WL-TR-91-3016

AD-A250 592

Variational Formulation and FiniteElement Implementation of Pagano'sTheory of Laminated Plates

R. S. Sandhu, W. E. Wolfe, and H. H. ChyouDepartment of Civil EngineeringThe Ohio State University DTIC.

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1 -MAY 27 1992July 1991

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11. TITLE (Include Security Classification)

Variational Formulation and Finite Element Implementation of Pagano's Theory ofT.mivtntpid P] tAR

12. PERSONAL AUTHOR(S)R. S. Sandhu, W. E. Wolfe, and H. H. Chyou

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16. SUPPLEMENTARY NOTATION

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)FIELD GROUP SUB-GROUP Composite Laminates Laminated Plates

I Finite Element Methods Variational Methods

19. ABSTRACT (Continue on reverse if necessary and identify by block number)

Pagano's theory of laminated plates is restated in the form of a set of self-adjointdifferential equations. Identifying consistent boundary operators, generalvariational principle is stated following standard procedures for linear coupled self-adjoint systems of equations. Extensions to relax requirements of differentiabilityof various field variables are indicated along with specializations to reduce thenumber of free field variables. One specialization is implemented in a finiteelement computer program and used to solve several example problems of free-edgedelamination specimens.

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FOREWORD

The research reported herein is part of a research program supported by the Air

Force System Command/ASD under Grant No. F33615-85-C-3213 to the Ohio State

University, and describes one of the several alternative approaches developed to evaluate

stresses in free-edge delamination specimens in order to define cumulative damage. Dr.

George P. Sendeckyj of Wright Laboratory/Flight Dynamics Directorate was the Program

Manager. At the Ohio State University, the research was carried out, under the

supervision of Professors Ranbir S. Sandhu and William E. Wolfe, by Hui-Hluang

Chyou, Graduate Research Associate in the Department of Civil Engineering. The work

reported is essentially based on Mr. Chyou's doctoral dissertation. The Instruction and

Research Computer Center (IRCC) at The Ohio State University and The Ohio

Supercomputer Center (OSC) provided the computational and documentation facilities.

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CONTENTS

FOREW ORD ............................................. iii

SECTION I: INTRODUCTION ................................. I

BACKGROUND .......................................... 1Introduction ............................................ 2

SECTION II: LITERATURE REVIEW ............................. 5

Theory of Laminated Composite Plates ........................... 5Classical Thin Plate Theory (CPT) ................................ 6Higher Order Theory .......................................... 7Discrete Laminate Theory ............................... 10

Analysis of Free-Edge Delamination ............................ 11Need for Solution Procedures- ................................ 17

SECTION HI: PAGANO(S THEORY OF LAMINATED PLATES ............ 20

Introduction ........................................... 20Equilibrium of an Elastic Solid .............................. 20

Generalized Equilibrium of a Two Dimension Plate ............... 20Kinematics ........................................... 22Constitutive Relations of a Monoclinic Material .................... 23An Approximate theory ................................... 24

Equilibrium Stress Field ................................ 24Pagano's Equations of Equilibrium ............................. 26Constitutive Relations for Generalized Displacements ................. 27

Equations of Displacement at the interfaces and Interface Conditions .... 30Interface Displacement Equations ........................ 30Interface Continuity ................................ 31Prescribed Tractions ................................ 31Prescribed Displacements ............................. 32

SECTION IV: VARIATIONAL FORMULATION OF PAGANOYS THEORYOF LAMINATED PLATES ......................... 33

Introduction ........................................... 33Basic Variational Principles ................................. 34

Boundary Value Problem ................................ 34Bilinear Mapping ..................................... 34Self-Adjoint Operator .................................. 35

V

Gateaux Differential of A Function .............................. 36Variational Principles For Linear Operators ......................... 36Coupled Problems ........................................... 37

Self-adjointness of Pagano's Equations ................................. 39Introduction ................................................. 39Equilibrium Equations ........................................ 40Constitutive Equations ........................................ 40Operator Matrix Form of the Field Equations ....................... 41Displacement Continuity Conditions .............................. 45Discussion ................................................ 49

Self-adjoint Form of the Field Equations .............................. 49Displacements at the Interfaces .................................. 50Constitutive Equations for Generalized Displacements ................. 53Adjointness of the Field equations ............................... 62

Operator matrix !At' ..... . . ... ............................... ... 62Operator matrix [-= l) ...................................... 64Operator matrices rCf k) and IC(k) .............................. 65Operator matrices I A fk) and I T j f.... ........................ 66Operator matrices [Bfk) and .B... ............... ........................ 67

Consistent Boundary Operators ..................................... 68Prescribed Boundary Conditions .................................. 80

Variational Principle ............................................ 83Variational Formulation for Linear Coupled Self-Adjoint Problem ...... .83Governing Function for the Laminated Plate ....................... 84The Set of Admissible States ................................... 90Proof of the Variational Theorem ................................ 93

Extended Variational Principles ..................................... 110Some Specializations ......................................... 139

SECTION V: FINITE ELEMENT FORMULATION ...................... 149

Finite Element Discretization ...................................... 149Finite Element Formulation .............................. 151

Selection of the Interpolation Scheme ........................... 166General Considerations . ................................. 166Interpolation function for the Ileterosis Element ................. 169

Computer Implementation .................................. 170

SECTION VI: EXAMPLES OF APPLICATION ...................... 174

Introduction ........................................... 174Delamination Coupons .................................... 174

Four-Ply Laminates ................................... 174Cross-Ply Laminate ................................ 177Angle-Ply Laminate ................................ 183

A Multi-Ply Delamination Coupon .......................... 192Numerical Evaluation ............................... 192

vi

SECTION VII: DISCUSSION ................................. 201

REFERENCES ............................................ 203

vii

FIGURES

1. (a) Local, (b) Global Coordinate Systems of Heterosis Element .......... 168

2. (a) 6x14 Element Mesh, (b) 6x18 Element Mesh ..................... 176

3. Distribution of Z-stress Along 90/90 Interface ....................... 178

4. Distribution of Z-stress Along 0/90 Interface (N=2) ................... 179

5. Distribution of Z-stress Along 0/90 Interface (N=6) ................... 180

6. Distribution of YZ-stress Along the 0/90 Interface ................... 181

7. Transverse Displacement Across Top Surface ........................ 182

8. Distribution of X-stress Along Center of Top Layer (45 degree) forN=2 .................................................. 184

9. Distribution of X-stress Along Center of Top Layer (45 degree) forN=6 .................................................. 185

10. Distribution of XY-stress Along Center of Top Layer (45 degree) forN=2 ................................................... 186

11. Distribution of XY-stress Along Center of Top Layer (45 degree) forN-6 .................................................. 187

12. Distribution of XZ-stress Along 45/-45 Interface for N=2 .............. 188

13. Distribution of XZ-stress Along 45/-45 Interface for N6 ............... 189

14. Axial Displacement Across Top Surface for N=2 ..................... 190

15. Axial Displacement Across Top Surface for N-6 ..................... 191

16. Finite Element Mesh for 22-ply Laminated Plate .................... 193

17. Distribution of Z-stress at Midplane ............................... 194

18. Through-the-thickness Z-stress Distribution Along the Free-Edge ........ .197

19. Through-the-thickness Z-stress Distribution at the Centroids ofElements Along the Free Edge ............................... 198

20. YZ-Stress Distribution at the Interface -25.5/90 ..................... 199

21. Effect of Mesh Refinement on YZ-Stress Distributions at the Interface-25.5/90 ........ ........................................ 200

viii

TABLES

1. A Comparison of the Assumptions in Some Higher Order Theories ....... 9

2. A Comparison of Various Methods for Solving the FED Problems ........ 17

3. A Comparison of Theories by Pagano, Chang, Hong, and Dandan ........ 19

4. A Comparison of Numerical Results (Through-the-thickness) at theFree Edge ........................................ 196

ix

SECTION I

INTRODUCTION

1.1 BACKGROUND

Most criteria for development of failure in materials are based upon

stress/strain/energy distribution. Therefore, in order to model cumulative damage

processes in laminated composites, we feel that reliable procedures for evaluation of

distribution of stress/strain/energy in the material must be available. This requires an

adequate theory governing the behavior of laminated composites along with appropriate

methods for solution of the boundary value problem. The current research program

covered development of theoretical framework as well as approximate solutions This

report covers one of the alternative approaches investigated viz., Pagano's theory of

linear elastic composite laminates based on the assumption of linear variation of

in-plane stresses over the thickness of a layer or sublayer (each lamina being further

divided into sublayers, if desired), satisfying equilibrium equations exactly, evaluating

strains from stresses, and evaluating generalized displacements by integration of strain

components and their moments up to a certain order. This section provides an

introduction to the problem and describes the scope of the work as well as the

organization of this report.

1.2 Introduction

Considerable research effort has been devoted to the development of analytical

procedures for the analysis of the behavior of composite materials. This has resulted in

a variety of laminated plate theories and solution methods including, among others,

classical thin plate theory [Stavsky 19611, higher order theories [Whitney 1973, Nelson

1974, Lo 1977, Reddy 19841 and discrete laminate theories [Srinwas 1973, Sun 1973,

Pagano 1978, 1983].

Classical thin plate theory (CPT) based on Kirchhoff hypothesis assumes that

transverse shear deformation is negligible. For analysis of laminated composites, it is

well known [Whitney 1969, Pagano 1969,1970, Srinivas 19701 that use of CPT leads to

underprediction of the transverse deflection. This is due to the fact that the ratio of

shear to Young's modulus is lower in most composite materials than in conventional

isotropic materials. Also, the error grows with an increase in plate thickness

Higher order theories [Whitney 1973,1974] which include higher order shear modes

lead to improved estimates of in-plane stress distributions. However, higher order

theories based on assumption of second and higher order polynomial distribution of

in-plane displacements over the depth- of the plate, have two critical deficiencies. The

first is their lack of capability to describe local deformation precisely. Due to this, it

is difficult to avoid error in calculating in-plane stresses around laminar interfaces,

especially, when shear rigidities of adjacent laminae are quite different [Sun 1973, Lo

19771. The other deficiency is the violation of equilibrium of the plate because stress

continuity at the interface is, in general, not satisfied. The need to eliminate these

deficiencies has motivated the development of several discrete laminated plate theories

[Srinivas 1973, Sun 1973] in which variation of directional properties within the

laminate is properly incorporated. As the discrete laminate theory not only removes the

drawbacks of higher order theories noted above, but also allows different boundary

2

conditions to be specified in each layer, it has been able to accurately describe the

mechanical behavior of most laminated plates. The discrete laminate theory results in

better estimates of in-plane stress distribution [Sun 19731 However, this theory, in

general, involves a large number of field equations, and consequently makes the

problems quite complicated.

Since the boundary value problem of a structure constructed with composite

laminates is extremely complex, approximate techniques are often used •o obtain

solutions. The approximate techniques based on the discrete laminated plate theory can

be classified as displacement-based or stress-based approaches. The analytical solutions

of displacement-based approach include the approximate elasticity solutions [Puppo and

Evensen 1970, Pagano and Pipes 1973), modified higher order theory [Pagano 1974),

boundary layer theory [Tang and Levy 1975], Pagano's theory based on the restatement

of Reissner's theory for each lamina [Pagano 19781, and its simplication the global-local

model [Pagano 19831 Numerical solutions include use of the finite difference method

[Pipes and Pagano 1970, Altus et al. 19801 and the finite element method [Wang and

Crossman 1977,1978; Raju and Crews 1981; Whitcomb et al. 19821. The stress-based

finite element approaches include Pian [19691 Rybicki [19711, and Spilker [19801.

Recently, Chang [1987] developed a finite element procedure based on minimization

of potential energy and ensuring continuity of displacements as well as tractions. This

development was for stress analysis of free-edge delamination specimens under uniform

longitudinal strain and is not applicable to damage cumulation in laminated plates of

arbitrary geometrical configuration. Hong [1988] developed a consistent shear

deformation theory in which the shear forces in each lamina depend upon the shear

deformation of all the laminae. Hong's analysis was for dynamic response of laminates

and ensured continuity of tractions at interfaces. However, Hong assumed the

transverse displacement to be independent of x3 coordinate i.e. the thickness of the

lamina is assumed to stay constant.

3

The objective of the current research program is to develop a procedure capable of

providing good estimates of interlayer and intra-layer stresses so that cumulative

damage criteria based on stress distribution can be used to predict damage. Pagano's

theory [1978] assuming the in-plane stresses to vary linearly over a layer or sublayer

had been shown to give excellent results for four layer laminates. However, the

soultion procedure employed by Pagano could not be applied to a larger number of

layers. The finite element method has proved to be a powerful tool for obtaining

numerical solutions to boundary value problems including the bending and stretching of

the plates. To apply this method to obtain numerical solutions and to develop

alternative solution strategies for Pagano's theory we felt it necessary to write the

governing equations in a self-adjoint form so that the general procedures for

development of variational principles for coupled linear problems could be applied. A

brief review of earlier work on the laminate theory is given in Section IL. Section III

contains a summary of Paganods equations. In Section IV it is shown that Pagano's

equations must be restated in a modified form to constitute a self-adjoint system. A

modification, involving a reduction in the number of field variables is proposed.

Variational principles governing Pagano's theory are developed including various

extensions and certain useful specializations. Section V describes a finite element

formulation. Section VI contains some examples of application.

4

SECTION II

LITERATURE REVIEW

2.1 Theory of Laminated Composite Plates

In essence, plates are three-dimensional solids. The advantage of being able to treat

them as two-dimensional problems has been the primary motivation for the

construction of plate theories. As the use of fiber-reinforced composite laminates is

being extended to various engineering fields, a considerable amount of work has been

done over the past few decades to develop a reliable theory of laminated plate.

Recently, AM-Ghothani [1986] and Hong [19881 presented reviews of the earlier work on

this subject. Existing theories may be categorized into three groups:

1. Classical thin plate theory

2. Higher order theories

3. Discrete laminate theories

The first two theories assume the displacements in a single power expansion of the

out-of-plane coordinate through the thickness of the laminate, whatever the number of

layers. On the other hand, the third group of theories treats each layer as a homogene-

ous, anisotropic plate and combines field equations of each layer through proper conti-

nuity conditions between layers.

In this Section, a brief review of some of the laminate theories described above is

presented as an update complementary to the ones by [AI-Ghothani 19861 and ]long

[19881 Throughout, standard index notation is used, in which Latin indices take on the

range of values 1,2,3 and Greek indices take on values 1 and 2.

2.1.1 Classical Thin Plate Theory (CPT)

Classical plate theory of composite laminates follows the same philosophy as

employed in the homogeneous isotropic thin plate theory. In developing the theory, the

displacement field through the thickness of laminate is assumed to be such that the

plane of crosssection before bending remains plane and perpendicular to the midsurface

of the plate during deformation. In addition, it is assumed that the variation of

lateral displacement through the thickness and the stress normal to the midsurface are

negligible. A mathematical representation of these assumptions is

ux(x) U ) - xw. (1)

u 3(xi) w(x•) (2)

Cr33 = 0 (3)

where u* are the components of in-plane displacements of the midplane. With this

displacement field, the kinematic relations are

e = 1uo .u + Xw (4)2 3

6, 3 =0 (5)

E3 3 =0 (6)

Apparently, the first complete classical laminated plate theory is due to Reissner

[19611 In analyzing an angle-ply laminate of two layers, it was noticed that coupling

of bending and stretching exists unless the material properties are symmetric with

respect to the midplane. Stavsky [1961] further investigated this phenomenon for a

multi-layer plate. Dong, Pister and Taylor [1962] extended this approach to the

analysis of anisotropic laminated shells.

The classical laminated plate theory neglects the effect of transverse shear

deformation, implying infinite shear rigidity, i.e., leading to overestimation of plate

stiffness. As a result, the theory gives an underprediction of lateral deflection.

6

Naturally, the error becomes larger as the thickness of plate increases. It was pointed

out [Pagano 1969, 1970, Srinivas 1970] that the transverse shear effect is more

pronounced in composite laminates because the ratio of shear to Young's modulus is

lower for these materials than for conventional isotropic materials.

2.1.2 Higher Order Theory

As noted above, it is clear that realistic transverse shear variations cannot be

achieved by theories based on the assumption of plane cross section. This assumption

has resulted in an inaccurate prediction of in-plane stress distribution, especially, for a

laminate made of layers with different material properties or differently oriented

material axes. The need to include the effects of such local deformation has

stimulated development of theories that use higher order terms in the assumed

displacement field.

Most of these theories are based on an assumed displacement field. As indicated

by A1-Ghothani [1986] these theories can be discussed within the framework defined by

the following displacement field assumption.

u.(x.) = u(x•) + x3 •(x•) + x2 pi(X,) + x 3 i(x) (7)

In the expression (7), it is worth r .:ig that the terms including even and odd powers

of x3, represent the symmetric and antisymmetric modes, respectively, of the in-plane

displacements through the thickness. Of course, powers higher than the third can be

included. Theoretically, as the degree of the polynomial increases, the displacement

components can be approximated as closely as one wishes. However, there are practical

limitations to the degree of the polynomial that can be used. Also we shall see that

in composite laminates there is in general discontinuity in gradient of displacements at

the interfaces of dissimilar layers. This makes polynomial approximation very

difficult.

7

Whitney [1973] first proposed a higher order theory with quadratic polynomial

functions for in-plane displacements (u.) and linear functions for out-of-plane

displacement (u3) to represent the first antisymmetric shear mode and non-zero normal

strain. Whitney [1974] also presented another higher-order theory, in which linear

variation of in-plane displacements and quadratic variation of normal displacement

through the thickness were assumed. This was used to analyze laminated cylindrical

shells with moderate radius-to-thickness ratio under static loading. Nelson [1974] used

quadratic functions for both the in-plane and the out-of-plane displacements. Such a

displacement field would model the effect of normal strain more precisely. In the

higher-order theory proposed by Lo [19771 cubic functions for in-plane displacements

and quadratic function for normal displacement were assumed. With this displacement

field, it was claimed that the level of truncation is consistent in the sense that the

transverse shear strains due to in-plane displacements and normal displacement are of

the same order in x. From the application of the theory to thick laminated plates in

cylindrical bending, it was shown that accuracy of in-plane stress distribution through

the thickness could be considerably improved, except at the interfaces of the laminate.

Although higher order theories were, to some extent, successful in incorporating the

effect of higher shear modes, the solution process was costly because more field

variables were involved. To overcome such difficulties, a simple higher order theory

was developed [Levinson 1980] using higher order terms for the in-plane

displacements, but assuming u3 to be constant over the thickness of the plate, ie.

u(x) u°(x)+x 3)+ (X )+x2,p (X )+x3 (x ) (8)

u 3(X) = w(xB) (9)

This idea was first proposed by Levinson [1980] for the homogeneous, isotropic

plate and extended to the laminated composite plates by Bert [19841 Imposing the

stress free plate surface conditions on this displacement field, two field variables

8

qio and f were eliminated to give

u u° + x 3 •-o • "(.)x 2(,.+W,.) (10)

u = w(xA) (II)

where h denotes the thickness of laminate. Equations (10) and (11) still include the

effect of higher-order terms. This theory was used [Bert 1984, Reddy 1984] to analyze

angle-ply and cross-ply laminated plates and it was reported that accurate results were

obtained in predicting transverse shear stresses. A comparison of various theories is

given, in summary form, in Table 1.

Table 1: A Comparison of the Assumptions in Some Higher Order Theories

I In-plane I Out-of-plane I Out-of-plane normalI strains I shear strain I strain

--- ------------------------------------ 4-----------------------------------------------

Whitneyl Iet al. I Quadratic I Linear I constant[19731] I I. 4---- --------------------.------------------------------------------------

Whitneyl I Iet al. I Linear I Quadratic I Linear(19741 1 I I

S4-----------4-----------------------4------------------------------------------------Nlelson I I Iet al. I Quadratic I Quadratic I Linear[19741 1 I

+------------------------------------+------------------------------------------------

Lo I I Iet al. I Cubic I Quadratic I Linear[1977) I 1 I

S--------------------------------------------+------------

The higher order theory has two critical drawbacks. First, it is incapable of

accounting for local deformation accurately because local deformation depends on the

stacking sequences of layers and elastic properties of each layer [Sun 19731 Second, it

9

violates equilibrium of the plate because stress continuity in the interfaces of the

laminate is not satisfied. This theoretical deficiency can be fatal in certain

applications, e.g., in-plane stress analysis or evaluation of natural frequencies of a

hybrid laminate.

2.1.3 Discrete Laminate Theory

The need to eliminate the deficiencies of higher order theory has motivated

development of several discrete laminated plate theories [Sun 1973, Srinivas 1973,

Pagano 1978]. Common procedure for derivation of those theories is to treat each layer

as a homogeneous, anisotropic plate and to combine the governing equations of each

layer by using interlaminar continuity conditions to obtain global governing equations

of the laminated plate. For a homogeneous, orthotropic laminated plate, exact

three-dimensional elasticity solutions [Pagano 1969, 1970] revealed that in-plane

displacements are, layerwise, almost linear through thickness, thickness stretch is zero

even for thick laminates, and the transverse shear stresses are close to parabolic over

each layer.

Basically, the discrete laminate theories described above are based upon the same

philosophy, i.e., assumption of layer-wise linear variation of the in-plane displacements

even though the final form of the governing equations is different. Also they do not

satisfy plate equilibrium. Stating that theories based on assumed-displacement field are

not reliable for stress analysis of laminates especially where the stress gradient is large,

Pagano (19781 developed a discrete laminate theory based on the assumed-stress field

using Reissner's variational principle. In-plane normal stresses were assumed to be

linear through the thickness of each layer and other stress components were obtained

from three-dimensional equilibrium equations. In this theory, all six stress components

are, in general, non-zero and continuity of stresses in the interfaces of the laminate is

exactly satisfied. Displacement continuity conditions are also satisfied.

10

In practical applications, a laminate may be composed of numerous layers. Even

though discrete theory may provide a reliable tool in predicting precise local behavior,

the problem becomes intractable as the number of layers become large. To overcome

this difficulty, Pagano [1983] developed the global-local model in which the cross

section of a laminated plate is divided into local and global portions. For the local

domain Paganos theory [19781 is used while a higher order theory is adopted for the

global domain. A variational principle was used to obtain governing equations of the

plate. This dual model is expected to relieve the burden of handling extremely

complicated problem, giving precise stress resolution in the local domain [Pagano 19831

However, in dividing the region into the local and the global domains, technical

difficulty remains because the critical portion is not always known in advance.

As an improvement upon the discrete laminate theory, Hong [1988] developed a

consistent shear deformation theory of laminated plates and applied it to vibration and

transient response. Hong [1988] ensured interlaminar traction continuity in his theory.

2.2 Analysis of Free-Edge Delamination

Due to the presence of singular interlaminar stresses near the laminate

free-boundary, edge delamination is observed to occur under incremental axial strain.

Delamination can be simply interpreted as separation of laminae from each other in the

laminate, and can occur under static, impact or fatigue loading conditions. Based on

the discrete laminated plate theory, there are many computational techniques developed

to calculate the stress components in laminated composites based on either

displacement-based or stress-based approaches to predict free-edge delamination.

Investigations of the free-edge problem were carried out by Puppo and Evensen

[1970] using a composite model essentially consisting of a set of anisotropic layers

separated by isotropic adhesive layers. It was assumed that the isotropic layers,

11

developed only interlaminar shear stresses, between the anisotropic layers. It was

reported that a sharp rise of the interlaminar shear stress could be observed in finite

width laminates. However, the simplicity of these elastic formulations precluded

calculation of the transverse normal stress and the problem became more complicated

when more layers were involved.

In an attempt to approximate the interlaminar normal stress, a simplified

formulation was developed by Pagano and Pipes [19731 for the free-edge problem in

laminate elasticity. The strategy was to use solutions along the longitudinal midplane

of the laminate based upon classical laminated plate theory in conjunction with an

assumed distribution of o'7,. One could then compute the force and moment resultants

caused by the interlaminar stresses on any plane z=constant through consideration of

static equilibrium. The maximum interlaminar normal stress at the free-edge was thus

expressed in terms of the transverse stress in the y-direction calculated from the

laminated plate theory. This assumed distribution, however, was based solely on statics

considerations and contained no description of the influence of material and geometric

parameters on the interlaminar normal stress [Pagano 1973].

Another approximate elasticity solution proposed by Pipes and Pagano [1974] was

based upon displacement-equilibrium equations for an anisotropic elastic medium.

Assuming the transverse stresses in the y, z directions to vanish, the equations were

written in terms of the single variable U (axial displacement function). This yielded

components of displacement, strain as well as remaining stress fields in the form of

sinusoidal-hyperbolic series. However, violation of stress equilibrium in the transverse

directions as well as neglect of the interlaminar normal stress constituted major

drawbacks of this scheme.

Pagano [1974] derived another approximate method for determination of distribution

of the interlaminar normal stress only along the midplane of a symmetric, finite

12

width laminate. The approach was based upon a modified version of a higher order

theory proposed by Whitney and Sun [1973], which recognized the effect of shear

deformation through inplane rotations as well as the thickness strain in the assumed

displacement field. However, like the approximate theories discussed previously, none

of them was able to determine the complete stress field near the free-edge.

A boundary layer theory for laminated composites in plane stress was developed

by Tang and Levy (19751 from the three-dimensional theory of anisotropic elasticity.

By expanding the stresses, displacements, body forces and surface tractions in power

series of the half-thickness of a lamina in the equations of equilibrium, compatibility

and boundary conditions, a sequence of systems of equations was obtained. The

complete solution was obtained by combining solutions of the interior domain based on

the classical lamination theory and those from boundary layer and matching a set of

appropriate boundary conditions. This formulation provided a way to obtain analytical

solution for estimating interlaminar normal as well as shear stress distribution, but

became Loo complicated with increasing number of layers.

In order to have displacement as well as stress continuity, a mixed formulation is

sometimes used. Unlike the elastic approximations discussed previously, Pagano [1978]

developed an approximate theory for a general composite laminate based upon an

application of Reissner's variational principle. The assumption was that the inplane

stresses are linear in the thickness coordinate while the transverse stresses derived from

equilibrium consideration are cubic. Substitution of stress components based on the

differential equations of equilibrium and the strain energy density of an elastic

anisotropic body into the Reissner's variational principle, integration with respect to z,

and setting the first variations equal to zero yields the appropriate field equations and

the boundary conditions. The field equations, which consist of the elastic constitutive

relations and the differential equations of equilibrium, must be satisfied within each

13

layer. If a laminate or a single lamina is viewed as an assembly of N sheets, each

having a finite thickness and required to satisfy force and moment equilibrium, the

analysis leads to a set of 23N algebraic and ordinary differential equations which had

to be solved simultaneously. Based upon the assumption that the stress field in a

free-edge delamination coupon is independent of the longitudinal axis, Pagano [19781

further specialized the theory to the free-edge problem by reducing the stress field

determination to the solution of a one-dimensional problem. However, the number of

layers considered in the solution process could not exceed six. The manner in which

singular behavior was described could not consider larger number of layers.

Pagano [1983] introduced a global-local model, which was able to define detailed

response functions in a particular, predetermined region of interest while representing

the remainder of the domain by effective properties. This reduced the number of

variables in a given problem. In this model, for the global region of the laminate,

potential energy was utilized, and the displacement components were based upon the

assumption given by Whitney and Sun [19731 The Reissner variational principle

described in Pagano (19781, however, was applied for the local region in which a

thickness distribution of the stress field satisfying equilibrium equation within each

layer was assumed. A variational principle was then used to derive the governing

equations of equilibrium for the whole system. However, for application to

delamination of laminated composites, it is sometimes hard to identify the location

where the delamination will occur in the delamination process.

Pipes and Pagano [1970] used the classical theory of linear elasticity to formulate

the problem of free-edge delamination of a strip under uniform axial strain. Allowing

for material symmetries and uniform extension, the transverse components of

displacement were assumed to be independent of the longitudinal coordinate. The three

coupled elliptic equations for the displacement functions were solved using a finite

14

difference solution technique to approximate the interlaminar stresses. Delamination

was assumed to be primarily due to the high shear stress near the free-edge and the

interlaminar stress field was found to be an edge effect which was restricted to a

boundary region approximately equal to the laminate thickness.

A three-dimensional finite difference analysis was carried out by Altus, Rotem and

Shmueli [1980] to examine the free-edge stress field. The displacement equilibrium

equation was solved by using central difference method while for displacement or

traction-free boundary conditions as well as interfacial continuity conditions, either

forward or backward difference scheme was applied. Convergence of the solution was

expected providing a reasonable displacement field was assumed initially. Although a

complete stress field was available due to three-dimensional characteristics, an iteration

scheme could be a serious inconvenience.

Wang and Crossman [1977] used 392 constant strain triangular elements with 226

nodal points to model the laminate boundary region through a crosssection of

quasi-three-dimensional boundary value problem with orthotropic material properties

only. The functional dependence of the assumed displacement field was of the same

type as in Pipes and Pagano's analysis [19701 The traction-free boundary conditions

cannot be satisfied in this analysis.

A quasi-three-dimensional finite element analysis was carried out by Raju and

Crew [1981] using eight-noded isoparametric elements. In order to approximate the

stress singularities, a polar mesh was introduced near the intersection of interface and

free-edge and a log-linear relationship between stress components of or and oa and

the radial distance for the singular power was postulated. The power of singularity

was determined by fitting a straight line to log-linear plots of stresses calculated from

several mesh refinements near the interface of the free-edge and the radial distance.

15

Whitcomb, Raju and Goree [19821 further pointed out that the disagreement for

both magnitude and sign of the interlaminar normal stress distribution among various

numerical methods could be attributed to the unsymmetric stress tensor at the

singularity. In their approach, the problem was modeled by eight-noded isoparametric

elements. It was concluded that finite element displacement models were capable of

giving accurate stress distributions everywhere except in the region within two

elements of a stress singularity. In this analysis, the traction-free boundary conditions

cannot be satisfied.

Chang [1987] used pseudo-three-dimensional finite element analysis to solve

composite coupons by satisfying the traction-free boundary conditions, continuity of

displacements and tractions at interfaces. However, the stress-equilibrium relations were

not satisfied. Dandan [1988] developed a finite element analysis of laminated

composite axisymmetric solids and used it to solve for stress distribution in laminated

composite coupons. However, in this work, continuity of tractions was not satisfied.

Rybicki [1971] used a three-dimensional equilibrium finite element analysis

procedure, based upon minimization of complementary energy, to solve the free-edge

stress problem. However, this method involved very large matrices and was

computationally expensive, and even at that did not yield a continuous stress field.

In Pian's hybrid model [19691, stress equilibrium in the interior of the elements as

well as displacement continuity along interelement boundaries are ensured, but the

interelement stress continuity is satisfied only in a weighted integral sense. Following

Pian's formulation, Spilker [19801 developed a special hybrid element for the edge-stress

problem in cross-ply laminates. In his work, the assumed stress field was made to

satisfy exactly the continuity of traction across interlayer boundaries as well as

traction-free conditions along exterior planes of the laminate. A comparison of various

methods for solving the FED (Free-Fdge Delamination) problems is given in Table 2.

16

Table 2: A Comparison of Various Methods for Solving the FED Problems

Ref Author method of analysis calculated stresses

28 Puppo & Evensen Elastic approximation (YyTX2 Txy

26 Pipes & Pagano Approximate elastic solution aX 7 XYTXZ YZ

41 Tang & Levy Boundary Layer theory O' Xa YZTyzY

20 Pagano Reissner's variational ax(Ty azTyzTxzTxy

principle--mixed method

22 Pagano & Soni Global-local model ax yrz7y7x1x

25 Pipes & Pagano Finite difference method axOyO'zTyzTxzTxy

43 Wang & Crossman Finite element method: xOr y 0rzTJTyzTX3y

constant strain triangle

Finite element method:45 Whitcomb et al. 8-noded isoparametric rxry(TzTyzTxy

element

32 Rybicki Finite element method: O'xCrzTy2Txz"xy

equilibrium stress approach

36 Spilker Finite element method: ay Z yz

hybrid assumed stress model

5 Chang Finite element method: a X r'zTZ Tyz T xy

Q23 element

6 Dandan Finite element method: aY a a T 7 Tx y z yz xz x y

Axisymmetric element

2.3 Need for Solution Procedures

Recent development in the analysis of composite laminate coupons under uniform

extension indicated that the high interlaminar stresses near the free edge are mainly

responsible for delamination failure, [Pagano and Pipes 19731 Before delamination can

be predicted on the basis of a stress-based failure criterion, it is essential that a highly

17

reliable estimate of interlaminar stresses be available for the given situation. i lowever.

it has been difficult to obtain solutions for the stress field because of the anisotropy as

well as heterogeneity of the material, and the difficulty in satisfying traction-free

boundary condition in a solution procedure based on the displacement formulation.

Some of the solution techniques are only applicable under certain conditions. For

this reason, a complete stress distribution is hard to obtain. Although results calculated

from various approaches have demonstrated similarities in some cases, discrepancies do

exist in the magnitude as well as sign of the computed interlaminar stresses near the

free edge of laminate coupons. One possible source of these discrepancies is that, in

these methods, the continuity conditions for displacements and tractions across laminate

interfaces along with traction-free boundary condition along free-edges characteristic can

only be approximated to a limited extent. Chang [1987] solved the problem of a

free-edge specimen by a pseudo-three-dimensional finite element procedure based on

minimization of potential energy formulation while satisfying continuity of tractions

and displacement. However, this theory is not general enough to apply to a laminated

plate. Also the stress-equilibrium relation of composite layers is violated. Dandan

[19881 used axisymmetric elements to solve the problem of free-edge coupons while

continuity of tractions was not satisfied. Table 3 gives a comparison of theory of

Pagano [1978] with those of Chang [1987], Hong [19881, and Dandan [1988].

In Hong [19881 theory, the interfacial transverse stresses are not considered as the

direct stresses in the variational formulation. The tractions at the interfaces are

assumed to be continuous in the sense of interpolating traction components from one

interface to another. The traction-free boundary conditions also are not satisfied.

Pagano's [1978] approximate theory for a general composite laminate, based on an

application of Reissner's variational principle to define the six stress components, has

been the basis of solutions to some problems and has been used as bench mark for

18

Table 3: A Comparison of Theories by Pagano, Chang, Hong, and Dandan

Displacement I Traction I Application I ApplicationI Continuity I Continuity I to I toI I I Coupon I Laminated Plate

---------------- - ------------------------- ------- -----------------

Paganol yes I yes I yes I yes[197811 I I I..-----------------------------------------------------------------

Paganol yes I yes I yes I yes[198311 I I

+------------4-----------------4--------------------------4-----------------------------

Chang I yes I yes I yes I no[1987]1 I I I.. ------------------------.-..... -4--------------------------------------------------

Hong I yes I yes I yes. I yes[1988] I I I

4.--------------4--------------------------------------------------

DandanI yes I no I yes I no[198811 I I I

validation of finite element procedures. This theory appears to be an excellent

candidate for successful determination of stresses in composite laminates. The finite

element method can handle a problem of complex boundary conditions in a

straight-forward manner and modern computers can easily handle a large number of

algebraic equations. To apply the finite element method effectively and to develop

alternative strategies, it is desirable to write the governing equations in self-adjoint

form so that the general procedure for development of variational principles for

coupled linear problems can be applied.

19

SECTION III

PAGANO'S THEORY OF LAMINATED PLATES

3.1 Introduction

In this section, equations of Pagano's theory [1978] of laminated plates are

summarized. Starting from the equilibrium of an elastic solid, equations of generalized

equilibrium in two dimensions are introduced. Pagano's theory [1978], based on an

assumed equilibrium stress field, is stated along with derivation of constitutive relations

for generalized displacements.

3.2 Equilibrium of an Elastic Solid

For a three-dimensional solid, the differential equations of equilibrium for linear

elastostatics are

" + f. f= 0 (12)

where rToj are the components of the symmetric Cauchy stress tensor and f, are the

components of the body force vector per unit volume.

3.2.1 Generalized Equilibrium of a Two Dimension Plate

We consider a plate of uniform thickness h in which the plate is assumed to be

homogeneous, linear elastic. For the Cartesian reference frame used, the origin is

located in the midsurface of the plate (x1-X2 axes) with x3 axis normal to this plane,

but the range of X3 is limited to the thickness of the plate i.e, 13 = ± k.2

To reduce the equilibrium to an equality in two dimensions, the equilibrium

equations are integrated over the transverse dimension. Equation (12) and its first

20

moment are integrated over the plate thickness. Neglecting the body force components:

2 2

fh odx 3 + ft a03 ,3 dx1 = 0 (13)

2 2

I Ib

2 2

(f 306a dx 3 + f or 33.3adx 3 -- 0 (14)

-h -b

2 2

2 i

Mf - 0 xd 3 + a dx 3 0 (14)

.t_-

12 12

2

f f +a(0"d3 f 0" 3) xx 0 (15)

h

221

12

vPfro rcdx 3 d (16)

12

a- fr cdh (17)

0 32

N, + tm W 0 (218)&3~ 03

22

h+-( + ) -V =0 (23)M•, 2 -3 + 3

where V., N.B, M. are components of the force resultants.

3.3 Kinematics

For small deformation, the kinematic relations for linear elasticity are:

! (u w+ u. ) (24)

where r is the symmetric strain tensor. For a laminated plate subject to bending and

stretching, in order to reduce the problem to one in two dimensions, the functional

dependence of the displacements upon the transverse coordinate x3 is made explicit.

Often, the in-plane displacements are assumed to vary linearly within the plate.

Mathematically, for a plate, this can be expressed as

u.(xi) = Vo(xP) + x3 i;o(xP) (25)

where u. are the components of inplane displacement vector; v. are the displacements

at the midsurface of the plate; and ý. are the rotations of the crosssection of the

plate. Substituting (25) into (24), the strain-displacement relations for the plate become

e + x (26)

=-3 (u2 3 +u (27)

e33 '= U3.3 (28)

where we define:

e(o) I (- + v- (29)

Eo - •(v. +~ ;.) - (.)Co

2(30)

22

Many approximate theories use (25) or higher order polynomials in x3 as the starting

point. In Pagands theory, (24) is used as the basis of evaluation of generalized

displacements from strains defined through constitutive relations by the assumed

equilibrium stress field in each lamina. No independent assumptions regarding variation

4 of displacements over a layer or over the laminate are made. This is discussed in

Section 3.6.

3.4 Constitutive Relations of a Monoclinic Material

For linear elastic material, the general stress-strain relationship is

0.ij Ef Eijkl ek, (31)

where C., E ,k are the components of the infinitesimal strain tensor and the isothermal

elasticity tensor, respectively. In the absence of body couples, due to symmetry of 0.,,

and e., and the existence of an energy function

E•t kl- EW - Eklij - E ijk (32)

and the number of independent constants is 21. For a monoclinic plate, with

symmetry about x3 O, the number of independent constants is 13 and the reduced

stress-strain relationship can be written in the form

0a E•, 6et, + E,- 33E3 3

0"3 = 0"3o, =2E,313EP3 (33)

033 =E E + E3e3 33or 3&iO' 3333 33

Inverse relationship of (33) can be written as

0,1 oy6(7y6 + S*P330.33

ES3 E3 - = 2S5 •0• (34)

E3 3 $33*0" j$ + 833330"33

where S •k are the components of the elastic compliance tensor with the properties

Skji = S kij f Sjk i S ijik

23

3.5 An Approximate theory

3.5.1 Equilibrium Stress Field

Equations (16) through (18) express V. , N.s , M., as force resultants of the

components of the stress tensor, determined over the plate thickness. The inverse

relationship, ie., stress distribution for given V. , N.,, M, is not uniquely defined.

However, if an assumption is made regarding distribution of some of the components

of Tij , the distribution of the others can be determined. For homogeneous plate,

Reissner assumed linear distribution of o',,, over the thickness. If oa.= a. + HRx 3,

substitution in (17) and (18) give

_ No (35)

and

H 1 (36)

i.e.

ac, am + 12Mx 3 (7)

The first two equilibrium equations, in the absence of body force, in (12) are

01 *,.0 + 0&3.3 = 0 (38)

Integrating (38) over the thickness of the plate,

0 tor3 = (--o"r.,)dx3 + oa3 (39)

Ii2

Substitution of (37) into (39) gives

X

S 12x( - h a-- (40)73 (------M )dx 3 + a" - (x 3 +

h h3 2 (02

Substitution of M.,,, from (23) into (40), we have

24

12x [Vo- (O+ -.f)] _h A"o3 02 ~dx3 + -c (x +-)0 (41)

3 J h 3 22

From (41), we have

3 [V.- _Sj[1-( )2]+ a- +(X + _)hP (42)r3 " 2h 2 h *3 3 2 h

Here we define4-

P o3 o"3

and

S. - -L0, -•32 0o3 o3

From the third equilibrium equation in (12), integrating along the x3 axis yields

a 33 =33 I(cr. 3 .0 )dX3 (43)

2

Substitution of (42) into (43), equation (43), combining with (43), yields

2 3+o+a) 2 X 3 X0-333 33 X ... (°. - 3.)( - ) + (0-+• + 0"•3) [.a - a]332 a3.o ot3,a 2h 8 (73a *. h 2 4

3 - -a4x3 (44)2 33 "33 h 3h 3

Following Reissner's assumption that in-plane stress components are linear functions

of x3 coordinate, Pagano assumed the in-plane stress distribution to be given by (37)

with (35) ie.

ack + 12x 3M (45)

h h3

but derived expressions for 0.3 and 0"33 in a form somewhat different from Reissner's.

25

3.6 Pagano's Equations of Equilibrium

Defining

h2

N33 f f 033 dx 3 (46)

h2

h

2

substitution of (44) into (46) and (47), with consideration of (22), yields

2

N t=o-h + -o.) + h 0--.) (48)233 3•' 33 1 203o o,3.)

M33 ("• + + 0T3 ) + "•-,3 - a-) (49)33 120 01. alo 10 O33 33

Equation (41), with (35) and (22), gives

+ ) 2 3V 420-r.• + °' - 0-3)13" +(03 + a. (12 L3 1) + -- (1_± 3-) (50)

3 O h 2 2-h h 2

Substituting (48) and (49) into (44), we have

-!C+12 3

3 3 _12 + _1(+ - 3]3) 40x3 6x

33 4"33 33)[ 4(33 33'h"h

3N3 3 4x 2 + 15M3 3 2x3 8x 3

h 2 + 2 h h3

Adding (22) and L times (49)h 2

20M3+ h + 0",,) 0(2

V +0 23 + ((r3- )- .(r +03a-= (52)Voo+ h 2 + 033 - r33) 6(0a3,a +

h2Adding (22) and times (49)

6M0 h +

Vo- + 2 +5+) -o3)--(a. + 0, o (53)h 23 33 2 o3.o o3,

26

These two equations may be used to replace (22) and (49). Summarizing, Pagano's

equations of plate equilibrium are:

N.,• + W0 -o ) 0 (54)03 *3

20M3 3 + -4k) +(k) h (.+(k) -4k))=0 (55)

V=2 + (a33 - 33 6 + a3,a 0 (55)h6

h -V +.!!(a- + 3a) 0 (56)

AA 2 *3 a

We note here the introduction of stress resultant M33 . This and the force resultant N33

are defined by (47), (46) and, for the assumption of a.o1 linear over the plate, by

approximation equations (48) and (49). Equation (49), as we have shown, may be

replaced by (53) yielding the following equations as completely defining N,, M3 3 for

Pagano's theory:

h(ao + a+) 2

N +- -i.3.(03, ,& , - )=o (57)33 2 12

60M 33 +

V +60 5 (0- - )or, 1(a-* + or ) (58)233 33 2

3.7 Constitutive Relations for Generalized Displacements

Substitution of (45), (50), and (51), taking account of (34), into the variational

equation of Reissner Complementary Energy and integration with respect to x3 leads to

the appropriate field equations e-d boundary conditions. In the derivation of the

governing equations, the integration with respect to X3 gives rise to weighted average

displacements at the surface of each layer. Pagano introduced the notation

27

- h f h dx3

-h2 xf =5 f dx3 (59)

-h 2 h h

h'/2 4x 2=f f -3 - dx_h/ 2 h2 h xs

where f may represent any function of x3 The constitutive relations given by Pagano

[19781, neglecting the expansional strains, were derived by integrating with respect to.7 3,2X, the quantities Es., x 3 ,,, E33, x3E33 , X32 33, X3E 33 , E-,3 , x 3E,.3 , and xE"3 . In evaluation of

the integrals, repeated use of integration by parts and substitution for stresses o,,, 0r33

for equation (50). (51) was required. The resulting equations are:

h

f -dx => u 2 + So 3 3N 33 ) (60)

_h

2

h

h

x e Nx + S* NS(1

f 333 ----> Ul3 - U3 -- S 3 3 aN y + S.3333 (62)

2

h

fhXE3x --- 3=U + I1- - S3fM - 25333M3 (63)

2

4XEdx=>6U+-S 3 ~ u -hN,+S No (62)-S

f h33 3 = 63 33N 3333 33+

_h

2

S3(u+ - -is (64)

28

h

2

h(U + U-)3 _(2 S

3633 dx 3 => U3• 8S- #v _MA÷ u-Ls6

333 3 3 -( 3 3 5h 3 3oMp 7hS 3~I3 333

_h

2

hh65

fS (h.+ - ( "-) (67)105 333 33 3

2

3 dx 3 => u 8 v (u+ - UD (66)

2

h

> 1- h + 67fx3 6 3 dx > -U -- u. -s. (.0 + - a ...o + o 6-f 333 4 3.k20 3 303 3 03 2 0

h2

f 2 edx =>t u 8L (a+~ + a- + -Ls VX3-33 3,* isO,3P3 03 03 h "3#3 0

2

- (u+ - u. (68)

The above equations contain quantities u+ and u.- which are apriori unknowns.

Combining (63) and (65) to eliminate u+ + u- gives

38S-33-3-Ls M +A- 5 M -LS (0.+ a-o3) (69)33 3 5h 33* A 7h 3333 33 35 3333 33 33

Combining (62) and (64) to eliminate u+ - u- yields

6u3 =2S3 oN,~ _L2 N h 5 l (+ +o0-) (70)63 33oN A 3333 33 5 3333 33 33

Combining (66) and (68) to eliminate u+ - u. gives

- 7-+ 8 + ) 2

3,a-ah 15 o33 03 03 5h 33 (71)

Equations (60), (61), (69), (70), and (71) are Pagano's constitutive equations for

displacement functions in terms of force resultants and surface tractions.

29

3.7.1 Equations of Displacement at the interfaces and Interface

Conditions

Equations (46) through (71) apply to each lamina in a laminate. The positive or

negative signs of the superscript denote, respectively, the top or the bottom surface of

the lamina.

3.7.1.1 Interface Displacement Equations

The displacement at the interface can be obtained [Pagano 1978] by combiningsome of the equations (60) through (71). Equation (67) gives u: + uZ in terms of

displacement functions and surface tractions as

+ u_" h u + +2h S (0-+ '3)(72)

a 0 2 3.o 3 , 303 A3 033

Equations (66) and (68) both involve u+ - u-. Combining these two equations gives

u - u = 2h[---3u + I u + 5 - 3 1 " h S (a+ + o.3) (73)

A8 3,& 8310 2h *'3P3 P4 S V5 $o3#3

From (72) and (73), the in-plane displacements at the interfaces are:

u 3 h(-!l 1 - 3 _ h ,Uor:=--( 3. -•- 3.° -- U•)- (-u - "-Uo)CV 8 8 2h c

(4cr_ - o )h V+ 4S% 3 -3 A0] (74)

3 30 10

u '=h(u3 13 h i8S3.[ U 30 h 4 + 2

-(4a - a +3)h V4S 3 P -3] (75)

a303 30 10

Clearly, other combinations of (66) and (68) are possible. Combining (62) and (64) to

eliminate N. yields

u+ u3 = 3u* -IS N +h-Ls (a+ o+ ) (76)- 5 ."3333 3- 10 -'3333( 33 33

30

Combining (63) and (65) to eliminate M., leads to

u;+ + u - 1 + S (o o)Lh + - o(0) (77)

From (76) and (77), out-of-plane displacements at the interfaces are:

u3 .- (5ý -iui)+ -!u3 + S333 [-o+7hN -37- 1h (78)3 4 3 2 ~ 70h 33 33 33 M33)

u- .3 (5a - 3) - 3u -$3333 [(6o•3 + or3 )h-2 7hN + 30M (79)U " 3 2 3 70h 33 33 33 + 33

3.7.1.2 Interface Continuity

Using superscripts in parentheses to denote the identifying lamina number, the

condition of continuity of displacement and traction at the interfaces implies that for

k=1, 2, 3, ... . N-I,

a -Ak) a qk~l) (80)03 i3

u.<)(>-t /2) - u<k+i)(ti/2) (81)

where t, is the thickness of the kth layer.

3.7.1.3 Prescribed Tractions

On interfacial planes, tractions or displacements may be specified [Pagano 1978] in

the case of a cracked or unbonded interfacial region. Symbolically, if tractions are

specified,

-(k) -(k)i3 i3

(82)+(k' I) +(k+ I)

0"i3 i3

for k=1, 2, 3, , . N-1

31

3.7.1.4 Prescribed Displacements

Pagano [1978] allowed for specification of displacements at interfaces i.e., for k=I,

2, 3. .. N-1,

u(kt i-t/2) U=k)Sk

(83)

Sk-ti I

The field equations, which consist of the elastic constitutive realtions, equations of

equilibrium, and equations of continuity in displacements and tractions, must be

satisfied within each layer for the composite laminated plate.

32

SECTION IV

VARIATIONAL FORMULATION OF PAGANO'S

THEORY OF LAMINATED PLATES

4.1 Introduction

To implement the theory described in Section III in a finite element analysis, a

self-adjoint form of the governing equations along with consistent boundary conditions

is desirable so that Ritz type variational formulation can be employed. In this section

we outline the basic procedure developed by Sandhu and Salaam [1975] and Sandhu

[1976] for variational formulation of linear problems and then proceed to apply it to

Paganos theory of laminated composites.

It is shown that Pagano's equations as originally stated and described in Section II1

do not readily lend themselves to a self-adjoint formulation. A modification, reducing

the number of field variables, is introduced to ensure a self-adjoint formulation. A

general variational principle for the problem is stated. Extended variational principles

are developed using self-adjointness, in the sense of the bilinear mapping used, of the

operator matrix. Specializations to reduce the number of field variables by requiring

that some of the field equations be satisfied exactly are derived. The general approach

provides a basis for development of consistent finite element approximation procedures.

33

4.2 Basic Variational Principles

In order to develop variational formulations for the multivariate problem of

laminated composites, it is necessary to introduce some definitions and procedures. We

summarize here the approach introduced by Sandhu [19761 for a self-adjoint linear

operator.

4.2.1 Boundary Value Problem

A typical boundary value problem is defined by the set of equations

A(u) =f on R,(84)

C(u) = g on S

where R is an open connected bounded region in an euclidean space, S is the boundary

of R , and R is the closure of R. A and C are linear bounded operators. The operator

A is the field operator and C is the boundary operators

A:WR-V R (85)

C : Ws-Vs (86)

where VRVs are linear spaces defined over the region R and S and WR,Ws are in

general, dense subsets in VR,Vs , respectively. Operator A and C are linear implies

A(au+bv) = aA(u) + bA(v) for all u, vEWR (87)

C(au+bv) = aC(u) + bC(v) for all u, yEWs (88)

where a ,b are arbitrary scalars.

4.2.2 Bilinear Mapping

A variational formulation of the problem seeks to set up an equivalent problem so

that the search for u0 EW for known f corresponds to the search for a function F

whose stationary points are solutions to the given equations. The function is based on

use of suitable bilinear mapping B, and B. such that

BR : VR X V -- S (89)

34

Bs : V s X Vs-,S (90)

where VR,Vs and S are linear vector spaces. To each ordered pair of vectors

u , vEV , B assigns a point B(u,v)ES, such that:

B(u , v) = B(v , u) (91)

B(au +bu 2 , v) = aB(u 1 , v)+bB(u2 v) (92)

B(u , av+bv2)- aB(u v,)+bB(u v2) (93)

where a ,b are scalars. BR is said to be nondegenerate if

BR(u , v) - 0 for all vEV if and only if u = 0 (94)

A variety of bilinear mapping have been used [Sandhu and Salaam, 1975].

423 Self-Adjoint Operator

An operator A is said to be adjoint of operator A:W-V with respect to a bilinear

mapping BR( , ) : V X V-S if

BR(u , Av) = B R(v A'u) + Ds(v ,u) for all yEW ,uEV (95)

where Ds(v, u) represents quantities associated with the boundary S of R. If A ,A*

are linear, Ds(u, v) is bilinear in its arguments. If A* = A, this A is said to be

self-adjoint. If A is a self-adjoint operator, D(u , v) is antisymmetric, i.,

D,(u ,v) = -DO(v , u) (96)

A self-adjoint operator A on V is symmetric with respect to the bilinear mapping if

W = V and

Bs(u,Av) = Bs(v , Au) (97)

Two operators A, E are equal on V if they have the same domain and range and

A(u) - E(u) for all uEV

35

4.2.4 Gateaux Differential of A Function

The Gateaux differential of a function F : V-S is defined as

AF(u) = lim [F(u+Xv)-F(u)] (98)

provided the limit exists. The quantity vEV is referred to as the path, X is a scalar,

and if u,vEV then u+XvEV. If AvF(u) exists at each point in a neighborhood of u,

(98) can be equivalently written as

AVF(u) = d F(u+Xv),x=o (99)dX O

If the Gateaux differential is linear in v and there exists G(u)EW, a linear vector

space, and a bilinear mapping B : V X W-.S such that

AvF(u) = B(v , G(u)) (100)

G(u) may be regarded as the gradient of F at u. If B is continuous and

non-degenerate and G(u) can be identified with the residual P(u) at u , F(u) is the

potential of the operator P. This is the basis for setting up variational formulations.

42.5 Variational Principles For Linear Operators

For boundary value problem with homogeneous boundary conditions, Miklhin

showed that for self-adjoint, positive definite operator A, the unique solution u,

minimizes the function

fl(u) = BR(u , Au)-B R(u , 2f) (101)

Here the inner product was used as the bilinear mapping. Conversely, u, which

minimizes the function of (101) is the solution of the problem defined by the set of

(84). Gateaux differential of (101) yields

A~jl(u) lim-L-B(u+Xv, A(u+Xv))-B(u+Xv , 2f)-B(u , Au)+B(u , 2f)]

= B(u , Av)+B(v , Au)-2B(v , f)

= 2B(v, Au-f) = 0 (102)

36

if A is symmetric

In addition to the symmetry of the bilinear mapping, only linearity and

self-adjointness of the operator A are assumed in writing (102). The Gateaux

differential evidently vanishes at the solution u, such that Au.--f - 0 In order that

vanishing of the Gateaux differential at u = u, imply Au--f = 0, the bilinear

mapping has to be non-degenerate. To prove the minimization property, the bilinear

mapping has to be into the real line and the operator must be positive. However, in

general, it is only necessary to use vanishing of the Gateaux differential as equivalent

to (84) being satisfied. For nonhomogeneous boundary conditions, Sandhu [1975] showed

that for a linear self-adjoint A and C consistent 'with A, an equivalent function to

the set of field equations is stated as

(1(u) - BR(u , Au-2f)+Bs(u , Cu-2g) (103)

Consistency of boundary operators with field operators is considered in the following

sub-section. Sandhu [1975] pointed out that appropriate boundary terms should be

included in the governing function even if they are homogeneous.

4.2.6 Coupled Problems

If u is not a single field variable but consists of n dependent field variables, then

a linear coupled boundary value problem may be written explicitly as

n

F. A u -f on R i = 1, 2..... n (104)j-I

n

" Cuu = g on S i =- , 2 ... . n (105)ji!

Here A,, is an element of the matrix of field operators and C,, is an element of the

matrix of boundary operators such that:

Au : WR 4VR i, j = 1, 2,. . n (106)

37

C, : Ws-'Vs i, j = 1, 2,. . n (107)

where W., and W. are subspaces of VR, and VS, respectively. (103) is completely

analogous to (104) through (105) if u is regarded as an n-tuple u={u,, i=l,...n} and A

is a matrix of operators. Similarly C,g,f have extended definitions. Then uEVR

where V. is the direct sum VR - VR,+VR2+ ... VR. A bilinear mapping B, on

V3 is defined by

n

B R(u , v) = BR(ui, vI) (108)i-l

where BR, is a bilinear mapping defined on VR. The matrix of operators A,, is

self-adjoint with respect to the bilinear mapping if

n n

"B,,(u,, A ivi)=aa(v, , F A.uj) + Ds(v, u), i = 1, 2, . (109)ji -Ij- I j-1

The matrix of boundary operators is said to be consistent [Sandhu 1976] with the

self-adjoint matrix of field operators if D. in (109) satisfies

n

D(v. , u.) = B (v. CC u)-TBS (u. , Cv,) (010)

Substitution of (110) into (109) results in

n nl n nEB,(u. Av,) = BR1(V.. EA A •)+B,(v 1 . Cu)-EB (u3 ,v) (,,,)

j-l j- I j"l j-I

Sandhu [1975] showed that the Gateaux differential of the function defined by

(103), along with the extended definitions (108) through (111), vanishes if and only if

the (104) and (105) are satisfied. The functions approximating the field variables are

required to obey certain continuity requirements so that they are admissible as possible

solutions of (104) and (105) ie. each function uj lies in the domain of the set of

operators iAj , i - 1, 2,.. n H. however, in seeking approximation to the correct

38

solution by the finite element method, the region under consideration is discretized into

a finite number of elements and the field variables are represented by functions which

satisfy the continuity conditions only piecewise within each element. If the continuity

conditions along the interelement boundary are not satisfied, internal discontinuity

conditions, [Sandhu 1976] need to be introduced in the form

n

(C,, u) = g'. on R. (112)J=

where a superscripted prime denotes the internal jump discontinuity along element

boundary R, embedded in the domain R and g' are the specified values of the jump

discontinuities. Sandhu and Salaam [1975] showed that this condition can be included

explicitly in the governing function by simply adding a term and defining the bilinear

map over R as the sum of maps over individual elements.

4.3 Self-adjointness of Pagano's. Equations

4.3.1 Introduction

The field equations for a single lamina include the equilibrium equations and the

constitutive relationships. These are given, respectively, by (21) through (23) and by

(60), (61) along with (69) through (71). There are five equations of equilibrium and

ten constitutive relationships. For a laminate, however, the interfacial displacements

and tractions are additional, apriori unknown, field variables. The six surface

displacement components are given in terms of mechanical and kinematic variables by

(74), (75), (78), and (79). The field variables must also satisfy the continuity

requirement expressed by (80) and (81). Satisfying the constitutive relations (74), (75),

(78), (79), the continuity equations are replaced by relationships between mechanical

and kinematic field variables. In the following we restate Pagands equations for a

laminate and examine their self-adjointness.

39

4.3.2 Equilibrium Equations

From (21), (22), and (23), we have, for the kth layer,

V (W + W(k) _ 0 .-4k) 0 (113)

r,r 33 33

1 r N(k) + (crk) _-(k)) 0 (114)2 I ,r3 r3

-r M(k) v(k) + tk (0.+(k) + 7-(k)) O (115)2 OP r T r3 r3

where

r7 = (k r. + So .- ) (116)

4.3.3 Constitutive Equations

Equations (60) through (68) are nine sets of constitutive equations for the plate.

Equations (62) and (64) are used to set up (70) and (76) and can therefore be replaced

by the latter. Similarly (63) and (65) can be replaced by (69), (77) and (66), (68) by

(71), (73). Restating (67) as (72) and replacing (72) and (73) by (74), (75), Pagano

used (60), (61), (69) through (71), and (74) through (77) as the 16 constitutive

equations. Equations (60), (61), and (71) rewritten, for the kth layer, are:

--4k)I ) I1 (k) (k)2-r 2 " APO NA + -S 1Ap33N33 (117)

A(k)I 3u r 12 Wk .(k)+ 12 .(k) (k)

22 tk 3 #APO~ A

3 #SP33M33k tk tk

.(k)

3 ,(- ' ^'W U 24 W(k) 2 W +(k) Ak))-(U 3 . ) + 3 - Sp3 r3-V - 0-p3r3 (Or3 + (TO (119)4 .. 0 tkc 5tk p pr333 r

where

r 2 =(Sp + 8i (120)

40

4.3.4 Operator Matrix Form of the Field Equations

Combining the equilibrium equations (113) through (115), the definitions (48), (49),

and the constitutive equations (117) through (119) along with (69), (70), appropriately

modified to apply to the kth layer, the complete set of equations for the kth layer

can be written as:

[A fk){u,)(k) + [B'fk){(o"-(k) + [C'Ik){o"}k) = 0 k=l , 2 , 3 , , N (121)

where [A']"), [B'Ik), [CI(k) are operator matrices and {u'}(k), {cr}+k), {o"}k) are sets of

field variables, respectively. Explicitly,

41

[Aick) =

oo o 0 o 2'o

0 0 00 -F0 0 0 0 -2 2

0 0 000 0 0 0 0 0

4 S(k) BS(k)0 0 0 I0 0 0 0-__ 0

5tk 7 tk

00 1 2 s(k) 12 W(k)

33o#---S3333 0 010 (122)

(k) (k)

-12 0 0 0 0 _p3_3 0 0 022 tk tk

0 0 0 00 0 1 0 0 0

0 --- r2 0 00 0 0 _ 4 s (k s2 sW 0, t3 t3

k k

0 0 0 00 0 0 0 1 0

0 - & 00 0 0 0 0 24 s(k)

OP a t k p3),3

42

-8 0

0t k0

2

0 -1

0 ~tk s (lo

0 3333

o tk s(k)

5 3333

[B'k)= (123)

0 0

tk _ _ tk

12 2

0 0

3 2_tIi t~k

120 o 10

5 Sp3-,3

43

S 0)yo

2

0

0 tk s (k)

35 3333

tk S(k)0 5-3333

[Ci]k) = (124)

0 0

t 2 t

12 2

0 0

3 2

120 ov 10

2 .(k) 0-- '" p3"y3 0

±(k)y3

=k =1, 2, 3, .,N (125)

0a33

44

--(k)u

2

. (k)U3-.

tk

"""3 3-(k)-(k3{(u 3 "u3

3N(k) 4)

tU} = 6u Ak) (126)3

M(k)

N(k)33

V(k)

M(k)M33

V W)

The ten sets of equations in (121) are, respectively, (114), (115), (113), (69), (70), (117),

(48), (118), (49), and (119).

4.3.5 Displacement Continuity Conditions

From the first two equations in (81), we have

-4k) +(k+ l)u~-( u~ (127)

Substitution of (74) and (75) into (127) leads to

-(k)(k) I t k.kA 1 _k) S(k) (4(7 0-- 3)) V(A '~ ( 1?~ !i~k ~-!-u )--u~-~i~k)~S~) k)~ 4k- - )

kt8(3.o -i83.'- 2-itU. )- 4-U3.o-2c- )-4 *3 30 10

45

S 3 -(k ' 0 1 -- 4k ý 1) 3 .( , ) t k , 1 1( -- 4k 1)•-tk 4 Iu -- u - -uO )- 4 ,3o 2 --

k418 3.o 8 3.o 2 k u

-(4o -(k)a (k1)t )k.. V(k- 0

+ 4S -_ ] (128)o303 30 10

From the third equations in (81), we have

-(k) (k-t 1)u3 = u3 (129)

Substitution of (78) and (79) into (129) leads to

4 u k)_--k) S( 3 3 (k) -(k- )t 2 - W(k) +30M

4 3 2 3 70t k 33 33 k k 33 33 -o^ (k> 4k[(ý() 0)3 3t (k 1)k

4 3 2u3 7333, , ) -- (k) )- .(k. I)*1

+ tU3 [ -(k)6 + -(k+ O)2 7tN N4)--3OM (k+I] (130)70t k+I T33 33 3 • k i 3 33

Adding displacement continuity equations (128), (130) to (121) above, the complete

set of equations can be written compactly in the form

[L)(k) {r}(k) = (0} (131)

where

[Cfk) [A jk) [B'fk) [o] o0)[L] k)=] (132)

f{ ,}-(k- 1)

Iu'}(k)

{rl(k)= {-0ry)-(k) (133)

{u'}(k+ 1)

l(a,)-k+ 0I

with

46

2 t Sk)015 k r3p3

[Hit]k-1)= (134)

0 - tSW

"70 k 3333

-8 tk tk k 88 aO 0 0 0 S(k)

'Y 2 8 24 .- AP ar 5 r3p3

[Hj k)-= (135)

0 0 -1 _3 1_ 0 -- 533---L 0 333 0

2 4 10 7 tk

8 t (t 5rk)+ 5 (k+,1) 0k r3p3 k+I r3p3

[H 3k)-=" '(136)

0 3 (t kS(k) +t+ S(k+ 1)3 0 •k k3333 tk+153333)

Lk- 1 0 t- tk4IO 0 0 0 2(k4 1)

24 2 8 24 - 0 0 0r 5 5r3p3

[H_4 k+1)_ (137)0 0 1 -- 2 0-3333 -0 -333_._. 0

2 4 10 7 tk+I

S2 t •(k+ 1) 0

--5 k+ l-r3p3 0[H J + 0-[ (138)

0 ~ 1 t k+S(k+ 1)

70 k+1 3333

The two sets of equations (132) correspond, respectively, to (121), and to the pair

(128), (130). For the laminate, combining equations for the layers (k), (k+1) and

(k+2), we get

47

[Ht• 31 1 [H 41 k [lt JO) [0] [01 [01 [01

[C'P•) [A'JO [B'3fk) [0] [01 [01 101

H It -1) [H,2 k) 3 [H 3 P) [HJfk+1) [HStk+1) 101 [01

] (k= [0] [0] [C'k" [A]Jkl ) [B f(k+1) [01 [0] (139)

[0] [0] [HIt k) [t 2](k+ 1) [H 31(k*1) [H 41 k42) [H.S1(k+2)]

[0] [0] [0] [0] [C,k+.2 ) [AJk42 ) [B,]k.2)

[01 [o] [o,] [o] [11 f k4 0 [10, -2) [11 •ký2:

~3{u'}(k)1(7,}-(k)

{ri(k) - u(k+=) (140)

O.,}-(k+2)

{u'}(k+2)

For this matrix of operators to be self-adjoint, a sufficient condition is that [A'Ilk) and

[H]k) be self-adjoint and [B'lk) be the adjoint of [H2]ck, [H4]tk) of [C'lk) and [H11 k) of

[1lj+0k). This is not true for the above formulation. As the operators are all linear,

it would be possible to follow Tonti's [19671 approach and write the variational

formulation as a generalization of Mikhlin [1965] least square method. However, this

type of formulation when used with the finite element procedure, would require base

functions with a high degree of regularity. This would be prohibitively expensive to

use. An alternative is to use u-k), and uk as field variables instead of the

3 -- (k) )combinations - (u - ) and 3ua() -- u. This would yield a self-adjoint form.

48

However, a simpler strategy, based on reduction of the number of field variables, is

presented in the following section. This approach will lead to a generalized Ritz type

variational formulation convenient for development of finite element approximations.

4.3.6 Discussion

Pagano [19781 used seven equilibrium equations, (113) through (115), and (48), (49),

ten constitutive equations, (60), (61), (69), (70), (71), and six interfacial continuity

equations (80), (81), to solve for 23 field variables as given in (125) and (126).

Actually, there are only five equations of equilibrium and, therefore, there can only be

five corresponding displacement field variables. The. quantities N, and M33 must be

regarded as entities introduced for convenience and completely defined by (48) and (49).

Using (48) and (49) to eliminate N3 and M3., the number of local mechanical

variables reduces to eight requiring exactly eight constitutive equations. Noting that

*

u3, u3 may be regarded as defined completely by (69) and (70), substitution in the

remaining eight constitutive equations viz. (60), (61) and (71) gives exactly eight

equations for the eight local kinematic variables viz. u(..8), u.,.A) and uA derived

from the five global variables u,, u* and u3 . The total number of field equations

[Pagano 19781 is thus reduced to 19 in five independent displacement field variables,

eight mechanical quantities and six interfacial tractions and displacement components.

The set of interfacial displacement continuity equations and field equations constitutes a

self-adjoint system. The precise form is derived in the following section.

49

4.4 Self-adjoint Form of the Field Equations

In this section we restate the governing equations of Pagano's theory, after

elimination of N3. and M3, in a self-adjoint form. Essentially, this consists of using

(48) and (49) as definitions for N33 and M3 and to eliminate them as field variables

from the system of equations.

4.4.1 Displacements at the Interfaces

Substituting for N33 from (48) into (70) leads to

3 + -S (o-' + 0-3) + h2 (o- -- (141)2 -U3 '1 S 33o 4.'P43333(33 3 " 20 3333 r3r 0 r3,r

Differentiation with respect to xP gives

hu* = h N +L12 (+ + )+ hLS (c - (142)-, 12 33&$B Ap 24 -- 3333( 33'P 33,p 120 3333 r3,rp 0r3.r)

Substituting for M33 in (69) using (49) gives

3 u u)=6S M +-9hS W -0 )+- (of + WV (143))(3.3 - M 9h + - ,- h2 +

2 - 3 •h• 33o# '0 70 3333 33 033 • 3333 r3.r + 0r3,r)

Differentiating with respect to xP gives3 ^ 1-fI 1•3• ~ 2S + -

h(Tu -3 ) = )1S M + -h (+83,p8- 10 280 $3333("33,P 3 3 ,p

+ -• 33 -+ + -) (144)84 333or3.rP r3.rp)

Substituting (142), (144) in (74) and regrouping terms:

+ -3, h + 2u---- r N -+-S3,.3r3- M -S VP 2 P 2 P 12 33i A. 10 33oYA oA.P 5 p3r3

h3 S - 0+ - h + h-c

120 3333.(or3,rp r3,rp) + 3Sp 3 .3 r3 0 r3

h 3 + a.- + h (o+G-

-T 840 33( r3.rP 0 ,.r.P 5 p3r3 r3 r0

3 2 (a+ - h 2 , + (.0' -LS+ 0--(145)

2 8 0 h 3 3 3 3 33,P "3 3.p) 4 33 33,p 33,p

50

Similarly, (75) with (142) and (144) gives

Up I •-U 3- h-- S33&N 1 S M + 1Sp3r3 V,

3 2 2 2 1 + - h 10 + 5-' - 33V•'3r --'3,p + -'pr Tr"'3

h3 S (or+ - h + -s 7120 3333 r3.rp r3,rp 3 p3r3 0r3 r3

a- h S (++r

840 333ar3,rp r3,p 5 pr 3 r

+ -. Lh2S 67+ - a- h2 '+ + -

280 3333 333,p 3 -33p 3 3 333p (P36,

Substituting for N33 and M, from (48) and (49), with (141) and (143), (78) and (79)

give

43 3 3 2).033p 5h' 3 3aM '0"4"3 3 3 3 33 33

+ 1--'S" n73333733)2 1 +

+ h 2S f3 (+ +(7 )- ( -7 )A (147)23333-80 r3,r 03,r)' 24(O r3,r-- r3,r

U= I(u3- u3)--IS NMNap + 6 h W + a-)3 4"3 3 3h8 3 3*Mc-4 3 3 3 3 33 33

+ 33 33 . 3 - (3)

2 3 + (114+ hS3 r33 3 3 3 [-(r +C )--L(a )]r(148)290 r3.r r32r 4 r3,r- r3.r

rVk)Following Pagano [1978, we introduce v, and q) through the relationship

v--k) U4k

_. ___ (149)P 2

*(k)

3u _ (150)P tk

Using (149) and (150), (145) through (148) may be rewritten as:

51

(k +k-4k kik - t ks(k) NWk 1 L(k) M(k) 2 S(k) W"'P 2T 2 ' " 12 33cpwp~ p- T0 33ap tBpp " p3r3 r

(k (),+(k) -_ -(k) k s(k) (+(k) --(k))

120 3333 r3'rp r3.rp + T"p3r3 r3 -- r3

tk ( + +k - W) t•) 4 (k) -(k)4 (a333 (r3,rp r3,rp + p3r3 r3 r3

23 t2 s(k) (a +(k) a-(k), tk s (k) (a +(k) +a-(k), 11

-80 k 3333 33,p 3 3,p 24 - 3333O33.p 3 3.p

S(k) tk -(k) tk 4(k tk (k) (k) I.(k) (k) 2 (kUP 2 2 P 12-334 N.p + T O0 33a3 o.sp + 3 p3r3 ,

t3k• (k) 0+k -(k) • k •(k) (0,+(k)_ o-(k))

120 3333 r3.rp r3,rp 3 p3r3 r3 r3

t (k)o +(k) -(k))_ tk(k) +(k)_ -4k))

+ - (o + c a+40 3333 r3,rp r3.rp 5" p3r3 r3 r3

3 2.•t(k) +0-(k) -c(k)) ) W (+(k) -(k)) (152)

280 k 33 33,p 33.p 24 33.p 33,p

k( •1 -(4k) I 6(k) A)k) MW(k) tk S(k) (+(k) + -(k)

_== V3 ::t'•o33o6 + N k 33op Mc•'t" -3333- 33 + "33)

U3: 2 3 T2o 33~ A Stk 43 3 3

+ 1 7 W (7 (k) - -k))14+ ]• k 3333 033 - 33

+ t2S(k) [3 +(k). () 1+a- (k (153)tk 23333 8 -80 (r3.r3,r r3.r -4- r3.r r3.r

where

-4k) =3 --4 'k) (54)v 3 = (U4 -u (

52

4.A2 Constitutive Equations for Generalized Displacements

The eight constitutive equations for the kth layer are giver. by (60), (61), and

(71). Substituting (48) and (49) into (60), (61), we have for kth layer, with (149)

and (150)

1 ( ) 1-4k)= I (4k) 1 (W I Wk tk (aýk) +a-k) t (k)) (15.522V i(,4P)=t~ k S AO N t k S P3 312 3 33 12 r3,r r3,r

2 31 - - k) _ 3 •k) -1 2 (k) . (k) _ 1 2 ,S(k ) t , 4(k ) -0 0) , t k , (k) - -(k XrI2 tUpk,' -- U•.o -- ••ta• f -• •uP33 - G33 - "33 (a" 1-- [ r3.r Or3.rj (156)

2 ~ U~p 3 3p33 1 0 3 120 r,k tk tk

(71) can be rewrittei as

-,40+.--rk) =3(-(k) ai(k)) .3 U+k _24 (Ok) VWk 2s(k) (0"+(k) -(kv3.* 4 u" P- )+ t U -+ ~ rV 5 Sp33- 0r3 + a 3§ (1:57)

3, , 3P k P tk

Combining (113) through (115) and (155) through (157) the equations for the kth

layer, are

[Al]k)Iul(k) + [BkP){or}k) + [Cy k)i{a}(k = 0 k=1 , 2 , 3 , ,. N (158)

where [Aick), [B]fk), [C(k) are operator matrices and {u}(k), {0o-k), {o}-•) are sets of field

variables, respectively. Explicitly,

53

0 0 0 2Ir 0 0

o o o o o0 o 0 0 i -82 -a

o 0 0 0 0 -ar

[Afk) -- (159)

-L s(k)

0 -IF, 0 0 02sk2 t 3 #Papk

rp ap St k p3r3

-80

tkk

-- 0

t k 02

0 -1

[Bfk) t ks(k) Is(k) (160)

12 MiP33 Or 2 Mp3 3

I S(k) & 6 s(k)T 0lp 3 3 O r 5tk up33

2 s(k)0

54

80

"ta

k 02

0 1

[cY-k k s(k) _ 1S(k) (161)

"12 323 2 p33

1 S(k) 6 S(k)1 0uP33 0 y 5tk lAp

3 3

5 p33 0

a *W")3±(k)1a'±t)= k 1 , 2, 3, . ,N (162)

033

-(k)

v3,

N(k)

'M (k)V 3Y

v(k)

The operator matrix (Alk) is self-adjoint in the sense of (111).

Substituting (151) through (153) into (81) leads to, for k=l, 2, ... N-1I

[Ahyk){o}-(k - + }+ ( 1o-(k)+[k I){u)(k+ )+-l[+k(r-(k+ )= 0 (164)

where

55

t.k tk S• k Is~k) 02 s (k)

i'Y 2 12 o03 3 ap 10 & 3 3 Op- -5 p3-y3[•,o _-(165)

0 0 -1 1 (k) 6 (k) 02" ,33 " tk -30'33

t_ 2 _(k)

YP -2 12 03 3 0-- 10 of

3 3 ap -5 p3-y3

[Cfk) = (166)

o 0 i !(k) 6 S. o32 oP3 Sýtk

. sk) 6 (k)=11 12~

= (167)

.4k) .4 k)j

21 22J

with

=(k) ),(k) t 3_Y + (I 1 1)t s (k)

120 8ý40 *3 3 3 33kp~ 3 5 k ph3)

+ (_' 8 _ ,_. ) S(k+l) 2 + (.1 +S(k+()

120 84o 33 33 kl aph 3 52 k+ 1pýy3

.(__) + _1 W•k t2&_0_ 3 1 -,(k-1) t 2 a"12 "(280 24 "s"3333 k ap -- 8"0 + "j4-"3333 k'+l a"

--(k) _ 3_1 + ,•S(,0 t2 a +• 1 + ý (k+ Ot2 ,= 21 280 -24' +"'3333 k' +- 2 4" 280"-S3333 k+l O

-4k) (1 17 )(k) t 1 17 )-(k+ 1)22 4 140 3333 k 4 140 " 3333k-+1

and

A11 A 121

(Afk) (168)

with

(W 1 1 (k) 1 a21_I _~ (k)All - --20 840 )sk3333pt k --- 3 5 k p3-y3

56

AWk) ( 3 + ( + )S(k) t2 a_12 280 24 3333 k ap

Wk _ _Sk 26A 1 24 280 333 k ft

(k) W 1 17 .- (k)

22 4 140 333 k

[ f I ) = " 1 2

(

[• " (169)

21 22

with

ý k+" 1 ) 1 " I(k4I)t3 a !2 "-1 s(k÷l0

11 1 i2-0840 333 k-I ape3V 3 5 k- I p3y'3

?ý2k+ 1) 3 1 A (k+1).2 a

12 280 24 3333 k-1 a

•k+1) (1 + 3 )S(k+1)t 2 221 24 280 3333 k80

k 1) 17 (k+t

22 4 140 3333 k4l

The two equations in (164) restate (81) using (151) through (153).

The quantities {or}*") and {lo}(N) are given for a problem and appear as forcing

functions defined as:

&(0)

{or}(I) - {0 }.4N) = (170)

&(0)-()c33 C33

Define, 0QY") and {Q}-(N) as

[Q 11 ) = [At')o-{}(o)

57

, 1 ) 3(o) + 1 ( ) 3 1 (1) 2-(0)

12 4 33Ir3,r 3 5 S p31-3 at.3 +-280 +2i4 )S33 I a3 3,p=[ (171)

! 1 3 ,..(l) t (o)+,1 17 .(1) &(o)I

24 _ 280 3333 t 'r3.3j,,+ • 140) 3333tj 33

and

[Q](N-,) = [--•N)1 1 .-(N)

( 1 (N) 3( .N) 1 (N)- N) 3 )SIN) 2&(N)

120 840 ' - 3333t N ar3."p + -- 3 +5 )tN p3r3Tr3 2+-80 24j 33333tN 33 .P (172)

(I1 3)S(N) 2 ^(N) (N) 1 17 )&(N)

24 280 - 3333t N r 3 1r+S 3 3 3 3 t 14 10 33

where superscript (0) denotes the top of the 1st layer and (N) denotes the bottom of

the Nth layer. Combining (158), (164), (171), and (172), the complete set of field

equations for the laminate is:

[XI {Y} -- Z) (173)

where

58

[AV' [BY") 0 0 0 0 0

A]O'[ 0 --f' 2) [NY o o o

o [CfY2 [AfY2 [B)f2 o o o

o [APf2) 2) -l ) [.f3) [-Au 3) 0

0 0 0 [ct3) [AY3) [BY3) 0

o o 0 [A [3) [Bil) [=f3) [-A4)

0 0 [C]14) [ABf) [BY 4) 0 0 0

[A•• [B•• 0 0 0

o 0 o 0 [f N-1) [J-N1) 0

[c]d -1) [A](NfN-) [BfYN-) o

0 [A] N- 1 - 1)[-=YN-1) [CIN)

0 0 0 [CYN) [AIN)

59

p)(1

icT} )

0

{UI(2) 0

}-(2) 0

o)(3)

0

0

0

0

{U}(N){Q(I

-(0)a -y

[cI" 1)&(0)1a.33j

60

&(0)

tj-(0)

T Or

-(o)7 33

(174)

* l2 s Ap 33 oT y3.,y + 2 S u 3 3 0r33t1 (1) -(o) 6g • m (o)(1 4

10 s jp33 G3, 5t S lAp33 33

10 o~3 c( ) -6 So) 0

:5 " p3y3 73

&(N)

{p}(N) = [BIN) I&(N)I

O*33

-(N)

2 -3

--&(N)

33

tN s(N) -(N) +1 (N) -(N) (175)

12 pap33 ),3,1 2" A 3p33

1 s(N) (N) 6 s(N) &(N)

(N) '(N St sp3 3 33S•up33 *y3,"/ - 5t-" N p 3 3

2 S(N) &(N)

5 P3 ),3 V3

61

4.4.3 Adjointness of the Field equations

For the operator matrix [X] to be self-adjoint in the sense of (111) a sufficient

condition is that [AT", [E=k' be self-adjoint and that [C--kl, [Crk); [Alk), [M]k);

[Brk), [RjkI constitute adjoint pairs. Using the symbol <, >R to denote inner

products i.e.,

= f fgdR<f ' g> R f R

where f, g are functions defined over R, we establish the following relationships for

the field variables associated with the kth layer:

4.4.3.1 Operator matrix [ A f k)

Considering the operators 4 and A of the operator matrix [Ark), taking inner

product of an arbitrary function in the domain of F2, with I- r , application2

of Green's first theorem [Kreyszig, 19791 gives

W9k W-(W--( - <• ;I W 1 , (W)< -! 'Y A 1 4N , P > R ( ") : - ,Y T IF -I O > R (k )

= -<N(k) W r (k)> + <(k) N W(k)>

- <N(k) A(k)W(k)> +< W (k) k> (176)Azp 41 y ~), , 7A•T , v s(k)

Here RIk) is the configuration of the kth layer and S(k) is its boundary. Similarly,

considering the operators A('), -A(') and taking inner product of (,- an arbitrary

function in the domain of F2 , with -1 1,02) Green first theorem [Kreyszig, 1979]

gives

-- (k A~)M~)k))1 (k))

= _<M(k) IF A(k)> +<0M >) ,A) ,.< (k)2

,uMP' , 2 '> R(k) W + <s W

62

= <:M ,A 2 >()+M 3T (k) -(k) (k) -(k) •(k) (77<MAP , A 520-y > R(k) + < Moq ,_o . S(k) (177)

For operators A(k) and .AW Green's first theorem [Kreyszig, 1979] gives, for an arbitrary

function of v in the domain of R(k),

< iL) ,A V(k)v > -2 (k) < V >( 36 ) ROL ) 3 & Rk)

(kk)

= v-k) W (-) > + <V(k) 7o 0(0)>I v 3,a R(k) 3 SOL)

Operaors-(k) , A*k) ,-- W are tensors which are symmetric in the sense of (97). For

(k ) k)

"sAk)N > (k) sWP AJUP 44cr "'4'•'R W <W AP t tk >Iop < N.0• t k Mp Rk W

-N 00 A W)N(k)> (179).i • •'"44 AAu •ROL)

(W AM(k)> (k) M2 S(k) W k)AS .55 >R <Mk -pr Mk> W=<M~,1S~k

'up ) SS a R()--- 'up 3 - a Rk ac 3 /AP R Wtk tk

= M(k) A(k) •(k)> 1OA0 A 55 p (k) 180)

-(k) (4Sc)

"<VW Akvk> )<Vpk S v(k) > < VW) 24S 3), 3 v(k) (.

p , A 66 -y R(k)-" p 5 t tk Y Rp <V St

= < V~(Ic) A~(k)>S' 66 p R(k) (181)

The pair A•k), "'Aý consists of linear algebraic operators which are trauispose of each

other. This fact along with (176) through (181) satisfies the requirement of

self-adjointness of operator matrix, [A0k).

63

4.4.32 Operator matrix [ I ki

Considering the operator matrix [=l] , the diagonal operators 22) is symmetric in

the sense of (97) because, for &rkI , ok defined over•-k -(k)=W -(k)• - W -(k) - -(k)_ 12

<33 ' -22 a"33 >R(k)= < (k33 ' 22 C33 > R(k) (182)-- =ioe k) -W -(k)

For the operator taking inner product of &-(k , defined over ,with a,3

the Green's second theorem [Kreyszig, 1979] gives

,4-(k) -(k) - -(k) I I (k) t3+ +(k-l) 3 )a-(k)p3 I -/3 •R W<0'p3 120- 840 3333 k S3333t k4 I )r3,py

+ (+!XtkS) W+tk Sk4 1 ))ao-(k)> W3 5 " k p3.3 y3 Rk ,)

<()-(-(k (k) 3.-(k-)3 &-(k)3 120 840 3333tk 3333 k I- p3,py

+ I 1 _ t S(kW + t s(k+ ) & k)>+(-3 5X k p3ay3 k+I p3 y 3 ),3 R(k)

+ <-(k) 1 1 )rS(k) t 3 +S(k+ ) t 3 - (k)>Y< 3 7)y), 120 840 3333 k 3333 k+I sp3'psk)

- )(k) _ 1 I W t 3 +S(k+1)t 3 ]0 -4k)>

a<,3 7 ' 120 840 3333 k 3333 ktI p3.p S(k)

-(k) .(k) -(k)y3 ' - I & p3 R(k)

-(k) (__ 1 XrW(k) 3 +(k+ ) 3 ]-(k)>+ <o&3 ' 12 840 8 3 3 3 3 tk+S 3 3 3 3tk++o p3p>SW(k)

-k) I 1 S(k) t3.+S(k+ ) t 3 ]&-(k)

y3 ' 120 840 3333tk 3333 k+l p3,p >s(k)

For the off-diagonal pair =(k) and SLk) setting up an inner product of =, 2 (k) and an

arbitrary .4k) in the domain of -k gie

< -(k) -Jk) -(k)_ < -(k) ( 3 - S t2+_<k 2 (k+ 1) 2 -- -(k) >p3 ' -12 "33 >IR(W)= p3 ' 2 24 3333 k -3333 tk+l o33,P R W)

"2033 2 24 3333 k 3333 k+I p3,p R(k)

<& W 1 + (k) 2 -,(k Wl) 2 ] W >

P3 kP 280 24 3133t k _3333 k 133 sk)

64

_~~~~~~~ -()•k•() r(3..I••k 2 •(k+l)t2 ]aT-(k) -00()> 14

" <(T33 ' 21 p3 R(k) < 280 24 ' 3333 k-- 3333 k+I 33 ' p3 7OP> (k) (184)

Equations (182) through (184) ensure the self-adjointness of operator matrix []m.

44.3.3 Operator matrices [Cfk) and [ICk)

Considering the adjoint operators C &' , C"; C'- , dk ) of matrices [C](k) and [(Cik),

taking inner product of Nl , an arbitrary function in the domain of C}, with

C4kok)-,*, Green's first theorem [Kreyszig, 19791 gives

14p I ۥ4 0 r3 RM)- p ' 12 pjp330 "r3,r R(k)

<+ k(k) ,(k) k < Wk) tk W(k) (k)

< (7*(k) t k S WN W> +<(T +(k - kSW W>W0 ' f2- cv833 a'Ar R(k) j• 0 r ' 1-2- P33 '6 s"

<or +(-k) & W >(k) <7 +(k) tk (k) (k)

03 14 , 0 R(k) 0"r3 1 T2 , 033N s (185)

Similarly, for the inner product of an arbitrary function Msin the domain of c',

with Cd)-r•k), Green's first theorem [Kreyszig, 1979] gives:

<-(k) dk) +(k) k).+(k)MAP 510 r3 >R(k)- < 'P ' -- O SAp cr3,r >R(k)

+(k) 1 ,(k) M(k) +(k) 1W(k) W(k)>r3 ' 10T o,33 o,,r >R(k)+ < (Tr3 71r' " To S*P33Map SW

W +( k)M(k) +(k) 1 k W

r3 ' 15 a Rk) r3 r' 33 S

Each of the six pairs of algebraic operators viz. d' 2 1 12

k , k C k)' , C ;; d) ; , r7k consists of operators which are

transpose of each other. This fact along with (185) and (186) ensures the adjointness

of the pair [Crk) and [Ck).

65

4.43.4 Operator matrices [A fk) and [Tjk)

Considering the operators V, and A,, of the operator matrices [A [AT),

application of Green's second theorem [Kreyszig, 19791 to the inner product of or,).

defined over, Aý) with 7-(),- implies

W .() 7-(Y3.-k <d+k) , 1 )3 -(k) I . 1, --(k) Wr-(k)p3 ' I1 r3 1(k): p3 " 120- 840 3333 k r3,rp-3-5 k p3r3 r3 W

< o -(O k) ( _ _ ) (k) t3 +(k) 1_ 1 + _I )t (k) W &+(k) >120 840 33333 k p3.pr 3 5 k p3r3 p3 R(k)

401k) , 1 1 3-(k) W -(k)+ <Or3 120 840 k 1 ' 3333 pa.P >sk)

1k) 1 1 •3 (k) ti+(k)_- <(T7 W (-- )t3 S W d-)

3 120 840 k 3333-p3.p p(k)

-1k) (k) +(k)Ar' A I ! r3 >R W

<& +(k) 1n 1 _3s(k) W (-k)>

+<OUr3 120 8 4 0 k 3333 p3.o s(k)

- 1k) -W I 1 -)t3s(k) 1+(k) >rr3 1r -'1 2- 0 - 4 0-- k-3333 p3.p (k)(

Application of Green's first theorem [Kreyszig, 19791 to the inner product of (k), an

arbitrary function in the domain of Aý2' , with V-2 (-, gives

<a6+k) ?&) -Wk) ,+(k) , 3 _1(k) W 2 -k)p3 ' (l 33 >iW(k) <V p3 ' 8-0 2ý4T 3333 k0T33,p> R(k)

-4k) , 3__ _ 1 j (k) t 2d&<k) -<-(k) ,3 1_L)t2S(k) aW-(k) >- -<a33 280 24 3333 k p3,p R(k)" W 33 ' 280 24 k 3333p3 7)p>s(k)

-Wk) W(k) ,k) (k) 3 1 1 2 W(k)W <a"33 'A21-r3 > R W)+<0 33 280 24 )tk 3 3 3 3 d&r3 77r>s(k)

Similarl, for .rk)an

Similarly, for &., an arbitrary function in the domain of '2 and I') 0k3)

application of Green's first theorem (Kreyszig, 19791 gives

<r(k) -. ?k) -(k)W > ' (k) - 1 3 -- (kW 2 -(k)r33 21rr3 R(k) 33 ' T24 +280 3333 k "r3.r R(k)

-ak) I 3 (k) W 2 &(k)>

kp3 24 280 k3333 k33.p RIL)

66

+ <(-1 +_3 t2 k)WS 0 o-(k) +<k)>24 280 k 3333 y3 7) 33 s

= -(k) A(k)3k)t 2+ 3 a. -(k +(k)

p3 , A-12 &33 > RW+ <('--8 2- 4 k3333 3 3 '7), 33 >>s(k) (189)

Operators A ) •nd A21 are identical scalars and for &,ak) , and o,-k), arbitrary functions

defined over of R("), implies

<,+(k) -7k) 40) < +W1(k _ 17 (k) t2C-(k)

0"33 ' '3.22 33 R() 33 , ' 40 33331k33 R)

-(k) 1 17 W(k) 2 +(k) +(k)

< T33 4 140 , - 3333tk a33 •>R W--< a 33 A 22 &33 >R() (190)

Equations (187) through (190) ensure the adjointness of the pair consisting of [At") and

4.43.5 Operator matrices [B fk) and [B Pk)

Considering the pair of operators B() ai:d E) belonging respectively to the

operator matrices [BIk) and VB"), the inner product of an arbitrary function M'., in

the domain of B14 , with 134 1o-.c3, the Green's first theorem [Kreyszig, 1979] gives

< rq(k) W -W < jq(k) tk (k) -Wk)

= _<O-(k) tk LS(k) N•(k) > 4-< -(k) tk.L(k) lN(k)>B 41 0 12 -,333" > RW(Li up-- 3Or3 O )r,, > - )(k)

-W k- tk WWk) N((k)<7-- k , ek,) N W >k)' +<a', -W•) , k-- S W3 No W- > (191)

For the pair of operators B(E) and "s, the inner product of an arbitrary function NlI-,

in the domain of r, with B o0-3 gives, upon application of Green's first theorem

[Kreyszig, 1979]

< M(k) B(k) W-(k) (k) > s(k) W(W ) -

< pM 51 -y3 R(k) P ' " p3 3 ")3.> R(k)

<p3 0S 3 M > RS 0M>

67

= -(k ) k ), IM Wk>)+-<k ) - (k ) 1 W(k ) W W((k )

),3 15 > &A k)+ W O,3 7) " 33 S (192)

The elements in the six pairs of symmetric operators viz. B W k; B•k I, k).;

B IT••, f); VkB); B , B; B•, B) consist of algebraic operators such that

each element is transpose of the other. This property along with (191) and (192)

ensures the adjointness of the pair of operator matrices [BY') and [W-h).

4.5 Consistent Boundary Operators

The general mixed boundary-value problem of linear elastostatics consists in finding

a state i.e., a set of displacement, strain, and stress fields, which satisfies the governing

field equations in a given spatial domain and meets the specified boundary conditions.

A variational form of the boundary-value problem exists if a function over the space

of admissible states can be defined such that its Gateaux differential along arbitrary

paths vanishes only at the solution state. The set of admissible states is the collection

of all possible iY} in the domain of the operator matrix [X]. Closure of this

collection includes the exact solution. A variational formulation for coupled boundary

value problems was proposed by Sandhu and Salaam [1975] for the case in which the

the operator matrix [X] is self-adjoint with respect to the bilinear mapping used and

the boundary operator is consistent with [XI in the sense of (110). The adjointness of

[X] has been shown in previous section. In the following section the consistent

boundary operators associated with the adjoint pairs of non-zero operators in [X] with

respect to the bilinear mapping used in the sense of (111) are identified.

Following (111), considering non-zero operators in [X] and taking inner product of

a typical {(i5k) with the corresponc:'-; set of equations, we have

<- lk) 1), k)+ ) (k)+[Bfk){a} (k) R)

= <o)-(k , [1 C {jk)})W> fk)+ <{uI<k , [A>fk){)(k)>,+ <{k 7) , [+10>rR [R

68

+ <{cV jk k. [E(k)l(k)> +( < j U}(k) , [D](k{iji}(k)> <k) 1o.}-(k) [fk)}k>

_ <{ }(k) , [F]fk){o,-(k-)+[Dy(k){u}(k)+[Ef k){a.}(k)> (> ) (193)su

Setting

WVa

0

(a)}- 00

0

0

the left hand side of (193) is identified as

<r(k) a-(k- 1)+N(k) -(k) (194)1 0 a3 +Nt • hp,A•O' a3 •Rk

Using (176)

<i(k) -(k-1) N..(k) -(k)>< o 3 ANoA-O 3 R(k)

-(k-I) k) -(k) (k) > <N(k) (k) >&3 a > R(k) + ,,3 o R(k) a* (0,1) R(k)

(k) (k)+ , < N V. )p> s(k) (195)

To obtain the consistent form (111), we rewrite the boundary term as

<N(k) Wk) > - <N(k (k) > W •_<N( (k) _ (196)<,N,*A , N 0 >s (k) ,> < N v 710> 4 (k)

where ,k) U Sk)=_5k) and Sk) fl s)=o.Proceeding similarly with

0

0

0

0

0

we have

69

A (k) tk -(k- I) (k) (k) tk -(k)

0 , 2 ora3 oM-- a 2 o3 > R W

-(k-I) tk (k) + -(k) tk (k) +<M (k) (k)- <ora3 2 o> k 3 , 2 > •R(L) 'M, -'•(o,3)>R W

(k) (k) (k)+ <V - > R(k) + < T > s() (197)

As before, the form of (111) is realized by writing the boundary as

(k)(k) ) (k) W _ W (k) (k) (198)

-where =0 and S( =.

0

0-kk)

V3

0

0

0

yields

(k. (k )+V( k -()-(k-i1) W -k)(k< 3 7 3 ,,cr 3 >R(k)= <OT*33 v ~3 > R W ( 3 -Vk)> R(k)

+<V W 3 > W (k)>"+<V (07 ) 3 ;W> (199)

The form of (III) is realized by writing the boundary terms as

<V(k) (k) = < .(k) W(k) < (k) (k)<Vo071 ril 3 > s k) = v Tový3 >. - < ,-r3 oV >..-S~k, (200)o 3 S6

where SkU S.S(k) and S(k )n S(k)0

For

0

0

0(k)

0

0

70

we have

1k) t

k (k) -(k-I1) 1 (k) -(k-I) -(k) I (k) (k)

tSk) -S(k) -(k) + S.k O.(T V)-S

-1 P3 3 0"y3," T P+33 33 RkW

=.<-(-"k-l1) tk(k 00 (k N W .• -(k-l) 1 s (k) N (k) >' -y"1 "2 - -% ,+ 3 3 ' ' o r+ , .y -> R ( k ) -- -< a 3 3 0 (0"o • 3 3 P•+ R W )

+ <--) -(k)a' o12. Nk R(k)

+ <N (k) I s (k) WOP "k . Rupp J'• R(k)

- (k) tk s((k) (k) > +<( -(k)

< 3 ) "3"-2 ,P33"o+A., -R (k)-- "(33 ' "2" .9+33"'0 6"R W

+(<) -k) - Nk) +(k) k (k) -(k)N

+<N(k) tk _k) +<N(k-) t k

+ <N ' •-+sa0 5 (ks) (201)

To obtain the form of (111), we rewrite the boundary term as

<N,) ---o'iv >s(k)=<v ,-,q N>s(.)--<ro I lov• >s k) (202)

For

0

0

00

M(k)

0

we have

< M(k) 1W(k) --(k-+). 6 W(k) + 12 (k) WM(k)*A+6 "1"0'00+33"-y,3,,y 5t" k ,033 "33 "W(o,P)"1 3" 'oAupM .,pk tk

1 W -(k) 6 S,(k) a-(k) >

71

= -(k 1) I (k) (k) -< k-k ) 6 S(k) M (k)><( ,3 10 -- • So033 M apY,> Rk+ < 33 ' ý " k 0-, 33 o'A R k)

+ •k) •(k) ..* 40 i. > R,. R 3(k)

+ <:M~') 12-(k) --(k)* <M t3 S DMP M P >RL

tk

-(k) I (k) (k) 6 (k) (k)+ <),3 '-1O :.33J M -y>R(k) 0" ' 3 tk- S033 M>R(k)

(k) k) +<M (k) 1 (k) -(k)

OP ~ ~ ~ 1 0 oS33 7 19y")3 > (k)

(k) S (k) -(k-I) (203)

<M TO> 5 1- (k)(03

To obtain the form of (111), we rewrite the boundary terms as

< ( npi>TP(k) -.rk) - < ,. k ) -(k) < --(k) k)>(2 4- •p~Y MO >(sk)--< l(k) , 7)}#< > •C) (204)•#,-' ca s(k)- *A.

For

000o0

V (k)p

we have

(k) 2 (k) -(-(k-I) _ k)--k) 24 (k) (k) 2 S(k) (T-(k)>

k) S()() 24 5 (k) (k)5 p3y3 )3 P 3.p5t k p3),3 5 p3y3 y3 R

<(k1-) 2 (k) 4k)> +<(k) Vk)> <4k) v(k)>53 3

)'p3 p P p R(k) 3' o<o >(k)

+ (k) 24v(k) 3 Vs(k) > (k)+<a-(k) 2 S(k) , 3 Vk)> (k)

y'Y 5t"k p3S--p 'y) )3 '-5 p3-y3 p R )

-<4) (k) (205)3 -' oVo >s(k)

T,. obtain the form of (111), we rewrite the boundary terms as

72

-5)-4k) (k) 0k) <4V)<V 3 ,-?),V > S(k) , --. V a > Sa 7) , " 3 >(

a ~ s

Taking inner product of a typical {&)"}) with the corresponding set of equations in

(173), we have< {.}..(k [Ak){.}-k_ )+[Bfk)fu}(k) [•Ik){o,.}-k)•.[Ck+ l)tu)(k÷l)+[-Ajk+ l){O.-(k+I)•> Rk

< <0-}-(k- 1) [Xfk){}--(k)> R W + <{u)(k) [Bfk){(r&'-(k)> (k)

+ <{0-}-k, [=_.k,,•&.-(,)> N4+<U}1), [Ck-i'){}-4k)>, * R(k)

+ < R f ) I

* < (k ) , [Ajk4 'i{d*}-(k)> R~) +<i.-k1 , [Ufk){d.}(k)> SWk

+ <1o}--k), [4,k){&}-(k)> W+ <to,}-(k+I) [0 ](kýJ){&})-k)> (k)

- <{•}-(k) [ 6 1 (k){o.-(k- I).+[Efk){u(k)+[W$k){0 T'(k)+_[flk+ I){ul(k~f+L)..[kI o--k " i (I)>

<(& ,k SW

(207)

Setting

the left hand side of (207) becomes

-(k) (k) -(k-) Wk) -(k-0)<&013 , AlI0-P3 +A- 1 2 -T3 3

-•k)+ tk k).it-S(k) N+(k) 1(k) M(k) W 2 W vW

- 2 ' 12 .o33 A 10,- P33 o') 5 • 3• 3 p

.. A(k) -(k) .,(k) -4k)+ -- I1I0-p3 += 12 033

+ k'+) .tk-!-l kk ) tk+I-(k+ ) .(k+I) 1 ,,(k ),,,(k+0) 2 S (k4+)VN(k+0)

I I p3 12 33 R k)

This can be rewritten using (182) through (192) as

-(k 1) -4k) -(k) -(k I) -•4 )•-4k)<a -y 3 A /iI0-Tp3 >R W•"< a33 ' 21 p3 R W)

73

= -- -- m! a |

-- 3k) R(k)" <V - 3 R(k)

+ <Nk) tk W

V~t ' 2-'-3R•30•,k - R

"+ <(k) tk s(k) ~-(k)+ 0 T2~ &0..~ 3 3 y3.v Rk

(k) Ws(k) -(k)" <M OP To s33 & y3')> R(k)

(k) 2 W(k) -(k)"p 5 p3y3 a 3 > R(k)

-(k) =(k) -.-(k) I 1 k) -(k) (k)

+ <a-y3 "11 ap3 > R0+<33 Z 21 y3 R( )

--4k+1) -(k)+ < y3 R(K)

+ 1) tk~. &-(k)R+ < 2 -,3 >' RW

(k+l) tk+l (k-l)~-(k)<Nt 1i2 , P33 r,3"), Roo+ N , 1) (k-1) >-(k)

"+ <M a 1- s ,A330&),3,), >" R00

"<V(k- I) 2 s(k+ 1)&-(k) >p '--5 p3-y3 -y3 R>W

-(k+l1) _(k+ 1)~--(k)_ -(k.l) .(kl) -(k)-

+ <a )y3 A All ap3 > R W)'+'<0,33 A 21 a y3 >R(k)

-(k) i _ kN I (kl0 (k l) tk W(k ( k)+ <4 k+' N' + "M )s +( -ýN -ToMk 33>sk

+ _ I I u-(k) t3 .+5 (k* 0)t

3 X<& k) . -(k)3 -k) -- (k)

120 840 3333 k 3333 k4l -r3 ' pap>S ()- 'Y3 7) orp3,p S J

+ <(1 3I- .-XS(k) t 3 _ (k+1)t 3 )&-(k) (r-(k) >24 280 3333 k 3333 kl -y3 ' 33 71), s(k)

+ I I - -(k-l) 3 -(k) -(k+1)> 0 <-(k-1) -k)>120 840 3333 kl 1'y3 71), ' ap3.p s(k) y 3 : , p3.p sk

_(k _ )s 3 [<& (k W 3 (k- 1) > _ -(k-l1) (• (k) J

120 840) 3333 k y3 T}, 'p3,p Sfk) y

3 7) , p3,p

74

+ -(k) _13 I (k) S ,,-(k- )><o'3 ama 280 24 k 3333 33 S(k)

-4k) _( 1 3 _2• •(k+l) -(k+l).

+ <&y3 1-y -4 24 280 1k 333330"33 > SW

For

6-1 = 033

we have

-( 3(k ) .(k ) -(k - 1) . _(k ) -(k - l1)

< , A 21 i 'y3 +A 22 033

--(k) I (k) (k) 6 (k) (k)

-33 3 2 :-5tk - 333 cq

+ =k) -(k) -(k) -(k)-21 3 +=22 033

-(k+l) 1 (k-1) N(k-1) 6 (k+l) (k+l)+v +-S N ~+ 6 S M >+ 3 T2 A33 oA T I 33 'A8 R~ W

+ -r(k + )) 4 ÷ D - - k * + -( k + l ) •+ 21 0") 3

22 a"33 >R(k)

Using (184) and (189) through (191), this can be rewritten as

12,3 1 33 R(k)" ' 33 22 "33 R W

-4k) --- (k)+ 0<33 >R(k)

+ <N(k) k -.- (k)o+ 3 2" CW033 33 >R W

() 6 •(k) •-(k)_* k < •-5tk S00330& 33 > R (k)+ < Mo, , Sdk

)-k) -A) --- (k) =(k) -(k)+* <03 -& > (k)+<o(>(3k) >, 12 33 R- 33 22 33 R

1(.) -(k)+ <v-"3 ' &3k> R(k)

(k-1 2) (k+l) -(k)+ < • N "up S *133 &33 > RW)

75

* <Mt(ki 1) 6 S (k-* >)_-(

*A 5t~ k- 0'3 33 Rk

< G -(k + 1) _(k + 1 ) ~- (k ) _ -(k -+ 1 ) A(k + i) -; ( 3k )+ <O\ 3 A 12 &"33 > R(k)+< W 033 A 33> R(k)

* <&-(k) _- 1 + 3 X ,,t2S(k) _t (k- )) -(k) >+2433 28 0• k 3333 k*1 3333 'y3 7* s W

+ -(k) -- 3 +.I)t2+2 S(k1) a-(k+0) >

2033 '-- 24 k 13333 -y3 7) s(k)

<&-(k) 3 1 t2(k) -(k-1) >

33 280 24 k 3333"y3 Th>S)s

The consistent boundary operators for N layers can be written collectively as follows:

[W] {VI = {G} (208)

where

76

[Df ") 0 0 0 0 0 0

[Ef II [*)I I [pf2) 2) o 0 0

o 0 [Df1) 0 0 0 0

0 [of 2)IEf2) 2 [ 3) [of 3 ) 0

0 0 0 0 [De 3)o 0 0

o 0 0 193) [1) [0,3) IpJ41 p 4)[w]= 0 0 0 [D](4)0 0 0 0

o 0 of 4) [-Ef ) [q,14) 0 0 0o0 0 0 0 [•f-1) WN-I; o

0 [D0N-1) 0 0

o [01(N-1) [V (-N1) [fN-"1) [frfN)

o 0 0 0 [Dt')

77

jU)(2) g}u)(2

(a)-Q) g) (2)

0.) (3 ) igoP()

IU)(N- 1) {g}l(N- 1)

io I(N ) {goI(N-1)

{u}(N) {gu)(N)

W.k

"- 3 (209)Wk

78

0 0 0-i0# 0 0

0 0 0 0 -- n 0

[DWk) 0 0 0 0 0 -- ). (210)

7)0 0 0 0 0 0

0O"1# 0 0 0 0

0 0 1). 0 0 0

0 0 0 tk S~) W 1 S(k)[L(k) 0 0 '2 -- O 3 3 "h)), 1 .9 3 3 7.)-, (211)

0 0 0 0 0 0

0 0 k (k) , (k) 0[ = 1 T2 SjSc 337)-y To ), (212)

0 00 0 0 0

0 03

_ 1 1 ,_3S(k) 1 + 3 ,2 (k)

[.j(k) _ 120 840 'k 3333 'Up0 24 280 k 3333 7'Y (215)

0 0t-()t 3 W(kI)]. _. (1+ 2 ,-(It --t2 2(k

[q]k) _ 120 840k3333+Py 24" 280 280 k33 k 3-33t 2h11

0 0

(216)

The vector [G] consists of valu.,s of the variables specified at various points on the

boundary. Explicitly, typical equations in (208) are:

79

[0'(k{o,}(k-|)+[L•](k{u(k,+t~y(k{O,}(kI•+[--j~k)ju}(k-+I.{[V](k, I),{O- (k-1 = gf k • o n S k[qjk~j(T)-(k { ,[(k) onk) Sk>~j

(217)

and

[Dk){u}(k)={g}') on S.k) (218)

4.5.1 Prescribed Boundary Conditions

From (208), the couplings of the field variables in the boundary terms are

realized. For existence of a variational formulation for the boundary value problem in

the form proposed by Sandhu [1975], the boundary conditions must be specified in the

consistent form represented by (217) and (218). The boundary terms will occur in the

governing function (103) in the form:< U u}k), k)I (k ) (k) . < ),-k [f ja (kI [f)U()

[D] {u}-2{gu} ek>+<o}()[]k{)}(-)t]k{}

+ [4 vfk){cr(k)+[PIkku}(k 1+[fP Oj{oTk- I)- 2 {g,,W > > (219)

{g.)}k), {g}L(k) are values specified on appropriate portions of the boundary of the kth

layer. Explicitly, (219), is

-(k) (k) (k) (

C, '-'.8oM.-2gl>~+ <•<k) .(k) (Ic)

<$ A 9 -OMo- 3 4>3

+<--4(k) W,(k Wk)

"+ <v 3 ,-- 0 -2gs > SIS)

+ N(k) -4k) .(k)>

"2

<M*o 70, 94• - Sg. sk)

"+ < V4k) (k),V (k)_2W>S6

"< .-(k) [._( (-k~ ) I -4k -+t 1+ 3 )t~2 S(k) " .r-4k-1)),3 120 840 k 3333Ic 3p 24 "0 33337"y 33

80

1 tk 3(k) jN (k) -- 3 S(k) 1.(k)

+ [(__-+ 1 Xt3s +t3 S 3k+1)3 .r -4k)120 840 k 3333 k4l 3333"y'-p3.p

. 3 v s )(k4 1 -(k) 2 iWW-(k)t+[ I+ (kAl) ]N,(k+l)

24 2i 8-0 k+i 3333 k 3 3 33)n•y

3 3 12 B3v or8

A+[_ s(k+1) ,(k+1)+,_( 11 1 3 s(k+l) -(k-# 1)

10 G0337'.Y P + ( 120 -EZ.)k+ I3333% )p3,p

+ [(-L- 3 )t2ý S(k4 1) ]-(k+') 2 g(k)_ (220)" --24 280 k I 333372)33 - a

Thus, the specified boundary conditions are:

-- on

7 1-- on --"q-gs on

(221)7k) =g(k) on (SWA on 2

ok) (k) (k)7)=g4 on S,0~ 4

-4k) Wk WkV3 71-g 6 on S6

and the continuity conditions are:

) t " 1 3 ( k ) - -( k - . r 1 ) + 3 ) t z W - ( -L - -I )

120 8 4 0 k 3 333 3Jp.p 24 280 k 3333T)'Y]" 33

+ tk (k) IN(k) +[ W W12 0- 0"be33 ")-yJ. *A8 T11 ol 033" y•Jap

+ I 1 1 -- 3-(k) +tl ,,,(k+ i ),, -- (k)

120 840 k 3333 k+l 3333.'p3,p

+ 1 + 3 xt2 S(k-') 2 W -(k).rtk.Z+(k4l) k I)k+1)

+4 2-80 k+1 3333- kS3 3 3 3 )rnyc33 "-iL2.&3"P•37))J0

+ I1S(ki),n JM(k+l) +[ (- - I \_3 s(k+0) k -(k.l)

10 "•• • 120 8 4 0 k 1 333371), p3,p

81

+ -(k I•3 g on (222)

Jump discontinuities in the field Nariables can be introduced as conditions at

internal boundaries. Indeed, as finite element bases generally have limited smoothness

across interelement boundaries, even if there are no discontinuities in the physical

problem, the smoothness condition of the physical problem needs to be introduced as a

set of homogeneous discontinuity conditions. Similar to the format for the conditions

on the boundary, these are:

-.. N _(k) _, ,(k) s(k)on S,

-4V k),10=g(k) on S W

(223)_(M~k)). _ .(k) s(k)

0 2 oS 2i

(;Ok)) 7 -g (k) on 3,

V 1).=9 6 on S56

120 840 'k '3333h -YPI 24 280 kL 33337)'Y33

+ (- k, s(k) Y [(k) Y

12 -033=g V o C033 O

+ I ,() S t(k) 3 S N4l) -(k)

120 840 k 3333 k".I••" 3 33 rpl33p

"1 3(k) ",2 SU(kI),__ W 2(k)y "ak)-,[jk.I 5 (k+0) YN(k+i)-24 280 k+1,S 33 3 ] 3333y33 12 S.03 y"• [- N 1 I 3M(kl) +[_3 1(k_ L)t+ S -(k-1) ( -k+1

24 803 &P 120 W k 33333-ka33Y 33T ,

+[(_L_ 3 t2 •(k+l y -(k+l)._, 00 Wk)

24 2 " ok--" '(Ikl3333))y 0'33 =g on S (224)

82

The homogeneous discontinuity condition, i.e. vanishing g', , represents the internal

continuity constraints for the physical problem. In allowing for jump discontinuities,

generalizing Sandhu's [1975] assumption for the problem of elastostatics, one could

require

(k)f S~k 0 (225)

S3 nf sk = 0 (226)s(k) (¢k)

S5 ni 6i 0 (227)

where S•n, a = 1, 2, . , 6 are surfaces imbedded in the region R kW. However, Bufler

[1979] showed that Sandhu's assumption is unnecessary. The relationship between

S1 k), i= 1, 2, . . 6 which are subsets of Sok) the boundary of R k) have already been

determined. {S•,k), ,÷k) , i = 1, 3, 5 } constitute pairs of complementary subsets of

S~k), k = 1, 2, .. , N.

4.6 Variational Principle

4.6.1 Variational Formulation for Linear Coupled Self-Adjoint

Problem

For a boundary value problem with n independent field variables, defined by

(112), (104) and (105), the governing function is [Sandhu, 1976]

n n n

f=E[<u. EA,juj - 2fi>R + <U,- Ecjuj 2g,>•ij-I j-1

n

+ <u , •'(Ci u Y-2g,'>s (228)j'8

83

4.6.2 Governing Function for the Laminated Plate

The set of equations (173) along with the boundary conditions, (221) and the

continuity condition (222), including the jump discontinuity conditions in (228) gives

Wu o)=2 <{(u1') , {(P)>())>R(+2 <{U}(N) {pj(N)> RI

NN+~<{U}(k, [Cfk){oF)(k1)> (k+p <{(U} . fAk){Ul(k)>(k

k-2 k-I

N-I

k-I

N-i N-i

k-I k',I

N-I N- I

k-I k=-2

N-2

k-I

+2 <{Io (N-1 ) [Q](NI) > R()+2<((i,)-()9[f)

N

+E <{u}(k , DkI(k2g ()>

k- I !.

+ <{o~ ),[l'I I~hI I i.-I )+[lFl 'IU1( 2)+[gfi 2)1O})- 2)-2{g0.'I) > SO1)

N- 2

+ E <{OIc(k [Oi'k){oYl(kI )+[EPk){u)k)+[*fk){or)-k)

k-2

+ <{ayY(N-1), [oN-){cff(N- 2 )+[EfN-I){U1(N-I)+[ 4 4 N-I ){ 0 .}-(N- I)+[FIN){U)(N)

-2(g)(N >s(N1

N

k-I

84

+ <4o} (- ' ({,)")+[Of;k(<i <2')).

+ Z < 1. , 1< 4 0-'( , [ O ] • ,)( { o .) -( , •- ,) ) .+ [ E , ,•( I u }( k ,•) .+ [ O f,] ( f,0 ( ) -( k •) ) ,

k-2

+[k+ '(U){uk+ ))' +[•f+ )(10,}•ý+ ))'-2{g'O)(k) > s

+ <lay (N-,), [0fN- ,)({o•- 2)y +[n -, lk{u }(- I )) +I+4 -,)({•.}AN-,)).+<ra'IN N), •+N\- (-)

trJf N)U (ju)(N)'-2g'•g•j(N ;> J(229)

where R(") is the two-dimensional region of the kth lamina and S(") symbolically

represents appropriate portions of the boundary of R(k). Ju), {(r}-k) denote the set

{u)(k)=l•k), i), kN), kM,)V• and 1O1 k)=jCr ), 0 j, k = 1, 2,.N

respectively. S.) represent appropriates subsets of internal boundaries in the region.

Substitution of (159), (160), (161), (165), (166), (167), (168), (169), (171), (172), (210)

through (212), and (214) through (216) into (229) gives the explicit form of the

functional, including the jump discontinuities, as

Wu , ()=2<vý2< , , > (2)+2 3R033 > R(I)

2<0) ( (0) +I1 ( 0) (o)>

Si-2 P-33o y3,y 2 .• ,33 "33 R (D

(0) 1 0() (0) 6 -( (o) (1) 2((i) (0)>*2MU To' " cP3.330",3,-y,'ýt51 I So•330"33 >R(,) -t 2< P ,-3p3)y30-y3 'lRt

+ 2<v> -+24(N--N) _(N)> + N) tN (N) 2V --4N) .(N) >

'2v ' ,3 R(N) Y Y3 >N)+ 2V 3 ' 33 R(N)

+ 2 <N(N) tN S (N) a(N) + I -(N) ( ()_>(N00 1T2 OP33 Y3' 2 '03333 R(N

+ (N) (N) (N) 6 S(N) (N) (N) 2 (N) (N)

12< M 0 0 To V3 3- St 033 33 >R (N )2 P ()3 a > R(N)

85

N

+) -(k- 1) k) tk "-(k-I -4k) -(k- 1)wy 33 R(k) 'y3 WR(k 3 33 R(k)

k-2

"+ <N(k) tk s (k) 0.--(k- 1)+!s W) -(k- 1) >o ' 12 o1 33 ,3.-y -2 C, 33 33 R~k)

"+ <M W) I| .. () -(k- ). 6 W(k) -(k-I)>00 10" *033 ),3 .y " tk Por33 33 R(k)

"+ <V•'(k)p " 3-ss(k ),30 > R)

N

-- k) "W:< p.' Mak) W <ý())

k-I

N(k) ,-4k) + _ (k) ,(k)* <N 0 V (o,.A) o.)AupN APR(k)

+ <M(k) .k)+s M(k) <V(k) -7k)--(k. 2 4 -(k) vWk) W,•5 r(-48) 3 Aup Azp R(k) ")l I-•) -V ,),'l P3),3 p •Rk)

t k V 3. t-k 'N-I

+,y.<; E- > +<• ,_,.,> +<v> ->•' y3 R(k) )" ' 2 ,y3 R(k) 3 - 33 R(k)

k-I

+ <N(k) k (k) -(k) I (k) -(k)"e <2 - S3cr,+- Or3 > (k)

"+ <M(k) 1 s(k) .--k) - 6s(k) 0 -(k)>10,' 10 &03

3 tk3,- A33 33 R)

"+ <V(k) 2(k) -(k)p ' -- P3'y3v3-v3 > R(k)

N-I

'r 2 12 .0 33 A), TO 33M "aA-rs v3-~, -y> ,Rkk-I

-(k) .- k) -(k) -A(k) -(k)+ - 3

= 11 4")y3 += 12 a33 >R(k)

. O.-(k) -- k+1). tk+n -I k+ 1) t k-l (lk+l)(] ) l- I (k-1). -k+i) 2 --(k+ •-tJ ) _-y3 ' 2 l y -1-2"0-1330 -Y &.A'33M .y- 5" p

3V3

P R(k.1)

-(k) _-(k) 1 W (k) 6 W ,,(k)+ <(733 ,- V 3 +"T•' N33N -o.K.0--Sk33 M0>R(k)

86

-W --Ak) -40) =..k) -Wk"+ <0"33 ,210"y 3 + -22(733 >R(k)

- --(k) 1)k 1+) (kL)N (k+)+ 6 (k+0)M(k+l) >"+ <0"33 ' 3 + T - A33 p -ýtk+ I

N-1 -(kO.• ) W ~o-'(k-I) , Wk (k- l1) _ - -Wk W k~ .-(k-0) . W k -(k -1) _ ,

(<*)3 II y3 +A12 a33 > 33 A 21 Oy3 +A 220"33

k-2

N-2

+Ef ,3 ?ý.11 a -y3 "I' 12 (T"33 > R(k+D

k-1

<--Wk) '(k0I)O. -(k* I).4k+0(k+i)-k)

33 ' 21 y3 22 33 R(k1)

+ 2 <iCr-(N-')N , [xN){i}(N)>R(N)

+ R{i)

N

,--<3N. -2g(I k) >

k-I

* <-k) W(k) (2W+-• -, 9g3 > 4 k)

+<v-k), 7 V(k) 2 ~ >(kL)•

(k) -(k) (k)

+ <N • , 2-gg2 >W)

W T, k (k)>+ <M(k) -)-(-2 4 W (hip p oS

4<v(k) (k) . (k)_

"7) .V 3 - 9g 6 S 6k)

+ <a --(0( ti () (0)+[ 1 S() NO)

-y3 1 2 .3-( ) 3--(2) , A 3---(a)

+ [I-+-Xt3 S(1 +t3S2 S T I120 840 1 3333 2 3333')yrp3,p

+ [ +i 3 X t2S (2)

2 (1) -- --- (1) t 2 (2) .,(2)

24 280 2 3333 1 3 3 3 3 )V1J(13 3 +T2m-s33 y o$

87

+ I S(2) )M(2)+[ 1 LL 2 ) ]-(2)

10 t' " 333''y 120 840 N33331)), p3,p

1(_4 3 , 2S(2) ](T-(2) (1)>+ 24 2 8 0 2 333377- 33 -- g S 1>

N-2+-{ < (k) , 1 1 t3 (k) -(k-I)+Ef<-),3 '-•120 840 k 333371sy p3,p

k-2

+ . ( 32-+~-2 (k) W -(k-)-r )+[ -k S(k) ] [ k) jN)k)

+ 24 280 k s333371- 3 12 &P3 -yo 0 P3 yx

1 1 3_(k) _t3 S(k+l)n 3 -(k)

120 840 k 3333 k+1 3333 -Y p3,p

+ 1 3 _2 (k l) t2 5 (k) 2 W ] -(k)+_ tk+i 5 (k+-J) IN(k4 )+ [(-L'4- 80 k Is3333-- k s3333 12 33 12 -. 0337))'I o.

I (k+1)7)1 1 x_ 3 (kN4) 1 -(k-l)

[ 1-O.331yfi 120 840 k) I s333371y p3,p

+ 1 3 )t2 S(k4 ,) ]cT-(k+I) 2 (k)>"24 280 k+i 3 3 3 3 ), 3 3 g"

+ <(-4N-I) 1 -(L--)t3 s(N-1) ioý-(N-2)

-y3 120 840 -N- 333371,y p3,p

+ , 1+-3)t22 S(N-1) 1 0.- (N-2)

24 2-80 N I 33337)y 33

+ I ý (N-1) 10(N-I) 1 .(N-I)

12 SP3))]a oP31ym.

+ I + I Xt3 --(N-1). 3 (N) I -(N-I)

120 840 N- !3333 N 3333)y p3,p

1 - _ - 3 v 2 (N) _2 t(N- ) -(N-l) N N(N) (N)

24 2 80 N 3333 N-I 33 3 1 37

+ _ (N)( N) (N-1)+ -S. 833,l ]MN)•2-ga S(N-1)

N+ Et{<v-=() ,(N (k))'-2g A) >

k-I I 1

+ <o:) .- (k))'-28 ,(k)3-Op8 3 S)31

88

+v < 3 ,-71.(V, )-2g's >

"+ <N, W o-~),2,k2i

"+ <M, (k) (ýk5- 2 g (k) >

W (j)(k [> 0~ 3 7K~

+ 2 8V 40)' 6

1- 320 ) , [- t, (

[-y-3 S124202 3333- u 10 y+.Z33 y OP

"[li( 2)+- xtsl +t3s()))xa-1+ 0 ~ Y O~120 840 1 33 23333 Iy" p3,p

+ 24 280 23 3333y 333 nyo'ý)'[1 A3TX A'

10 Q37-'Ma) -12 84 0 - 8- 33 2 y p3,p

N-2

D <r-k) f __ (__)t3v ) Kr-k

k-+ , ~ 120 40 3333), p,

24 280 k333), 3

_L [ S.k 71 KN(k))Y+[-L 1~ KM(kL

12(P3 - p 1 A3 37,k+)'~Y-()

+ 3 L+L~2 S(k+ 1) -t2(kW -(k) tk k ) (41[(+-.X t~3Yy~3 2o~324 280 k~ 3333 k 3337Y(3 )'+1-L- 12 03),IO

+ _ (k.1) (k+ 1) 1 tx3 S(k+I) XY-(k +

I10 SA337v4KMO )1 1[20 - L-1~k1 333371-yK P3.,

+ [1L 3 t2 S(h-s)24 280 k+1 33337)Y 33-k+)'2g(

89

+ <a--(N-1) [4 {1 1t3 S(N-1) yX -(3-2))y3 '+ 120 840 3tN-133337)•/)lp3'

+ [- + 3 t2_. Is(N-) yN -4N-2)1,)24 280-1 'N 33337)y-33

+ tN-1 (N-1) N(-,+ s(N-1) M(N- 1)),12 0--•-•3371-y• *A 10 L-P337)-yt P

+ 1""+__LXt3 S (N-I)+t 3 S(N) )7) -(N-1)

120 840 N-I 3333 N 3333 TyJp3,p

tN"+ [ 1_ + _ 3 ,Xt2(N) _t 2 S(N-1))Kn -(N- I q) (N)

24 28 33 N-I 3333 'Y Y3333 NoqI

"+ 10 [ S. ) M(N))'y2 ,(N-')> (230)10 A3 -Y 4 g ' eiN-1)

We shall show that the Gateaux differential of this function vanishes if and only if

the field equý-ions as well as the boundary conditions of the problem are satisfied.

4.6.3 The Set of Admissible States

The governing function defined in the previous section is defined over the linear

vector space U of the ordered set of functions u={f), k, i •k), N3, Vk) and

0-0, or, k, j - 1, 2, . . N-I I so that each function belongs to the intersection of

the domains of the set of operators which act on it. The domain of definition M of

any operator A is the set V such that for any u, vEV, the inner products

<u, A v> and <v , A u> exist. Thus if we denote the space of function of which

derivatives up to order q are continuous by C', an admissible state is the set of

U= ,k), k), -V , Mk , V and -j)a,-(') k, j - 1, 2, .,.,N-1 } such that

<V, W Y> I(k) exists. To ensure this, it is only necessary that

a--(k) EC2 (231)-y3

a-(kL) EC 223,3

0"33 ECf• (232)

V(k)EC1 (233)

90

M W)EC' (234)

N W)EC1 (235)Oot

V-k) 1 (236)

(k)ECI (237)

v 3 EC' (238)

However, to ensure that all the differential equations can be satisfied by an element in

the set of admissible states, it is necessary that various field variables have appropriate

smoothness. The equilibrium equations (113) through (115) require that:

1. 1 -kEC

2. or k).

3. V< )EC', at least one order of continuous differentiability higher than that of

-(,k)

4. MAJEC2, at least one order of continuous differentiability higher than that of

o.73 tre3 and one order higher than that of V.'

5. N' EC', at least one order of continuous differentiability higher than that of

-4k)

From constitutive relations, (155) through (157), for ol'EC-

1. 1.,-(k) E' one order of continuous differentiability higher that that of a-(k).

Hence, N(k)EC 2 from (5) above.

2. v(L)EC 3 , one order of continuous differentiability higher than that of N) and

at least the same as that of -ry

3. ik)EC3 , one order of continuous differentiability higher than that of M1 and

-4k)at least the same as that of y"•3

91

4. V3 EC4 , one order of continuous differentiability higher than that of V'." and

i and at least one order higher than that of a-•k)

The continuity equations, (164), require that:

V(k), -ý'ý"VE 0 (239)a , v3 C

V(k) , r o., -(EC° (240)

00 033

and

0-- k)EC2 (241),/3

Combining the requirements of continuity, equilibrium, and constitutive equations, we

have the following restrictions upon the field variables so that the differential equa-

tions can be simultaneously meaningful:

vI(k) EC3 (242)

iok) EC 3 (243)a

v-3 )EC (244)

N(k) EC 3 (245)

M(k) EC 3 (246)OIP

V(k) EC2 (247)

C-(k) EC2 (248)y 3

0-3"3) EC' (249)

We note here that the domain of the differential operators defined above is contained

in the domain over which the governing function (230) is defined. In the above dis-

cussion Co may be regarded as the space of piecewise continuous functions over the

domain R(k). This is important for finite element approximations.

92

4.6.4 Proof of the Variational Theorem

It is important that even if there are no internal jump discontinuities, i., if

g'1 -, -g'. vanish, the jump terms in the functional (u , ar) be introduced in the

formulation [Sandhu and Salaam 19751 If these terms are excluded, and approximants

which have interelement discontinuities are used, a built-in source of error in the

approximation may exist.

The adjointness relations hold only if the functions have appropriate smoothness

properties over RWk. If there are internal discontinuities, the relations have to be

restated. The functions may be sufficiently smooth for the divergence theorem to be

applicable over subregions or elements but not over the entire region. Thus if a

partition of RWk) into subregions or elements is available such that all S•) are included

in the intersection of the closure of the elements, application of the divergence theorem

to the sum of the inner products over n elements will lead to equations of the type

n nf n

E<U , Av>e <v, Au> ) + E <V, Cu>k)e-1 tI C-! (. e-I (.

where n is total number of elements; R~k) is the region occupied by the eth element; C

is a boundary operator, SI) is the boundary of the R•)1 . Clearly, summation over all

the elements leads to

E<U, Cv> <U , Cv> + f (uCv)'dSWk (2:50)

where SPk) is the boundary of the region R(k) or of its finite element approximation;

S',") is the union of interelement boundaries and the superscripted prime denotes a jump

in the quantity.

To evaluate Gateaux differential of fO(u, o) along arbitrary paths in the space of

admissible states following the definition given in (98) through (100), it is necessary

and sufficient to consider paths in the space of admissible states of each field variable

93

separately e.g. in (98) the v could be arbitrary in the set of admissible states of one

variable and have zero components in the spaces of admissible states of the remaining

variables. Thus considering paths exclusively in the space of •k), denoting an

arbitrary path by i•, the Gateaux differential of f) with respect to ) along path

X. is

f=<x--W ) -(k- 1 ,() + - -(k) _T-(k). -(k-1)_Ax-,4k)a• < ,Oor3 No. -'O3 -a3 aO*3 R(k)

(k) _-k)

-k ()) W _ (k)+ <x0 ,-)ON -2g( >

(k) --(k)+ ,,'• 3xk 5 (k)

s2

+ <X-(: .. (k)v.,(k)_

Noting that

WNk ---k) < N-_.(k) --4k), N.(k) --4k) N(0-40k (,sk)<N0 , (a,$) > R W=-1 A'A X( k) -N<r ,, x o ) •> fS-J (k) (N a X, ndS

3i

and that S(k)U " )=S(k) and Sk') n sv')=0

< -(k) (k) -(k) -(k-1) <(k) (k) (k)A 4 )il =2 x , N 00 .0'-- a o 3 +!"O " 3 >iROL) + 2 < a - lP NO P-- g i > s( k

+ 2<i"(k) ,.N(k).(k) (251)

Noting that inner product is a nondegenerate bilinear mapping, i.e,

<U, v>R - 0 for all v <--> u - 0 on R

and that arbitrary i. could be selected such that j) vanish on any two of R(k), S(k)

and S.k ), vanishing of Ak)fl for all i) implies

94

W(k) -(k). -(k- o R(k)No.-Oa _TO3 =C 0 onRDAP4 o3 , 3

(k ) S(k)-- N =g 1 on

and

- ),(N k g I on Sl(k)

Similarly, Gateaux differential of ni with respect to . along path 2 is

C- -- k) tk -(k- )...k) W(k) tk -k). tk -(k) tk -(k )>

<2 '2 0-3 2,• - ÷y .o3 T2 o3 T O 3 2 k)

(k) --4k)+ <Mop -Y(CP) R>kR

(k) -- k)+ < , ,y' > R(k)

-4k) (k) Wk)+ <Y. , -- }•M•9--g 3 > Sk)

"+ <M O )9, >(l)>

-4k) -- (k)"+ <y ,-ro(MO)-2g 3

4k) -- (k),, 2,>k)

Noting that

(k) -4k) (k) -4k) Wk) -4k) (M W 0k)<M ,-y(..,) > R(k)--<,MoP,P, I, y> .R(k)-- M#,P Y". T•>SC)- fk W # (MpY. T/•IAdS

and S- U _.=(k) 5 k) n s(L)=k -- <-W W _a . (k) ,(). tk -(k).+ -(k-lk .I -->k +2 10k) (k) >

Ahfl=2<y* , MOM V+ -!(a4k)o3 k -43 00 Y _Po-

2 <o--o3 (,.. 2 < ,, 3(k)k+2 < Y , -r7(~)OK O~ ) 3> SV

Vanishing of AZk)fl for arbitrary ý(k) implies

95

MC() V(+. tk(0, -k()+. -(k I)) 0 on R(k)2 o3 &3

_71 (k) (k) o (k)

A= g3 on S3

and

M ,(k) Wi)-- S3 on 3i

Gateaux differential of fi with respect to )k) along path ik) is

f -4k)k) k -1) v(k) -( -ik) +--(k- 1)>

z 3 "33 o+ o 33 33 33

+ <v(k) _k) >"Va ' 3,o >R(k)

-(k) W_ (k)"+ <z 3 1,-oVo- 2g 5 >S4sk)

Wv) -(k)" o 'V ")z 3 >• s~k)

-4k) ,.,(k), 2 ()"+ ' -7r(v 0 ) }-2g5(k >

+ < -k)

Noting that

VW ) _-k) W -v(k) (k) -•k) (v(k) -k) W)

< a x 3-*)> R~)=<V 0",z 3> Itk <VI 1 ek-Jfe ) z l)

s(k) n s}o

A fl=2<z•3 V a--(k) W -(k- 1 -(k)_ .<-4k) -,nV(k)-gk), V 10,0 033 -033 >R~k"' W-' V -g +2 3)

+ 2< <k-() _ .n.((k)x,_,(kL)>(2 )+ <i,-ao(V'-g's s•) (252)

Vanishing of &,,fl for arbitrary i") implies

V W+a.-(k-t) -(k) , 0 on R0.0 33 33

(I) _ C(k ) W (k)Ch g5 on S.

and

96

_7 (V(k) = g,(k) s(k)av = on a5i

Gateaux differential of fl with respect to N W along path n(k) is

( () tk .(k) W -(k-l)+L s(k) - v +32 (k) k+(k)

0 1 2 P3 y3,v 2 33 33 (0P tk rPp APP

t Ž(k) W -(k) 1 W(k) W)+ I(k) k) 1 S(k) -(k-1)>- "T 2 oS& 33 or'),3), 2 "• orP 33( 33 2 "• oa(P33 0 "33 + T Pq33 0 "33 >R 00

+ < --(k) (k) n,a ap noO•(k)

-(k) tk Wk) (k)+<',(T y , T2-3.A33 nj,,,> R W)

-(k-1) tk (k) (k)+ "y3 2 -- A o,33n A*,y R-Y

) ---(k) -2 W+ <n•e ap),"Ov. -- 92 >'~k+ <nv-(k) (k) S2

4' ) W k)

Wk) t k(k) "-(k) tk k) -(k-1)"o < -12"*331•)yO"),3 " S12.-337)-)r V3 v > S(k)

"+ <n W-n)A(gk),,, 2 g.(k)>

-4k) W(~y"+<,0, -<V(n a) > ,•)11

"+ < n (k) tk S Wk ) ,,O-(k) ),.'+ k •(k) -'(k-l)x,

+ , 1Tý 2 S, (0 - Y'3 s~1, 2 P33 -y A 33•)•yy3 (k)

Noting that

--4 (kW W)> --k ( --(k) 0<(k) -- (k)> + r ,(k) "(k)y,.(k)< vo# , n(.,) >•--n ,v+e)#) o#},• () (n)•#

R , (*A.) >R(k) +<n.,7). V. > k) f )O'p a

-(k) tk (k) k) (k) tk Wk) -4k)-( )3 S cri33n,,, > W=-<n, -F S. > W13 ' ]2" . 3 f,' R~k -- A•, v 2 Ae 33 ),3,),y Rk

Wk tk (k) _ -(k)-+ <n a, -S 71~<)a >s)T2 .0337 -,-y3 s(k)

97

+ f s(n/ (k) tk s(k) -(k) d (k)

SIk ' 12 .03)-() )

< -•ki) S n(k (k <n-• k) tk S(k) _(-)

-(k) tk k(k) -(k- I)nay33 12' f2P3 >s W

< (nk W l S(k) -(k- 1 ) > d(k)

fsP 12 " 3337yO'0 y3

WI t k (k ))

and 2 ; sý n s =0

tk (k) -(k- ) tk W(k) -(k) -4k)

as -2 1-2-

+ Ik S W NkW+ I S (k) (a-(k- 1) +01.-k)) >+ A ~p 'P T .033 'O33 33 R Wtk

(k) -4k) (k)>

" 2<n,, ' noav, -g2 >s~k

2W (k no k),_g,(k)•

"2<n O -P-gy 2 >(k) (253)S21

Vanishing of A.,fl for arbitrary n W implies

-sk W (k- 1) -(k)ý) 7 _ 1 (k) +S (k)_ 1S(k) (&-(k-l)+.o.k) 0 on R W

1 2 033 (a , 3.y - 3, ((O) + 'k o p up _2 o,033 33 33

--4k) (I) (k)= g2 on2

and

g=(k) on p

W

Gateaux differential of il with respect to M) along path mW is

W I W -(k-, 1) 6 (k) -k-() -4 ) +241 k) M(k)

I (k) -(k) 6 S(k) -4k) 6 (k) -(k) 6 (kW -(k-)+ -o33 33 o - 033 -3s CT - 033 33 ,+,C)

98

)k) W""4k , m c8,p :> iR Wk

O-00) 1 W.k Wk"+ <c ,3 To -- 03•33m CA- > R W)

*-(k-l) 1 (k) (k)" <a. Sy T a03In kf.,> R

+ ,• W k) 2

-k) (k)+ <$ ) (k).

<m Wk I s(k W -(k)+ _L W (k) -(k-1) >

,1 TO (-,-k))0 -y ,k-

+ <m , " jL•, (/--+g V > 5 kk)>0 10 a10~i

+ <Wk ((k)-,,>

1 (k) IO.-4k). I W(k ,-4k-I* < M1'O' To s *337)-y' (( 3 )' 1"0 " *P33"r-y'•,3 l)> Si(,)

Noting that

<$-ik) (,) . <m(k) --W.(k) W + (k) nik)> +f (k (M(k) -- W(k) 5 (k)

*A.-(k I(alp ) ROO ap t {k) R 0,_ s(k) >-+k) M.

-y3 T- 'O 0 0,33 rr* 01,Y ' R ~k - mW,* 10 cwP33 -y,3,y R ~

f (MW 1 WS 0 -W )>

rei( k) 1 TOkS.P3371yO.-<k),dsk

<:o.k--) I W(t Wk W I Wk • -(k-1)_< a -y- -3 0 3 3 I n 0 0 ", > R W) < - m CI , T o's' c A3 3 0 "T y 3 , ' > R (k )

W0 W1 0(k O-(k- 1)

< m *A , -F -O s 33")q yO'-3 >S Wk

f (M(k) I k W .- (k-1)), (k)

Si

9 9k ) s T k 3 -( k - )- 99 S 7)

and VtI U S(, S(f)n s.ko

_L (k) -(k- 1) . (k) (k) k)

+ 12 (k) _.k), 6 •(k) , -(k-1) -(k),,.-L "•"oA8upM JAzp'•• +-S33 .O 33 -- 0"33 )>, W~k

t3 op 5~ t k R3k k

"+ 2 <m W) -,-(k) (W) >W +k (k) >(( 5)

4i2<+, ltqo -- 4> s(IL)25

Vanishing of A f for arbitrary m., implies

1 W(k) 1) -(k) k) 12 (k) ,,(k) 6 (W) -(k ) 0 -k) o k-S, (0 , 4, + -SAMP +- o -o 0 onR

1 0 33 y3,y y,3,,'-'/ (00 ) t ip- • LL p 5 -t 0o 33 33 33ILk k

",n o = gW on OJ

and

_gok (kL) (kL)v = 94> on sdi,

Gateaux differential of f with respect to V(k) along path q(k)

Wk<q(k) 25 (k) _-(k-L1)_•4k.._( (k) +•48 S() v(k)

S qp ' 5 03"y3 -y3 p p 3,p StI --p3y 3 v

2 W(k ,-(k) 2 c S(k) --(k) 2 S(IL) -(k-I)>- p3y3 y3 -3 p3y-3-y3 5 p3-y3 y 3 R(k)

-(4k) (k)"+ <v 3 , q.er> R(k)

+ <q p, ,v 3 926 > ýk

-(4k) (k)"+ <V 3 , 7).q >sk)

"+ <q ), 3

"+ <v -113aJ>

Noting that

100

-- k) (k) -4k) (W -k) (k) f--4k) Wk . (k)< V 3 q Q~ "•' ,. > - < v 3,a , q O > " R W + < 3 , q u 71 .> S + s.,k) v 3 q u -q ,), Ia

W =g $ SSW (k--and U Sk)6 k n ss.,l s=0

A Q =2 <q (k) 2 (k) a-(k-). -(k),-"k)_V--k)+. 24 S(k) v(k)>

q W P =5 p3vy3 -y3 -y3 p 3,p Stk p3-y3 y R(k)

(k) -4k) (k)

2<qp '(v) V)3 6SOg (255)•6i

Vanishing of for arbitrary qP implies

2I(k) W (k- )+ -4k)) -r-(k)k) 24( W (k) RWko

-4 k) Wk (k)7333, 3 = 96'on S6

and

pV3 =3 on 6i

Gateaux differential of Q with respect to c•k a t k

-(k) , k+l)+tk+l;kl) 2 (k-0v(k+3 )_ V--4k) tk: 5(k) 2W.rk) ••--<T),3) 2 Try 3'",), yTw--p3"/3--p

-- k) tk ;Wk)_ tk S(k) N 1 W(k) M W(k 2 (k) v W

- (k) -(k) - (k) - (k) tk-, - M(k+{) tk! Z(k+) V(k-1)

+ -110"I),3 12 33 2 2 -- 12 Ao33 .A

41 (K+I) (k-j) 2 •(k+I),(k A ( -(k- k) -(k -).- S~ SGM -) S 'zV~4)+A k)'.+A 12cT33"10 *033 *j.y-y 5- p3y3 p 11 y'3 12 33

+ -k+1) - -(k+1)+,tk+1) -(k+l) >+ I "y3 12 33 5 (k)

+ T-(k) tk+J •(k+0) (k+0) 1 44 1(~ )l (k, 1)+ < y3,),', 12 "0A33 No,P + " -F o S .33 M O

tk (k) t W(k+) { (k k) M (k) >T2 033aP 10 .0."- " ° o- S 83 3N ' o t " "1 3 3 M& 0> R (k )

1 01

-(k) =(k) -(k)' y "l1T y3 > R W

• (k) =(k) -(k)+ <01 33,-17 y33 *'2Iy3 -R(k)

+ <a33 A17 K)

-(k+1) (k+1) -(k)+ <a 33 , A2 1 Ty 3 >R(k)

+ <aO.-4-) -,k)-(k)> W-y)3 11 ), lr 3 R~

+ <(r-(k- 1) -7ýk) 7-(k) >+ *33 , 21 -y3 ( R(k)

+ < T - k(k) 1-(k-,).(k[( _ ) 3 )t2 s (k) .- (k- )

+ •a' 120 840 k 3333)p3.p 4 280 k 333371r 3

+ tk(k) •N(k)+1s(k) (k)

12 S.- 0 33 1 y 10 s331YN.A

+ I + I 1 t s W• k -t3 --(k+ 1))n y • _-(k)

+ 12 0 840 k 3333 k+1 3333 yp3,p

3 2 (k+1) ^ (k) -(k) tk+ s(k- l) ].(k 1)l+ +[-1 S-+;33 3333)1-0"33 [-•- 0337)-

24 280 k 33 12

+ I (k+1)7) IM(k+l)(-(--- L )t•+3 s(k+1)3 ]-(k+{

10 -. P33 -yJ *l120 840 k 1 333371)J p3,p

+ 3_ _ --2 -(k+l) ",_-(k+{) Wk)

+ [(" 42 8 0 )k+i 3333"yJ33 -- 2gcr >(k)

-(k) ,- 1 _ 1 )t3 s(k+1) ] -(k" 1)"+ < p3,p '[ 120 840 k' 33337)'j *y3

"+ 1_ 1_ . 3( +t 3 S(k+1)--l-(k)+ .( 1 1 3s(k) ---W(k- 1)>120 840 k "' 3333 k4I J 3333 )yJ3 120 8 4 01-"k 33337y)ay3 S(k

-(k) r 1 1 -3._(k) s -(k-I) +[.. I 3•2s(k) 2 ( -(k-I),I"+ ,-y3 '- 120 840 k 3333'y p3,p 24 T280 ks3333-y 33

+ W n))N+[ _L W..', W

12 '03 )o & 10 ;37y .

+ I + I [ -X -k) +t 3 S(k+0) 7) lcr-(k)y

120 840 k 3333 k+l 3333 Y p3.p

102

r+ [(_ l+ 3 Xt2+ S(k-' ) -t 2 s~ W yc-(k)X0l, T'+[t (k+ ISU ) yNdk'1)),

1 3 -2 -(k'i) 2o-kt, -oIKNk)> ~

24 280k 333k337)Y3 12o3,y

(k [) 4(k)1 ) 1) 1 " (k+I1) 7 (k+ 1)

+ pIp. IM-- )-' +[--- )tk'+l 3333333), Yap0"y

10 P337) A 12-0 40ki 4 331-03,

+ [(_---- )t 2 S (kI)7+t (ko1l)Y' 2 (k)>-24 280 k•I•3333 33 +•3SS•J )

-k)_ p 1 _1 1 3)3 (k--l) ,_*+ <Tpp 2 840 k+ I S)33

+ [(-LI+-!_LXt3 3S(k) + t 3 S (k-1))7 Kxa-k)),

120 840 k -3333 k+l 333 7), )3

+ [4 1 __ _t S k a(k-I ) )120 840 •k 33337)), Yor3 s! ;>(k)

Using the divergence theorem:

-(k) (t k+I N(k+1) IM (k+l))s(k' 1)>< 3-y'12 NOP 10 GO &P3 R W)

V3 . '' 12 'O~* 10 "~ u.833 R~k)

+ < l,-(k) _(k+t1l N(k+1)" 1 M(k+ l)M(k+ 1)S>y3 '---2 •.1, "F 1 .1, o a033 I 5k)

_ "-~ ( 1k .1 k' k I) ]d(k+ 1))Sl(k+ 1)_.ik

)•,371 (), 12 OP( 10)4. P aP3 s.kl\(~),(k)

k ( (k) (k )i (k+1)+1M(k+I))S(k (k)

+, (' r >))d

< _(--- ,.. ( kt)• )S( -1Mk) ,> kfy3 k y~l 12 Op'P 10 arP p or 33 Rik

S -(k) ( tW N(k)+LM(k))(k)112--•O 10 RP e S 5(k)

+ <7 (-(k)( tk N(k) + _ .LM(k))S(k) >.,y- 7t)Y 12•3 O-- P• 1--0 M # aP 33 >S(k)

-: O ( k ) - A ~) -( k ) =: T ( k ) = , k ( -( k ) >

< -y3 3>R(k) <- y3 7 > R W

103

XSk +(k+ O3 o1) (k -(kt) -(k)

+ ____)S()t3+S kJ t 3 -4&k) -(k) >1S120 840 333 k 3333 k+I f y3 11- 1 p3,) I3h 3p()

+ ~ ~ 1,r+sk) 7) ITr-(&k)7) dS~kdSk

-4k) 4(k) -4k -(k) .(I) --(k)<33 '21 T-y > R < T y3 = 12a3 >&k

<- ,-(k) + 3 x(k) t3-(k+I) t3 )'r- Wk>-y3 7i-(T 4 2 8~0 3333 k3333 k-I' 33

r _ + 1 X t 3(k) (k+ 1) t3 r (k) (-kfh) 24 280 333 k

-(.~k-+ ) A(k+ 1), - (k) = (< ) k+l 1(~)<o'y If T y Wý< 3 7II1o y3 R W)

+ (l I-)s(k+l)t3 [< 0 r-(k41I) -(k) >-(k) (4I

120 840 3333 k+l 'Y 7),, ' P. 0 <7 y3 7)~ y , 0 P3.p>S

1 1 I--s(k+l,)t:, 3 (0 -(k+ 1) 7)Y-(k))'d(k)__ I -(k) -~(k+ 1))'dS(k)]120 840 33t kf1 4 k) 'Y3 ,p f lk 'r. 7)- 'p3

y3 I I -y3 R ~ = 3 11 -y~3 R W)

120 840 333 k -3 7 1) %'p,p > S~) < 3 l)y ' O0 p3,p> l

+ 1-- 1--)S k) I (o'-(k 7)T (k)yd S(k)... (T.-(k) 0 -(k- '))ds'I)120 840 3333k: f W~ -y p3,p i f W yk~3 7)-y ap3.p

33 ' 21 1,3 it(k) 'y 12 33 R(k)

+ <,r (k) ( 3 _I )t2 s~k W IT-k-1)>,3 77,' 280 2g 4 k 3 3 33 3 3 S(k)

+ f k) 280 24 k 3333 71y) ,~3 33 i S

-0.(k+I1) A(k+ 1) -(k) -4-k) (k+J) -(k.I)<T33 A 21 7 y3 >R(k) <7 -3 ' 3ý 2 (T33 >R(k)

104

+< ;(k)T) 1 3 t2 S(k4) -(k+ )0+

3 , 4 280 k+ 1 3 3 3 3 " 3 3 >

+ I 3 )t2 S(k+1) -k)-(k*1) >s 2 280 k+1 333371/33 ;33

Substituting these relations into the expression for the Gateaux differential we have

-(k) Wk) -(k-1)+W -(k- 1)A 3"(k)(=<--y3 A IIy3 12 33

v--k)+1t k)+ (k) tk (k) 1M(k) 2 (k) (k)

'Y 2 033" 12 A- 10 -A' :5 p3),3 P

--(k) -(k). .(k) -(k)+ = 1 3 +=12 T33

_-(.k+l)_ -k+l -rk+ 1) l~(k+1)( t k+J (k+l)- I (k~l)ý_._ 2SU4t) (k~l)

+ -k ). -(k+ ).-rk)) "7-(k+M 1)>2 I y 2 3 12 3 0R k)

+ -2 ( ,r - (k ) [4 _ L I 1 t S3 -(k ) ]a - (k - 1 ) . , 1 . 3 -_ - 2 --( k)W--- (k - 1 )+ j

S. ) +[ S nR(k)

+ [-L(k) + •3(k)+-3 -(kl)•--(k)12 P33)' OP 10 *P3 3 y P

+[(_I_+-IXt3(k W +t3S(k+1)~ ]a-(k)* -120 840 k S3 3 3 3 k+I 33333n' p3,p

+ R I + 3 xt2 S(k+i) t 2S(W ]CY.-(k)+[ tk. IS(k+1) ].(k+1)24 280 k+I 3333- k 3333)nlf 33 12 - P337yJ•o

* [ s(k+ 1) N .(k+ 1)+ [4 -- 1 1 --3 S(k+ 1) s7i -(k+ 1)10 aA3 1) O 12 84 k 1 333 pp

+ 1 3 )2 5 (k+1) ---(k+1) (k)24 280 k+I 3333

7)1y33 -. o s(k)

+ 2<r-<k) [_( 1 L)t 30(k) 1 )r 1 + 3 _ t 2S(k) 0 -(k- I,

y3 ' 120 84)0 k 3333 71-rx p3.p 24 280 k 33337)yJ 33

+ tk (k) -W _ - k IW - 4(k),+ - .0''3 j33 7•))JkN o~p) '+ - *P''• •33 Y N .N o )12 1 ) 'r 10

+ I +_LvXt3-(k) 3 (k1120 840 k,3333 kI 3333)''.) p3,p

105

+ [(_L + 3i.)t2, (k- 0_ 2 S(k) ~-(k)y+[tj N-1 ' k- 1)24 280 k Is 3333 t k 3333 )T lo 3 12 S,03),XO

10 S.37- a1 Vp 120 840 k I S333371 .Yx p3.p

+ [(1L 3 )2 S N41)~ 7)xo*-(k+I )y...gL() > (256)24 280 k+i 3333 y 33 S"k)

Vanishing of A ,,,fl for arbitrary T~k) implies

(k W (k a 1 I) + -(Ik-I)A11cr., 3 12A c 3 3

-- k)t k)Sk c Nk) W (k) ) 2 (k) V(k)

-V 2 y 12 y' 0 p33 p

+ II y3 +=12 4733

" (IcI+ 0 k+ I Ok41) +S (k+I)( tk.I 'N(k+ ) 1_?*(k4 oI>2 S(k4 I)V(k4 1)2 v aP33 12 OB 10 0A 5 P3 ),3

" t l ( (+)7+ a-I+ 1) = 0 on R W)

120 840k 33337)y] p3.. 2[-4+ -28 k 3333y) 33

+ [t~ k(c W , ]N)+[ f )12 .0337)yV 'c 10 Sý37- -P'

*~~~ [(_+IX3 S(11) t3 kl), a()+'120 840Xtk 3333 +k+I 3333 y P3.p

+ [(_L+ 3 )Xt2 S(k- 1tz(I) 2 W ]a -(k)+f t k+ I SW 1*) ]N(k- 1)24 280 kc4Is 3333- kIc 3333)' V3 3 12 '*P3371V ov

+ Oý37y* 10 (-12-0 84-0 kcI 337)yp0

+ I..._...3..1X)2 S (k+ 1) -k~+ 1) g W~ on Sýk)24 280 Ic+I 3331) 3 a

and

1 _NSi) l -4-),[_L _~ 2 (k) x 0 .- ( - 1)y

120 840) k 33337)' p3.p 24 280k3371Y3

106

,3n(k)-- 3 (k-I).v-(k)y,

+ [,-Lk -0'-t k 3 3 3 3 t 33120 840 UP+33

+(120 840 N333k 1 3333 ), 3.p+ 4-[(_I +i t21 (k~l) 2 .k)W -(ký,_ Ok+n S(k+l) v.(k+l)).24 2" -• k 1 3333--k 3333) ryX!O'33 12t-- 033 '-Pa

+ ( - ,.(k + 1 )) j M . (k + 1)I r , | | _ • 3 ^ S (k + I ) X - -(k + l ) '•o

+ [ I_ ._•3 2 (k÷,) X -(ký)y, ,(k) (~k)" 24 280 -k+ I 33337)-y 0'33 1 g• o

Gateaux differential of il with respect to 0,k along path " is

-(k) -=4k1l)+ 1(k+0 (k) 6 6 (k40 (k+1) I (k) k) 6(k) M(k)A 4k)fl <7r33 , S. + 33 )NkA +Tt7S-.s0kiM. +-IS N0 -- Go - M 33'

23 -t 4 'ý ~3 tk

-- k) --4k) -4kl). I W (k) 6 (k) (k)-v -v 3 +V 3 20- o .a - .k .t aM,"

* I.S(k1I)N(k-i)+ 6S(k+1)m(+)>2 P" a o tk .03 O L)

-(k) k -(k) k) -. - (k ) W -(k-1 () W -(k-1)* < 733 z 210"y3 +='22 033 +A21" y3 A22 d33

-+ t k+i), -(k+l) -- ",t k+l) "-(k+l) -

2I ).3 22 0"33 >it(k)

-(k) Ak) --(k) -(k) -- k) -(k)+ <ry3 '=12 T33 > R~ W +<a33 ' 22 33 •>R(k)

-(k•3 1) (k+ I) -(k) -(k. 1) (k+1) -(k)+ <Oa)3 A 12 733 R(k) 0"33 , A 2 2 733 > R00

+ <o.--(k- 1 ) Vk) 7 -)(k) > +(k - 1-y3 ' I12 /33 •R(kt) '' 33 ' 22 33 •R(k)

-Lk) 21 3 2 S(+I) --(k+ 1)+ <733 ' 280-• " ,)k+1 333371-y)TOy3

"+ [ l+ 3 St2 (k+)t2S(k) 2 -W -(k)24 280 k+l 3333t k 3333)n-yOT-,3

"r(_ 3) •.t2SO2(k) -k-0l)>

24 280 k 3333T1), 3 -y )

107

+ < -(k) , ( 1(-- + 3 •~2ý S(k-1) x,-(k~l))v

'33 24'2go k2 1333371)/ 73 )

+r(_L+ .. ý Xt2 5 (k- 1) 2 W(-kk+ (24 280 k+• S3 3 33- k S 3 33 3)7"•)yOy

r+ [(L 3 )t2^k S ny -(k - I ,>+ 24 280 k' 3333Y',3'

Using the relationships

-(k) .(k) -k) -k) -k) -(k)<O-y3 2•12T33 > R k)-W 33 ' -- 210"•y3 > R(k)

*-(k) +( _ +. 3 )t2S(k) -t S(k+1))-(k)_

+ <T33' 24 280 k 3333 k--I 3333"-y3 ')-7>ýk

+ f )(_ +. 3 _ t2 (k) W(k+t)) -(k)a -(k) )... (k)+ J (24 280-----t k 3 3 3 3-- tk+I 3 3 3 3 ),., 3 3 -y3 ) I

< -(k4I) A(k+ 1) - )-(k)(k) -- k+1) a-(k+ )>

y3 12 33 R(ft)- 33 21 y3 R1k)

+ -(i) • 3 ( - 1 2 S(k4 ) -(k 1)s_33 280 24 'k+i 33330-y3 7)y> JO

+ f ((__3-+L)t2 , (k+I) -(k) 0-(k+ I,>

Sr 280 24 k+I 333371'y'r33 ' y3 5(,k)

-(k-I) -kk) - k) ,) A~k)--(k-1) >< c 3 y3 12 7"33 •> R"')- • 33 ' " 21 -y3 R(k)

S -(k) - 3 _ _ t S W( ) -4k- 1) >+ <7 33. 280 24 kkS3 33 o3 -y3 11,s~k

+1f , W -(k) -(k- >+ k) 280 24 k 33337"•1y 33 Oy3 ) 5k)

-4k) , (k) -4k) .< T -(k) , 4k) -(4k)<033 r22733 >it ) 33 , 22 33 R (k)

-(k+1) 441) -(k)_ _ -(k) •-kl)a-(k)•033 •22 "33 >Rk)--i 7"33 P22 "33 •Rtt

and

S-(k -1) - 5 k4i) -(k) -(k) (k+I) -(k-I)>33 , 22 33 Rk) 733 22 033 R~k)

we have

108

•,)fn=2<1••) ,~ +%"•)')A,,,) '

33-k) ' 21(k ),( 3 22,(k 33

. (k) k) (k) -(1k)+ --"=2I<"'33 "A'2220-33

- )o1 1 (k.,) (k+W ) 6 W .W(k+I k+1)+v 3 T+-. 33 N t *+----3 3 3 M&P

21 ")o3 ̂ -22)3

-k (k-) I(k) -(k- S) M(k)+t~+A 21 a3 22 ""2 33

k+), 1•(k+).,(k+) 6 Sk) (k)mplies

-v 3 2 u•#33a. - ot#33 a•+ ( (k- 1) +A-(k). ,(k)-(k021 21 3 y22 33--Wk+)- I S k+|) N(Wl)_ 6W , (kl)..kl

+ 3 &0£o33 Pe "P-ý-%P ',+33-'*P o

-rXk+) -(k) -A)---(k+ )_-k )R+ A21 0- y3 +220, 3 r3 -0o

Equation (230) represents the basic function governing the behavior of laminated

composites. Use of this function is possible for both nonhomogeneous and homogeneous

problems along with discontinuity conditions. This function is completely general, in

the sense that it admits v-(k) k) W k) , M(k) V( r as field variables and

there is no requirement that admissible variables identically satisfy any of the field

equations or the boundary conditions. We note that the function is defined on a space

which includes the set of admissible states as a subset. The space is defined by (242)

through (249). This directly represents an extension of the space of admissible states

in which the approximate solutions are often sought.

109

4.7 Extended Variational Principles

The functions in (230) must belong to the domain of definition of each of

A(k) i, j = 1, ... n . This implies restrictions of smoothness on the choices of vj in

approximate solution schemes. Some of these restrictions can be relaxed using extended

variational principles based on elimination of some of the operators. These extensions

were first discussed by Prager [1967], and Pian [19691 Sandhu [1975,1976] proposed a

general scheme for these extensions using the self-adjoint property of the operator

matrix. In the context of the finite element method, it is clear that the extension of

the admissible space provides greater freedom in selection of approximating functions.

To apply these ideas to the present problem, recall:

•k)(k) -k(k) (k) -)S, -t/l13 R(k)-- l ,f V(CWA) R(k)

W <Nk -() -(k) + (;k ) W* < N , 0vot > <V , Nr*A > W

_ .(khmean W-'k)•, + <["",--kjmrean (N(k)

+ <[NLk# , 9 (k~y> <sk) ]ma ' (7 )" >> k) (258)

Here [N.,]"'. [ are the mean values across the internal surface Sk of the

quantities in the brackets.

-4k) - (k) >M(k) ,k). MPA R (k)=- ,• '-M,p .S) R(k)

W k) k) (k)

+ <M $k)00> + 4 , M 7'

+ <M(k) - k) (k)

<[M~kmean t )Wk) > +k) , (mean9M)'v?)> (259)

where [M.•.""', [-xk)?-" are the mean values across the internal surface !k).

-4k) Wk Wk -(k)< ,V3 > R(k)- < V V I, > R(k)

(k) -- k) -4k) (k)* < V , 7 .V 3 > S W+ < , V ar 7 >Ssk

6

+ <(Vme , (Ak))> k,+ <( k> k)" (260)3 () 3 (Vo)r)> 20

110

where [Vr•I-, (rs""-" are the mean values across the internal surface Sý. Equations

(258) through (259) can be used to eliminate

1. or 6j)

2. or Ka)

3. 0.0or v3

from the general variational formulation. For example, using (258) the set of admissi-

ble states for Nk) is extended from C' to Co. Various combinations give rise to distinct

extended formulations. To illustrate application of the general procedure to the analy-

sis of laminated composites based on stress formulations, we shall only state six exten-

sions leading to some useful formulations. Other formulations can be constructed on

the lines indicated by AI-Ghothani [19861 in the context of a unified approach to the

dynamics of bending and extension of moderately thick laminated composite plates.

Elimination of N'k, from (230) gives

' • y3 It(l) 'wY , (D3 33 R(Q)

+ <• tI ̂ (1) (0) +ISMO (0)>

* 12 a" 330 . 3,, 2'033 33 R(I)

2<Mi) 1 S () a(o) 4 6 S(i) a(o)> +2V 2(1 (o)_>

O T a03 )y3ly -t-P 3 3 33 R (I) + 2<p( )

(N) (N) tN (N)> + 2 <-(N) (N)* 2< -a > N4 ) 2V -."y3 R(N) y ' 2 v"y3 R(N) 3 33 R

+ 2 <N(N) tN S(N) a(N) +1 (N) (N)>00 12 *P3 ),3,), 2 S.33r

4 2<M ) .s)S(N) (N) -6 -S(N) (N)> + 2 <V(N)- 2.. S(N) a(N)>1 0 *P33 "y3,--r i •33" 33 R(N) p -- p3' 3 )3 R(N)

N

O,3 W T0ry3 3 ' "33 R(h)

k-2

lii

W(k) 2

tk s(k) -(k- I).+ I W(k) -(k- I)>

" <N 1 2 . 033 y3, 2 , 933 33 R (k)

<M (k) s (k) -(k-])._6 --(k) -(k-I)>

v(k) (k) -(k- I)" -P p3y3 3 >R(

N,q : -(<; k), M .(k) Wk) < _-4k) v(k) :

+(k 3 >• • , (k)

k-I

(k) <k) . (k) I . (k) W (N )- / 2< *A V(*,A)> k ;k)"I ,t A3 tk ,PA A#plz R (k)

(kk

_ 4k) + 12~S~k) M(k)><•M k) ' "•(•,) k3 O3•p uIp R

+<V W) 4.k) .--(k) 4"24 s (k) v (k) >J

.), , Y 3, --ay- + •k p3-y3 p Rk)tk

-IE<_)k) -0 0> + k) k --(k)> -(k) - -(k)av31 k) ' 2 ")3 R(k)3 CT-03 3 R~k)

k-I

WNt) 2 tk k(k) 0W-4k) 1 S(k) (k) >"O,-i- -l2• a 3 .3 3y + 2 3 3 33 R W

M.k) 1 5 (k) W -k) 6-(k) W -(k)>

W(k) 2(k) W -(k)>" <V - p3y3 ay3 R(k)"

N-IN-1 -(k) -- •k),tk-( k)l k(k Ik)_ 2 (k) v(k)>

+7(< -v ",' 2T 10 33 R

k-I

<•O-(k) A()o.-4k).t .-- k)o-(k) •

+ p3 ' I 1 y3 +- 12 33 R(k)

-o (k) _, ( - k- ) - tk.1 _O k+ l)

+ <(T3 .v -

1 s (k+ 1) kýo _ 2 S(k+1)V(k+ )> 1)>

- o•A,),"-5 p3Yo 5-y3 p R333

-(k) -(k) I (k) W( 6kW)+ <(0"33 ,-_v +"-S% N-s--k.5 (k) M.kp•>RW)

112

- ,(k) - ) a - ( k ) . .. ? k ) a - ( k ) > W<0-33 '21 )-3 -22 33 R(k)

-(k) --(k+1) 1 (k+l) (k+). 6 s k'I) m(k 1)>+ <o33, v 3 +" 'S0 3 3 N "- +- - A3 3 M >R (kj)

3k4, I

N-I-(k) (k) -(k- 1) (k) -(k-1) 4 -(k) (k) --(k-) 1 (k) -(k-1)

p AII0-y3% +A 1 2 o33 >+ <33 A2 aO ,3 +A2 2cr33 >R(kd

k-2

N-2+Zl< a -'(k) -r'(k+ 1) a--(k+l)__-r'ký-1) (r-(k+ 1) >U1+.<-y3 11 -y3 1s÷ 2 "33 R ,k-I

+ -(k) -(k-f i) -(k+I)0+7"-(k 1) a-(k" 1) ><033 , 21 -y3 "1/22 "33 >R (k D)}

+ 2 <(r}"-(N-l), [LfN){orj(N)>

+ 2 <{-}-', [At',{o-}°>>R

N(kk) )+E(<,-2g, >~k

k-I

+ < k) .. (k) (k)

+ V " -iMv#-- 2 g3 > Sk

+ 2 ~k) -(k) (k)_

-2V 3 > ()S2

+ <M(k) -VWk) W()

* <v(k) --(k) 2g(k)_>. ' 7)aV3 -- 96 > S(lk)

-0) 1 0 () + 3)

120 840 13333 2 3333 )yl 3,p

+ [(-4+ 3 xt2 (2) t2S(1) l(a .-(1)2+ 28- 2 S3 3 3 3 - 1 3 3 33"1y "33

113

[ ]M((2) .2). - 12 I 0t3

10 t] 0"33 )]&[A ( 120 840 p233333)p'p3,p

1 3 t 2(1) 1 (2).. (. )>

2428 2 3 3 3 3 ,c 3 3 ~.~s 1

N-2-(k) [ 14 _ - )t3s (k) 't -(k-l)+[(., I+ 3 ,,t2 ( LW -- (k-1)+El 3 120 840 y3k 33337)-fOp3"p 24 280 k 3333 1

7-y'33

k-2

"" [- s (k) )M(k)+I(2.1+ I Xt3s(kW +t3 s(k+ 1),n ]ar(W

10 P33 -y oto 120 840 k 3333 k+l 3333 )) p3,p

" 1 [ . 3 ) 2 s (k+i) 2S(k) 7 ] -(k)+ [ 28--0t k 5 3333 k S 3 3 3 3

7 )y 33

"1 (k+l) +[k ( . 1 ---(1 ] 3 s(kN41) ) -(k4I)

+'10 n033) OP 120 840 k+I 333371yOp3.p

+ [( 3

-)t2 S (k- 1) -(k1) 2 g(k)>24 280 k 1 3333"'Y) '33 a s•k)

+ -(N- 1) t3j(N-) -- N-2) [(. , + 3 ,_2 )(N-1) I2 8N-2)

+[-L [2S, 7)N]a __N(N_)

103 120 840 N 24•2• 0

+ 1 1 3 1(N-I)t3(N) -(N-I)

+ [I_-+-1-Xt3 S (NI+t3S(N )7)11]a-4-120 840 N•• 3333 N 3 33 y p3,p

+ I [ 3 ,Xt2-(N) _. 2 5 (N-J)l ---(N-I)

24 280 N + 3333- N-1 3333 )y33

+ [1 (N) .(N) (N-1)10 S.31)Y" .- g > sq (N1

N( )(k)

k-I si

+ < Lk) , m(k)V, ,g(k) >

+ 2<N"=) ,--d). ,i(k)>

SMS(k)

114

"+ <V k), - (2 -S~k'-7JIM'(kY

"+[_L+ t (1 t3 s (2) ,'

120 840 1 s3333 t2s33) yp3,V

"+ [(_L+ 3 )(-t2 s(2) -t 2 s () ý y 1())24 280 2 3333 1 3333 -y 33

+10 s oAr -yap 120 840' 2 333371y" p3,p)

_1(I3 )t2S(2) XY -(2 )'2g()>

+ 24 280 2 333Y12g3

N-2<a-(k) 1 1 [4._I)3 (k) -(-1

y3 120 840 k 3333Th-yyp3,pk-2

24 K 2 gO kS 3 3 33vY33u

+ I .(k)( b'-

+ R[(.!-+_.!--t3Sk s W ~ "> t3Sk1)rlxa.(k))y120 8w k 3333 k+l 3333 Vp3,p

+ 1 3 _X2 S (k+ 1) t 2 S W )n X-(k)

[ 2 4 + 2 8 0 tk+l 3333 tk 3333" y'33

+ [_c(k+1) YXi(k+1)Y,+p. 1~!-~-- _)t3' N-1) ycr3 k J)1+ o3 3 71J, OPi~ 120 840 k I 33337 'Y P3.p

+ 24 280 k+ i'3333 -y~ 33 S11

+ <a [4N ) (11.- 1 .)t3 s (N- ) 0 -(N-2))y'3 120 84 N-1 333311y PIP

+ [(- 1+ 3 ~ 2 s(N-1) v))Xa (N-2)

+ XM s(N-1)+tsNhy,)

120 840 N 1 3333 N 3333 )l p3.p

+ [-l 3 xt2 (N) -t2 s(N-I), ,k-(N-I).,

24 280 N 3333 N-1 3333 )y,33

+ [1 (N) VM (N)y 2 .(N-)0

10 3- g& >S sN-I

Clearly, n1 is defined over an extension of the domain fl insomuch as W need

only belong to CO and not to C'. In finite element procedures this relaxation of

continuity (Pian 1969] is quite important in allowing lower order interpolations or

simpler approximating functions to be used. Elimination of M() from fl, gives

(0)l) t•<--(1) (0()>

fl (u , o)=2 <V--') , a" 3 >()M+2 < ,y ' > (1)>y3R "y3 3 33 R(I)

+ <N() a ( (0) + 1 S(1) a (0)>+ 2< 12P , "2-o,'33 V3,7 "2" P33 33 RMI)

(2<M1) 1 S,() (o) S 6 o() (o> +2<V(1) _2S) oo)>""+"MOP10P3307"'y -K1*33"33 R(1) + '5 p3y3 py3 R(

-(N) (N) N) tN (N) -- N) (N)+ 2<v -Ocr3 > R(N)+2< 2 - ,3 >N + 2 < '3 -- a33 > R(N)

+ 2 < N(N) tN (N) Cro(N) +s 1 S(N) a.(N)>

1p 2 '&P3 v3y 2 &P3 33RN

(N) 1 S(N) (N) 6 S(m(N ) . .. (N) 2 S(N) (N)+2<M , 5(),N)/ 3 +2 <5"(N), 70"33 >RN10 &,633y3,t- --' 33"33 R(N) p " p3y 3

ýN

NNEy < V-(k) .a-(k- 1) > ýk) k .-(k- 1) >-4-k) -(k - 1)>

Y 7y3 R' V 2 V3 R(k) 3 ' 33 R)

k=2

W(k) 2 (k) -(k-) 1 k(k) -(k-1)1 < Nap 2-yS303,ay +"2 "S &',3 3o" 3 3 >R W

S (k) (k) -(k-I) 6 (k) -(k-1)* ,3 <M &ATO"b•33"3, a "ly+ gtS.33 (T33 >R W)

W 2 W -k3 R(k)

N

k) 3 -- 0a R(k) 3 &'*o. lk

k-I

116

W(k) -(k)-2<N OP, V (A,) >R(k)

Wk I1 k+ <N~k 1s (P) A(k)>,

k

k < (k)-2<M >(k

M(k) 12 (k) W) + W ,k)_--4k) (k) ,,(k)

+<M , >,•- +V 3'- - t, 5tkps)a p >Ro

-- ---(k)t __ . a). p ).,<' (: ,p R O 7 .~ ) t • V 3 R,k

k-I

+<N(k) 2 tk (k) -(k) 1 (k) -(k)<N -.'s a33" 3,+ 2:•,P330"33 >R(k)

OP 12 oP33 ),3,+S-y O3 >~k

+ <M 2, 2-(k) -(k) 6-s(k) o -(k)>* P M . "T o ,330"y3.y-t 5k A33 "33t R(k)

+ <V(k) 2 W(k) -(k)>J

p 3 5 P3.y 3O*-y3 R"

N-O -4k , k;O2 - S~ 3kV) W (k

k-I+ •{<O4-(k I-( )+t-) v 12s 33 V(k) .

+L.2'3 -V' ~ -- 5" 3V3 P Rek+)-(k) -(k) - (k) -(k)__-(k))+ <3p3 = II'y3 +" "12 S ro33 > R(k)

-(k)k) ) - .k) , k ) ->k)

+. <O"33 2I 1 y3 +22O 33 1 (k)

-- k) -4k+I) I (k+i)N(k+l) 6 (k+ k1)• ,

-V +-S N +-ýS 'M,

<a"33 23 T" 33 e tk • 33 A R W

N-I ( ) k) -'(k--)k+ <O.-(k) -()-(k-l) - (k -(k-i) 6 .t O3(3k+) m W1).y > j203 >~

+ I<aP3 , All, y3 +A"12 33>+<a 3A21 3 22 33

k-2

N-2

+E ~ -(k) , 9k+ 1) ol-4kj ) -xk4 1) a (k4 1)>(k1yr3

' I y3 12 33 Rk-I

117

+ (k -. k+l)o -(k+l k- 1) -k 1

+ 2 < fcr}-N1) , WN)1Il(N)> R(N

+ 2 <{r', [Af'){cr1( 0'>( 1R

N

+E(<(k) 2 g1 (k)>k-I VSlk

- k) -2 (k) > k

* <V 3 , -7).V. -2gs >Sk

W -4~k -k)_ (k)>

* ' 2<M V -g42 k

Wk -k) Wk

a 3 S6

+ 0 <c 1 (_.k+_ALXt3S(1) +t3S(2) -0)-y3 120 840 1 33 2 3333 yp,

+ [(-L-+--_.-Xt 2s(2) -t2 s~ (1) -0<)24 280 2 3333 1 3333 v ]C33

12 84 2 33 337)), p3,p

+ [(__ 34 t s~3 1 (2 ]-2-2g(')> s

24v 280 j 2 333 24 280 so)3'y"3

k-2

+ 1(~-A-+ 1 3 (k W t3s (k+l) )nloa -k)120 W XkS33333 k+J 3333 vp3,p

+ 1 3 L~2 s(k+1) -t2 s(k) -(k)

24 280 k4 1 3333 k 3333 qyT33

120 840 k I 33337)y p,-(4

+[(_L _ 3 )t2. Sk'I -(k+l)- g(k) >-- ~~1 3333~1cT3 -2., ~k

24 280 k 3 3

+ (-I < (-I -(N-2) 1 3tl 2 (N 2),n -(N-2)[__ __ t2y3 120 840 N 33yP3IP 24 280 N-I 33337),y33

+ [(_!-_+ _LXt3 _ s(N1) +t 3 s (N) ) a.(-1120 84 N-I 3333 N 3333 )'PIP

+ 1 3 2S~t2(N) .2 S(N-1k T-N (N-I1)>24 20 N3333 _N-I 3 3 3 33,,uny 33 )-2g >( )

N

+Ei<V-, -2g(Vk -2)~ > ~

+ 2M, SVSk

* < VM~ ki -2g~~ (Ic)>3 S4

+ 21 3 N((2) q A~) > a 1~

W >

* < 120 ,70(V<)-284u p3.p

*~ <(T -0) , [ _.+-L t 1 ()ts +tS2 K k)71 -1-3 120 840 1 3333V 2p33 ),"P.

+ [_4__+.L_ )t~S3 s() ) a(2-)y,

24 2802333y

N119

+ 1 + -1Xt3s(k) +t3 S(k+1) -(k)S,

120 840 k 3333 k I 3333)'IT'),Op3,p)

+ I + 3 Y t2 S(k- ) 2S(k) -- v -(k(k)+ 280 " k+1 J3 3 3 3 tkS3 3 3 3)7yJ O*3 3 )

+ - 1 __ ~3 S(k+l)n K 0.-(k+l).*+L120 8 4 0-'kI 3 333Ty• p3.p

24 280 k+1 33337ivyO33 )y. k)>• I 3 •2 c(k+l1 (~)• ,,k•

-(N-1) 1 1 3 (N-1) -(N-2),." <(7 y3 • 120 840 N_- I33337j)-p.p

"+ [(__+ 3 )t2 S(M ,) 0 (-N2)).24 280 N- 13333 7)) 33

"+ [(1 +

1 Xt3 s(N+t3 S(N) )71 -(N I

120 840 N 1 3333 N 3333 )Xo' p3lp

"+ . + -3Xt2 S (N) -t2 S(N- ) I )-( .2 (N- I)

24 280 N 3333 N-I 3333 33 y- 3 S!,N)

The domain of n2 is an extension of the domain of fl1 to include MNf'• which

EC* but may not be continuously differentiable. Elimination of from n2 gives

n 3(u',a= ' ') 024 1 j( 0 2v() 1(0),>SY 2,3 L T' 7 +y3 R>(1) 3 033 S)

-2 <N) t 1(No)a 0 + (') a(0)>+ f T2 &P '•'a330.t3,)l 2 *033 33 R (1)

+ 2< 1 -1) (0) 6 (t) (o)> 2 <V) 2(N) o*A TO P330-y3,)," tI%0330'33>,R(I) + --"By303 •R(

' y3 R(S' Tory3 R(N) 3 3 R(N)

+ 2 <N( ) tN (N) T(N) ,(W) (N)>12 -'&033 '- 3 .-y÷V'•o33033 >2 ("

+ 2<M (N) I S (N) (N) 6 ,(N) _(N) >. + 2 <V(N) 2 -(N) (N) >+ rp TO *0 P330Uy3., - aP%33'r33 p R(N) p -- Sp3-y30y

3 R(N)

N

+j< -k -4k-1) tka -(k-1) -(k) -(k -i)R ' 2 -/3 R(k)

3 * 3 3 >Rk-2

120

+ < -2tks(k) -(k-I)+ s(k) 1 W -(k-1)' -12-- 33 ),3.), 2

M(k) 2 S(k) 00 -(k- 1)+ 6 W -(k -1)>* 1M* OS03 47 y3- +t - .33 cr3

W vk 2 o.(o -(k- 1)>k

p - p3y3 r,3 > }

N

+ Ej <R, V W>

k-I

(k) --(k)2<N OP ,v(a.p) >~k

<N(k) I S(k) N(k)>Go•8 ' k 'P A Pp AP R~k )

W k

- 2<Gr .P>R(6

< 3 O P R(k)tk

W 2 <V --) ) +(k W)._k),24,(k)v(k)> )- < 3., > R(k)+ < , 4r.y +"t•k p3),3 3 p >R(k)]

N-i

T' - 3 R(k) 2T Y3 R(k) 3 '0 3 3 RTk)

k-I

+ <N(k) 2t (k) -(k) I (k) -(k)* ,<N P -_12033 -y3,), 2+ "j 33033> R(k)

W M k 2 ^(k) -(k) - 6 s (k) a,-(k) >* <M "'TO0330"y 3

.-- Stk •O33 33 R)

* < v~k) 2 .(k) -(k)-

p ' 5 p3.30"r3 > R(k)

- 00< k)_ 2k. ) 2 Y .(k) 3 (k)

k-I

-(k) ,k) -4k) ,(k) -(k)+ <a p3 = €110a),3 +--i2O"33 >R(k)

-(k) -4-( ).tk~I , kJ)_.. (k-1)vU-1)

+ "y,3 2 T p3Y3-- R(h. )

121

-W _O ) --4k) I s (k W N k) 6 W(k W(*< , 33,V 3 + T' o 0a3A•--St k %0l33M-0 18 Rtk

-- .k) -,(k) -W(k) .,k) -(k)* <or'33 , =21 O-3 +i'-"2 20- 3 3 >R(k)

_ - Wk -4-'( * )+. l .(k+ I)N,(k+ i) 6_._ S,(k -- 1) M N+ 1) •>+ <a033 , V 3 2 N•33 pW 5tk+ I P3 3 P R(k> d 4

N- I.I ( - (k) W(k -(k- W) k -(k-1):>• -Wk W(k •kl-(k- ) W (k- 1)

A 1p3 A II-y3 +A12 033 >+(33 A 21 y3 +A22 a33 >R()

k-2

N-2 -W{]r k -.-4-k+l) (kN )_0 + l)4 -kN)- ,

+EI y3 A Ifl 0"3 "'12 a"33 R(k.

k-I

+ -< W --t k+1) (-(k+I)_- )"(k4 1) >33 21 -y 3 22 33 R(k+1)'

+ 2 <( 0-}-(N-I), [N){ 7.)(- > R(N)

+ 2 <{)-}I) , [At 1)(a)(0-)> Ro

NS -(), -2g(k)>

a g >ek)

k-I

+ <•k) W(k)S4 2g 3 > k)

-4k() (k)

" 2 <N(k) -W(k) (Ig)>&P a 2

"+ 2<M 7p$-9 I)g()>

W(I) -(k) (k)"+ 2<V ' 7, -96 1sOVs

"<0--() 1(__+ 1 Xt3S(1) +t3.s(2) )7 ]or-(!)

+ y3 120 840 I 3333 2 3333"'-yp3.p

+ [ . I1 3 - - S --( 2 ) - t 2 -s ( I) - ( 0 )

24 280 2 3333-1 3333 y 33

122

I I '-L~3s42) q-)+H120 84023331,pp

+ [( 3 )t2 (2)ln (2) (1)>

24 280 2 337) 3 2,>s

N-2

<a ) ,3 120 840 k s33337), p3,p 24 280 ks33 33'Ty9 33-

k-2

-+ I-2 +-2-Xt3sk W 3 (k+1)~ -(k120 80 k3333 k+l 3333J -Y p3,p

+ 24 280 k+I 3333 k3 371Y 3

120 84- 337) I

+ -(- -L)t2. s(l ]-(k+ 1) 2 (k)>24 280 k4 I 3333)-y 33 a2~>

-+ 1o4) [~ 1 )t 1 3N-0)~ ]a<N-2)+[(( 1+3_ 2 (N-I1 (N2+ < '3 -p 120 8ý40- JN- I a3333'y- I 24 280 )t 1J3333 Vra33

+ I + (3 s(N- J) +t3 s(N) ~-4N- 1)

120 84 N-i 3333 N3333nyp,

[I1 + 3YX2 (N) 2(-I]-N+24 2 80 ~tNS3 3 3 3 tN )7) )-2g3 3 /1)I

N

+E(;ýk ,-2g' >s))k-I 1

.-4k) - (k)+ <V 3 ,-2g 3

+ 2< 40, (;ok))_(k)>

+3 4<Velk)

W 3 6Sk),(k)

+ (IV) 1 9 6 3 S(I) 3()-)

+ <a- ( (-L+-IXtl S) +t 3S () n10 -)f3 120 840 1 13 2 3333 -y p3.p

123

+ -_ ] . _ 3 ,, S (2 ) - t 2 S ( 1) X-- , -( I) ,+ 24 280 2 3333 1 3333)ýn) 33

+ 4_1_1 t3S(2 ()120 840 3332) 33 (2p3,p

+ [(-L-3-)t2 2S(2) "Oa'-(2)y- 2 g'(i)>24 280 2 3 3 3 3 '" 33 g0 S•1 )

N-2 +Z O-0k) )t3 1 W -,3(() k- I.v€ /3 ' 120 840 k3333" ' p3.p /

k-2

+ [(L+-•3)t2S SW(k) " y-(k-1)

24 280 k 3333)Y"33

+ [(_I_+ ILXt3S(k) +t3 S(+l)n " -(k)"

120 840 k 3333 k+1 3 3 3 3 )'Y p3.p

+ [(1+_ 3 LXt2? SL. -• t) 2-2 -- W(k).,24 280 k" 13333 k 33337)•y 33

+ 1 j___-3- S(k+I) a -(k+•I

120 840 " k • 3333"/yJOp3.p

" 1___It2 S 2 (k-1)1) Xa-(k+')'2 4,() >

-24 280 k+l 13333 yO33 a) 2g k

+ < -(N-,) I_( . 1 3 ) ( N-I) I-(N-2),

+ [( _L+ 3_ 2 (N-I) -(N-2).,24 280 n- • ' 3333 7

1yl'33

r+ [(_ .L _ vt33 S(N-1)_+t3 S(N) y v -(N- I1)•,

120 840 N• I 3333 N 33333)1), ,p3.p

+ [(L+ 3_.L •t2 (N) 2 Y(N-I) -N-I) 2g (N-I) (261)24 280•- N••[ N3333--tNI 3333), -,'33 a-Z 4 N-1._)(21

The domain of definition of fl 3 is an extension of the domain of n 2 insomuch as V(21

ECO and not necessarily EC'.

Alternatively, extended formulations which do not contain the derivatives of

kinematic variables >), , and v-'t) can be obtained. Elimination of () from

(230) gives:

124

4u (0) I (0)> + -1) (0)>J(u o-)=2<v a 0" >RM <0 ,3>)+2<v¢ tr 33 >)>

' 2oI ~~ 2 • or •3 33R(12 )3 R

+ 2<MN() t, s(M)a(0) +_1 s(1) a(0) +* 1 T2 P33u v3.•-y t T P33 33•- RS()

+ 2<Nji) 1 s M(T (o) + 6 s (i) (T o) > 2<V (i) , 2 ,(i) ((o() >'i FO Pi33 "y3y 3t-1 P33 "33 R(l) + p ' 5 p3 y3 -S13 i(l)

+ 2 < V-N) ) ) > + 2 <;ON) t(N)> 3(N) -- " (N) >'Y3 R(N -2 v3 R(N)+ 2 <3 T3 3 >~

(N)N_) tN (N) (N) 1 (N) (N)NN -S aT + -IS.30Qs 12 03 -'3,-y 2 3 3 > R(N)

+ (2 <MN) I (N) (N) 6 5(N) (N)> <v(N) 2 (N) (N)>*~ ~ 2 <"OA o SaJ330"y)3 .,y- Tt S- QJ • t333 c"33 > R(N)' '2 < p s -- ' p3),3 -y3 > R(N)

N

k-2

+ <N(k) k -- k- 1)-(k- )+ --(k) -k-1)>)ak-1 "

3 ' R2 - y333 R 3)3

0,8' 12 433 ),3,y 2 0~•.330"33 R(k)

"+ <M k) s k)+6 5 k - T.33 33> R(k)*P T .03 y3.r Stk

" <V(k) 2 (k) -(k-I)

N

+k-2 a<P R(k), > a c ,- > R(k) 3 0 ,,0>R(k)

k-k

+ <N~k) I(k) N(k)

+ < MW 4k) 12.2) W (k)> +<V(k) _4k)_-<k) +24 (k)

+ p' (04 3 -Sp M ' _y V 3 ,y + -S p33ma3vp lR(h

N- I

y 23 R(k) S &3 R(k) 3 -0 3 3 R(k)k-I

N() t k -(k) I (k) -(k)

, No -- 5 (kCT +-S CT0 12 *0"V

3 3 2y3,y 2 033"33 R(k)

125

" M(k) I W(k) - 6 -(k) -(k)>+ o TO .l " 330"y3.*y,-- k 33 33 SRt W

" <V (k) 2 S(k) (7r-(k) >R

p 5 p3y3 y3 R(d)

N 00r ) --(), tk $-( k) ) k W WN 1(k) k) _2 (k) V(k)

k-I

*o-(k) J k) --(k) -(k) -<k)>+ <(T p3 2 aIy3 + 2120733 > oR(k)

< O-(k) v --W k 0 + tk +! ;O •k- 1) tk ~l s (k~ l) N (k* 1)+ "< 3 ' Y) --2"-"") -T--'0"33 A8,),

T 1 .0 )3 3 0. 5 •3)V3 P Rikk- )

k) -- (k) _L W(k) 6 W (k)* <0O"33 , -v3 + 2 - o033N, -- • S.P 3 3 .MVo> R(k)

_ -(k) -Jk) -(k) k) -( k)+ (O 33 , = 210"y3 +"-220"33 > R(k)

-(k) -- k+I) k I .(k-I) (k+1) +6 (k+I1) (k+1)+ <033 , V 3 +T' "o033 ap "5s"k+ aP 33 "m1 R(k+ )N

N-i

+ E <~ -(3k) .(k) -(k-0l) -- k '(k-1) - -(k) 00k o-(k- l1) - .Wk -(k- 1) >

+E <p A If r -3 +A x12 ("33 > <(T 33 , 21 (7y3 +[A 220"r33 >R(d)

k-2

N-2+ I{<o-(k) L- 1k) Or-(k+ 1)_-4-k+I) Ol-(k+l1) >')3 11 AII 7"3 1'A 2 "33 "•R(k-1l)

k-I

<033 ' 21 -,3 22 33 "R(k-

+ 2 <{(}•-1) )" > R(N)-

+ 2 < ' , [AJIc(O) >•O)

N

k-I

+••k) Wok W k)

+ , M 2g3 >4k

126

<--4 k) _ W-k W k)

2<N(k) (k).

- 2<No ' g 2 > s(2k)

W ý 0 -2<M(k) (k) 9 (k)

+ (k) -(k) (k)W

<V. , v)V3 -2g 6 >s(k

+ <c(jI3-) .( +[ •(') n .N()+ 12 3 t-- p 1- • 3o 33 ), *A

+ , -L + ,,--L t3s(i) +t3 .. (2) . -q -0l)

120 840 1t3333 2 3333) -y p3,p

,, 1 . 3 )(t2 (2) 2 -(1) t- -- ). 2 .(2) •,2)

+ [-+LX 2s~ -t s ))IhfcV-")+[ -S(7, IN(224 280 2 3333 1 3333)YT33 T- P337)-y -

+ [_1 ^s(2) •(2) 1 1 3 (2) -. <2)

10 P3371-trp 120 840 333)y7l

+ 1 2(2) -(2) (1)

24 280 [•- - 33337(T33 so)

N-2+E{ <aT-W(k) [ 1 -.--. t-<k-1)_ 1 .3 (k) ]--(k-1)

-y3 120 840 k-333 •7)'Yp3lp 24 280 k-33337y33k-2

+ tk s(W) n ] ,(k) _, S(k) L- (k) +[ ( l - 1 ,t3 3 (k) - 3 s(k+ ) -) -(k)033 Y *A 10 P33 yo 120- 3333 k+I 3 3 33y p3,p

+ [(_+ 3 .Xt2 (k)t2 s(k) , -(k)+tk+,S(k+1) ,(k+l)

24 280 k 1 3333 k 3333)n) 33 12 a33 yotp

+ _( kL ) (k+) (k+1)+r 1 1 .3 s (k+l) ]a-(k+l)

10 3" 120 84 0S + 0 k+i33333)Y p3.p

+[( 1 3 )t2 (k-l) I-(k+I) 1 (k) W

24 280 k I s333371))a33 -& Sk

<T-(N-1) r L - I t3 s(N-I) -(N-2) 1 3 ,t2 (N-0 ])-(N-2)+ 3,3 -,-[4-120 840 N_ I s333 •p3.p 24 28-- -0 N -•I•333371y 33

S t[ I-s(N_- ) •(N-{) r 1 s(N-1) IM(N 1)

12 ." 33v , IN 0 t[ Toa 33T)y-ý

127

+ 1 + 3 -(N-) 3 s(N) l -- (N- 1)

120 840 Nt- 3333 N 3333)1 lp3,p

+ [(!+ 3 xt2S(N) 2 s(N-I N N (N) 1) I(N)24 280 N S33333 333--,333 33 1-I03311-Y p

+ 1 (N) ha(N) 2 (N-1)+ [' P33f) - go"Pl)> s(N-I)

N' -00I•,p "-" I (k)

k-I Ii

+ k) -- (M(k))' -2g(k)>

3i

(x( (y2 k)>+ <v - 0))'23 0 5

- 2 <N(k) 9(k) )a ' s 2 is•

"+ <M W ,.-,k )'-2g> (k)>

"+ <VW n,--((k) 2̂ g (k)>S, • t 3 6- z s • :'

+ 0<0.) 1 (1) 0(,-

)3 -'12 P33 7VyN 0 • 0 +S 33

" [(__-+ Xt3 S3(1) +t -0)

120 840 1 3333 2 3333 -yloyp3.p

"+ 1 I 3 2-(2) t 2S() )T) Y A -(I)),+[ t2 S(2) IN (2)I ,

24 280 2 3333 1 3333 "y 33 12 33 " &P

+ _1 -(2) •..(2)), .p -- 1 1 3.3S(2) 4~-2) ,

+ 1005337)Y & 120 m- 2 333'1 v,, p3,p'

[( I 3 )t2S(2) Y -(2) ._...(I)>

+ --24 280 2 33337)'Y 33

N-2+E ( <a -(k) [-3 "s W')-y3 ' 120 840 tk 33337),'" p3,•

k-2

+ +( 1 =2 )t2 sk) WT-(k- I))

24 280 k 3333)y" 33

128

• t ^() ",,.(k)-+, I •(k) ". (),

+ [!-�-�- ý3-q)vyN )3+f33-3)r M312 *"v 10 O3

+ I + I t3 k +, N,)(k))

L 3 -,y2 ,(k+I) 2 - , W ,(k)s(k+1), k )

[(-24 280 k+1 3 33)33'1y33 12 o"7(VK

+ G (N I rr 1 1 ,,.3_ )t3+i-i -.o.(k+-1) y,

10 -"33-y } 12-0 8i4 k3 I 33337)v p3 p3.p

k 1 3 -2 ( -) v

24 2--8 0 -k _ i3 3 3 3 'yi, O3 3 )W

+ + I-( ' 1 )t3 -(N-2))S(N) 1I ,- - ),y3 120 840 Ni 337' Kap.

+ [(-_L+_3 L)t2 s(N-1) X -(N-2),)24 280 N_ 1 33337)) 033

"+ [ I s(N 1)7)Y[ 1M(NN-l),12 %P33 ., 10 -- 33 as

"eRiIy+ 1 1.3 (N-E) fro 3 J (N) e[(+-KtS +t.S 1') X r (N-J+ 120 840 N-I 3333 N 33331 '() o p3)

"+ (2<+3, Xt2 (N) 2 +(,) (N)> 2<v(N)24 280 N 3333-tNW 3333/I1 33 +L 12(N)

"+ [2_LS( 3 71 )M(N) 2gNI>NII

The domain of j, is an extension of the domain of ai requiring Vf' EC'O and not

+k)

neCessarily EC'. Elimination from J, gives:

-41) (0)=< v> +2< ip") 11 (0) -41) (0)

( )y3 R( Y ' 2 <v)3 '3

+ 2 < N( 1 2 D t. 1 ()(20)

F2 6 (I)(o)> ' 5 p%,330(0> >(RO

2< 1 10() () 6 () ()>O +3+- ,2 < VO - 2 (1) (0)

+2< N)_-(N) > +< O) N (N) >-2<(N) ,_ (N)>I+ 2 03> R(N)V aNc,3() +v 3 33

2, ' R (N) 1 (N)

129

(N) tN (N) (N) I (N) (N)>+ 120 * "oP33 r3,3, ' 2 &P33"33 R

(N) I (N) _(N) 6 ,(N) (N) (N) 2 (N) (N)+ 2 <Mp &A TO s 030-3- 3; *,303 R MN)< p ' p3y3 a -3 >R (N)

N

.<v•, ' R k"• ' ; 2 3 R(k) 3 33 )k-2

" <N(k) tk s(k) -(k-I) 1 (k) -(k-1) -

.p 12 " P330" ).3,-y 2 T S'aP33a"33 > R(k)

"+ <M•(k) W 1 (k) -(k-I) 6 s(k) -(k-I)0>cep T " ' 0o•33 0"y3,,y 5t" k '033 0"33 R W )

- <WV(k) 2(k) 2 .- (k-)> )"p ' 5 p3y3 Y3 R(kd

N

ofe' a R k) ctJR(k of R W) 3 a(. R(Wk-I

+ 1 .1 5 (k) (k)

* <M(k) I25 (k) M(k)> +<V(k) (k)-k)+24S(k) ..+k)>

+ '3 "•NP•I•S p R100 )' -'y 3 k•P3y3 p R

tk

N-I"-'<;k > 0.0+ +<;, t,,o. -W > +<v -- .. ,)>W

k-I

(k) tk 5 (k) --(k) W (k) -(k)* <NWP I "o,033 ' 1 y3,2 + is" e33 33 > RW

* <Md(k) Wk a -Wk) 6 s(k) -(k)0+ 10' 10 33 "y3.- Stk *,33"33 R(k)

kk

* ,Vk -- 2 s0, O-(k) 4 W>

p; 5"P3.),3 y,3 R()'ý

N-{1 -40 (k)+ tk -k)+ tk(k) W k ' (k) 1W (k) 2 WS 2 T 12 " e33 N 10--' 33 oPy-' Sp3")3

p R(k)k-I

-(k) Jk) -0) --(k) -W+ <(Tp3 ' a -y3 += 12 "33 > R(k)

130

•O-Wk v-4k+ 0).+ tk+ I k+l V- k+I ,•(k4 1)N(k+ 1)

+ 2-3 , T V2 1-2 o"33 oP.),

I (k+l) (k+ k+I)VIk4 )>"T "S*33 S,•,-t - 5 p3y3 p R(k+0)

- W _ -4 v k ).{ I W. k W,]. k 6 s(k W m 0 0(k )+ <0"33 V 3 +2 T 33" N, - A33 "3 Ro)

0 k

---Wk) Z-(k) -Wk) .,k) -W k)+ <01 33 21, a 3r,3 +=22 033 R(k)

-W +) k+,). 1 •(k+ 1)],T(k+ 1) + 6 s(k+ ) m U4(k 1) >

+ <0" 33 2 3 +" T-ý-----S. 3 33 M3 >R(kJ

N-I+.. { O -k W /,A(k)o.-(k- 1) _-(k -(k- )> { 1k) -W (k) -(k-l).1() W -(k-1) _

,(<a p3 I I, ay3 "k+A 12(" 3 3 >+<a"33 A 2 1 a -y3 +A"22c" 3 3 >Rk)J

k-2

N-2.+Ff<O.( -Wr"(k+ 1) c-(k+ 0.-.&,(k1) Or-k+ 1) >_ +"-y3 11 -/ )3 1"A 2 "33 )R~tZ

k-I

S-W(k -"k+0) -(k+l). 'k+l) "-4k+l)_

+ <T , 2ý T +ý > 133 21 -y3 22 33 R NJ

+ 2<{o7)-(N-0) , f&N){o(N)> R(N)

+ 2 <{o1(1), [Af 1)a){(°)> Ro)

N< ,V -- 7)A 00--g W > W~k

k-!

k) (k) (k)

,k--'rpM•.8--3 13 4 k)l-4k VIO W

+ <• (k) 3 a k)2g >(

_2 N(Ic) (k)_- 2<N W W4 > 5k

+ <V , ,,7)V3 2g6 >

131

+ <o§5) o) (1 .[-iso' n ]MIM)-3 12 o 0337- v N 0 +[ To-3

+ I _ _t 1 +t3S(2) 7 ý1120 840 13333 2 3333 yop3,p

+ [1L 3 X2 (2) t2 ( 1)" ~I<)t(2) .(2)S2 3333 1 3333''yV33 -ILaP337 or

1+ %P-33 'y *P+ 12 0 84 0 2 S3 3 3 3 37ŽP3 .pl

+ K(1 _ 3 )t2S(2) 7)]o-2 )-2g()>

24 280 2 3 3 3 31 ~ 33 ~ e

N-2

+E f <0, -(k) [)-t3 s W ) . 7-r~(k-I)[_ I1 3 )2s() ]7(-1-y3

' 120 840 k 33337, +[---) 2420k3331,3k-2

+ L [~~s u YIN.+ S-L(k)[I (k) JM(k)+[(l+ IXt3(kW +t 3 S~~ ),]a-k)

12 33) To- 033 up 120 840 k 33k+I 3333 Vp3,p

+ I(+_L+~.Xt 2 s (k+1) t 2 s(k) -(L t k.+ s W 1) eN k+ 1)

24 280 k+I 3333 k 3333'* 1 y'r33 +[ 2 337).Yl up

1 _ (k.I) (k+ 1) 1 -1)3 kl a(4

+ [(LX (k+r 120 840+ 1) 1 W

+"24 2 8 0 'k--I s333371~Y'33 -2g,,> S(k)

+ < 1-(N-1) 4 _ _I )t3_ s (N- 1) ~-(N- 2) 1 3__) 2 (N1 k--2+ y~3 ' 120 840 N-1I 3333* T

)Vy'ap3.p 2 8 33),3

120~~2 280 N-I 3333tN3333'yyp33

+ L N)2I(- -(N 1) + Ns()D

+rI(±+)...XtN)_2g (N--1)vt~N >)~ N10 243 *,up N 7 33 eN 1333Dv~

+ rls(NL Nk) (N-I)

k-I I

132

-(k) * (k)(k

- 2<N (k),.0 >~~

- 2< 4k:) ( 4k)) > g(k>

1 (1) 3~2 v(1) 0+ <( 2 2 3 3 3 3 yJmOp)p

+ ((i-+~!.X .033 7)-f Y1 )33'+[IyI33 )'I 0S3 .i~

+ (IS(2 () +tylS)y (2) -7 -. ) aS3 3 3)12 840 1 333 V 20 840 y

+ 1 3 2 (2) 2() -)L 2 2" K24 2 80 )(2Js333V- IS337-ylg3 )' ~12 .3) YN,

N-2

Y3 120 840 k33-yp3,p

k-2

24 280 k 33337) 33(c-I)

+ [ti, W7 tN WY+-LS(k) 7) IM(k)Y,

+ I _~tSk +t 3 s (k+ 1) -c(k)y120 840 k 3333 k+1 3333 V"ýO p3.p

+ 3 L--x2 S(k+t) t2 S(k) ). j7tk)+[1Ljs(k+ 0) X~~)[ 4- 280 klS3 3 3 3 tkS3 3 3 3 ), c3 3 )+ 12 .Sa 3 3 7? -y a

+ [_L s (k1), IKM(k+l)Y+14~ 1 - 1 _)t 3 ý s (k1) I -4~k~l)y,10 ,#37) or 120 840 k' 1 333) Cp3.p

133

r)t \2 --(k l) (7 -(k+ 1)'- 2 g ,(k) •

--24 280 k•• 3 33 37)0" 33 )-gF > () }

+ <a.-(N-I) , ( I ) (N-1) X-(N-2)

-y3 ' 120 840N-I- 33337),J p3op

+ I 3 -_2 _ g(N-1) ,, -4N-2) ,

24 280 N-1 333379' 33

.[-(N(N-1) YN(N-I).+(N-I) M(N-)Y12 0331- v~ 1O0337)-y

+ 1 ._. 1 3X S (N-)+t 3-S(N) 7 Y -(N-1),

120 840 N-1 3333 N 33333 )JOp3.p

+ 3 ,,_ 2 (N) _.2 (N-1 - )N-) tN S(N) N (N),24 280•-tNS3 3 3 3 -tN3I33)SyO33 12 o•A33T)), G-P

r , -n S( ) M ( ,)•'-2(N -(N- 0)+ 10 P33 y ap•l- g N-2

The domain of definition of J 2 is an extension of the domain of J, to admit

p.k)ECI instead of requiring r.'EC'. Elimination of v from J2 gives:

-(1) (0) ,) t 1 (0) + 1) (0)+3 ' 12 "yR)-/3 >R(0 3 a33>R(1)

+ 2<, N,,) o1) O) +1 1) o(O)>T2 "%",33 o',30,y T e,33c 33 i> l)

+1< ( •(1) o(0) + 6 ^(j) (o)_ V(o 2 (o (o)_

"10MS.( +" 3 -v St1 7> < 5 ~(7>OP8 T•"O P33 y)3.'y) , A1: 330"33 'R() p 2 P ,- ':p3.r),30"3 M(I

--(N) ~(N) , <ON) ,tNG-(N)> + 2 4N) ~(N)>"+ 2<v-) --a',> (•+2<a> +<v --y3 R(N) ' T -3 R(N) 3 ' 33 R(N)

+ 2<N• tN (N) (N) 1 (N) (N)>1 2'"pT2 b330,0 y33 , ':ý33033 'R(N)

+ <,(N) 1 (N) (N) 6 (N) (N) .. () 2 (N) (N), S o "e30y,-T - (7•303 > iRN'!• +<•V -3':p3-y30"T y3 >R(N)

1) k) tk-(k ) 0 - "3W -(k -)-y3 >(k) 2 3 W 3 3 3 (k)

k-2

134

*A 12 *033 y3.- 2 &P333 3 (10

1 <(k)I -(k- 1) 6 Wk -Ak-1)+ p To~k -S*34y + +t-k %0303 > RO

+ < 5k W -(k-1)>p -T pý)3"y3 >3 I(k)

+<M~ N J2.S(k) N(k) > .k)24S)Vk>

+ <M -S p W+<

tkk

LIt 1 3 R~k R 2 3 (k 3 33 JRkk-I

+ Nw, tk~k Wk -00 sW -Ak)+ 9 < 1p-2 &.4330 y 2 S.433C33 > 1 (k)

+ < 4k) I (k) l-(k) -- 6 S(k) o..(k) >TO F.433 y3y ýtStk043 33 R(k)

+ < V~k 2 (k a, -(k) >P ' 5 p3 , 3 -y ~

v, 2 v 12 a#33 *IL- 10 .033 Ay~ p3-y3 Wk-I2

R

+ -(k) Z,~ ()-k 0+ <47 p3 11Or3 12 Ia33 > (k)

-y<O3~ ' 2 V 12 .P,33 rzAy

10 S*033 &A sK P30 p 1 (k.))

*. <O* 33 , V3 + js .33. -P *03 ,$>

-(k) U.k) -(Ak)+ Jk) -(k)*< a33 , 2ic -3 +=22 O'33 >R(k)

135

-W -4k 1(k) 1 s(k 1) N (k+1)+ 6 s(k+1)M(k+ 1)>33k , V 3 +T -OCP33 A S "--k+l; &P33 op > *R~k+l))

N-i+-{<O -k) W(k) -k1). W) -(k- 1) > - W ( k).-(k-I) 22 3-(k3-)

,p Al3 ' iO"3 -AI20 33 > 33 A 21 O), 3 +A"220"33 >R0)k-2

N-2+Ef <a•k -W k+ 1)or-(k+1)+.-rk+i1)or-(k+l1) >

,y 1 1i O 'y 3 "l • 1 2 ( 3 3 • R U k4 1

k-I

+ <(k) -W k4 ).a(k 4i)+0 k+7 )o.-(k+÷) >33 ' ,2 v3 22 33 R(k+ld

+ 2< {yo}-N-0) -N){ 0_,(N) > (N)

+ 2 < {())( , [AY1'){oa)()> >€'•

N

k-I

+ 2 <•k 4k 7 g) W> 4k)

+ k)&P(k) (k)9

- 2<V& , g6 > )

<or -y3 ' t- P3*•33 7)'P1o•J3'ypI

12 Gp 10 %P33 aP

+ [( 1 4- Xt3 (1) +t3 (2) a-(I)+ K -M--1 -- X S33 3 3 2 S-33 3 )34"yip3U

+ [(_-+-3Xt 2S 3 s (2 t2I)(+)t _g2) n* 2)24 280 2 33331 1 3333"33 12 33

1 s-(2) 7)1 1 (2) -(2)

10 P3 - - 120 840 2 33337)Yp,

136

+ [(.-L- 28 t23a3Y g ~

N-2 +E < (k) )t iW <- 1) 3[_L _.t 2s(k).V .- (kI),y 20 840 k 3333~7)7"p3.p 24 280 k 3333y ~33

k-2

+ W-.±' W 1(Sk W n k)+E(_I+_-Xt3 W +t3 S 'P),)1. JcT''12 oýP337No(T s 10 P337 Q' 120 840 k 3333 k.1 3333 p3,p

+ I [(+....Xt2 s (k+ 1)-t 2( s .W)+ t+I(k+ 1) N(124 80 k+1 3333 k S33 33 )rrj,,ju3 3 12

+ 24 280 k4 I'33337)-y 33 -2, >scr

3( I) I- )t S(NA) ia o.-(N-2)+[(__I+__t2 (N- 1) Iff-(N-2)

+<OY3 -y3 H 3 1 840 -gO -I 33337) p3' 24 280 N_-I 3333 71-y33

12 0337 yIap 1O0 r331).P

+ [(__L+ 81 X~ 3 N )3() -N

I (--+_.LXt2 (N) 2 ~N- 1)~ tN ~(N) N24 280 N 3 3 33 tN-1 3333 33 12 7Y

+ [ oL33N 7) N(N)2g(')10 c V3 'y -

N-- 4) ,_-q(N(k) ,g(Ik)

k-I R

+ 24 k.) 9 - m(),g,)

- 2< Nk k

Wk (k)- 2<Ma OP94 > (k

137

(k) .(k) ,

- 2<V 9g >

+ <a" , , IMy3 12 033 ovA 10 V3 y w

+ 1(_ .ý_ _ t 1 3(1) t3S(2) a-(0)),120 MXt1S3 3 3 3t2S 3 3 33 hi)7a'x 3 .)

+o 24 28 3333 1 3333n X3 3 12+[-a s 7 ) YN

+ (2) 1) 1M2)+ 3(2) -(2).

+24 280 2 3333 7) y,'r33 ) -2

N-2 2 tSk (-1)

k-2

+ 1( 2+_AL)t2 s W ) X7(-24 280 k 333 ,3

+ [(_~.L+. ~ Xt3S(k) +t 3 s (k+ 1) ).nlj0.- (k))120 40k 3333 k-IA 3 333 p3.p

+ [i....+Ž....X2 ~k4).....~s~) ~o.(k),+[tk+IS(k*I)~ N(k+ 1))

24 280 k' 1 3333 k 33 y 33 12 P~3371),

+ _ (k+I) (k+ 1) * 1 1 3 (k+1) o-k l ,

-SP37y'* )'+1-( --- ao)tk IS337p'

10 [(L 3 )t+ s(k 1 20 840 -kk+ 333 )'_gk >

+24 280 k 337 33 (cr

+ (N-~I 1) a -+ <a y3 120 840 337yyp,

+ 4 280~ + ý-3S3333 7)Va' 3 3 )

_S(~I>NN -n y IN - I( m )

138

, , 1 - - x l . ( N 1 ) 3 -S • -( N ) X r-- , , 1• -) )+ 120 840 NI•- 3333 N 3 3 3 3)7)'y p3,p

"+[_ 1+ 3 Xt2 (N) _2 ((N-1),.[ tN (N) XN(N)'24+ 20 NS 3 3 3 3 N-la3333 ), XO-033 1 ["A33" -A

" I[-S(N) (NI)M(N,'2 (N-1)10 %P3 My-ap 'r>SI(N

The domain of definition of J3 is an extension of the domain of J2 to which

3k)EC instead of V-()EC.

4.7.1 Some Specializations

Forcing some of the field equations and/or boundary conditions to be satisfied

identically, the number of field variables is reduced and some interesting specializations

of the extended variational principles are realized. Combining (184), for layers k=I to

N-1; each of (187) and (189), for layers k=2 to N-I and (188) and (190), for layers

k-i to N-2, the following self-adjointness relationships arise.

N-I N-I-0 k - (k) -(k) - -(k) - ) -(k)<a y3

' 12 a33 >R W <(TO33 ' Z21a y3 >R(k)k-I k-I

N-I+E<0-(k) 3 1 W(k) 2 S(k+1)2 ]7 -(k)>

p3 '280 24 3333k 3333tk+I p033 >s(k)k-I

N-I+ - /r ( 0r-,k) @-I+_.X (k) t2--(k+l)

2 -- -(k)yds(k) (+ f S(k)T p3 280 24 S3333 kI 3333 k+)ýI pC33 (262)

k- I

N-I N-IS<00-(k-I) k) -(k)> N -% < (k) A(k) o-(k-I)

"IIy3 11 P3 R W) E P3 II y3 R(k)k-2 k-2

N-2S 0-(k) , 1 1 3 (k+I) -(k+1)>+ [,: <a . 1 -8"40 k*I S 33337?y00p3Jp >(kL-)

k-I

N- IS ,(k) 1 1 3 (k) -(k-1)>

- or /3k 120 8 4 0 k S3 3 3 3700 p3,p s(k3 )k12

139

N-2 ( k) I k;3 O(k.I) (kI1)),d (k, )

+ (k+I) , 120 840/ k iS 3 3 331#0p 3,p )

N-I

E f (r 1 1)t3s(k) .- (k-1), (k-I1)

k-2 s~k) '/ 120 8A0 k S3333 yp3.p dS (263)

N-2 N-I~ 0.- (k) 4r~k+0) --(k+l)> _" -(k) Wk) -(k-1)

E <y3 ' 12 O°33 >R(k+l)--E <a 33 A 2 1 O3 R(k)

k-I k-2

N-I

+ <a-(k-1) ,(3 I t 2kW -(k)>+ y3 280 24 Ylk 3333-1-y0033 Sýk)

k=2

N-I

J (a -(k-1) 3 1 (k) -(k) k (264)k,-.2Jsi k) 0y3 280 24 k Tk033337y033

N-I N-I

<0T-(k) A A (k)-k-1)> _-< -(kT 1) a k)o.(k)>" "y3 12 33 R(k)= F. 33 ' 21 '3 R( )

k-2 k-2

N-I+ <.-(k) -- 3 1 ,(k) 0 -(k- I)>

y3 ' 280 'k3333'y033 S (k)

k-2

N-IE f (00-(,L) L3 1 ,_2 (k) -(k- I )',(k)

J 9110 -y3 • 280 24 k 3333y33 i (265)

N-2 N-Iv -(k) k- k1) -(k+1)> -(k) W6 -(k-I)<33 ' 33 22 0a33 > R(k+). <a033, A 2 2C0 3 3 > R(k) (266)

k-I k-2

Substituting (262) through (266) into (261) to eliminate :,k) X2 , A-k2 and

S)flC 3(u , a0) becomes

+ 2<N() --0012 333) > (()+,(u ,0)=2<V-(') ,0 a+> o)+2<;s) ,t0 (o) > +2<v - a

' 10 R ' 2 y3 RSt,) 3

t ) 1o0) _1 40) (0),p "••33 00.-,y't At3303--()

+ -< M (1) 1 S (1) a o .0 + 6 •(1) (o) + 2< V (i) ,_ 2_.S(1) (0o) >

TO P r'"-~j33 0y3.'y t"5 .0 3 3or 33 >R (1) + P 5 p3),3 a y3 > R(1)

140

+ -4N), %3> + 2 <4N) tN o(N) -(N) (N)>v 'Y 3 ' 2 3> R(N) ' 33 R(N)

+ 2 <N(N) tN (N) (N) +1S(N) (N)12 ' ,,330"y23,-y" •ot,8330"33 > R(N)

+ <M(N) I (N) r(N) 6 (N) (N)> .. .. (N) 2 (N) (N)>+ 2<k 10 s O-I33 10 ),3,)--3i- SP330"33 > R(N" •' <Vp 5-- P3" 3 a3y3 R(N)

N

F2 k)0- > (+4'), * 0 > + <) ' 0"y3 >R k) gama 2 y )3 R(k) 3 '33 R(k)

k-2

(k) tk (k) -(k-1) 1 A) -(k-I)-s ( +Is (k)(<Na 1-2 03 y3,,y 2 "P33 33 ROk)

+ <M I (k) k) 0 -(k-0)+ 6 5 (k) -(k-I)>

"+ <v(k) 2. 333k) 5(3k-I)

Po 2 W -(k 1) k" "<V p ls p3y)3 a ),3 >R(kd

N--( ) R(k)

k-I

+ <N~, W S(k) Nk)>&A• , k PAPf•' ;L ROL)

k )

+ 2<M*s, -'4•k)> R(k)

+ <M(k) 12W(k) M(k)>

orp 3 P'AP AP R(k)tk

+ < (k) ---- k) -,k))>+ V2<V ), ,V )3 R(k)

WV 24 W( (k)+ <V ýt-p3y3V > ROOk

N- IN1 <ýk) __ac,-(k)> <~k) t k -4k) -4k) -4k)'-'°'*,3 > (k)+ < • -'0 y3 >Rtk) < v3 - 33 >R•,)

k-I

+ < W(k) tk (k) -(k) W -(k)+ o, --- T3 "-2of+33("33 (>R

141

" " M(k) I W(k) -(k) 6 s(k) a(k)>ckA "1'ckA330")3,y -ýtk 0o330"33 >R~k

"+ <V , 2WSk) 2W -(k)> )p 5 -- ' p3),3 -y3 Rka

N-1-- k) .,A ) -(k) - -(k) -k) -(k) -(k) .(k) -(k)"t+•-.{<:Op-3k I I liO-y3 > R(k)+t2<a '33 21 '210-y3 > R 00 < 33 '•22 a"33 >R~kJ

k-I

N-IY12(k)o A (kl)>I k).+ 2 <T(3) Ak) -(k-)> + -(k () -(k>

p I I A y3 R 3 2 1 ),3 R(k)+ 33 W A 2 2o"33

k-2

N-2

-3 ' 21 0y3 R k+)

+ 2<[O-] (N,-1), [A]N)[Io.N)> (N

+ 2 <[o..•1>, [A](I)1c°0)> (,)

N

a-k ( k)k-I S

+ k) (k)+ , -2g 5 9 >•k

4(k) -(k) (k)"+ 2<V3 , 295 -g > 5,k)

(k) r(k) (k) >"+ 2<M ~ ,o ) 3~-g4

+ 2 <V(k) -(k) (k)""' 2 l)V 3 -g 6 >S(

(t3S+) +t 3S (2)(

+ 2<cO-I [(<1 + 1- 1 3 3 3 3 2 3333 >,y3 120 84 2 p3,-ga" >s(I)

N-22<a-(k) 1 - (k) _-1k-I)

/3 ' --- 120 840 k3 33337)-yO'p3.ok-2

142

1 s(Ic) .. t3 -(k+0)•

+ R + k 33 3 3 kmI 3333J .I-(k) (k) >[1-0 84 2 " " p-g° S W

+ 2<a---(N- I) [ 1 1 --3 S(N-I) --- (N-2)),3 ' 120 840)N 3 3 v33 p3.p

.t3 S(N-1)+ () 3 AN)+[1+ R 1) NI 3333 tA•:3333) 7_N_) N-)>

120 8 -40 2 1 p3,p~ a (~I

N< -k) (k)+y o -2g 3> '(s)

+ <•;) ,()

+ 2<N;k) , - -k ,(k) > W

- <M(k) l-4k)-g)>(k)

+ 3< OP 7 -+- (v 2 333 7-0v() (n•§' ,_,(k)>+z 2< ' 3 16 >)

(t3 -g i) 3 3 ,(2)+ <O-(1) 1 ) 1 l3333"+t 2 S3333) , -01),, >()

120<0840 2 v p3.p'-,3 ' (' -0" F'46 2 "• 'r3") c S,()

N-2+ E { 2 < 0 O .- ( k ) , -t 3 , .S ~ WO - k l •3 I-y3 120 840 'k 3333 -y p3,p

k=2

+ 1( W + W(t3s(k +t3 S(k+I)) -* .( ) k,• k33333 k.+1 3333) -71 X -(g) , .o (k)_

f 12" 0 + 8 14' 0 2 ' kY )) -g ck S>) 3

* 2 < r.-(N-I) 1 -14 1 1 .3 s(N-I) .-(N-2),"y3 1-f20--'•K-I 3333T8-yJ 0p3,p

3 (N-I) 3-gN)

+ [ +. . tNI-IS 3 3 3 3 +tý 33S 3 )3 Vya.-(N-I), ,(N-1)>T120 840" 2 p-I) (267)

For the boundary value problem considered, if the set of admissible states is restricted

to one that identically satisfies the constitutive equations, (155) through (157), the

functional f)4 is specialized to

143

!n' 3 , ) V >2 v3 ' >(,)+2<- cr > (1)' , v 3' ,3 R.y y R• .3 33 •>R(

_2<--N) O 3 R(M ) .2 N)'( tN i(N)> +2 4N) (N)_-2+2<v a'Y R() y T 2 3 R(N) 3 33 R

N

E2( 'Y Oy3 > R W) )y 'T y3 R W+ <v 3 (r O33 >R(k)

k-2

N

N(k) W I (k)

+ <M( ,2.S(k) M(k)>

3 C' 3 ALP ALP ~tk

+ <V(k) 24s(k) v(k)W>1+< ' 5t k sp3),3 vp> R(dY

N-It

N * 13 R(k) ", y3 )3 ' <0033 -4R(k)

k-I

N-i

-(koj ) .Ik) -(k)> k -(k) 4(k) -(k) -<k) 4Jk) -(k)>""Ellp I y3 > R(k)2< 33 , 21 (T-y3 •> R W + <(T33 ' --220"33 •> d

k-I

N-I-.1 2 <c,-(k) (k) -(k-I) -(k) (k) -(k-) -(k) -- k-I)

+ ,'p3' AIol 3 > +2<a3 A21cy > R+ 2 <ok 'A a > Rp3(k) 33 , 210y3 R(k) 33 ' 22 33 R(k

k-2

N-2 -3k) -,4k+l) -(k+l)+E12<a33 7ý21 r-y3 >R(k+])

k-I

+ 2 <[O']-(') , [0(1)04'(°)> 1O)

N

+y<-lk) ,2(,k)>

k-I

+ <$k) 2 (k)W

(-'g k)

*4-2g >

144

(k) --(k) (k)* 2 < No ,v0vv g 2 >

+ 2<M -k >

W --() k) ()_ ,>+ 2<V a %V3 -g 6 >s(,

S(t3S() +t 3 s (2)+2o-0) 1( __+I)t:3333 2 3333;.•rO n

y3 V 120 840 2 7), p3,p g, >S(o)

N-2 2< a-<-(k) I--1 1 3(k) ]-(k-1)y3 120 840 k 333371yp3,p

k-2

W 3 (k) +t3 S(k4 1)

+ [.----- k 3333 k4 1 3333 ](a_(k) (k) }120 840 2 1) p3,p-g, S(k)

+ 2<7-(N-1) 1 1 3 (N-I) C-(N-2)

y3 T 120 840 -t'N-133337)) p3,p

(tt3 s (N- 1) +t3 S (N)+ [( 1+ )+ 3 ()N-1 3333 N 3333 ]0 .-(N-I) (N-I)>

"120 840 2 p3'p S(N-1)

N

< t<-Vl) , o2g(k) >

a Iv(3 g (k)_

k-I 1,

+ rk < -,2g(k) > Ik)

-4) k)" 3, -2g 5 >s )

+ 2<N ( ,-(k)g,, ,(k)"" k 2 )l•7) -- 2 > k)

+ 2<M~ W (-k),, ,(k)>

W <V- ( k)., .(k) > .+ 2<V 7. (V 3 v - g6 S I)ý

W s (S) +t 3S(2) )

1 1 3333 2 3333 1 )y,3 ' 120 840 2 y P3.p

145

N-2 +E 2or•k ,-1 1 .3(k) y -(k-i).,

+EI2<a-y3 120 840 k3333'qYxP3,(k-2

,3(k) +t 3 S (k + 0) - I

[(ru1..l. 1) 'tk 333'3 k+I 3333' .,ok)-.,(k) }+ L')+-- 10- cp 3 ,p)'g ,> W~120 840 2 Js

-2 <o(N-1) a -(N-2),+ 2<r 'y3 [--120 840 N 1 33337yop3.p

( _ 3 _ (N-1).+t 3 (N) -N+ [(- + 1 N-1 3333"N 3333 .x -(N-1),_,,(N-l) N (268)

120 M4 2 p3, s.

Here N(), Vk are not independent field variables but defined completely by

jtk, k) andork) through the constitutive relations (155) through (157). Even if the

physical problem has no discontinuities i.e., g'tk) vanish, the discontinuity terms must beincluded. These vanish if ; ýk) and (T _(k are restricted to being continuous

across all internal boundaries. Satisfying the displacement boundary conditions (i.e. the

last three conditions in (221) and in (223)) identically, the traction-free boundary

conditions and assuming no physical discontinuities (i., g', g'3, g'. and g'.) vanishing,

0s(u, or) leads to

-0 (0 ,0 t 1 '(0)o> -+1) (0)n 6'(u. or)=2<v-- , o3>R(1)+2< ' 3 () 2<v 3 9 (T333>RW)

N (-))tN (N) -(N) (N)>- N (')> +2 a >(N) 2 <V3 ,o33 (N)

SY3 R(N)W ' 2 , 3 R 3R 3

N(-1 t k -(k-1) -(k) -(k-J)>SR(k) + <R 3 3

k-2

+ N (<N k(k) 0SWa-4k-1) +I (k)a- 11: 12 ' oT332 P

33 v "+2S33T33 i (t)

k-l

* < 4k 1,,W -<k-,).+ 6 .(k) a.-,k-,1)>ap 1 &#33 , "ly Stk +5 33 33 R W

+ < V(k) 2 5 (k) -W 1) >p ' - p3gamma3 y3 R(k

146

N-I ( -4k) . . k) W -k)

k-I

10N W 00 --W I_ W - Rk)(--"k")bo, k S330".),),' iS&0330'33 >R2 W

k-I

+ <M(k 1•k) I OW.- -W) 6 S.., a -W(>)_*A 10 a33 -.3.,-' 5ti ,,44333 33 >R0

+ <V p "-p3"),3 a),3 >Rlk)'

N-I

.4 {< V--k) o-(,k)> +<0 • ,(k tk .- (k) _-(k) .- (kW•

) 1 -y3 R (k) ),3 RW 3 ' 3 3 R 2d

k-I

N-I

.E < .- (k) --= ( 3k) • - (k ) .. -•( (k) .,(k) -Wk) -Wk ,-,Wk -Wk_ "

,3 3 -1" '-y3 >R A +<a 33 2233 >2233

21 3 R(R R

k-l

N-1" -W'( WA(k*o -(k-1) -(k) W -(k- 1 ) -W (W -(k-1)

, p3 I II y),3 R(k) 33 21"y3 >R(k)' <a 33 22033 R(k)k-2

N-2

-W k+ (kl)

• "3 3 2 1 - 3 R lk +

k-I

+ 2 <[o(k--) , g lklo)f)> R(N)

+ 2<[o'-('), [AY('j)3o333)> 2)

N

k=2

I a I() e1k)7+ 4 . 3 -'4,

v-4k) (W)"" <v3 5 >,•.)

- 3 S (M .t 3 -s(2 ) ,"- 2<a-(0) 1 1 tl3333 t2 3333' 40)+ y 3 -[(-•+[" :12 1],3ý.)> >1l)

N 2+T.,, 2 <0"(k) r_ _I,_L~3s(k ) .,-ny (k- 1)),

< -/33 ' 12 "•O-8"40 k 3-333T)jl P3,O

k-2

147

+ 1( i)ý(tS 3 3 SW +t3 1 S U 1)12() 840 (k 2 +t r4KcZ'~Y)k+ [(V "I ) 'k 3333 k 1 -333 Y -(k) )' >

-(-) 1t (N 3 ) (N2+ 2<a0", [4 1--( 3

+120 840 13333 Y'rp3,P

W- 3 (N-I)+ 3 (N)

+ [(1 + 1 )'tN-IN3333 N 3333') (269)"120 840 2 yp3.p )> 1)

Because the mechanical quantities, N"k ,' M(k) and V(k) in On are completely defined by

the constitutive relations (155) through (157), the independent field variables are

, ), and a"-k* The number of independent field variables, therefore, is

8N+3 (N= number of layers) for this specialization. This is less than the 13N used by

Pagano [1978] for obtaining numerical solutions. For other cases, where any of the

force resultants may not satisfy the constitutive relations, these would have to be

included as field variables. Thus for the mogt general case, there would be 16N+3

field variables.

The specialization defined by fl,, having the least number of field variables, was

used to develop a finite element solution procedure described in the next section.

148

SECTION V

FINITE ELEMENT FORMULATION

5.1 Finite Element Discretization

In the finite element method, the region R is replaced by a collection of m

disjoint open subregions called elements {R e = 1, m } such that in a sequence

of refinements

Ji = i U e (270)e1l

and the subregions have the property that

SnRf - 0 if e • e f (271)

These elements are connected at a finite number, N, of nodal points. We assume that

a finite element representation of R is available such that {(SI, , S2) , (S , S,) and

(Su, S.,)} are contained in the union of intersection of element boundaries.

Over an element, let approximation to the unknown field variables be, in matrix

form, as follows

v. =4[*]T aa). (272)

where [4i] , the set of base functions, is a row vector and lav}e is a column vector of

coefficients. Evaluating the function, and its derivatives up to a certain order, at

nodal points yields

{V.J T(a). (273)

149

where {(a}. is the vector of nodal point values of the functior and its derivatives up

to the order selected, and [f4teT is the matrix of base functions and their appropriate

derivatives, if required, evaluated at each nodal point. The rows and columns of

must be linearly independent. If square, the matrix is invertible. Hence, we can

write

-[A]l {-,c}e (274)

where (A] f[Ot

Substitution of (274) into (272) leads to

V [, ] [ATl 1},

= [(41 T (275)

where

[+,, - [#].' [AT1 (276)

For the kth layer, we have

V40) T Wkv•- t ,-) (277)

where [•t]J can now be regraded as a set of interpolating functions relating nodal

point values of a function and its derivatives up to a preselected order, to the value

of the function ;<o ) at an arbitrary point within the element e. Similarly the other

field variables are approximated, in the finite element procedure, as

i k) T (k)

o.(k) T - )k)3 #,V (278)

a-(k) 4 T1 1-(k)-y3 yX [,3•cry)}e

O.(k ) T" -Wk

33 [4133 -e C15

150

5.1.1 Finite Element Formulation

In the present investigation, the specialized functional I6(u , or) was used as the

basis for setting up an approximate numerical solution scheme. fl(u , ai) is completely

defined by the kinematic variables, V , ) ), ;V-, and the stress variables, in n

deriving the governing function l1(u , a) the following assumptions were made:

1. Constitutive equations are identically satisfied, ie., N., Mk, V?) are not

field variables but completely defined by i), Vk) and the stress vari-

ables o,4").

2. Displacement boundary conditions are exactly satisfied, i.e., V", 3j, ' are

restricted to the set exactly satisfying the last three equations of the set (221).

3. Stress boundary conditions are satisfied at the free edges of the plate.M)K) k)

4. No jump discontinuities in the force quantities N(") M• , V.)a exist in

the interior of the plate.

5. -- , ) satisfy the last three of the discontinuity equations (223) iden-

tically ie., if the physical problem does not have kinematic discontinuities,

V, ), V are continuous in the interior of the plate.

To explicitly write fl(u , or) in terms of the free field variables, it is necessary

to use the constitutive relations (155) through (157) to eliminate N(k, MWk) and V(kI.

Upon appropriate rearrangement, (155) through (157) yield:

tt

N(k)=(ek) dt -tk) tk W(k) ( -(k-I) a-k) _tk W(k) , _-(k) -(k-i))} (279)

3 2 13M(k)_Clk) X Ik k) , k W k-) 1 (k) -0c -<k-I -(k) (280)

tk a, _, _ __:LS oWP- 'paOr 12 Q fo>"• P3 33 33 120 of33 -v3,y 3y(

W _ 5tk C•k) c' Vk)+-r(k)+tk -lk-( ) -k) (48

24 3p3 3 -y y 12 P3 p3

151

where I [S.,,]' and [C, 3A] = [Sv 3pd1 . To recover continuous N("), Mk),, v("), it

is necessary that V, •, o ik)_ be continuously differentiable. Otherwise,

N'&, M., V") will be discontinuous at inter-element boundaries. However, this

requirement of differentiability makes the finite element scheme unwieldy and is not

necessary for convergence of sequence of solutions, ordered by mesh refinement, to the

correct value. For this reason, in the finite element scheme discussed later in this

section, simpler interpolation schemes yielding discontinuous force resultants at

inter-element boundaries was adopted. Substituting (277) and (278) into (279) through

(281), N( ,.M', and V( can be expressed in terms of shape functions and generalized

coordinates associated with V, , I v and ark). The governing functional

fl,(u*, a) after discretizing the spatial domain R into m disjoint elements has the form:

m

0 6(u , a)=" 11(u , a') (282)e-I

where

f =tk (•/ r o-1"-)+I 51kW -(k-I-,,

e E *f 12 333 2yl 0 3 3 3 3 s-"ek-I R

+ " M(k) [1_ (k) W -(k-). 6 (k) _-(k- I

+ JRe *A 1 0'S.0330"y3,), + t'Ik*033'"33 1 " t e

-f V(k) 2 (k) W -(k- dR}fR P 5 p3-y3 -y3 ee

NNW -4) W k) dR + R R (k)R (k) ( Ok

k-I WA 0A)ef eD QP e J Ree , eN

+Nf N• tk W( -(k) +I W(k -(k "

CepR " 1 2 Pr 33 r3 ,y 2 & 3

SMf (k) [ I (k) -(k) 6 -(k) 0(4k*A 10e S.033 t' ,a0"3,,--'St" o P330"33 )dRe

1k

152

f vk) 2 s(k) (T-(k)dR},W Re p -5 P3),3 y3 e

-4k) 4r-k-)R I 2 r k) t k -(k-)d -+k 2f-(k-"d

N

+•{-2 r k) ork-)dR. + 2 f ik) tk_-(k)dR _ 2 -4k) -ok).l }

k-I JR -JR.R - 2 JRa 3 3 33

N-I+ETfk2R ,-k)k) -Ak) -(k) -Ak) 'W -- k) -Rk) -3k)

C,.r ,,2(7Rk)da+2 fo a2 2O',3 dR(+ a a-( - d1 RekJR 3 11 -3 Re 332J, efR;3 22 33

E -(4k) A -(k -1) -- )dR + 2 -(k) A (k) 12.-(k- 1)

k-2 Ro p3 II y3 e JR 33 21 )3 e

* r -(k-J) -,k) a-(k)~ ide2f( (k) AW( -(k-1 J

J R 3 3 21-y3 +Re+33 3 22 33 e

2 r 2 a-N-1) AN),°-4N)dR + 2 7-•(N _1) cff ao-N)dR1, p3 11 ),3 e 3JRe -y3 12 33 e

+ 2 r -(N-I) AtN)a-(N)dR + 2 cr -(.N-1) 7N 4-(N)dRJ 1

0 33 21 ),3 JRe 33 22 33e

* 2fa c' 1 ) A (1) a 0 dR +2 a c-40) A (1 a 0 ) dRf.p3 I I y3 eC JR 0 y3 12 33C

* 2 a .-0I) A (1)ar(0) dR +2 a -0I) A (1)ol( 0 ) dRJRe 33 21 y3 e JR 33 22*33 e

N

Vlk) g(k) $-14 ) (k)-Ef g dSe+2f g 3 )dS3ek-I snls. s 3 f s.

+ 2 ,4k) g W Sg ) -d

+ ( 1 '+ ('*)")t13333(0"p3p- 1dS120-f ) "3 -y 1 3333"p3.p i

+c -(1~ + ii)f )'7) t2S3 s (2) a4 1))YdS.120 W .3 -y 333 p3.p i

N 2+E (2( 1 1~f ,, (-( k). t3--(Wk) -(k ),dS,

120 840 r3 , -k 3333 p3,pk12

153

1 1A01k) 3 () t3 SkW 0-(k)),dS

+ (- 8+ -- f (or 3 'ytkS3 3 33 p3.p

+ 0+ -f t 3k) .t3

4-1) 01 -(k I(I-• 8-• Jstr3 'Y k 1 3333 p3"p) •i

120 80 840 S3 3 S120 &4 fS, y3 'Y N l 33330"p3.p° (Ii

+( +1 + 1 . )r (&,-(N-I) 3 ^(N-I) -(N-I ,,dS

120 840 J y3 Tt-I S3333 Op3.p

+ I + 1 f ((N )t 3 S(N) -(N1) Yds (283)1f 2=0 840 J 3 ThrN 3333 p3.p

Substituting (277) through (281) into (283), the spatially discretized governing

functional is obtained.

S W -k-1)QT (k)

3 (kC33 3y 24 Vy) + 3( e 144 ) ey3 e

.(k) (kT ()

, :-(k-1)r9 K .{ -(k-l)+ WT W K( . 1-(-1)to, y3e 144 Vy3). aVe v33 331e

-a ):k-l )T K3 3 3 3 k-1) !fa -(k) T3333 (a -k-l

33 4 '(733)e 33 C 334 e

T,(k) T K~)

+ {( - -(k)T .. }a - _3 3 { -(k-I){0 . -e(k-,l 33:. k 1- -)

V3 424 24

* m( l(k)Tk(k) " la ()-(k- I) 1-(k- 33)T 3 (or )-(k-1)

SyeD 3 3"e 33% e 100 3 e

* l -3k) "! 3 3

V,3 ) C-(k- ) a - (k k- I)r KV3 3 X 1 "(k- l)

+1033 1 V3" -- O 1200

- fa 4):TL ( K)3~~-WO)-y3 1200 -y3 e

+ ,(k)T.k) T K-(k-l) r }-(k- )T 3 K() W ,-(k-1)D033 33. 33 e 3333533

154

T 3 (k)

+ { -(k)T 3~K (k) io )-(k 1){ 0 -t--(k OtI) o )-333 1)33 t 25 3333 33 -y3 100 33

- fa 1 A(k9 y33 3 f{ ).-(k- Oi ){~(k)TQ~) fa )-(k- 1)-Y3 e 100 33e 3c wY3 -y3 e

- iý)ek)TK (k) { ), (k-1){0 r }-:k- W)T(k) :kir }~()K~ W 1-(k-) 1)4,010*'3)o3e p3e p3-y3 -y3 p3e p31'3 -y3 e

N T )ck- I)TK(k) { (k)*~~v ak~k c(' 0 33e 33v a

k-I

- f )(k TK(kviv 0}(k4 {0 . h)-(k)T Wj- 1(k)33 e 33 y3 e -y3v.

- to _(k- 1 T) K~ (k)vj

+ (k) T KW(k) (k){~ -0 -k- 1)T K(k) (k)

. o 33 e 3D'

+ 0 f~( )k)T K~ W~ (k) 0 1-(k- ITK (k) wf. 1(k)

33 33DO';) oc y3). -y3D o

+ . -(k- I)TR(k) W~}k+ 0 ~ k)Rk ~ 1 k- ~ for -y3C c~ 3. y

V ) (k '(k) {~}k+~lk) TK(k){ 1 (k)3~* e WWIoe

+ I .-}(k-I)T (k) {V }(k)+{ 0 3( k R¶, (01 k)

N t (k) Rk )kTK(k) e

+[ -{;)()T (k K )(k) 1;ý(k)+{a W 1 k ) -(k)Oe I)3 )e 3e 24 q.).

-(-IT W (k) Io )-(k)T(k) (k

+ _(V )(OT~K 33 W f k) +a)-k } T )3 3 (0 )-(k)E 3e 243 y3 e Y33 14 y3 e

k(k)

4 )-}-{k- W )-(k) (k) -(k)

y3 144 y3 e

(~)T ((k)

+ f{.)k T K (k {a I (k)_4 . )-(k- OT 3333 . )oc v33 33e 33 e 4 33.e

155

- la, }ik)T K(33 , 3 la I-k+O )-) {Lfa )-k

33e 4 33 e '3 e 24 33 e

- { e-(k- ))K 3 -(k)-y3 24 3e

+ W-3 -y 33 e 100 ,3 e

-a 1 ~kT K W~, (k)k

()-(k) 33y () )k T 3-3 (k) ~

+ 1O33 e 250 333 y33e -a3)e 1200 )3 e

+{ T W() 3 ~-y3 e 1200 y33e

- {V3")k TK Wto-)-k +fo (k)OT 3KWl)-ke W33 33 e 33 e 25 333 33T e-

f- ): 1 .k-1() T 1(a k){0 )-{ (k- I~ }T-(k)

y3ec 10 p33 ye pe py 3

N (k k)T W to )-~ ;,(k) (kT W fi (-(k

[ ap3)e (1-20 ý4-0 p)3 y3)e 3'p 5 p3, y3 e

k-I

(a)(kl)T(+)D (k+1) {. -k)-(k)T. 1 (+-1)D (k+ Oj 1 -k)

p3 e T20 8 i40 p3-y3D -y3 p3 e 3 5 p3), y3 3e

-2{cr ~ )D)~ 3 ,4}k)+ 2{ 0 )Dk 24'80333o-33 24 280 33y y (l'3(ek)T(i+24 D280 I33ay3)-y3

+ ' _y(k)T1 + 17)D(k) -(.~k)+o -(<k) T 1+17) (k-+ 1)(0 1 k)

33~ 4 140 333 43e 3) 140 3333 33

N- -, {o }(k )T( 1 9... k) {~ -(k 1) WTr} k) (k)(k 1p. 120 840 33D 3epe 3 5 p~ 3

k-2

1:56

,-ýk)T 1 3 WD k to . y -•k- l) 2 y" -k-J )T ( 1 3 )D ~k W O I--W"+ 2{o-33}e (24 280 )33-y3 -y3 e -24- 28330e T4-280 33,y3 -y3 e

" ^ 2( -y k) Tr(1_ 17 , k)D W la )-(k - 1)+ 2 3 3 4 140 3333 333e

Nj 2t(j) v o" R- )+ 2 ; }()(k)k) 6R t -(k,)+ 2 (v (k)PR (o 1 -k-I) }, y3la-y3e e 0y3 ),3"e 3 e w33 33e

k-I

N

+ { ) T( Rv k)+ 2 g (}k)Rk ) 6R W }-(k) 2{V (k)T {k r R o (k)-- " v)3 o

3e ee • 333 y3"e 3 e w33 33 e

k-I

^ -(N-I)T, 1 I (N) (N) (N I)T I 1 (N) , - (N)"+ 2fo-p3 e 120 840 )D p3,y3Dy3e -y3) p3 e ,-•---3-)3 p3,3".y3 3e

-+ e 280 24 y333D 33 e

+ 2o -(N- I)T( 3 Io x( ) --(N)2 - )-(N- I)ýT, 1 7 D(N)'O A-N)" 21a3 280N 2 ()+ 2 1a ~(~)T .)DN i33 80 24D33e 4 140 3333 33

" 2{C ),1)T( _ 1 ) (1) 01r )(o)-21cr )-(,)T(I I)D(l) (a )(o)p3 e T2•0 -850 p3y3DI y3 e p3 e 3 :5 p3)

3 -y3 e

""1 _,T 1 3_)D(l) (a" 1(°)+2(o"- )-(0T(-1 3 ,)D(') to" •(0)

+ 1 y3cel)T(, 24 2-0 )333D 33 e 33 e 24 280" 33,y3 13)e

" 210- )_(,)Tr(1- 17 ),)Dl) {a" )(o)+ 2o33)e ' -Z-40" 3 3 3 3 e

N (k)T W 'r (k)T W(k T+2{ R(k)T (

2(V-c, R vn~f oe Rq* on ]J wk-I

+ a Y4I)T 1 1• (i) " -()"-y33 If 120 840 y3p3J p3 e

+ (a )T|-L +1- 1 -K( 2) (c" 1-41)e,3 120 840 y3p3J p3 e

N-2 "(k)T 1 1 ,(k) i" )-(k- 1)

E -2 - y3)C 120 840 )2E y30P3J p3 ek-2

,+ ( - -40T)T- L + 1 W .(k W l "-(k)+ {cT0 120 840 y3p3J p3 e

157

+ 1 . -(k)T 1 . (ký I) --(k)

/y3 e 12 -84•0•)K y3p3J ( p3)e

21- )-~y}(N I)T (- -1.. 1. )K (N- 1)•O -(N -2)

"e 120 840" y3p3J p3 e

+ (or C-(N- 1 )T -I+ 1- )K(N-I)o0 f -N- )")3" c 120 840 y3p3J p3e

+o fa -(N- I)T 1 +-1 )K (N) to- )-N-1)(24+",3 e 120840 -y3p3J p3 e (284)

where

K(k) f [DV tk (k) S(k) W IdR (285)

v)'3 J R 12 [DO:P- 33 y3,yfe(

(k) tk cek) S(k) ITdR (286)f3 Re l3.yly 12 '"pu3[*, w ee

K(k) f r[r.tL, tkck) S(k) [*,I, (287)1v33 JRe 2 u a633

(k) f tk k (k) S D). tdR

K f3 3 R, tit't l- 5 p p 3 3[Ive (288)33v 2R 2 /'

K =k ff IitS33]etkdl •Sj,335 3[lt' 3,3IIedRe (289)

(k) -'k) (k) (k) T"

[(k) ]t3 Sk 33[Wt3 I'dR (290)" "J3333 fRe k 'p p3 3S 33 c

(k f(3ek) (W) (k) [ dR (291)

K y JR ['0,3.e Mpo5

A'p33S.33 y3e R21

K f r ' i r 2 C(k) S(k) Sk) r, ]TdR (292)"K-J333 fRe [0y3,ye k Apor ,p33SoA33[33t e

3

[ ] ,-_ tk d(k) W T) (293)

KR 3 WC 120 up0p Or 3 3 [D*,Je •(24

158

K(k) [D,] tkdek) S (k) [4 TFd (295)

D033 f~e 12 jp~ *3. 3 3~ 3 c'3 e

(k) r k S k 3f. # I dR , (299)33 f~ e [~P33e 3 0 ka up33L e e

W- ~~LkCk)(k TKy3 fil Re Y~ 14) 2P*S)30yI'dR (397)

K5k tk T~)J0edR, (303)p3 y3f NRe'pp]e- p

(k) r kCk) TKv f Re NrDwJet 24o J 3D*VtdRe (304)

K~k f [DO,] kdk) J&,,,dR0 (305)

Wk r k d) 1TdKwSwy f Reyk 2 p3-y[ .)

Wk ( t [ k) T 37K 00 3 f JRe p*,leS3 24 P 3'y#dR. 33

(k) r ,Y.t (k)[#]T=w J NRe b# 24p 3 -y3Nr03 IdRe (308)

=0 f Re 24 3),tS3 3 ~ 3 C~ (w09

W L1.y3 ) qP T S (06

10 k3 159

3 IR Y]3,te k ) 33 3e (310)

D(k) f t ]•t 2S(k) r- (311)3 R "*33e k 3333 "•3,ye e

Dv333D R i3 JtS W T (312)

(k) JR [. (k) [# ]TdR

"(33)3D fRe [*33e k 3333 ),3 (313)

f= fR 0, [4.y3]ftdRe (314)e

R(k) k [ 4•w• ,dRe (315)

wY3 R yefR, 1R(k) = [,] [W,TdR 316)

y3w R e 12

R~3 = R r-.J 3 ] (317)

+,3 Re

(k) f T(31)

R),30 JR ,3..12 ,d (319)

R W,, f g?)dS., (320)Sv f Se

R f _3 g(kdSe (321)

R W [=wt g(k)dSf (322)

The spatially discretized functional (284) can expressed in matrix form as

ne = -IU)T [KL (U), + 2 {U)T (F). (323)

where

160

[411 1 jfli)1741) o o o o o

1 1 (1 )T [at"' [of" o 0 0 0 0

Wyj )p 1() [AJf' I [(f2) [ýf 2) 0 0 0

o 0 IA2T [1f2) [py2) 0 0 0

o 0 [t]2T [f 2) [.f 2) [,yf) [ff]3 0

o 0 0 0 [1 (f3 )T [all) [](3)

Ike=0 1 3)T 1(p3)T [AIff 0 0 0

0 0 0 1 1 4 ) [V N3)( 1 0 0

0 0 0 0 [N- V [Pf1 NI1 0 0

* WN- 1 )T [ON- 1) [ 1,N) [,fN

[*YN [aN) pf N)

o [nlfN)T ,~fNJ 1 ~

161

(o.)(0){0)

luk 1) {FJ=

I) {R}(Nto)

to,) (3){to)

and

162

v-4k)1V

{u}(k)= $k) k , 2, N

V3

- k = 1, 2, N-I

"-(k0, t9'33l

-*Y3

(N

{cTY(N) = °•'

(k)-Rvn

{R)(k) , -R(k) k 1, 2, 3, N (324)

-R(k)

Superposed bar on a quantity in the following denotes average of the quantity itself

and its transpose. Here, elements of the [K], are explicitly,

(- I + 1 )K(l) +K(I) ( 1 + 1 )K(l)

144 1200 ),,y p3y 24 100 y33

[ .(325)I ( --- L)KO . I •K)K~

24 100 "3y4 2 3334

[ -Y ) I (326)

(O) (11)~21 'r 2 2 j

-L(/ - I .()K ..M , -- M 1V) +( 1 ) D(l)144 1200 "3y3t p3,3 120 840 p3,3D 3--- p3•• 3

163

(_) _1 _ (I) 1 33 (1)12 24 100" )333 24 280 "333

(2) ..4 _.._ 1_)K(i) _ 1 3 -)D--)21 24 100l 333 2T4 -20 33y3D

(1) - ( I 3 3;¢(t) __( - 17 )D( I)

r22 4 25 3333-4140 3333

+ . ' L)K(N) +K (N) _ + 1 )K(N144 1200 y 3y3 p3-3 24 100 y333

[]N) = (327)

-41 + 1 L)K(() 1+ 3 )K(N)"24 100 33,y3 4 25 3333

(N) (N)~I 1 121

[r](N) = (328)(N) (N)I21 22

•(N) 1 1 V(N) .,()-- 1 1 •-(N) + 1 1 r(N)

"""j-20-0"',y3)3 +Ka-• P3 33-)1- 144 1203v h 120 84 p3-y3D 3 5 p3-y3

221 T4 1-f0 33Y3 24 280 33y3

lrN.) - 0-- 3 )K(N) -. _1- 17 )D (N)

22 • 25 3333 -4 140 333

K(k) 0vv

[offk) 0 (.Kk) +K ()) K(k) k = 1, 2, ... .N (329)

0 K (k) K (k)

0. WW

(k) +R -00V-,3 -W3 -- v33

[o-)-(K (k) +5K (k) K 00i•3 k .- 1, 2,. . N (330)D)'Y3 #Y'3 D033

K(k)wy3 Rw33

164

i () K ) , ,(k) ' K(k) ]

4( lkl+~ 3 -(K W +5K~k W K W

"p)k = 1, 2,.. N (331)-K 33v -K -R33w

33v 33DO

11 1X 21[) k -I,2, 3,. N-I (332)

X(k) x W~

[) k = 2, 3..... N-1 (333)

C=P2 C22

where

• = (-+. I.1 .L.X(k) +K(k+1))+.K(k) W +(k- W)

11 144 1200 'gmnma3),3 /3y3 p3.y3 p3-y3

+ 1 (_L ++k*) )l+!XDk) +D(k+l))

120 840 p3.J 3D p3y3D 3 5 p3")3 p3")3

1 (~+ I lXKk) ,e+K 1)120 840 -fpJ tpJ

1 3 r%(k+1)-X(k) - 1+( 1 2XK(k+1) K(k) )+G 1+ XD(k) Dk)

12 24 100 -24333 2•,3338 T T8" y3 3 3 -y333-

X(k) - (-.L+ 1 XK,(k+l)_Kk) )+(. +_3XD(k) W D(k+ )O

21 24 100 33-y3 33)3' 24 280 33)y3 -- 33'y3"

X~t (I 3 .. (~l)..() . /I 17 r.(ktl) r(k)

W~k - (! + -I-XK (+' )+K W) )- -!+-!7XD (+'+D W"22 4 25 3333 3333 4 1 3333 3333

(k) _ __L_ I Kk) +K(k) -- 1 1L)Jk) +( l!)D Wk)

1 144 1200 -3-3 p3y3 120 840 p3y3D 3 5"p3y3

+( I I)K ek)+ 120 840 -3p3J

C k) ( 1 1 •)K(k) .. ( 1 3 )D(k),2 24 100• y333 24 280 y

e .k) _4, 1 _ L)K W) +(_L_ 3 )D W)

21"-T4 -100 333 24 280 33,3

W) (1_3 )K(.) -4 -1 17 ) 65W22" "4-25"3333-- 4 140 3333

165

The spatially discretized governing function for the global system is

m

n,= n = -{U}T[K]U}+2fu}T{F) (334)e= i

where {U) is the vector of values of the field variables at the global nodal points, {F}

is the set of corresponding load vectors and [K] is the system matrix corresponding to

[K]t for an element. Vanishing of the differential of fn, in (334) with respect to (U)

gives the set of equations

[KMU} = IF) (335)

where

In

[K] = E [K], (336)e= I

and

m

{F} - E [F]t (337)C-I

5.2 Selection of the Interpolation Scheme

5.2.1 General Considerations

The basic requirement of selections of element to be used in the numerical solution

procedure is that over each element the first order derivatives of , , , 33

and the second order derivatives for 0r4 ) include, at least, constant values. Also, •,

33 (, _T and a(" should be continuous across interelement boundaries. This

would require, for triangular elements, v) ik, vk) and oa-") to be linear or higher

order. oi5ll needs to be at least quadratic. For rectangular elements, bilinear

interpolants for V Y , vk and o '-W would be necessary and higher orderfor 47k).

interpolants would be required for oY.

166

The Heterosis element introduced by Hughes [1978] is a higher order element

satisfying all of the above requirements. This element has been found [Hughes 1978]

to be very good for isotropic plates. It has been used for laminated plates by Hlong

[1988] and no comparative studies of its effectiveness in comparison with new

possibilities are available. For the present application, this element was used.

In this section, the interpolation functions of the Heterosis element and the

differential operator matrices involving in the (285) through (322) used to implement

the finite element analysis are summarized. The finite element computer program

incorporated the Heterosis plate bending element [Hughes 1978] without using

reduced/selective integration technique. This element is a variant of isoparametric finite

element and, therefore, element matrix can be formed following the usual procedure for

isoparametric element formulation. However, the Heterosis element differs from other

isoparametric elements in using different interpolation schemes for lateral displacement,

4) , )kV3 , and transverse normal stress, cr.3 . In-plane kinematic field variables •., 2

_-(k)

and transverse shear stresses Or,3 are approximated by nine-node Lagrange interpolation

functions while the lateral displacement, ;;3 and transverse normal stress or-, are

approximated by quadratic functions for eight-node isoparametric element. Figure I

shows the geometry and the nodal points of the Heterosis element.

167

t

S(1.01)

(0. 0) 9 / I"^

SS

1 5 J2

(a) Local

Y

(b) Globl

Figure 1: (a) Local, (b) Global Coordinate Systems of Heterosis Element

168

5.2.2 Interpolation function for the Heterosis Element

Interpolations functions of eight-node isoparametric element and nine-node Lagrange

element in terms of natural/local coordinates (s,t) and their derivatives with respect to

s and t are as follows:

(1-sX 1-tX-1-S-t) (1 -tX(2s+t) (0 -sX(2t+s)(1+sXi-tX-1+s-t) (1-tX2s-t) (1 +sX(2t-s)1+sX 1+tX(-1+s+t) (1+tX(2s+t) (1-4 sX(2t+s)(i-s)(+tX-1-s+t) aN 1(1+tX2s-t) 8N 1 (1-sX2t-s)

NN _,SXt (338)_-is-sX-t -4 s(l -t) at 4 ~2(,__S2)(382 (1+SX,-t 2 ) 2 (1_t 2 ) -4 t(1+s)

2 (1-sX,-t2 ) -2(0I-t 2 ) -4t(I-s)

St(i+s)(t-i) t (2s+1) (t-1) s (2t-1) (s+1)st(1+s)(1+t) t(2s+l)(t+1) s(2t+l)(s+l)st (S-1) (t+1) t (2s-1) (t+1) s (2t+l) (s-1)

L- 2t (I-S2)(t-1) _O - 4st(I-t) 1_ 2 (2t-1) (IS2) (339)

2t (,_S 2 ) (t+1) -4st (t+1) 2 (2t+1) (,_S 2)

2s(s-1) (1-t2 ) 2(2s-1) (I-t 2 ) 4 st (1-s)

4 (1-S2)(1-t 2) 8s (t2 _1) 8t(S2-1)

'(2s-1)(2t-1) 2t (t-1) 2s(s-1)

(2s+I) (2t-1) 2t (t-1) 2s(s+1)

(2s+lIX2t+l) 2t(t+l) 2s(s+l)

2 (2s-1)(2t+I) 2 2t(t+I) 2 2s(s-I)OU =I 4s(I-2t) IL I 4t(0 -0 01. =- I 4 (,_S2) (340)

Cpa 4 _4(sl s2 4 40-' tt 4-ssl

-4t (2s- 1 4 (I-t 2 ) 4s(I-s)I6st 8 (t 2 _1) 8 (S2_I)

169

Here, N and L denote interpolation functions for eight-node isoparametric and the

nine-node lagrange elements, respectively.

5.3 Computer Implementation

Since the field variables are interpolated over an element in natural coordinates

(s,t), it is necessary to set up the relation of the global coordinates and natural (local)

coordinates for evaluation of the element matrices defined in (285) through (322). We

consider a mapping of global coordinate system (x,,x,) to local coordinate system (s,t).

We assume that this mapping is one-to-one and onto. By chain rule, the derivative in

each coordinate system is related by

&a a a= a or = -J (341)

a a a_&at ay ay at

where Jacobian matrix J and its inverse are defined as

,j=C C and J_ I at O (342)

at at at CIS

Here, IJI is the determinant of Jacobian matrix. Following the concept of isoparametric

formulation, global coordinates are interpolated over an element as

x * T x (343)

where * is the vector of interpolation functions used for field variable. x is the

vector of global coordinate values at nodal points.

Using (338) through (343), the differential operator matrices in (285) through

(322), explicitly, are

H_ =k1 = H k , .) =ILLOI (344)

H Hk , =H =N (345)

170

T

jDO}T=fT LT p (346)

),r fT..~T Y- R (347)

{o-3,y T T TT 1 ( -T X (348)*p 7

V3 V p

a2 2

a 2 a2

ii L P1 - TPR p Ox O

R= N, NT_ ,y

YL, L, tL. - LL _L 1 L y

Ox2 2~ + st TyI~

+ (--e LL _- L LT T + 2 -y L, , L jL 1)y

+ yL[tyLt 1 2 2yT[L, yj Is - yT(LL'+,L,')yjý L-J CIS 3 Tt 3 i

and

I LT+L, T T-"I

B - XT [L.,L, T,ýLT-L, L, -_L,L,QJ2 - St t 'Itt 't St

171

a0 = [AT T x-BxT T x+I T(L ,T_'L )I2 [-x L, L.X-----x_ L, L, X+- (. TxI

ayt t 2- t ts • - os

[ . . T B r T TT L T T

+--T L,tL,sx+ x L,L,sX + -xT(L, LT-L, L,T)x]-

IT a 2• XT TX 2

" -LX L,L x-~-- TO LIE LT(L,.L, T+LIL ThO

J 2- Itt Io - s t2 2 tjI Sasat

= AT T .B T x. T T( T TOxOY , X - -L,L a

LT , LL, -- y LLx - L,( L, -L, U _)x]0

T 02 T2T--. T a 1 T T T) 21t-y xC2 2 ±YTL L x T V (L,SL,t +L,tL,T )x S .,&a

a• T. T .B T-- T I XT (L, IT -,L LT)Y]A2 A T--T B T.

L A L ,tLty-+-x LsL.,y - X t(L LtLL tt

+ T T B T T 1T( LIT TA LtL,, .Ly - -2 -xL, L,t+IS

_T 1~x LT T +..LXT( ,T+LT)yLj2- L I" Y- L,-LtAL

as 2 s t i t Is "t

In element matrices given in (285) through (322), integrands are functions of local

coordinates (st). Therefore, the surface integration extends over the natural coordinate

surface. Since, in general,

dR = Jdsdt (350)

integration in the two coordinate systems is related by

_f fn xxt = F(st) IjIdsdt (351)

For numerical evaluation of the integrals, Gaussian quadrature implies:mm

f F(xy)dR = EEF(s,,,t ) Iti IWi) (352)

172

where rn is the number of Gaussian quadrature points and W,, are the corresponding

weighting values. Here, it should be mentioned that in the Hteterosis element

numerical integration was performed. In the matrices given in (285), through (322),

the highest order of numerator of the integrands in (s-t) is order of nine and the

highest order of denominator is order of six. To properly perform the numerical

integration, it is necessary to use as many Gaussian quadrature points as possible until

convergence is attained. It is quite expensive to perform the numerical integration

when the number of Gaussian quadrature points is more than three. Therefore,

three-point Gaussian quadrature was used in the examples described in the next section.

173

SECTION VI

EXAMPLES OF APPLICATION

6.1 Introduction

The finite element formulation of the modified Pagano's [1978] theory developed in

the previous section was used to obtain solutions to displacement and stress fields in

delamination coupons. Three examples were solved. The first two examples involved

four-ply symmetric laminates. Both cross-ply and angle-ply coupons were considered.

The purpose was to validate the finite element model by comparing the numerical

solutions with those from Pagano's [1978] analysis. The third example consisted of

studying stress distribution in a multi-ply laminate subjected to uniform stretch. A

theoretical solution for this case was not available. The results were compared with

Chang [1987] who used a continuous traction element for the problem of 3-D elasticity

specialized to the coupon with a potential energy minimization procedure. A stacking

sequence of [(25.5/-25.5),/90t, was used.

6.2 Delamination Coupons

6.2.1 Four-Ply Laminates

In this section, analysis of two long symmetric laminate strips made of

graphite-epoxy materials, with fiber orientations of [45/-45], and [0/90]. under uniform

displacement in the longitudinal direction is described. The relation between laminate

width and thickness was 2b=16h following Pagano [1978] In the analysis each ply

was idealized as a homogeneous, elastic orthotropic material. For comparison purpose,

the material properties assumed, following Pagano's work [1978], were:

174

E 1 =2OX10Xpsi

E2 2=E33-2.1x 106psi

G 12 =G 13 =G 23 =0.85X10 6psi

V 122v 13-=V 23=0.21

The subscripts 1, 2 and 3 denote the longitudinal, the transverse and the thickness

directions respectively. The 6x14 and 6x18 finite element meshes as shown in Figure

2 were used to discretize a coupon. This corresponds to N=6 and N--4 respectively in

Paganos analysis [19781 Numerical results based on the finite element model were

compared with Pagano's [1978] analytical solution.

The value of N in the following figures corresponds to the number of sub-layers

used in Pagands theory. Thus, N=6 indicates that each physical layer of thickness h

was modeled by three sub-layers each of thickness h/3, while N=2 denotes that each

physical layer is treated as a unit as stated in Pagano [19781

175

(a)

(b)

Figure 2: (a) 6x14 Element Mesh, (b) 6x18 Element Mesh

176

6.2.1.1 Cross-Ply Laminate

Distribution of o, along the width on the central plane of the [0/901, laminate

for N=6 shown in Figure 3, indicates a sharp rise near the free-edge boundary.

Solutions obtained from the finite element model agree with Pagano's N-2 and N-6

solution over the entire width of the laminate.

Figure 4 and Figure 5 show the variations of a,, along the interface between the

0 and 900 plies for N=2 and N=6. Due to the presence of the discontinuity in elastic

properties, a singular stress behavior would be expected at the free edge.

Values of T., along the interface between the 0/90 layers, calculated from the

finite element model ( Figure 6), showed that the refinement through the thickness is

necessary to have satisfactory agreement with those calculated by Pagano's method.

Comparative results for the variation of transverse displacement along the top

surface of the [0/901, laminate are shown in Figure 7.

177

o F.E. (6X1t4) N=6 CHYOU

- PRGRNO N=6 (1978)

cr)

CY)OCD(

C)Do

MCr

Wr)

'0.00 0.20 0. 140 0.60 0.80 1.00'V/B

Figure 3: Distribution of Z-strew Along 90/90 Interface

178

o F.E. (SX1L4) N=2 CHYOU

- PRGRNO N=2 (1978)

cc,

00

ZCW

Cl)

Lii*

C)

101.00 0 1.20 01 .14 0 1.60 0 .80 1 .00

Figure 4: Distribution of Z-stress Along 0/90 Interface (N-2)

179

oD F.E. (SX1'4) N=6 CHYfiU

- PRGRNM MI=6 (1978)

-cc

.0

Cr)

F-0

f%4

0.0 0 .20 0 .40 0 .60 0 .80 1.00

Figure 5: Distribution of Z-stress Along 0/90 Interface (N-6)

1 80

(D F.E. (6X14l) N=6 CHT@U

- PRGRNO N=6 (1978)

zpCc'

I.-

CD"

N

(U) C 0

C3)

9300 0.20 1.4Li 0.60 0.80 1.00T/IB

Figure 6: Distribution of YZ-stress Along the 0/90 Interface

( F.E. (6X18) N=2 CHYOU

+ PRGAN0 N=2 (1978)

to

z

cr)~CoCO

(n.

zo

Co

~-4-

0

9.00o 0.20 0.0 0.60 0.80 1.00l/B

Figure 7: Transverse Displacement Across Top Surface

182

6.2.1.2 Angle-Ply Laminate

Figure 8 and Figure 9 show the distributions of o,, along the width of the

laminate a. the center line of the top (450) layer for N=2 and N=6. The distributions

of % along the width of the coupon at the middle of the top (45 degree) layer for

N=2 and N=6 are shown in Figure 10 and Figure 11. The results obtained using the

finite element model agreed with Paganos solutions for N=2 and N=6 across the entire

width of the laminate.

A comparison of the shear stress (rx) distributions along the interface of the

45/-45 layers fo. N=2 and N=6 ( Figure 12 and Figure 13), indicated that the

solutions of the finiue element model had sharp rise toward the free-edge similar to

Pagano's solutions with N=2 and N=6.

For the axial displacement distributions across the width of the top surface for

N-2 and N=6, the finite element results compared well with Pagano's N=2 and N=6

solutions ( Figure 14 and Figure 15).

183

oD F.E. (6X18) N=2 CHYOU

- PRGRNO N=2 ('1978)

CO

cc.

U 0

.............. ........ ...... ..................

.. ......+. .....

XC)

CDj

LO

"cb.o00 0'.20 0'.140 0.60 0.80 1 .00'(/B

Figure 8: Distribution of X-stress Along Center of Top Layer (45 degree)for N=2

184

+ F.E. (6X1'4) N=6 CHYrOU

- FRGRNO N=6 (1978)

r-4

OD ++

LLJ C

CD

cuJ

tru

A0

00%.oo. 00.20 0.110 0.60 0.80 1.00

T/B

Figure 9: Distribution of X-stress Along Center of Top Layer (45 degree)for N-6

185

CD F.E. 16X18) N=2 CHYOU

- PFIGRNO N=2 (1978)

Q

0: CO

Cc

LLJ(CNJ

(\5

c00C3

.. ..................(.. ................. ..... ...................0.. ........ .........

X0\.. ~ ..... ............

Cý 00.20 0.LiG 0.60 0 .80Y/IB

Figure 10, Distribution of XY-stress Along Center of Top Layer (45 degree)for N=2

186

F.E. K6X1L1] N=6 CHTOU

- PRGRNO N=6 (1978]

cu

~CY

ILO

LLJC N

(NJ

00

.. ... .... ......-. ... ...... ..... .......... ..... ..930 02 Yl .0 '8

... ... .. ... ... ... ..... ... .... ... ..../.. .....

Fiur 11 itiuino Y-tesAogCne f o ae 4 erefor .N...-........6.. ... ..

18

o F.E. '(6X18) N=2 CHYOU

- PRGRNO N=2 (1978)

In

Cco

+LU.

Cr)

LUJ

Cr0

Y. 00 0'.20 0.140 0 .60 0.80 1.00T/B

Figure 12: Distribution of XZ-stress Along 45/-45 Interface for N-2

188

(D F.E. (6X1'4) N=6 CHTOU

- PRGRNO N=6 (1978)

zC

Zcc

CD)

+11

LLJ~F-4

YC3

X)

4,0002L'U '.6 .010

LiB

Figue1: Dsrbto fUsrssAog4/4 nefc o -

Cf189

o) F.E. (6X18) N=2 CHYOU

- PRGRNO N=2 (19783

zCEcc

.. .. .. .

.. .... ...

r3j.

9300 0.20 0.40I 0.60 0.80 100T/D

Figure 14: Axial Displacement Across Top Surface for N-2

190

oD F.E. (6X1'4) N=6 CHXOU

- PRGRNfO N=6 (1978)4C

z

'CECD.F-

C3

,C) 0

S00 0:. 20 0.140 0.60 0.80 1.00Y/B

Figure IS: Axial Displacement Across Top Surface for N-6

191

6.2.2 A Multi-Ply Delamination Coupon

Analysis of the four-ply laminate specimens described in the previous section

demonstrated the validity of using the proposed finite element procedures in solving

delamination coupons. In this section, application of the finite element method to

investigate the stress fields in the multi-ply laminate is described and the results are

compared with those by Chang [1987]. The structure of the laminate with

predetermined fiber orientations used in the present investigation was, following Sandhu

and Sendeckyj [19871

Stacking Sequence Width Ply thickness Plies

[(25.5/-25.5)5/90], 1.0 in 0.00505 in 22

The material used in the study was AS4/3501-6, graphite-epoxy, and the elastic

constants were [Sandhu and Sendeckj 1987]

Ell =19.26x1O6 psi

E22,1.32X10 psi

G ,=0.83x 10 psi

V 12=0.35

6.2.2.1 Numerical Evaluation

A 64 element model shown in Figure 16 was used to discretize the delamination

coupon. Interlaminar stress field within the delamination coupon for an applied

longitudinal average unit strain was computed.

The distribution of oa along the midplane of the multi-ply laminate, shown in

Figure 17 indicates that the finite element model predicts a sharp rise toward the free

edge similar to Chang [19871

192

64 Element Model

Figure 16: Finite Element Mesh for 22-ply Laminated Plate

193

F.E. PRESENT METHOD

F.E. CHRNG(1987)

V

zo

U)

*L .......

U)0LU

U)

CEO

LA) +

1 0.0 0 .20 0.40 0 .60 0 .80 1 .00

T/B

Figure 17: Distribution of Z-stress at Midplane

194

Figure 18 shows the through-the-thickness stress distributions of oa calculated from

the present model and the continuous traction Q-23 element [Chang 1987] at the free

edge of the laminate specimen. Figure 19 shows the through-the-thickness stress

distributions of o7, at the centroids of elements along the free-edge. It is observed that

the stress field based on Q-23 element (Chang 19871 is not smooth at all compared

with that by the present model. The slope discontinuity of a., at the interfaces in

Chang's analysis indicates limitations of the displacement-based model in satisfying

traction-free conditions. Table 4 gives a comparison of numerical results obtained by

the present study and Chang [1987] at the free edge (through-the-thickness) of the

22-layer delamination coupon.

The stresses (7 at the interface of -25.5/90 calculated by the present model and

Chang [1987] are shown in Figure 20. Figure 21 shows the effect of mesh refinement

on the a., distribution along the interface of -25.5/90 calculated by Chang's analysis

[19871 The oscillating pattern of orP is present both in Cbang's analysis and the

present study. Chang [1987] found that refinement of mesh near the edge eliminated

the oscillation. The same could be expected with the present model. However, due to

limitations on core and auxiliary storage on the available computational facilities,

refinement of meshes could not be implementated at this stage.

195

Table 4: A Comparison of Numerical Results (Through-the-thicknems) atthe Free Edge

Z/H Chyou I Chang [1987]

z-stress/(unit strain x 106) Iz-stress/(unit strain x 106)S4--------------------------------+-------------------------------------------------

0.0 0.00000 1 0.000000.5 -0.15750 -0.154601.0 -0.31500 1.891301.5 -0.25191 -0.428252.0 -0.18882 0.939022.5 -0.14466 -0.259793.0 -0.10050 1.664703.5 -0.08347 -0.270344.0 -0.06644 1.072904.5 -0.05523 -0.262355.0 -0.04402 1.652405.5 -0.02552 -0.25146

6.0 -0.00703 1.134806.5 0.03743 -0.255227.0 0.08188 1.940707.5 0.17704 -0.203458.0 0.27220 1.433908.5 0.48780 -0.177569.0 0.70340 3.058109.5 1.19430 1.82050

10.0 1.68520 -0.0638110.5 1.97660 5.8932011,0 2.26800 4.46050

196

F.E. PRESENT METHOD

+ F.E. CHANG (1987)

S _ _ _ _ ............ _ _ _...

... ......a ............... ........................

.. ...a . ............ .... ....... ...

.a ....

.. ........... ..........

...... ..... ............

.. ...............

. ... ...........

.. .. . .... .....

......a....

a;... .

a. o 0 0 22 9.e .0 70ZZSwS(.)/IExTAN

Fiue1: ~og-tetikns -trs itrbto AogteareEg

19

F .E. PRESENT METHOD

+ F.E. CHANG (1987)

..... ..... ..... ......

... ............ ....... .... .............. ...

.... .. .. ...... .... ....... . .......

. .. ... ........ ......... ... ... ..... . .... ............. .......

.... ...... 00...... .......ZZ S R ( .8.........) ./ .......E ..............

Figure l9~. Trog-h-hc es Zstes isiuto atte etris fEle ent Aln the. Fre E......dge . ...... .............

...... ....... 1 98. ......

oD F.E. PRESENT METHOD

+ F.E. CHRNG (1987)

Co

CE

C3

LUJ

ýCO

LUa:

0h o 0'.20 0.40 0.60 0.80 1.00

Figure 20. YZ-Stress Distribution at the Interface -25.5/90

199

CD FINE MESH CHANG (1987)

+ CORI9SE MESH CHRNG (1987)

CE

CD~

LU

(0

V,)LUJ

I-V( 0

C)

9y.O D 0'.10 0.20 0.30 0.40 0.50'(/B

Figure 21: Effect of Mesh Refinement on YZ-Stress Distributions at theInterface -25.5/90

200

SECTION VII

DISCUSSION

The problem of analysis of composite laminates has been investigated. Paganods

theory has been examined carefully, rewritten in terms of a reduced number of field

variables and stated in a self-adjoint form so that a general variational formulation

could be developed. The resulting functional was specialized for implementation in a

finite element model for stress and deformation analysis in laminated composite plates.

The procedure was applied to study of stress fields in free-edge delamination specimens.

As originally stated, Pagano's [1978] theory used seven equilibrium equations, ten

constitutive equations, and six interfacial continuity equations involving 23 field

variables. An important feature of the present research was to rewrite the equations

of Paganos theory in terms of fewer field variables and to state the equations in a

self-adjoint form. Physically, there are only five equations of equilibrium and,

therefore, there can only be five corresponding displacement field variables. The

quantities N, and M33 introduced by Pagano could, therefore, be eliminated by writing

explicit expressions for these in terms of stresses (termed equilibrium equations by

Pagano). In this manner, the number of local mechanical variables reduced to eight

requiring as many constitutive equations. The total number of field equations in

Paganos theory is thus reduced to 19 in five independent displacement field variables,

eight mechanical quantities and six interfacial traction and displacement components. It

is shown that the system of interfacial displacement continuity equations and field

equations is self-adjoint in the sense of inner products.

201

Boundary operators consistent with the field operators for the problem have been

identified following the procedure outlined by Sandhu and Salaam [1975] and Sandhu

[1976. For self-adjoint matrix of operators and consistent boundary operators a general

variational principle was derivedallowing for possible internal discontinuities. Extensions

to relax the requirement of differentiability of certain field variables have been

developed, along with specializations to reduce the number of independent field

variables.

For finite element implementation the interpolants can be restricted to satisfy some

field equations and boundary conditions identically. One specialization in which the

generalized displacements are continuous across boundaries of subregions or elements but

generalized forces need not be continuously differentiable was used to develop a finite

element program. This program was verified against Pagano's [19781 four-ply laminated

free-edge delamination specimens, and also applied to a 22-ply specimen. The theory

and the numerical procedure developed are quite general and applicable to laminated

composite plates with arbitrary boundary conditions including internal boundaries e.g.

holes. Discontinuities, e.g. delaminations, can be easily included by retaining

corresponding terms in the governing functional. The general variational theory could

form the basis of several alternative finite element schemes which could be used to

define bounds to the solution.

As at present developed, the finite element computer program requires enormous

amount of storage. This is largely due to the large number of algebraic equations

with large bandwidth. Available equation solvers and storage strategies cannot handle

the problem in an economical fashion. It appears necessary to develop efficient

equation solving procedures taking advantage of the multi-banded nature of the system

matrix. Global-local strategies to reduce the size of the problem and solve it in two

or more steps should also be investigated. Different finite element interpolations could

be implemented to obtain results in more economical fashion.

202

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*U.S. GoVernment Pkith Off0:19W--8W127I62= 206