Constitutive Modelling of Anisotropic Sheet Metals Based on a Non-Associated Flow Rule

Constitutive Modelling of Anisotropic Sheet Metals Based on a Non-Associated Flow Rule

Modelleren van het constitutief gedrag van anisotroop plaatmateriaalgebaseerd op een niet-associatief vloeimodel

Mohsen Safaei

Promotor: prof. dr. ir. W. De WaeleProefschrift ingediend tot het behalen van de graad van Doctor in de Ingenieurswetenschappen: Werktuigkunde-Elektrotechniek

Vakgroep Mechanische Constructie en ProductieVoorzitter: prof. dr. ir. P. De BaetsFaculteit Ingenieurswetenschappen en ArchitectuurAcademiejaar 2012 - 2013

ISBN 978-90-8578-596-5NUR 978, 971Wettelijk depot: D/2013/10.500/29

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De auteur geeft de toelating dit doctoraatswerk voor consultatie beschikbaar te stellen, en delen ervan te kopiëren uitsluitend voor persoonlijk gebruik. Elk ander gebruik valt onder de beperking van het auteursrecht, in het bijzonder met betrekking tot de verplichting uitdrukkelijk de bron te vermelden bij het aanhalen van de resultaten van dit werk.

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The author gives the authorization to consult and copy parts of this work for personal use only. Any other use is limited by the Laws of Copyright. Permission to reproduce any material contained in this work should be obtained from the author.

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Copyright © M. Safaei Gent, May 2013

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Promotor

prof. dr. ir. Wim De Waele

Ghent University

Faculty of Engineering and Architecture

Department of Mechanical Construction and Production

Examination Committee

prof. dr. ir. Patrick De Baets (Chair) Ghent University

prof. dr. ir. Wim De Waele Ghent University

prof. Jeong Whan Yoon Swinburne University of Technology

prof. Sandrine Thuillier Université de Bretagne-Sud

prof. dr. ir. Patricia Verleysen Ghent University

dr. ing. Steven Cooreman ArcelorMittal R&D

dr. ir. Stijn Hertelé Ghent University

Research Institute

Ghent University

Department of Mechanical Construction and Production

Laboratory Soete

Technologiepark 903

B-9052 Zwijnaarde

Belgium

Tel. +32 9 331 04 79

Fax. +32 9 331 04 90

[email protected]

http://www.soetelaboratory.ugent.be

http://www.soetelaboratory.ugent.be/

Dedicated to Shima

Acknowledgement

First and foremost, I would like to express my gratitude to my supervisor Prof. Wim

De Waele for his strong support, patience and constant availability for technical

discussions. I feel motivated and encouraged every time I attend his meeting.

This work was made possible by the financial support of the Ghent University Special

Research Fund (BOF) grants nrs. 08/24J/106 and 01J10608.

I would like to thank several colleagues beyond Ghent University who in one way or

another contributed in this research. I would like to express my particular gratitude to

Prof. Jeong-Whan Yoon form Swinburne University of Technology in

Australia for sharing part of his invaluable knowledge. Collaboration with

him was a great experience. I would like to thank him for being my

inspiration.

Prof. Shunlai Zang from Xi’an Jiaotong University in China for being a nice

friend and his unconditional helps and quick responses.

Prof. Myoung-Gyu Lee from Graduate Institute of Ferrous Technology in

Korea for his contributions in various steps of my research. I would like to

thank him for his enthusiasm in research.

I cannot say thank you enough for their supports and helps.

Most of the experimental results in this research would have been impossible without

the technical support of some people and organizations.

My thanks go to:

ii

ArcelorMittal R&D (OCAS) and Kris Hertschap for materials and various

experimentations on sheet metals. My thanks also go to Klaas Poppe and

other engineers and technicians of ArcelorMittal who helped me during

experimentations.

Gaëtan Gilles and prof. Anne-Marie Habraken from University of Liège for

shear tests and their hospitality.

I am also grateful for the various discussions with Dr. Tom Stoughton (General

Motors, USA) on non-associated flow rule and, more importantly, on originality of

research ideas.

I would like to thank Prof. Rob Wagoner, Prof. Kwansoo Chung, Prof. Salima

Bouvier and Prof. Tudor Balan who openly answered my questions.

My time at laboratory Soete was made enjoyable in large part due to nice friends and

colleagues. I would like to thank all of them: Patrick De Baets, Rudi Denys, Gusztav

Fekete, Yeczain Perez, Zamaan Sadeghi, Jacob Sukumaran, Matyas Ando, Vanessa

Rodriguez, Patric Neis, Tan Dat Nguyen, Stijn Hertele, Matthias Verstraete, Jeroen

Van Wittenberghe, Koen Van Minnebruggen, Timothy Galle, Hanan Al Ali, Diego

Belato Rosado, Felicia Jula, Reza Hojjati Talemi, Jan De Pauw, Yue Tongyan,

Wouter Ost. My thanks also go to all other colleagues in laboratory Soete, Tony

Lefevre, Chris Bonne, Rudy Desmet, Jonathan Vancoillie and Georgette D'Hondt.

Most of all I would like to thank my wife Shima for her unyielding devotion and love,

support, encouragement and quiet patience. My deepest appreciation also goes to my

parents Asadalah and Farkhondeh, my brothers Reza, Ali and Majid and my only

sister Azar for their faith in me and allowing me to be as ambitious as I wanted. I

would like to thank my lovely nieces, Nasim and Niloofar for gifting so much

laughter. I am so grateful to my new family, Taaleh, Mohamad Hasan, Vahid and

Shiva for their unending support.

Mohsen Safaei

Gent, April 2013

Summary

Optimization of material forming processes requires an in-depth knowledge of

material constitutive models and methods to implement these into user-friendly

numerical tools. Restricting the discussion to polycrystalline materials, constitutive

models can be developed based on two distinct approaches. In the first approach,

which is referred to as crystal plasticity, the polycrystalline behaviour is described

based on the behaviour of each individual crystal. In the second approach, called

phenomenological approach, the average behaviour of all grains directly determines

the global material behavior. Without a doubt, both approaches have proven their

numerous advantages. However the phenomenological approach has persuaded

researchers in industry and academia whenever simplicity and simulation speed have

been the major concerns. Compared to experimental trial-and-error, a faster

optimization of the metal forming process can be achieved by numerical simulation

techniques such as finite element (FE) methods. Based on a variety of generated FE

simulation outputs the optimization of the forming process can be carried out more

effectively.

In general, this dissertation deals with phenomenological constitutive modeling and

its implementation in FE simulation of sheet metal forming processes. More

specifically, considering the constitutive modeling, this dissertation focuses on the

anisotropic behavior of sheet metals. Sheet metals undergo severe plastic deformation

during manufacturing processes such as cold rolling. This introduces a preferential

orientation to the grains and thus a variation of mechanical properties at different

orientations is to be expected. Due to its important impact on the distribution of

stresses and strains, the shape of formed parts is influenced by the anisotropic

material behaviour. Most of the available anisotropic models are based on the

Associated Flow Rule (AFR) hypothesis. This hypothesis states that the anisotropic

iv

yield function determines both the onset of yielding and the plastic flow direction.

The AFR approach , which has been used for long time in material constitutive

models, is with various limitations. For instance, this model cannot be used to

simultaneously describe pressure sensitivity and zero plastic volume change upon

plastic deformation. In addition, modeling highly anisotropic materials is a challenge

for most AFR based models.

Accordingly, the concept of non-associated flow rule (non-AFR) is adopted in this

dissertation. In the non-AFR approach, elastic limit and plastic flow direction are

addressed independently. Two non-AFR models such as Hill 1948 and Barlat’s

Yld2000-2d are thoroughly evaluated in this work. To generalize the constitutive

model for applications involving cyclic loading or load reversal, a very recently

developed mixed isotropic-kinematic hardening model is combined with the

anisotropic yield model. This hardening model is capable of predicting Bauschinger

effect, transient behavior and permanent stress shift upon load reversal. The model

has been implemented into a user material subroutine for the finite element software

package Abaqus using FORTRAN programming language and considering the fully

implicit backward Euler scheme.

Accuracy of the implemented model is verified comprehensively in terms of cyclic

hardening, directional yield stresses and Lankford coefficients. Next, its capabilities

with respect to the simulation of challenging anisotropic material behavior are

evaluated. Validation of the implemented model in terms of cyclic hardening is

investigated for aluminium alloy AA5754-O which is used for automobile structural

parts. On the other hand, the highly anisotropic aluminium alloy AA2090-T3 (used in

aerospace industry) and a challenging fictitious material are used to compare AFR

and non-AFR based models. Evaluation is performed based on prediction of yield

stresses and Lankford coefficients at various uniaxial tensile specimen orientations

with respect to the rolling direction as well as one balanced biaxial stress condition.

To prove the capability of the implemented non-AFR and AFR based models with

respect to simulation of metal forming processes, cylindrical cup deep drawing are

simulated for the mentioned highly anisotropic materials. Cups made from these

materials respectively show 6 and 8 ears. These numbers can only be predicted by

advanced anisotropic models such as Barlat’s Yld2004-2d or high order polynomials.

The importance of using accurate hardening models for the prediction of springback

is well established. Accordingly, the effect of a recently developed mixed isotropic-

kinematic hardening model on the final shape of deep drawn cups was investigated.

The outstanding accuracy of simulations of highly anisotropic material behaviour

using non-AFR based yield models is shown in this dissertation. Nonetheless,

implementation of such models combined with a mixed hardening model in a fully

implicit backward Euler scheme is a laborious task. Therefore, to alleviate the degree

of difficulty of implementation, a simplifying approach is proposed which gives rise

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to a considerably easier fully implicit scheme without significant loss of accuracy.

Another part of this dissertation is attributed to the implementation and evaluation of

various numerical integration schemes. Both implicit and explicit schemes have been

implemented. The choice for explicit schemes could be motivated when (for instance

in industry) simple integration and high simulation speed are desired. However, it

remains essential to investigate the accuracy of the final results.

Characterization of hardening and anisotropy of a deep drawing interstitial free steel

(grade DC06) is also covered in this dissertation. Optical measurements based on

digital image correlation are compared with conventional extensometer

measurements. The optical measurement system was shown to allow for a thorough

characterization of anisotropy evolution.

Following the experimental observations of anisotropy evolution, an evolutionary

anisotropic model based on non-AFR Yld2000-2d is presented which simulates the

distortional hardening and instantaneous Lankford coefficients with high accuracy. A

prerequisite for the parameter identification of this model is the equivalence of plastic

work. An advantage of this model is that it can be implemented based on any

numerical implementation scheme developed for a non-evolutionary non-AFR. It

must be noted that describing anisotropy evolution in terms of both Lankford

coefficients and distortional hardening has been rarely performed. The majority of

proposed models in this field is based on input data at a limited number of tensile

specimen orientations, and mostly considers either Lankford or yield stress distortion

for a specific type of material.

In short, the current dissertation focuses on constitutive modeling and more

specifically on non-AFR based anisotropic models. It can be concluded that the non-

AFR based Yld2000-2d anisotropic model combined with a mixed isotropic-

kinematic hardening function constitutes an outstandingly strong model to be used for

highly anisotropic sheet metals. Various implicit and explicit implementation schemes

were described and compared in this dissertation. A simplified non-AFR model is

suggested and evaluated and finally, an evolutionary non-AFR model is proposed

which can be used when anisotropy of the sheet metal is dependent on the level of

plastic deformation.

Samenvatting (Dutch summary)

Optimalisatie van omvormprocessen noodzaakt een grondige kennis van constitutieve

materiaalwetten en methoden om deze te implementeren in gebruiksvriendelijke

numerieke tools. Voor wat betreft de klasse van polykristallijne materialen, kunnen

constitutieve modellen ontwikkeld worden op basis van twee onderscheiden

methodieken. Een eerste methodiek is de zogenaamde kristal-plasticiteit. In dit geval

wordt het polykristallijne gedrag beschreven uitgaande van het mechanisch gedrag

van elk individueel kristal. De tweede methodiek behelst de zogenaamde

fenomenologische modellen waarbij het gemiddeld mechanisch gedrag van alle

korrels het globaal materiaalgedrag bepaalt. Zonder twijfel, kunnen aan elke

methodiek meerdere voordelen toegeschreven worden. De fenomenologische

methodiek heeft meerdere onderzoekers in de academische en industriële wereld

weten te overtuigen in geval eenvoud en snelheid de belangrijkste drijfveer zijn.

Numerieke methoden zoals eindige elementen simulaties laten toe om, in vergelijking

met experimentele trial-and-error, een veel snellere optimalisatie van het

omvormproces te realiseren. Zulk een optimalisatie kan inderdaad effectiever

uitgevoerd worden op basis van verscheidene uitvoerparameters verkregen uit eindige

elementen simulaties.

In algemene termen behandelt dit proefschrift fenomenologische constitutieve

materiaalwetten en hun implementatie in eindige elementen simulaties van

plaatomvormprocessen. Met betrekking tot de constitutieve modellering, focust het

proefschrift meer specifiek op het anisotroop mechanisch gedrag van plaatmateriaal.

Plaatmateriaal ondergaat zware plastische vervorming tijdens de productieprocessen

zoals het walsen. Dit veroorzaakt voorkeursrichtingen in de korrelstructuur en dus kan

een variatie van de mechanische eigenschappen volgens de oriëntatie in het plaatveld

verwacht worden. Dit heeft een belangrijke impact op de verdeling van spanningen en

viii

vervormingen, en aldus zal de finale vorm van componenten hierdoor beïnvloed

worden. Het merendeel van de beschikbare anisotrope vloeimodellen is gebaseerd op

de hypothese van associatief vloeien. Deze hypothese stelt dat de anisotrope

vloeifunctie zowel het begin van plastisch vloeien als de richting van plastische

vervorming bepaalt. Niettegenstaande associatief vloeien reeds heel lang de regel is

bij de ontwikkeling van constitutieve materiaalwetten, gaat dit gepaard met een aantal

beperkingen. Bijvoorbeeld laten zulke modellen niet toe om gelijktijdig de

afhankelijkheid van hydrostatische druk en het feit dat geen volumeverandering

optreedt tijdens plastische omvorming te beschrijven. Daarenboven vormt het

modelleren van sterk anisotroop materiaalgedrag een uitdaging voor de meeste

modellen die op deze hypothese zijn gebaseerd.

Daarom spitst dit proefschrift zich toe op het concept van niet-associatief vloeien. In

zulk een geval worden de overgang van elastisch naar plastisch materiaalgedrag en de

richting van plastisch vloeien onafhankelijk van elkaar beschreven. Twee voorbeelden

van niet-associatieve modellen, namelijk Hill 1948 en Barlat Yld2000-2d, worden in

dit werk uitvoerig geëvalueerd. Om zulk een constitutief model te veralgemenen naar

toepassingen waarbij cyclische belasting of belastingsomkering optreden, wordt een

recent ontwikkeld gemengd isotroop-kinematische verstevigingswet gecombineerd

met het anisotroop vloeimodel. Deze verstevigingswet laat toe om zowel Bauschinger

effect, overgangsgedrag als blijvende spanningsverschuiving bij belastingsomkering

te beschrijven. Het volledige model werd geïmplementeerd in een gebruikersmateriaal

subroutine voor de eindige elementen software Abaqus. Deze implementatie werd

uitgewerkt in de FORTRAN programmeertaal en gebruik makend van een volledig

impliciet achterwaarts Euler schema.

De nauwkeurigheid van het geïmplementeerde model is uitgebreid geverifieerd voor

wat betreft cyclische versteviging, richtingsafhankelijke vloeispanningen en Lankford

coëfficiënten. Ook werden de mogelijkheden met betrekking tot de simulatie van

uitdagend anistroop materiaalgedrag bestudeerd. Validatie van het model met

betrekking tot cyclische versteviging werd onderzocht op basis van de

aluminiumlegering AA5754-O welke typisch gebruikt wordt voor

automobielonderdelen. Anderzijds werden de sterk anisotrope aluminiumlegering

AA2090-T3 (gebruikt in luchtvaart) en een uitdagend fictief materiaal gebruikt om

associatieve en niet-associatieve modellen te vergelijken. Deze evaluatie is gebaseerd

op de voorspelling van vloeispanningen en van Lankford coëfficiënten

overeenkomstig verschillende oriëntaties met betrekking tot de walsrichting

(uniaxiale trekproeven) en ook een gebalanceerde biaxiale spanningstoestand.

Om hun geschiktheid voor de simulatie van omvormprocessen aan te tonen, werden

de geïmplementeerde associatieve en niet-associatieve modellen gebruikt om het

dieptrekken te simuleren van cylindrische stukken vervaardigd uit de hogergenoemde

sterk anisotrope materialen. Bekertjes vervaardigd uit zulke materialen vertonen

ix

respectievelijk 6 of 8 oren. Deze aantallen kunnen enkel voorspeld worden door

geavanceerde anisotrope modellen zoals Barlat Yld2004-2d of hogere orde

polynomen. Het belang van geschikte verstevigingswetten toe te passen voor de

simulatie van elastische terugvering is alom gekend. Ten dien verstande, werd het

effect van een recent ontwikkelde, gemengd isotroop-kinematische, verstevigingswet

op de finale vorm van diepgetrokken bekers onderzocht.

De uitgesproken nauwkeurigheid van niet-associatieve vloeimodellen voor het

simuleren van sterk anisotroop materiaalgedrag wordt aangetoond in dit proefschrift.

Desalniettemin blijft de implementatie van zulke modellen in combinatie met

gemengde verstevigingswetten in een volledig impliciet achterwaarts Euler schema

een veeleisende taak. Om de moeilijkheidsgraad van de implementatie te verzachten,

wordt een vereenvoudigde aanpak voorgesteld die aanleiding geeft tot een aanzienlijk

eenvoudiger volledig impliciet integratieschema zonder beduiden verlies van

accuraatheid. Een deel van dit proefschrift is gewijd aan de implementatie en

evaluatie van verscheidene numerieke integratieschema’s. Zowel impliciete als

expliciete schema’s werden uitgewerkt. De keuze voor een expliciet schema kan

gemotiveerd worden wanneer eenvoud van integratie en hoge simulatiesnelheid

gewenst zijn. Uiteraard blijft het vanzelfsprekend om de accuraatheid van de

uiteindelijke resultaten te beoordelen.

Karakterisering van het verstevigingsgedrag en van de anisotropie van een

dieptrekstaal kwaliteit DC06 wordt tevens beschreven in dit eindwerk. Optische

vervormingsmetingen gebaseerd op digitale spikkelcorrelatie worden vergeleken met

conventionele metingen op basis van een extensometer. Er werd aangetoond dat de

optische vervormingsmetingen een doorgedreven karakterisering van de evolutie van

anisotropie toelaten.

Volgend op deze experimentele beschouwingen, werd een evolutionair anisotroop

model ontwikkeld gebaseerd op niet-associatief Yld2000-2d. Deze is in staat om de

evolutie van versteviging en anisotropie als functie van plastische omvorming met

hoge nauwkeurigheid te simuleren. Een vereiste voor de parameteridentificatie van dit

model is de equivalentie van hoeveelheid plastische arbeid. Een belangrijk voordeel

van dit model is dat het kan geïmplementeerd worden op basis van om elk numeriek

integratieschema dat werd ontwikkeld voor niet-evolutionair en niet-associatief

vloeien. Er dient opgemerkt dat de beschrijving van de evolutie van anisotropie

(zowel wat betreft Lankford coëfficiënten als vloeispanningen) slechts zelden wordt

gedaan. Beschikbare modellen zijn meestal gebaseerd op experimentele gegevens

overeenkomstig een beperkt aantal oriëntaties en beschouwen meestal ofwel Lankford

coëfficiënten ofwel vloeispanningen voor een specifiek materiaal.

Samenvattend, focust dit proefschrift op de modellering van constitutieve

materiaalwetten en meer specifiek op niet-associatie anistroop materiaalgedrag. Er

kan besloten worden dat het niet-associatief Yld2000-2d anisotroop vloeimodel

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gecombineerd met een gemengd isotroop-kinematische verstevigingswet een

uitstekend model vormt voor de simulatie van omvormprocessen van sterk anisotrope

metalen. Implementaties op basis van verscheidene impliciete en expliciete

integratieschema’s werden uitgebreid vergeleken. Een vereenvoudigd niet-associatief

model werd voorgesteld en geëvalueerd. Tot slot werd een evolutionair niet-

associatief model uitgewerkt dat kan toegepast worden voor materialen waarvoor de

mate van anisotropie wijzigt tijdens de plastische omvorming.

Contents

1 Motivation and objectives ..................................................................................... 1 1.1 Introduction ................................................................................................. 2 1.2 Objectives and motivation ........................................................................... 3

1.2.1 Accurate description of severe anisotropy .............................................. 4 1.2.2 Combination of a strong anisotropic yield function with mixed

hardening definition ................................................................................ 4 1.2.3 Description of the evolution of anisotropy during the deformation

process.. ................................................................................................... 5 1.2.4 Implementation of advanced material models into commercial finite

element software ..................................................................................... 5 1.3 Overview of this dissertation ....................................................................... 6

2 Continuum plasticity: some basic concepts ........................................................ 11 2.1 Introduction ............................................................................................... 12 2.2 Stress and strain tensors ............................................................................ 12 2.3 Tensor invariants ....................................................................................... 14 2.4 Deviatoric stress ........................................................................................ 16 2.5 Hooke’s law............................................................................................... 17 2.6 Incompressibility hypothesis ..................................................................... 19 2.7 Various measures of strain ........................................................................ 19 2.8 Simple shear test as an example ................................................................ 22 2.9 Co-rotational rate of Cauchy stress ........................................................... 23 2.10 Summary ................................................................................................... 25

3 Hardening models ............................................................................................... 27 3.1 Introduction ............................................................................................... 28 3.2 Hardening of metals .................................................................................. 28 3.3 Isotropic hardening .................................................................................... 29

xii

3.4 Kinematic hardening ................................................................................. 31 3.4.1 Bauschinger effect ................................................................................ 32

3.5 Mixed isotropic-kinematic hardening ........................................................ 34 3.5.1 Transient effect ..................................................................................... 34 3.5.2 Permanent softening .............................................................................. 35 3.5.3 Work hardening stagnation ................................................................... 39

3.6 Physically based hardening models ........................................................... 40 3.7 Summary ................................................................................................... 41

4 Anisotropic yielding ........................................................................................... 45 4.1 Introduction ............................................................................................... 46 4.2 Introduction to yield functions .................................................................. 46

4.2.1 Lankford coefficient .............................................................................. 47 4.2.2 Associated flow rule ............................................................................. 48

4.3 Isotropic yield functions ............................................................................ 50 4.3.1 Tresca 1864 ........................................................................................... 50 4.3.2 von Mises 1913 ..................................................................................... 51 4.3.3 Hershey 1954 and Hosford 1972........................................................... 52 4.3.4 Barlat 1986 (Yld86) .............................................................................. 52

4.4 Anisotropic yield functions ....................................................................... 53 4.4.1 Hill’s family of yield functions ............................................................. 53 4.4.2 Barlat’s family of yield criteria ............................................................. 59 4.4.3 Banabic’s family of yield criteria .......................................................... 68 4.4.4 Cazacu and Barlat’s yield criteria ......................................................... 69 4.4.5 Polynomial yield functions ................................................................... 69

4.5 Summary ................................................................................................... 70 5 Non-associated flow rule .................................................................................... 77

5.1 Introduction ............................................................................................... 78 5.2 Limitations of associated flow rule (AFR) ................................................ 78 5.3 Non-associated flow rule (non-AFR) ........................................................ 79

5.3.1 Background ........................................................................................... 79 5.3.2 Concept ................................................................................................. 80

5.4 Non-AFR based yield models ................................................................... 82 5.4.1 Non-AFR version of Hill 1948 ............................................................. 82 5.4.2 Non-AFR version of Yld2000-2d ......................................................... 83 5.4.3 Evaluation of non-AFR models versus various AFR models ............... 83

5.5 Non-AFR and stability .............................................................................. 84 5.6 Summary ................................................................................................... 87

6 Fully implicit backward Euler integration scheme ............................................. 91 6.1 Introduction ............................................................................................... 92 6.2 Return mapping algorithm ......................................................................... 92 6.3 Elasto-plasticity with non-AFR ................................................................. 93

6.3.1 Kuhn-Tucker complementary criteria ................................................... 96

xiii

6.3.2 Elastic-plastic or continuum tangent modulus ...................................... 96 6.4 Stress-update algorithm ............................................................................. 98

6.4.1 Fully implicit backward Euler ............................................................... 98 6.4.2 Newton-Raphson iteration scheme ..................................................... 101 6.4.3 Algorithmic or consistent tangent modulus ........................................ 104

6.5 Summary ................................................................................................. 106 7 Validation and evaluation of the UMAT implementation of anisotropic yield

models ............................................................................................................... 109 7.1 Introduction ............................................................................................. 110 7.2 Validation of the developed UMAT subroutine ...................................... 110

7.2.1 Model validation in terms of hardening .............................................. 111 7.2.2 Model validation in terms of anisotropy ............................................. 116

7.3 Evaluation of various AFR and non-AFR models ................................... 117 7.3.1 Comparison of various hardening models ........................................... 117 7.3.2 Comparison of various anisotropic yield models ................................ 118 7.3.3 Spatial representation .......................................................................... 122 7.3.4 In-plane flow direction ........................................................................ 127

7.4 Cup drawing simulations ......................................................................... 128 7.5 Summary ................................................................................................. 134

8 Simplification of the numerical implementation of the non-AFR model .......... 137 8.1 Introduction ............................................................................................. 138 8.2 Discretization of rate elasto-plasticity equations ..................................... 138 8.3 Impact of simplification on equivalent plastic strain rate ........................ 141

8.3.1 Full non-AFR method ......................................................................... 141 8.3.2 Un-scaled simplified non-AFR method .............................................. 142 8.3.3 Scaled simplified non-AFR method .................................................... 145

8.4 Metrics for error analysis ........................................................................ 151 8.5 Impact of model simplification on cup drawing simulation .................... 155 8.6 Summary ................................................................................................. 156

9 Comparison of stress-integration schemes ........................................................ 159 9.1 Introduction ............................................................................................. 160 9.2 Integration schemes suitable for an explicit time integration FE code .... 160

9.2.1 Classical forward Euler’s method (CFE) ............................................ 160 9.2.2 Next Increment Corrects Error (NICE-h)............................................ 167 9.2.3 Convex Cutting-Plane (CCP) algorithm, a semi explicit approach ..... 168

9.3 Some remarks on using the explicit stress-update integration in implicit time

integration code. ......................................................................................... 172 9.4 Comparison of stress-update schemes ..................................................... 172 9.5 Summary ................................................................................................. 180

10 Evolutionary non-AFR anisotropic model ........................................................ 183 10.1 Introduction ............................................................................................. 184 10.2 Material characterization ......................................................................... 184

xiv

10.2.1 Experimental procedure .................................................................. 184 10.2.2 Hardening ....................................................................................... 186 10.2.3 Lankford coefficients ...................................................................... 190

10.3 Experimental observations on anisotropy evolution ................................ 191 10.3.1 Yield stress ..................................................................................... 191 10.3.2 Lankford coefficient ....................................................................... 196

10.4 Anisotropic models ................................................................................. 200 10.4.1 Non-AFR Hill 1948 model ............................................................. 200 10.4.2 Non-AFR Yld2000-2d model ......................................................... 204

10.5 Evolutionary anisotropic models ............................................................. 206 10.5.1 State of the art ................................................................................. 206 10.5.2 Evolutionary non-AFR Yld2000-2d model .................................... 207 10.5.3 Implementation using scaled simplified approach .......................... 217

10.6 Summary ................................................................................................. 218 11 Conclusions ....................................................................................................... 221

11.1 Summary and main conclusions .............................................................. 222 11.1.1 Accurate description of severe anisotropy ...................................... 222 11.1.2 Combination of strong anisotropic yield function with mixed

hardening definition ........................................................................ 223 11.1.3 Implementation of advanced material models into commercial finite

element software ............................................................................. 223 11.1.4 Description of the evolution of anisotropy during the deformation

process ............................................................................................ 224 11.2 Future works ............................................................................................ 224

Appendices

A Fully implicit backward Euler scheme .............................................................. 227 A.1 Newton-Raphson iteration scheme .......................................................... 227 A.2 Consistent tangent modulus ..................................................................... 231

B Parameter optimization ..................................................................................... 235 B.1 Introduction ............................................................................................. 235 B.2 Normalized yield stress ........................................................................... 236

B.2.1 Uniaxial direction ................................................................................ 236 B.2.2 Balanced biaxial condition .................................................................. 237 B.2.3 Out-of-plane direction ......................................................................... 238

B.3 Lankford coefficient ................................................................................ 238 B.3.1 Uniaxial direction ................................................................................ 238 B.3.2 Balanced biaxial condition .................................................................. 239

B.4 Error function .......................................................................................... 240 B.4.1 AFR Yld2000-2d model ..................................................................... 240 B.4.2 Non-AFR Yld2000-2d model ............................................................. 240

Publications............................................................................................................... 243

Chapter 1

1 Motivation and objectives

2 Chapter 1, Motivation and objectives

1.1 Introduction

Sheet metal plastic forming is the most common metal shaping process used in history

(Marciniak et al., 2002). Its range of applicability covers automotive, aerospace,

packaging, home appliances and marine. Historically, empirical methods were the

dominant approach of evaluating, testing and improving the sheet metal forming

processes. However, during the last three decades the application of finite element

simulations has seen an increasing trend in various steps of sheet forming processes

from design to testing (Zhou and Wagoner, 1995). This consequently demands further

improvements in terms of efficiency and accuracy. Efficiency of simulations has been

dramatically improved due to the advent of fast computational resources and parallel

calculation techniques. On the other hand, realistic material constitutive models which

are an indispensable part of any accurate simulation still need improvements due to

the widespread application of high-strength steel, aluminum alloys and complicated

forming processes.

The material constitutive model affects the prediction of forces, springback, final

shape, wrinkling and failure of the product. The evolution of instantaneous yield

stress during plastic deformation and the direction dependency of mechanical

properties are respectively dealt with by hardening and yield models. Yielding can be

described at different scales: microscopic or macroscopic (phenomenological).

Description of all microscopic phenomena which occur in a forming application is

obviously impossible (Barlat, 2007). Therefore a phenomenological model for

simulation of large plastic deformation is preferable.

Sheet metals exhibit either isotropic or anisotropic yielding behaviour. An isotropic

yield surface is assigned to a material with identical mechanical properties at different

orientations. Various isotropic yield functions are available such as von Mises (1913),

Tresca (1864), Hosford (1972) and Hershey (1954), Barlat and Richmond (1987),

Bishop and Hill (1951), Bassani (1977) and Budianski (1984). However, sheet metals

are prone to anisotropic (direction dependent) behaviour. This is because sheet metals

(generally) undergo severe plastic deformations during manufacturing processes such

as cold rolling. This introduces preferential orientations to the grains. Consequently,

the material obtains a direction dependent mechanical behavior. Material anisotropy

highly affects the distribution of stresses and strains and consequently the shape of the

final parts, their thickness and possible instabilities such as wrinkling for a deep

drawn part.

Starting from Hill’s quadratic anisotropy model (Hill, 1948), various yield functions

have been proposed to describe the initial anisotropy of metallic sheets such as Barlat

et al (1997; 2003; 2005; 2007), Banabic et al (2005), Cazacu and Barlat (2002, 2004),

Cazacu et al (2004; 2006), Hu (2007), Bron and Besson (2004), Karafillis and Boyce

(1993) and very recently Barlat et al (Barlat et al., 2011). The choice of yield function

3

depends on material, experimental constraints, required accuracy and the FEM code

used (Flores, 2006).

In order to accurately describe both yielding and plastic flow behavior of sheet

metals, the coefficients of the above anisotropic yield functions commonly need to be

optimized explicitly or iteratively from experimental tensile, shear or bi-axial yield

stresses and Lankford coefficients. During a metal forming process, the discussed

yield surface experiences translation and/or expansion. This, for instance, could be

described by incorporating an appropriate mixed isotropic-kinematic hardening

function.

As mentioned above, various constitutive models have been proposed to describe the

material behavior more accurately. However, more improvements can be expected.

Moreover, commercial finite element simulation software packages are not always

equipped with the state of the art material constitutive laws.

1.2 Objectives and motivation

The ambition of this dissertation is to contribute to the advancement of fundamental

knowledge in the field of sheet metal plastic forming for the own research group and

the entire research community. This comprises a thorough and critical review of

available models for description of hardening and anisotropic yielding, including

improvements of constitutive models and introducing such models into commercial

finite element simulation. Availability of such numerical tools could ultimately lead

to a reduction in lead time in the design of sheet products and manufacturing tools,

the optimization of process parameters and an increased quality of the final product.

To reach the global ambition, a number of specific goals have to be realized within

the scope of this dissertation:

Accurate description of severe anisotropy;

Combination of strong anisotropic yield functions with mixed hardening

definition;

Description of the evolution of anisotropy during the deformation process;

Implementation of advanced material models into commercial finite element

software.

All analytical and numerical routines discussed in the following chapters (UMAT and

VUMAT integration schemes, model parameter optimization, …) have entirely been

developed and implemented in the framework of this dissertation.


1.2.1 Accurate description of severe anisotropy

Traditionally, most anisotropic yield functions have been based on the associated flow

rule (AFR) hypothesis. This approach is based on the normality hypothesis that

describes the equality of plastic potential function (which determines the flow

direction) and yield function (which determines the transition from the elastic to the

plastic regime). Several studies evaluated the accuracy of AFR based yield functions

for the description of various levels of anisotropy. For instance, Yoon et al (2007)

reported that the quadratic Hill (1948) and non-quadratic Yld2000-2d (Barlat et al.,

2003) yield functions can only predict 4 ears for a deep drawn cup made of an

aluminium alloy AA2090-T3 which exhibits 6 ears in experiments. Cvitanic et al

(2008) reported similar limitations for the same material based on Karafillis–Boyce

(1993) anisotropic yield function. Therefore it can be concluded that it is difficult to

describe a highly anisotropic material by means of a model in which an identical

formulation for yield function and plastic potential function is used (Stoughton,

2002).

During the last decade, more attention has been paid towards the development and

implementation of non-AFR based models for metal plasticity and more research

studies tend to make use of the advantages of this approach (Stoughton, 2002;

Stoughton and Yoon, 2004; Stoughton and Yoon, 2006; Yoon et al., 2007; Cvitanic et

al., 2008; Stoughton and Yoon, 2008, 2009). This approach removes the artificial

constraint based on which plastic flow direction and yielding are determined from

identical functions and thus two separate functions are considered. This research will

present an extensive literature study of both AFR and non-AFR based models and will

provide a systematic comparison of various models based on these two opposite

points of view.

1.2.2 Combination of a strong anisotropic yield function with mixed

hardening definition

Various metal forming operations experience reversed loading conditions during

which phenomena such as Bauschinger effect, transition, stagnation and permanent

stress shift can be observed in the hardening curve. Upon tool removal (e.g. removal

of punch in a deep drawing process), this highly influences the elastic recovery and

thus the final shape of the product. Isotropic or kinematic hardening definitions alone

are not capable of accurately describing the mentioned phenomena. Therefore, a

combination of an advanced description of anisotropy and an appropriate mixed

isotropic-kinematic hardening definition is essential. Among the most accurate

hardening models we can name the two-surface models of Yoshida and Uemori

(2002) and the physically based model of Teodosiu and Hu (1995). These models

have been proven to yield accurate prediction of elastic recovery. Nonetheless, the

5

degree of complexity of implementation and parameter optimization of these models

roars in accordance with the degree of capabilities of the models. Therefore, it seems

advantageous to seek for a mixed hardening model with good accuracy and

convenient formulation. For instance, Taherizadeh et al (2010) combined a non-AFR

Hill 1948 yield function with Chaboche mixed isotropic-kinematic hardening law.

However, some phenomena observed in the hardening curve at load reversal cannot

be accurately predicted by the classical Chaboche model.

1.2.3 Description of the evolution of anisotropy during the deformation

process

In simulation of sheet metal forming processes based on phenomenological models, it

is a general conception that the material anisotropy, in terms of Lankford coefficient

or yield stress directionality, is preserved during plastic deformation. In other words,

it is assumed that the anisotropic behaviour is insensitive to the level of plastic

deformation. Literature shows that initial anisotropy in terms of both r-value and yield

stress evolves during plastic deformation (Hu, 2007; Hahm and Kim, 2008).

Description of the texture evolution during plastic deformation by means of crystal

plasticity is however a common approach (Li et al., 2003; Duchene and Habraken,

2005). Various researchers attempted to make use of the simplicity of

phenomenological models for description of distortional hardening. For instance,

Stoughton and Yoon (2005), Abedrabbo et al (2006a, b), Hu (2007) and Zamiri and

Pourboghrat (2007). However there are limitations assigned to any of these models. In

short, combination of high accuracy and simplicity still has to be improved in an

evolutionary model.

1.2.4 Implementation of advanced material models into commercial finite

element software

The majority of commercial finite element codes is not equipped with state of the art

constitutive models. For instance, the commercial finite element code Abaqus still

lacks a strong non-quadratic anisotropic yield model. Even more, only the classical

model of Chaboche is the most advanced built-in hardening model in this software

package. Therefore, knowledge of implementation of more advanced material models

into finite element code is essential. Considering those commercial finite element

codes which provide more built-in advanced material models, access to and reception

of material codes from their developers are not generally possible. Concurrent with

the advancements in computational resources, more complex material models are

developed for simulation of more complex deformation processes and material

behaviors. In other words, simulation time remains a challenge. Therefore a thorough


comparison of simple and complex integration schemes with respect to computational

speed and simulation accuracy is critical.

1.3 Overview of this dissertation

Chapters 2 to 4 are dedicated to the modelling of material mechanical models

representing hardening and anisotropy, and constitutes the bibliographic part of this

manuscript.

Chapter 2 describes the prerequisites of continuum plasticity. This chapter is meant as

an introduction for less experienced readers. The majority of the discussed concepts

are repeatedly referred to in later chapters.

In chapter 3, concepts of various hardening models are discussed. Although the focus

of this work is on the modelling of anisotropy, a thorough understanding of hardening

is indispensable. A review of isotropic, kinematic and mixed isotropic-kinematic

hardening models is given and their capabilities are presented.

Chapter 4 describes the concept of anisotropy. The associated flow rule (AFR)

hypothesis is described and a comprehensive review of various AFR based

anisotropic yield models is given. Advantages and disadvantages of different models

are described.

Chapter 5 is the transition from the bibliographic part to the author’s main

contributions. It introduces the concept of non-AFR in detail; drawbacks of AFR are

discussed, as well as advantages of non-AFR. The choice for a non-associated flow

rule framework is motivated and two non-AFR based models are defined and

explained in detail. These non-AFR versions of Hill 1948 and Yld2000-2D will be

used in the remaining part of the PhD work.

Chapters 6 to 10 discuss the original contributions of this dissertation, such as

implementation and evaluation of several numerical integration schemes, and

experimental observation and simulation of distortional hardening. Chapter 6

elaborates on the elasto-plastic formulation of a general non-AFR anisotropic model

combined with a recently proposed mixed hardening. The fully implicit backward

Euler integration scheme is introduced and its implementation into a user material

subroutine for Abaqus is discussed.

Chapter 7 provides a comprehensive verification of the implemented non-AFR

anisotropic yield model including mixed hardening. The finite element

implementation of this model was further evaluated based on cup deep drawing

simulations.

In chapter 8 a simpler numerical implementation of the non-AFR model is discussed.

The proposed simplification is advantageous when a complex mixed isotropic-

7

kinematic hardening model is considered. The effect of the simplification on the

accuracy of the model responses is comprehensively evaluated in this chapter.

Chapter 9 describes and compares various stress integration schemes (both explicit

and implicit). Accuracy of developed models in a deep drawing simulation is

examined.

In chapter 10 experimental observations of distortional hardening are discussed. An

evolutionary anisotropic model is introduced and its accuracy is evaluated based on a

comparison with experimental results.

Chapter 11 summarizes the main dissertation findings and gives some perspectives

with respect to future research.

In appendix A, the fully implicit backward Euler scheme for non-AFR anisotropic

yield and mixed hardening model is discussed in detail.

Finally in appendix B, the optimization schemes for parameter calibration of AFR and

non-AFR based general non-quadratic anisotropic models are discussed.


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Chapter 2

2 Continuum plasticity: some basic

concepts

12 Chapter 2, Continuum plasticity: some basic concepts

2.1 Introduction

This chapter introduces some basic concepts of continuum plasticity required for an

in-depth study on constitutive modeling which follows in the next chapters. Different

notational conventions for stress and strain tensors, various strain definitions with an

example for a simple shear test, basics of kinematics of large deformations, and

concepts such as deviatoric stress and zero plastic dilatancy are described. Moreover,

isotropic elasticity formulation for both three-dimensional and plane stress problems

is provided. Comprehensive reference books written by Khan and Huang (1995),

Belytschko et al. (2000), Crisfield (2000; 2001) are recommended for an in-depth

study.

2.2 Stress and strain tensors

The true stresses and true strains corresponding to an arbitrary set of orthogonal

Cartesian axes can be described in either tensor or vector (Voigt) notations.

Considering the tensorial notation for a three-dimensional stress and strain state we

can write

[

] (2-1a)

[

] (2-1b)

The tilde sign is used as notation for second order tensor. For plane stress conditions

these change to

[

] (2-2a)

[

] (2-2b)

Considering plane stress conditions, the imposed constraints to stress and strain

components in normal (through thickness) direction give rise to difficulties in the

numerical integration algorithm. Therefore an alternative notation is commonly used.

The above 3*3 tensors reduce to 2*2 tensors and the strain in normal direction is

explicitly updated based on the in-plane components at the end of the numerical

integration scheme (see section 2.5). Consequently this reduced notation is presented

by

13

[

] (2-3a)

[

] (2-3b)

Using Voigt notation is advantageous over tensorial notation for describing plasticity

equations and, moreover, it is the convention used in the commercial finite element

(FE) software ABAQUS. However, attention has to be paid to the fact that the shear

strain in Voigt notation is the engineering shear strain and is twice the tensorial

shear strain value.

(2-4)

In addition, the order of stress and strain vector components in ABAQUS/Implicit is

inconsistent with that in ABAQUS/Explicit. For three-dimensional elements in

ABAQUS/Implicit the stress and strain storage schemes respectively take the

following forms

[ ] (2-5a)

[ ] (2-5b)

In case of three-dimensional elements in ABAQUS/Explicit

[ ] (2-6a)

[ ] (2-6b)

For plane stress problems the following representations apply for ABAQUS/Implicit

[ ] (2-7a)

[ ] (2-7b)

Considering ABAQUS/Explicit, the stress and strain storage scheme is slightly

different such that

[ ] (2-8a)

[ ] (2-8b)

It is observed that the tensorial shear strain component is stored and not engineering

component.


2.3 Tensor invariants

Tensor invariants of a symmetric tensor are characteristics that are not changed by

any linear operator such as a transformation to a different reference system (Lubliner,

1990).

Fig 2-1 illustrates a degrees rotation of one sheet metal orthotropic frame (xx-yy) to

another orthotropic frame (11-22). The transformations of stress and strain tensors (in

Voigt notation) between these two Cartesian reference bases are respectively defined

in Eqns. (2-9a) and (2-9b)

Fig 2-1: Illustration of a degrees rotation of a Cartesian reference base.

[

] [

] [

] (2-9a)

[

] [

] [

] (2-9b)

Considering the tensorial notation, for instance using (2-3a), the (clockwise) rotated

stress tensor

is calculated by

(2-10)

where the “.” denotes dot product of two second order tensors (i.e.,

) and

the rotation matrix

is

[

] (2-11)

The principal stress invariants are given by

22

11

yy (transverse)

xx (rolling)

15

( )

(

( ))

(2-12)

( )

where the “:” denotes double contraction of two second order tensors (i.e.,

). Einstein summation convention for repeated indices has been used. The stress

components , and are the principal stresses.

For plane stress problems the stress invariants are obtained as

(2-13)

For both three dimensional and plane stress problems the principal invariants (

and ) can be encapsulated in the characteristics equation described in Eqn.(2-14).

The roots of the cubic and quadratic characteristics equations, respectively for 3D and

2D plane stress problems, define the eigenvalues (principal stresses). The

characteristics equation is described by

(

) (2-14)

With being the principal stresses and for 3D problems and for 2D

problems. The tensor is the second order identity tensor

[

] (2-15)

and are orthonormal bases and is the Kronecker symbol

{

} ( 2-16)

The symbol denotes tensor product (dyadic product) such that

(2-17)


Recasting principal invariants into the polynomial characteristics equation for third

rank tensor (3D problems and solid elements) results in

(2-18)

And for plane stress problems

(2-19)

2.4 Deviatoric stress

According to experimental tests carried out by Bridgman (1947, 1952), the

hydrostatic stress has no influence on plastic deformation and yielding of the material.

Hence, many yield criteria use deviatoric stress instead of Cauchy stress. This is done

either by direct input of deviatoric stress instead of Cauchy stress or by using Cauchy

stress as the input and transforming it to deviatoric stress within the yield function as

is the case for well-known yield models proposed by Barlat et al. (1997; 2003; 2005;

2007) that will be discussed later in section 2.5.

Deviatoric stress is determined by

(2-20)

and

(2-21)

where is the hydrostatic stress.

An alternative way to determine deviatoric stress from Cauchy stress using matrix

operation in Voigt notation is

(2-22)

where T is a transformation matrix. For plane stress and three-dimensional problems T

respectively becomes

[

] (2-23a)

17

[ ]

(2-23b)

2.5 Hooke’s law

At very small strains, the stress-strain behavior of metals is almost reversible and

linear. Beyond this elastic range, the material is in the so-called yielding condition

(elastic-plastic condition) in which permanent or inelastic deformations occur. Using

additive decomposition of elastic and plastic strain rates, the strain increment is split

into elastic and plastic components

(2-24)

where

,

and

are increments of respectively total strain, elastic strain and

plastic strain. This additive decomposition is unambiguously valid for infinitesimal

strain tensors. For finite strains, it must be noted that the elastic and plastic strain rates

of deformation are additively decomposed only when the elastic strain is defined by

an appropriate logarithmic strain measure, and is considerably smaller than the plastic

one (Dunne and Petrinic, 2005). Using Hooke’s law, the increment of Cauchy (true)

stress (

) is a function of elastic strain increment in both elastic and elastic-plastic

regions. Hooke’s law is written as

(2-25)

where the fourth order symmetric tensor

is the elasticity tensor. The generalized

relation for a multi-axial stress state is

(2-26)

With and being Lame constant and shear modulus respectively. Tensor

is the

fourth order identity tensor (Simo and Hughes, 1998).

(2-27a)

(2-27b)

[ ] (2-27c)


where and are elastic modulus and Poisson’s ratio.

The elasticity tensor in Eqn.(2-25) can be redefined to Eqns.(2-28a) and (2-28b)

respectively for three-dimensional and plane stress problems

[

]

(2-28a)

[

] (2-28b)

From Eqns.(2-7b) and (2-28b), one may notice that the through thickness strain value

is not directly incorporated in the definition of elasticity and strain tensors for plane

stress problems. That is due to the fact that the most convenient approach to deal with

the zero stress and non-zero strain in the thickness direction in plane stress problems

(e.g. using conventional shell elements) is an explicit definition of elastic-plastic

strains in the through thickness direction based on the in-plane strains. Through

thickness elastic strain is determined by the in-plane strains and stresses respectively

in Eqns. (2-29a) and (2-29b).

(2-29a)

(2-29b)

On the other hand, based on the assumption of constancy of volume the through

thickness plastic strain increment can be expressed as

(2-30)

Finally, from Eqns.(2-29b) and (2-30) the total through thickness strain increment

becomes

( 2-31)

The reason for using incremental values instead of total values is that the incremental

values are determined by the numerical algorithms and subsequently these increments

are added to their corresponding values from previous steps within the algorithm.

19

2.6 Incompressibility hypothesis

Experimental observations of Bridgman (1947, 1952) showed that plastic deformation

takes place with negligible volume change. Accordingly, zero permanent volume

change or so-called zero (or negligible) plastic dilatancy has been the basic

assumption for almost all metal plasticity theories (Khan and Huang, 1995).

In equation form, zero plastic dilatancy states that during plastic deformation

( )

(2-32)

2.7 Various measures of strain

Different definitions for strain exist in continuum plasticity. To this end, let us define

the concept of deformation gradient that is essential for definition of different strains.

First, consider an imaginary continuum body consisting of particles continuously

distributed in space. Fig 2-2 presents this continuum body at both undeformed (initial)

and deformed (current) states. At the deformed state the material has undergone rigid

body movement, rigid body rotation and stretch due to external force. As seen in Fig

2-2, the infinitesimal vector embedded in the undeformed state is transformed to

in the deformed state. The stretch and rigid body rotation can be described by

the second order deformation gradient F.

(2-33a)

(2-33b)

(2-33c)

In other words, the deformation gradient is a linear operator that relates every

infinitesimal line such as in the initial state to the corresponding infinitesimal line

in the deformed state (Khan and Huang, 1995). The deformation gradient does

not include rigid body translation.


Fig 2-2: A continuum body in undeformed state undergoing deformation to the

deformed state.

According to the polar decomposition theorem, the second order tensor of

deformation gradient can be decomposed into the product of a rotational tensor

(orthogonal) and a stretch tensor (symmetric positive-definite) such that

(2-34)

and

being respectively right and left stretch tensors and

is the rotation tensor.

is an orthogonal tensor in a sense that

(

and

and are

symmetric in a sense that

and

. Right and left stretch tensors are

related to each other by (Simo and Hughes, 1998)

(2-35a)

(2-35b)

So-called right and left Cauchy-Green tensors are respectively defined as

(2-36a)

(2-36b)

21

As opposed to the tensor

which is not necessarily symmetric, the

tensor is

always symmetric. Right and left Cauchy-Green tensors are related to each other by

(Simo and Hughes, 1998)

(2-37a)

(2-37b)

The following expression can be derived to relate the reverse of left and right Cauchy-

Green tensors to the deformation gradient (as will be required for different definitions

of strain later in this section)

(

)

(2-38a)

(

)

(2-38b)

The tensor

is also called Finger deformation tensor (Lubliner, 1990).

From Eqns. (2-36a) and (2-36b) one may notice that for a rigid body motion (in terms

of both translation and rotation) when the right and left stretch tensors become

identity tensors, the Cauchy–Green tensors give non-zero strain values.

Therefore proper definitions of strain that solely depend on stretch, such that zero

strain is obtained for rigid body motions, are preferable. Accordingly, alternative

strain measures such as Euler-Almansi, Green-Lagrange and true strains have been

defined.

The Euler-Almansi strain is defined as

(2-39)

The Green-Lagrange strain

(

) (2-40)

The true (or logarithmic or Hencky) strain

(2-41)


Consequently, these strain measures result in zero strain for the case of rigid body

movement.

2.8 Simple shear test as an example

The aim of this section is to determine the true strain value using deformation

gradient tensor for a simple shear test on an element. Fig 2-3 schematically presents

the initial and the deformed element shape.

Fig 2-3: Schematic drawing of a simple shear test.

For a simple shear test the deformation gradient is obtained as

[

] (2-42)

where

(2-43)

for small deformations. From Eqn. (2-36b) the right Cauchy-Green tensor can be

written as

[ ] (2-44)

Finally, the Green-Lagrange strain tensor and its corresponding shear strain

component respectively are

(

)

[ ] (2-45a)

(2-45b)

dInitial state

b Deformed state

Y

X

xx

yy

xy

23

Considering the true strain tensor, as shown in Eqn. (2-41) a natural logarithm

operator has to be applied to the reverse of the left Cauchy strain tensor determined in

Eqn. (2-46).

[ ] (2-46)

However, to perform an operation on a tensor, the operation has to be applied on the

eigenvalues or principal values. In other words, the tensor has to be transformed into

its principal coordinates by finding the eigenvectors and eigenvalues and then

applying the operator (natural logarithm in this case). Transforming the tensor back to

its former coordinate system will complete the operation. Therefore Eqn. (2-41) is

recast into

(

)

(2-47)

where the matrix

and diagonal matrix

respectively contain the eigenvectors

(principal directions) and the eigenvalues (principal strains) of

.

Diagonal matrix

[

( √ )

( √ )

] (2-48)

The matrix M with the eigenvectors is

[

( √ )

( √ )

] (2-49)

Finally, the true shear strain value is calculated as follows

( √ ) [

( √ )]

√

( √ ) [

( √ )]

√ (2-50)

2.9 Co-rotational rate of Cauchy stress

A tensor

is called objective or frame indifferent when it rotates as follows

(2-51)


where

is the rotated matrix. However, as opposed to the Cauchy stress ( the rate

of Cauchy stress is not objective as shown in

(2-52a)

(2-52b)

To solve this fallacy, an objective stress rate is required to properly take the material

rotation into account for calculation of Cauchy stress tensor. Various definitions for

objective (corotational) stress rates exist such as Jaumann, Green-Naghdi and

Truesdell. The Jaumann and Green-Naghdi stress rates are used in Abaqus solvers as

shown in Table 1 (Hibbitt Karlsson and Sorensen Inc).

Table 1 Objective stress rates in Abaqus solvers Solver Element type Constitutive model Objective rate

Abaqus/Standard

Solid (Continuum) All built-in and user-defined

materials Jaumann

Structural (Shells, Membranes,

Beams, Trusses)

All built-in and user-defined

materials Green-Naghdi

Abaqus/Explicit

Solid (Continuum) All except viscoelastic, brittle

cracking, and VUMAT Jaumann

Solid (Continuum) Viscoelastic, brittle cracking,

and VUMAT Green-Naghdi

Structural (Shells, Membranes,

Beams, Trusses)

All built-in and user-defined

materials Green-Naghdi

To determine the Jaumann and Green-Naghdi stress rates some important concepts

concerning the kinematics of deformation have to be defined. First, consider the time

rate of change of the deformation gradient:

(

)

( 2-53)

and are respectively the velocity gradient and rate of deformation gradient, and

is the velocity. The velocity gradient can be decomposed into a symmetric and an

anti-symmetric (skew) part.

( 2-54)

with the symmetric part

and the skew part

being respectively the rate of

deformation and the continuum spin.

( 2-55a)

25

( 2-55b)

Finally, the Jaumann rate of Cauchy stress

is determined by

( 2-56)

The Green-Naghdi stress rate

is

( 2-57)

where

( 2-58)

It must be noted that all the quantities in the remainder of this work are measured with

regard to the orthotropic co-rotational reference frame. In addition, in this dissertation

the conjugate stress of logarithmic strain, true stress, is accepted as the only definition

of stress, unless stated otherwise.

2.10 Summary

In this chapter we discussed some basic concepts of continuum plasticity. These

concepts will be used repeatedly in next chapters. For instance, the concept of

deviatoric stress is used for the definition of anisotropic yield functions. Hooke’s law

is used for elasto-plastic formulations. The incompressibility hypothesis is used for

the derivation of through thickness strain that is needed for calculating Lankford

coefficients. Regarding the stress and strain definitions, we always use true values in

the remainder of this dissertation. However, shear strain is always considered as its

engineering value unless stated otherwise. No further effort is needed to convert the

corotational rate of Cauchy stress from its Cauchy value. This is due to using shell

elements in Abaqus finite element code. In other words, when using shell element in

this code, the coordination system (in which stress and strain are defined) always

coincides with the material orthotropic orientations.


Bibliography









1009-1039.



896.

Belytschko, T., Liu, W.K., Moran, B., 2000. Nonlinear finite elements for continua

and structures. John Wiley, Chichester ISBN 0471987735

Bridgman, P.W., 1947. The Effect of Hydrostatic Pressure on the Fracture of Brittle

Substances. J. Appl. Phys 18.

Bridgman, P.W., 1952. Studies in large plastic flow and fracture with special

emphasis on the effects of hydrostatic pressure, 1st ed. McGraw-Hill, New York,.

Crisfield, M.A., 2000. Non-linear finite element analysis of solids and structures

Wiley, Chichester ; New York.

Crisfield, M.A., 2001. More plasticity and other material non-linearity-II, Non-linear

finite element analysis of solids and structures. Wiley, Chichester, UK pp. 158–164.

Dunne, F., Petrinic, N., 2005. Introduction to computational plasticity. Oxford

University Press, Oxford ; New York.

Hibbitt Karlsson and Sorensen Inc, Abaqus theory manual ABAQUS User's Manuals

Version 6.10. Pawtucket, Rhode Island, USA.

Khan, A.S., Huang, S., 1995. Continuum theory of plasticity. Wiley, New York.

Lubliner, J., 1990. Plasticity theory. Macmillan and Collier Macmillan, New York.

Simo, J.C., Hughes, T.J.R., 1998. Computational inelasticity, Interdisciplinary applied

mathematics. Springer, New York ISBN 0387975209

Chapter 3

3 Hardening models

28 Chapter 3. Hardening models

3.1 Introduction

This chapter explains the concept of hardening and presents a review of commonly

used phenomenological hardening models in the framework of quasi-static loading

conditions. Common features of load reversals are described and capabilities of

different hardening models in modeling these features are discussed. In Section 3.2

the concept of hardening is defined. In Section 3.3 and 3.4 isotropic and kinematic

types of hardening are discussed respectively. Due to the fact that none of these

models alone is sufficiently accurate to describe hardening behaviour, in Section 3.5

the concept of mixed hardening is defined and various classic and advanced mixed

hardening models are discussed. Lastly in Section 3.6, a physically based hardening

model is briefly discussed.

3.2 Hardening of metals

The dislocation glide or slip on crystallographic planes and directions is known to be

the reason of work hardening in metals at low temperatures (Kocks et al., 2000). As

deformation proceeds, a gradual lattice rotation is caused by the dislocation slip and

next the dislocations accumulate at microstructural obstacles resulting in an increase

in the slip resistance for further deformation. This increase in slip resistance during

plastic deformation is characterized as the hardening of the material (Cardoso and

Yoon, 2009). The part of the stress-strain curve representing plastic deformation is

also called the flow curve. Because of hardening the flow stress increases with the

total amount of plastic dissipation or a corresponding measure of accumulated plastic

strain. When the material is unloaded, the instantaneous flow stress becomes the new

yield stress (Barlat, 2007).

The total hardening behavior can be decomposed into two basic types of hardening

that respectively deal with translation and expansion of the yield surface. The

motivation for such consideration is that an accurate description of hardening upon

load reversal is highly necessary due to its significant effects on stress and residual

stress distributions, which are important in calculating springback and loading

capacity of sheet metal parts. Distortional hardening is yet another type of hardening

behavior based on which the shape of the yield surface evolves during plastic

deformation. In Chapter 10, distortional hardening is experimentally evaluated and an

evolutionary hardening formulation is proposed.

The isotropic hardening function, , determines a uniform increase of the yield

surface’s size in all directions (Fig 3-1 -b.) and the kinematic hardening function

(back-stress), , translates the yield surface in stress space (Fig 3-1 -c.). An isotropic

hardening function cannot describe the distinct features of a cyclic hardening curve.

Translation of the yield surface was experimentally observed in sheet metals

subjected to cyclic loading (Zhao and Lee, 2001). However, if only a kinematic

hardening function is used, both hardening in reversed loading and springback are

29

underestimated. This effect is reversed if only isotropic hardening is considered.

Therefore a proper combination of isotropic and kinematic hardening functions,

which simultaneously accounts for both expansion and translation of the yield

surface, is indispensable to predict the stress-strain state of a material under both

proportional and reversed loading conditions. A combination of isotropic and

kinematic hardening functions is further referred to as a mixed isotropic-kinematic

hardening function (Fig 3-1 -d.). The concepts of isotropic, kinematic and mixed

isotropic-kinematic hardening are depicted in Fig 3-1 .

Fig 3-1 Concept of isotropic, kinematic and mixed isotropic-kinematic hardening.

3.3 Isotropic hardening

Using an isotropic hardening model is reasonably effective to simulate monotonous

processes in which the load direction does not change (Chung et al., 2005). Several

commonly used isotropic hardening laws are presented in the following.

Ludwick (1909) (3-1a)

Hollomon (1944) (3-1b)

( ) Voce (1948) (3-1c)

Swift (1947) (3-1d)

Prager (1938) (3-1e)

These hardening laws make use of the following material parameters: is the initial

yield stress, K is a strength coefficient, is a strain hardening exponent, Q is a

q

d)

c)b)

Mixed isotropic-kinematic hardening

Kinematic hardeningIsotropic hardening

yy

xx

xx

yy

xx

yy

xx

yy

Initial yield surfacea)

-q


saturation parameter with equal to the saturated value of stress at very high

plastic strain, is the rate of saturation, the initial plastic strain and

E is the elastic modulus.

Among the isotropic hardening functions given in Eqns. (3-1a) to (3-1e) the Voce and

Swift laws have been widely used as hardening definitions for sheet metals. The Voce

hardening law is mostly used for steels which generally exhibit a saturating hardening

behavior in a sense that the rate of increase of stress decreases with additional plastic

deformation and turns into zero at very high plastic strains. On the other hand, the

Swift hardening law is preferred for most aluminium alloys that exhibit a non-

saturating hardening behavior. A combined Swift-Voce (CSV) law incorporates both

saturating and non-saturating hardening functions through the use of a weighting

factor . This can result in an excellent fit to both steels and aluminium alloys.

The combined Swift-Voce model is expressed as

( ( )

) ( ( )) (3-2)

where is the weighting factor.

The Swift, Voce and CSV laws are compared in Fig 3-2 for an interstitial free DC06

deep drawing steel. The experimental data correspond to a uniaxial tensile test on a

specimen with a 15° orientation to the rolling direction.

Fig 3-2 Hardening curves predicted by Swift, Voce and CSV laws for DC06 deep

drawing steel.

It must be noted that the reason that a proper isotropic hardening function generally

gives appropriate results for applications involving monotonic loading is that the

functions presented in Eqns. (3-1a) to (3-1e) are capable of providing a suitable fit to

31

the uniaxial tensile stress-strain curve for different orientations with respect to the

rolling direction.

Many metal forming processes involve reversed loading (e.g. imposed by draw beads

in deep drawing) or other non-proportional loading conditions (e.g. imposed by a

sequence of stamping processes). In case of using an isotropic hardening function for

such load reversal conditions, the hardening in reversed loading is overestimated

resulting in an exaggeration of the predicted springback and residual stresses (Zhao

and Lee, 2002). Using a kinematic hardening function might be an alternative. This is

discussed in the following section.

3.4 Kinematic hardening

Load reversal is a common phenomenon in sheet metal forming operations such as

bending and unbending at die shoulder and punch in a deep drawing or stamping

process (Zang et al., 2011). Fig 3-3 shows a forward and reverse deep drawing

simulation which involves the load reversal phenomenon (Yoon et al., 2004). It is

well known that upon load reversal the Bauschinger effect, transient behavior and

some degree of stress shift (permanent softening) can be observed. In addition, for

mild steel sheets, the abnormal evolution of the hardening curve (stagnation behavior)

may be observed (Yoshida and Uemori, 2002), Fig 3-4. These phenomena, that are

explained in following sections, cannot be predicted by an isotropic hardening

function.

Forward deep drawing (first step)

Forward deep drawing (second step)

Reverse deep drawing (first step)

Reverse deep drawing (second step)

Fig 3-3 Finite element simulation of forward and reverse deep drawing tests proposed


at NUMISHEET’99 conference by Yoon et al (2004).

3.4.1 Bauschinger effect

This feature observed for several metals and their alloys appears due to the micro-

residual stresses existing in the material resulting from strain incompatibilities

between grains during hardening (Barlat, 2007). The Bauschinger effect is

macroscopically characterized by a reduced yield stress of the material upon load

reversal, typically tension followed by compression (Cardoso and Yoon, 2009). For

instance, yielding upon load reversal happens at instead of as shown in Fig 3-4. It

has been reported that the Bauschinger effect can change the magnitude of springback

by a factor of two (Wagoner et al., 2012). A linear kinematic hardening function takes

this Bauschinger effect into account. However, using a non-linear kinematic

hardening component improves the shape of the hardening curve (Thuillier and

Manach, 2009).

Fig 3-4 Schematic of hardening curve for a reversed loading condition.

The Bauschinger effect can be quantified by the Bauschinger ratio which is calculated

as (Lee et al., 2007)

(3-3)

where is the flow stress or instantaneous yield stress at the beginning of load

reversal and is the initial flow stress after load reversal (i.e., and are stress

values respectively corresponding to and in Fig 3-4.). In addition, the size and the

c

a<2 2

TransientStagnatio n

Permanent

softening

33

translation of the yield surface upon reversed loading can be evaluated by the

isotropic ( ) and kinematic hardening values ( ) respectively

(3-4a)

(3-4b)

To predict the Bauschinger effect, Prager proposed the first linear kinematic

hardening model with one material parameter (Prager, 1956)

(3-5)

In Prager’s model, the yield surface translates in the same direction as the plastic

strain rate due to the fact that the evolution of back-stress (

) is collinear with that of

plastic strain. Due to this linearity, difficulties appear when this model is used for

cyclic loading which involves load reversals (Chaboche, 1986). In addition, as

described by Ziegler, Prager’s model does not generate consistent results for 2D and

3D cases (Ziegler, 1959).

To overcome this inconsistency, later, Ziegler proposed another linear kinematic

hardening function in which the yield surface translates radially from the center

(Ziegler, 1959)

(3-6)

denotes the yield function (or in other words the size and shape of the yield surface)

and is a hardening parameter.

As they use only one parameter in the kinematic hardening function, the Prager and

Ziegler models cannot describe the hardening non-linearities which play an important

role in the mechanical behavior.

The Bauschinger effect is reproduced by various classical hardening models such as

Mroz’s multi surface theory (1967, 1969), and the two surface models of Dafalias and

Popov (1976) and Krieg (1975) and Armstrong and Frederick (1966). Major

difference between these models is the definition of the generalized plastic modulus

i.e.,

in

(Ristinmaa, 1995). In Mroz’s model the translation direction and

the generalized plastic modulus are defined at the first step and then by applying the

consistency rule (which accounts for preserving the stress on the yield surface) the

magnitude of translation of the yield surface is determined. In the Armstrong-

Frederick model, on the other hand, the direction and magnitude of translation of the


yield surface are determined first and then through the consistency condition the

plastic modulus is calculated (Geng and Wagoner, 2002).

The next section focuses on mixed hardening models that can account for hardening

non-linearities.

3.5 Mixed isotropic-kinematic hardening

3.5.1 Transient effect

From the above-mentioned hardening non-linearities, the transient behavior is the

smooth transition from the elastic to the plastic region in the hardening curve with a

rapid change of strain-hardening rate (segment corresponding to in Fig 3-4).

Classical hardening models proposed by Armstrong and Frederick (1966) and by

Chaboche (Chaboche, 1986, 1991) consider this evanescent strain memory effect

observed in cyclic loadings by adding a relaxation term (also called recall term) to

Ziegler’s linear kinematic hardening function, both of which evolve independently.

The Armstrong–Frederick type non-linear kinematic hardening functions dominated

before 2005 and remain prevalent (Wagoner et al., 2012). To improve the prediction

of both Bauschinger effect and transient behavior, the Armstrong–Frederick

kinematic hardening function combined with an isotropic hardening function was

proposed and used by different authors (Chung et al., 2005; Cao et al., 2009).

The Armstrong–Frederick kinematic hardening function is defined by

(3-7)

where and are hardening parameters and

. Integrating this model for

uniaxial loading gives

⁄ ( ) (3-8)

where ⁄ and denote the saturated value of back-stress and rate of saturation,

respectively.

Later Chaboche (Chaboche, 1986) proposed superposing multiple Armstrong &

Frederick functions as one model which may result in a considerable improvement of

results in most cases. In this case

∑

(3-9)

35

where N is the number of back-stress components. The Chaboche kinematic

hardening function is commonly used with the Voce isotropic hardening law. The

Voce isotropic hardening function is written as

(3-10)

with

(3-11)

where the superposed dot denotes the rate of a variable. Assuming uniaxial loading,

the Voce model becomes

( ) (3-12)

In cyclic loading conditions the Voce isotropic hardening stabilizes to with a

saturation rate of after a certain number of cycles (Chaboche, 1986). In the Voce

model the yield surface expansion rate decreases monotonically, which properly

generates a convex hardening curve for stable materials. For materials with non-

saturating hardening behavior the Voce model can better be replaced by other

hardening laws such as the Swift law. Assuming the Chaboche kinematic hardening

function with Voce isotropic hardening law for a uniaxial tensile test results in

⁄ ( ) (3-13)

3.5.2 Permanent softening

The permanent softening or the permanent stress offset, i.e. between and in Fig

3-4, is the feature following after transient behavior. This effect is caused by partial

dissolution of dislocation cell walls performed during forward deformation (Zang et

al., 2011). The magnitude of the permanent softening is known to change with the

magnitude of pre-strain (Hahm and Kim, 2008).

According to Chaboche (1986), the Armstrong–Frederick models can also be

considered as a two-surface model. Subsequently, to include the permanent softening

effect, the bounding surface describing the limiting state of stress, can evolve

according to a kinematic or mixed isotropic-kinematic hardening model, resulting into

two-surface models proposed by Geng and Wagoner (2000; 2002), Yoshida and

Uemori (2002, 2003) and Lee et al. (2007). Considering the model proposed by Geng

and Wagoner (2000) with three material parameters, the bounding surface develops

according to a mixed hardening law to expand and translate simultaneously. Fig 3-5

schematizes the concept underlying the Geng-Wagoner hardening model. In this

model, the evolution of the yield surface resembles that of the Armstrong–Frederick


function with the recall term replaced by a vector connecting the centers of the yield

and bounding surfaces. Moreover, the transient behavior is modeled via a translation

of the yield surface and the permanent softening effect is produced by the bounding

surface evolution.

Fig 3-5 Schematic representation of the Geng-Wagoner hardening model (Geng and

Wagoner, 2002).

One of the computational drawbacks of the two-surface model proposed by Geng and

Wagoner (2000) is the procedure for updating the distance of the stress state point (a)

and the mapping point (A) on the bounding surface shown in Fig 3-6, leading to

inconsistencies (overshooting problems) for complex loading conditions (Khan and

Huang, 1995).

Fig 3-6 A schematic view of the two-surface model and two gap distances.

This problem typically appears when the material is elastically unloaded and then

reloaded again, (Lee et al., 2007), as illustrated in Fig 3-7. To avoid the overshooting

problem, Lee et al. (2007) proposed a practical two-surface model that can

incorporate general anisotropic yield surfaces as well as the combined isotropic and

kinematic hardening for both yield and bounding surfaces. They defined a reverse

δ

OO

σΣ

A

a

Bounding

surface

Yield surface

σyy

σxx

37

loading criterion such that the new initial gap distance, the initial in Fig 3-6, is

updated only when the reverse loading criterion is satisfied (Lee et al., 2007).

Fig 3-7 The overshooting problem in the two-surface model (Lee et al., 2007).

Having introduced the two-surface models, to capture the permanent softening in the

framework of a one-surface cyclic model, Chun et al (2002a) proposed a modification

to the isotopic hardening part of the mixed isotropic-kinematic hardening function

with two term Chaboche model. In their model the second Chaboche term is

described by a Ziegler’s linear kinematic hardening. This second term turns into zero

in case of load reversal thus requiring a loading/unloading criterion in Chun’s model.

A loading/unloading criterion for a general plane-stress case can be formulated using

the stress tensors at the previous and current time steps. For example, the state of the

loading at the current step is defined as reversal if the angle between these two stress

tensors is between 90° and 270° (Chun et al., 2002b). Nonetheless, Zang et al (2011)

described that this loading criterion must be carefully formulated otherwise an

incorrect stress might be obtained. An improved version of Chun’s model is discussed

in the next section.

Recently, Zang (2011) inspired by the work of Chun (2002a), proposed a one-surface

cyclic hardening function that predicts the Bauschinger effect, transient behaviour and

permanent softening. In addition, in his model the loading/unloading criterion is no

longer required. Analogously to the Chaboche model, Zang’s model generates the

saturating hardening after some cycles (Fig 3-8).


Fig 3-8: Cyclic loading generated using Zang’s model. The saturating hardening

behavior is properly generated at a high number of cycles.

In Zang’s model the isotropic hardening function is described in the following form

( )

⁄ ( ) (3-14)

It is noticed that the first and second terms in the right side of the Eqn.(3-14)

constitute the Voce isotropic hardening law, and the third term is the integrated form

of the one-term Chaboche kinematic hardening function for a uniaxial loading

condition.

The kinematic hardening model consists of a two-term Chaboche function in which

the first term is a non-linear Armstrong and Frederick function and the second term is

Ziegler’s linear kinematic hardening.

(3-15a)

(3-15b)

(3-15c)

where , and are material parameters. The non-linear term in Eqn. (3-15b) is

solely associated to the transient behavior. The second term, Ziegler’s linear

kinematic hardening law in Eqn. (3-15c), is used to generate a constant stress offset. It

must be noted that considering Zang’s mixed hardening model, the introduction of the

modified isotropic hardening law Eqn.(3-14) improves the capability of the

39

constitutive model in modelling transient behaviour since it causes the back-stress

to be only associated to the transient behavior (Zang et al., 2011). The term can be

extended to better describe saturating behaviour (Zang et al., 2011). For uniaxial

loading, Zang’s model can be integrated to

( ) (3-16)

where denotes back-stress in uniaxial tension. For uniaxial compression, due to the

change of stress direction, we have

(3-17)

being back-stress during reversed loading. In general, the integrated form of the

tensor ( during uniaxial and reversed loading can be written as

⁄ (

⁄ ) ( ) (3-18)

where gives the flow direction (i.e. +1 and -1 respectively denote forward

and reversed loading) and

and respectively denote the value of and

accumulated plastic strain at the onset of load reversal (Chaboche, 2008). Similarly,

for the second term of kinematic hardening we write

(3-19)

with

the value of at the onset of load reversal. Considering the load reversal

( , the combination of Eqn.(3-18) and Eqn.(3-19) gives the total kinematic

hardening denoted by

⁄ (

⁄ ) ( )

(3-20)

3.5.3 Work hardening stagnation

This phenomenon appears as an abnormal shape in the hardening curve in the

reversed loading condition, corresponding to the segment in Fig 3-4. This

effect is very apparent in mild steels whereas it vanishes for high strength steels and is

less pronounced in most aluminium alloys (Yoshida and Uemori, 2002). It has also

been found that the magnitude of the stagnation phenomenon depends on the

magnitude of pre-strain, i.e. the length of the stagnation plateau increases with an

increase in pre-strain. To the best of our knowledge, this phenomenon can only be


described by the two-surface model of Yoshida and Uemori (2002, 2003) and the

dislocation-based microstructural model of Teodosiu and Hu (1995).

Yoshida and Uemori (2002, 2003) proposed a two-surface hardening model that

gained considerable popularity mainly due to the relatively low number (6) of model

parameters to be determined. More importantly, their model can predict work

hardening stagnation as well as transient behaviour, Bauschinger effect and

permanent softening. In Yoshida-Uemori model, the yield surface only moves

kinematically within a bounding surface of which the evolution is controlled by a

mixed isotropic–kinematic hardening model. This translation of the yield surface is

obtained by the superposition of two non-linear kinematic hardening functions

leading to modeling the Bauschinger effect and transient behavior. In addition, the

isotropic hardening of the bounding surface describes the global work hardening.

Permanent softening and work hardening stagnation are reproduced by the kinematic

hardening and non-isotropic hardening region of the bounding surface. Despite its

high efficiency, limiting the equivalent plastic strain to the von Mises definition is a

theoretical drawback of this model.

As opposed to the two-surface models of Geng and Wagoner, overshooting is no

longer a problem in the Yoshida-Uemori model. That is because the relative

translation of the yield surface with regard to the bounding surface is controlled by a

function of the difference between the sizes of the two surfaces and it saturates to the

gap between the two surfaces.

3.6 Physically based hardening models

Nonetheless the considerable capabilities of the Yoshida-Uemori model in prediction

of features of cyclic hardening curves, it cannot describe the behavior of the

hardening curve for more complex industrial processes such as multi-step operations.

Therefore, due to the fact that hardening is essentially due to the dislocation

microstructure and its evolution (Haddag et al., 2007), the physically-based models

can be expected to even more enhance the predictive capabilities of hardening

models. The dislocation-based microstructural model proposed by Teodosiu and Hu

(Teodosiu and Hu, 1995) shows significant strength in prediction of all the

aforementioned features in cyclic loading as well as those particularly observed in

case of orthogonal strain paths. This is due to the introduction of physically-motivated

internal variables that describe the evolution of the persistent dislocation structures

(Haddag et al., 2007). Fig 3-9 presents a comparison of the level of predictability of

the models of Chaboche and of Teodosiu and Hu during a load path change. The

significance of using physically based models is pointed out in works of Bouvier et al

(Bouvier et al., 2005; Bouvier et al., 2006), Flores et al (Flores et al., 2007) and

Haddadi et al (Haddadi et al., 2006).

41

Fig 3-9 Comparison between Teodosiu and Hu’s microstructural hardening model and

Chaboche’s classical model for a reverse shear test on mild steel at 10% and 30% pre-

strain (Haddag et al., 2007).

3.7 Summary

In this chapter we presented the common phenomena observed in the hardening curve

upon load reversal. It is assumed that the yield surface expands and translates during

proportional and reversed loading respectively. This translation and expansion of

yield surface is taken into account by splitting the hardening curve into kinematic and

isotropic components. We provided a brief review of various hardening models

among which the Chaboche model has been preferred for use in finite element

simulations for a long time. A modification to the mixed hardening of Chaboche-

Voce as suggested by Zang et al (2011) was discussed. In chapter 5, this model will

be combined with two anisotropic yield criteria. The review of hardening models in

this chapter also covers models of Geng and Wagoner (2000), Yoshida and Uemori

(2002) and Teodosiu and Hu (1995). In Table 3-1, the capabilities of discussed

hardening models with respect to the prediction of different phenomena of cyclic

loading are provided. In addition, we add a newly published work of Barlat et al

(2012) for comparison in that table. His model is a homogeneous anisotropic

hardening model that captures all the phenomena discussed in this chapter. Due to

complexity of this model, we only describe the capabilities of this model in Table 3-1.

Table 3-1 Characteristics of various hardening models

Bauschinger

effect

Transient

Behavior

Permanent

softening

Work hardening

Stagnation

Voce, Swift and CSV - - -

Prager (1956) - - -

Ziegler (1959)

Chaboche (1986) - -

Zang (2011) -

Geng-Wagoner (2002) -

Yoshida-Uemori (2002)

Teodosiu and Hu (1995)

Barlat (2012)


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Chaboche, J.L., 2008. A review of some plasticity and viscoplasticity constitutive

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Chapter 4

4 Anisotropic yielding

46 Chapter 4, Anisotropic yielding

4.1 Introduction

Sheet metal forming simulations rely on an accurate description of the anistropic

yielding behavior. This chapter starts with a brief introduction on yield functions and

an overview of some well-known isotropic yield functions. The main part of this

chapter presents a review of various anisotropic yield functions. Classical yield

functions developed by Hill, more modern yield functions developed by Barlat,

Banabic and Cazacu, and some polynomial-based models are described. Hill’s and

Barlat’s families of yield functions are described in detail because of their

acknowledged contributions to the development of new anisotropic yield functions

and also because of the large popularity of these models for finite element

simulations. Section 4.2 introduces the concepts of Lankford coefficient, normality

hypothesis and associated flow rule. In section 4.3, various isotropic yield functions

are described. Then section 4.4 extensively discusses various anisotropic yield

functions. For instance, sections 4.4.1 and 4.4.2 discuss the Hill and Barlat families of

anisotropic yield functions respectively. In addition, in section 4.4.3, the contribution

of Banabic in introducing new non-quadratic yield functions is explained. At the end,

models of Cazacu and some polynomial anisotropic models are discussed in sections

4.4.4 and 4.4.5.

4.2 Introduction to yield functions

Aggregates of single crystals constitute the crystallographic structure of most metals.

Considering a single crystal, considerable anisotropy of mechanical properties such as

different yield stresses at different orientations is observed. The mechanical

anisotropy at crystal level turns into isotropy at macro-scale level in a polycrystalline

aggregate with a sufficiently random distribution of crystal orientations (Neto et al.,

2008). In other words, the average behavior of all single crystals represents the total

material behavior. Thus an isotropic yield function seems to be a sufficient

assumption for the description of macroscopic behavior for finite element simulations.

However, sheet metals undergo severe plastic deformations during manufacturing

processes such as cold rolling. This introduces a preferential orientation of the grains.

Therefore isotropy is no longer the appropriate assumption to represent the

mechanical behaviour of a rolled sheet metal. Moreover, the anisotropic behaviour

has been known to have a great influence on the shape of the specimen after the

deformation. Earing at the rim of a deep drawn part is an example of distinct

anisotropic behaviour.

Focusing on the material constitutive models in general and more specifically on the

yield function, there are two major approaches to describe this behaviour for

polycrystalline materials. The first approach is crystal plasticity and the second one is

the phenomenological approach. In the first approach, the behaviour of one grain or a

distribution of grains is used to describe the polycrystalline behaviour (Arminjon,

47

1991; Van Houtte, 1994; Gambin and Barlat, 1997). In the phenomenological

approach, on the other hand, the average behaviour of all grains determines the global

material behaviour. According to Barlat (1991), using a phenomenological yield

function has advantages over its microstructure based equivalent. For instance (Barlat

et al., 1991):

a) They are easy to implement in FEM and lead to fast computation ;

b) They can describe global anisotropy whereas microstructure based models

account for crystallographic texture;

c) They are easy to adapt for different materials.

Many phenomenological yield functions have been successfully proposed for use in

finite element codes to simulate the isotropic or anisotropic mechanical behaviour of a

material. The different yield functions generally make use of different combinations

of yield stresses and Lankford coefficients to represent a multi-dimensional surface

determining the transition between elastic and plastic deformation. First some basic

concepts needed for the formulation of yield functions are introduced.

4.2.1 Lankford coefficient

Anisotropy is generally described on the basis of the Lankford coefficients (also

called r-values) and/or the yield stresses along the orthotropic (rolling and transverse)

and diagonal directions of the metallic sheets. The Lankford coefficient at any

orientation with respect to the rolling direction is determined as the ratio of width to

through thickness plastic strain (increments). Due to practical difficulties in

measuring the through thickness plastic strain in sheet metals, this value is

conventionally determined using the incompressibility hypothesis of metals. Lankford

coefficient ( ) at degrees from the rolling direction (RD) and through thickness

plastic strain increment (

) are given by

(4-1a)

(4-1b)

Two definitions based on Lankford coefficients, which are widely used in industry are

normal and planar anisotropy respectively given in Eqns. (4-2a) and (4-2b)

(4-2a)

(4-2b)


Table 4-1 Anisotropy coefficients for various metals

Material

Deep drawing steels (DC01-DC07) 1.30 to 2.00 Up to 0.70

Stainless steel 0.70 to 1.10 -0.25 to 0.20

TRIP steels 0.90 -0.03

Aluminum alloys 0.60 to 0.80 -0.60 to -0.15

Copper 0.60 to 0.80 -

Brass 0.60 to 1.00 -

Zinc alloys 0.20 to 0.60 -

Titanium alloys 2.00 to 8.00 Up to 4

A higher normal anisotropy, , results in more resistance against thinning and is thus

preferable for deep drawing applications. A higher planar anisotropy, , can be

observed by more pronounced earing in a deep drawn cup. Planar and normal

anisotropy coefficients for various materials are presented in Table 4-1 (Grote and

Antonsson, 2008).

4.2.2 Associated flow rule

The foundation of most anisotropic yield functions has been based on the Associated

Flow Rule (AFR) hypothesis which states that the flow rule is associated with the

yield criterion. In AFR based models, the yield function is also the potential for

plastic strain rate. In other words, the AFR hypothesis reflects the normality rule

based on which the gradient of a continuously differentiable yield function determines

the direction of plastic strain rate. Before proceeding, let us define yield criterion and

material orthotropy. The yield criterion is given by

( ) ( ) (4-3)

, and respectively denote yield criterion, yield function and isotropic

hardening. In the light of sheet metal orthotropy we have (Soare et al., 2008)

( ) ( ) (4-4a)

( ) ( ) (4-4b)

And for plane stress conditions

( ) ( ) (4-5)

The normality hypothesis is shown in Fig 4-1. In this figure the stress components are

normalized with respect to yield stress in rolling direction.

49

Fig 4-1 Concept of normality in AFR ( is yield function;

is Cauchy stress;

is

plastic strain rate direction).

The normality rule is given by

(4-6)

where

(4-7)

is the plastic multiplier factor (compliance) to be determined by using a loading-

unloading criterion, and the second order tensor

is the plastic strain rate direction in

AFR approach. According to Bishop and Hill (1951), the normality hypothesis was

theoretically valid for polycrystalline materials. In addition, Hecker (1976) described

that the normality hypothesis is reasonable for most single phase like materials based

on an extensive review of experimental yield surface results.

Moreover, the AFR hypothesis was strengthened by experimental observations of

Bridgman (1947, 1952). He performed a series of tensile tests on metals in the

presence of very high hydrostatic pressure and noticed that this pressure had no

influence on the yielding of the material. In addition, a negligible permanent volume

change was shown to exist (Khan and Huang, 1995). Due to the absence of pressure

sensitivity in the plastic deformation, only the deviatoric stress is involved in the

formulation of the yield function. On the other hand, the zero plastic dilatancy (zero

permanent volume change) will not be violated by using the same formulation for

yield function and plastic potential function (equivalence of yield and plastic potential

functions).


Accordingly, under the assumption of AFR and in the light of material orthotropy,

starting from Hill’s quadratic anisotropy model (Hill, 1948), various yield functions

have been proposed to describe the initial anisotropy of metallic sheets. Examples are

Karafillis and Boyce (1993), Barlat et al (1989; 1991; 1997; 2003; 2005; 2007),

Cazacu and Barlat (2002, 2004), Bron and Besson (2004) , Banabic et al (2005) ,

Vegter and van den Boogaard (2006), Cazacu et al (2004; 2006), Hu (2007), and very

recently Aretz and Barlat (2012). In order to accurately describe both yielding and

plastic flow of sheet metals, the coefficients of anisotropic yield functions commonly

need to be optimized explicitly or iteratively from experimentally determined tensile,

shear and/or bi-axial yield stresses and Lankford coefficients. It is worth noting that in

the last two decades the isotropic plasticity equivalent theory generalized by Karafillis

and Boyce (1993) has been a popular approach in the development of new yield

functions. Recently, Soare and Barlat (2010) have proved that these orthotropic yield

functions obtained through a linear transformation method are homogeneous

polynomials, which brings potential benefits for numerical implementation and

development of new yield functions. In the following section, various yield functions

are briefly described. Hill 1948 and Yld2000-2d functions are described in detail as

they form the basis of further investigation of non-AFR in following chapters.

4.3 Isotropic yield functions

4.3.1 Tresca 1864

The Tresca isotropic yield function proposed in 1864 is known to be the oldest yield

criterion. Tresca assumed that the material undergoes plastic deformation at a critical

shear stress. In terms of principal stresses the Tresca yield function becomes

[| | | | | |] (4-8)

Under plane stress conditions

| |

| |

| |

(4-9)

In Fig 4-2 two- and three dimensional representations of the Tresca yield function are

presented.

51

Fig 4-2 Tresca isotropic yield surface in normalized stress space.

4.3.2 von Mises 1913

The von Mises isotropic yield function was proposed in 1913 and is the second oldest

yield function. The von Mises isotropic yield function is widely used and given by

(

)

(4-10)

where is the deviatoric part of

. Fig 4-3 compares the 2D representations of Tresca

and von Mises yield functions. For plane stress conditions

√

(4-11)

Fig 4-3 Tresca and von Mises isotropic yield surfaces in normalized stress space.


4.3.3 Hershey 1954 and Hosford 1972

An identical non-quadratic isotropic yield function was independently proposed by

Hosford (1972) and Hershey (1954)

| |

| |

| |

(4-12)

where m is a constant that depends on crystallographic structure. For FCC and BCC

materials m is respectively 8 and 6. The Hosford yield locus resembles the shapes of

von Mises and Tresca when m is equal to 2 or infinity respectively. It can be seen that

the presence of shear stress is not accommodated in this model. Hosford (1985)

unsuccessfully attempted to add shear stress to his in-plane isotropic model. Since the

model was not based on stress tensor invariants he only obtained a proper in-plane

isotropic function when (Barlat and Lian, 1989).

4.3.4 Barlat 1986 (Yld86)

Barlat and Richmond (1987) generalized the Hosford isotropic yield function and

proposed a plane stress isotropic yield function that takes the shear stress components

into account. This was achieved by using stress invariants and instead of

principal stresses in the yield function. Their so-called Yld86 yield function is defined

as

| |

| |

| |

(4-13)

where

(4-14a)

(( )

)

(4-14b)

This model showed excellent agreement with the Bishop and Taylor yield surface

obtained for isotropic FCC metals (Barlat and Richmond, 1987). It can be observed

that when the stress components coincide with the material orthotropic directions the

Yld86 function reduces to the Hosford function.

Besides the discussed models, there are more isotropic yield functions such as Bishop

and Hill (1951), Bassani (1977) and Budianski (1984). For instance, using the model

of Bishop and Hill (1951), the isotropic yield function for FCC and BCC metals are

different and lie between the yield loci of Tresca and von Mises.

53

4.4 Anisotropic yield functions

4.4.1 Hill’s family of yield functions

Hill proposed various anisotropic yield functions among which the 1948 version is

still commonly used mainly due to its simplicity and user friendliness. Therefore the

anisotropic yield function Hill 1948 is described in detail in the following.

4.4.1.1 Hill 1948

One of the first phenomenological anisotropic yield functions was proposed by Hill

(1948). Von Mises (1928) had already proposed an anisotropic yield function but for

single crystals. Hill’s first anisotropic yield model is a generalization of the von Mises

criterion and due to its quadratic nature can predict two or four ears for a deep drawn

cup. The parameters of the Hill 1948 quadratic function can either be calibrated using

directional plastic strain ratios (referred to as r-based Hill 1948) or using directional

yield stresses (referred to as S-based Hill 1948).

The S-based Hill 1948 model requires uniaxial yield stresses corresponding to rolling

direction, diagonal direction, transverse direction as well as the balanced biaxial yield

stress, respectively written as and . The balanced biaxial yield stress can

be determined by a viscous pressure bulge test (VPB) or a cross tensile test. The S-

based Hill 1948 yield function is defined as

(

)

(4-15)

, and are yield function parameters. These parameters can be directly

determined as follows

(

)

( (

)

(

)

) (4-16)

(

)

(

)

Eqn.(4-15) can be recast into a tensorial form which is more suitable for determining

the first and second order gradients (these are used in the calculations involved in a

fully implicit backward Euler integration scheme which will be described in next

chapters). Accordingly, Eqn.(4-15) is written as

( ⁄

) (4-17a)


[

] (4-17b)

Then the first derivative is obtained by

(4-18)

As mentioned above, implementing the implicit backward Euler integration scheme

requires the derivative of

with respect to

. This can be calculated as follows

(Crisfield, 1997)

⁄

(

) (

)

(4-19)

Due to the fact that the parameters of the yield function are directly calibrated using

input yield stresses, the predicted r-values can be obtained as closed form

expressions. So

(

)

(

)

( ) (4-20)

(

)

(

)

Or in general for any orientation with respect to the rolling direction the

corresponding r-value is obtained by

( )

( ) ( ) (4-21)

The Lankford coefficient for the balanced biaxial stress condition ( ) is defined as

(4-22)

For S-based Hill 1948 function this Lankford coefficient becomes

(4-23)

55

The directional yield stress at an angle with respect to the rolling direction is also

determined in a closed form expression as

√ ( ) (4-24)

For a balanced biaxial stress condition the yield stress ( ) is

√

(4-25)

The r-based Hill 1948 yield function follows the same formulation as the S-based but

with different inputs for parameter identification. The parameters of this function are

calibrated using the Lankford coefficients corresponding to rolling direction, diagonal

direction and transverse direction respectively denoted by and .

(

)

(4-26)

denotes the r-based Hill 1948 yield function and , and are material

parameters.

( )

( )

(4-27)

( )( )

( )

The normalized yield stresses can be calculated as follows

(

)

( )

( )( )

(4-28a)

(

)

( )

( )

(4-28b)

(

)

( )

(4-28c)

By changing the subscript y to p the Eqns.(4-21) to (4-25) can be applied for the r-

based Hill 1948 yield function.

As seen above, the Hill 1948 formulation is simple and follows a very convenient

approach for parameter identification. As already mentioned above this model can

predict only 2 or 4 ears. More importantly, the r-based and S-based versions are very


weak respectively in prediction of yield stresses and Lankford coefficients. Besides,

accurate prediction of , the Lankford coefficient at balanced biaxial state, cannot be

guaranteed (Safaei et al., 2012b).

The Hill 1948 function is widely used for steel applications. Considering aluminium

alloys, this yield criterion reveals the inability of predicting first and second

anomalous behaviours. The first anomalous behavior means that for metals exhibiting

normal anisotropy it can be experimentally observed that

(Woodthorpe and Pearce, 1970). Experimental tests performed by Woodthorpe and

Pearse (1970) for rolled aluminium, showed that was always around 1.1 and

the Lankford coefficient varied between 0.5 to 0.6. However, for the same

condition, Eqn.(4-28c) results in

(4-29)

In other words, for , the Hill 1948 yield locus falls inside the von Mises

ellipse.

The other weak point of Hill 1948 is that it cannot represent the second anomaly

meaning that when then and vice-versa. This is due to Eqn.

(4-28b) that leads to

then and reciprocal (4-30)

4.4.1.2 Hill 1979

To overcome the improper prediction of a yield locus inside that of von Mises for

metals possessing so-called first anomalous behaviour, Hill proposed a non-quadratic

yield function in 1979.

| |

| |

| |

| |

| |

| |

(4-31)

For seven parameters (including ) exist that have to be determined by seven

experimental inputs such as both Lankford coefficient and yield stress at rolling,

diagonal and transverse directions and one additional input (Lankford coefficient or

yield stress) corresponding to a combined loading condition. Hill showed that there

are many possible parameter combinations for which the prediction of first anomalous

behaviour can be ensured. For instance, considering planar isotropy (normal

anisotropy) the ratio of balanced biaxial yield stress to uniaxial yield stress along the

rolling direction becomes

57

(

)

( )

(

( )( )

) (4-32)

According to Eqn.(4-32) the first anomalous behaviour can be well accommodated

under conditions and and also when . Note that as

opposed to the Hosford model (see Section 4.3.3) the parameter is no longer

associated to the crystallographic structure and can even be a non-integer.

Hill described that, depending on the area of application, not all terms of Eqn.(4-31)

are required and it can be truncated. For instance, different versions for plane stress

conditions can be derived from Eqn.(4-31) among which the case referred to as case

IV (Hill, 1979) with the constraints is the most widely used and

is defined as

| |

| |

(4-33)

The model parameters of this function are calibrated based on experimentally

determined normal anisotropy

(4-34a)

(4-34b)

For a balanced biaxial stress state this leads to

(

)

(4-35)

From Eqn.(4-35) it can be derived that the prediction of first anomalous behaviour is

accommodated when for and for .

Moreover, analogously to Hill 1948, the Hill 1979 function has a simple format and

provides an analytical expression for effective plastic strain. However, the case in

which

and

cannot be described by Hill 1979 and even more, the

inclusion of shear stress is absent in this model. In other words, this yield function

works only when the directions of principal stress coincide with the material

orthotropic directions.

4.4.1.3 Hill 1990

Hill proposed a new plane stress yield function in 1990 (Hill, 1990) to take into

account the presence of shear stress, i.e. when the directions of principal stress do not

coincide with the material orthotropic directions.


( ) | |

|( )

|

|

|

[ (

) ( )

] (4-36a)

and

( ) ( ) (4-36b)

where is yield stress in pure shear state. The value of m depends on yield stress at

45°, balanced biaxial yield stress and Lankford coefficient at 45° according to

( ( )) ( ) (4-37)

Coefficients a and b can be expressed as analytical expressions on the basis of yield

stresses:

(

)

(

)

(4-38a)

|(

)

(

)

| (

)

(4-38b)

Hill also described that the coefficients a and b can alternatively be expressed in

terms of Lankford coefficients. However, Lin and Din (1996) concluded that the yield

surface predicted by Eqns.(4-38a) and (4-38b) is in better agreement with that

modeled by Taylor’s crystal plasticity theory.

Hill 1990 yield function can represent the first and second order anomalous

behaviours. In addition, the in-plane yield stresses and Lankford coefficients are well

predicted, and a higher number of experiments are incorporated for parameter

calibration as compared with previous functions proposed by Hill. However, this

model requires improvements in terms of user friendliness and simulation time (Lin

and Ding, 1996).

4.4.1.4 Hill 1993

Besides the improvements of the Hill 1990 yield function, there are drawbacks shared

by all yield functions proposed by Hill. For instance the condition

unavoidably results in

. However, the presence of materials that exhibit both

and

has been experimentally proven (Stout and Hecker, 1983; Safaei

et al., 2012a). Therefore, a yield function cannot predict both directional yield stresses

and Lankford coefficients unless it can incorporate all associated experimental inputs

59

in its parameter identification procedure. Accordingly, Hill (1993) introduced a new

yield criterion that takes and into account in the analytical

description of its parameters c, p and q.

[( )

( )

]

(4-39)

Hill 1993 can be described as a simple and user-friendly model that has the ability of

modeling first and second order anomalous behaviour by possessing high flexibility

due to the number of experimental input data incorporated. Nonetheless, the inclusion

of shear stress is absent in this model. Moreover, the variation of in-plane Lankford

coefficients and yield stresses cannot be modeled due to taking rolling and transverse

directions as the only in-plane orientations for experimental inputs. The predicted

yield surface has poor agreement with polycrystalline theory of Taylor or Bishop-Hill

(Banabic, 2009).

4.4.2 Barlat’s family of yield criteria

4.4.2.1 Yld89

Following the isotropic yield function Yld86, Barlat and Lian (1989) proposed a new

anisotropic yield function (so-called Yld89) based on a linear transformation of stress

tensors. In this approach, the linearly transformed stress tensors are substituted in an

isotropic yield function, for instance the Hosford 1972 function. This approach has

been shown to be very effective and is extensively discussed in a paper of Barlat et al

(2007).

The Yld89 yield function can be considered as a generalization of the Hosford 1972

function. On the one hand, the Hosford 1972 model provides convexity for .

On the other hand, applying linear transformation has the advantage of preserving the

convexity (Eggleston, 1958; Rockafellar, 1972). Therefore the Yld89 yield function is

unconditionally convex for .

Since unconditional convexity became a weakpoint of Barlat’s next anisotropic model

(Yld96 in Section 4.4.2.3). Convexity is required for the stability of the yield surface

such that it assures uniqueness of strain rate for any given stress state (Drucker, 1951,

1959). In other words, a yield function is convex only when it presents a smooth

surface with no vertex. In addition, we know that a yield function is convex when

its Hessian matrix is positive semi-definite; that means its eigenvalues are positive or

zero. The fourth order Hessian matrix

is obtained by

(4-40)


The Yld89 yield function is defined by

| |

| |

| |

(4-41)

where

(4-42a)

(( )

)

(4-42b)

and , , and are material parameters. The parameter simply defines

the ratio of to . The exponent is associated to the crystallographic structure,

as in Hosford 1972, and equals 6 and 8 respectively for BCC and FCC metals. For

preserving the convexity the following conditions must be met

and (4-43)

The first approach for identification of parameters , and is taking the uniaxial

yield stress at 90° and two independent critical shear stresses and such that

respectively the stress tensors [ ] and [ ]

are imposed to the material (Barlat and Lian, 1989). However, experimental tests to

obtain and are difficult to perform. Therefore Barlat and Lian (1989) proposed

using Lankford coefficients as an alternative due to the convenience of their

experimental measurement. In this case, and will be identified explicitly based

on and . The coefficient , however, needs to be calculated by means of

numerical iteration such that the discrepancy between predicted and experimentally

measured values is minimized. Based on Yld89 function in Eqn.(4-41) the

Lankford coefficient is predicted by

( ) (4-44)

and are components of the derivative of the yield function for a uniaxial

direction at 45° with regard to the rolling direction. Interestingly the Yld89 functin

can be reduced to Tresca and Hill 1948 yield functions when equals 2 and infinity,

respectively.

From the advantages of the Yld89 function one can point out the easy parameter

identification except for p, and suitable results for moderately anisotropic metals

(Geng and Wagoner, 2002). As disadvantages, the poor prediction of balanced biaxial

yield stress for highly anisotropic metals and the requirement of numerical treatment

61

for finding parameter p can be mentioned. More importantly, the variation of in-plane

yield stresses and Lankford coefficients cannot be simultaneously predicted.

4.4.2.2 Yld91

Barlat in 1991, proposed a generalized version of his previous model to consider

three-dimensional problems (Barlat et al., 1991). The Yld91 yield function is defined

as

| |

| |

| |

(4-45)

and are eigenvalues of the symmetric transformed stress tensor

[

] (4-46)

where

( ) ( )

( ) ( )

( ) ( )

(4-47)

and are function parameters. The constants and should be

calculated by Newton-Raphson iteration from uniaxial tensile yield stress data at 0°,

90° and normal (through thickness) direction. Shear test results are required for the

remaining parameters. The parameter should be for convexity but in practice

. The Yld91 function can be reduced to a plane stress yield function in which

only and remain, which can be optimized based on yield stress data obtained

from uniaxial tensile tests at 0°, 45° and 90° and from a through thickness disk

compression test. The viscous pressure bulge test or cross tensile test can be used as

an alternative for through thickness disk compression test.


The yield surfaces predicted by the Yld91 model are in good agreement with those

obtained by Taylor and Bishop-Hill. This model is also easy to implement in FEM

code. However, calculation of strain rate (normality in AFR) is found to be lengthy

but at least straightforward (Barlat et al., 1991). The Yld91 model was found to result

in a poor yield stress prediction for aluminium-magnesium (Al-Mg) alloy sheets

which underwent a high cold rolling reduction (e.g. 80% cold reduction) prior to

solution heat treatment (Barlat et al., 1997). This drawback is shown in Fig 4-4.

Fig 4-4 Yield surface for material Al-2.5%Mg, 150 m grain size, 80% cold reduction

before solution heat treatment (annealing). Experimental data, Taylor-Bishop- Hill

polycrystalline and Yld91 yield function predictions (Barlat et al., 1997).

4.4.2.3 Yld94 and Yld96

To remedy the inability of the Yld91 function to accurately predict the yield surface

of Al-Mg alloys sheets specifically in pure shear conditions ( ), Barlat et al

(1997) proposed the Yld94 yield function that improved the prediction of pure shear

yield stress without affecting other in-plane yield stresses (see Fig 4-5).

| |

| |

| |

(4-48)

and are eigenvalues of the symmetric transformed stress tensor

.

where are functions of ( ) direction cosines between the

principal axes of anisotropy and the principal axes of .

(4-49)

For plane stress problems, six material parameters and (embedded in the

definition of the transformed stress tensor ) where must be determined by

Newton-Raphson iteration from experimental mechanical tests giving

63

and . Barlat mentioned that the Yld94 function does not always

improve the yield surface prediction compared with the Yld91 function. Even though

that in-plane yield stresses can be predicted accurately, the variation of in-plane

Lankford coefficients lacks considerable accuracy.

Fig 4-5 Yield surface for material Al-2.5%Mg, 150 m grain size, 80% cold reduction

before solution heat treatment (annealing). Experimental data, Taylor-Bishop-Hill

polycrystalline, Yld91 and Yld94 yield function predictions (Barlat et al., 1997).

To improve the performance of the Yld94 function, Barlat et al (1997) proposed

another yield function (called Yld96) for both three dimensional and plane stress

problems in which the are no longer constant but depend on the orientation of the

principal axes with respect to the material orthotropic orientations. Considering the

plane stress case, there are seven material parameters that should be experimentally

identified and . Assuming all material parameters equal to

unity, the conditions and respectively correspond to

Tresca and von Mises isotropic yield functions. The higher value results in a

decrease of the radius of curvature of the rounded vertices near the uniaxial and

balanced biaxial stress states, Fig 4-6.


Fig 4-6 Variation of Yld96 yield surface with respect to exponent m (from outermost

to innermost surface respectively m= 2,6,10,14 and 200).

This model can well predict the variation of in-plane yield stresses and Lankford

coefficients (Yoon et al., 2000). For plane stress implementation in FE code, the

Yld96 function does not provide any particular problem and leads to good simulation

results. However, numerical difficulties may be problematic for the 3D case due to the

relative convexity of the Yld96 function (Barlat et al., 2003). Another drawback is

difficulty in calculating the strain rates analytically for FEM simulations.

4.4.2.4 Yld2000-2d

To overcome the drawback of the Yld96 function (relative convexity) and also to

obtain a better prediction of in-plane variation of yield stresses and Lankford

coefficients, Barlat and coworkers (2003) proposed a new plane stress Yld2000-2d

yield function. This model gained considerable popularity mainly because of its

accurate prediction of yield stresses and Lankford coefficients at rolling, diagonal and

transverse directions as well as balanced biaxial stress state. The non-quadratic

Yld2000-2d yield function is based on a linear transformation of two unconditionally

convex functions and of deviatoric stress tensor.

[

( )]

(4-50)

where

(

) , (

) (

) (4-51)

and

are the principal values of the linear transformation on the stress

deviators and respectively. Similar to the Hosford function and all earlier Barlat

65

models, the coefficient m is associated to crystallographic structure and is 6 for BCC

and 8 for FCC metals respectively. The linear transformation of is applied on the

deviator stress by

[

] [

] [

] (4-52a)

[

] [

] [

] (4-52b)

where and are linear transformation matrices

(4-53a)

(4-53b)

and

[

] (4-54)

As seen in Eqn.(4-53a) the transformation can be applied directly to the Cauchy stress

rather than to the deviatoric stress tensor by means of and matrices. In

Eqn.(4-53a) and (4-53b) the and matrices are

[

] (4-55a)

[

] (4-55b)

And their components are given by

⌈⌈⌈⌈

⌉

⌉⌉⌉

⌈⌈⌈⌈

⌉

⌉⌉⌉

[

] (4-56a)


⌈⌈⌈⌈

⌉

⌉⌉⌉

⌈⌈⌈⌈

⌉

⌉⌉⌉

⌈⌈⌈⌈

⌉⌉⌉⌉

(4-56b)

The principal values of transformed stress deviator denoted by and

can be

calculated as

(

) ⁄ √

((

) ⁄ ) (4-57a)

(

) ⁄ √

((

) ⁄ ) (4-57b)

Similar functions apply for and

.

Eight experimental results such as yield stresses and Lankford coefficients

corresponding to rolling, diagonal and transverse directions and to balanced biaxial

stress state ( ) are required to determine the

coefficients. However, if no 8th experimental measurement is available, one may

suggest (therefore

) or (therefore

) (Chung et

al., 2005). The parameter identification procedure requires approaches such as

Newton-Raphson iteration. The optimization procedure is described in Appendix B.

The Yld2000-2d model has been proven a robust yield function and has been

successfully implemented into FEM by many authors, leading to good

correspondence between simulation and experimental results (Chung et al., 2005;

Yoon et al., 2005; Yoon et al., 2006; Lee et al., 2007; Ahn et al., 2009; Yoon et al.,

2010; Park and Chung, 2012).

4.4.2.5 Yld2004-18p

Recently, Barlat et al (2005) generalized the plane stress Yld2000-2d yield function to

consider six-component stress states for three-dimensional problems. This yield

function has been shown to be very powerful in terms of modeling highly anisotropic

materials such as aluminium alloy AA2090-T3; for instance 6 and 8 ears were

simulated by Yoon et al (2006). The variation of yield stresses and Lankford

coefficients can be simulated excellently by the Yld2004-18p model.

The Yld2004-18p function is defined as

[

∑

]

(4-58)

67

where and are linearly transformed stress tensor deviators that can be

determined by

(4-59a)

(4-59b)

and are linear transformation tensors. Each of these tensors include 9 material

parameters . Due to different representations of the stress vector in

Abaqus/Implicit and Abaqus/Explicit (see Section 2.2) it is required to define and

for each case separately. Accordingly, the definitions for UMAT

(Abaqus/Implicit) are (Yoon et al., 2006)

[⌈⌈⌈⌈

]

⌉⌉⌉⌉

[⌈⌈⌈⌈

]

⌉⌉⌉⌉

(4-60)

and for VUMAT (Abaqus/Explicit)

[⌈⌈⌈⌈

]

⌉⌉⌉⌉

[⌈⌈⌈⌈

]

⌉⌉⌉⌉

(4-61)

is an operator that simply converts Cauchy stress to its deviator.

[⌈⌈⌈⌈

]

⌉⌉⌉⌉

(4-62)

For use in 3D cases, there are 18 material parameters ( ) that have to be

optimized by numerical methods such as Newton-Raphson iteration (described in

Appendix B). Tensile yield stress and Lankford coefficient at each 15° from rolling to

transverse direction as well as at balanced biaxial stress state provide sixteen

experimental inputs for parameter identification. To obtain the Lankford coefficient at

balanced biaxial state, the disk compression test proposed by Barlat et al is preferred

due to less errors compared to the viscous pressure bulge test (Barlat et al., 2003).

The other inputs for parameter identification could be out-of-plane yield stresses at

45° tension in TD-ND and ND-RD planes or simple shear tests at TD-ND and ND-


RD planes (Barlat et al., 2005). Since this is very challenging from a practical point of

view, it is recommended to perform this parameter identification based on

polycrystalline simulations. In case such simulations are not available, Barlat et al

suggested that all out of plane yield stresses can be assumed equal to that

corresponding to rolling direction and for pure shear tests

( ) (Barlat et

al., 2005).

4.4.2.6 Yld2004-13p

Barlat et al (2005) reduced the Yld2004-18p function to a simpler version suitable for

cases when not enough experimental results are available. The so-called Yld2004-13p

function is shown to be convex for .

[

(

)

]

(4-63)

The coefficients in Eqns.(4-60) and (4-61) have to be adjusted such that .

For 3D and plane stress conditions, respectively 13 and 9 parameters are necessary.

Barlat et al (2005) discovered that the so-called locking-effect troubles the

optimization procedure when 12 and 8 parameters are respectively taken for 3D and

plane stress cases.

4.4.3 Banabic’s family of yield criteria

Banabic et al (2000) proposed the plane stress BBC2000 yield function by adding

weight coefficients to the Hershey isotropic yield function. Banabic et al (2005) and

Butuc et al (2002) discussed that the BBC2000 model can be described as an

extension of the Yld89 model. This yield function has 7 coefficients and thus 7

experimental tests ( and at three uniaxial directions and at balanced biaxial state)

are required for parameter identification. Moreover, very simple constraints imposed

on the model’s exponent (similar to m in Barlat’s models) assures convexity of this

non-quadratic yield function. Banabic et al (2005) improved the BBC2000 function to

the BBC2005 yield function that takes an additional experimental input ( at balanced

biaxial state) for parameter identification. Barlat et al (2007) proved that the Yld2000-

2d and BBC2005 functions are actually similar. However, Banabic et al (2009) stated

that the development procedures adopted were different in a sense that the BBC yield

functions are developed by adding weight factors to Hershey’s and Hosford’s yield

criteria but the Yld2000-2d yield function was based on a linear transformation

approach. Recently, Comsa and Banabic (2008) proposed the BBC2008 yield

function that shares some features with Barlat’s Yld2004-18 and Yld2004-13p

functions in a sense that all require the same experimental data for parameter

identification and can be used in 3D and plane stress cases. However, the BBC2008

69

function can be based on more parameters and consequently requires more

experimental inputs. This feature has been added to a very recently proposed Yld2012

yield function proposed by Aretz and Barlat (2012) which can take one or more

additional linear transformations compared with Yld2004-18p model.

4.4.4 Cazacu and Barlat’s yield criteria

Cazacu and Barlat (2004; 2004) proposed a new yield function based on the linear

transformation approach to consider the strength differential effect observed as

yielding asymmetry in pressure insensitive materials such as hexagonal close-packed

(HCP) metals including titanium, textured magnesium and magnesium alloys. The

strength differential effect is observed when yield stresses in tension and compression

are different. In materials exhibiting this behaviour, the plastic deformation occurs by

both slip and twinning and due to the polarity, the material response depends on the

sign of the stress (Cazacu and Barlat, 2004; Barlat et al., 2007). Plunlett et al (2006)

implemented the improved Cazacu model into finite element code to account for the

rapid change of yield surface due to texture evolution during monotonic loading.

4.4.5 Polynomial yield functions

Hill (1950) proposed a general framework of plane stress yield functions based on a

polynomial expression as given by Eqn.(4-64)

∑

(4-64a)

(4-64b)

where are function parameters, and are non-negative integers and is the

n-th order homogenous yield function. Polynomial functions are very attractive due to

convenient calculation of derivatives which is needed for implementation into FE

code.

Gotoh (1977) proposed a fourth order polynomial with 9 material parameters that can

model anomalous behavior. In his model, . The advantage of his model

was the fact that the variation of in-plane Lankford coefficients and yield stresses are

well predicted. The large number of experimental tests (including uniaxial, biaxial

and pure shear) required for optimization of 8 parameters (i.e. 9 for compressible

materials) may be considered as disadvantage for this model. It must be noted that this

problem might be relieved by the progress of recent experimental techniques. In

addition, a large number of experimental data is indispensable for describing a highly

anisotropic material.

Hu (2003, 2005) proposed two polynomial-based yield functions for plane stress and

3D applications, respectively. He described that the Lankford coefficients and not the


balanced biaxial yield stress influence the shape of the yield surface. Leacock (2006)

discovered that Hu’s models may cause oscillations in both yield stresses and

Lankford coefficients outside the points used in the parameter identification. To

overcome this drawback, Hu (2007) proposed an improved version of his previous

models. He assumed that yield stress at 22.5° and 67.5° are linear functions of initial

yield stress at 0°, 45° and 90°.

Recently Soare et al (2008) proposed 4th, 6th and 8th order polynomial functions

(respectively called Poly 4,Poly 6 and Poly 8) respectively with 9, 16 and 25 model

parameters. Their models were originally developed for plane stress problems but can

be extended conveniently for 3D conditions. They elaborated on the convexity of the

models and clarified the parameter identification procedure. Poly 4 function with 9

parameters was dominant to Yld96 function. Both Poly 6 and Poly 8 results were

dominant over Yld2004 function for aluminium alloy AA2090-T3.

4.5 Summary

In this chapter, a review of various isotropic and anisotropic yield functions was

provided. Among the anisotropic models, Hill’s and Barlat’s family of yield functions

were described in detail and the advantages and disadvantages of these models were

discussed.

If anisotropy must be considered and simplicity of the model is the priority then the

Hill 1948 model would be the preference. According to the characteristic of this

model, it is not recommended to be used for highly anisotropic materials and most

aluminum alloys. Other Hill anisotropic models lack simplicity and efficiency.

The chronological improvement in the models proposed by Barlat was also illustrated.

Among the models he proposed, Yld96, Yld2000-2d and Yld2004-18p gain

considerable popularity due to their great accuracy. The difficulties in derivation of

first and second order gradients of these models (required for development in finite

element as user material subroutine) could be a concern. Among these models, the

Yld2004-18p can describe the variation of directional yield stresses and Lankford

coefficients at each 15° from rolling to transverse direction. Therefore, this model is

recommended when accuracy is a concern (for a highly anisotropic material).

If accuracy is desired together with simple derivation of model derivatives, then high

order polynomial-based anisotropic models can be recommended. Due to the

polynomial nature of these models their derivatives can be obtained simply. Some

considerations during parameter identification must be taken into account to ensure

convexity of these models. According to the results of Soare et al (2008), the results

of the Poly4 (4th order polynomial) anisotropic model is very close to Yld96. He also

showed that the results of Poly6 and 8 models are very close to Yld2004-18p.

Therefore, polynomial models could be considered as combination of simplicity and

accuracy.

71

All discussed models were based on normality and AFR hypotheses. In the next

chapter, the concept of non-AFR is described and the Hill 1948 and Yld2000-2d yield

functions are converted into non-AFR versions. The improvement of using non-AFR

instead of AFR based models will be discussed in Chapter 7.


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Approach. International Journal of mechanical Sciences (DOI

10.1016/j.ijmecsci.2013.04.003).

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develop anisotropic yield functions with applications to sheet forming. International


Soare, S., Barlat, F., 2010. Convex polynomial yield functions. J Mech Phys Solids

58, 1804-1818.

Stout, M.G., Hecker, S.S., 1983. Role of geometry in plastic instability and fracture of

tubes and sheet. Mech Mater 2, 23-31.

Van Houtte, P., 1994. Application of plastic potentials to strain rate sensitive and

insensitive anisotropic materials. International Journal of Plasticity 10, 719-748.

Vegter, H., van den Boogaard, A.H., 2006. A plane stress yield function for

anisotropic sheet material by interpolation of biaxial stress states. International


von Mises, R.V., 1928. Mechanics of plastic deformation of crystals. Zeitschrift für

Angewandte Mathematik und Mechanik 8, 25.

Woodthorpe, J., Pearce, R., 1970. The anomalous behaviour of aluminium sheet

under balanced biaxial tension. International Journal of Mechanical Sciences 12, 341-

347.


Yoon, J.-H., Cazacu, O., Whan Yoon, J., Dick, R.E., 2010. Earing predictions for

strongly textured aluminum sheets. International Journal of Mechanical Sciences 52,

1563-1578.

Yoon, J.W., Barlat, F., Chung, K., Pourboghrat, F., Yang, D.Y., 2000. Earing

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Yoon, J.W., Barlat, F., Gracio, J.J., Rauch, E., 2005. Anisotropic strain hardening

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in a drawn cup based on a new anisotropic yield function. International Journal of


Chapter 5

5 Non-associated flow rule

78 Chapter 5, Non-associated flow rule

5.1 Introduction

This chapter discusses the concept of non-associated flow rule and its advantages over

its AFR counterpart. In section 5.2, some disadvantages of an associated flow rule

(AFR) approach for finite element simulation of sheet metal forming are highlighted.

Then, in section 5.3, the concept of non-associated flow rule is introduced and its

advantages over AFR are discussed. Section 5.4 introduces two non-AFR based

anisotropic yield models and briefly compares AFR and non-AFR based anisotropic

yield models. Finally, in section 5.5, the stability issue of non-AFR is discussed.

5.2 Limitations of associated flow rule (AFR)

In the previous chapter, various phenomenological anisotropic yield functions in the

framework of associated flow rule hypothesis have been introduced. In AFR, which is

one of the cornerstones of classical plasticity theory for metals, the yield function

determines both yielding and flow direction (plastic strain rate) simultaneously.

However, various studies described the inability of the AFR concept in dealing with

highly anisotropic materials. For instance, (Cvitanic et al., 2008) showed that Hill

(1948) and Karafillis–Boyce (1993) anisotropic yield functions cannot predict both

directional r-values and yield stresses simultaneously for AA 2008-T4 and AA 2090-

T3. Park and Chung (Park and Chung, 2012) reported that Hill 1948 and Yld2000-2d

generate poor accuracy for directional r-values and yield stresses for AA 2090-T3 and

AA 5042. Yoon et al (Yoon et al., 2007) showed that Hill 1948 is unable to predict

the exact numbers of ears in deep drawn cups made of AA 2090-T3. Therefore it can

be concluded that describing a highly anisotropic material in terms of both plastic

strain rate and yielding behaviour with an identical function for yield function and

plastic potential function is difficult (Stoughton, 2002). This can, for instance, be

fulfilled by using more complicated yield criteria with a large number of parameters

such as the Yld2004-18p model developed by Barlat et al (2005).

From a physical point of view, experimental tension and compression tests on iron

based metals and on aluminium, reported by Spitzig and Richmond (1984), revealed

the (linear) dependency of yield stress on the superimposed hydrostatic pressure.

They also showed that an associated flow rule over-predicts the plastic dilatation in

the presence of superimposed hydrostatic pressure. Therefore the AFR approach is

unable to deal with zero plastic dilatancy and pressure sensitivity because zero plastic

dilatancy requires the plastic potential to be a function of the deviatoric stress only,

and must therefore be insensitive to pressure (Stoughton and Yoon, 2006). Similar

observation as made by Spitzig and Richmond was reported for geologic materials by

Lade et al (1987) and the invalidity of AFR approach for application to porous,

granular, and geologic materials has been proven.

79

5.3 Non-associated flow rule (non-AFR)

5.3.1 Background

The non-associated flow rule approach removes the artificial constraint of equality of

plastic potential and yield function enforced by the AFR assumption. Consequently,

two separate functions for yield function and plastic potential function are adopted. In

other words, the yield and plastic potential functions respectively describe the elastic

limit and plastic strain rate direction independently. This assumption leads to various

advantages and flexibility such as

A non-AFR approach could be the answer for description of simultaneous

pressure sensitivity and negligible plastic dilatancy.

A larger number of experimental data are used for calibration of the

parameters of yield and plastic potential functions resulting in a better

agreement between simulation and experimental data, e.g. better prediction

of yield stress and Lankford coefficient at multiple in-plane orientations.

Furthermore, the inability of Hill 1948 quadratic yield function for modeling

the first order anomalous behaviour in balanced biaxial tension can not be

because of the quadratic order of the formulation, but is rather due to the

restriction forced by the equivalency of the plastic potential function and

yield function (Yoon et al., 2007).

Considering a highly anisotropic material, large gradients on the curvature of

the AFR yield surface may cause convergence problems. The non-AFR

approach, by using two separate functions, reduces the curvature and

improves the convergence (Stoughton and Yoon, 2006).

Considering the various advantages offered by non-AFR, recently its popularity has

increased and more attention is paid to the development and implementation of non-

AFR based models for metal plasticity. However, the number of publications in this

field is few. For instance, the distinguished work of Stoughton (2002) proposed a non-

AFR based Hill 1948 quadratic formulation that accurately predicts both direction-

dependent Lankford coefficients and yield stresses at rolling, transverse and diagonal

directions. In addition, difficulties in description of the first and second order

anomalous behaviours for metals with low r-value were resolved in his model. The

reported efficiency in prediction of direction-dependent Lankford coefficients and

yield stresses was due to sufficient degree of freedom for choosing material

parameters that could match to the input experimental values (Stoughton and Yoon,

2006). Continuing his previous model, Stoughton together with Yoon (2004)

developed a pressure sensitive non-AFR model that predicted the strength differential

effect observed in tension-compression tests. Cvitanic et al (2008) developed a non-

AFR model based on Hill 1948 quadratic and Karafillis and Boyce non-quadratic

yield functions combined with isotropic hardening (Swift law) which showed an


improved prediction of the heights of deep drawn cups made of an aluminium alloy.

Stoughton and Yoon (2009) proposed a non-AFR based distortional hardening model

that resulted in an excellent prediction of the hardening curves corresponding to the

rolling, diagonal and transverse directions as well as balanced biaxial stress state.

Improvements in the prediction of cup height and springback of U-bend using non-

AFR with mixed isotropic-kinematic hardening have been reported in recent work of

Taherizadeh et al (2010). Recently, Park and Chung (2012) proposed an analytical

approach to achieve a symmetric consistent tangent modulus for non-AFR based

models, which is suitable for FEM software that cannot deal with asymmetric

matrices.

5.3.2 Concept

The yield criterion for both AFR and non-AFR based models is defined by

( ) (5-1)

The normality rule is the fundamental assumption in AFR plasticity. This rule

describes that the plastic strain rate is directly associated to the first gradient of the

continuously differentiable yield function , i.e. normal to the yield surface. This

results from the assumption that the maximum plastic work occurs only when the

plastic strain increment is normal to the yield surface. To prove the normality for AFR

let us start with the principle of maximum plastic work. This principle describes that

the stress must be restricted to the yield surface and, in addition, should be such as to

maximize the increment/rate of plastic strain (Crisfield, 1997).

(5-2a)

{

} (5-2b)

This can be dealt with creating a Lagrangian function (a mathematical approach for

finding extrema named after Joseph Louis Lagrange) which includes the constraint

and adds a Lagrange multiplier times the yield criterion F

( )

(5-3a)

(5-3b)

Eqn.(5-3b) defines the normality rule

(5-4)

81

where

(5-5)

is also called the plastic multiplier and the second order tensor

is the first

gradient of the yield function.

Fig 5-1 Concepts of associated (left) and non-associated (right) flow rule.

However, in the non-associated flow rule, the normal to the plastic potential

function , describes the plastic strain rate direction. Then the normality rule is

described by


(5-6)

where

(5-7)

Simply, due to equality of and in the AFR hypothesis one obtains

(5-8)

An example of AFR and non-AFR based models and their associated plastic potential

function and yield function are depicted in Fig 5-1.

5.4 Non-AFR based yield models

5.4.1 Non-AFR version of Hill 1948

Considering the non-AFR version of Hill 1948, its yield function is based on the S-

based Hill 1948 function. This function is described by Eqn.(5-9) and only requires

yield stresses and for parameter calibration.

(

)

(5-9)

, and are yield function parameters that can be calculated as follows

(

)

,

( (

)

(

)

) , (

)

(

)

(5-10)

The plastic potential function is represented by the r-based Hill 1948 function. It

should be noticed that the difference between the two versions of Hill 1948 is the type

of experimental inputs needed for parameter identification. Considering r-based Hill

1948 function, these inputs are directional r-values and .

The plastic potential function is

(

)

(5-11)

where the parameters , and are given by

,

,

(5-12)

83

5.4.2 Non-AFR version of Yld2000-2d

Considering the non-AFR version of the Yld2000-2d model, its yield function is

described by

[

]

(5-13)

where

,

(5-14)

and

are the principal values of the linear transformation of the stress deviators

and .

,

(5-15)

⌈⌈⌈⌈

⌉

⌉⌉⌉

⌈⌈⌈⌈

⌉

⌉⌉⌉

[

] ,

⌈⌈⌈⌈

⌉

⌉⌉⌉

⌈⌈⌈⌈ ⌉

⌉⌉⌉

⌈⌈⌈⌈

⌉⌉⌉⌉

(5-16)

The parameters of the Yld2000-2d yield function , , are optimized based on

directional yield stresses at every 15° from rolling to transverse direction

( ) and the balanced biaxial yield stress ( . The plastic

potential function of the non-AFR based Yld2000-2d model follows the same

formulation of the Yld2000-2d yield function as defined in Eqn.(5-13). However, it

requires Lankford coefficients for unidirectional loading in different orientations

( as well as that of balanced biaxial loading ( for

parameter optimization.

5.4.3 Evaluation of non-AFR models versus various AFR models

An in-depth evaluation of the non-AFR versions of quadratic Hill 1948 and non-

quadratic Yld2000-2d models is provided in Chapter 7. However, one may already

notice that a considerable improvement can be expected from the non-AFR models

compared with their AFR counterparts. For instance, the AFR r-based Hill 1948

model can result in an accurate prediction of Lankford coefficients at 0°, 45° and 90°

and the AFR S-based Hill 1948 model results in an exact fit to the initial yield stresses

at these directions and at the balanced biaxial stress state. For the non-AFR version,

however, both yield stresses and Lankford coefficients at 0°, 45° and 90° as well as

yield stress for balanced biaxial stress are exactly predicted (Safaei et al., 2012a, b).


Due to the absence of a direct link between yield stress and Lankford coefficient, the

first and second order anomalous behaviours can be modeled with the non-AFR

quadratic Hill 1948 model. Considering the non-AFR Yld2000-2d model, it will be

shown in Chapter 7 that in-plane yield stresses and Lankford coefficients at each 15°

from 0° to 90° are in excellent agreement with experimental results. As a result,

prediction of 6 and 8 ears becomes possible using this formulation (Yoon et al., 2006;

Park and Chung, 2012; Safaei et al., 2012c).

5.5 Non-AFR and stability

Stability of constitutive models is critical for finite element applications. The general

stability requirements for rate and temperature insensitive material models for finite

element simulation are (Stoughton and Yoon, 2006):

1. Positive rate of plastic work ( ) and change of equivalent plastic strain

( .

2. Unambiguous definition of the rate of change of all state variables for both

strain-rate and stress-rate controlled boundary conditions

3. The strain state must remain unchanged for any closed-loop loading path that

does not expand the yield surface.

4. The net amount of work on any closed cycle of strain must also be positive

for all possible deformation paths.

Violation of any of the above four conditions can lead to illogical bifurcations such as

yield point phenomenon, problems in numerical convergence and/or irrealistic

predictions.

Drucker (1959) described a class of stable material models that cover all four

requirements and discussed the relation between AFR and stability. His postulate

states that for any stress state the second order work state is always positive

(5-17)

Drucker also showed that when Eqn.(5-17) is satisfied, consequently the first order

plastic work rate is always positive

(5-18)

Accordingly plastic deformation is always a dissipative process. Drucker showed that

a material constitutive model which is based on AFR is always stable and satisfies all

requirements mentioned above. Consequently, Drucker’s postulate ensures maximum

plastic dissipation and that leads to convexity. In Fig 5-2, the relation between

convexity and maximum plastic dissipation is shown.

85

Fig 5-2 Relation between maximum plastic dissipation, convexity and normality in

AFR method (Yld2000-2d as a convex yield surface).

However, Mroz (1963) showed that AFR is a sufficient requirement for stability but

not a necessary one. As discussed earlier, there are classes of metals that cannot be

described by AFR and, thus, the non-AFR concept is required. Therefore Drucker’s

postulate seems to be incompatible with non-AFR. For instance, in Fig 5-3 at a stress

state denoting (

) and , non-convexity of the non-AFR

Yld2000-2d model is shown.

Fig 5-3 Non-AFR Yld2000-2d as a non-convex anisotropic model.

Stoughton and Yoon (2006) described that AFR is not the only postulate for stability

and provided a broad framework of material models including non-AFR based models

Convexity of associated flow rule

26-Apr-13 12

Convex

Convexity of non-associated flow rule

26-Apr-13 13


with the same level of stability among which we briefly describe the

loading/unloading and non-singularity constraint.

The classical loading/unloading criterion for a work hardening material is defined by

purely elastic

and purely elastic (5-19)

and ideally plastic

and elastic-plastic

Fig 5-4 Schematic of various loading conditions in an AFR model.

Fig 5-4 depicts concept of various loading conditions assuming AFR approach. The

ideal plastic behaviour raises instability problems and does not represent a proper

material behaviour for FE simulations. Excluding this case, the loading/unloading

criterion can be written in the following form for non-AFR models

purely elastic

and

purely elastic (5-20)

and

elastic-plastic

Stoughton and Yoon (2006) proved that by imposing this constraint the rate of

change of the effective plastic strain is uniquely defined. Considering AFR models,

the singularity (for stress controlled applications) happens when the slope of the stress

versus plastic strain curve is zero. Therefore this can be avoided by a simple

constraint

87

(5-21)

For the case of non-AFR models, another constraint must also be imposed to avoid

such singularity

(5-22)

This results from the governing equation on plastic compliance factor which

will be derived in the next chapter.

(5-23)

There are other constraints that are required to analytically achieve the stability of

non-AFR models (Stoughton and Yoon, 2006). However, from the point of view of

implementation in FE software, the constraints mentioned above are the most

important.

5.6 Summary

In this chapter, the concept of non-AFR which removes the artificial constraint of

equality of yield and plastic potential functions was described. By using two

independent functions the zero plastic dilatancy and pressure sensitivity can be

described simultaneously. Moreover, compared to AFR models a more accurate

description of anisotropy is possible by non-AFR models due to incorporating

additional experimental data for parameter calibration. Regarding the stability of non-

AFR, the general requirements for stability for rate and temperature insensitive

material models were briefly described. Violation of any of these stability conditions

can lead to instability problems. Drucker (1959) showed that a material constitutive

model which is based on AFR is always stable. However, Mroz (1963) showed that

even though AFR is a sufficient requirement for stability, it is not a necessary one.

Furthermore, hydrostatic pressure sensitive metals cannot be described by AFR based

models as shown by Spitzig and Richmond (1984) and, thus, the non-AFR concept is

essentially required. In other words, Drucker’s postulate seems to be incompatible

with the non-AFR assumption that is required for modeling of materials that possess

pressure sensitivity and incompressibility characteristics. Stoughton and Yoon (2006)

described that AFR is not the only postulate for stability and they provided a broad

framework of material models including non-AFR ones with the same level of

stability.


Bibliography



1009-1039.

Crisfield, M.A., 1997. Non-linear finite element analysis of solids and structures.




646-687.

Drucker, D.C., 1959. A Definition of a Stable Inelastic Material. J. Appl. Mech.

Trans. ASME, 26 101-107.



Sciences 193, 281-297.



Lade, P., Nelson, R., Ito, Y., 1987. Nonassociated Flow and Stability of Granular

Materials. Journal of Engineering Mechanics 113, 1302-1318.

Mroz, Z., 1963. Non-associated flow laws in plasticity. Jl. De Mechanique 2, 21.

Park, T., Chung, K., 2012. Non-associated flow rule with symmetric stiffness

modulus for isotropic-kinematic hardening and its application for earing in circular

cup drawing. International Journal of Solids and Structures 49, 3582-3593.

Safaei, M., De Waele, W., Zang, S.-l., 2012a. A rate-independent non-associated

constitutive model for finite element simulation of sheet metal forming, in: Khan, A.

(Ed.), Plasticity and its Current Applications, 18th International symposium,

Proceedings. Numerical Engineering Analysis and Testing (NEAT), Poerto Rico, US,

pp. 61-63.

Safaei, M., De Waele, W., Zang, S.-l., 2012b. Evaluation of Associated and Non-

Associated Flow Metal Plasticity; Application for DC06 Deep Drawing Steel, in:

Merklein, M., Hagenah, H. (Eds.), ESAFORM 2012. Trans Tech, University

Erlangen-Nuremberg, Germany, pp. 661-666.

Safaei, M., Zang, S.L., Lee, M.G., De Waele, W., 2012c. Evaluation of Anisotropic



10.1016/j.ijmecsci.2013.04.003).

Spitzig, W.A., Richmond, O., 1984. The effect of pressure on the flow-stress of

metals. Acta metallurgica 32, 457-463.



89

Stoughton, T.B., Yoon, J.W., 2004. A pressure-sensitive yield criterion under a non-

associated flow rule for sheet metal forming. International Journal of Plasticity 20,

705-731.





1817.




Yoon, J.W., Barlat, F., Dick, R.E., Karabin, M.E., 2006. Prediction of six or eight ears

in a drawn cup based on a new anisotropic yield function. International Journal of






Chapter 6

6 Fully implicit backward Euler

integration scheme

92 Chapter 6, Fully implicit backward Euler integration scheme

6.1 Introduction

In Chapter 5 the concept of a non-associated flow rule was presented and its

advantages for describing the anisotropic behaviour of sheet metals were discussed.

To implement the non-AFR approach into finite element code, the governing elastic-

plastic rate equations have to be integrated in order to calculate the incremental values

of stress and strain and to incorporate the effects of load path. To this end, different

integration schemes will be discussed in Chapter 9. For implementation in finite

element software, the integration schemes should be converted to a user material

subroutine which is a code written in a programing language such as FORTRAN or C.

The user material subroutines UMAT and VUMAT are respectively used for

Abaqus/Standard and Abaqus/Explicit. Abaqus/Standard and Abaqus/Explicit

respectively work based on implicit and explicit time integration schemes. In either of

them, it is possible to implement any type of implicit, explicit and combined stress

update schemes. The user material subroutine receives stress, strain and other

information corresponding to the previously converged step from finite element

software for each element. Then its task is to find the updated stress and strain data

based on an imposed strain increment. The difference between several integration

schemes lies in the numerical techniques they use to resolve the output data.

The most favourable integration scheme, because of its high accuracy, is the so-called

fully implicit backward Euler method. This one is adopted in this chapter. The

approach presented is capable of dealing with any first order homogeneous yield and

plastic potential function. For the purpose of extending the range of applicability of

the non-AFR model to applications including load reversal and not only proportional

loading, it is proposed to employ the non-AFR model with assemblage of a mixed

isotropic-kinematic hardening model proposed by Zang et al (2011). This proposed

one-surface hardening model was presented in Section 3.5.2 and can predict the key

phenomena observed during load reversal such as Bauschinger effect, transient

behaviour and permanent softening. Moreover, it can be simply reduced to a classical

Chaboche and Armstrong-Frederick model or even to isotropic hardening. The

linearization approach presented in this chapter is provided in more detail in

Appendix A.

6.2 Return mapping algorithm

Most commercial finite element codes are established based on the strain-driven

boundary condition in which a defined total strain rate is imposed to each element and

successively the stress history is obtained from the strain history by means of an

integration algorithm (Simo and Taylor, 1986). In other words, the total strain

increment has to be decomposed into its elastic and plastic components and the

effective plastic strain rate and unique stress rate associated to that plastic strain rate

tensor have to be determined. Finally the determined variables have to be integrated

93

over a very small time increment resulting in a unique stress solution corresponding to

the determined plastic strain tensor.

Different return mapping algorithms have been proposed to solve the rate constitutive

equations for a given set of initial conditions. In this class of integration schemes, first

the stress is assumed to be outside the yield surface and using different return

mapping techniques it is driven back to be located on the new yield surface. For

instance Wilkins (1963) suggested the first radial return scheme for J2 plasticity. The

fully implicit backward Euler integration scheme gained considerable popularity due

to its unconditional stability and moreover its quadratic convergence rate. The

quadratic convergence is due to the inherent characteristic of the Newton-Raphson

iteration method employed in this scheme. In this approach, the increments in plastic

strain and all internal variables and

are calculated at the end of the step

denoted by and the yield condition is enforced at the end of the step

(Belytschko et al., 2000). In other words, the fully implicit backward Euler integration

scheme is implicit in terms of plasticity parameter , flow direction

, as well as

and . This integration scheme requires second order derivatives of yield and plastic

potential functions giving rise to difficulties in the development of the update scheme.

Alternatively, if one seeks a compromise between computational time and accuracy,

the so-called semi-implicit backward Euler integration scheme may be used, as it is

only implicit in terms of the plasticity parameter thus bypassing the need of second

order derivatives. The flow direction and plastic modulus are from the previous step,

i.e.

and

where subscript n denotes the previous step. By imposing the

yield condition at the end of the step, the consistency condition is satisfied thus

avoiding the drift of stress from the yield surface.

In contrast with the fully implicit scheme, the semi-implicit approach is not

unconditionally stable and accuracy and stability may be a concern. There are also

various simple integration schemes such as forward Euler algorithm (non-return

mapping technique) and cutting plane algorithm developed by Ortiz and Simo (1986).

However, the simplicity of the integration scheme generally limits the user towards

using considerably smaller time increments to reduce the risk of falling beyond the

yield surface. In this work the fully implicit backward Euler scheme is adopted

because of its high accuracy and unconditional stability.

6.3 Elasto-plasticity with non-AFR

As indicated in the previous chapter, in a non-AFR based model the continuously

differentiable plastic potential and yield functions are incorporated in the material

constitutive model to address respectively the direction of plastic strain rate and

yielding of material. This results in violation of the normality hypothesis, but on the

other hand it adds considerable flexibility and strength to the model. The stability


issues have been extensively investigated by Stoughton and Yoon (2006, 2008) and

were briefly described at the end of the previous chapter.

Recall the definition of the yield criterion

( ) (6-1)

In case of the non-AFR Hill 1948 model, the yield function is identical to the S-

based Hill 1948 function. For the non-AFR Yld2000-2d model, it takes the

formulation (5-13) with directional yield stresses ( ) and

balanced biaxial yield stress ( for parameter optimization. The function only

serves as an elastic limit predictor.

As opposed to the normality rule for AFR, the plastic strain increment in a non-

associated flow rule is described by the normal to the plastic potential function

(6-2)

with the plastic strain rate direction given by

(6-3)

is the plastic multiplier factor to be determined by the loading-unloading criterion,

and the second order tensor

is the plastic flow direction. The plastic multiplier

factor, , is also called compliance factor, plasticity parameter or Lagrange

multiplier. The plastic flow direction,

, is determined by the continuously

differentiable plastic potential function which is the r-based Hill 1948 function or

the Yld2000-2d plastic potential function respectively for non-AFR Hill 1948 and

non-AFR Yld2000-2d. The plastic potential function of non-AFR Yld2000-2d

requires directional Lankford coefficients ( as well as the

balanced biaxial Lankford coefficient ( for parameter optimization.

Equivalent plastic strain and plastic strain tensor

are related using the principle

of plastic work equivalence

( )

(6-4)

Euler’s theorem for any first order homogenous function states

(

)

( ) (6-5)

95

This equation is regularly used in treatment of elastic-plastic formulations. For

instance, applying Eqn.(6-5) to the yield and plastic potential functions one obtains

(6-6a)

(6-6b)

where the second order tensor

is the normal to the yield surface (

!), i.e.

(6-7)

Substituting Eqn.(6-2) into Eqn. (6-4) and next applying Euler’s theorem Eqn. (6-6a)

( )

( )

( )

( ) (6-8)

This definition ( ) was used by Cvitanic et al (2008) for a model

including isotropic hardening. However, for linearization of the stress-update

algorithm in case of complex mixed hardenings this leads to a very laborious

numerical description in which the rate of change of effective plastic strain ( ) will

be a complex function of rate of change of the compliance factor ( ). An alternative,

assuming in Eqn.(6-8) can save computational cost and development effort

because Eqn.(6-8) in that case simplifies to

(6-9)

It must be noted that this simplification does not lead back to AFR. This is due to the

fact the simplification applies only to Eqn.(6-8) and that the normal to the plastic

potential function (and not to the yield function) is always considered for plastic strain

rate direction (Eqn. (6-3)). This assumption has been adopted by Stoughton (2002),

Stoughton and Yoon (2006; 2007; 2009) and Taherizadeh et al (2010). As will be

shown in Chapter 8, with specific care and by employing a scaling approach, the

simulation results such as cup profile and height, will be considerably close to

experimental values. Consequently, for sake of convenience and brevity of

description of the constitutive model only the simplified approach, Eqn.(6-9), is

presented in the remainder of this chapter. Both the simplified and the non-simplified

stress update schemes were implemented in user material subroutines and are

extensively compared in Chapter 8.


6.3.1 Kuhn-Tucker complementary criteria

The stress-update algorithm starts with evaluating whether the current stress state is in

the elastic or in the plastic domain. An elastic process implies that the stress state is

inside the yield surface and thus and . When the stress state of the

material is nested on the yield surface ( ), then besides the yield criterion a

complimentary postulate is required for determination of loading or unloading state of

the deformation process. The loading/unloading condition can be defined by the

Kuhn-Tucker complementary condition (Simo and Hughes, 1998)

and (6-10)

With additional plastic deformation (loading condition) the plastic compliance factor

will be non-zero and moreover the stress state of the material will remain on the

yield surface ( and ). On the other hand the stress state can still be on the

yield surface ( ) without plastic deformation occurring such that . In this

case that is called pure elastic or elastic unloading, the rate of change of yield stress

surface will be equal or less than zero, . The case where is referred to as

neutral loading. It’s worth noting that the above postulates only apply to work

hardening materials for which no strain localization happens (Yoon et al., 2007). One

can note that the consistency condition that simply restricts the stress state inside or

on the yield surface is already included in the Kuhn-Tucker criteria.

6.3.2 Elastic-plastic or continuum tangent modulus

Let us consider effective plastic strain and all back-stress tensors as internal variables

to generalize the development of a stress-update algorithm such that it is easily

adapted for different hardening models. To this end, the incremental change of

effective plastic strain and back-stress tensors in Zang’s hardening model can be

rewritten in a form such that

(6-11a)

(6-11b)

(6-11c)

with

(6-12a)

(6-12b)

(6-12c)

97

and

are respectively increments of equivalent plastic strain, first and

second terms of the two-term Chaboche kinematic hardening law and and

are their conjugate plastic moduli.

Imposing the consistency condition to the yield criterion in Eqn.(6-1) gives

(

) (6-13)

The above equation is derived from a Taylor’s truncated series with and

neglecting higher order terms. For further analysis, the additive decomposition of

strain increments into elastic and plastic components gives

(6-14)

where

and

are respectively rate of elastic, plastic and total strain.

Now, the Hookean elasticity relation for hypo-elastic materials is used to relate stress

to elastic strain at any moment of elastic-plastic deformation

(6-15)

where the symmetric fourth order tensor

describes the isotropic elasticity of the

material. Combination of strain incremental additive decomposition in Eqn.(6-14)

with the Hookean elasticity relation in Eqn.(6-15) simply relates stress increment to

plastic strain increment

(

) (6-16)

Substituting the incremental hardening Eqn.(6-11a) to Eqn.(6-11c) into the

consistency condition Eqn.(6-13) and using Hooke’s law Eqn.(6-16) and the non-

associated flow rule in Eqn.(6-2), after some manipulation finally yields

(6-17a)

(

)

(6-17b)


with

(6-18)

is the rate of the plastic compliance factor. The scalar denotes the tangent of the

isotropic hardening function with respect to equivalent plastic strain

(6-19)

The stress and strain increments are related by the elastic-plastic tangent modulus

(also called continuum tangent modulus). Early integration schemes (Hinton and

Owen, 1980) extensively relied on the use of the continuum tangent operator. This

operator,

, is obtained by using the incremental plastic multiplier from Eqn. (6-17)

and Eqn.(6-16), which after some manipulations yields

(

) (

)

(

)

(6-20)

For large time steps, the combination of continuum tangent operator and stiffness

matrix degrades the convergence rate. Therefore, in the remaining, the algorithmic or

consistent tangent operator is introduced which is consistent with a quadratic

convergence rate of the fully implicit backward Euler integration scheme.

6.4 Stress-update algorithm

6.4.1 Fully implicit backward Euler

The fully implicit backward Euler integration scheme for non-AFR with mixed

hardening including the two-term Chaboche kinematic hardening is defined as

(6-21a)

(6-21b)

(6-21c)

(6-21d)

(6-21e)

(

) (6-21f)

99

(

) (6-21g)

The plastic strain increment at the end of the time increment is

given by

(6-22)

and

(

) (6-23a)

(6-23b)

(6-23c)

with

(6-24a)

where

and

respectively denote the trial stress of the

elastic predicator step and the plastic corrector. During the elastic predicator step, it is

assumed that stress is purely elastic in a sense that the total strain increment

substitutes the elastic strain increment and stress lies on the yield surface or

which is the yield surface at the trial stress state as shown in Fig 6-1.

(6-25)

Subsequently during the plastic corrector or so-called relaxation step, the stress is

returned to the yield surface at time increment which is notated as (see

Fig 6-1). This is performed using the Newton-Raphson method based on linearization

of the set of equations Eqn.(6-21). Once the solution converged, the stress at the end

of the step is updated as

(6-26)

The initial guess has a considerable influence on the convergence of the iteration

scheme. Therefore, after the elastic predicator step, the stress is first returned to an

imaginary surface shown in Fig 6-1.


(6-27)

With

(

)

(6-28)

Then, the stress

is the initial guess for linearization based on Newton-Raphson

iteration.

Fig 6-1 Illustration of backward Euler return mapping scheme for one step.

It is worth noting that the convergence of the Newton-Raphson method inherently is

faded at large strains, and as mentioned above the initial guess has a considerable

influence on the convergence of the iteration scheme. The quadratic convergence rate

is achieved only if the approximation of the initial value is within the radius of

convergence. Therefore, to assure the convergence at larger strains, the multi-stage

return mapping method based on the incremental deformation theory proposed by

Yoon (1999) has been applied in the implementation of the return mapping algorithm.

Consequently, each step is divided into sub-steps. The yield criteria associated to

each of the sub-steps are

(

( )) ( )

(

( )) ( ) (6-29)

101

(

( )) (

)

(

( )) ( )

with

(6-30)

where

is the prescribed value at j-th sub-step and .

Using this sub-stepping algorithm, a new series of equations is made for any sub-

potential residual. This procedure is schematized in Fig 6-2. At the beginning of the j-

th sub-step the stress is driven to the auxiliary yield surface

.

(6-31a)

(6-31b)

(6-31c)

(6-31d)

with

being the initial guess stress to be sent to the integration scheme.

Fig 6-2 Schematic of backward Euler return mapping scheme using sub-stepping

technique.

6.4.2 Newton-Raphson iteration scheme

Linearization is done with respect to the effective plastic strain

assuming

the total strain is constant. Using Newton-Raphson for linearization of any equation

such as at the k-th iteration leads to


(

)

(6-32)

where

(6-33)

and being the iterative change in at k-th iteration. Note that each sub-step

denoted by includes a complete set of Newton-Raphson iterations. To apply the

linearization, the update expressions in Eqn.(6-21a) can be cast into the following

forms suitable for the Newton-Raphson iteration scheme.

(6-34a)

(6-34b)

(6-34c)

(6-34d)

(6-34e)

Linearization of the above and

gives

(6-35a)

(6-35b)

(6-35c)

(6-35d)

[ ]{ }

(6-35e)

Note that the complete definition of these equations with detailed interpretation is

provided in Appendix A. The system of equations in Eqn.(6-35) can be written in

matrix form

103

⌈ ⌉

{

}

{ } { } (6-36)

with

{ }

{

}

, { }

{

}

(6-37)

Therefore

{

}

⌈ ⌉{ } ⌈ ⌉{ } (6-38)

Substituting Eqn. (6-38) in Eqn. (6-35e) gives the incremental change of the

compliance factor

[ ]

[ ] (6-39)

At the end of the k-th iteration the parameters

,

and

are determined and their corresponding accumulated values are updated and sent for

the next iteration. The iteration continues until convergence to the updated yield

surface is obtained within an acceptably small tolerance. Subsequently, the

constitutive solutions are passed into the FE solver.

(6-40a)

(6-40b)

(6-40c)


(6-40d)

(6-40e)

(6-40f)

6.4.3 Algorithmic or consistent tangent modulus

The continuum or standard tangent operator relates the stress to total strain rates.

However, according to Belytschko (2000) the continuum tangent operator can

generate a spurious loading and unloading condition during the abrupt transition from

elastic to plastic state. In addition, the consistent (or algorithmic) tangent modulus is

required to preserve the quadratic rate of asymptotic convergence inherent to the

Newton-Raphson iteration nested in the fully implicit backward Euler algorithm

(Simo and Hughes, 1998). Analogously to the previous approach for finding the rate

variables at time step n+1, the consistent tangent modulus is obtained by linearization

of the constitutive equations to relate the stress increment to total strain increment at

the time t+1. By following the same approach as described in section 6.4.2 but

assuming the total strain as non-constant and residuals ({ }) as zero, after many

manipulations the closed form of consistent tangent modulus is obtained. It is worth

noting that the non-symmetric consistent tangent modulus converts to the standard

tangent operator by reducing the step size to zero. Furthermore, both consistent and

standard tangent moduli turn into elastic stiffness matrix when no plastic loading

occurs.

To calculate the consistent tangent modulus, we write the set of Eqns.(6-21) in rate

form so that

(

) (6-41a)

(6-41b)

(6-41c)

(6-41d)

(6-41e)

(6-41f)

Substituting Eqn. (6-41b) in Eqn. (6-41a) and solving for and we can write the

system of equations in matrix form

105

{

}

⌈ ⌉{ } ⌈ ⌉{ } (6-42)

where

{ } {

} (6-43)

Substituting Eqn.(6-42) into the incremental consistency condition Eqn.(6-41f) gives

[ ]

[ ] (6-44)

⌈ ⌉ ⌈ ⌉

⌈ ⌉

⌈ ⌉ ⌈ ⌉ ⌈ ⌉ (6-45)

Substituting Eqn. (6-45) into Eqn.(6-42)

{

} [ [ ]

[ ] ] {

} (6-46)

Finally, the consistent tangent modulus is obtained.

⌈ ⌉{ } [ ] ⌈ ⌉

[ ] ⌈ ⌉{ } (6-47)

It must be noted that the second order tensor

only affects the convergence rate

and not the accuracy. This convergence rate plays an important role in computation

time specifically when large time steps are used in FE simulation.

The final step is to calculate the through thickness elastic and plastic strain

components. For plane stress conditions (such as when shell elements are used), the

elastic and plastic strain components for the through thickness direction are explicitly

determined at the end of the converged step

(6-48a)

(6-48b)

with being Poisson’s ratio.


6.5 Summary

To introduce any new kind of constitutive model into FE code, a proper integration

scheme has to be chosen. There are various techniques among which the return

mapping method is favoured due to improved accuracy compared with forward Euler

method. Several integration schemes will be introduced in Chapter 9. However, in this

work the technique known as fully implicit backward Euler is adopted due to

significant accuracy, high convergence rate and applicability in Abaqus/Standard to

avoid dynamic response of material which is critical for unloading applications. This

approach encompasses a linearization scheme based on Newton-Raphson iteration

which brings the quadratic convergence rate. Using this approach requires the

development of a consistent tangent modulus to preserve the quadratic convergence of

Newton-Raphson iteration. The fully implicit backward Euler integration scheme

described in this chapter was implemented into a user material subroutine for the

commercial FE code Abaqus. In the next chapter, these developments will be

evaluated based on prediction of directional yield stresses and Lankford coefficients

and various results of cup drawing simulations.

107

Bibliography

Belytschko, T., Liu, W.K., Moran, B., 2000. Nonlinear finite elements for

continua and structures. John Wiley, Chichester ISBN 0471987735

Cvitanic, V., Vlak, F., Lozina, Z., 2008. A finite element formulation based on

non-associated plasticity for sheet metal forming. International Journal of


Hinton, E., Owen, D.R.J., 1980. Finite Elements in Plasticity: Theory and

Practice. Pineridge Press, Swansea, Wales.

Ortiz, M., Simo, J.C., 1986. An Analysis of a New Class of Integration

Algorithms for Elastoplastic Constitutive Relations. International Journal for

Numerical Methods in Engineering 23, 353-366.

Simo, J.C., Taylor, R.L., 1986. A Return Mapping Algorithm for Plane-Stress

Elastoplasticity. International Journal for Numerical Methods in Engineering 22,

649-670.

Simo, J.C., Hughes, T.J.R., 1998. Computational inelasticity, Interdisciplinary

applied mathematics. Springer, New York ISBN 0387975209



Stoughton, T.B., Yoon, J.W., 2006. Review of Drucker’s postulate and the issue

of plastic stability in metal forming. International Journal of Plasticity 22, 391-

433.

Stoughton, T.B., Yoon, J.W., 2008. On the existence of indeterminate solutions

to the equations of motion under non-associated flow. International Journal of


Stoughton, T.B., Yoon, J.W., 2009. Anisotropic hardening and non-associated

flow in proportional loading of sheet metals. International Journal of Plasticity

25, 1777-1817.


constitutive model with mixed iso-kinematic hardening for finite element

simulation of sheet metal forming. International Journal of Plasticity 26, 288 -

309.

Wilkins, M.L., 1963. Calculation of elastic-plastic flow, Other Information:

Orig. Receipt Date: 31-DEC-64, p. Medium: X; Size: Pages: 63.

Yoon, J.W., Yang, D.Y., Chung, K., 1999. Elasto-plastic finite element method

based on incremental deformation theory and continuum based shell elements for

planar anisotropic sheet materials. Comput Method Appl M 174, 23-56.






Zang, S.L., Guo, C., Thuillier, S., Lee, M.G., 2011. A model of one-surface

cyclic plasticity and its application to springback prediction. International

Journal of Mechanical Sciences 53, 425-435.

Chapter 7

7 Validation and evaluation of

the UMAT implementation of

anisotropic yield models

110 Chapter 7, Validation and evaluation of the UMAT implementation

7.1 Introduction

In the previous chapter, the stress-update scheme based on fully implicit backward

Euler method for non-AFR based yield models and mixed hardening was described.

This algorithm has been implemented in the commercial FE code Abaqus/Standard by

means of a user material subroutine (UMAT). This chapter is devoted to the

verification, application and evaluation of these implementations.

In Section 7.2, the systematic validation of the non-AFR Yld2000-2d model with

mixed isotropic-kinematic hardening is presented. This validation is performed in

terms of anisotropy and hardening predictions. The corresponding FE simulation

results are compared with an explicit program to ensure sound development of the

code. In Section 7.3, various implemented AFR and non-AFR based models are

compared and results in terms of yield stresses and Lankford coefficients are

provided. Finally, in Section 7.4, cup drawing simulation results are presented.

7.2 Validation of the developed UMAT subroutine

Validation of the developed UMAT subroutine in terms of hardening for uniaxial,

cyclic and shear loading conditions was carried out by comparison with numerical

data obtained by explicit programming. Explicit programming was performed in the

symbolic manipulation software package Mathematica® and is based on a simple

forward Euler approach with increments of equivalent plastic strain equal to 10E-6.

The explicit program was developed to generate the cyclic hardening behaviour of the

aluminium alloy AA5754-O. This alloy is being used in automobile structural

members and shows considerable stress shift (permanent softening) upon cyclic

loading. However, its mechanical behaviour in terms of both normalized yield stresses

(normalized with respect to the rolling direction yield stress) and Lankford

coefficients at different orientations shows a negligible level of anisotropy. The planar

anisotropy of this alloy was experimentally determined as 0.065. Consequently, the

aluminum alloy AA5754-O was chosen to verify the accuracy of the developed

subroutine in terms of cyclic hardening. Various hardening models (isotropic,

kinematic and Zang’s mixed hardening models) were programmed into the UMAT for

FE simulation. However, for purpose of validation of the UMAT, only the mixed

isotropic-kinematic hardening model of Zang (2011) is considered. This hardening

model is compared with isotropic and kinematic hardening models in Section 7.3.1. It

is worth noting that the UMAT contains all of the mentioned hardening models.

To verify the non-AFR Yld2000-2d model with mixed isotropic-kinematic hardening

in terms of anisotropy, the highly textured AA2090-T3 aluminium alloy was selected.

This material exhibits severe anisotropic behaviour. For instance it has a planar

anisotropy equal to -1.125. Besides, cup drawing experiments revealed that this

material develops 6 ears (Chung et al., 1996; Yoon et al., 2000). It is at this point

worth mentioning that the majority of the classical anisotropic yield functions cannot

111

predict more than 4 ears; the polynomial yield function proposed by Gotoh (1977) is

one of the exceptions. In addition to the AA2090-T3 alloy, a fictitious material (FM8)

introduced by Yoon et al. (2006) is considered for evaluation of the non-AFR

Yld2000-2d model. This material exhibits a considerable anisotropic behaviour in a

sense that two maxima are observed in the Lankford coefficients between rolling

direction and transverse direction.

In Fig 7-1, the directional yield stresses and Lankford coefficients for the materials

mentioned above are presented.

An optimization code was developed in Mathematica® to perform the model

parameter identification based on an inverse approach. The optimization procedure is

described in detail in Appendix B.

Fig 7-1 Comparison of directional Lankford coefficients (left) and normalized yield

stresses (right) for aluminium alloys AA2090-T3 and AA5754-O and for fictitious

material FM8.

7.2.1 Model validation in terms of hardening

Fig 7-2 depicts the single element FE tension test at degrees orientation with regard

to the rolling direction. Applying a compressive load in the uniaxial direction

provides tension-compression (T/C) loading to simulate a cyclic hardening curve.

This section discusses numerical T/C tests for aluminium alloy AA5754-O in the

rolling direction only. Other orientations will be considered in the next section for

evaluation of anisotropy using other materials. It must be noted that all hardening

models considered for AA5754-O are combined with the non-AFR Hill 1948 yield

model. The experimental directional yield stress ratios and Lankford coefficients are

adopted from Lee et al. (2007) and are respectively presented in Table 7-1 and Table

7-2. The parameters of the non-AFR Hill 1948 yield and plastic potential functions

are presented in Table 7-3. Isotropic elasticity is assumed for all studied materials

with Young’s modulus =70GPa and Poisson’s ratio =0.33.


Fig 7-2 Schematic illustration of a single element tensile test with material orientation

along degrees with regard to rolling direction.

Table 7-1 Experimental/Input normalized yield stresses

0° 15° 30° 45° 60° 75° 90° b

AA2090-T3 (Exp.) 1.000 0.961 0.910 0.811 0.810 0.882 0.910 1.035

AA5754-O (Exp.) 1.000 - - 0.923 - - 0.938 0.996

FM 8 (Input) 1.000 1.020 1.045 1.050 1.045 1.020 1.000 1.000

Table 7-2 Experimental/Input r-value coefficients

0° 15° 30° 45° 60° 75° 90° b

AA2090-T3 (Exp.) 0.212 0.327 0.692 1.577 1.039 0.538 0.692 0.670

AA5754-O (Exp.) 0.760 - - 0.710 - - 0.790 -

FM8 (Input) 0.600 1.000 0.750 0.300 0.750 1.000 0.600 1.000

Table 7-3 Non-AFR Hill 1948 model parameters

AA2090-T3 0.175 2.238 0.427 0.637 2.571 1.207

AA5754-O 0.432 1.349 0.978 0.556 1.576 0.975

FM8 0.375 1.000 1.000 0.500 1.314 1.000

In Zang’s (2011) mixed hardening model the isotropic hardening function is described

in the following form

( ) ( )

⁄ (

) (7-1)

The kinematic hardening component of Zang’s model consists of a two-term

Chaboche model in which the first term is a non-linear Armstrong and Frederick

model and the second term is Ziegler’s linear kinematic hardening.

(7-2a)

(7-2b)

22

11

xx (RD)

yy (TD)

ll0

113

(7-2c)

where , , are material parameters. Table 7-4 provides the hardening parameters

for the aluminium alloy AA5754-O.

Fig 7-3 shows the results of the numerical T/C test with compression starting at 0.078

pre-strain in tension for both the implicit UMAT subroutine and the explicit

Mathematica® program. The total back-stress and isotropic hardening components as

well as the total true stress of Zang’s hardening law are also plotted in Fig 7-3. The

excellent agreement between the curves obtained by UMAT and Mathematica® is

evident. This guarantees that the developed UMAT accurately represents the

implemented hardening model.

Table 7-4 Hardening parameters

AA5754-O AA2090-T3

& FM8

Parameter Unit Isotropic

(Swift)

Kinematic

(two-term Chaboche)

Mixed

hardening

(Zang)

Isotropic

(Swift)

σ0 MPa 94.8 94.8 94.8 279.6

k MPa 452.6 - - 646

n - 0.34 - - 0.227

ε0p - 0.01 -

0.025

Q MPa - - 126.4 -

b - - - 16.1 -

c1 MPa - 1997.3 4665.3 -

γ - - 23 212 -

c2 MPa - 409.6 204.8 -

As an additional verification, the isotropic hardening of aluminium alloy AA2090-T3

predicted by the non-AFR Yld2000-2d model (UMAT) is compared with the

analytical Swift curve. The experimental directional yield stress ratios and Lankford

coefficients are respectively provided in Table 7-1 and Table 7-2. The parameters of

the non-AFR Yld2000-2d model are presented in Table 7-5.

Table 7-5 AFR and non-AFR Yld2000-2d model parameters

m AA2090-T3

Potential ( ) -0.856 1.154 -0.293 0.326 0.683 0.482 0.752 1.024 8

Yield ( ) -0.713 2.037 1.629 0.69 0.552 -1.057 1.255 -1.263 8

AFR 0.488 1.377 0.754 1.025 1.036 0.904 1.231 1.485 8

FM8

Potential ( ) 2.946 -2.946 0.399 1.421 -1.421 -0.399 1.213 -1.822 8

Yield ( ) 0.814 1.002 2.129 0.571 -0.34 1.224 0.975 -1.123 8

AFR 0.958 0.958 0.968 1.016 1.016 0.968 0.863 1.012 8


Fig 7-3 Tension-compression results for aluminium alloy AA5754-O obtained from

UMAT and Mathematica® code.

Recall from Eqn.(3-3) that Swift isotropic hardening law is defined by

(

) (7-3a)

(

)

(7-3b)

and are material parameters. These parameters are presented in Table 7-4. In

Fig 7-4, the results obtained with both UMAT and Eqn.(7-3) are compared. Again,

excellent accuracy of the implemented material subroutine is proven by exact fit to

the analytical hardening curve.

Fig 7-4 Uniaxial hardening curve for aluminium alloy AA2090-T3 in rolling direction

as predicted by UMAT and theoretical Swift law.

Considering the validation of a developed constitutive model implemented by a user

material subroutine into FE code, it is highly recommended to evaluate the hardening

115

curve under more complex loading conditions such as simple shear test (Dunne and

Petrinic, 2005). The smooth simulated stress against plastic strain adds to the degree

of reliability of the model even if no experimental result is available for such complex

loading condition. This statement is only valid when the model has proven its

accuracy under proportional loading conditions such as those presented earlier for

AA2090-T3 and AA5754-O. To this end, we performed a numerical simple shear test

on a single element model for aluminium alloy AA2090-T3 based on the non-AFR

Yld2000-2d model. The single element test is schematically illustrated in Fig 7-5. In

Fig 7-6 the results of this single element shear test are presented. In the same plot, the

denotes the engineering plastic shear strain. The smooth predicted stress

components are apparent in that plot and add to the reliability of the model. Note that

the simulation of these curves includes the effects of anisotropy.

Fig 7-5 Schematic illustration of a single element simple shear test with material

orientation along RD.

Fig 7-6 Normal and shear stress components versus plastic shear strain for numerical

simple shear test on aluminium alloy AA2090-T3 at RD predicted by UMAT.

Simple shearPure shear

xx (RD)

yy (TD)

xx

yy

U1

Initial state

b

Deformed state

yy (TD)

xx (RD)

tan()=U1/b

xy

xx

yy

U1Initial state

b

Deformed state

tan()=U1/b

xy

0.0 0.1 0.2 0.3 0.4 0.5-300

-225

-150

-75

0

75

150

225

300

p

Shear

str

ess (

MP

a)

xx

yy

xy

shear at 0° U1=0.5 mmAA2090-T3


Considering the shear test, the normalized shear stress at yielding is analytically

derived by . This value is obtained through (i.e. see Appendix B)

( ) (7-4)

On the other hand, the results of the virtual single element shear test (Fig 7-6)

obtained by finite element simulation using UMAT gives .

Considering pure shear test i.e. (Barlat et al., 2003), both UMAT

and an inverse approach in Mathematica® determine the initial yielding at

.

The next section provides the results of verification in terms of anisotropy.

7.2.2 Model validation in terms of anisotropy

The directional normalized yield stresses and Lankford coefficients predicted by FE

simulation and analytical approach for aluminium alloy AA2090-T3 are presented in

Fig 7-7 and Fig 7-8 respectively. The FE simulation results are based on the non-AFR

Yld2000-2d model and Swift isotropic hardening law. The analytical results are

obtained by Mathematica®. The experimental normalized yield stresses and Lankford

coefficients (r-values) at different orientations for this alloy are presented respectively

in Table 7-1 and Table 7-2. These mechanical properties are reported in Chung et al.

(1996) and Yoon et al. (2000). The experimentally obtained values for

and were used for optimization of the AFR based

Yld2000-2d model parameters.

Fig 7-7 Distribution of normalized tensile yield stresses for aluminium alloy AA2090-

T3.

The experimental values for and

respectively were used to optimize the model

117

parameters of Yld2000-2d plastic potential and yield functions. The optimized

parameters of the yield and plastic potential functions are available in Table 7-5.

Fig 7-8 Distribution of Lankford coefficients for aluminium alloy AA2090-T3.

Comparison of predicted normalized yield stresses and Lankford coefficients is

presented in Fig 7-7 and Fig 7-8 respectively. The excellent fit between FE simulation

and analytical approach verifies the finite element result denoting the soundness of

the developed non-AFR based UMAT. Originally, this subroutine has been developed

for the non-AFR Yld2000-2d model (and non-AFR Hill 1984 model) with Zang’s

mixed hardening. However, this model can simply switch to any combination of AFR,

pure isotropic hardening (Swift, Voce or combined Swift-Voce) and two-term

Chaboche kinematic hardening by simple user inputs.

7.3 Evaluation of various AFR and non-AFR models

In Section 7.2, the validity of the finite element model was shown. This section

mainly aims to compare the non-AFR Yld2000-2d model, the non-AFR Hill 1948

model and their AFR counterparts. But, first, the non-AFR Hill 1948 model combined

with various hardening laws is studied.

7.3.1 Comparison of various hardening models

For the aluminium alloy AA5454-O, experimental tension/compression results at

rolling direction and different amounts of pre-strain (0.025, 0.050 and 0.078) are

available from Lee et al. (2005). These test conditions were simulated using different

hardening models: a) isotropic hardening according to Swift, b) two-term kinematic

hardening of classical Chaboche and c) Zang’s model for mixed isotropic-kinematic

hardening. The isotropic and mixed hardening laws have been described earlier in

Section 7.2.1. Material parameters of the implemented hardening laws are available in

Table 7-4. Furthermore, all simulations are based on the non-AFR Hill 1948 model.


Fig 7-9 Comparison of experimental and simulated tension/compression hardening

curves at different pre-strains for aluminium alloy AA5754-O.

Predicted and experimental tension/compression curves are plotted in Fig 7-9. It is

seen that the kinematic and isotropic hardening laws result respectively in an under-

and overestimation of the stress upon load reversal. The stress overestimation by

using isotropic hardening is simply due to missing the Bauschinger effect. As seen in

the same plot, both under- and overestimation are increased at higher pre-strains. It is

also noticed that the permanent softening effect is more pronounced at higher pre-

strains. As opposed to the isotropic and kinematic hardening definitions, application

of Zang’s mixed hardening model leads to simulated stress versus plastic strain curves

that are very close to the experimental ones. The effect of various hardening models

on cup drawing simulations of aluminium alloy AA5754-O will be presented in

Section 7.4.

7.3.2 Comparison of various anisotropic yield models

Predictive capabilities of various anisotropic yield functions in terms of directional

normalized yield stresses and Lankford coefficients are compared in this section. The

aluminium alloy AA2090-T3 and the fictitious material FM8 represent the selected

anisotropic materials. The experimental/input normalized yield stresses and Lankford

coefficients are available respectively in Table 7-1 and Table 7-2. Both associated and

non-associated versions of Hill 1948 and Yld2000-2d models were chosen as

anisotropic yield functions. The parameters of AFR and non-AFR Hill 1948 models

can be found in Table 7-3. In Table 7-5 parameters of plastic potential and yield

functions of the Yld2000-2d model are presented. The obtained simulation results are

also compared with simulations based on the Yld2004-18p anisotropic yield function

with 18 material parameters, reported in recent work of Yoon et al. (2006).

Simulated normalized yield stresses for AA2090-T3 and FM8 respectively are plotted

in Fig 7-10 and Fig 7-11. The predicted values from all developed models are

119

presented in Table 7-6. It is observed that the anisotropic predictive capability in

terms of normalized yield stress obtained by the yield function of Yld2000-2d (non-

AFR Yld2000-2d) is dominant over other models including Yld2004-18p. Moreover,

non-AFR Hill 1948 and AFR Yld-2000-2d are only accurate at 0°, 45° and 90°. In the

same figures, a weak prediction of normalized yield stresses by non-AFR Hill 1948

potential function is observed.

Fig 7-10 Distribution of normalized tensile yield stress for aluminium alloy AA2090-

T3.

Fig 7-11 Distribution of normalized tensile yield stress for fictitious material FM8.


Table 7-6 Simulated normalized yield stresses

0° 15° 30° 45° 60° 75° 90° b

AA2090-T3

Experimental 1.000 0.961 0.910 0.811 0.810 0.882 0.910 1.035 Potential Yld2000-2d ( ) 1.451 1.449 1.380 1.281 1.379 1.711 1.955 1.546 Yield Yld2000-2d ( ) 1.000 0.960 0.910 0.811 0.810 0.881 0.910 1.035 AFR Yld2000-2d 1.000 0.957 0.867 0.811 0.821 0.876 0.910 1.035 Potential Hill's 1948 ( ) 1.000 0.941 0.857 0.849 0.964 1.256 1.531 0.963 Yield Hill's 1948 ( ) 1.000 0.946 0.856 0.811 0.826 0.878 0.910 1.035

FM8

Input 1.000 1.020 1.045 1.050 1.045 1.020 1.000 1.000 Potential Yld2000-2d ( ) 0.618 0.615 0.661 0.691 0.661 0.615 0.618 0.554 Yield Yld2000-2d ( ) 1.000 1.020 1.045 1.050 1.045 1.020 1.000 1.000 AFR Yld2000-2d 1.000 1.012 1.037 1.050 1.037 1.012 1.000 1.000 Potential Hill's 1948 ( ) 1.000 1.024 1.079 1.109 1.079 1.024 1.000 0.894 Yield Hill's 1948 ( ) 1.000 1.012 1.037 1.050 1.037 1.012 1.000 1.000

Fig 7-12 Distribution of Lankford coefficient for aluminium alloy AA2090-T3.

In Fig 7-12 and Fig 7-13, the simulated directional Lankford coefficients of

respectively AA2090-T3 and FM8 are presented. The predicted values from all

developed models are presented in Table 7-7. Similarly to the observations made for

the normalized yield stress predictions, it is evident that the plastic potential function

of non-AFR Yld2000-2d generates the highest accuracy of prediction compared with

all other studied models. For instance, in Fig 7-12 for aluminium alloy AA2090-T3, it

is observed that the peak and trough (local minimum) respectively at 45° and 75°

orientations are accurately predicted by the Yld2004-18p and non-AFR Yld2000-2d

models. As will be shown in the next section, accurate prediction of these peak and

trough results in an accurate prediction of the number and height of the ears in a deep

121

drawn cup. It is also observed from Fig 7-13 that the non-AFR Yld2000-2d model

excellently predicts the input Lankford coefficients of the challenging fictitious

material FM8.

Fig 7-13 Distribution of Lankford coefficient for fictitious material FM8.

Table 7-7 Simulated Lankford coefficients

0° 15° 30° 45° 60° 75° 90° b AA2090-T3

Experimental 0.212 0.327 0.692 1.577 1.039 0.538 0.692 0.670 Potential Yld2000-2d ( ) 0.232 0.286 0.771 1.539 1.142 0.539 0.680 0.677 Yield Yld2000-2d ( ) -0.713 2.037 1.629 0.689 0.552 -1.057 1.255 -1.263 AFR Yld2000-2d 0.211 0.406 1.063 1.577 1.348 0.888 0.692 0.670 Potential Hill's 1948 ( ) 0.212 0.436 0.998 1.577 1.722 1.182 0.692 0.306 Yield Hill's 1948 ( ) 1.753 1.964 2.285 2.254 1.829 1.331 1.117 1.570

FM8

Input 0.600 1.000 0.750 0.300 0.750 1.000 0.600 1.000 Potential Yld2000-2d ( ) 0.600 1.000 0.750 0.300 0.750 1.000 0.600 1.000 Yield Yld2000-2d ( ) 0.505 0.493 0.864 1.911 2.981 4.503 6.060 0.051 AFR Yld2000-2d 0.600 0.515 0.364 0.300 0.365 0.514 0.600 1.000 Potential Hill's 1948 ( ) 0.600 0.525 0.375 0.300 0.375 0.525 0.600 1.000 Yield Hill's 1948 ( ) 1.000 0.953 0.860 0.814 0.860 0.953 1.000 1.000

A thorough comparison of AFR Yld2000-2d and non-AFR Hill 1948 models based on

all results provided in Fig 7-10 to Fig 7-13, Table 7-6 and Table 7-7 leads to the

following three conclusions. Firstly, both AFR Yld2000-2d and non-AFR Hill 1948

models generate accurate and similar results for 0°, 45° and 90° orientations.


Secondly, considering the results for other in-plane orientations, the AFR Yld2000-2d

model generates a higher accuracy. And third, as opposed to the non-AFR Hill 1948

model, the AFR Yld2000-2d model provides better results for the balanced biaxial

stress state.

7.3.3 Spatial representation

Two-dimensional representations of various yield and plastic potential functions are

presented in Fig 7-14 and Fig 7-15 respectively for aluminium alloy AA2090-T3 and

fictitious material FM8. These functions include von Mises, Hill 1948 yield and

plastic potential functions, Yld2000-2d yield and plastic potential functions. The

black dots in the yield function representations denote the experimental/input values.

To provide a more complete image, 2D representations including the shear stress

component are also presented. Moreover, the three dimensional representations of

AFR Yld2000-2d and non-AFR Yld2000-2d model components (plastic potential and

yield functions) for AA2090-T3 and FM8 are shown in Fig 7-16 and Fig 7-17

respectively. From these plots, it can be concluded that a reasonable degree of

complexity can be modeled by the non-AFR formulation as compared to the AFR

models.

123

Fig 7-14 Two-dimensional representation of various yield and plastic potential

functions for aluminium alloy AA2090-T3.


Fig 7-15 Two-dimensional representation of various yield and plastic potential

functions for fictitious material FM8.

125

Fig 7-16 Three-dimensional representation of AFR and non-AFR Yld2000-2d models

(plastic potential and yield function) for aluminium alloy AA2090-T3. Small dots

denote experimental yield points at seven uniaxial directions and balanced biaxial

stress state.


Fig 7-17 Three-dimensional representation of AFR and non-AFR Yld2000-2d models

(plastic potential and yield function) for fictitious material FM8. Small dots denote

input yield points at seven uniaxial directions and balanced biaxial stress state.

127

7.3.4 In-plane flow direction

According to the non-associated flow rule concept, the outward normal to the plastic

potential function determines the plastic strain rate direction and consequently the

corresponding Lankford coefficient. The outward normal direction to the yield

function (in case of an AFR model) or plastic potential function (considering a non-

AFR variant) is schematized in Fig 7-18. At an in-plane loading direction of

degrees with respect to the rolling direction, the plastic strain rate direction makes an

angle with the rolling direction. The variation of as function of for aluminium

alloy AA2090-T3 and fictitious material FM8 is shown in Fig 7-19 and Fig 7-20

respectively.

It is noted that, in general, the classical models of von Mises and Hill 1948 present a

simple variation of plastic strain rate direction. For both materials the Yld2000-2d

yield and plastic potential functions can predict a more detailed variation.

Fig 7-18 Schematic illustration of plastic strain rate direction (

) at an in-plane

loading direction of degrees with respect to the rolling direction.


Fig 7-19 Evolution of plastic strain rate orientation as function of loading direction

obtained by various yield and plastic potential functions for aluminium alloy

AA2090-T3.

Fig 7-20 Evolution of plastic strain rate orientation as function of loading direction

obtained by various yield and plastic potential functions for fictitious material FM8.

7.4 Cup drawing simulations

The results of cup drawing simulations for aluminium alloys AA2090-T3 and

AA5754-O are presented in this part. For alloy AA2090-T3, different anisotropic

models (AFR and non-AFR versions of Hill 1948 and Yld2000-2d) were studied and

results were compared with experimentally determined cup profiles. Regarding the

alloy AA5754-O, the cup drawing simulations are solely based on the non-AFR Hill

1948 model combined with different hardening models. No experimental cup drawing

result are available for this alloy. However, as we will see later, the simulation data

are interesting in a sense that the under- and overestimation of springback due to the

use of kinematic and isotropic hardening in simulations of U-bend tests (Lee et al.,

2007) is reflected as under- and overestimation of the simulated cup profile.

129

The cup drawing process is schematized in Fig 7-21 and the tool dimensions are given

in Table 7-8. A sheet thickness of 1.6 mm was assigned to the specimens in all

simulations. In the light of material orthotropy, only one quarter of the sheet was

modeled. The blanks of aluminium alloys AA5754-O and AA2090-T3 are modeled

using respectively 2147 and 3800 first order reduced integration quadrilateral shell

elements with respectively 10 and 15 Gauss integration points. The initial meshes are

shown in Fig 7-22. The difference in number of elements was due to using two

different computational sources for the simulations.

A blank holder force of 5.5kN for the quarter model (corresponding to approximately

1% of the initial yield stress in rolling direction) was implemented. This force was

found to be high enough to avoid wrinkling in the rim area. Coulomb friction with a

friction coefficient equal to 0.1 was assumed.

Typical simulation results are shown in Fig 7-23 which shows the evolution stresses

during four different stages of the cup drawing process.

Fig 7-21 Tool and blank geometries for cylindrical cup drawing.

Table 7-8 Tool dimensions (in mm)

Dp Dd Db rp rd g t

97.46 101.48 158.76 12.7 12.7 2.7 1.6


Fig 7-22 Initial mesh for aluminium alloys AA2090-T3 (left) and AA5754-O (right).

Fig 7-23 Simulation of cup deep drawing process for aluminium alloyAA2090-T3

using the non-AFR Yld2000-2d model. The legend denotes the equivalent stress.

Three hardening definitions were evaluated based on cup height prediction of

aluminum alloy AA5754-O. As before, the hardening definitions are a) isotropic

hardening according to Swift b) two-term kinematic hardening of classical Chaboche

and c) Zang’s mixed isotropic-kinematic model. The cup height predictions using the

non-AFR Hill 1948 model are plotted in Fig 7-24 for a quarter model. From this

figure, the over- and under-estimation of the flow curve respectively obtained by

isotropic and kinematic hardening as was observed in Fig 7-9 is also reflected in the

cup profile. Due to a more realistic prediction of tension/compression behaviour,

Zang’s mixed hardening definition avoids these under- and overestimations.

Interestingly, the same trend has been observed for the U-bend springback test for an

identical material using isotropic, kinematic and an accurate two-surface hardening

model (Lee et al., 2007).

131

Fig 7-24 Cup height prediction for aluminium alloy AA5754-O using non-AFR Hill

1948 model and three hardening definitions.

Fig 7-25 shows simulated and experimental cup height profiles of aluminium alloy

AA2090-T3 using AFR and non-AFR versions of Hill 1948 model considering Swift

isotropic hardening law. The experimental cup profile exhibits four big ears located

at 45°, 135°, 225° and 315° orientations and one small ear at 180°. However, none of

the presented variations of Hill 1948 is able to predict more than four ears. That is due

to the insufficient number of material parameters. The predicted troughs (local

minima) for all three Hill 1948 models are weak except for S-based Hill 1948. On the

other hand, respectively Hill’s non-AFR based and R-based models predict a cup

height closest to the experimental ones.

In Fig 7-26, cup height profiles predicted by AFR and non-AFR versions of the

Yld2000-2d model are presented. For sake of comparison, the non-AFR Hill 1948

model is also plotted in this figure. Among these models, only the non-AFR Yld2000-

2d model predicts the small ear at 180° in addition to the big ears and therefore

presents the correct number of ears in accordance with experimental results. The

predicted height of the large peaks slightly improves for the non-AFR Yld2000-2d

model as compared with the non-AFR Hill 1948 model. Similar to the S-based Hill

1948 model in Fig 7-25, the AFR Yld2000-2d model underestimates the cup height at

the main 4 peaks. However, the same model predicts the troughs closer to the

experimental ones when compared with R-based Hill 1948 and non-AFR Yld2000-2d

models. Fig 7-27 illustrates the predicted final cup shape, von Mises stresses and

effective plastic strains for aluminium alloy AA2090-T3 using the non-AFR Hill 1948

model and AFR and non-AFR versions of Yld2000-2d model.


Fig 7-25 Cup height prediction for aluminium alloy AA2090-T3 using AFR and non-

AFR Hill 1948 models.

Fig 7-26 Cup height prediction for aluminium alloy AA2090-T3 using AFR and non-

AFR Yld2000-2d and non-AFR Hill 1948 models.

133

Fig 7-27 Final cup shape and distribution of von Mises stress (left) and effective

plastic strain (right). From top to bottom respectively, non-AFR Hill 1948 model,

AFR Yld2000-2d model and non-AFR Yld2000-2d model.


7.5 Summary

In this chapter, first the verification of the developed subroutine and implemented

models was presented. This is an indispensable step prior to applying them for larger

simulations. The methodology encompasses verification of the model in both

hardening and anisotropy predictions. Next, a comparison was made of simulation

results using AFR and non-AFR versions of the quadratic Hill 1948 model and of the

Yld2000-2d model in terms of one-dimensional normalized yield stresses and

Lankford coefficients. It was shown that excellent results are achieved using the non-

AFR 2000-2d model. It was also shown that a same order of accuracy as obtained by

the 18 parameter model Yld2004-18p can be achieved by the non-AFR Yld2000-2d

model.

In the last part, results of cup drawing simulations for aluminium alloys AA5754-O

and AA2090-T3 were presented. Cup drawing simulations for alloy AA5754-O using

isotropic, kinematic and mixed hardening definitions show that the over- and

underestimation of the hardening curve generated by isotropic and kinematic

hardening respectively, is directly observed in the predicted cup height. For

aluminium alloy AA2090-T3, the predicted cup height for isotropic hardening and

using the non-AFR Yld2000-2d model shows considerable improvement compared

with the AFR Yld2000-2d model as well as other presented models. It was shown that

only the non-AFR Yld2000-2d model can predict the exact number of 6 ears as

observed in experimental results. However, a slight improvement compared to the

AFR Hill 1948 model was observed by using its non-AFR counterpart. It is worth

noting that the simulation of cup heights can be further improved by choosing optimal

coefficient of friction and blank holder force.

135

Bibliography

Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J.,

Pourgoghrat, F., Choi, S.H., Chu, E., 2003. Plane stress yield function for

aluminum alloy sheets - part 1: theory. International Journal of Plasticity 19,

1297-1319.

Chung, K., Lee, S.Y., Barlat, F., Keum, Y.T., Park, J.M., 1996. Finite element

simulation of sheet forming based on a planar anisotropic strain-rate potential.



University Press, Oxford ; New York; ISBN 0198568266.

Gotoh, M., 1977. A theory of plastic anisotropy based on a yield function of

fourth order (plane stress state)—I. International Journal of Mechanical Sciences

19, 505-512.

Lee, M.-G., Kim, D., Kim, C., Wenner, M.L., Wagoner, R.H., Chung, K., 2005.

Spring-back evaluation of automotive sheets based on isotropic-kinematic

hardening laws and non-quadratic anisotropic yield functions: Part II:

characterization of material properties. International Journal of Plasticity 21,

883-914.

Lee, M.G., Kim, D., Kim, C., Wenner, M.L., Wagoner, R.H., Chung, K.S., 2007.

A practical two-surface plasticity model and its application to spring-back

prediction. International Journal of Plasticity 23, 1189-1212.

Yoon, J.W., Barlat, F., Chung, K., Pourboghrat, F., Yang, D.Y., 2000. Earing

predictions based on asymmetric nonquadratic yield function. International


Yoon, J.W., Barlat, F., Dick, R.E., Karabin, M.E., 2006. Prediction of six or

eight ears in a drawn cup based on a new anisotropic yield function.


Zang, S.L., Guo, C., Thuillier, S., Lee, M.G., 2011. A model of one-surface

cyclic plasticity and its application to springback prediction. International

Journal of Mechanical Sciences 53, 425-435.

Chapter 8

8 Simplification of the numerical

implementation of the non-

AFR model

138 Chapter 8. Simplification of the numerical implementation of the non-AFR

8.1 Introduction

As opposed to an associated flow rule (AFR) approach, independency of plastic

potential and yield functions is adopted in a non-AFR approach. This inherent

characteristic of a non-AFR model imposes a specific constraint (but not equality)

between equivalent plastic strain and plastic compliance factor. Unavoidably, this

leads to a laborious effort for FE implementation of constitutive models when a high

number of internal variables (such as kinematic hardening or damage parameters) is

involved. Consequently, this chapter is devoted to studying the conditions at which a

non-AFR model can be simplified to provide convenience of the numerical

implementation scheme without significant loss of accuracy.

Section 8.2 discusses the discretization of rate elasto-plasticity equations in which the

presence of kinematic hardening is not considered. In section 8.3, a simpler

implementation of non-AFR is described. In this approach a scaling method (further

referred to as scaled simplified) is proposed that guarantees the equality of equivalent

and longitudinal plastic strain in the rolling direction. Even more, the proposed

approach results in a minimal discrepancy for other orientations when compared with

the full method.

All presented approaches were implemented into commercial FE code Abaqus using

user material subroutines based on fully implicit backward Euler method. Section 8.4

provides the error analysis of directional hardening predicted by different non-AFR

approaches including the proposed scaled simplified and the full approaches. It must

be noted that the main difference between the different approaches is found in the

relation of equivalent plastic strain and compliance factor. Finally in section 8.5, the

different approaches are compared based on the cup drawing simulation of highly

textured aluminum alloy AA2090-T3.

8.2 Discretization of rate elasto-plasticity equations

Application of the so-called scaled simplified non-AFR model rather than the full

expression (non-simplified) is motivated in this section. The aim is to alleviate the

laborious task of a fully implicit backward Euler stress update algorithm with the

lowest possible error. But, let us be reminded of the prerequisite formulations. The

non-AFR model accounts for a relation between plastic strain increments (

and a

plastic potential function based on the following relation

(8-1)

where

(8-2)

139

is the plastic multiplier factor and the second order tensor

is the plastic flow

direction. In other words,

denotes the normal to the plastic potential function .

The principle of plastic work equivalence yields the relation between equivalent

plastic strain and plastic strain tensor.

(8-3)

being the yield function. Euler’s theorem for any first order homogeneous function

states

(8-4)

Applying this on or results in

(8-5a)

(8-5b)

The second order tensor

is the normal to the yield surface

(8-6)

Substituting the non-associated plastic flow rule Eqn.(8-1) in the principle of plastic

work equivalence Eqn.(8-3) and applying Euler’s theorem Eqn.(8-5a) results in

( )

( )

( )

( ) (8-7)

Application of Eqn.(8-7) is motivated by the accuracy of this model since no

simplifying assumption is considered. However, with special care (to be discussed in

this chapter) Eqn.(8-7) can be reduced to

(8-8)

It must be noted that this simplification does not lead back to AFR. This is due to the

fact that the normal to the plastic potential function (and not yield stress function)

always has been considered for plastic strain rate direction.

Eqn. (8-8) results in a simpler implementation of the non-AFR constitutive model and

has therefore been adopted by Stoughton (2002), Stoughton and Yoon (2006; 2007;


2009) and Taherizadeh et al (2010). However, this assumption weakens the reliability

of the model since it violates the principle of plastic work equivalence thus possibly

leading to a large discrepancy of the FE results. It must be noted that the application

of the reduced formulation as defined in Eqn.(8-8) can only be motivated in the case

of a fully implicit backward Euler integration scheme described in Eqn. (8-9). This is

due to the necessity of calculation of derivatives of all plastic moduli such as in

Eqn.(8-7). For other integration schemes such as forward Euler, cutting plane, semi-

implicit and semi-explicit this is not the case.

As the fully implicit integration scheme has been adopted, the increments in the

plastic strain and flow direction are attributed to the current step denoted by .

This is indicated in the following integration scheme written for the non-AFR model

with isotropic hardening and general anisotropic yield function.

(8-9a)

(8-9b)

(

)

(8-9c)

(

) (8-9d)

( ) (8-9e)

This set of equations has to be linearized with respect to at time increment

. The differences between right and left sides of Eqn.(8-9a) to

Eqn.(8-9e) define corresponding residual functions. The values of these residuals are

minimized within a very small tolerance by means of a Newton-Raphson iteration

scheme. More specifically, to include the full expression of Eqn.(8-9c) in the fully

implicit integration scheme a residual function is defined

(8-10)

Constructing the truncated Taylor expansion of leads to

(8-11)

The superposed dot denotes the incremental change during the k-th Newton-Raphson

iteration. Using Euler’s rule

(8-12a)

141

(8-12b)

Alternatively, if the simplified expression is used, then Eqn.(8-9c) can be substituted

by a linear function in which the derivative of can be eliminated

(8-13)

Apparently, the Eqns.(8-10) and (8-11) are omitted as a result of this simplification.

To this end, a special constraint referred to as scaling in this chapter has to be taken

into account to minimize the discrepancy between results of full and simplified

expressions.

Regarding the details of the integration scheme, Cvitanic et al (2008) presented the

fully implicit scheme for a non-associate flow rule in the case of isotropic hardening.

Taherizadeh et al (2010) and Safaei et al (2012) developed the fully implicit scheme

for mixed isotropic-kinematic hardening. Due to the fact that in this chapter only

isotropic hardening is considered, the integration scheme is similar to that of Cvitanic

et al (2008) so the reader is referred to that paper for further details.

8.3 Impact of simplification on equivalent plastic strain rate

For the sake of convenience, the following terminology is agreed upon

in Eqn.(8-7) full method

in Eqn.(8-8) with original un-scaled simplified method

in Eqn.(8-8) with scaled scaled simplified method

Based on these approaches, the non-AFR constitutive model is evaluated and results

are presented in the following subsections.

8.3.1 Full non-AFR method

From Eqn.(8-7) one obtains

(8-14)

If in Eqn.(8-14) is substituted into the non-associated flow rule in Eqn.(8-1), the

following equation is obtained

(8-15)


It would be advantageous to consider all equations in a normalized stress space;

normalized with respect to the yield stress in the rolling direction denoted by . In

the remainder of this section, the notation refers to this normalized stress. Therefore

the normalized uniaxial stress state with only a non-zero stress component in rolling

direction is represented by

(8-16)

Substitution of Eqn.(8-16) into Euler’s theorem Eqn.(8-5a) leads to

(8-17)

In normalized stress space yielding occurs when

(8-18)

Substitution of Eqn.(8-16) and Eqn.(8-18) into Euler’s theorem Eqn.(8-5b), leads to

(8-19)

If Eqn.(8-17) and Eqn.(8-19) are substituted into Eqn.(8-15) for a uniaxial stress state,

the following equation is obtained

(8-20)

In short, the full expression automatically equalizes the longitudinal plastic strain in

rolling direction (

) and the equivalent plastic strain ( ).

8.3.2 Un-scaled simplified non-AFR method

The un-scaled simplified model is based on elimination of the term in

Eqn.(8-9c) with no additional stipulations. Substitution of into the non-

associated flow rule in Eqn.(8-1) leads to

(

)

(8-21a)

or alternatively

(

)

(8-21b)

143

From Euler’s theorem in Eqn.(8-5a) and in case when yielding occurs, one

obtains that the first component of

for uniaxial stress along the rolling direction

does not equal 1

(8-22)

Consequently, combining Eqn.(8-22) with Eqn.(8-21a) for uniaxial stress state along

the rolling direction, leads to

(8-23)

Therefore, if no constraint is enforced to the un-scaled simplified approach, this

method results in a wrong relation between equivalent plastic strain and longitudinal

plastic strain for uniaxial stress along the rolling direction.

Considering a general stress state, determination of the ratio of equivalent plastic

strain obtained through the full method to equivalent plastic strain from the un-scaled

method, , can be calculated as the ratio of Eqn.(8-15) to

Eqn.(8-21a). After some manipulations

(8-24)

Fig 8-1 plots the variation of parameter for different stress states

at the first quarter of the Yld2000-2d yield surface (non-AFR Yld2000-2d model)

applied to aluminium alloy AA2090-T3. A cylindrical coordinate system is

introduced, Fig 8-2, to enable a more straightforward evaluation of the variation of

. In Fig 8-3 the parameter values are plotted for different shear

stress levels. This plot illustrates that the values and thus the

plastic potential are always less than one for aluminium alloy AA2090-T3 (note

that when yielding occurs). Consequently, substituting Eqn.(8-17) into Eqn.

(8-21a) for a uniaxial stress state along the rolling direction, leads to

(8-25)

From Eqn.(8-25) one can conclude that the un-scaled simplified approach

overestimates (overshoot) the equivalent plastic strain and consequently a difference

between experimental and simulated hardening curves for the rolling direction is to be

expected (see Fig 8-4 and Fig 8-5).


Fig 8-1 Variation of β for the un-scaled simplified non-AFR Yld2000-2d model

applied to aluminium alloy AA2090-T3.

Fig 8-2 Schematic of a cylindrical coordinate system.

Fig 8-3 Variation of β in function of the cylindrical coordinate for un-scaled

simplified non-AFR Yld2000-2d model applied to AA2090-T3.

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

xy

xy

xy

xy

xy

un-scaled simplified

xx

=1 ,yy

=0 ,

Yield surface

at

yy

xx

Material:

AA2090-T3

yy

xx

0 15 30 45 60 75 900.4

0.6

0.8

1.0

1.2

1.4

xy

xy

xy

xy

xy


AA2090-T3

(°)

145

Fig 8-4 Simulated stress versus longitudinal plastic strain (

) at rolling direction

using full and un-scaled simplified methods for the non-AFR Yld2000-2d model

applied to AA2090-T3.

Fig 8-5 Simulated stress versus equivalent plastic strain ( ) at rolling direction using

full and un-scaled simplified methods for the non-AFR Yld2000-2d model applied to

AA2090-T3.

8.3.3 Scaled simplified non-AFR method

A remedy to the observed shortcomings of the un-scaled simplified method would be

enforcing a scaling constraint for which at rolling direction

applies. This

can be accomplished by applying a scaling of the parameters (i=1-8) of the plastic

potential function

(8-26)

where

(

)

(8-27)


being the scaled plastic potential function, the i-th parameter of the plastic

potential function and k is the scaling factor. This scaling factor is equal to 1.451 for

the aluminium alloy AA2090-T3. This approach is referred to as scaled simplified

method.

Subsequently, the relation between plastic strain and equivalent plastic strain is recast

to

(8-28)

where

(8-29)

therefore

(8-30)

The flow rule takes the following form

(

)

(8-31a)

By substituting Eqn.(8-27) into (8-30) and recalling Euler’s theorem in Eqn.(8-5b) for

uniaxial tension along the rolling direction we can write

(8-32)

The constraint in Eqn.(8-32) is the outcome of the scaling procedure. Finally

Eqn.(8-28) for uniaxial tension in the rolling direction reduces to

(8-33)

leading to the equality of equivalent plastic strain and longitudinal plastic strain at

rolling direction what was not possible to achieve without scaling.

Similar to Eqn.(8-24), the the parameter can be obtained from

(8-34)

147

Fig 8-6 compares the variation of and for

different stress states at the first quarter of the Yld2000-2d yield surface (non-AFR

version). Fig 8-7 shows the top view of the same plot which simply allows to

conclude that the shape of the yield and plastic potential functions are not changed

upon any kind of simplification.

Fig 8-6 Comparison of for scaled simplified and un-scaled simplified non-AFR

Yld2000-2d model implementations for different values of shear stress and applied to

an aluminium alloy AA2090-T3.

Fig 8-7 Top view of Fig 8-6.

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

xy

xy

xy

xy

xy

Scaled simplified method

Un-scaled simplified method

xx=1 ,

yy=0 ,

Yield surface

at

yy

xx

xx Material:

AA2090-T3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 scaled simplified


xy

xy

xy

xy

xy

AA2090-T3

Y

Y

xx


In Fig 8-8, the and values are plotted for

different shear stress levels in a cylindrical coordinate system. In this plot it is shown

that the parameter equals 1 at uniaxial tension in rolling direction.

Fig 8-8 Variation of β in function of the cylindrical coordinate for scaled and un-

scaled simplified non-AFR Yld2000-2d model implementations for aluminium alloy

AA2090-T3.

In Fig 8-9, the scaling procedure is illustrated for 0° and 76° orientations of in-plane

stress state. In this plot, the superscripts “sc” and “un” respectively denote the scaled

simplified and un-scaled simplified approaches. In this plot, it must be noted that

(value of scaled plastic potential function) equals 1 at the rolling direction.

Fig 8-9 Examples of scaling procedure applied to in-plane stress states at 0° (right)

and 76° (left) orientations for aluminium alloy AA2090-T3 and non-AFR Yld2000-2d

model.

0 15 30 45 60 75 900.4

0.6

0.8

1.0

1.2

1.4

xy

xy

xy

xy

xy

scaled simplified


AA2090-T3

(°)

149

As a consequence of the scaling method, overshooting of equivalent plastic strain and

stress at rolling direction that was present in the un-scaled method is not a concern in

the scaled approach (see Fig 8-10 and Fig 8-11).

Fig 8-10 Simulated stress versus longitudinal plastic strain (

) at rolling direction

using full, un-scaled simplified and scaled simplified methods for the non-AFR

Yld2000-2d model applied to AA2090-T3.

Fig 8-11 Simulated stress versus equivalent plastic strain ( ) at rolling direction

using full, un-scaled simplified and scaled simplified methods for the non-AFR

Yld2000-2d model applied to AA2090-T3.

Nonetheless, for stress states other than this uniaxial one, the curves of versus

might deviate from those obtained using the full expression. For all methods including

full, scaled simplified and un-scaled simplified the equivalent versus uniaxial plastic

strains at each 15° from rolling to transverse direction as well as balanced biaxial

stress state are plotted in Fig 8-12. As proven higher, the equivalent and uniaxial

plastic strains for scaled simplified and full expression at rolling direction completely

match with each other and a very good agreement is also present for the data at 15°


and 30° orientations as shown in Fig 8-12. However, a slight deviation is observed

from 45° (less than 9% error) onwards and reaches a maximum at 90°. Interestingly,

the deviation of the un-scaled simplified method is much more pronounced. As will

be shown in Section 8.5, even though some deviation is observed from 45° to 90°

orientations for the scaled simplified approach, the results of cup deep drawing

simulations are very similar for both full and scaled simplified expressions.

Fig 8-12 Equivalent versus uniaxial and equi-biaxial plastic strains at different

orientations using scaled simplified and full expression approaches for the non-AFR

Yld2000-2d model and applied to aluminium alloy AA2090-T3.

151

8.4 Metrics for error analysis

In this section the error of the three approaches discussed in the previous section is

quantified for various loading conditions momentarily. First the formulation is

described from which the directional hardening can be extracted.

Formulating the non-associated flow rule in generic form for uniaxial tension

(8-35)

where and are respectively plastic strain and plastic flow direction for

uniaxial tension at degrees with respect to the rolling direction. The flow direction

in any orthogonal system can be projected to the uniaxial direction denoted by

subscript .

(8-36)

Combination of the three different approaches described earlier with Eqn.(8-35) for

uniaxial direction gives

(

)

(8-37)

(

)

(8-38)

(

)

(8-39)

Consequently, the uniaxial stress at degrees with respect to the rolling direction,

for the different approaches are described by:

Full approach:

(

)

(8-40)

Un-scaled simplified approach:

(

)

(8-41)

Scaled simplified approach:

(

)

(8-42)


For balanced biaxial stress the derivatives of plastic potential and yield functions are

respectively

(8-43)

(8-44)

Therefore the balanced biaxial stress for the different approaches is described by:

Full approach: (

)

(8-45)

Simplified un-scaled: (

)

(8-46)

Simplified scaled: (

)

(8-47)

where

(8-48)

Finally, the error between experimental and predicted stress-strain data by any of the

studied models is defined as

( )

(

)

( )

(8-49)

where ( ) is the experimental hardening curve at degrees with respect to the

rolling direction and true uniaxial strain . The parameters of the experimental

hardening curve originally from Rousselier et al. (2009, 2010) are provided in Table

8-1 for different orientations. Stoughton and Yoon (2009) proposed a criterion for

measuring the error at all orientations momentarily. The average RMS error over a

uniaxial true plastic strain range of is

( ) √

∫ (

)

(8-50)

The average RMS error for seven uniaxial directions is

√

(( (

))

( (

))

( (

))

( (

))

( ) (

) (

)

) (8-51)

153

Table 8-1 Swift hardening parameters for aluminium alloy AA2090-T3

Orientation

0 646.015 0.025 0.227

15 630.085 0.032 0.247

30 596.989 0.031 0.245

45 532.190 0.029 0.242

60 530.188 0.030 0.243

75 577.705 0.031 0.245

90 596.989 0.031 0.245

EB 821.355 0.059 0.369

Taking the error of predicted balanced biaxial stress into account

( )

(

)

( ) (8-52)

where ( ) is the experimental balanced biaxial stress-strain relation. Similarly, the

RMS error over a range is

( ) √

∫ (

)

(8-53)

Finally the overall model error including all considered uniaxial and balanced biaxial

stress states is

√

((

) (

) ) (8-54)


Fig 8-13 RMS error for prediction of stresses by full, scaled simplified and un-scaled

simplified non-associated flow models for 7 uniaxial and one balanced biaxial stress

conditions applied to aluminium alloy AA2090-T3.

155

Fig 8-14 Accumulated error in prediction of stress for alloy AA2090-T3 in uniaxial

loading conditions.

Fig 8-15 Accumulated error in prediction of stress for alloy AA2090-T3 in uniaxial

and balanced biaxial loading conditions.

8.5 Impact of model simplification on cup drawing simulation

The impact of the proposed scaled simplification of a non-AFR model is evaluated by

FE simulation of cylindrical cup deep drawing of aluminium alloy AA2090-T3. The

non-AFR Yld2000-2d anisotropic model using both full and scaled simplified

expressions is studied and results are compared with experimentally determined cup

profiles. The user material subroutines were developed for ABAQUS/Standard using

a fully implicit integration scheme.

The tool dimensions are similar to the ones described in the previous chapter. In the

light of orthogonal symmetry, only one quarter of the sheet was modeled. The blank


was meshed using approximately 700 first order reduced integration quadrilateral

shell elements with 11 Gauss integration points. The simulated cup profiles are

plotted in Fig 8-16. It is shown that the impact of simplification, even for a very

anisotropic material such as aluminium alloy AA2090-T3, is negligible. As a

conclusion it is proven that simplification of the non-AFR model, taking the scaling

of the plastic potential function into account, leads to very similar simulation of cup

profile as obtained using the full expression.

Fig 8-16 Simulated cup profile using full and scaled simplified expressions of non-

AFR Yld2000-2d model.

8.6 Summary

Two methods for simplification of the relation between equivalent plastic strain and

compliance factor in a non-AFR model have been described. It was shown that if the

non-AFR is simplified without scaling the plastic potential function, this results in a

wrong definition of equivalent plastic strain. This inaccurate definition was shown to

lead to overestimation of Cauchy stress for aluminium alloy AA2090-T3 due to over-

prediction of equivalent plastic strain. However, it was shown that this can be

significantly improved if the plastic potential function is scaled based on the data at

uniaxial stress state. If the scaling is carried out based on the ratio of yield to plastic

potential functions then the equality of equivalent plastic strain and compliance factor

can be assured resulting in a more convenient implementation scheme. FE simulation

of uniaxial tensile tests at different orientations for the non-AFR Yld2000-2d model

showed that for both full expression and scaled simplified approach the equivalent

and uniaxial plastic strains at rolling direction completely match and the good

agreement still holds for 15° and 30°. However, slight deviation is observed at 45°

and reaches a maximum at 90°. Nonetheless, the cup drawing simulations of alloy

AA2090-T3 using the scaled simplified approach show a very similar cup profile as

compared to the one obtained by the full expression. These results indicates that

implementation of the scaling technique gives rise to a reliable alternative for the full

expression.

157

Bibliography



646-687.

Rousselier, G., Barlat, F., Yoon, J.W., 2009. A novel approach for anisotropic

hardening modeling. Part I: Theory and its application to finite element analysis of

deep drawing. International Journal of Plasticity 25, 2383-2409.

Rousselier, G., Barlat, F., Yoon, J.W., 2010. A novel approach for anisotropic

hardening modeling. Part II: Anisotropic hardening in proportional and non-

proportional loadings, application to initially isotropic material. International Journal


Safaei, M., Zang, S.L., Lee, M.G., De Waele, W., 2012. Evaluation of Anisotropic


Approach. International Journal of mechanical Sciences (accepted).







1817.








Chapter 9

9 Comparison of stress-

integration schemes

160 Chapter 9. Comparison of stress-integration schemes

9.1 Introduction

In Chapter 6, the numerical formulation and implementation into a user material

subroutine of a fully implicit stress integration scheme were presented for non-AFR

based yield models and mixed isotropic-kinematic hardening. Simulation results

obtained using Abaqus/Standard, including verification of hardening and anisotropy

and cylindrical cup deep drawing were discussed in Chapter 7. In this chapter, various

stress update schemes recommended for explicit time integration in FE code

Abaqus/Explicit are discussed and compared. Even though that only isotropic

hardening is assumed as material model for FE simulations in this chapter, the stress

update scheme for a mixed isotropic-kinematic hardening model is described. The

main reason is that implementation of mixed hardening is slightly different from a

pure isotropic model when fully implicit and convex cutting plane algorithms are

considered.

In Section 9.2.1, the first order classical forward Euler’s method is described. Section

9.2.2 provides an improved forward Euler’s method that generates more accurate

results compared with the original forward Euler’s approach. In Section 9.2.3, the

convex cutting plane method is described. In Section 9.3 some remarks are given on

using fully implicit integration schemes is a explicit time integration scheme finite

element solver. These models together with the fully implicit backward Euler’s

scheme were implemented in Abaqus/Explicit as user subroutine (VUMAT) and

results are compared for cup deep drawing simulations in section 9.4.

9.2 Integration schemes suitable for an explicit time integration FE code

In the following sections, all variables are considered at the current step

unless stated otherwise.

9.2.1 Classical forward Euler’s method (CFE)

The classical forward Euler’s scheme (further called CFE) can be considered the most

simple integration scheme. In this method, all internal variables (equivalent plastic

strain and kinematic hardening components) and the flow direction at current step

denoted by are considered with respect to the previous step denoted by .

Accordingly, the increments at the current step can be written as

(

) (9-1a)

(9-1b)

(9-1c)

161

(9-1d)

with

the plastic flow direction and and denoting the plastic moduli (plasticity

parameters)

(

)

(9-2a)

(9-2b)

(9-2c)

In Chapter 5 it was shown that the expression in Eqn.(9-1c) is achieved as following

( )

( )

( )

( ) (9-3)

As seen in Eqn.(9-2c) kinematic hardening is assumed with the general plastic

modulus being a function of

and . Including even more kinematic hardening

components is straightforward as will be discussed hereinafter. The plasticity

parameter if only isotropic hardening is assumed.

The compliance factor , that is to be determined by the consistency condition, is

the only variable based on which all the aforementioned equations will be solved. The

update scheme begins with calculating the trial stress for a given discrete strain

increment

(9-4a)

(9-4b)

The yield criterion is subsequently checked

(

) (

) (9-5)

would denote that deformation is in the elastic regime and the trial stress

replaces the current stress

whilst all other variables remain unchanged and the

simulation passes to the next step. Otherwise, in case the current step is in

the plastic regime and the stress and all internal variables will be updated on the basis


of the compliance factor as elaborated in the following. Fig 9-1 illustrates the

described steps for the CFE method.

Fig 9-1 Geometric interpretation of classical forward Euler’s method (CFE).

By means of a Taylor series expansion (with higher orders neglected) the yield

criterion in Eqn.(9-5) can be recast

(9-6a)

Assuming

(9-6b)

then

(9-7)

with

(9-8a)

(9-8b)

Taking into account that

and the normal to the yield function

(9-9)

B

C

xx

n(n)

n(n)

Ce

(n)

(n+1)

n C

e n

(n)

yy

A

F (n+1)

F (n)

163

this results in

(9-10)

Using the equations given by Eqns. (9-1a) to (9-1d) and Eqn.(9-8a) to (9-10), after

some manipulation the compliance factor is found

(9-11)

Subsequently, this compliance factor is used to update the increments defined in

Eqns.(9-1a) to (9-1d). Next, the integration scheme to obtain all variables at the end of

the current step can be written as

(9-12)

(9-13)

(9-14)

(9-15)

If

was located on the yield surface (presence of plastic strain) then all stress-

update methodologies presented above can be simply used.

Fig 9-2 Geometric interpretation of transition from elastic to plastic regime and

locating the intersection A in CFE method.

Conversely if the stress in the previous step was located inside the yield surface as

illustrated in Fig 9-2 then the intersection with the yield surface during the elastic

F (A)

F (n+1)

X

B

C

xx

n(A)

n(A)

Ce

(X)

(n+1)

n C

e n

(n)

yy

A


increment (denoted by A in the same plot) has to be known to determine the flow

direction. In other words, the stress

that denotes the stress on the surface A has to

be determined (Crisfield, 2000).

(9-16)

where is the ratio to be determined using Newton-Raphson’s iterations.

is the

stress at the previous step that was inside the yield surface.

The criterion to investigate whether the location of the stress at the previous step

was inside the yield surface is

(9-17)

Finding the intersection A requires the following yield criterion to be satisfied

(

) (9-18a)

(

) (

) (9-18b)

Taylor expansion of Eqn.(9-18a) around gives

(9-19)

where subscript k denotes the iteration number. Therefore

(

)

(9-20)

being the incremental change in and

(9-21)

Substitution of Eqn.(9-21) into Eqn.(9-20) gives

(

)

(9-22)

165

Note that the initial guess of is very determining and an improper guess may

discard the chance of convergence. Accordingly it is highly recommended to use the

initial guess denoted by

(Crisfield, 2000)

(9-23)

Then would be updated iteratively by

(9-24)

This procedure continues until the yield condition at point A described in Eqn.(9-18)

is satisfied. In that case

(9-25)

and Eqn.(9-1a) to Eqn.(9-14) can be applied. The numerical steps describing the CFE

integration scheme are presented in Table 9-1.

The advantage of using CFE is its significant simplicity. In addition, the need for

computing the second order gradients of and is bypassed. Finding the closed

form of second order gradients is a laborious task for complex yield functions and

increases computational costs such as simulation time. However, the underlying

assumptions in the CFE technique bring a series of vulnerabilities and drawbacks that

have to be considered carefully. For instance, the plasticity parameters and flow

direction solely depend on the state of the previous step and consequently the

consistency constraint is not satisfied for the current step. Furthermore, the

assumption is not true due the fact that the consistency is not satisfied in the

previous step too. This describes the conditional stability of the CFE. The accuracy,

on the other hand, is highly dependent on the size of the time increments. In other

words, the time increments should be considerably small because otherwise the stress

may drift away from the yield surface over many steps (Dunne and Petrinic, 2005).

Having mentioned some drawbacks of the CFE, the simplicity of this integration

scheme has been shown to be very persuasive if used in explicit time integration FE

codes such as Abaqus/Explicit with very small time increments (roughly speaking, in

the order of 10-7 for a deep drawing application). Using such small time increments,

or using higher simulation time in other words, has been shown to be sufficient to

generate accurate results in terms of the prediction of yield stress and anisotropy

coefficient (Lankford coefficients) (Cardoso and Yoon, 2009).


Table 9-1 Numerical algorithm for CFE integration scheme

1) Input

, ,

,

,

,

,

2) Calculate trial stress for a given

,

3) Check yield condition

(

) (

)

4) If then set

,

,

,

,

Go to step 5

Else

If

Find intersection A in Fig 9-2.

End if

Calculate

(

)

Update the increments of

,

,

and

Update

,

and

End if

5) Go to the next step

167

9.2.2 Next Increment Corrects Error (NICE-h)

As mentioned in the previous section, the main source of inaccuracy of the CFE

method is the lack of fulfillment of the consistency condition. In other words, even

though the consistency condition has not been imposed to step that hypothetically

was located in the plastic regime, is assumed and furthermore is

assumed but no numerical iteration is performed to assure this stipulation for the

current step ( .

Halilovic et al. (2009) proposed a slight change to the CFE method based on which

the error on from the previous step is taken into account for the calculation of the

yield criterion in the current step. According to this method which is called NICE-h (h

representing the degree of truncated Taylor expansion), one can write

∑ (9-26a)

(9-26b)

All procedures are identical to the CFE method except the fact that is not

disregarded in Eqn.(9-26a). Higher order terms such as in the right side of

Eqn.(9-26a) can also be considered in CFE.

Following similar steps as described for the CFE method in Table 9-1, finally the

compliance factor for the NICE-1 scheme is obtained

(9-27)

The considered in the nominator is the only difference between NICE-1 and CFE

methods. Even though no numerical iteration is performed to assure the yield

condition or , higher accuracy for NICE-1 compared with CFE was

reported by Vrh et al (2010). They examined different integration schemes for a

complex non-linear loading path and showed that using the same number of

increments the NICE-1 scheme is the fastest solution with acceptable accuracy.

Halilovic et al (2011) compared NICE-2 and NICE-1 methods (i.e. 1 and 2 denote the

order of Taylor expansion) for a cup deep drawing simulation and reported a

maximum residual (value of ) of 1E-5 versus 2.23E-3 respectively for NICE-2

and NICE-1 at 50% of punch displacement. Vrh et al (2010) reported that the same

level of accuracy as for a classical backward Euler’s scheme (further called CBE) was

obtained in 1/10th of simulation time.


9.2.3 Convex Cutting-Plane (CCP) algorithm, a semi explicit approach

As opposed to the explicit update schemes described above, in return mapping

algorithms such as fully implicit CBE, all variables at the previously converged step

( ) are updated at the end of the current converged step ( ). The CBE method

guarantees high convergence accuracy by means of solving a (generally non-linear)

system of equations. The quadratically converging Newton–Raphson method is

normally adopted as an efficient scheme to solve the system of equations (Neto et al.,

2008). The iteration continues until the desired accuracy is achieved or in other words

the value of the yield function is sufficiently close to zero.

With respect to accuracy, the value of (the convergence accuracy or as it is

further called, the residual) is set to approximately 10E-10 multiplied with initial yield

stress. This is implemented in the UMAT for Abaqus/Implicit discussed in chapter 6

by means of the TOL parameter

(9-28)

Unavoidably, when complex hardening definitions or non-AFR models are used, the

algorithmic/consistent tangent modulus continuously changes and the stress-update

algorithm should accommodate these non-constant moduli. However, development

effort for this tangent stiffness is cumbersome and, furthermore, requires second order

gradients. The second order gradients are difficult to calculate considering complex

non-quadratic yield and plastic potential functions.

Ortiz and Simo (1986) proposed a return mapping scheme to bypass the need for

second order derivatives and reduced the system of equations with 1+k unknown

variables to a system of equations with only one unknown to be solved by Newton-

Raphson iterations in which k denotes the number of internal variables i.e considering

isotropic hardening k=1. The CCP return mapping algorithm belongs to the elastic-

plastic operator split methodology that involves two consecutive steps (Ortiz and

Simo, 1986). In the first step, the current state is assumed to be totally elastic (the

plastic part is frozen). If the stress is located inside the yield surface then the current

step is known to be elastic. Otherwise, the second step or plastic corrector, attempts to

bring back the stress onto the yield surface. This is a similarity between CCP and

classical implicit backward Euler’s method. However, the CCP is an explicit approach

in a sense that flow direction and plastic modulus are based on the initial iteration and

the consistency condition is also enforced at the initial state (Simo and Hughes, 1998).

Fig 9-3 shows the geometric interpretation of the CCP technique.

169

At the start of the CCP algorithm, the yield condition is checked based on initial/trial

values

(

) (

) (9-29)

and

(9-30a)

(9-30b)

(9-30c)

where subscript denotes initial/trial state. Superscripts and respectively

denote previous and current steps.

If the following condition is satisfied

(9-31)

then the stress state is located inside the yield surface and the step is elastic.

Otherwise, the yield condition is linearized around the current values of stress and

internal variables

with i denoting the iteration number. Assuming all the

variables are known at their initial state (Ortiz and Simo, 1986), the only variable that

should be determined iteratively is the compliance factor .

The yield condition at each iteration step can be written as

(

) (

) (9-32)

To find the incremental change of the compliance factor, the Newton-Raphson

iteration scheme is employed such that

(

)

(9-33)

and

(

)

(

)

(9-34)

with


(9-35a)

(9-35b)

(9-35c)

(9-35d)

where

is the iterative change of the compliance factor . Substituting Eqn.

(9-35) into (9-34) and then into (9-33) one obtains

(9-36)

Subsequently, stress and internal variables at each iteration step are updated by

(9-37a)

(9-37b)

(9-37c)

The yield condition is checked and if the desired accuracy within the range of

assigned iteration numbers is satisfied, then the step is complete. The CCP stress

update scheme is described in a step-by-step fashion in Table 9-2.

Fig 9-3 Geometric interpretation of convex cutting plane (CCP) stress update scheme.

(n+1)

(k)

F (n+1)

(k)

F (n+1)

(k-1)

F (n+1)

(2)

F (n+1)

(1)

(n+1)

(n+1)

(k-1)

(n+1)

(2)

(n+1)

(1)

F (n)

xx

Ce

(n)

yy

trial

=(n+1)

(0)

171

Table 9-2 Numerical algorithm for cutting plane integration scheme

1) Initialize

,

,

,

,

2) Calculate the hardening moduli, and yield function

(

)

(

)

2-1) Check the yield condition

(

) (

)

2-2) If then

,

,

Go to the next step

Else

3) Calculate the increment of the compliance factor

4) Update variables

and go back to 2

6) End if

7) Go to Next step


9.3 Some remarks on using the explicit stress-update integration in implicit

time integration code.

One should note that all aforementioned explicit integration schemes can generate

acceptable results when used with very small time increments in an explicit time

integration FE code such as Abaqus/Explicit. It is clear that the mentioned explicit

techniques are conditionally stable but using such small increments will alleviate the

degree of violation of the consistency condition. For instance, Cardoso and Yoon

(2009) reported that the forward Euler’s integration scheme generates the same degree

of accuracy as can be achieved by the backward Euler’s method when these

approaches are used in explicit time integration with small time increments. On the

other hand, they found large discrepancies in the result of CFE when used in implicit

time integration with large time steps.

Using explicit stress update schemes in a FE code with implicit time integration

schemes, such as Abaqus/Implicit, has some disadvantages. For instance, using very

small time increments is not computationally efficient for a complex deformation

simulation. More importantly, the algorithmic tangent modulus is required for a good

convergence to guarantee the computational efficiency of the simulation in terms of

time. However, development of consistent moduli is a laborious task that is only

reasonable to perform when fully implicit backward Euler is employed.

9.4 Comparison of stress-update schemes

In this section the described stress-update schemes are compared for the non-AFR

Yld2000-2d model with isotropic hardening. Simulations of cylindrical cup deep

drawing were performed in Abaqus/Explicit, applying a user material subroutine

VUMAT allowing for the following approaches

1. Classical backward Euler’s method, CBE; a fully implicit approach.

2. Convex cutting plane, CCP; a semi explicit approach.

3. First order classical forward Euler, CFE; a fully explicit approach and

4. First order forward Euler with next increment correct error, NICE-1; a fully

explicit approach.

In addition, the CBE approach is also implemented by UMAT in Abaqus/Implicit for

comparison with other implemented subroutines.

Two highly anisotropic materials, fictitious material FM8 and aluminium alloy

AA2090-T3 respectively showing 8 and 6 ears in a completed deep drawn cup are

selected to compare different integration schemes. The geometry of the tooling is as

described in chapter 7. All other simulation configurations are similar unless stated

otherwise. Parameters of the non-AFR models for those materials were optimized

with an inverse approach and are available in Table 7-5 in chapter 7. The punch

173

stroke is 60 mm and simulation time for Abaqus/Explicit was set to 0.05 and for

Abaqus/Implicit 1 with fixed time increments equal to 0.001.

Apparently, no iteration is involved for CFE and NICE-1 schemes. However,

considering the VUMAT, 20 and 10 Newton-Raphson iterations are considered

respectively for backward Euler (CBE) and cutting plane (CCP) algorithms. A large

number of iterations is used for CBE in UMAT so that the assigned tolerance

(TOL=10E-8) is obtained. It was found that the time cost of simulation in explicit time

integration code (Abaqus/Explicit) when requesting the same tolerance is extremely

high, thus the number of iterations is limited to 20 for CFE in VUMAT.

Fig 9-4 and Fig 9-5 respectively show the distribution of stress and equivalent plastic

strain at 50% of the total punch stroke (top view is presented). The complex cup

profiles illustrate the great capabilities of the non-AFR Yld2000-2d model for the

prediction of highly anisotropic material behaviour. For the same materials, in chapter

7 excellent results in terms of directional yield stress and Lankford coefficients were

reported for the non-AFR Yld2000-2d model.

Fig 9-4 Distribution of Yld2000-2d effective stress values for FM8 and AA2090-T3

materials at 50% of the total punch stroke. The results are obtained using UMAT for

non-AFR Yld2000-2d model with CBE scheme.


Fig 9-5 Distribution of equivalent plastic strain for FM8 (left) and AA2090-T3 (right)

at 50% of the total punch stroke. The results are obtained using UMAT for non-AFR

Yld2000-2d model with CBE scheme.

Fig 9-6 and Fig 9-7 present the residuals (value of yield stress , for instance see

Eqn.(9-32)) at 50% of the total punch stroke for FM8 and AA2090-T3 respectively.

As expected, the CFE approach generates the worst accuracy (higher residuals).

Interestingly, the accuracy of the NICE-1 approach is better than the implicit

backward Euler scheme with 20 iterations. This proves that the NICE-1 scheme can

be considered as an attractive approach for simulations involving monotonic loading

conditions (no stress reversal is engaged). More investigations are required to judge

about the reliability of the NICE-1 scheme in load reversal. Among the considered

models, the convex cutting plane scheme generates the best accuracy.

In Fig 9-8 and Fig 9-9, the distribution of von Mises stresses is presented. The

distribution is clearly scattered for the forward Euler scheme (CFE). NICE-1 and

convex cutting plane approaches predict very similar von Mises stress distributions.

The difference between CBE in UMAT and VUMAT, as mentioned earlier, is due to

the guaranteed accuracy that can be directly requested from the CBE (UMAT)

method at large increment sizes when Abaqus/Standard as an implicit time integration

code is used. Unavoidably, in Abaqus/Explicit the time increment size should be

considerably small to preserve dominance of quasi-static against dynamic response

(i.e. millions of iterations are imposed in Abaqus/Explicit). Performing a large

number of Newton-Raphson iterations for CBE (VUMAT) for each of millions of

increments to guarantee the consistency is time costly. This is why the results of CBE

in Abaqus/Explicit are worse when compared with the same algorithm in

Abaqus/Implicit.

175

VUMAT

classical backward Euler

(CBE)

VUMAT

NICE-1

VUMAT

classical forward Euler

(CFE)

VUMAT

convex cutting plane

(CCP)

Fig 9-6 Distribution of residuals (yield function) for FM8 using non-AFR Yld2000-2d

model (values below zero denote elastic deformation and are not considered).


VUMAT


(CBE)

VUMAT

NICE-1

VUMAT


(CFE)

VUMAT


(CCP)

Fig 9-7 Distribution of residuals (yield function) for AA2090-T3 using non-AFR

Yld2000-2d model.

177

UMAT


(CBE)

VUMAT


(CBE)

VUMAT

NICE-1

VUMAT


(CFE)

VUMAT


(CCP)

Fig 9-8 Distribution of von Mises stress for FM8 using non-AFR Yld2000-2d model.


UMAT


(CBE)

VUMAT


(CBE)

VUMAT

NICE-1

VUMAT


(CFE)

VUMAT


(CCP)

Fig 9-9 Distribution of von Mises stress for AA2090-T3 using non-AFR Yld2000-2d

model.

179

Fig 9-10 and Fig 9-11 show the predicted cup heights for the materials AA2090-T3

and FM8 respectively. Interestingly, the results of all integration schemes used in

Abaqus/Explicit are very close. The results of CBE in Abaqus/Implicit are slightly

different from those obtained using Abaqus/Explicit.

Fig 9-10 Cup height predicted by various integration schemes for AA2090-T2 using

the non-AFR Yld2000-2d model.

Fig 9-11 Cup height predicted by various integration schemes for FM8 using the non-

AFR Yld2000-2d model


In Fig 9-12, simulation times for different integration schemes are presented. It is

shown that the CBE (UMAT) approach needs the minimum simulation time

compared to all implemented approaches. Conversely, when implemented into an

explicit time integration scheme, CBE (VUMAT) leads to a very long simulation time

(see Section 9.3 for explanation).

Fig 9-12 Cup drawing simulation time using different integration schemes.

9.5 Summary

This chapter has been devoted to the implementation of different integration schemes

such as simple classical forward Euler (CFE), next increment correct error (NICE-1)

and convex cutting plane (CCP). CFE and NICE-1 are fully explicit schemes in which

no iterations are involved. Therefore implementation of these approaches is a simple

task. The CCP, on the other hand, requires Newton-Raphson iteration to iteratively

bring the stress back to the yield surface. This approach is considered as a return

mapping scheme. For the purpose of comparison, the classical backward Euler’s

scheme was included, as another return mapping scheme.

It was shown that the CFE method results in the lowest accuracy of all studied

approaches. However, considering a simple change to the CFE scheme generates

much more accurate results (NICE-1 technique) in a sense that it yields results close

to the CBE approach with 20 iterations. It was illustrated that the CCP scheme is the

most accurate technique taking into account the time cost among all the models that

were implemented in Abaqus/Explicit.

However, it must be noted that the CBE method naturally yields a very high accuracy

(defined by the user) by means of Newton-Raphson iterations as far as a proper initial

181

guess is given to the iteration. If so, even at large strains the requested accuracy can

be guaranteed. However, requesting a large number of iterations for too many

increments in a sufficient step time (as is necessary in Abaqus/Explicit to preserve the

quasi-static response) is too time expensive. Therefore the CCP scheme even with

half number of iterations as compared to the CBE scheme, can be a better choice at

least for applications involving monotonic stress paths such as cup deep drawing.


Bibliography

Cardoso, R.P.R., Yoon, J.W., 2009. Stress integration method for a nonlinear

kinematic/isotropic hardening model and its characterization based on polycrystal

plasticity. International Journal of Plasticity 25, 1684-1710.

Crisfield, M.A., 2000. Non-linear finite element analysis of solids and structures



University Press, Oxford ; New York.

Halilovic, M., Vrh, M., Stok, B., 2009. NICE-An explicit numerical scheme for

efficient integration of nonlinear constitutive equations. Math Comput Simulat 80,

294-313.

Halilovič, M., Vrh, M., Štok, B., 2011. NICEH: a higher-order explicit numerical

scheme for integration of constitutive models in plasticity. Engineering with

Computers, 1-16.

Neto, E.A.d.S., Peric, D., Owens, D., 2008. Computational methods for plasticity :

theory and applications. Wiley, Oxford.

Ortiz, M., Simo, J.C., 1986. An Analysis of a New Class of Integration Algorithms

for Elastoplastic Constitutive Relations. International Journal for Numerical Methods

in Engineering 23, 353-366.


mathematics. Springer, New York.

Vrh, M., Halilovic, M., Stok, B., 2010. Improved explicit integration in plasticity.

International Journal for Numerical Methods in Engineering 81, 910-938.

Chapter 10

10 Evolutionary non-AFR

anisotropic model

184 Chapter 10, Evolutionary non-AFR anisotropic model

10.1 Introduction

In Chapter 4, a review of various classical and advanced anisotropic yield functions

was provided. In Chapter 5, the non-AFR Hill 1948 and Yld2000-2d models were

fully described. In Chapter 7, it was shown that the non-AFR Yld2000-2d model can

be considered as an alternative for the AFR Yld2000-2d and Yld2004-18p models.

However, all of the described anisotropy models were based on constant model

parameters. In other words, it was assumed that the initial anisotropy is preserved

during the plastic deformation. In this chapter, experimental characterization of DC06

steel (interstitial free deep drawing grade) is provided. Thanks to the accuracy and

feasibilities brought by an optical measurement system, it is shown that the anisotropy

(in terms of both yield stresses and Lankford coefficients) is changing with plastic

deformation. More importantly, a simple formulation is presented to describe the

evolutionary anisotropy based on non-AFR Yld2000-2d model and polynomial

interpolation. The presented model enables the yield and plastic potential functions to

change as function of plastic deformation.

First, in Section 10.2, the experimental procedure is described and basic calculations

of stress and strain are given. Next, in Section 10.3, the presence of distortional

hardening and the evolution of Lankford directionalities in the experimental results

are shown. In Section 10.4, the capabilities of non-AFR Hill 1948 and Yld2000-2d

models in describing the anisotropy at various degrees of plastic work are evaluated.

This evaluation is based on separate sets of parameters at discrete increments of

equivalent plastic strain. So far, no continuous evolution is included in the models.

Finally, in Section 10.5, the evolutionary non-AFR Yld2000-2d model is described.

Except some laborious numerical treatments for consideration of the principle of

plastic work equivalence to be carried out for parameter identification, the current

evolutionary model is very simple and no additional effort is required for the stress

integration scheme when compared with the integration schemes described in

chapters 6 and 9 or in general, any integration scheme for isotropic hardening and a

non-AFR model.

10.2 Material characterization

10.2.1 Experimental procedure

Tensile test specimens were extracted from an 0.8 mm thick DC06 steel sheet at 0°,

15°, 30°, 45°, 60°, 75° and 90° orientations with respect to the rolling direction. The

tensile samples were provided by ArcelorMittal Global R&D (OCAS). The geometry

of the dogbone specimens was in accordance with ISO 6892-2 (central section

12.5mm wide and 195 mm long). Tensile tests were carried out on a servo-hydraulic

testing machine (MTS 810) with hydraulic wedge grips. The gauge length of the

mechanical extensometer (MTS632) was 50 mm. The average strain rate was 1.8E-4

185

(1/s) for longitudinal strain up to approximately 0.02. Subsequently, tensile loading

was continued slightly faster with average strain rate 1.9E-3 (1/s).

Besides a conventional mechanical extensometer, the Digital Image Correlation (DIC)

technique was used to measure strains in longitudinal and transverse directions. The

DIC setup is shown in Fig 10-1. This technique is suitable for full-field, non-contact

measurement of 3D deformations. A stochastic pattern of black and white speckles is

produced at the specimen’s surface by spray painting. Software based on a dedicated

image correlation algorithm is used to process the images recorded by two

synchronized digital cameras (Fig 10-1). The used digital image correlation system

was Q-400-3D from LIMESS. The accuracy in strain measurement around 0.001 can

be obtained by commercial DIC algorithms these days. In all the post processing

steps, 25 and 5 were chosen as subset and step sizes, respectively.

Fig 10-1 DIC test setup.

A considerable advantage of using DIC as compared to a mechanical extensometer

for strain measurements is that data can be extracted beyond the onset of necking.

Another advantage is that strain can be measured at any point on the surface of the

specimen during post-processing. For instance, in Fig 10-2 the measured distributions

of longitudinal and transverse true strains are shown along length and width directions

of the sample at various deformation stages. The localization of strain or diffuse

necking occurs at maximum tensile load (longitudinal strain of approximately 0.2).

The extensometer is practically inutile beyond this point. However, up to 0.80

longitudinal true strain was measured by DIC. This strain corresponds to failure of the

sample. To evaluate the accuracy of this technique, strains measured by both

specimen

hydraulic gripCCD camera

illumination


extensometer (50 mm gauge length) and DIC system (center of specimen) are plotted

in Fig 10-3. An approximate difference of only 0.003 was measured at 0.2

longitudinal strain. Considering the advantages of non-contact optical measurements,

this discrepancy can be overlooked. Therefore the strains reported in the following are

these measured by the DIC technique.

Fig 10-2 Variation of longitudinal (left) and transverse (right) true strains measured

by means of DIC at different stages of a tensile test.

Fig 10-3 Comparison of true strain measured by optical and mechanical techniques.

10.2.2 Hardening

The classical method to characterize a hardening curve for uniaxial loading includes

calculating the true stress (σ) and true longitudinal strain ( ) from the corresponding

engineering values ( and ).

(10-1a)

(10-1b)

0 20 40 60 80 100

0.28

0.42

0.56

0.70

0.84

0.98 0% Length location 100%

defo

rmation

pla

stic

Length

tru

e s

train

Length location (%)

0 20 40 60 80 100

-0.54

-0.36

-0.18

0.00

0.18

0.36

width location 0%

Wid

th t

rue s

train

Width location

pla

stic d

efo

rmation

(%)

100%

187

Regarding the DIC method, the Lagrangian strain is calculated using commercial

image correlation software (VIC-3D®) and converted to true strain. The true stress is

now obtained as the ratio of force to the instantaneous cross sectional area, which is

found by incorporating the incompressibility hypothesis for metals as discussed in

Eqn.(2-32). Values of true strains ( ) in longitudinal ( ), transverse ( ) and through

thickness ( ) directions are used to calculate the instantaneous values of width and

thickness.

(

) (10-2a)

(

) (10-2b)

(10-2c)

Using the incompressibility hypothesis

(10-3)

it can be found that

(10-4)

The instantaneous cross sectional area is calculated as

(10-5)

where the cross sectional area factor is

(10-6)

Finally, true stress is obtained by

(10-7)

where is the applied uniaxial load.

Longitudinal and transverse (width) plastic strains are calculated by

(10-8a)

(10-8b)


A Young’s modulus of =200 GPa and Poisson’s ratio of 0.3 were assigned as

elastic properties.

Fig 10-4 True strains can be determined based on a single point measurement (black

dot) or using a virtual extensometer (white line).

Two different methods were applied to investigate whether the non-uniformity of

strain at the center of the necking area, as shown in Fig 10-2, has any influence on the

calculation of the instantaneous cross section as required in Eqn.(10-7). In the first

approach, transverse and longitudinal true strains are extracted at the center of the

necking area (single point in Fig 10-4). In the second method, the transverse strain is

extracted from a virtual extensometer (line in Fig 10-4) that measures the width of the

central section of the specimen (the necking area) and the longitudinal strain is taken

as the average value for this line. At first sight, the latter approach appears to be more

realistic because it samples the entire width. Interestingly, comparing both approaches

as presented in Fig 10-5 reveals a negligible discrepancy (less than 0.5E-3).

Consequently, the single point measurement at the center of the specimen is used for

strain measurement by the DIC technique in the remainder of this work.

Fig 10-5 Discrepancy of calculations of the area factor using point and line

measurements at the specimen’s center.

Finally, the true stress versus true strain is calculated based on Eqns. (10-7) and

(10-8b). Fig 10-6 shows the obtained curves for seven orientations from 0° to 90°. At

first sight it is noted that these hardening curves might resemble an isotropic material

due to the fact that they hardly can be distinguished. However, to preserve the

189

generality of the model which shall be discussed in section 10.5 of this chapter, the

directional hardening curves are assumed independent from each other.

Fig 10-6 Experimentally measured hardening curves for DC06 at seven orientations

(slight discontinuity due to slight increase of strain rate described in Section 10.2.1).

Curve fitting based on Swift’s hardening law (Eqn.(3-1d)), Voce law (Eqn.(3-1c)),

and combined Swift-Voce law (Eqn.(3-2)) was carried out and the parameters are

respectively presented in Table 10-1, Table 10-2 and Table 10-3. In Section 3.3 an

excellent fit for steels and aluminum alloys was shown to be achieved by the

combined Swift-Voce hardening law. Similar comparison is also is brought in Fig

10-7 which shows that this hardening law avoids saturation and non-saturation trends

as generated respectively by Voce and Swift laws.

Fig 10-7 Examples of curve fitting to stress-strain data measured at 0° and 45° using

Swift, Voce and combined Swift-Voce (CSV) hardening laws illustrating the

excellent fit of CSV.

It is noticed that the experimental hardening data in Fig 10-7 are plotted up to 0.3

plastic strain which is beyond the diffuse necking (which occurred at 0.2 strain). This

is possible as shown by Iadicola (2011) who combined the full field optical technique


with X-ray diffraction method and proved that the uniaxiality condition for the tensile

test is not breached even at a strain value equal to twice the strain at which diffuse

necking occurs. Nonetheless, this should not be taken as a general rule but, simply,

could be the justification to use higher stress-strain data for a better curve fitting

based on simple tensile test results.

Table 10-1 Parameters of Swift hardening law.

0° 580.436 0.002 0.285

15° 587.393 0.002 0.283

30° 584.845 0.004 0.288

45° 582.459 0.003 0.286

60° 577.100 0.003 0.284

75° 582.834 0.003 0.283

90° 580.941 0.003 0.281

Table 10-2 Parameters of Voce hardening law.

0° 131.662 267.960 11.353

15° 136.180 266.771 11.533

30° 135.604 272.111 11.877

45° 136.142 265.572 11.151

60° 135.753 260.701 11.516

75° 136.372 264.441 11.533

90° 138.814 263.458 11.275

Table 10-3 Parameters of combined Swift-Voce (CSV) hardening law.

0° 539.542 0.012 0.326 0.848 557.223 34.822 29.247

15° 669.128 0.092 0.535 0.649 408.32 25.302 14.290

30° 488.056 0.000 0.390 0.468 288.963 13.258 209.536

45° 581.667 0.024 0.946 0.644 457.245 22.122 325.168

60° 617.050 0.257 1.000 0.593 410.000 21.844 82.687

75° 586.481 0.042 1.000 0.624 454.761 21.635 298.764

90° 637.222 0.062 0.937 0.568 381.940 21.593 239.494

10.2.3 Lankford coefficients

Based on Eqns. (10-8a) and (10-8b) the width to length plastic strain ratios are

calculated for each orientation. The obtained results are plotted in Fig 10-8.

191

Fig 10-8 Experimentally determined width to length plastic strain ratios at seven

orientations.

As described in Section 4.2.1, the Lankford coefficient (also called r-value or plastic

strain ratio), is defined by the ratio of width to thickness plastic strain increments.

However, due to practical difficulties associated with the direct measurement of

thickness strains in sheet metals, this quantity is calculated based on the

incompressibility hypothesis and using increments of plastic strains at tensile loading

direction and at direction 90°+ , denoted respectively by and

.

Correspondingly, the Lankford coefficient at orientation was defined in Eqn.(4-

1a) as

(10-9)

In accordance with ISO 10113 the linear regression of width versus length true plastic

strain data (between specified lower and upper limits of plastic strain) is used to

calculate the Lankford coefficient. In this study we chose 0 and 0.3 respectively for

lower and upper limits of plastic strain when a fixed Lankford coefficient is

considered. If the gradient to (the linear curve fits of) these curves is , then the

Lankford coefficient is obtained by

(10-10)

10.3 Experimental observations on anisotropy evolution

10.3.1 Yield stress

In the majority of the yield functions (and for all anisotropic models discussed in

chapter 4) it is assumed that the directional normalized yield stresses (stress

directionalities) are not sensitive to additional plastic deformation. Therefore the yield


surface expands proportionally in all directions (considering isotropic hardening) with

increase of plastic strain and the shape of the yield surface is constant regardless of

the level of plastic work. For instance, considering the sheet metal DC06, the constant

normalized yield stresses at a very small plastic strain 0.001 were assumed for the

entire plastic deformation, Fig 10-9.

Fig 10-9 An example of fixed normalized yield stresses.

However, a closer look to the hardening curves at, for instance, 0°, 45° and 90° as

shown in Fig 10-10 reveals a distortion on the proportionality of the hardening curves

with plastic deformation. This implies that the shape of the yield function will change

with deformation. In other words, the variation of Lankford coefficients can be

expected.

Fig 10-10 Distortion of hardening proportionality with increasing plastic strain.

193

Calculation of directional yield stress requires the principle of plastic work

equivalence. This principle describes that yielding at various stress states happens

when the stress state is on the same plastic work contour. This principle is

schematized in Fig 10-11. In other words, the true stress versus true longitudinal

plastic strain at each orientation must be converted to true stress versus equivalent

true plastic strain ( .

Fig 10-11 Principle of plastic work equivalence

First, the equality of equivalent plastic strain ( ) with longitudinal plastic strain (

)

in the rolling direction must be reminded. Referring to chapter 8 (section 8.2), for a

non-AFR model it was shown that

(10-11)

When yielding occurs in the rolling direction then

(10-12)

From Euler’s theorem

(10-13)

Substitution of Eqn.(10-13) and (10-12) into (10-11)

(10-14)

w0

P (

p

) = w

P (

p

)

°0° (RD)

p

Longitudinal true plastic strain at °

w

pTru

e s

tress

Longitudinal true plastic strain at 0°

= Equivalent plastic strain

p

T

rue s

tress

w0

p


Accumulated longitudinal and equivalent plastic strains are calculated by

∫ (10-15a)

∫

(10-15b)

As shown in Fig 10-11 the plastic work at any plastic strain in the rolling direction is

∫ (

)

(10-16)

with ( ) representing the hardening curve in rolling direction.

Next, the yield stress at degrees orientation (denoted by ) corresponding to the

same amount of plastic work has to be determined. Hereto an iteration scheme

such as Newton-Raphson is employed to find the longitudinal plastic strain at

degrees orientation that results in a plastic work equal to . The stress

corresponding to directional strain

is determined based on CSV hardening model.

In Fig 10-12, a plot of plastic work versus equivalent plastic strain is illustrated.

Fig 10-12 Plastic work versus equivalent plastic strain.

A code was developed (in Mathematica® v9) which calculates the yield stress for

starting from 0 to 0.3 at each 0.002 increment for 0°, 15°, 30°, 45°, 60°, 75° and 90°

orientations. Considering the computation time, all the results are generated in less

than 10 minutes.

In Fig 10-13, it is shown that the trend of directional yield stresses (normalized with

respect to rolling direction) at the onset of deformation denoted by MPa

and corresponding to

is largely distorted at higher values of plastic work.

195

Fig 10-13 Evolution of yield stress anisotropy for various amounts of plastic work

(MPa).

Fig 10-14 shows that the trend irregularly changes with additional plastic work but

after around .1 the differences remain very small. Consequently, two

statements can be made here. First, it is concluded that the initial yield stress ratio is

not a good criterion to determine the anisotropic coefficients of a yield function when

distortional hardening is not considered. Second, the model which is presented in

section 10.5 is capable of predicting even those small and irregular changes in the

plastic work dependent yield stress directionalities. This is a considerable advantage

when dealing with highly anisotropic materials. It must be noted that completely

constant normalized yield stresses is a rare characteristic in sheet metals and every

material exhibits a degree of distortional hardening.

Fig 10-14 Instantaneous variation of directional normalized yield stresses with respect

to equivalent plastic strain.


10.3.2 Lankford coefficient

In section 10.2.3, the conventional method of calculating the Lankford coefficient was

described. An example of a linear curve fit ( ) to the experimental width to

length plastic strain ratio is shown in Fig 10-15.

(10-17)

Fig 10-15 Linear and 3rd order polynomial fits to the experimental width to length

plastic strain ratios.

Corresponding parameters for seven orientations are given in Table 10-4. To use

Eqn.(10-10) one should note that . Corresponding fixed Lankford coefficients

at seven orientations are plotted in Fig 10-16.

Table 10-4 Parameters of linear fit.

0° 0.0001 -0.6421

15° 0 -0.6421

30° 0.0005 -0.6463

45° -0.0001 -0.667

60° -0.0001 -0.6936

75° 0.0002 -0.7164

90° 0.0002 -0.7168

197

Fig 10-16 Prediction of directional Lankford coefficient using a linear fit function.

It would be more advantageous to measure the instantaneous instead of the constant

Lankford coefficients. For instance, one may suggest using a polynomial function

(Safaei and De Waele, 2012). Correspondingly, a third order polynomial fit (further

called Poly3) would be

(10-18)

Accordingly, in Fig 10-15 the third order polynomial fit to the experimental width to

length plastic strain ratio is also shown for the rolling direction. Parameters of Poly3

are provided in Table 10-5.

An appropriate evaluation of the curve fit function can be achieved by comparing the

residual values between curve fit model and experimental data. The residuals

corresponding to the linear and Poly3 fits are plotted in Fig 10-17. The absolute

residual value of the linear fit is about 0.005. However, the Poly3 fit makes an

excellent fit to the experimental data by producing a maximum absolute discrepancy

of 0.001.

Table 10-5 Parameters of Poly3.

0° 0.0008 -0.6763 0.1343 0.0174

15° 0.0005 -0.6619 0.0597 0.0941

30° 0.0008 -0.6606 0.0153 0.1404

45° 0.0003 -0.6808 0.0249 0.1241

60° 0.0003 -0.7102 0.0425 0.107

75° 0.0005 -0.7309 0.0292 0.1243

90° 0.0005 -0.7277 0.0088 0.1459


Fig 10-17 Residual of Poly3 and linear fits to experimental width versus longitudinal

plastic strains.

It must be noted that the derivative of the function which determines the final

Lankford coefficient is significantly sensitive to the residual. This sensitivity is

evident from Fig 10-18 which shows instantaneous Lankford coefficients versus

longitudinal plastic strain obtained by the Poly3 fit. A significant decreasing trend in

the Lankford values is observed. Similar convex-like evolution of Lankford

coefficients as seen in Fig 10-18 has been obtained very recently by viscoplastic self-

consistent polycrystal formulation (VPSC) (An et al., 2013)

To accommodate the use of these evolutionary curves in a plasticity model, the

directional longitudinal plastic strains must be converted to equivalent plastic strains.

This conversion was carried out using the methodology described in section 10.3.1

(using the principle of equivalent plastic work). Consequently, the Lankford

coefficient versus both longitudinal and equivalent plastic strains at seven orientations

is plotted in Fig 10-19. It is noticed that the difference is negligible. This is due to the

fact that the hardening behavior of the sheet metal DC06 is very close to an isotropic

material. However, this simplification is not taken into account and the approach in

this work is based on independent hardening curves. The evolution of Lankford

directionality with increase of plastic deformation is shown in Fig 10-20. It is

observed that the evolution of Lankford coefficient from 0° to 90° changes with

plastic deformation.

199

Fig 10-18 Prediction of directional Lankford coefficient with respect to longitudinal

plastic strain using a polynomial (Poly3) fit function.

Fig 10-19 Prediction of directional Lankford coefficient with respect to equivalent

plastic strain using a polynomial (Poly3) fit function.


Fig 10-20 Evolution of Lankford coefficient at various orientations illustrated for

increasing equivalent plastic strain.

10.4 Anisotropic models

In this section the capabilities of non-AFR Hill 1948 and non-AFR Yld2000-2d

models are evaluated. Evaluation is performed in terms of prediction of Lankford

coefficient and normalized yield stress at every 15° from rolling direction to

transverse direction. The model parameters are calibrated at various levels of plastic

work separately. Therefore, the continuous evolution is not yet considered in the

models. In other words, the aim of this section is to investigate whether those

anisotropic models have enough predictability to be modified into an evolutionary

anisotropic model.

10.4.1 Non-AFR Hill 1948 model

In Section 5.4.1, the non-AFR Hill 1948 model was fully described. In Section 7.2.2,

it was shown that this model generates accurate results in terms of Lankford

coefficients at 0°, 45° and 90° orientations. However, the prediction for balanced

biaxial stress state and other in-plane orientations is not guaranteed. Similarly, in

terms of yield stress prediction, only the 0°, 45° and 90° uniaxial orientations and

balanced biaxial state are guaranteed. In summary, this model might not be the best

choice for highly anisotropic materials.

In this work regarding the sheet metal DC06, the Lankford coefficient at balanced

biaxial stress state is assumed to be similar to that of 0° direction. The yield stress at

201

the balanced biaxial state is assumed to be the average of these corresponding to the

0° and 90° orientations.

In this section, the parameters of both plastic potential and yield functions are

calibrated at small equivalent plastic strain increments. Parameter values are given in

Table 10-6. In Fig 10-21 the evolution of parameters with respect to equivalent plastic

strain are plotted.

Table 10-6 Parameters of non-AFR Hill 1948 model

0.001 0.676 1.519 0.929 0.451 1.209 0.785

0.050 0.663 1.528 0.913 0.502 1.535 1.006

0.100 0.649 1.533 0.899 0.494 1.479 0.975

0.150 0.635 1.535 0.887 0.494 1.476 0.975

0.200 0.620 1.532 0.878 0.495 1.480 0.98

0.250 0.606 1.526 0.87 0.496 1.476 0.982

0.300 0.591 1.516 0.864 0.494 1.460 0.977

Fig 10-21 Evolution of of non-AFR Hill 1948 model parameters.

For most model parameters a saturating trend is observed from an equivalent plastic

strain onwards (also in Table 10-6). Continuous evolution is observed for

the parameters of the plastic potential function. Fig 10-22 and Fig 10-23 illustrate the

shape of respectively plastic potential and yield functions in normalized stress space.

Fig 10-22 indicates that the shape of the yield surface at large strains is stagnated. In

Fig 10-23 the correlation between moderate changes of parameters and shape of

plastic potential surface can be perceived. The normal to (gradient of) the plastic

0.00 0.05 0.10 0.15 0.20 0.25 0.300.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8DC06

p

p

p

y

y

y

Hill

1948 c

oeff

icie

nts

Equivalent plastic strain


potential function is very sensitive to any (even slight) change in the shape of the

surface.

Fig 10-22 Two-dimensional representation of Hill 1948 yield function at various

equivalent plastic strains.

Fig 10-23 Two-dimensional representation of Hill 1948 plastic potential function at

various equivalent plastic strains.

In Fig 10-24, the predictions of normalized yield stresses are shown. As stated at the

beginning of this section, the non-AFR Hill 1948 model only guarantees accuracy at

0°, 45° and 90° among all in-plane orientations. The investigated DC06 steel is prone

203

to almost isotropic yield stress behavior after an equivalent plastic strain .

Therefore the accuracy of non-AFR Hill 1948 model enhances at higher plastic

deformations. The prediction of Lankford coefficients at various levels of plastic

deformation is shown in Fig 10-25. Analogously to the yield stress prediction, the

accuracy of the model enhances with increase of plastic strain.

Fig 10-24 Prediction of yield stress by non-AFR Hill 1948 model at different levels of

equivalent plastic strain.

Fig 10-25 Prediction of directional Lankford coefficients by non-AFR Hill 1948

model at different levels of equivalent plastic strain.


10.4.2 Non-AFR Yld2000-2d model

In Section 5.4.2 the non-AFR Yield2000-2d model was fully described. In Section

7.3.2 its excellent prediction in terms of both Lankford coefficient and normalized

yield stress for all seven in-plane orientations and balanced biaxial stress state was

shown.

Similar to the previous section, in this section the parameters of plastic potential and

yield functions are calibrated at various plastic work increments. The parameters are

given in Table 10-7 and Table 10-8. As seen in Table 10-7 and Table 10-8, the

optimized parameters at various equivalent plastic strains (or plastic works) are not

scattered and they are optimized to be close to each other. To obtain such

compatibility, first parameters are optimized at an arbitrary equivalent plastic strain

such as . Then parameters at other equivalent plastic strains must be

optimized given the parameters at as initial guess. Therefore, the obtained

parameters are close and no scattered results will be obtained. This is a crucial

prerequisite because finding a pattern in the parameters is the key point to succeed

with an evolutionary anisotropic model which is based on parameter interpolation.

More experimental inputs such as plane stress tests at RD and TD are needed to

guarantee the uniqueness of parameters.

Table 10-7 Parameters of non-AFR Yld2000-2d yield function

0.001 1.471 0.367 0.4 0.827 -0.834 -1.452 0 -1.562

0.05 1.117 1.051 1.016 0.925 -0.923 -1.271 0.862 -1.669

0.1 1.163 1.014 1.09 0.916 -0.843 -1.348 0.934 -1.56

0.15 1.159 1.048 1.377 0.924 -0.631 -1.294 1.021 -1.387

0.2 1.16 1.035 1.241 0.932 -0.769 -1.296 0.983 -1.486

0.25 1.153 1.007 1.08 0.941 -0.881 -1.25 0.842 -1.641

0.3 1.178 0.913 0.88 0.95 -0.936 -1.259 0.574 -1.701

Table 10-8 Parameters of non-AFR Yld2000-2d plastic potential

0.001 0.35 1.043 0.202 0.485 0.599 -0.185 0.549 1.543

0.05 0.336 1.025 0.178 0.5 0.613 -0.196 0.539 1.553

0.1 0.302 1.007 0.153 0.521 0.635 -0.214 0.494 1.576

0.15 0.349 0.979 0.23 0.516 0.623 -0.152 0.649 1.532

0.2 0.358 0.97 0.248 0.535 0.64 -0.138 0.691 1.508

0.25 0.302 0.94 0.242 0.559 0.676 -0.164 0.701 1.513

0.3 0.324 0.947 0.297 0.564 0.681 -0.126 0.722 1.49

205

In Fig 10-26, the directional yield stresses predicted by non-AFR Yld2000-2d model

are presented and compared with the corresponding results obtained with the non-

AFR Hill 1948 model at various equivalent plastic strains. As expected, the

predictions from the non-AFR Yld2000-2d model are excellent at all orientations. An

exception is noticed for the very small plastic deformation (almost initial state). The

dominance of the non-AFR Yld2000-2d model over the non-AFR Hill 1948 model is

clear. In Fig 10-27, a similar comparison is presented for Lankford coefficients. Again

an excellent prediction by the non-AFR Yld2000-2d model is observed.

It must be noted that in the current section, the predictions have been carried out

based on independent sets of parameters. In other words, the models themselves do

not yet take into account the evolution. The comparisons were presented to show the

capacities of the models at various levels of plastic deformation.


Fig 10-26 Comparison of predicted directional yield stresses by non-AFR Yld2000-2d

and non-AFR Hill 1948 models at various equivalent plastic strains.

Fig 10-27 Comparison of predicted directional Lankford coefficients by non-AFR

Yld2000-2d and non-AFR Hill 1948 models at various equivalent plastic strains.

10.5 Evolutionary anisotropic models

10.5.1 State of the art

Several researchers have developed methodologies to incorporate variation of

anisotropy into their finite element simulations. For instance, Stoughton and Yoon

(2005) proposed a non-associated flow model in which the hardening at 0°, 45°, 90°

uniaxial stress and balanced biaxial stress state are explicitly incorporated into the

Hill 1948 yield function. A significant improvement was reported, especially for the

prediction of biaxial hardening for stainless steels 719-B and 718-AT, and aluminum

alloys AA5182-O and AA6022-T4 even when compared with the non-quadratic AFR

Yld2000-2d anisotropic yield function. Abedrabbo et al (2006a, b) used third and fifth

207

order polynomial functions to predict the variation of anisotropy coefficient with

respect to temperature. Hu (2007) introduced a formulation in his yield criterion in

which directional hardenings were described explicitly. Zamiri and Pourboghrat

(2007) introduced the evolution of Lankford coefficients in the r-based Hill 1948

model. Aretz (2008) proposed a yield function based on linear transformation of stress

tensors in which the yield stress at different orientations is determined by an

equivalent plastic work theorem incorporated in a user material subroutine. Wang

(2009) updated the Yld2000-2d model parameters with effective plastic strain using

sixth order polynomial functions. Very recently Yoon et al (2010) used the plastic

work equivalence theorem as an iterative manner to find the yield stresses and model

parameters for Yld2000-2d and CPB06ex2 (Plunkett et al., 2008) anisotropic yield

functions. Darbandi and Pourboghrat (2011) implemented the evolution of Lankford

coefficients into the AFR Yld2000-2d model.

Nonetheless efforts made, none of the above models combines simplicity and

accuracy. For instance, the model of Aretz (2008) requires iteration inside the

material subroutine, which slows down the simulation speed. The model suggested by

Yoon et al (2010) only predicts hardening at 0°, 45° and 90° orientations. The model

proposed by Wang (2009) lacks efficiency due to incorporating only 4 stress states.

The model of Hu (2007) only works if Swift hardening is assumed and moreover,

only a specific type (saturating) of evolution of Lankford coefficients can be modeled.

In the model of Zamiri and Pourboghrat (2007) the distortion of hardening was not

considered and lacks efficiency due to limitation of r-based Hill 1948 and

incorporating only four Lankford coefficients.

10.5.2 Evolutionary non-AFR Yld2000-2d model

In this section, the non-AFR Yld2000-2d model discussed in chapter 4 is converted

into an evolutionary model to simply incorporate the evolution of both Lankford

coefficient and yield stress directionalities at seven uniaxial and one balanced biaxial

stress conditions. Simplicity of the model which accommodates its use in the finite

element code Abaqus is a priority. For instance, the integration scheme is similar to

the one used in chapter 5 or any of those described in Chapter 8.

The methodology to constitute the evolutionary non-AFR Yld2000-2d anisotropic

model is as follows:

1. Find accurate descriptions (e.g. CSV in Eqn.(3-2)) for the hardening curves

corresponding to 7 uniaxial states plus one balanced biaxial one.

2. Determine Poly3 curve fits, Eqn. (10-18), to the Lankford coefficients at 7

uniaxial stress states plus one balanced biaxial one.

3. Find directional Lankford coefficients and normalized yield stresses at small

plastic work increments or equivalent plastic strains using an inverse


iterative method and imposing the principle of plastic work equivalence.

This is the most challenging part.

4. Optimize the parameters of both yield and plastic potential functions of the

non-AFR Yld2000-2d model. Select a set of parameters for these two

functions at an arbitrary plastic work value as initial guess for the parameters

at other quantities of plastic work.

5. Apply a 4th order polynomial fit (Poly4, defined in Eqn.(10-19) ) to the

parameters of plastic potential and yield stress functions.

The Poly4 is defined by

(10-19)

In summary, each of the 16 parameters of the non-AFR Yld2000-2d model are fourth

order polynomial functions of equivalent plastic strain. The parameters of Poly4 for

yield and plastic potential functions of the non-AFR Yld2000-2d model are given

respectively in Table 10-9 and Table 10-10. It was found that optimization of Poly4

parameters to the parameters of yield and plastic potential functions, yields

appropriate accuracy for increments equal to 0.05 equivalent plastic strain.

Table 10-9 Parameters of Poly4 for yield function parameters.

1.48 0.36 0.40 0.83 -0.82 -1.45 -0.01 -1.55

-12.01 22.28 14.20 3.18 -8.02 6.05 26.31 -8.56

133.84 -232.41 -75.48 -34.13 146.62 -76.01 -240.97 154.91

-572.44 963.5 160.63 145.13 -757.16 358.90 937.83 -798.1

829.56 -1387.02 -164.67 -207.18 1177.86 -552.26 -1351.81 1237.66

Table 10-10 Parameters of Poly4 for plastic potential function parameters.

0.357 1.043 0.209 0.483 0.596 -0.179 0.559 1.539

-1.831 -0.468 -2.236 0.795 1.005 -1.795 -2.512 1.063

26.149 2.340 31.360 -8.805 -13.013 28.133 34.751 -12.788

-125.299 -18.124 -138.189 45.547 67.472 -132.518 -122.279 45.836

190.311 39.785 205.462 -73.306 -106.901 201.671 134.071 -55.893

In Fig 10-28, the excellent fit of Poly4 to the parameters of Yld2000-2d yield stress

function (Table 10-7) is shown. Identically, Fig 10-29 shows the excellent fit of Poly4

to the Yld2000-2d plastic potential parameters (Table 10-8). These results indicate

209

that a 4th order polynomial curve fit is highly accurate for simulation of parameter

evolution.

Fig 10-28 Polynomial fit (Poly4) to the parameters of the yield function of non-AFR

Yld2000-2d model.

In Fig 10-30, the evolution of the yield function Yld2000-2d at various levels of

equivalent plastic strain is presented. Shown are the contours of the three dimensional

yield surface corresponding to different levels of shear stress (0.07 increment). In Fig

0.00 0.05 0.10 0.15 0.20 0.25 0.301.00

1.35

1.70 0.00

0.65

1.300.2

0.7

1.2

0.8

0.9

1.0-1.2

-0.8

-0.4 -1.50

-1.39

-1.28

-1.17-0.2

0.3

0.8

-1.8

-1.5

-1.2

Para

mete

rs o

f Y

ld2000-2

d y

ield

str

ess function (

non-A

FR

)


Poly4

Input parameters


10-31, the yield surfaces are presented in one plot which visualizes the evolution

more clearly.

Fig 10-29 Polynomial fit (Poly4) to the parameters of the plastic potential function of

non-AFR Yld2000-2d model.

Previously, in Fig 10-14 it was shown that the level of distortion of the hardening

decreases after except for orientation. Accordingly, in Fig 10-31 it is

0.00 0.05 0.10 0.15 0.20 0.25 0.300.23

0.35

0.47

0.59 0.900

0.957

1.014

0.10

0.23

0.36

0.49 0.4

0.5

0.60.5

0.6

0.7 -0.3

-0.2

-0.10.40

0.57

0.74

1.5

1.6

1.7

Poly4

Input parameters

Para

mete

rs o

f Y

ld2000-2

d p

ote

ntial fu

nction (

non-A

FR

)


1

2

3

4

5

6

7

8

211

shown that those small variations in the directional hardenings after result in

negligible changes in the shape of the yield surface.

Fig 10-30 Yield surface contours at 0.001, 0.1, 0.2 and 0.3 equivalent plastic strain

and different levels of shear stress.

Fig 10-31 Yield surfaces at various equivalent plastic strains.

In Fig 10-32, the two-dimensional representations of evolutionary plastic potential

function at various equivalent plastic strains are illustrated. Again, the surface

contours correspond to increments of 0.07 shear stress. The change in the shape of


these surfaces is observed with respect to increase of plastic deformation. In Fig

10-33 plastic potential surfaces at various equivalent plastic strains are compared in

one plot.

Fig 10-32 Plastic potential surface contours at 0.001, 0.1, 0.2 and 0.3 equivalent

plastic strain and different levels of shear stress.

Fig 10-33 Plastic potential surfaces at various equivalent plastic strains

In Fig 10-34 and Fig 10-35 respectively the yield and plastic potential surfaces are

illustrated in three-dimensional normalized stress space.

213

Fig 10-36 presents the simulations of evolution of directional normalized yield stress

with regard to plastic deformation at each orientation based on the evolutionary non-

AFR Yld2000-2d model. Similar results for evolution of directional Lankford

coefficients using the evolutionary non-AFR Yld2000-2d model are given in Fig

10-37. Based on these plots, the significant predictabilities of evolutionary non-AFR

Yld2000-2d model in terms of evolutionary directional Lankford coefficients and

yield stresses are evident. It is worth noting that the slight changes in shape of the

plastic potential function (Fig 10-33) result in an accurate prediction of changes in

Lankford coefficients at various levels of plastic work.

Fig 10-34 Three dimensional representation of yield surfaces at 0.001, 0.1, 0.2 and 0.3

equivalent plastic strain.

It has to be reminded that this evolutionary anisotropic model automatically updates

its parameters for every equivalent plastic strain based on a fourth order polynomial

function.

It is interesting to evaluate the results of this approach in terms of yield stresses and

Lankford coefficients with similar results obtained by the separate sets of parameters

that were used for Poly4 calibration (as presented in Table 10-7 and Table 10-8).


The ideal result would of course be a perfect correspondence. To this end, Fig 10-36

and Fig 10-37 show the evolution of respectively normalized yield stresses and

Lankford coefficients by evolutionary non-AFR Yld2000-2d model and non-AFR

Yld2000-2d model which is updated by independent sets of parameters given in Table

10-7 and Table 10-8. Considering the qualitative complexities in the yield stress

directionalities at each level of plastic deformation, the evolutionary non-AFR

Yld2000-2d model interestingly predicts very well the directionality of those obtained

experimentally and also of these predicted by independent sets of parameters given in

Table 10-7 (Fig 10-36). A difference in absolute values can be observed for

equivalent plastic strain 0.2, but this is a normal consequence of using a polynomial

approximation of the real evolutionary data.

Fig 10-35 Three dimensional representation of plastic potential functions at 0.001,

0.1, 0.2 and 0.3 equivalent plastic strain and different levels of shear stress.

Similarly, from Fig 10-37 it can be clearly concluded that the evolutionary non-AFR

Yld2000-2d model remarkably reproduces the experimental results and those obtained

by independent sets of parameters given in Table 10-8.

215

Fig 10-36 Predicted evolution of yield stress directionality by evolutionary non-AFR

Yld2000-2d model (Poly4) compared with experimental results and results obtained

using indepenent parameters (Table 10-7) at various equivalent plastic strains.

Fig 10-37 Predicted evolution of Lankford coefficient directionality by evolutionary

non-AFR Yld2000-2d model (Poly4) compared with experimental results and results

obtained using indepenent parameters (Table 10-8) at various equivalent plastic

strains.

In Fig 10-38, evolutionary non-AFR Yld2000-2d and ordinary non-AFR Yld2000-2d

models are compared in terms of evolution of directional normalized yield stress. The

(ordinary) non-AFR Yld2000-2d model has constant parameters and only can capture

initial anisotropy. Parameters of the ordinary non-AFR Yld2000-2d model are those

corresponding to in Table 10-7 and Table 10-8. A similar comparison for

evolution of directional Lankford coefficients (r-values) is presented in Fig 10-39.

These plots prove that the current evolutionary non-AFR Yld2000-2d model can

accurately simulate the evolution of distortional hardening and directional r-values

with plastic deformation.


In conclusion, the evolutionary non-AFR Yld2000-2d model is highly capable of a)

accurate prediction of stress and Lankford coefficient directionalities and b) following

the evolutionary behavior of material during plastic deformation with great accuracy.

Fig 10-38 Predicted evolution of yield stress directionality by evolutionary and

ordinary (fixed parameter) non-AFR Yld2000-2d models at various plastic works.

217

Fig 10-39 Predicted evolution of r-value directionalities by evolutionary and ordinary

(fixed parameters) non-AFR Yld2000-2d at various plastic works.

10.5.3 Implementation using scaled simplified approach

Special attention is required to implement this evolutionary model based on the scaled

simplified method described in Section 8.3.3. Recall from Eqn.(8-26) that scaling is

applied by


(10-20)

The scaling factor must be updated only once at beginning of each iteration using the

equivalent plastic strain from previous step. Subsequently, each parameter is updated

using Eqn.(10-19).

10.6 Summary

In this chapter, the distortional hardening with increase of plastic deformation was

experimentally investigated. For the interstitial free steel DC06 (deep drawing grade),

the level of hardening distortion is remarkable between the initial state of the material

and an equivalent plastic strain of 0.1. Minor changes in the hardening were observed

after that level of deformation. In addition, it was experimentally observed that the

Lankford directionality highly evolves with plastic deformation. If an anisotropic

model from Hill’s or Barlat’s families would be chosen, then unavoidably prediction

of evolution is impossible due to the constant model parameters. Previous

developments of evolutionary anisotropic models lack combination of accuracy and

simplicity. Significant capabilities of the non-AFR Yld2000-2d model in prediction of

stress and Lankford directionalities were proven in chapter 7. Therefore, it would be

highly advantageous if one could introduce the evolution of parameters by a simple

approach. Generally spoken, introduction of the evolution of parameters is challenged

due to the lack of appropriate patterns in the evolution of parameters at various plastic

work levels. However, using a simple numerical technique, it was possible to

optimize the parameters in a way that they change around one specific set of

parameters. In a next step, a 4th order polynomial function was used to describe the

evolution of parameters with respect to equivalent plastic strain. It was shown that

this simple technique results in strong simulation of evolution of Lankford coefficient

and yield stress directionalities.

219

Bibliography

Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006a. Forming of aluminum alloys at

elevated temperatures - Part 1: Material characterization. International Journal of


Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006b. Forming of aluminum alloys at

elevated temperatures - Part 2: Numerical modeling and experimental verification.


An, Y.G., Vegter, H., Melzer, S., Triguero, P.R., 2013. Evolution of the plastic

anisotropy with straining and its implication on formability for sheet metals. Journal

of Materials Processing Technology (under publication).

Aretz, H., 2008. A simple isotropic-distortional hardening model and its application in

elastic-plastic analysis of localized necking in orthotropic sheet metals. International


Darbandi, P., Pourboghrat, F., 2011. An evolutionary yield function based on Barlat

2000 yield function for the superconducting niobium sheet. AIP Conference

Proceedings 1383, 210-217.



Iadicola, M.A., 2011. Validation of Uniaxial Data Beyond Uniform Elongation. AIP

Conference Proceedings 1383, 742-749.

Plunkett, B., Cazacu, O., Barlat, F., 2008. Orthotropic yield criteria for description of

the anisotropy in tension and compression of sheet metals. International Journal of


Safaei, M., De Waele, W., 2012. Plastic Strain Induced Anisotropy In Sheet Metals,

Advances in material processing and technology, Wollongong, Australia.

Stoughton, T.B., Yoon, J.W., 2005. Sheet metal formability analysis for anisotropic

materials under non-proportional loading. International Journal of Mechanical

Sciences 47, 1972-2002.

Wang, H., Wan, M., Wu, X., Yan, Y., 2009. The equivalent plastic strain-dependent

Yld2000-2d yield function and the experimental verification. Computational

Materials Science 47, 12-22.



1563-1578.

Zamiri, A., Pourboghrat, F., 2007. Characterization and development of an

evolutionary yield function for the superconducting niobium sheet. International

Journal of Solids and Structures 44, 8627-8647.

Chapter 11

11 Conclusions

222 Chapter 11, Conclusions

11.1 Summary and main conclusions

The following subsections summarize the main conclusions of this dissertation, with

respect to the research goals that have been set forward in the first chapter.

11.1.1 Accurate description of severe anisotropy

This work mainly emphasized on the advantages of non-AFR based anisotropic yield

models for the description of severe anisotropy in sheet metals. It was investigated

whether the removal of the artificial constraint of equality of yield and plastic

potential functions in non-AFR based models can generate more accurate results as

compared to their AFR counterparts. The expectation was that due to incorporating

additional experimental data for model parameter calibration, the non-AFR approach

would allow to model severe anisotropy with higher accuracy.

To this end the non-AFR Yld2000-2d and non-AFR Hill 1948 models were

implemented into the commercial finite element software Abaqus (versions 6.10 and

6.11). To this end a user material subroutine was developed based on a fully implicit

backward Euler integration scheme. First the verification of the developed subroutine

was presented. Subsequently, a comparison was made of simulation results obtained

by AFR and non-AFR versions of the mentioned yield models. It was shown that

excellent results are achieved using the non-AFR Yld2000-2d model. It was also

shown that the same order of accuracy as obtained by the 18 parameter AFR based

Yld2004-18p model can be achieved by the non-AFR Yld2000-2d model.

Interestingly, we showed that the non-AFR Yld2000-2d model produced excellent

accuracy for the severely anisotropic fictitious FM8 material.

Even more, to evaluate the improvements of results in more demanding loading

conditions, results of cup deep drawing simulations for aluminium alloy AA2090-T3

were presented. A deep drawn cup of this material exhibits 6 ears. This can only be

predicted when the variation of in-plane anisotropy is predicted with high accuracy.

Considerable improvement was observed by applying the non-AFR Yld2000-2d

model when compared with its AFR based counterpart as well as other presented

models. It was shown that only the non-AFR Yld2000-2d model can predict the exact

number of 6 ears as observed in experimental results. Even more, prediction of 8 ears

observed in cup deep drawing of a challenging fictitious alloy was also reported in

this work based on the non-AFR Yld2000-2d model.

223

11.1.2 Combination of strong anisotropic yield function with mixed hardening

definition

Once anisotropy was described with significant accuracy, the generalization of the

model allowing inclusion of an accurate mixed isotropic-kinematic hardening

definition was presented. This generalization was performed despite the fact that

springback prediction was outside the scope of this research project. Nonetheless, the

impact of hardening definitions on the prediction of cup height was the major concern

of this part. Accordingly, cup deep drawing simulations for aluminium alloy AA5754-

O using isotropic, kinematic and mixed hardening models were presented. It was

reported that the over- and underestimation of the hardening curve generated by

isotropic and kinematic hardening respectively, is reflected in the predicted cup

height.

11.1.3 Implementation of advanced material models into commercial finite

element software

So-called user material subroutines (UMAT) for Abaqus were developed in

FORTRAN based on a fully implicit backward Euler method for the models described

in Sections 11.1.1 and 11.1.2. Various codes were developed in Mathematica® to

generate analytical results for cyclic loading conditions and uniaxial and biaxial

tensile tests. The analytical results together with experimental data were used to

extensively evaluate the developed subroutines and the strength of the developed

material models.

A part of this work was devoted to introducing a simpler version of the non-AFR

model (compared with original version) to alleviate the effort of a fully implicit

scheme. The proposed method is based on a simplification of the relation between

equivalent plastic strain and compliance factor in a non-AFR based model. It was

shown that when the non-AFR model is simplified without scaling the plastic

potential function, this results in a wrong definition of equivalent plastic strain. It

must be noted that such simplification of the non-AFR yield model without further

treatments (i.e. scaling) is physically erroneous and violates the equivalence of plastic

work. Furthermore, it was shown that this simplification leads to an overestimation of

Cauchy stress for aluminium alloy AA2090-T3 due to over prediction of equivalent

plastic strain. Accordingly we showed that this discrepancy can be improved

significantly by applying a scaling on the size of the plastic potential function. If such

scaling is carried out on the simplified model, then the equality of equivalent plastic

strain and compliance factor can be assured. This technique was evaluated for the

non-AFR Yld2000-2d model and it was shown that the proposed scaled simplified

approach highly improves the accuracy of simulation results when compared with the

un-scaled simplification. In addition, the cup drawing simulations of alloy AA2090-


T3 using the scaled simplified approach show a very similar cup profile as compared

to the one obtained by the full (non-simplified) expression. These results prove that

using the scaling technique, the simplified approach can be a reliable alternative of

full expression for FE simulation.

Also with respect to implementation of the non-AFR model in a user material

subroutine, various implicit and explicit integration schemes were deployed and their

results were compared for a cup deep drawing simulation of materials AA2090-T3

and FM8. The residuals of the various models were compared at an identical punch

stroke. It was shown that the accuracy of a classical forward Euler scheme is the

worst amongst all studied approaches. However, considering a simple change to this

integration scheme generates much more accurate results (next increment correct

errors, NICE-1 technique) in a sense that it gives close results as compared to the

classic backward Euler approach with 20 iterations. It was reported that the convex

cutting plane scheme is the most accurate technique (taking into account the

computational time cost) among all schemes that were implemented in

Abaqus/Explicit.

11.1.4 Description of the evolution of anisotropy during the deformation

process

Lastly, we elaborated on experimental observations of distortional hardening,

evolution of Lankford coefficients and a model which can properly simulate those

experimental observations. The results of experimental tensile tests performed on an

interstitial free deep drawing steel DC06 confirm that an evolution of anisotropy can

be expected during plastic deformation. To include the prediction of this evolution in

the non-AFR Yld2000-2d model, a simple approach has proposed. The patterns

observed in the parameter evolution of plastic potential and yield functions were

described by fourth order polynomial functions. It was shown that such interpolation

properly simulates the pattern in the parameters of plastic potential and yield

functions. An accurate description of the evolution of the model parameters

consequently results in a reliable prediction of instantaneous r-values and yield

stresses.

11.2 Future works

This dissertation concluded that anisotropic behaviour of sheet metals can be

well predicted by the non-AFR Yld2000-2d yield model. The developed user

material subroutine has been generalized such that it is able to use any yield and

plastic potential function. However, in this work it was reported that the non-

AFR Yld2000-2d is a considerably strong model and can predict severe degree

of anisotropy with significant accuracy. Similar observations were reported for

225

advanced high strength steels such as DP600 (Taherizadeh et al., 2010) and

TRIP780, DP590 (Mohr et al., 2010). Therefore it seems that the need for an

advanced anisotropic model for proportional loading conditions and constant

anisotropic behaviour is satisfied. Furthermore, the evolution of anisotropy is

also predicted by a user-friendly evolutionary anisotropic model based on non-

AFR Yld2000-2d. Experimental analysis of evolution of hardening and r-value at

balanced biaxial test is foreseen for future work. Optimization of Zang’s

hardening parameters of DC06 using experimental data of simple shear test is

recommended as the next step.

The presented evolutionary model was a continuation of the works of Yoon et al

(2010) and Darbandi and Pourboghrat (2011). However, the model presented in

this dissertation does not take the effects of strain path changes into account.

Evolution of the yield surface under cyclic and cross strain path changes were

respectively considered in very advanced models of Barlat et al (2011) and

(2012). Application of non-AFR in those models is expected to combine the

accurate modelling of strain-path change and anisotropy, simultaneously.

The extension of the developed non-AFR based model to consider the strength-

differential effects (as observed for hexagonal closed packed materials) and also

pressure sensitive yield functions could be recommended for future works. For

instance, the non-AFR can be used for the recently proposed model of Plunkett et

al (2006) that introduced the evolution of yield surface due to texture evolution in the

CPB05 (Cazacu et al., 2006) anisotropic model for hexagonal closed packed

materials.

Other suggestions for further extension of the model capabilities are the

consideration of rate and temperature dependent effects into the developed non-

AFR model.

Finally, introduction of suited damage laws into the hardening definitions in a

non-AFR based constitutive model is recommended.


Bibliography

Barlat, F., Gracio, J.J., Lee, M.-G., Rauch, E.F., Vincze, G., 2011. An alternative to

kinematic hardening in classical plasticity. International Journal of Plasticity 27,

1309-1327.

Barlat, F., Ha, J., Grácio, J.J., Lee, M.-G., Rauch, E.F., Vincze, G., 2012. Extension

of homogeneous anisotropic hardening model to cross-loading with latent effects.

International Journal of Plasticity.



Darbandi, P., Pourboghrat, F., 2011. An evolutionary yield function based on Barlat

2000 yield function for the superconducting niobium sheet. AIP Conference

Proceedings 1383, 210-217.

Mohr, D., Dunand, M., Kim, K.-H., 2010. Evaluation of associated and non-

associated quadratic plasticity models for advanced high strength steel sheets under

multi-axial loading. International Journal of Plasticity 26, 939-956.

Plunkett, B., Lebensohn, R.A., Cazacu, O., Barlat, F., 2006. Anisotropic yield

function of hexagonal materials taking into account texture development and

anisotropic hardening. Acta Mater 54, 4159-4169.






1563-1578.

Appendix A

A Fully implicit backward Euler

scheme

(for non-AFR anisotropic flow and mixed isotropic-

kinematic hardening)

This appendix provides a detailed description of the fully implicit backward Euler

integration scheme for non-AFR anisotropic flow and mixed hardening model

described in chapter 5.

A.1 Newton-Raphson iteration scheme

The update expressions in Eqns.(6-21a) to (6-21g) can be written in the following

forms suitable for the Newton-Raphson iteration scheme

(A- 1)

where

(A- 2)

228 Appendix A, Fully implicit backward Euler scheme

Linearization of the above and

gives

(A- 3)

[ ]{ }

where

( )

( )

(A- 4)

( )

( )

( )

( )

where means first derivative of with respect to .

( )

(A- 5)

and

(A- 6)

and

229

(A- 7)

and

(A- 8)

The system of equations in (A- 3) can be written in matrix form

⌈ ⌉

{

}

{ } { } (A- 9)

where

⌈ ⌉

[

]

(A- 10)

{ }

{

}

, { }

{

}

(A- 11)

Because effective plastic strain (

and compliance are linearly related

(A- 12)

Therefore


{

}

⌈ ⌉{ } ⌈ ⌉{ } (A- 13)

Using(A- 4) to (A- 8) in (A- 10) and determining the ⌈ ⌉

⌈ ⌉

(

)

(A- 14)

where

(A- 15)

and

(A- 16)

and

(A- 17)

and

(

)

231

(A- 18)

and

(

) (A- 19)

Substituting (A- 13) in the last part of (A- 3)

[ ]

[ ] (A- 20)

⌈ ⌉

⌈ ⌉

⌈ ⌉

⌈ ⌉ ⌈ ⌉

⌈ ⌉

where ⌈ ⌉

denotes the row of associated to (first, second, third and fourth

row of respectively for , and

). Note that

⌈ ⌉

(A- 21)

⌈ ⌉

When is determined, ,

,

and

are updated using (A- 3) and

subsequently the internal parameters are updated as follows

(A- 22)

A.2 Consistent tangent modulus

The continuum (standard) elasto-plastic tangent operator relates the stress to total

strain rates. However, according to Belytschko (Belytschko et al., 2000) the

continuum (standard) elasto-plastic tangent operator can generate spurious loading


and unloading condition during the abrupt transition from elastic to plastic state. In

addition, the consistent (algorithmic) tangent modulus is required to preserve the

quadratic rate of asymptotic convergence inherent to the Newton-Raphson’s iteration

nested in the fully implicit backward Euler algorithm (Simo and Hughes, 1998).

Analogously to the previous approach for finding the rate variables at time step (n+1),

the consistent modulus is obtained by linearization of the constitutive equations to

relate the stress increment to total strain increment at time (t+1). By following the

same approach described in the previous section but assuming the total strain as non-

constant and residuals ({ }) as zero, after many manipulations the closed form of

consistent tangent modulus is obtained. It is noticed that the non-symmetric consistent

modulus converts to the standard tangent operator by reducing the step size to zero.

Furthermore both consistent and tangent moduli turn to elastic stiffness matrix when

no plastic loading occurs.

We write the set of Eqns.(5-21a) to (5-21g) in rate form so that

(

)

(A- 23)

where

(A- 24)

Substituting (A- 23)_2 in (A- 23)_1 and using (A- 24) and solving for and

{

}

⌈ ⌉{ } ⌈ ⌉{ } (A- 25)

233

where

{ } {

} (A- 26)

Substituting (A- 25) into the incremental consistency condition (A- 23) _6

[ ]

[ ] (A- 27)

⌈ ⌉ ⌈ ⌉

⌈ ⌉

⌈ ⌉ ⌈ ⌉ ⌈ ⌉ (A- 28)

Substituting (A- 29) into (A- 25)

{

} [ [ ]

[ ] ] {

} (A- 30)

Finally the consistent tangent moduli is obtained

⌈ ⌉{ } ([ ] ⌈ ⌉)

[ ] ⌈ ⌉{ } (A- 31)


Bibliography

Belytschko, T., Liu, W.K., Moran, B., 2000. Nonlinear finite elements for continua

and structures. John Wiley, Chichester ISBN 0471987735


mathematics. Springer, New York ISBN 0387975209

Appendix B

B Parameter optimization

This appendix provides a detailed description of the parameter identification for AFR

and non-AFR anisotropic functions based on the error minimization technique.

B.1 Introduction

In order to optimize the model parameters of the yield function an error function is

minimized. To this end, the method of calculating Lankford coefficients and

normalized yield stresses at various stress states must be described.

First, one should note that the order of tensor components in this appendix is identical

to the order used in the UMAT for Abaqus/Standard for which the stress vector is

written as follows

[ ] (B- 1)

In addition, from Fig 2-1, the distinction between material orthotropic frame (xx-yy)

and a rotated frame (11-22) must be reminded.

Recall the yield criterion and a general non-quadratic yield function

( ) (B- 2)

For Barlat yield functions such as Yld96,Yld2000-2d, Yld2004-18p and Yld2004-13p

we can write

(

)

(B- 3)

236 Appendix B, Parameter optimization

with

(B- 4)

Therefore the yield function is recast to

(B- 5)

B.2 Normalized yield stress

B.2.1 Uniaxial direction

For an in-plane tensile test applied to a sample taken at degrees orientation from the

rolling direction, the corresponding stress tensor expressed in the material’s

orthotropic frame can be represented by the following vectors of rank 6 and 3 for 3D

and plane stress case, respectively (Barlat et al., 2005)

(B- 6)

where for 3D case

[ ] (B- 7)

For plane stress case

[ ] (B- 8)

where is the flow stress in the corresponding direction. Considering the AFR

Yld2000-2d model, is 0°, 45° and 90°.

As opposed to the Barlat family of yield functions, some other yield functions require

that the deviator stress should be directly given to the model such that

(B- 9)

where

[(

) (

) (

) ]

(B- 10)


237

[(

) (

) ]

(B- 11)

Finally, the flow stress can be normalized with respect to the flow stress at rolling

direction ( )

|| ||

(

)

(B- 12)

where || || is normalized yield stress, k is equal to 2 for Yld2000-2d and Yld2004-

13p models and 4 for Yld2004-18p model.

B.2.2 Balanced biaxial condition

For in-plane balanced biaxial tension, the stress is represented by

(B- 13)

For 3D case

[ ] (B- 14)

For plane stress condition

[ ] (B- 15)

When deviator stress has to be directly given to the yield function then

(B- 16)

where

[

]

(B- 17)


[

]

(B- 18)

Finally, the normalized flow stress || || is

|| ||

(

)

(B- 19)


B.2.3 Out-of-plane direction

For 3D functions such as the Yld2004-18p model the out-of-plane normalized flow

stresses || || might be necessary

|| ||

( ) (

( ))

(B- 20)

where for 45° tension at (TD-ND) plane is

[ ] (B- 21)

When deviator stress is required

[ ] (B- 22)

And for 45° tension at (ND-RD) plane

[ ] (B- 23)

When deviator stress is required

[ ] (B- 24)

For a simple shear test at (TD-ND) plane

[ ] (B- 25)

For a simple shear test at (ND-RD) plane

[ ] (B- 26)

B.3 Lankford coefficient

B.3.1 Uniaxial direction

The Lankford coefficient at the corresponding uniaxial direction is

(B- 27)

239

Using the incompressibility hypothesis

(

) (

) (B- 28)

Substitution of Eqn.(B- 28) into Eqn. (B- 27) leads to

(B- 29a)

(B- 29b)

(B- 29c)

Also

(B- 30)

Recall that is the normal to the yield surface in AFR.

Applying Euler’s theorem to the first order homogenous yield function , at

uniaxial tension direction we have

(B- 31)

Here is an auxiliary uniaxial tensile stress at direction (i.e. ).

Therefore

(B- 32)

Finally, the Lankford coefficient at degrees orientation with respect to the rolling

direction is

(B- 33a)

|| ||( ) (B- 33b)

B.3.2 Balanced biaxial condition

Calculation of Lankford coefficient at balanced biaxial condition is straightforward


(B- 34)

B.4 Error function

In general, the error function which should be minimized is defined by

∑ (||

||

||

|| ) ∑ (

) (B- 35)

Or in a simpler form

∑ ( ||

|| ) ∑ (

) (B- 36)

The superscripts and respectively denote experimental and simulated values,

p is the number of all experimental yield stresses, q represents the number of

experimental Lankford coefficients and is a weighting factor. In the above equation

represents the set of model parameters (Barlat et al., 2005).

B.4.1 AFR Yld2000-2d model

Considering the AFR Yld2000-2d model the error function is minimized using

experimental uniaxial data at and 90° and balanced biaxial state (i.e.

).

B.4.2 Non-AFR Yld2000-2d model

In this case, the parameters of yield and plastic potential functions are optimized

separately.

Considering the yield function, is set zero. Subsequently, optimization is performed

based on experimental yield stresses at and and

balanced biaxial condition (i.e. ).

Considering the plastic potential function, the Lankford coefficient at

corresponding uniaxial direction is

(B- 37)

And the coefficient at balanced biaxial condition is

(B- 38)

241

Subsequently, the minimization of error function in Eqn.(B- 36) is performed with

set to zero and using experimental Lankford coefficients at

and and balanced biaxial condition (i.e. ).


Bibliography



1009-1039.

Publications

A1 - Peer reviewed journal publications included in Science Citation

Index

Safaei, M., Zang, S.L., Lee, M.G., De Waele, W., 2012. Evaluation of Anisotropic



10.1016/j.ijmecsci.2013.04.003).

Safaei, M., Yoon, J.-W., De Waele, W., Study on the definition of effective plastic

strain under non-associated flow rule. International Journal of Plasticity (submitted).

Safaei, M., De Waele, W., Hertschap, K., 2012. Characterization of deep drawing

steels using optical strain measurements. Steel Research International,4, 403-406.

Safaei, M., Lee, M.G., Zang, S.L., De Waele, W., An evolutionary anisotropic model

for sheet metals based on non-associated flow rule approach. Computational

Materials Science (under review).

Safaei, M., Lee, M.G., De Waele, W., Evaluation of various implementation schemes

of asociated and non-associated flow rule based metal plasticity models.

International Journal for Numerical Methods in Engineering (in preparation).

A2 - Peer reviewed journal publications not included in Science Citation

Index

Khalili, K., Safaei, M., 2009. FEM analysis of edge preparation for chamfered tools.

International Journal of Material Forming 2, 217-224.

Safaei, M., De Waele, W., 2011. An implicit return mapping algorithm for anisotropic

plasticity with mixed non-linear kinematic-isotropic hardening. Mechanical

Engineering Letters 5, 17-29.

244 Publications

P1 – Publications in conference proceedings included in Science Citation

Index

Safaei, M., De Waele, W., Hertschap, K., 2010. Finite element analysis of influence

of material anisotropy on the springback of advanced high strength steel, in: Chinesta,

F., Chastel, Y., El Mansori, M. (Eds.), AIP Conference Proceedings. American

Institute of Physics (AIP), pp. 371-376.

Safaei, M., De Waele, W., Zang, S.L., 2012. Evaluation of associated and non-

associated flow metal plasticity: application for DC06 deep drawing steel, in:

Merklein, M., Hagenah, H. (Eds.), Key Engineering Materials. Trans Tech, pp. 661-

666.

C1 – Publications in conference proceedings

Safaei, M., De Waele, W., 2011a. A constitutive plasticity for non-linear mixed

hardening and Hill's quadratic yield criteria, in: Kurják, Z., Magó, L. (Eds.), Synergy in the technical development of agriculture and food industry (Synergy2011). Szent

István University. Faculty of Mechanical Engineering.

Safaei, M., De Waele, W., 2011b. Development of a continuum plasticity model for

the commercial finite element code ABAQUS, in: Van Wittenberghe, J. (Ed.),

Sustainable Construction and Design, 2 ed. Ghent University, Laboratory Soete, pp.

275-283.

Safaei, M., De Waele, W., 2011c. Towards better finite element modelling of elastic

recovery in sheet metal forming of advanced high strength steel, in: Van

Wittenberghe, J. (Ed.), Sustainable Construction and Design, 2 ed. Ghent University,

Laboratory Soete, pp. 217-227.

Safaei, M., De Waele, W., Zang, S.-l., 2012. A rate-independent non-associated constitutive model for finite element simulation of sheet metal forming, in: Khan, A.

(Ed.), Plasticity and its Current Applications, 18th International symposium,

Proceedings. Numerical Engineering Analysis and Testing (NEAT), pp. 61-63.

Safaei, M., Movahhedy, M.R., Khalili, K., 2009. FEM analysis of edge preparation

for chamfered tools, in: Arrazola, P.J. (Ed.), Proceedings of the 12th CIRP conference

on modelling of machining operations. Mondragon Unibertsitateko Zerbitzu, pp. 187-

193.

C3 – Conference abstracts

Safaei, M., 2010. Influence of contact and material models on springback simulation

in sheet metal forming, UGent-FirW Doctoraatssymposium, 11e. Universiteit Gent.

Faculteit Ingenieurswetenschappen.

Safaei, M., De Waele, W., Yoon, J.-W., 2013. Earing predictions using different

associated and non-associated plasticity models, in: Khan, A. (Ed.), 19th International

245

Symposium on Plasticity and its Current Applications, Abstracts. Numerical

Engineering Analysis and Testing (NEAT), Inc.

246 Publications

Constitutive Modelling of Anisotropic Sheet Metals Based on a Non-Associated Flow Rule

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