.432 0.3G23[GPa] 25.6 43.85G13[GPa] 25.6 43.85G12[GPa] 30.6 43.85e15[C/m2] 12.72 0e24[C/m2] 12.72 0e31[C/m2] −5.20 0e32[C/m2] −5.20 0e33[C/m2] 15.08 0ε11[pC/V m] 1.306E4 8.850ε22[pC/V m] 1.306E4 8.850ε33[pC/V m] 1.151E4 8.850ρ[Kg/m3] 7600 2768hPZT4[m] 0.001 − −hT [m] − − 0.003hTOT [m] − 0.004 −a[m] − 0.3048 −b = 2πRβ[m] − 1.82841 −Rα[m](at midsurface) − ∞ −Rβ[m](at midsurface) − 0.289 + 0.002 −

Table 8.15: Geometrical, elastic and piezoelectric properties of ring in PZT-4 and tita-nium for closed circuit vibrations problem.

tion is given by Heyligher et alii [184]: the first fundamental frequency in Hz is givenby imposing m = 0 and n = 4, 8, 12, 16, 20. An exhaustive comparison between the 3Dsolution and the refined and mixed models based on CUF is given in the Table 8.16.The same conclusions obtained for the free vibrations of the plate are here confirmedfor the shell: it is multilayered and this means LW theories mandatory for such inves-tigations. With respect to the plate case (where the first three fundamental frequenciesfor the waves number m = n = 1 was given), in this shell investigation the fundamen-tal frequencies are investigated for high values of waves number, which means highervibration modes investigation. The investigation of such higher modes leads to biggererrors in percentage with respect to the lower modes (m = n = 1) investigated in theplate case. Further investigations and omitted results for the other 2D models obtainedby CUF, can be found in [79], [193] and [225].

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ELECTRO-MECHANICAL ANALYSIS 245

Ti

PZT-4

a=0.3048

z

a

b

Rb

Figure 8.10: Multilayered piezoelectric ring: geometrical data. Closed circuit configu-ration.

8.4 Static analysis of plates including functionally gradedpiezoelectric layers

In this section the extension of CUF to functionally graded piezoelectric materials isverified. The refined models are obtained from the PVD-EM variational statement. Theproposed benchmark is a one-layered functionally graded piezoelectric plate, so no dif-ferences are exhibited between the LW and ESL models, for sake of brevity only the LWmodels are given. In order to verify the effectiveness of the proposed refined modelsas well as to evaluate their performance with respect to the First order Shear Defor-mation Theory (FSDT), the benchmark proposed by Lu et alii [226] is considered. Asingle-layered Functionally Graded Piezoelectric Material (FGPM) simply supportedplate is investigated. This is a functionally graded piezoelectric plate that has the PZT-4 as a reference material:

C011 = C0

22 = 139 GPa , C033 = 115 GPa ,

C012 = 77.8 GPa , C0

13 = C023 = 74.3 GPa ,

C044 = C0

55 = 25.6 GPa , C066 = 30.6 GPa ,

e031 = e0

32 = −5.2 C/m2 , e033 = 15.1 C/m2 ,

e051 = e0

42 = 12.7 C/m2 , ε011 = ε0

22 = 1.306× 10−8 F/m ,

ε033 = 1.151× 10−8 F/m ,

these properties change in the thickness direction z according to the following law:

C(z) = C0expβ zh ,

e(z) = e0expβ zh , (8.3)

ε(z) = ε0expβ zh ,

with 0 < z < h and−1.0 < β < 1.0. h is the thickness of the plate and the in-plane platedimensions are a = b = 1 m. Each variable given in the tables and figures is consideredin x = y = 0.25a.

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246 CHAPTER 8

n = 4 n = 8 n = 12 n = 16 n = 203D 31.27 170.42 407.29 745.21 1190.48

LM1(EM2) 32.53 176.42 418.30 756.73 1191.00Err(%) (−4.03) (−3.52) (−2.70) (−1.55) (−0.13)LM4(EM2) 31.64 171.59 406.87 736.07 1158.52Err(%) (−1.18) (−0.69) (0.10) (1.23) (2.68)LM1(EM) 32.39 175.66 416.50 753.47 1185.87Err(%) (−3.58) (−3.07) (−2.26) (−1.11) (0.39)LM4(EM) 31.64 171.59 406.87 736.07 1158.52Err(%) (−1.18) (−0.69) (0.10) (1.23) (2.68)EM2(EM) 31.65 171.68 407.10 736.53 1159.33Err(%) (−1.21) (−0.74) (0.05) (1.16) (2.62)EM3(EM) 31.65 171.64 406.98 736.28 1158.86Err(%) (−1.21) (−0.72) (0.08) (1.20) (2.66)LD1(EM) 33.28 180.51 427.99 774.25 1218.55Err(%) (−6.43) (−5.92) (−5.08) (−3.90) (−2.36)LD4(EM) 31.64 171.59 406.87 736.07 1158.52Err(%) (−1.18) (−0.69) (0.10) (1.23) (2.68)EDZ2(EM) 31.64 171.63 406.97 736.29 1158.94Err(%) (−1.18) (−0.71) (0.08) (1.20) (2.65)EDZ3(EM) 31.64 171.63 406.95 736.21 1158.75Err(%) (−1.18) (−0.71) (0.08) (1.21) (2.66)FSDT (EM) 30.72 166.61 395.06 714.75 1125.03Err(%) (1.76) (2.24) (3.00) (4.09) (5.50)CLT (EM) 30.72 166.67 395.37 715.74 1127.47Err(%) (1.76) (2.20) (2.93) (3.95) (5.29)

Table 8.16: Closed circuit vibrations problem for piezoelectric ring in Titanium andPZT-4, fundamental frequency in [Hz]. m = 0 and n = 4, 8, 12, 16, 20. 3D results [184]vs classical, refined and mixed models.

Lu et alii [226] gave an exact three-dimensional solution for two possible configu-rations, as illustrated in Figure 8.11. The sensor configuration has a mechanical loadapplied at the top:

P tz = Pz sin(

mπx

a) sin(

nπy

b) ,

Φt = Φb = 0 , (8.4)

with m = n = 1 and Pz = −1. The actuator configuration has an electric potential

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ELECTRO-MECHANICAL ANALYSIS 247

applied at the top:

Φt = Φz sin(mπx

a) sin(

nπy

b) ,

Φb = 0 , (8.5)

with m = n = 1 and Φ = 1. t and b mean the top and bottom of the plate, respectively.m and n are the chosen wave numbers in the plane. The three-dimensional solutions byLu et alii [226] were unfortunately only given in graphic form and, as a consequence,only a few significant digits are considered in the tables.

sensor actuator

Figure 8.11: Simply supported single layered functionally graded piezoelectric plate insensor and actuator configuration.

Tables 8.17-8.19 consider the sensor configuration for thin and very thin plates. Me-chanical and electrical variables are investigated for different values of the parameter β(which means different thickness laws for the material properties, β = 0 means "classi-cal" PZT-4). In order to obtain the three-dimensional solutions, higher orders of expan-sion are requested. FSDT gives very large errors with respect to the three-dimensionalmodel. By considering the variation of the parameter β, when this is different fromzero, the electric potential is never linear even if the plate is thin. The FSDT considersthe electric potential linear and this introduces a further error in the analysis of FGMplates. For thin plates, the LD2(EM) model gives quite good results, but for thickplates, as illustrated in Table 8.20, the use of the LD4(EM) model is mandatory. Forparticular variables, such as the transverse-normal stress σzz and the transverse-normalelectric displacementDz, the use of the LD4(EM) model is mandatory even if the plateis thin, see Tables 8.18-8.19. In the case of a sensor configuration, it is not possible toobtain the electric potential Φ by means of FSDT, because the degrees of freedom of thistheory are not sufficient to impose the short-circuit configuration (Φt = Φb = 0). By us-ing the LD4(EM) model, it is possible to obtain the quasi-3D evaluation for any valueof the thickness h and the parameter β. The variables plotted in Figure 8.12 are forthickness h = 0.1 and for each value of β, the results given by the LD4(EM) model arein agreement with the 3D-solution in Lu et alii [226]. The plots in Figures 8.13 and 8.14,clearly demonstrate the inefficiency of the FSDT model; the LD4(EM) and LD2(EM)models are very close for the thin plate (h = 0.01).

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248 CHAPTER 8

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

u10−11 u10−9

-1.0 0.650 0.647 0.650 0.463 - 0.695 0.695 0.462-0.5 0.247 0.243 0.245 0.185 - 0.280 0.280 0.1850.0 -0.031 -0.029 -0.029 0.000 - 0.000 0.000 0.0000.5 -0.197 -0.192 -0.193 -0.112 - -0.170 -0.170 -0.1121.0 -0.275 -0.273 -0.274 -0.170 - -0.256 -0.256 -0.170

w10−9 w10−6

-1.0 -0.251 -0.248 -0.247 -0.197 - -0.237 -0.237 -0.182-0.5 -0.196 -0.194 -0.193 -0.154 - -0.184 -0.184 -0.1420.0 -0.151 -0.151 -0.150 -0.121 - -0.143 -0.143 -0.1110.5 -0.119 -0.118 -0.117 -0.094 - -0.112 -0.112 -0.0861.0 -0.094 -0.092 -0.092 -0.072 - -0.087 -0.087 -0.067

Table 8.17: Sensor configuration. In-plane and transverse displacements u and w inh = 0 and x = y = 0.25. Thick and thin plates. 3D solution by Lu et alii [226].

The comments made for the sensor configuration are also confirmed for the actu-ator case. The same variables of the sensor case are investigated for the actuator inTables 8.21-8.24. It is clearly shown in Figure 8.15 that for particular variables, such asthe transverse shear/normal stresses σxz and σzz, even when a higher order model isapplied, the results are not completely satisfactory. The zero-homogeneous conditionsfor σxz and σzz at the top and bottom of the plate are difficult to reach. A possibleremedy to this situation could be the extension of Reissner’s Mixed Variational Theo-rem (RMVT) to FGPM plates, in which case, the transverse shear/normal stresses areconsidered as independent variables and evaluated "a priori". Figures 8.16 and 8.17confirm the inadequacy of the FSDT for these types of problem. However, Carrera’sUnified Formulation has been successful extended to FGPM plates in the frameworkof the Principle of Virtual Displacements (PVD). Refined models with higher orders ofexpansion in the thickness direction have been implemented; First order Shear Defor-mation Theory (FSDT) results has been obtained as particular cases. The use of refinedmodels (higher orders of expansion) is mandatory for FGPM plates, as the FSDT isinefficient for such structures. Future developments could be devoted to the analy-sis of multilayered plates including FGPM layers as well as the use of the Reissner’sMixed Variational Theorem to "a priori" obtain some transverse variables such as thetransverse shear/normal stresses σn and the normal electric displacement Dz. Fur-ther considerations about the investigation of FGPM plates are found in [42], [226] and[227]: in [42] a complete description of the 2D refined models is given, [226] givesthree-dimensional solutions for FGPM laminated plates in sensor and actuator config-urations, other lamination sequences are discussed with respect to those verified in thissection, in [227] the 3D solutions for the cylindrical bending of FGPM plates are given.

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ELECTRO-MECHANICAL ANALYSIS 249

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

Φ10−2 Φ10−1

-1.0 -0.596 -0.591 -0.592 0.000 - -0.588 -0.590 0.000-0.5 -0.492 -0.493 -0.492 0.000 - -0.490 -0.490 0.0000.0 -0.393 -0.393 -0.392 0.000 - -0.390 -0.390 0.0000.5 -0.299 -0.299 -0.298 0.000 - -0.297 -0.297 0.0001.0 -0.220 -0.217 -0.218 0.000 - -0.216 -0.217 0.000

Dz10−9 Dz10−7

-1.0 0.312 0.316 0.323 0.091 - 0.379 0.395 0.091-0.5 0.144 0.142 0.139 0.050 - 0.202 0.204 0.0470.0 -0.061 -0.059 -0.059 0.000 - -0.001 -0.001 0.0000.5 -0.267 -0.259 -0.259 -0.047 - -0.203 -0.205 -0.0471.0 -0.445 -0.432 -0.444 -0.091 - -0.380 -0.396 -0.091

Table 8.18: Sensor configuration. Electric potential Φ and electric displacement Dz inh = 0 and x = y = 0.25. Thick and thin plates. 3D solution by Lu et alii [226].

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

σxx σxx

-1.0 15.23 15.21 15.22 16.33 - 1509 1505 1631-0.5 13.05 12.97 12.94 13.95 - 1286 1286 13950.0 10.92 10.96 10.88 11.85 - 1088 1088 11850.5 9.138 9.156 9.042 9.989 - 909.1 909.4 998.71.0 7.529 7.504 7.401 8.323 - 745.5 748.2 831.6

σxy σxy103

-1.0 3.041 3.039 3.011 2.347 - 0.312 0.312 0.235-0.5 3.616 3.612 3.581 2.818 - 0.370 0.370 0.2820.0 4.247 4.241 4.207 3.345 - 0.432 0.432 0.3340.5 4.959 4.952 4.917 3.937 - 0.502 0.502 0.3941.0 5.830 5.773 5.739 4.604 - 0.582 0.582 0.460

σzz σzz

-1.0 -0.500 -0.511 -0.385 5.700 -0.500 -0.511 -0.381 -570.0-0.5 -0.500 -0.507 -0.445 -6.846 -0.500 -0.507 -0.441 -684.60.0 -0.500 -0.500 -0.516 -8.125 -0.500 -0.500 -0.512 -812.50.5 -0.500 -0.491 -0.596 -9.563 -0.500 -0.490 -0.593 -956.31.0 -0.500 -0.481 -0.681 -11.18 -0.500 -0.480 -0.679 -1118

Table 8.19: Sensor configuration. In-plane stresses σxx(−h/2), σxy(+h/2) and transversenormal stress σzz(+h/2) in x = y = 0.25. Thick and thin plates. 3D solution by Lu etalii [226].

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250 CHAPTER 8

h = 0.25 h = 0.15β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

uz10−10 uz10−10

-1.0 -0.200 -0.196 -0.189 -0.177 -0.780 -0.777 -0.767 -0.639-0.5 -0.157 -0.156 -0.151 -0.141 -0.615 -0.611 -0.602 -0.5040.0 -0.124 -0.124 -0.120 -0.111 -0.480 -0.479 -0.471 -0.3940.5 -0.098 -0.098 -0.094 -0.085 -0.375 -0.375 -0.369 -0.3061.0 -0.078 -0.076 -0.074 -0.065 -0.294 -0.293 -0.289 -0.235

Φ10−2 Φ10−2

-1.0 - -0.242 -0.240 0.000 - -0.396 -0.396 0.000-0.5 - -0.202 -0.199 0.000 - -0.331 -0.329 0.0000.0 - -0.161 -0.159 0.000 - -0.263 -0.262 0.0000.5 - -0.122 -0.121 0.000 - -0.200 -0.200 0.0001.0 - -0.088 -0.088 0.000 - -0.146 -0.146 0.000

σzz σzz

-1.0 -0.500 -0.511 -0.405 -0.912 -0.500 -0.511 -0.390 -2.533-0.5 -0.500 -0.508 -0.464 -1.095 -0.500 -0.507 -0.450 -3.0420.0 -0.500 -0.501 -0.532 -1.300 -0.500 -0.500 -0.520 -3.6110.5 -0.500 -0.492 -0.608 -1.530 -0.500 -0.491 -0.599 -4.2501.0 -0.500 -0.483 -0.688 -1.789 -0.500 -0.481 -0.682 -4.970

Table 8.20: Sensor configuration. Transverse displacement w(0), electric potential Φ(0)and transverse normal stress σzz(+h/2) in x = y = 0.25. Thick plates. 3D solution byLu et alii [226].

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

u10−9 u10−8

-1.0 -0.129 -0.125 -0.125 -0.029 - -0.126 -0.126 -0.029-0.5 -0.129 -0.128 -0.128 -0.030 - -0.129 -0.129 -0.0300.0 -0.129 -0.128 -0.129 -0.030 - -0.130 -0.130 -0.0300.5 -0.129 -0.128 -0.128 -0.030 - -0.129 -0.129 -0.0301.0 -0.129 -0.125 -0.125 -0.029 - -0.126 -0.126 -0.029

w10−9 w10−7

-1.0 0.327 0.325 0.323 -0.101 - 0.379 0.379 -0.002-0.5 0.140 0.139 0.138 -0.113 - 0.202 0.202 -0.0010.0 -0.074 -0.072 -0.072 -0.124 - -0.001 -0.001 -0.0010.5 -0.280 -0.283 -0.282 -0.133 - -0.204 -0.204 -0.0011.0 -0.480 -0.466 -0.466 -0.137 - -0.380 -0.381 0.000

Table 8.21: Actuator configuration. In-plane and transverse displacements u and w inh = 0 and x = y = 0.25. Thick and thin plates. 3D solution by Lu et alii [226].

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ELECTRO-MECHANICAL ANALYSIS 251

-0.04

-0.02

0

0.02

0.04

-4 -2 0 2 4 6

z

ux 10

-11

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-3 -2.5 -2 -1.5 -1 -0.5 0

z

uz 10

-10

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-8 -7 -6 -5 -4 -3 -2 -1 0

z

Φ 10-3

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-6 -4 -2 0 2 4 6

z

Dz 10

-10

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-15 -10 -5 0 5 10 15

z

σxx

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-6 -4 -2 0 2 4 6

z

σxy

β=-1.0β=-0.5β=0.0β=0.5β=1.0

-0.04

-0.02

0

0.02

0.04

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

z

σxz

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-0.5 -0.4 -0.3 -0.2 -0.1 0

z

σzz

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

Figure 8.12: Functionally graded piezoelectric plate with thickness h = 0.1. Sensorconfiguration, LD4(EM) model. 3D solution by Lu et alii [226].

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252 CHAPTER 8

Figure 8.13: Functionally graded piezoelectric plate with thickness h = 0.1 (left) andh = 0.01 (right). Sensor configuration, LD4(EM), LD2(EM) and FSDT (EM) models.Exponential β = 1.0.

10

Figure 8.14: Functionally graded piezoelectric plate with thickness h = 0.1 (left) andh = 0.01 (right). Sensor configuration, LD4(EM), LD2(EM) and FSDT (EM) models.Exponential β = 0.5.

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ELECTRO-MECHANICAL ANALYSIS 253

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

Φ Φ-1.0 0.181 0.183 0.185 0.250 - 0.188 0.190 0.250-0.5 0.210 0.212 0.213 0.250 - 0.219 0.219 0.2500.0 0.242 0.243 0.243 0.250 - 0.250 0.250 0.2500.5 0.272 0.273 0.273 0.250 - 0.281 0.281 0.2501.0 0.300 0.303 0.301 0.250 - 0.311 0.309 0.250

Dz10−7 Dz10−6

-1.0 -0.457 -0.455 -0.481 -0.355 - -0.461 -0.481 -0.355-0.5 -0.609 -0.606 -0.619 -0.456 - -0.613 -0.619 -0.4560.0 -0.787 -0.788 -0.796 -0.585 - -0.796 -0.796 -0.5850.5 -1.009 -1.004 -1.021 -0.751 - -1.011 -1.021 -0.7511.0 -1.257 -1.249 -1.306 -0.965 - -1.253 -1.307 -0.965

Table 8.22: Actuator configuration. Electric potential Φ and electric displacement Dz inh = 0 and x = y = 0.25. Thick and thin plates. 3D solution by Lu et alii [226].

h = 0.1 h = 0.01β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

σxx σxx

-1.0 -18.06 -18.04 -10.81 -5.862 - -187.3 -135.9 -59.58-0.5 -19.71 -19.71 -15.79 -5.766 - -206.3 -190.5 -57.970.0 -23.79 -23.68 -21.29 -5.722 - -250.2 -250.0 -57.220.5 -31.75 -31.75 -27.49 -5.791 - -337.7 -316.5 -58.341.0 -44.27 -45.61 -34.62 -6.063 - -486.5 -393.2 -62.50

σxy σxy

-1.0 -14.22 -13.80 -13.79 -2.050 - -131.5 -131.7 -20.50-0.5 -20.31 -19.99 -19.97 -3.452 - -188.0 -188.0 -34.520.0 -27.19 -27.16 -27.15 -5.722 - -250.6 -250.6 -57.220.5 -34.84 -34.56 -36.56 -9.399 - -309.2 -309.3 -93.991.0 -41.72 -41.00 -41.04 -15.37 - -348.8 -349.3 -153.7

σzz σzz

-1.0 0.000 0.002 0.611 32.75 0.000 0.000 0.063 327.5-0.5 0.000 0.003 0.995 54.18 0.000 0.000 0.102 541.80.0 0.000 0.005 1.545 89.40 0.000 0.000 0.159 894.00.5 0.000 0.008 2.316 147.3 0.000 0.000 0.238 14731.0 0.000 0.001 3.426 242.6 0.000 -0.002 0.351 2426

Table 8.23: Actuator configuration. In-plane stresses σxx(−h/2), σxy(+h/2) and trans-verse normal stress σzz(+h/2) in x = y = 0.25. Thick and thin plates. 3D solution byLu et alii [226].

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254 CHAPTER 8

h = 0.25 h = 0.15β 3D LD4(EM) LD2(EM) FSDT 3D LD4(EM) LD2(EM) FSDT

w10−9 w10−9

-1.0 - 0.013 0.011 -0.100 - 0.116 0.114 -0.100-0.5 - -0.023 -0.024 -0.113 - 0.029 0.027 -0.1130.0 - -0.063 -0.064 -0.124 - -0.070 -0.070 -0.1240.5 - -0.102 -0.102 -0.133 - -0.167 -0.167 -0.1331.0 - -0.136 -0.137 -0.139 - -0.252 -0.252 -0.138

Φ Φ-1.0 0.159 0.159 0.163 0.250 0.176 0.177 0.180 0.250-0.5 0.184 0.184 0.185 0.250 0.203 0.205 0.206 0.2500.0 0.209 0.209 0.209 0.250 0.232 0.234 0.234 0.2500.5 0.235 0.236 0.235 0.250 0.263 0.263 0.263 0.2501.0 0.262 0.262 0.260 0.250 0.292 0.292 0.290 0.250

σzz σzz

-1.0 0.000 0.021 1.301 13.10 0.000 0.005 0.882 21.85-0.5 0.000 0.036 2.131 21.65 0.000 0.009 1.438 36.100.0 0.000 0.073 3.335 35.75 0.000 0.016 2.237 59.600.5 0.000 0.130 5.055 58.90 0.000 0.028 3.364 98.201.0 0.000 0.201 7.565 97.05 0.000 0.032 4.989 161.7

Table 8.24: Actuator configuration. Transverse displacement w(0), electric potentialΦ(0) and transverse normal stress σzz(+h/2) in x = y = 0.25. Thick plates. 3D solutionby Lu et alii [226].

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ELECTRO-MECHANICAL ANALYSIS 255

-0.04

-0.02

0

0.02

0.04

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6

z

ux 10

-10

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-6 -4 -2 0 2 4

z

uz 10

-10

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

0 0.1 0.2 0.3 0.4 0.5

z

Φ

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-16 -14 -12 -10 -8 -6 -4 -2

z

Dz 10

-8

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-60 -50 -40 -30 -20 -10

z

σxx

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

-45 -40 -35 -30 -25 -20 -15 -10

z

σxy

β=-1.0β=-0.5β=0.0β=0.5β=1.0

-0.04

-0.02

0

0.02

0.04

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

z

σxz

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

-0.04

-0.02

0

0.02

0.04

0 0.02 0.04 0.06 0.08 0.1

z

σzz

β=-1.0

β=-0.5

β=0.0

β=0.5

β=1.0

Figure 8.15: Functionally graded piezoelectric plate with thickness h = 0.1. Actuatorconfiguration, LD4(EM) model. 3D solution by Lu et alii [226].

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256 CHAPTER 8

Figure 8.16: Functionally graded piezoelectric plate with thickness h = 0.1 (left) andh = 0.01 (right). Actuator configuration, LD4(EM), LD2(EM) and FSDT (EM) mod-els. Exponential β = −1.0.

Figure 8.17: Functionally graded piezoelectric plate with thickness h = 0.1 (left) andh = 0.01 (right). Actuator configuration, LD4(EM), LD2(EM) and FSDT (EM) mod-els. Exponential β = −0.5.

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ELECTRO-MECHANICAL ANALYSIS 257

8.5 Conclusions

This chapter considers the static and dynamic analysis of smart structures includingpiezoelectric layers with constant electrical and mechanical properties or materialsfunctionally graded in the thickness direction. Both plate and shell geometries havebeen taken into account, and refined and advanced models based on CUF have beenapplied. The static analysis has been accomplished for both sensor and actuator con-figurations, the dynamic analysis has been ivestigated for the free vibrations problemin case of closed circuit configuration. Functionally graded piezoelectric layers hasbeen discussed in the last section, in this case the use of classical theories such as theCLT and FSDT is inappropriate even if the structure is one-layered and thin. In mul-tilayered smart structures, the use of mixed models is mandatory in order to fulfil theinterlaminar continuity of the transverse variables such as the transverse shear/normalstresses and the transverse normal electric displacement. The use of layer wise theoriesgives the quasi 3D results. For the free vibrations problem, the LW models, with higherorders of expansion, are enough to obtain the correct values of the frequencies of thedifferent thickness modes, independently by the employed variational statement.

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Chapter 9

Conclusions and outlook

An exhaustive thermo-electro-mechanical analysis of multilayered structures (platesand shells) including isotropic, orthotropic, composite and piezoelectric layers hasbeen considered in this thesis. The layers have elastic, thermal and electrical proper-ties which are constant in the thickness direction or continuously varying in a chosendirection, as in the so-called Functionally Graded Materials (FGMs). The investigatedproblems involve several physical fields and they are defined as multifield problems(mechanical, thermal and/or electrical loadings).

A large number of refined and advanced two-dimensional models have been ob-tained using Carrera’s Unified Formulation (CUF). These are used to investigate bothanalytical and numerical solutions. CUF permits a general class of hierarchic two-dimensional models to be obtained which can differ in the variable description (Equiv-alent Single Layer (ESL) or Layer Wise (LW)) and in the order of expansion in thethickness direction used for the primary variables of the problem. CUF has also beenemployed to model the material properties of FGMs. The refined models have beendeveloped in the framework of the Principle of Virtual Displacements (PVD) withthe possibility of a priori modelling the displacements, the electric potential and thetemperature. The advanced models, also called mixed models, have been developedin the framework of Reissner’s Mixed Variational Theorem (RMVT), which permitsthe displacements, the electric potential, the temperature, the transverse shear/normalstresses and the transverse normal electric displacement, to be considered as indepen-dent variables. The obtained governing equations have been solved in an analyticalway using Navier’s solution. In order to overcome the limitations of closed form so-lutions, some analyses have also been conducted with the finite element method. Inparticular, the case of that finite element method for shell geometries, which is a newapproach, has been introduced.

The main results have been organized in three parts: mechanical analysis, thermo-mechanical analysis and electro-mechanical analysis.

In the first part, the refined and advanced models have been compared with sev-eral three-dimensional solutions obtained from the open literature. The importance ofhigher orders of expansion has been examined by considering multilayer compositeplates and shells, and sandwich structures. Moreover, for some cases, such as sand-

259

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260 CHAPTER 9

wiches with a soft core and carbon fibre layered structures, the use of LW approachesis mandatory. Mixed models are fundamental to obtain the continuity of transverseshear/normal stresses through the interfaces. The introduction of curvature does notmodify the capability of refined and advanced models. The finite element shell hasbeen validated by considering some well-known benchmarks, such as pinched cylin-drical shells and simply supported or clamped homogeneous plates with a concen-trated load. This new finite element shell is an extension of the degenerative approach.In fact, each variable of the model is initially written in a global reference system, butwith respect to classical degenerative approaches, a different transverse normal strainfrom zero is considered. The implemented finite element has nine degrees of free-dom for each node and an eight-node element is employed. A quadratic expansionof the transverse normal displacement (which means a linear transverse normal strainthrough the thickness) permits Poisson’s locking phenomena to be avoided. Finally,several plates and shells including FGM layers have been analyzed: classical models,such as the Classical Lamination Theory (CLT) and the First order Shear DeformationTheory (FSDT), are totally inappropriate for the analysis of FGM plates and shells.

The second part is about the thermo-mechanical analysis of multilayered plates andshells. The thermo-mechanical analysis is proposed in uncoupled form or by using afull-coupling between the mechanical and thermal fields. In the first case, the temper-ature is seen as an external loading. In order to determine this load, the temperatureprofile in the thickness direction can be assumed linear or can be calculated by solvingFourier’s heat conduction equation. The assumed and calculated temperature profilescoincide in the case of thin and homogeneous structures: the use of a calculated tem-perature profile to determine a correct thermal load is mandatory for moderately thickand/or multilayered plates and shells. In order to obtain a correct thermal stress anal-ysis, both refined models and correct evaluations of thermal loads must be taken intoaccount. In the full-coupling thermo-mechanical analysis, the temperature is a variableof the problem like the displacement. The PVD is extended by including the internalthermal work. Such an analysis made it possible to consider: - plates and shells witha temperature imposed on their top and bottom surfaces (the obtained results are veryclose to those of the uncoupled case with a temperature profile calculated via Fourier’sconduction equation. The great advantage is that the temperature through the thick-ness is obtained without solving Fourier’s equation); - plates and shells with an appliedmechanical load (the full-coupled model makes it possible to evaluate the effects of thetemperature on the mechanical field, the displacements obtained considering the ther-mal coupling are smaller than those obtained with a pure mechanical analysis and therelative temperature profile can be obtained); - a free vibrations problem (the inclusionof the thermal field leads to larger frequencies). The uncoupled thermal stress analysishas been extended to FGM structures and the related shells and plates did not havea liner temperature profile through the thickness even though the structure was verythin and one-layered.

Smart structures have been analyzed in the last part. These structures can be sub-jected to a mechanical load (sensor configuration) or to an electric potential (actuatorconfiguration). FSDT and CLT are inappropriate for the analysis of such structures,

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CONCLUSIONS AND OUTLOOK 261

and refined and advanced models must be employed to obtain a correct evaluation ofboth the mechanical and electrical variables. Higher orders of expansion are requiredand layer wise models are mandatory for multilayered smart plates and shells. UsingRMVT permits the transverse shear/normal stresses and the normal electric displace-ment to be obtained a priori in order to consider their interlaminar continuity: refinedmodels based on PVD do not offer this opportunity. The dynamic analysis has beenconducted by solving the free vibration problem of plates and shells in a closed circuitconfiguration (electric potential equal to zero at both the top and bottom of the multi-layered structure), the correct values of frequencies associated to a particular vibrationmode in the plane are obtained using higher orders of expansion. The use of layer-wise advanced mixed theories is mandatory to obtain the thickness modes, in terms ofdisplacements, electric potential, stresses and electric displacement. Piezoelectric func-tionally graded layers embedded in smart structures can be a very interesting solutiondesign, and the importance of refined models for such cases has been observed.

The benchmarks and assessments proposed in the thesis make it possible to vali-date the refined and advanced two-dimensional models based on CUF for the cases ofmultilayered structures and multifield problems. Classical theories, originally devel-oped for conventional structures, result inappropriate for such cases. The conductedanalysis makes it possible to investigate the main problems connected to these casesand to recognize the importance of refined/advanced 2D models. Analytical solutionscan represent useful assessments for future finite element applications and for fur-ther models proposed by other scientists. The new finite shell element, proposed forthe mechanical case, could be a candidate for a complete extension to the numerical,thermo-electro-mechanical analysis of plates and shells. For the extension to non-linearproblems, a further idea could be the implementation of a geometrically exact finite el-ement shell.

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Chapter 10

Conclusions et perspectives

Une analyse thermo-électro-mécanique de structures multicouches (plaques et coques)composées de couches isotropes, orthotropes, composites et piézoélectriques a été con-duite dans ce travail. Les couches ont des propriétés élastiques, thermiques et élec-triques supposées constantes dans la direction de l’épaisseur, ou variant continumentdans cette direction, comme pour les matériaux à gradient fonctionnel (FGM). Lesproblèmes étudiés mettent en jeu différents champs physiques et sont définis commedes problèmes multiphysiques.

De nombreux modèles bidimensionnels, raffinés ou avancés, ont été obtenus à par-tir de la Formulation Unifiée de Carrera (CUF). Ces modèles ont été utilisés pourobtenir des solutions analytiques ou numériques. CUF permet de décliner une hiérar-chie de modèles qui se différencient dans la description des variables (couche équiva-lente (ESL) ou layerwise (LW)) et dans l’ordre de développement dans l’épaisseur pourles variables choisies. La CUF a aussi été utilisée pour modéliser les propriétés matéri-aux des FGM. Les modèles raffinés ont été développés en utilisant le principe des dé-placements virtuels (PVD) en incluant a priori le déplacement, le potentiel électriqueet la température. Les modèles avancés, appelés aussi modèles mixtes, utilisent lethéorème variationnel mixte de Reissner (RMVT), qui permet de considérer le déplace-ment, le potentiel électrique, la température, les contraintes normales et le déplacementélectrique normal comme des variables indépendantes. Les équations d’équilibre ainsiobtenues ont été résolues dans le cadre d’une approche équilibre en utilisant les solu-tions de Navier. Afin de s’affranchir des limitations de l’approche continue, des analy-ses ont aussi été effectuées en utilisant la méthode des éléments finis. En particulier, lecas de géométrie coque a été abordé.

Les principaux résultats ont été présentés en trois parties: mécanique, thermo-mécanique et électro-mécanique.

Dans la première partie, les modèles raffinés et avancés ont été comparés avec dif-férentes solutions tridimensionnelles issues de la littérature. L’importance de l’ordredu développement a été examinée en considérant des plaques et des coques multi-couches et sandwichs. De plus, dans certains cas particuliers, sandwich à coeur mouet stratifiés à fibres de carbone, l’utilisation des modèles LW est obligatoire. Les mod-èles mixtes sont nécessaires afin d’obtenir la continuité des contraintes normales aux

263

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264 CHAPTER 10

interfaces. L’introduction des courbures n’altère pas l’efficacité des modèles raffinéset avancés. L’E.F de coque a été validé en considérant des tests classiques, tel lecylindre pincé, et des plaques appuyées ou encastrées soumises à une charge concen-trée. Ce nouvel E.F s’inscrit dans le cadre de l’approche dégénérée. Chaque variableest exprimée dans le repère global mais la contrainte normale transversale est sup-posée non nulle. L’E.F implémenté a neuf degrés de liberté (ddl) par noeud et unegéométrie à huit noeuds est utilisée. Un développement à l’ordre deux du déplace-ment transversal (signifiant une variation linéaire de la déformation normale transver-sale dans l’épaisseur) permet d’éviter le verrouillage dans l’épaisseur, dénommé "Pois-son’s locking". Finalement, différentes plaques et coques avec des couches FGM ontété évaluées: les modèles classiques tels que la théorie des stratifiés (CLT) et la théoriedu premier ordre (FSDT) sont totalement inadaptés pour l’analyse de ces structures.

La seconde partie est dédiée à l’analyse thermo-mécanique de plaques et coquesmulticouches. L’analyse thermo-mécanique est présentée sous ses formes découpléeet couplée. Dans le premier cas, la température est un chargement extérieur. Afin dedéterminer ce chargement, le profil de température dans l’épaisseur peut etre supposélinéaire ou peut etre calculé en résolvant l’équation de Fourier. Les températures sup-posées ou calculées sont identiques dans les cas minces et homogènes; la résolutionde l’équation de la chaleur est conseillée pour les cas semi-épais et/ou hétérogène.Afin d’obtenir une analyse en contrainte correcte, il faut calculer les distributions detempérature et utiliser un modèle raffiné. Dans le cas d’une analyse couplée thermo-mécanique, la température devient une variable au meme titre que les déplacements.Le PVD est prolongé en introduisant le travail interne de la température. Ce typed’approche permet de considérer: - des plaques et des coques avec une températureimposée sur les faces supérieure et inférieure (les résultats obtenus sont très prochesde ceux obtenus dans le cas découplé avec une température calculée par l’équationde Fourier. L’avantage consiste alors à ne plus résoudre cette équation); - des plaqueset des coques avec un chargement mécanique sur les faces (l’introduction de la tem-pérature entraine un réduction des élongations);- les problèmes de vibrations libres(l’introduction de la température entraine une augmentation des fréquences propres).L’analyse des contraintes thermique découplée a été étendue aux structures FGM et ladistribution dans l’épaisseur de la température n’est plus linéaire, y compris pour lesstructures minces et homogènes.

Les structures intelligentes ont été étudiées dans la dernière partie. Ces structurespeuvent etre soumises à des chargements mécaniques (configuration capteur) ou àdes potentiels électriques (configuration actionneur). FSDT et CLT sont inadaptés àce type de structures et les modèles raffinés et avancés doivent etre utilisés afin d’avoirune évaluation correcte des variables électriques et mécaniques. Des développementsd’ordre élevé sont nécessaires et une approche layerwise est conseillée pour les plaqueset coques multicouches et intelligentes. L’utilisation de RMVT permet d’obtenir lescontraintes normales et le déplacement électrique normal a priori sous une forme con-tinue: les modèles raffinés basés sur PVD ne le permettent pas. L’analyse dynamiquea été traitée en résolvant le problème de vibration libre de plaques et coques dans uneconfiguration circuit fermé, le potentiel électrique est alors égal à zéro en haut et en bas

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CONCLUSIONS ET PERSPECTIVES 265

de la structure multicouche. Des valeurs correctes de fréquence associées à un modede vibration dans le plan sont obtenues en utilisant des développements d’ordre élevé.L’utilisation de l’approche layerwise mixte est nécessaire pour obtenir les modes dansl’épaisseur, en terme de déplacements, potentiel électrique, contraintes et déplacementélectrique. Le cas de couche piézoélectrique avec FGM est intéressant en tant que so-lution de conception et l’importance de ces modèles a été observée.

Les tests et les hypothèses proposés dans ce travail permettent de valider les mod-èles raffinés et avancés basés sur la CUF dans le cas de structures multicouches et deproblèmes multiphysiques. Les théories classiques, à l’origine développées pour lesstructures conventionnelles, s’avèrent inadaptées dans certains cas. Les solutions an-alytiques proposées peuvent permettre de valider de nouveaux modèles théoriquesou numériques. L’E.F proposé dans le cas mécanique, peut etre étendu afin de pou-voir conduire des analyses thermo-électro-mécanique. L’extension aux problèmes non-linéaires pourrait s’orienter vers l’implémentation d’une approche coque dite géométrique-ment exacte.

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Curriculum Vitae

Name Salvatore BrischettoDate of birth 15 June 1979Place of birth Catania, Italye-mail [email protected]

Education

January 2006 - December 2008 Ph.D. student at Politecnico di Torinoand Université Paris Ouest - Nanterre LaDéfense

October 1998 - July 2005 Academic degree in Aerospace engineeringat Politecnico di Torino

September 1993 - July 1998 Higher education at Istituto TecnicoAeronautico "Arturo Ferrarin", Catania (Italy)

Professional experience

Since February 2006 Research assistantships at Politecnico di TorinoOctober 2005 - January 2006 Research fellowship at Politecnico di Torino

Teaching activities

Since March 2007 Teaching assistant of the course "Computationalaeroelasticity" at Politecnico di Torino

Since March 2007 Teaching assistant of the course "Structures foraerospace vehicles" at Politecnico di Torino

November 2007 - September 2008 Tutoring activity for MOSAIC (Master OfSystem engineering in Aeronautics for theInternational Community)