DEVELOPMENT OF PLASTICITY AND DUCTILE FRACTURE MODELS

INVOLVING THREE STRESS INVARIANTS

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Tingting Zhang

May, 2012

ii

DEVELOPMENT OF PLASTICITY AND DUCTILE FRACTURE MODELS

INVOLVING THREE STRESS INVARIANTS

Tingting Zhang

Dissertation

Approved: Accepted:

__________________________ ____________________________ Advisor Department Chair Xiaosheng Gao Celal Batur

__________________________ ____________________________ Committee Member Dean of the College Fred Choy George K. Haritos

__________________________ ____________________________ Committee Member Dean of the Graduate School Gregory Morscher George R. Newkome

__________________________ ____________________________ Committee Member Date Ernian Pan

__________________________ Committee Member Kevin Kreider

iii

ABSTRACT

It has been shown that the plastic response of many materials, including some

metallic alloys, depends on the stress state. Based on plasticity analysis of three metal

alloys, a series of new plasticity models with stress state effect is proposed. The effect of

stress state on plasticity and the general forms of the yield function and flow potential for

isotropic materials are assumed to be functions of the first invariant of the stress tensor

(I1) and the second and third invariants of the deviatoric stress tensor (J2 and J3). Finite

element implementation, including integration of the constitutive equations using the

backward Euler method and formulation of the consistent tangent moduli, are presented

in this thesis.

A 5083 aluminum alloy, Nitronic 40 (a stainless steel), and Zircaloy-4 (a

zirconium alloy) were tested under tension, compression, torsion, combined torsion-

tension and combined torsion-compression at room temperature to demonstrate the

applicability of proposed I1-J2-J3 dependent models. It has shown that the output

produced by the proposed model have better agreement with experimental data than those

produced by the classical J2 plasticity theory for the tested loading conditions and

materials.

Furthermore, the Gurson-Tvergaard-Needleman porous plasticity model, which is

widely used to simulate the void growth process of ductile fracture, is extended to include

iv

the effects of hydrostatic stress and the third invariant of stress deviator o n the matrix

material.

The experimental and numerical work presented in this thesis reveals that the

stress state also has strong effects on the ductile fracture behavior of an aluminum 5083

alloy. For the ductile fracture analysis, The Goluganu-Leblond-Devaux (GLD) model is

employed to describe the porous plasticity behavior of aluminum 5083. The effect of

stress triaxiality and Lode angle is analyzed and fracture locus is calibrated as a criterion

for void coalescence. The GLD model combined with the fracture locus can be applied to

predict the failure of aluminum 5083 specimens with experiencing a large range of stress

triaxiality and Lode angle. The numerical analyses agree with the experimental data very

well.

v

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my sincere gratitude to my advisor

Dr. Xiaosheng Gao for his encouragements, consistent help and, most important, his

valuable instructions in the academic area. The accomplishment of this degree would be

impossible without his support.

I would also like to show my thanks to my committee members: Dr. Fred Choy,

Dr. Gregory Morscher from Mechanical Engineering Department, Dr. Ernian Pan from

Civil Engineering Department, and Dr. Kevin Kreider from Mathematics Department.

Special appreciation is given to them for their time on reading and evaluating my thesis.

Their comments and suggestions have improved the quality of my research work.

I would also like to thank my colleagues, including Jun Zhou, Sunil Prakash,

Yifei Gao and others. Discussions on the academic issues with them always enlightened

me to approach the research objectives. Special Thanks are given to Stacy and Christina,

the secretaries of Mechanical Engineering Department, for taking care of my

administrative issues during the last five years.

Last but not least, I want to thank my parents for their endless love and support.

During the time when I was pursuing my Ph.D. degree, my maternal grandfather passed

away. If there were no consolation or encouragement from my parents on the overseas

vi

calls, I would be struck down by sorrow and sadness. I love them more and more as time

goes by.

vii

TABLE OF CONTENTS

Page

LIST OF TABLES ............................................................................................................... x

LIST OF FIGURES ........................................................................................................... xii

CHAPTER

I. INTRODUCTION.............................................................................................................2

1.1 Development of plasticity models............................................................................. 2

1.2 Research objectives on plasticity modeling .............................................................. 3

1.3 Stress state effects on ductile fracture ....................................................................... 4

1.4 Research objectives on the ductile fracture study ..................................................... 6

1.5 Outline of the Thesis ................................................................................................. 7

II. LITERATURE REVIEW .................................................................................................9

2.1 Basic Concepts and Definitions ................................................................................ 9

2.2 Review of Plasticity Models.................................................................................... 13

2.3 Micro Mechanics Models ........................................................................................ 18

2.4 Non-associated Flow Rule ...................................................................................... 24

III. MODELING OF PLASTICITY RESPONSES: INVOLING THREE STRESS

INVARIANTS ...................................................................................................................26

3.1 Introduction ............................................................................................................. 26

viii

3.2 Plasticity Modeling ................................................................................................. 28

3.2.1 Influence of I1 and J3 ........................................................................................ 28

3.2.2 Yield Function and Flow Potential for 5083 Aluminum Alloy and Zircaloy-4 32

3.2.3 Yield Function and Flow Potential for Nitronic 40 Stainless Steel .................. 33

3.3 Numerical Implementation of the I1-J2-J3 plasticity model .................................... 34

3.3.1 Stress Update .................................................................................................... 34

3.3.2 The Consistent Tangent Moduli ....................................................................... 36

3.4 Applications of the stress-state dependent plasticity models .................................. 39

3.4.1 Designing of torsion-tension specimen ............................................................ 39

3.4.2 Materials, Specimens and Experiments ............................................................ 52

3.4.3 Finite Element Procedure ................................................................................. 58

3.4.4 Plasticity response of an aluminum 5083 ......................................................... 61

3.4.5 Plasticity response of a Nitronic 40 .................................................................. 68

3.4.6 Plasticity response of a Zircaloy 4 .................................................................... 72

3.5 Concluding Remarks ............................................................................................... 78

IV. MODIFIED POUROUS GURSON MODEL ...............................................................80

4.1 Introduction ............................................................................................................. 80

4.2 Modified Gurson Model with I1, J3 Effects............................................................. 81

4.2.1 Modified Gurson Model Theory....................................................................... 81

4.2.2 Numerical Implementation ............................................................................... 82

4.3 Numerical examples using the Gurson-Tvegaard-Needleman model..................... 83

4.4 Conclusion............................................................................................................... 88

V. PLASTICTY AND DUCTILE FRACTURE ANALYSIS FOR ALUMINUM 5083 ..89

ix

5.1 Introduction ............................................................................................................. 89

5.2 GLD model.............................................................................................................. 90

5.3 Specimen geometries and finite element procedures .............................................. 93

5.4 Experimental and numerical results .........................................................................97

5.4.1 Effect of the stress state on the materials plastic response .............................. 97

5.4.2 Effect of the stress state on the ductile failure strain ....................................... 101

5.5 Concluding remarks .............................................................................................. 108

VI. CONCLUDING REMARKS AND SUGGESTIONS FOR FUTURE WORK ......... 110

6.1 Conclusions and future works on plasticity modeling involving three stress

invariants ..................................................................................................................... 110

6.2 Conclusions and future works on stress state effects on Ductile Fracture ............ 111

BIBLIOGRAPHY ............................................................................................................ 114

APPENDICES ................................................................................................................. 124

APPENDIX A. EXAMPLES OF FIRST ORDER HOMOGENEOUS FUNCTIONS OF

STRESSES ....................................................................................................................... 125

APPENDIX B: GLD ........................................................................................................ 127

x

LIST OF TABLES

Table Page

3.1 Chemical composition (in weight percent) ................................................................53

3.2 Elastic properties........................................................................................................54

3.3 Notch radii of the notched round bars........................................................................55

3.4 Groove radii of the plane strain specimens ................................................................56

3.5 Ratios of the applied tensile displacement and applied twist angle used in the

tension-torsion tests ..........................................................................................................56

xi

LIST OF FIGURES

Figure page

2. 1 (a) The Haigh-Westergaard stress space, and (b) the deviatoric plane ...................... 11

3. 1 Variation of )33/( 2/121 JIT and )2/(332/3

23 JJ with plastic deformation in the center element of the smooth round bar. ..................................................................... 41

3. 2 Variation of )33/( 2/121 JIT and )2/(332/3

23 JJ with plastic deformation in the center element of the E-notch specimen. .................................................................... 41

3. 3 Variation of )33/( 2/121 JIT and )2/(332/3

23 JJ with plastic deformation in the center element of the G-groove specimen................................................................... 42

3. 4 Dimensions of NT specimen. All dimensions in mm. ............................................. 43

3. 5 Dimensions of Lindholm specimen. All dimensions in mm. ................................... 44

3. 6 Comparison of gage sections in NT and Lindholm specimens (to scale). ................ 44

3. 7 Finite element meshes for the, (a) NT specimen, and the, (b) modified Lindholm specimen............................................................................................................................ 46

3. 8 Through-thickness distribution of T, and p at the mid-section of the specimen for 0.37p and pure torsion loading, (a) NT specimen, (b) Lindholm specimen. ............. 48

3. 9 Through-thickness distribution of T, and p at the mid-section of the specimen for effective plastic strain of 0.02, (a) NT specimen, (b) Lindholm specimen. ..................... 49

3. 10. Through-thickness distribution of T, and p at the mid-section of the specimen for effective plastic strain of 0.20, (a) NT specimen, (b) Lindholm specimen. ............... 49

3. 11 Through-thickness distribution of T, and p at the mid-section of the specimen for effective plastic strain of 0.44, (a) NT specimen, (b) Lindholm specimen. ................ 50

xii

3. 12 Through-thickness distribution of T, and p at the mid-section of the specimen when p reaches 0.14, (a) NT specimen, (b) Lindholm specimen.................................... 51

3. 13 Evolution of T, Lode parameter, and with increasing effective plastic strain, (a) NT, (b) Lindholm specimens. ........................................................................................... 52

3. 14 Sketches of a round tensile bar, a notched round bar specimen, a compression specimen with L/D = 0.75, a grooved plan strain specimen and a torsion specimen. ...... 55

3. 15 Typical finite element meshes for (a) a notched round bar specimen, (b) a grooved plane strain specimen, (c) a compression specimen with L/D = 0.75, and (d) a torsion-compression specimen with a central pin. ........................................................................ 60

3. 16 Comparisons of load vs. displacement and/or torque vs. twist angle responses between the experimental data and the J2 model prediction for aluminum 5083: (a) the round tensile specimen, (b) the compression specimen with L/D = 0.75, and (c) the torsion specimen (Experimental data were from [111]). .................................................. 62

3. 17 (a) Projection of the yield surface of aluminum 5083 on the -plane; (b) Plot of the flow potential; (c) The equivalent stress vs. equivalent plastic strain curve describing the strain hardening behavior of the material. ........................................................................ 64

3. 18 Comparisons of the predicted load vs. displacement and/or toque vs. twist angle responses using the calibrated I1-J2-J3 plasticity model with experimental records for aluminum 5083: (a) the round tensile specimen; (b) the compression specimen with L/D=0.75; (c) torque vs. twist angle response of the pure torsion specimen; (d) axial force vs. axial displacement response of the torsion-tension specimen (TT-15); (e) torque vs. twist angle response of the torsion-tension specimen (TT-15) (Experimental data were from [111]) . ...................................................................................................................... 65

3. 19 Comparisons of the numerical predictions and experimental records: (a) notched round bar (E-Notch); (b) plane strain specimen (G-Groove); (c) torque vs. twist angle response for torsion-tension test (TT-16); (d) axial force vs. axial displacement response for torsion-tension test (TT-16) (Experimental data were from [111]). ........................... 67

3. 20 Comparisons of the numerical predictions using the classical J2-flow theory and the proposed I1-J2-J3 model for the tension-torsion test TT16 with experimental results: (a) torque vs. twist angle; (b) axial force vs. axial displacement (Experimental data were from [111]). ....................................................................................................................... 68

3. 21 Comparisons of load vs. displacement and/or torque vs. twist angle responses between the experimental data and the J2 plasticity theory predictions for Nitronic 40: (a) the tensile specimen, (b) compression specimen with L/D = 1.5, (c) axial force vs. axial displacement response for the torsion-compression specimen, and (d) toque vs. twist angle response for the torsion-compression specimen (Experimental data were from[112]). ........................................................................................................................ 69

xiii

3. 22 (a) Projection of the yield surface of Nitronic 40 on the -plane, and (b) the equivalent stress vs. equivalent plastic strain curve describing the strain hardening behavior of the material. ................................................................................................... 70

3. 23 Comparisons of the predicted load vs. displacement and/or toque vs. twist angle responses using the calibrated I1-J2-J3 plasticity model with experimental records for Nitronic 40: (a) the tensile specimen, (b) the compression specimen with L/D = 1.5, (c) axial force vs. axial displacement response for the torsion-compression specimen, and (d) torque vs. twist angle response for the torsion-compression specimen (Experimental data were from[112]). ............................................................................................................... 71

3. 24 Comparisons of numerical predictions using the calibrated I1-J2-J3 plasticity model with experimental data for Nitronic 40: (a) load vs. displacement response of the compression specimen with L/D=0.75; (b) torque vs. twist angle response of the pure torsion specimen (Experimental data were from[112]). ................................................... 72

3. 25 Comparisons of the experimental data and the J2 model predictions for Zircaloy: (a) load vs. displacement response of the tensile specimen, (b) load vs. displacement response of the compression specimen with L/D = 1.5, and (c) torque vs. twist angle response between experimental data and the I1-J2 model prediction for the torsion specimen (Experimental data were from[112]) ................................................................ 74

3. 26 (a) Yield surface of Zircaloy, (b) Equivalent stress vs. equivalent plastic strain curve describing the strain hardening behavior of the material ........................................ 76

3. 27 Comparisons of the predicted load vs. displacement and/or torque vs. twist angle responses using the calibrated I1-J2-J3 plasticity model with experimental records for (a) the tensile specimen, (b) the compression specimen with L/D = 1.5, (c) axial force vs. axial displacement response for the torsion-compression specimen, and (d) torque vs. twist angle response for the torsion-compression specimen (Experimental data were from[112]). ........................................................................................................................ 77

3. 28 Comparisons of the experimental data with the numerical results computed using the calibrated I1-J2-J3 plasticity model for Zircaloy: (a) load vs. displacement responses of the compression specimen with L/D = 0.75, (b) torque vs. twist angle response of the pure torsion specimen (Experimental data were from[112]). ........................................... 78

4. 1 A cubic element. ........................................................................................................ 83

4. 2 Comparison of the 02022 /./ Duvs response and 02 /. Duvsf response predicted using a1 = a2 = b1 = b2 = 0 (dotted lines) and a1 = a2 = 610-4 and b1 = b2 = 0 (solid lines)............................................................................................................................................ 85

4. 3 Comparison of the 02022 /./ Duvs response and 02 /. Duvsf response predicted using a1 = a2 = b1 = b2 = 0 (dotted lines) and a1 = a2 = 0 and b1 = b2 = -60.75 (solid lines)............................................................................................................................................ 85

xiv

4. 4 Comparison of the 02022 /./ Duvs response and 02 /. Duvsf response predicted using a1 = a2 = 610-4 and b1 = b2 = 0 (associated flow rule; dotted lines) and a1 = 610-4 and a2 = b1 = b2 = 0 (non-associated flow rule; solid lines). ............................................. 86

4. 5 Comparison of the 02022 /./ Duvs response and 02 /. Duvsf response predicted using b1 = b2 = -60.75 (dotted lines) and b1 =-60.75 and b2 = 0 (solid lines), where a1 and a2 are taken as zero............................................................................................................ 87

4. 6 Comparison of the 22 0 2 0/ . /vs u D response and 2 0. /f vs u D response when 1 = 0.268 and 2 = 0.634 (dotted lines) with those when 1 = 0.4 and 2 = 0.4 (solid lines), where a1 = a2 = 610-4 and b1 = b2 = -60.75. .................................................................... 88

5. 1 Sketches of a smooth round bar, a notched round bar, a grooved plane strain specimen and a torsion specimen...................................................................................... 94

5. 2 Dimensions of the specimens (unit: mm): (a) notched round bar, (b) plane strain specimen and (c) torsion specimen. .................................................................................. 95

5. 3 Typical finite element meshes: (a) an axi-symmetric model for a notched round bar, (b) a 1/8-symmetric model for a grooved plane strain specimen. ..................................... 97

5. 4 (a) Measured, uniaxial, engineering stress-strain curve, (b) measured shear stress vs. shear strain curve. ............................................................................................................. 98

5. 5 Comparison of the true stress vs. true plastic strain curves (power-law) obtained using the smooth tensile bar data and the torsion test data. ............................................ 100

5. 6 Comparison of the numerical and experimental load vs. displacement curves for (a) the smooth tensile specimen and (b) the torsion specimen (Experimental da ta were from [118])............................................................................................................................... 101

5. 7 (a) Variation of the critical failure strain with stress triaxiality for =1; (b) (d) Comparison of the predicted and measured load-displacement responses for the notched round tensile specimens having three different notch radii respectively (Experimental data were from [118]). .................................................................................................... 103

5. 8 (a) Undeformed mesh for the E-notch specimen, (b) deformed mesh just before failure occurs, and (c) comparison of the predicted and measured load vs. diametral contraction response (Experimental data were from[118]). ............................................ 105

5. 9 (a) Variation of the critical failure strain with stress triaxiality for =0; (b) (d) Comparison of the predicted and measured load-displacement responses for three grooved plane strain specimens respectively (Experimental data were from[118]). ...... 106

5. 10 Variation of the ductile failure strain with the stress triaxiality for the aluminum 5083 alloy........................................................................................................................ 107

1

CHAPTER I

INTRODUCTION

In recent years, rapid developments in computational mechanics have enabled

engineers to be capable of analyzing complex structural components, assessing structural

reliability and optimizing structural designs. Consequently, the need for more accurate

material models become increasingly evident, particularly when minimizing design

margins becomes the approach for weight optimization or life-extension efforts. The

predictions of a numerical model can vary widely depending on the material models

employed and the numerical prediction is useless unless a proper constitutive model to

accurately describe the material behavior is provided. In this chapter, the development of

the plasticity models and ductile fracture models are briefly introduced, and the research

purpose of this thesis is illustrated. The structure of the thesis is outlined at the end of this

chapter.

1.1 Development of plasticity models

The scientific study of plasticity may justly be regarded as beginning in 1864

when Tresca published his results on punching and extrusion experiments and formulated

his famous yield criterion [1] . This yield criterion was then used by Saint-Venant [2] and

Levy [3] in their development of the theory for rigid-perfectly plastic solid. Another well

2

known yield criterion was proposed by von Mises [4] on the basis of purely mathematical

considerations. Later von Mises criterion was interpreted by Hencky [5] as plastic

yielding occurs when the elastic shear-strain energy reaches a critical value. Von Mises

also independently proposed the equations similar to Levys yield criterion for the rigid-

perfectly plastic materials. Other important contributions in the early development of the

plasticity theory include the works by Prandtl [6], Reuss [7], among others. Subsequently,

within the scope of elastic-plastic materials under small deformation, the notation of yield

in the stress space formulation was generalized to cover work-hardening materials and a

unified theory of plasticity began to emerge after World War II [8, 9].

To date, an overwhelming majority of the structural analyses employ the classical

J2 plasticity theory to describe the plastic response of metallic alloys. Although the J2

plasticity theory has shown great success in various applications, it has been found that

the classical J2 plasticity theory does not lead to satisfactory predictions for some

materials. It is the geomechanics community that has long recognized the so-called

pressure sensitive and Lode dependent yielding of many geomaterials and

incorporated the hydrostatic stress (pressure) and/or the Lode angle (related to the third

invariant of the stress deviator) into the yield functions of various plasticity models [10-

16] .

Experiments also showed that many other materials such as certain polymers,

ceramics, metallic glasses and metallic alloys exhibit pressure sensitive yielding and

plastic dilatancy [17-24]. With these experimental findings, a large amount of studies

have been concentrating on building the plasticity models with the hydrostatic pressure

3

and/or the Lode parameter effects [10, 25-32] in order to provide a better description of

the plastic responses of these materials. Recently, Chaboche [33] provided an extensive

literature review of the plasticity and viscoplasticity constitutive theories for metal alloys.

The literature review shown in the chapter II also provides the detail formulations of

some well known plasticity models that include the hydrostatic stress and/or Lode

parameter effects.

1.2 Research objectives on plasticity modeling

The main research objective on plasticity modeling area in this thesis is to build a

series of general plasticity models for isotropic materials, which are functions of the

second, third invariants of the stress deviator and the hydrostatic stress.

Then the finite element implementation of these proposed plasticity models

including integration of the constitutive equations using the backward Euler method and

the formulation of the consistent tangent moduli is presented. The derivation is based on

small-strain formulation. For finite strain plasticity, kinematic transformations are

performed first so that the constitutive equations governing finite deformation are

formulated using strains-stresses and their rates defined on an unrotated frame of

reference. Once the kinematic transformations eliminated the rotation effects on the rate

of the tensorial quantities, the stress updating procedure and the consistent tangent

stiffness formulation remain the same as those for small-strain formulation. Most

commercial finite element programs adopt this kind of treatment for finite strain plasticity

thus only small-strain formulation needs to be considered in development of a user

material subroutine.

4

As applications, the proposed plasticity models are calibrated and verified for a

5083 aluminum alloy, a Nitronic 40 and a Zircaloy 4. The detailed numerical and

experimental results are compared, and good agreement is achieved.

Furthermore, the Gurson-Tvergaard-Needleman porous plasticity model is

extended to include the effects of the hydrostatic pressure and Lode parameter in the

matrix material, and a few numerical examples are presented. The modified porous

plasticity model is expected to improve the accuracy in predicting ductile fracture process

of certain materials.

1.3 Stress state effects on ductile fracture

There is overwhelming evidence showing that ductile fracture has strong stress

state dependence. Research on the effect of the stress state on ductile fracture probably

can be traced back to the early work of Ludwik and Scheu [34], in which the authors

hypothesized that fracture of ductile metals was governed by a strength-strain curve.

They also recognized that the strength-strain curve could be obtained through testing

tensile specimens with circumferential notches of various depths and sharpness. The

exploratory work of Orowan [35] on notch brittleness was a pivotal point in the

development of the physics of ductile fracture, and since then the effect of plastic

constraint on ductile fracture has drawn much attention of the fracture mechanics

community.

The stress triaxiality parameter, defined as the ratio of the hydrostatic stress to the

equivalent stress, is often used to characterize the plastic constraint. High triaxial tension

5

subjected by the core of a plastically deformed circumferential notch explains why failure

starts at the center of the neck. The experimental work by Bridgman [36] showed the

strain to failure in a tension test could be greatly increased if the test was carried out

under pressure to reduce the stress triaxiality in the neck. Similarly, the influence of

superimposed hydrostatic pressure on the fracture mechanisms of copper, aluminum and

brass were studied by French and Weinrich [37-40]. In French and Weinrichs study, the

magnitude of the hydrostatic pressure increased the strength of the materials and changed

the fracture strain. Using axis-symmetric notched tensile specimens and flat-notched

plane strain tensile specimens, Hancock and Mackenzie [41] and Hancock and Brown

[42] demonstrated that the strain to initiate ductile fracture was a decaying function of the

stress triaxiality. A widely used ductile fracture criterion was provided by Johnson and

Cook [43] , in which a damage parameter was defined as a weighted integral with respect

to the effective strain and the integrand is the reciprocal of the effective failure strain as a

function of the stress triaxiality, strain rate and temperature. The experimental and

numerical studies of Mirza et al. [44] on the pure iron, mild steel and aluminum alloy

BS1474 over a wide range of strain rates and Bao and Wierzbicki [45] and Bao [46] on

aluminum alloy 2024 under quasi-static loading reaffirmed the strong dependence of the

equivalent strain to the crack formation on the level of stress triaxiality.

The common attribute of the above mentioned papers are that the materials

fracture is related to the stress triaxiality. What is missing is that the third invariant of

stress deviator, which is related to the Lode parameter, is not considered in the fracture

criterion. Studies by Kim et al. [47-49] and Gao et al.[50, 51] found that the Lode

parameter should be considered in order to distinguish the stress state with the same

6

triaxiality ratio, since the macroscopic stress strain response and the void growth and

coalescence behavior of a representative material volume significantly vary for each

stress state even though the triaxiality stays the same. Similarly, Wierzbiski and Xue [52]

proposed a macroscopic ductile failure criterion as a function of both the first and third

stress invariants, and the 3-D fracture locus based on this criterion was calibrated for an

aluminum alloy 2024-T351. Barsoum and Faleskog [53, 54] and Bai and Wierzbicki [55]

demonstrated that the stress triaxiality alone cannot sufficiently characterize the effect of

the stress state on the ductile fracture and the effect of the Lode angle needs to be taken

into account. More recently, Brunig et al. [56] proposed a stress state dependent damage

criterion for ductile materials based on a thermodynamically consistent continuum

damage model.

1.4 Research objectives on the ductile fracture study

In this thesis, the recent research efforts on modeling plasticity and ductile

fracture of an aluminum 5083 alloy are presented. The investigation reveals the strong

stress-state effects on the plastic response and the ductile fracture behavior. These stress

state effects can be described by using the stress triaxiality and the Lode parameter

(which is related to the third invariant of the stress deviator). In particular, it is found that

the stress triaxiality has relatively small effect on plasticity but significant effect on the

ductile failure strain. On the other hand, the effect of the Lode angle on ductile fracture is

negligible but its effect on plasticity is significant.

A porous micromechanics model, Gologanu-Leblond-Devaux (GLD) plasticity

model (Gologanu et al.,[57-59] ; Pardoen and Hutchinson [60]), and the computational

7

cell approach [61, 62] are used to perform the detailed finite element analyses of the

smooth and notched round tensile bars, grooved plane strain specimens and the

Lindholm-type torsion specimen [63]. Very good comparisons between the model

predictions and the experimental measurements are observed. The new findings of this

research challenge the classical J2 plasticity theory and provide a blueprint for

establishment of the stress-state dependent plasticity and ductile fracture models for

aluminum structural reliability assessments.

1.5 Outline of the Thesis

There are five chapters, two appendices and a bibliography of cited references

presented in the thesis.

Chapter 1 gives the history of plasticity theory and ductile fracture development,

provides the research objectivities of this thesis and presents the outline of thesis

structure.

Chapter 2 provides the detailed literature review about the stress state dependent

plasticity models, micromechanics models and the non associate flow rule.

Chapter 3 describes a series of new plasticity models, which is dependent on the

second and third invariants of the stress deviator as well as the hydrostatic stress.

Numerical implementations of the new plasticity models are also presented. As

applications, these plasticity models are calibrated and validated for three materials, a

5083 Aluminum, a Nitronic 40 and a Zircaloy 4.

8

Chapter 4 presents the extended Gurson-Tvergaard-Needleman porous plasticity

model to include the effects of the hydrostatic pressure and Lode parameter and gives a

few numerical examples.

Chapter 5 provides the research efforts on modeling ductile fracture of an

aluminum 5083 alloy by employing the GLD porous model.

Chapter 6 concludes the present thesis and gives the suggestions on the future

possible research work.

9

CHAPTER II

LITERATURE REVIEW

Great amount of theoretical work has been done on the plasticity and ductile

fracture areas. A literature review is provided in this chapter on development of the

plasticity theory with the hydrostatic pressure and Lode parameter effects. It is started by

a short introduction of some common used concepts and definitions in plasticity theories,

and proceeds with a detailed review about plasticity models with I1 and J3 effects,

micromechanics models and non-associated flow rule.

2.1 Basic Concepts and Definitions

Let ij be the Cauchy stress tensor with 1, 2 and 3 being the maximum,

intermediate and minimum principal stresses respectively. The hydrostatic stress (or

mean stress) can be expressed as

)(31

31

31

3211 iih I

(2. 1)

where I1 represents the first invariant of the stress tensor and the summation convention is

adopted for repeated indices. Let ij be the stress deviator with 1 , 2 and 3 being its

principal values, i.e.

10

ijhijij (2. 2)

where ij represents the Kronecker delta. The first, second and third invariants of the

stress deviator are given as

3213

213

232

221

1332212

3211

31)det(

)()()(61

)(21

0)(31

kijkijij

jiij

J

J

J

(2. 3)

The famous von Mises equivalent stress is expressed as

2 2 2

1 2 2 3 3 1 21 32e

J (2. 4)

A particular stress state can be represented by a point in a Cartesian coordinate

system with axes 1, 2 and 3, which is the so-called Haigh-Westergaard stress space as

shown in Figure 2.1(a). The diagonal of the stress space, denoted by a unit vector

3/)1,1,1(n , is called the hydrostatic axis. For any stress point P, a plane containing P

and perpendicular to the hydrostatic axis is called the deviatoric plane, which contains

line PN in Figure 2.1(a). The deviatoric plane passing through the origin is called the

plane. When it is viewed in the direction of hydrostatic axis, the projections of the 1, 2

and 3 axes on the deviatoric plane are shown in Figure 2.1(b). It is often more

convenient to use the cylindrical Haigh-Westergaard coordinates given below to describe

the stress state

11

2/32

3

2

1

227

3cos

23

3

JJ

PNJ

ONI

h

(2. 5)

where is the Lode angle. The Haigh-Westergaard coordinates, and therefore any

general stress state, are completely determined by the three stress invariants, I1, J2 and J3.

(a) (b)

Figure 2. 1 (a) The Haigh-Westergaard stress space, and (b) the deviatoric plane

The stress triaxiality ratio is defined as the ratio of the hydrostatic stress (or mean

stress) over the effective stress,

)3/(2)3/(/ 1 IT h (2. 6)

where is the effective stress. It is very clear that, for a given stress triaxiality ratio T,

there exists an infinite number of stress states and each of them corresponds to a point on

the surface of a cone with ON as the axis. To distinguish various stress states having the

1

3 2

P

N

ni N

P(1,2,3)

1

2

3 o

12

same triaxiality ratio, Lode angle is used as the second nondimensional parameter to

define a stress state.

There are many different definitions of the Lode angle (or Lode parameter) in the

literature. According to the work of Lode [64], the Lode angle , as the angle shown in

Figure 2.1(b), can be expressed as

12123

32tan

(2. 7)

Xue [65] defined the relative ratio of the principal deviatoric stresses as

31

32

(2. 8)

and an alternative Lode angle was expressed in the terms of ,

(2. 9)

Also this alternative Lode angle is related to the original Lode angle by

(2. 10)

Barsoum and Faleskog [53] defined another Lode parameter,, as

2 1 31 3

2

(2. 11)

This parameter can be easily connected to the original Lode angle by

13

3

tan (2. 12)

Bai and Wierzbicki [55] introduced a normalized third deviatoric stress invariant,

B , which is related to Lode angle through

3327cos 32B

J

(2. 13)

All the parameters presented so far are based on the condition 1 2 3 . The Lode

parameter used in this thesis is defined as

3327cos 3

2 2J

(2. 14)

2.2 Review of Plasticity Models

The most widely known theory of plasticity is the flow theory, which consists of a

yield criterion, a flow rule, a hardening law and the loading-unloading conditions. The

yield criterion determines the stress state when yielding occurs; the flow rule describes

the increment of plastic strain after yielding; the hardening law characterizes the

evolution of the flow stress with increased plastic deformation; and the loading-unloading

conditions determine if the stress path moves outward, inward or along the current yield

surface.

The most popular continuum plasticity model is the so-called J2-flow theory. This

theory assumes hydrostatic stress as well as the third invariant of the stress deviator has

no effect on plastic yielding and the flow stress and it is widely employed to describe the

14

plastic response of metals. However, as shown in chapter I, numerous experimental

works have shown that this assumption is invalid for many types of materials. For this

reason, many criteria, such as Mohr-Coulomb [66, 67] and Drucker-Prager [10, 25]

among others, had been proposed to include the hydrostatic stress and third invariant of

the stress deviator effects.

The Mohr-Coulomb theory was named in honor of Charles-Augustin de Coulomb

[66] and Christian Otto Mohr [67]. The Mohr-Coulomb yield (failure) criterion is

expressed as

tan( ) c (2. 15)

where is the shear strength, is the normal stress, c is the intercept of yield (failure)

envelope with the axis, and is the slope of the yield (failure) envelope with value

range from 0o to 90o . The Mohr-Colulomb criterion reduces to the Tresca criterion [1]

when =0o. On the other hand, if =90o, the MohrCoulomb model is equivalent to the

Rankine model [68]. This criterion is an extended form of the maximum shear stress

criterion, and it has made great success on modeling the failure in elastic range or small

strain plastic range. Besides, there were also several successful applications of this model

to predict ductile fracture. The distinctive characteristic of the Mohr-Coulomb criterion is

the explicit dependence on the Lode parameter, which is not considered in most ductile

fracture models. Bai and Wierzbicki [32] demonstrated the applicability of a extended

Mohr-Coulomb criterion to model ductile fracture of crack-free bodies.

15

The Drucker-Prager yield criterion [10] , which was introduced to model the

plasticity deformation of soils, is a pressure sensitive model to determine whether a

material has undergone plastic yielding. This yield criterion is expressed as

(2. 16)

where the I1 is the first invariant of the Cauchy stress tensor, J2 is the second invariant of

the deviatoric Cauchy stress tensor and A, B are the material constants determined by

experiments. If the yield stress, t, at uniaxial tension and the yield stress, c, at uniaxial

compression are known, the A and B can be expressed as

(2. 17)

With I1 term being considered, this yield criterion can describe the strength-differential

effect in tension and torsion. However, this form shows some insufficiency. For example,

if t approaches c, the value of B will be close to zero. At this situation, the hydrostatic

pressure effect cannot be described by this yield criterion anymore.

Different to the previous Drucker-Prager yield criterion [10], Drucker [25]

proposed a yield function that depends on the second and third invariants of the stress

deviator, by which the yield surface lies between the von Mises yield surface and the

Tresca yield surface. This Drucker criterion has the form

22332 kJJ (2. 18)

where is a constant, which lies between -27/8 and 9/4 to make sure the yield surface

satisfy the convexity condition, and k is a material constant.

16

Inspired by the extensive experimental results reported by Spitzig et al. [69, 70]

on the behavior of high-strength metals undergoing uniaxial tension and compression,

Brunig [26] presented a formulation of a generalized I1-J2 yield criterion to describe the

effect of the hydrostatic stress on the plastic flow properties. This I1-J2 yield condition is

written in the form as

0)1( 12 IcJ (2. 19)

where is a material constant, and the coefficients c are strain-dependent. It should be

noticed that this I1-J2 yield criterion is an extended form of Drucker-Prager yield model.

But in Drucker-Prager yield model, c is a material constant and the work-hardening

effects are not considered. Later, this I1-J2 plasticity model was applied by Brunig et al.

[56] to study the effect of stress triaxiality on the onset and evolution of damage in

ductile metals.

In Brunig et al. [27], the authors described an I1-J2-J3 flow theory in their

numerical simulation of the deformation and localization behavior of hydrostatic-stress-

sensitive metals. The new I1-J2- J3 yield condition is shown as

0)1( 3 312 JIcJ (2. 20)

where and are material constants, and the coefficients c are strain-dependent.

Hu and Wang [29] proposed a stress-state dependent yield criterion for isotropic

ductile materials. This yield criterion can be indicated by the three stress invariants as

17

CJ

JBJAI 3

2

321 (2. 21)

where A, B and C are three material constants, which can be determined through different

experiments.

In order to describe the constitutive response of a material that has pressure

sensitivity, Subramanya et al [71] employed an extended Drucker-Prager yield model to

study the roles of pressure sensitivity, plastic dilatancy and yield locus shape on the

interaction between the notch and a nearby void. The Extended Drucker-Prager model is

given as

31 1 tan1 (1 )( ) tan (1 ) 0

2 3h cq r

C C q

(2. 22)

where h is the hydrostatic stress, q is the von Mises effective stress and 3

327 / 2r J .

Also, c is the true yield stress in a uniaxial compression test (with initial value of 0 )

and and C are materials parameters. This yield surface in principal stress space is a

conical surface with a vertex in hydrostatic tension axis. The minimum value of C is

0.778 to make sure the convexity condition of the yield surface. When C=1, this yield

criterion degenerated to original Drucker-Prager yield criterion, and when C=1 and =0,

is reduced to the famous von Mises yield criterion.

There are also other contributions with regard to the yielding criteria including the

I1 and J3 effects. Kuroda [28] presented a phenomenological plasticity model accounting

for hydrostatic stress-sensitivity and vertex-type effect. Cazacu and Barlat [30] and

18

Soare et al. [31] extended Druckers J2- J3 yield function [25] to include plastic

anisotropy and applied it to simulate sheet forming. A most recent study by Bai and

Wierzbicki [55] discussed a pressure and Lode dependent metal plasticity model and its

application in failure analysis. Yang et al. [72] conducted five types of tensile tests on a

2A12-T4 aluminum alloy and modified the von Mises yield criterion to include the Lode

dependence.

2.3 Micro Mechanics Models

The microstructure of materials is very complex and the fracture of ductile

materials usually follows a multistep failure process: void nucleation, growth and

coalescence. Void nucleation usually happens at a quite low stress level, so that material

can be assumed to have voids at the start point of loading. Two common methods are

used to describe this fracture mechanism. One is the straight- forward approach by

modeling individual voids explicitly using finite element mesh. Void growth performance

can be accurately simulated by this method. However, it would require a large number of

elements to model the voids in a structural component. A huge number of finite element

elements are too much for the computational capability at present. So another approach

called porous continuum method was developed to study the ductile fracture for a

practical alternative. Recently, Benzerga and Leblond [73] gave a very detailed literature

review about the ductile fracture by void growth to coalescence.

The innovative works of studying micromechanics were due to McClintock [74]

and Rice and Tracy [75] by considering the growth of isolated voids. McClintock

investigated an isolated cylindrical-shaped void that was subjected to uniform remote

19

stressing. Rice and Tracy considered the growth of a spherical void in non-hardening

material subjected to remote uniaxial tension strain rate field. Not only has the void

grown in the radial direction, but also the shape of the void has changed. Later on, based

on homogenization assumption, constitutive equations for porous materials were

developed. The most well-know porous constitutive model is proposed by Gurson [76], in

which the yield criteria were approximated through an upper bound theorem of plasticity.

The matrix materials were considered as rigid-perfectly plastic and the plastic behavior in

the matrix obeyed the Von Mises yield criterion. Associated flow rule is employed with

yield function served as flow potential. Equations of the Gurson model are formulated in

terms of the average macroscopic Cauchy stress ij , with corresponding stress deviator

/ 3ij ij ij kkS . The void shape was considered as a sphere. The yield criterion of

Gurson model was expressed as

0)1(23cosh2 2h2

2

ffe

(2. 23)

where e is the von Mises effective stress ( 3 / 2e ij ijS S ), is the yield stress of the

matrix material, f is the void volume fraction and h is the hydrostatic stress. The yield

surface given by equation (2.23) turns into von Mises yield criterion at f=0. Associative

flow rule is employed and the macroscopic plastic strain is

ij

pij

(2. 24)

20

where pij are the rates of the plastic strain components,

is a positive scalar called the

plastic multiplier and ij are the macroscopic stress components. The matrix plastic strain

is related to the macroscopic strain by enforcing the equivalence of plastic work, ie.

pijp

ij f )1(: (2. 25)

In this model, the void volume fraction, f, is considered as an extra internal variable to

capture the growth of cavities.

For Gursons yield function, the complete loss of load carrying capacity occurs

when void volume fraction ratio equals one, which is greater than the experimental

observation. Tvergaard [77] introduced two more parameters, q1 and q2, to improve this

situation based on their bifurcation study results. Moreover, Tvergaard and Needleman

[78] modified the Gurson model by introducing a void volume fraction function, f*, to

model the rapid loss of load carrying capacity after the coalescence occurs. The Gurson-

Tvergaard-Needleman (GTN) model was expressed as

0)(12

3cosh2 2*1h2*

12

2

fqqfqe

(2. 26)

where e is the von Mises effective stress, is the yield stress of the matrix material, f*

is the void volume fraction function, h is the hydrostatic stress and q1 and q2 are model

parameters which have an effect on the shape of the yield surface. When q1 = q2=1, this

model becomes the original Gurson model.

To account for the final material failure, the function f*(f) was specified by

21

* 1

,1/ ( ),

c

cc c c f

f c

f f fq ff f f f f f f

f f

(2. 27)

where fc and ff are the void volume fractions at coalescence and failure respectively. It is

noted that the macroscopic stress carrying capacity vanishes when the void volume

fraction function f* achieves the ultimate value * 11/f q . According to the experiment

results from Brown and Embury [79] and the numerical model analysis from Andersson

[80], fc =0.15 and ff =0.25 are appropriate values.

The development of void volume fraction is partially caused by growth of existing

voids and partially caused by nucleation of new voids. So the evolution law for void

volume fraction is determined by:

nucleationgrowth fff )()( (2. 28)

where growthf )( and nucleationf )( represent the increasing rate of void volume fraction due

to growth and nucleation respectively.

The matrix is assumed to be incompressible, but the macroscopic response of the

material is compressible because of the voids. So growthf )( is caused by the total volume

change

pkkgrowth ff )1()( (2. 29)

For ductile material, void nucleation is caused by the cracking and decohesion of

inclusions or second-phase particles. Needleman and Rice [81] suggested that the

22

localization occurs at early stage of deformation. Due to complicated physical structure

of the inclusion and the microstructure, the void nucleation particles are not homogenous

in the solids. Various nucleation criteria [81-85] had been proposed. Chu and Needleman

[86] suggested a plastic strain controlled nucleation in which

, Afnucleation (2. 30)

where is the matrix plastic strain rate, and A as

2

21exp

2 NN

N

N

ssfA

(2. 31)

where sN and N are the standard deviation and the mean value of the distribution of the

plastic strain, fN is the total void volume fraction of void nucleating particles.

Despite of the great success of GTN model, a distinct limitation of the GTN

model is the assumption that voids are spherical in materials and remain spherical in the

growth process. Actually many materials, such as rolled plates, have non-spherical void

shape. Even for materials having initially spherical voids, the void shape may change to

probate or oblate shape after deformation depending on the applied stress state. In order

to overcome this disadvantage, Goludanu, Leblond and Devaux [57-59] proposed the so-

called GLD model, in which both void volume fraction and void shape evolve with

deformation. Since non-spherical voids were considered in this model, a preferred

material orientation existed and the plastic behavior became anisotropic. The Gologanu-

Leblong-Devaux (GLD) models yield function is expressed as

23

2 2' 22 2 1 cosh 1

0

hh

C q g g f g q g f

X (2. 32)

where ij are the macroscopic stress components, f is the void volume fraction, is the

yield stress of the matrix material, denotes the von Mises norm, represents the

deviatoric stress tensor, h~ is the generalized hydrostatic stress defined by

yyzzxxh 22 1~

(2. 33)

and X is defined as

zzxxyy eeeeee 3/13/13/2X (2. 34)

where (ex, ey, ez) is an orthogonal basis with ey parallel to the axisymmetric axis of the

void and denotes tensor product. S is the void shape parameter. The parameters C, , g,

and 2 are functions of f and S, and the heuristic parameter q depends on initial void

volume fraction, strain hardening exponent of the matrix material, void shape parameter S

and the macroscopic stress triaxiality factor T. Detailed description of the GLD model

can be found in chapter V. The GLD model can turn into Gurson model by setting the

void shape as sphere. It can be also reduced to the von Mises yield criterion when f=0 at

prolate void condition. For oblate voids, f=0 corresponds to a material with a distribution

of penny-shaped cracks.

Kim and Gao [87] developed a generalized approach to formulate the consistent

tangent stiffness for complicated plasticity models. Using the approach developed by

24

Kim and Gao, the GLD model was implemented into ABAQUS via a user subroutine to

study stress state effects on ductile fracture of aluminum 5083 in chapter v.

2.4 Non-associated Flow Rule

A common approach in the metal plasticity theory is the adoption of the

associated flow rule or normality condition, which can be interpreted as that the normal

to the yield surface is the same as the direction of the plastic strain increment vector.

Normality condition requires the yield function and the flow potential to be identical.

The normality condition has been confirmed by a lot of experiments, based on the

observations that some metals do not have pressure sensitivity in yielding and volume

change after large deformation is negligible. However it is found to be seriously in

inaccurate for geological materials, where the associated flow rule overestimates the

plastic volume changes. For these materials, the non-associated flow rule must be

employed [88-90].

In metal plasticity, non-associated flow rule were developed after the

experimental observations by Spitzig and Richmond [19] for iron-based materials and

aluminum, in which they found that while yielding is pressure dependent, plastic

deformation is essentially incompressible. Therefore, their results suggests that normality

concept, which overestimated the plastic volume increase for their materials, may not

always be appropriate in metal plasticity. Even for the materials without the pressure-

sensitive effects, Lademo et al. [91] provided an evaluation of three proposed material

models, found that an associated flow rule was inadequate to describe the anisotropic

25

flow properties observed in uniaxial tension tests and suggested the non-associated flow

rule.

Many studies, such as those by Mroz [92], Nemat-Nasser and Shokooh [93],

Doris and Nemat-Nasser [94], Nemat-Nasser [95, 96], Runesson [97] , Brunig et al. [27] ,

Stoughton [98] and Stoughton and Yoon [99, 100] , also indicated that appropriate

constitutive description of many materials can be achieved by using the less restrictive

non-associated flow rule. Brunig [26] and Brunig et al. [27] demonstrated that pressure

sensitive yielding and non-associated flow rule remarkably influence the onset of

localization and the subsequent localization behavior. Stoughton [98] proposed a material

model based on non-associated flow rule for sheet metals to describe the directional

dependence of uniaxial tension data. In order to account for the strength differential

effect, Stoughton and Yoon [99] proposed a non-associated flow rule based on a pressure

sensitive yield criterion. To model the ductile fracture process in solids, Ma and

Kishimoto [101] proposed a non-associated flow rule to characterize the yield and plastic

deformation of void-containing materials, where the yield function took the form of the

Gurson-Tvergaard-Needleman porous plasticity model [76-78, 102] while the flow

potential took a slightly different form. More recently, Cvitanic et al. [103] presented a

plasticity model based on non-associated flow rule and detailed finite element

formulations for sheet metal forming. Taherizadeh el al. [104] presented an anisotropic

material model based on non-associated flow rule and mixed isotropic-kinematic

hardening for simulation of sheet metal forming.

26

CHAPTER III

MODELING OF PLASTICITY RESPONSES: INVOLING THREE STRESS

INVARIANTS

3.1 Introduction

Plasticity describes the deformation of a material undergoing non-reversible

changes of shape in response to applied forces. Study of materials plasticity is an

indispensable part of the structural deformation research. In recent days, structural

analysis and integrity/risk assessments of high performance engineering components

often demand constitutive models that can accurately describe a material's plasticity

behavior. A complete plasticity model consists of a yield criterion, a flow rule and a

hardening law. The most popular continuum plasticity model is the so-called J2-flow

theory. In this theory, the second invariant of the deviatoric stress tensor controls

yielding and plastic flow. The hydrostatic stress (first invariant of the stress tensor) as

well as the third invariant of the stress deviator (Lode parameter) is assumed to have no

effect. The J2 flow theory has been widely employed to describe the plastic response of

various materials. For a material that obeys J2-flow plasticity, its plasticity behavior is

characterized by the (von Mises) equivalent stress-strain curve which can be obtained by

conducting a uniaxial tension test, a compression test, or a torsion test. The stress-strain

curve obtained from either one of these tests is then used to predict the materials p lastic

response under various states of stress.

27

However, increasing experimental evidence shows that a fundamental assumption

made in von Mises theory, that plasticity is only characterized by J2, contains varying

levels of inaccuracies over a wide range of materials. Hydrostatic pressure and Lode

parameter also play very important roles in the plasticity behavior for some certain

materials. As we discussed in chapter I and chapter II, there have been a lot of

contributions made to include the hydrostatic pressure and Lode parameter effects into

the plasticity models.

In this chapter, a series of new plasticity models is introduced to include the

hydrostatic pressure and Lode parameter (related to the third invariant of stress deviator)

effects as well as the second invariant of stress deviator. Based on the procedure

proposed by Avaras [105] and Kim [87], the finite element implementation including

integration of the constitutive equations using the backward Euler method and the

formulation of the consistent tangent modules is presented. These plasticity models are

implemented in to the finite element software ABAQUS via UMAT.

These new plasticity models are calibrated and verified for an Aluminum 5083

alloy, a Zircaloy-4 and a Nitronic 40 to simulate the plasticity behaviors under different

stress-state tests. In order to provide large range of triaxiality and Lode parameter values,

different types of testing specimens were employed. A torsion-tension specimen is

designed to provide stress states with low triaxiality values. Test matrix includes round

bar tensile specimens, notched round bar tensile specimens, compression specimens,

grooved plains train specimens, modified Lindholm torsion specimens, modified

28

Lindholm torsion-tension and torsion-compression specimens. The numerical plasticity

predictions using this new proposed plasticity models agree with experiments very well.

3.2 Plasticity Modeling

In this section, detail formulation of the new I1-J2-J3 plasticity model is presented.

The influence of I1 and J3 are also shown, and the appropriate yield function and flow

potential forms are listed for the three materials studied in this thesis. The yield function

and flow potential forms for these three materials are determined by comparing numerical

simulation results by using different yield function and flow potential forms with the

experimental data (three sets of experimental data: round bar tensile test, compression

tests and torsion test) to see which set of the yield function and flow potential forms leads

to best match.

3.2.1 Influence of I1 and J3

Let ij be the Cauchy stress tensor and 1, 2 and 3 be the three principal stress

values. The hydrostatic stress (or mean stress) can be expressed as

)(31

31

31

3211 iih I (3. 1)

where I1 represents the first invariant of the stress tensor and the summation convention is

adopted for repeated indices. Let ij be the stress deviator tensor and 1 , 2 and 3 be

its principal values, i.e.

29

ijhijij (3. 2)

where ij represents the Kronecker delta. It is obvious that the first invariant of the stress

deviator tensor is zero. The second and third invariants of the stress deviator tensor are

defined in Eq. (3.3) as

1

2 1 2 2 3 3 1

2 2 21 2 2 3 3 1

3 1 2 3

01 ( )2

1 ( ) ( ) ( )61det( )3

ij ji

ij ij jk ki

J

J

J

(3. 3)

For isotropic materials, the plastic behavior is often described by the stress

invariants I1, J2 and J3 and consequently, the general forms of the yield function (F) and

the flow potential (G) are expressed as functions of I1, J2 and J3. Eq. (3.4) describes the

yield condition

0),,( 321 JJIF (3. 4)

where is the hardening parameter.

When material deforms plastically, the flow rule is used to define the inelastic

part of the deformation [106]

ij

pij

JJIG

),,( 321 (3. 5)

30

where pij are the rates of the plastic strain components,

is the plastic multiplier which

is a non-negative scalar, and G is a function called the flow potential [107]. Eq. (3.5)

suggests that the directions and magnitudes of the rates o f the plastic strain components

are determined by the plastic flow potential and the plastic multiplier respectively. In

most cases of metal plasticity, the flow potential and the yield function are assumed to be

identical, i.e., F = G, which is known as the associated flow rule or normality rule. When

F and G are not identical for a material, this material is said to follow a non-associated

flow rule.

Various forms of F and G functions can result in different plasticity models.

While there is not a single functional form F and G must satisfy, here they are taken as

first order homogeneous functions of stress. The choice of F as a first order homogeneous

function of stress leads to a straightforward definition of the equivalent stress, Fe .

Appendix A gives six examples of first order homogeneous functions of stress

that are considered in this work to represent the yield function, F, and the flow potential,

G. These functions are generalized from existing plasticity models, e.g. the Drucker

model [25] and the Drucker-Prager model [10]. These functions depend on material

constants a1, b1, a2, and b2. The a1 and b1 values control the hydrostatic pressure and

Lode parameter effects on yielding respectively. The ranges of the parameters involved in

these functions are also given in Appendix A to ensure the convexity requirement is

satisfied.

31

For all six functional forms listed in Appendix A, if a1 = a2 and b1 = b2 the

material follows the associated flow rule. If a1 = b1 = a2 = b2 = 0, the plasticity model

reduces to the formulation of the classical J2-flow theory and e defined by Fe

becomes the von Mises equivalent stress.

The hardening parameter depends on the strain history. By enforcing the

equivalence of plastic work, i.e.,

pijijp (3. 6)

the equivalent plastic strain increment can be defined as

/pijijp (3. 7)

For strain hardening materials, the hardening behavior can be described by a

stress vs. plastic strain relation )( p , where tpp d . If a material follows the J2

plasticity theory, the stress strain curve obtained from a uniaxial tensile test defines

)( p . For materials with I1-dependent plasticity, the input curve must be modified as

discussed later in this chapter. Since the flow potential is taken to be a first order

homogeneous function of stress, Eulers homogeneous function theorem results in

GG

ijij

pijij

(3. 8)

From (3. 6) and (3. 8), the plastic multiplier and the equivalent plastic strain rate

can be related through

32

GF

Gpp

(3. 9)

If the material follows the associate flow rule (F = G), it is obvious that and p

are equal. For materials in which the non-associated flow rule applies, the equivalent

plastic strain rate defined by Eq. (3. 9) differs from by a factor GF / .

Because of the I1 term in the flow potential G, the plastic response becomes

dilatant, with the rate of volume change given by

6512

62

5512

62 3)/3( G

FIacGIacG pkk

pkk

(3. 10)

For materials considered in this study, the 5083 aluminum alloy, the Nitronic 40

stainless steel and the Zircaloy 4, the following forms of F and G functions are found

appropriate by comparing model output with test data and manually manipulating the a1,

a2, b1, and b2 parameters to produce what is believed to be the best match based on visual

inspection of the experimental data.

3.2.2 Yield Function and Flow Potential for 5083 Aluminum Alloy and Zircaloy-4

For aluminum 5083 and Zircaloy-4, the following first order homogeneous

function of stress is found proper for defining the yield function

])27([ 6/123132111 JbJIacF (3. 11)

33

where a1, b1 are material constants and c1 is determined by substituting the uniaxial

condition into Eq. (3.11) so that the equivalent stress defined by Fe equals to the

applied stress. This results in

6/1111 )1729/4(/1 bac (3. 12)

The flow potential takes a similar form, i.e.,

])27([ 6/123232122 JbJIacG (3. 13)

where ))1729/4(/(1 6/1222 bac .

If a1 = a2 =0, the plasticity model reduces to the form of the Drucker model [25]

and if b1 = b2 =0, it degenerates to the form of the Drucker-Prager model [10].

3.2.3 Yield Function and Flow Potential for Nitronic 40 Stainless Steel

For Nitronic 40, the following first order homogeneous function of stress is

proposed for the yield function

])33([ 3/1312/3

2111 JbJIacF (3. 14)

where a1, b1 and c1 are material constants. Again, the constant c1 can be found by

substituting the uniaxial condition into Eq. (3.14), which leads to

3/1111 )127/2(/1 bac (3. 15)

When the material is subjected to a uniaxial stress , the value of c1 given by Eq. (3.15)

ensures F .

34

The flow potential takes a similar form, i.e.,

])33([ 3/1322/3

2122 JbJIacG (3. 16)

where 3/1222 )127/2(/1 bac .

Again, If a1 = a2 =0, the plasticity model reduces to the form of the Drucker

model [25] and if b1 = b2 =0, it degenerates to the form of the Drucker-Prager model [10].

3.3 Numerical Implementation of the I1-J2-J3 plasticity model

Following the procedures of Aravas [105] and Kim and Gao [87] , we implement

the I1-J2-J3 plasticity model outlined in the previous section into ABAQUS [108] via a

user defined subroutine. Since ABAQUS/Standard is employed to carry all the numerical

analysis, stress update and consistent tangent moduli are needed to be provided in this

user subroutine (UMAT).

3.3.1 Stress Update

We adopt the backward Euler method to integrate the rate constitutive equations.

For the strain driven integration algorithm, the stresses and state variables are known at

the start of each increment and their values need to be updated at the end of the increment

corresponding to the total strain increment. The elasticity equations give

pkleijklTijpklkltekleijklttekleijklttij CCC (3. 17)

where

35

kltekleijklTij C (3. 18)

is the elastic predictor, t represents the time at the start of the increment, t+t represents

the time at the end of the increment, and the superscripts e and p denote elastic and

plastic components respectively. The total strain increment kl is known, and if the

linear elastic behavior is isotropic, the elastic moduli eijklC can be expressed as

klijjkiljlikeijkl GKGC

32 (3. 19)

where G and K are the elastic shear and bulk moduli respectively. Therefore, to update

stresses, the plastic strain increments need to be determined. The following outlines the

procedure for computing pij .

The yield condition and the flow rule are written as

0)(),,( 321 p

tttttttt JJIF (3. 20)

and

tt

ij

pij

JJIG

),,( 321 (3. 21)

Equations (3.20) and (3.21), when considered together with Eq. (3.17), result in

10 equations for and nine components of pij , among which 7 equations are

independent due to the symmetry of pij . If the state variable p is updated and thus

36

)( p tt is known, these equations can be solved iteratively for p

ij and using the

Newton-Raphson method. The iterative process follows these steps: 1) assume initial

0 pij and use Eq. (3.17) to estimatett

ij ; 2) use Eqs. (3.20) and (3.21) to solve for

pij ; 3) update stresses using Eq. (3.17); 4) repeat steps 2) 3) until convergence

conditions are satisfied.

To update p , consider the hardening law and the evolution equation for p

)()( pptp

tt (3. 22)

and

pijtt

ijp (3. 23)

At each iteration of pij , p and )(p

tt can be obtained by solving equations (3.22)

and (3.23) iteratively using the Newton-Raphson method.

3.3.2 The Consistent Tangent Moduli

Simo and Taylor [109] showed that use of the consistent tangent moduli

significantly improves the convergence characteristics of the overall equilibrium

iterations. The consistent tangent stiffness corresponding to the backward Euler

integration can be obtained by linearization of equation (3.17).

tt

kl

ijijklJ

(3. 24)

37

Since all quantities in calculating Jijkl are referred to time t+t, the superscript t+t will

be dropped from hereafter.

Equation (3.17) can be rewritten as

pkltpklkleijklpklkleijklekleijklij CCC (3. 25)

Differentiating (3.25) results in

)( pkleijklkl

eijklij CC (3. 26)

The relationship between ij and )(pkl can be found as follows.

Let h be the hardening modulus, i.e., ph / , then Eq. (3.23) can be

rewritten as

h

pijij

(3. 27)

By differentiating (3.27) and simplifying the resulted relation, the following

equation can be obtained

pijpmnmn

ijijp

mnmn

pij

hh

hh

22 (3. 28)

Differentiating Eq. (3.20) and combining the result with Eq. (3.28) give

ijpmnmn

pij

ij

pijp

mnmn

ij

hhJJIF

hh

)

),,(()( 2

3212 (3. 29)

38

Eliminating from the nine equations given by (3.21) results in eight equations,

such as

0),,(),,(

0),,(),,(

2221

32121

22

321

2211

32111

22

321

pp

pp

JJIG

JJIG

JJIG

JJIG

(3. 30)

Differentiating (3.30) leads to

ijij

p

ij

ppp

ijij

p

ij

ppp

GGGG

GGGG

22

2

2121

2

222221

2122

22

2

1111

2

222211

1122

(3. 31)

Eqs. (3.29) and (3.31) provide nine equations between )( pij and ij , which

can be summarized as

DK p (3. 32)

where T33312111T

33312111 ,,,,,, ,,,Pppp p , K is the coefficient

matrix of p and D is the coefficient matrix of .

From Eqs. (3.24) and (3.30), we can obtain

CDCDK eep 1 (3. 33)

39

where eC is a 99 matrix representing the elasticity tensor eijklC . Finally, the consistent

tangent matrix can be obtained by Substituting (3.33) into (3.26)

eeee CDCDKCCJ 1/ (3. 34)

3.4 Applications of the stress-state dependent plasticity models

In this section, the proposed I1-J2-J3 plasticity models are employed to study the

plastic response of Aluminum 5083, Nitronic 40 and Zircaloy-4. Designing of the

torsion-tension specimen is also shown here. These proposed I1-J2-J3 plasticity models

are calibrated and validated for these three materials and good comparison between the

numerical predictions and experimental data have been achieved.

3.4.1 Designing of torsion-tension specimen

In order to generate a wide range of stress states, special designed test specimens

are needed to be considered in our experimental and numerical analysis. In the study of

Gao et. al [110], round bar tensile specimen, notched round bar specimen, torsion

specimen and grooved plane strain specimens are employed to investigate the influence

of the hydrostatic stress and Lode angle on ductile failure of DH36 steel. With definitions

of triaxiality as )33/( 2/121 JIT and Lode parameter as )2/(332/3

23 JJ , round bar

tensile specimen has T value initially as 1/3 and increase subsequently after the onse t of

necking. Notched round bar specimens provide different triaxiality values according to

the notch radius. For both round bar tensile specimen and notched round bar tensile

specimens, the value is 1. Triaxiality T of torsion specimen is 0, and grooved plane

40

strain specimens have different triaxiality values according to different grooved radius.

For both torsion specimen and grooved plane strain specimens, Lode parameter is 0.

Aluminum 5083 is one of the materials studied in this thesis. In order to give a

clear view about how the stress states change with the loading history for different type

of specimens, detailed plots of triaxiality and Lode parameter for each type of specimens

are shown and analyzed here by taking aluminum 5083 as example. The geometries of

each specimen can be found in the later section of this chapter.

The stress state of a material point in the test specimens evolves as plastic

deformation increases. For aluminum 5083, Figures 3.1 3.3 show the variation of

triaxiality T and Lode parameter with loading history (measured by the equivalent

plastic strain, p ) in the element at the specimen center for the smooth round bar, the E-

notch round bar (notch radius 6.35 mm ) and the G-groove plane strain specimen (groove

radius 5.08 mm). For the smooth round bar, remains at 1 during the entire loading

history while T increases from 1/3 to about 0.45 before specimen fractures. For the E-

notch specimen, remains at about 1 and T increases from 0.71 to 0.8. For the G-groove

specimen, quickly decreases to 0 and remains at this level as plastic deformation

increases while T increases from 0.51 to 0.8 before failure occurs. For the notched and

grooved specimens, changing notch (groove) radius changes the level of T in the

specimen. Three different notched round bar specimens and three grooved plane strain

specimens are considered. The notch radii/groove radii for these specimens are shown

later in this chapter (Table 3-3 and table 3-4). Considering the entire loading history of

each specimen and the three different notch (groove) radii, the range of T experienced by

41

0

0.2

0.4

0.6

0 0.1 0.2 0.3

the center element is 0.71 T 1.6 for the notched round bar tests and 0.46 T 0.97

for the plane strain tests.

Figure 3. 1 Variation of )33/( 2/121 JIT and )2/(332/3

23 JJ with plastic deformation in the center element of the smooth round bar.

(a) (b) Figure 3. 2 Variation of )33/( 2/121 JIT and )2/(33

2/323 JJ with plastic

deformation in the center element of the E-notch specimen.

0

0.5

1

1.5

0 0.1 0.2 0.3

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

T

p

p

(a) (b)

T

p p

p

42

Figure 3. 3 Variation of )33/( 2/121 JIT and )2/(332/3

23 JJ with plastic deformation in the center element of the G-groove specimen.

As we can see from the all the figures above, the round tensile bars (smooth and

notched) and the grooved plane strain specimens cannot provide data points in the low

stress triaxiality range (T < 0.4). So a specimen is required to provide a low triaxiality

value and also with Lode parameter between 0 and 1. The low range of triaxiality can be

achieved using a simple notched tubular specimen tested in combined tension and torsion

by Faleskog et.al. [53, 54]. Here an alternative specimen design is developed, which can

be used for conducting combined tension-torsion tests to provide useful data points at this

range. This specimen is modified from the specimen that Lindholm [63] used for high-

rate torsion tests. The difference between this modified Lindholm specimen and the

original Lindholm specimen is that this modified Lindholm specimen is longer and the

ends are cylindrical rather than hexagonal.

Next, the investigation of the distributions of stress, triaxiality and Lode

parameter in the gage section of the modified Lindholm specimen is conducted. In order

to provide an effective comparison with established approaches, numerical analysis of the

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

-0.6

-0.3

0

0.3

0.6

0 0.05 0.1 0.15 0.2

T

p p

(a) (b)

43

modified Lindholm specimen is compared with the notched tube specimen used by

Barsoum and Faleskog [53, 54]. The dimensions of Faleskogs specimen (referred as NT

specimen) are shown in the figure 3.4. The dimensions of the modified Lindholm

specimen are shown in the figure 3.5. Even though the overall sizes of these two

specimens are close to the same, the gage sections are significantly different. The NT

specimen has both internal and external circular notches, while the modified Lindholm

specimen is notched only on the outside with a trapezoidal notch profile. The gage

section in the modified Lindholm is uniform over 2.54 mm (0.1 in), while the gage

section in the NT specimen is curved throughout. The gage section width (wall

thickness) for the NT specimen is 1.2 mm at the narrowest point, which is about twice the

thickness of the wall throughout the gage section of the Lindholm specimen (0.74 mm).

The thin wall in the gage section acts to minimize the variation in stress through the wall

thickness. The very tight tolerance on concentricity shown in figure 3.5 is intended to

ensure axisymmetric deformation in the gage section.

Figure 3. 4 Dimensions of NT specimen. All dimensions in mm [53,54].

44

Figure 3. 5 Dimensions of Lindholm specimen. All dimensions in mm [53,54].

Figure 3. 6 Comparison of gage sections in NT and Lindholm specimens (to scale).

A series of finite element stress analyses were conducted to compare the

distributions of stress and effective plastic strain in the gage section of the NT specimen

and the modified Lindholm specimen under different combinations of tension and

LINDHOLM NT

45

torsion. The stress state for both specimens will be characterized using the triaxiality

parameter, T, and the Lode parameter, [53, 54]. The Lode parameter can be gained by

re-arranged following the equation:

1 32

1 3

2

2

(3. 35)