Microstructure evolution in crystal plasticity : strain ...

Microstructure evolution in crystal plasticity : strain path effectsand dislocation slip patterningCitation for published version (APA):Yalcinkaya, T. (2011). Microstructure evolution in crystal plasticity : strain path effects and dislocation slippatterning. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR716655

DOI:10.6100/IR716655

Document status and date:Published: 01/01/2011

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 20. Apr. 2022

https://doi.org/10.6100/IR716655

https://doi.org/10.6100/IR716655

https://research.tue.nl/en/publications/cb384a47-cd34-46a3-976f-e0000d238ae8

Microstructure evolution in crystal plasticity:strain path effects and dislocation slip

patterning

This research was carried out under the project number MC2.03158 in the framework of

the Research Program of the Materials innovation institute M2i (www.m2i.nl), the former

Netherlands Institute for Metals Research.

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Tuncay Yalcınkaya

Microstructure evolution in crystal plasticity: strain path effects and dislocationslip patterning /by T. Yalcınkaya – Eindhoven : Technische Universiteit Eindhoven, 2011.Proefschrift.A catalogue record is available from the Eindhoven University of TechnologyLibraryISBN: 978-90-386-2729-8Subject headings: BCC metals / crystal plasticity / non-Schmid effects /plastic anisotropy / strain path change effect / Bauschinger effect /cross effect / microstructure evolution / non-convexity /phase field modeling / dislocation patterning / finite element method /non-convex free energy / strain gradient crystal plasticityCopyright c©2011 by Tuncay Yalcınkaya, all rights reserved.

This thesis was prepared with the LATEX 2ε documentation system.Reproduction: Universiteitsdrukkerij TU Eindhoven, Eindhoven, TheNetherlands.

Microstructure evolution in crystal plasticity:strain path effects and dislocation slip

patterning

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Eindhoven,

op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn,

voor een commissie aangewezen door het College voor Promoties

in het openbaar te verdedigen

op donderdag 20 oktober 2011 om 16.00 uur

door

Tuncay Yalcınkaya

geboren te Ankara, Turkije

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. M.G.D. Geers

Copromotor:

dr.ir. W.A.M. Brekelmans

Contents

Summary ix

1 Introduction 1

1.1 Crystal plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 A finite strain BCC single crystal plasticity model and its experimental

identification 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Slip mechanisms in BCC metals . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Violation of Schmid’s law in BCC metals . . . . . . . . . . . . . . . . . . 10

2.4 A BCC crystal plasticity model at material point level . . . . . . . . . . 12

2.4.1 Kinematics in crystal plasticity . . . . . . . . . . . . . . . . . . . 12

2.4.2 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Modeling some intrinsic properties of BCC single crystals . . . . . . . . 17

2.5.1 Orientation dependence . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 Example: α-Fe single crystal . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Example: molybdenum single crystal . . . . . . . . . . . . . . . 19

2.5.4 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 A composite dislocation cell model to describe strain path change effects in

BCC metals 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Dislocation substructure evolution . . . . . . . . . . . . . . . . . . . . . 28

3.3 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Modeling of microstructure evolution . . . . . . . . . . . . . . . . . . . 34

3.4.1 Monotonic deformation . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.2 Orthogonal change of deformation . . . . . . . . . . . . . . . . . 36

3.4.3 Reverse deformation . . . . . . . . . . . . . . . . . . . . . . . . . 37

v

vi Contents

3.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.1 Example 1: monotonic deformation of single crystals . . . . . . 38

3.5.2 Example 2: strain path change of single crystals . . . . . . . . . 39

3.5.3 Example 3: strain path change of polycrystals . . . . . . . . . . 40

3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Deformation patterning driven by rate dependent non-convex strain gradi-

ent plasticity 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Macroscopic view: material instability and microstructure evolution

in inelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Thermodynamics of strain gradient plasticity . . . . . . . . . . . . . . . 50

4.4 Particular choices of free energy functions . . . . . . . . . . . . . . . . . 54

4.4.1 Slip based strain gradient plasticity . . . . . . . . . . . . . . . . 54

4.4.2 Slip based non-convex strain gradient plasticity . . . . . . . . . 55

4.5 Non-convexity and patterning in phase field modeling . . . . . . . . . 57

4.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6.1 Numerical example 1: convex case - monotonic loading . . . . . 59

4.6.2 Numerical example 2: non-convex case - monotonic loading . . 60

4.6.3 Numerical example 3: non-convex stress relaxation of a 1D bar 66

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8.1 Finite element implementation of slip based strain gradient

plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8.2 Finite element implementation of slip based non-convex strain

gradient plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Non-convex rate dependent strain gradient crystal plasticity and deforma-

tion patterning 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Strain gradient crystal plasticity and finite element implementation . . 75

5.3 Latent hardening based non-convex plastic potential . . . . . . . . . . . 80

5.3.1 Conditions for plastic slip patterning . . . . . . . . . . . . . . . 80

5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Convex strain gradient crystal plasticity . . . . . . . . . . . . . . 84

5.4.2 Non-convex strain gradient crystal plasticity . . . . . . . . . . . 88

5.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Discussion and conclusions 95

Bibliography 99

Contents vii

Dankwoord / Acknowledgements 109

Curriculum Vitae 111

Summary

During deformation polycrystalline metals tend to develop heterogeneous plastic

deformation fields at the microscopic scale, as the amount of plastic strain varies

spatially, depending on local grain orientation, geometry and defects. While grain

boundaries are natural places triggering plastic slip accumulation and geometrically

necessary dislocations that accommodate the gradients of the inhomogeneous plastic

strain, the deformation localizes within grains revealing dislocation cell structures or

micro slip bands (e.g. clear band formation in irradiated materials). Across grains,

macroscopic plastic slip bands (Luders bands, etc.) exist as well. These intergran-

ular and intragranular deformation patterns are stated to be inherent minimizers

of the free energy (including the microstructurally trapped plastic energy). These

microstructures may macroscopically manifest themselves through softening of the

material or through plastic anisotropy in hardening under strain path changes. These

effects are crucial with respect to the mechanics of the materials under consideration

and should be taken into account in the constitutive modeling.

In this thesis, the computational modeling of microstructure evolution (with soften-

ing or plastic anisotropy) is covered in different crystal plasticity frameworks. The

scope is basically two-fold. First, in the chapters 2 and 3 the plastic anisotropy of

Body Centered Cubic crystals is studied from the onset of deformation due to an

intrinsic orientation dependence from non-planar dislocation core structures, to the

anisotropy upon a strain path change owing to resulting dislocation cell formation.

In this part of the thesis, after developing a proper BCC crystal plasticity framework

taking into account the intrinsic anisotropy, a composite cell model was established

where the evolution of dislocation cells was modeled under monotonic and non-

proportional loading histories. Here, the existence of a dislocation microstructure

is introduced into the model in terms of internal variables and the evolution was

described by phenomenologically based evolution equations. However, this phe-

nomenological approach is not able to incorporate the formation stage of the mi-

crostructure. Hence, the crystal plasticity framework called for an extension in order

to capture the evolution of the microstructure driven by the free energy of the mate-

ix

x Summary

rial.

In order to complete the missing link between the formation of the microstruc-

ture and its evolution in crystal plasticity frameworks, the second part of the the-

sis concentrates on the development of a non-convex rate dependent crystal plas-

ticity model, which reveals a rate dependent dislocation microstructure formation

and evolution together with macroscopic hardening-softening-plateau stress-strain

responses. To this end, non-convexity is treated as an intrinsic property of the plastic

free energy of the material. First, this non-convex contribution is incorporated into a

strain gradient crystal plasticity framework with a double-well character, which re-

sults in a computational routine partially dual to the Ginzburg-Landau type of phase

field modeling approaches (with high and low slipped regions representing the dif-

ferent phases). In this model, both the displacement and the plastic slip fields are

considered as primary variables. These fields are determined on a global level by

solving simultaneously the linear momentum balance and the slip evolution equa-

tion which is rederived in a thermodynamically consistent manner. In chapter 4, the

analysis is conducted in a 1D mathematical setting in order to illustrate the ability of

the model to capture the patterning of plastic slip. In chapter 5 and inspired by the

literature, the non-convexity originates from latent hardening in a multi-slip strain

gradient crystal plasticity framework. Hence, the 1D approach pursued in chapter 4

is extended to a 2D plane strain setting. Even though the phenomenological double-

well free energy function used in the 1D approach allows to track non-equilibrium

states during microstructure evolution, it does not rely on a physically based ex-

pression for non-convexity, but presents a generic formulation. Instead, chapter 5

concentrates more on the physical reasons of plastic slip localization, where a slip

interaction potential is analyzed and incorporated into the rate dependent strain gra-

dient crystal plasticity framework. The non-convexity due to the slip interactions is

explicitly illustrated and the possibility of deformation patterning in the material is

discussed in a boundary value problem. The last part of the thesis, chapter 6, presents

a discussion and conclusions.

Chapter one

Introduction

Abstract / The physics and the basic principles behind the general crystal plasticity mod-eling are explained. An overview of strain path change related anisotropy and dislocationmicrostructure evolution in crystal plasticity approaches is presented. The objectives andoutline of the thesis are given.

1.1 Crystal plasticity

A perfect metallic single crystal, which is characterized by a specific periodic ar-

rangement of atoms, can respond only in a reversible elastic manner in thermal equi-

librium with its surroundings when stressed monotonically well below the critical

levels that destabilize the crystal structure. Under an applied homogeneous stress,

the elastic response is homogeneous down to the atomic level. In contrast, the plastic

response is locally heterogeneous and requires crystal defects for its development.

The type and intensity of the plastic response depend on the character of the defect

state. For this purpose the crystal defects are introduced in a hierarchy of increasing

dimensionality from point, through line, to planar defects (see Argon (2008) for an

extensive overview). Among these, the line defects, i.e. dislocations, are regarded as

the principal carriers of plastic deformation. The crystallographic slip of dislocations

occurs on the most close-packed slip planes and in the most close-packed directions

which together form the slips systems. Depending on the specific arrangement of

atoms, each metal has specific slip systems. The first part of this thesis is focusing on

the body centered cubic (BCC) type of atomic arrangement.

Under an applied stress, the atomic lattice deforms elastically until the stretched

bonds near a dislocation break down and new bonds are formed. During this prop-

1

2 1 Introduction

GRAINS

DISLOCATION CELLS

CONSTITUTIVE MODELING OF DISLOCATION MOVEMENT

DISLOCATION CELL BLOCKS

Figure 1.1 / Crystal plasticity bridges the deformation of bulk material and themovement of dislocations (chapter 2). Clustering of dislocations is accomplishedvia non-convex strain gradient crystal plasticity (chapter 4 and 5). The phenomeno-logical evolution of dislocation structures is simulated via a dislocation cell model(chapter 3).

agating process, a part of the crystal gradually slips one interatomic distance with

respect to the other part. Instead of the ideal fictitious strength associated with the

movement of an entire slip plane, the dislocations enable only sections of the slip

plane to shear, resulting in the observed decimated strengths necessary for plastic

deformation.

The stress directly affecting the motion of dislocations is the projected shear stress

on the specific slip systems, which is also called the resolved Schmid stress. When

the resolved shear stress is larger than the resistance on the respective slip system,

glide is activated. The viscous type of crystal plasticity theories as used in this thesis

employs a power law relation between the rate of plastic slip and the ratio between

the shear stress and the slip resistance. Conceptually, all the slip systems are active

but the most favorable ones carry most plastic deformation. Crystal plasticity is a

mesoscopic modeling approach, bridging the stresses on the slip systems to the total

amount of plastic slip in the bulk material (see Fig. 1.1), eventually affecting the total

amount of macroscopic plastic strain.

In addition to their role of accommodating the plastic deformation in metals, the

work (strain) hardening behavior can also be attributed to the dislocations. The

distinct stages of strain hardening are related to their multiplication or mutual in-

1.2 Objective and outline 3

teraction processes. After moderate deformations, the formation of dislocation cell

structures (see Fig. 1.1) plays an important role in the hardening behavior of the

material.

Another important effect in the hardening of crystals is the gradient of the plastic

slip, requiring so-called geometrically necessary dislocations (GNDs). Once the ap-

plied load, or the material structure itself, triggers a gradient of the plastic deforma-

tion, a certain amount of GNDs will be necessary to preserve lattice compatibility

and to accomplish the required lattice rotation. Conventional crystal plasticity the-

ories, as used in the chapters two and three of this thesis, however, do not take into

account these effects. Their strengthening mechanisms are therefore inherently in-

capable of predicting scale dependent behavior, i.e. different mechanical responses

due to varying plastic strain gradients. Hence, in the strain gradient crystal plastic-

ity frameworks of the subsequent chapters, the gradients of the plastic slip enter the

plastic slip law together with a length scale parameter, where these do not only allow

for size effect predictions but also play a regularization role in the viscous formula-

tion of non-convex strain gradient plasticity.

1.2 Objective and outline

During metal forming processes, materials experience complex strain path histories

which result in plastic anisotropy, i.e. transient hardening or softening regimes oc-

curring in the macroscopic stress-strain response. This phenomenon plays an impor-

tant role in the metal deformation and the effect should be included in the constitu-

tive modeling. The physical origin resides in three distinct factors at three different

length scales: dislocation slip anisotropy, evolution of the dislocation microstructure

and textural anisotropy. The thesis focuses on the first two effects which are crucial

in early stages of the deformation and subsequent moderate straining.

In chapter two, a crystal plasticity model for body centered cubic (BCC) single crys-

tals, taking into account the plastic anisotropy due to non-planar spreading of screw

dislocation cores is developed. A comprehensive summary of the intrinsic properties

of these materials is presented and incorporated into the framework through a mod-

ification in the plastic slip law. In the numerical examples section, emphasis is given

on the intrinsic orientation dependence of the flow stress due to the non-Schmid com-

ponents of the stress field projected on the slip plane under monotonic deformation.

Attention is therefore given on the quantitative prediction of single crystal behavior,

for which experiments from the literature have been used.

Next, in chapter three, the anisotropy due to the dislocation cell structure evolution

is considered. A composite dislocation cell model has been combined with the BCC

4 1 Introduction

crystal plasticity framework to describe the dislocation cell structure evolution and

its macroscopic anisotropic effects. The computational framework departs from a

composite aggregate with a cell structure, consisting of a soft cell interior component

and hard cell wall components. The constitutive response of each component has

been obtained from crystal plasticity simulations, while a set of phenomenological

evolution equations for the cell size, the wall thickness and the dislocation density

captures the evolution of the microstructure under complex strain paths. Numerical

examples study both the intrinsic orientation dependence and the anisotropy due to

cell structure evolution.

Chapter four focuses on one of the origins of self-organizing dislocation structures

(driven by the deformation) rather than imposing the cell structure evolution as done

in the previous chapter. To this purpose, a rate dependent strain gradient plasticity

framework for the description of plastic slip patterning in a system with non-convex

energetic hardening is developed. Both the displacement field and the plastic slip

field are considered as primary variables. The slip law differs from classical ones in

the sense that it includes a stress term originating from a non-convex double-well free

energy, which enables patterning of the deformation field. The derivations and im-

plementations are performed in a single slip 1D setting, which allows for a thorough

mechanistic understanding, not excluding its extension to multidimensional cases.

The numerical examples illustrate both the homogeneous and inhomogeneous plas-

tic slip distributions as well as the stress-strain response in relation to the imposed

boundary conditions and the applied rate of deformation.

In chapter five the non-convex strain gradient crystal plasticity formulation is ex-

tended to the 2D plane strain case, including multiple slip systems. In order to

capture the effect of dislocation interactions on the non-convexity of the plastic slip

dependent free energy function, a more physically based free energy expression is

incorporated. Attention is focused on the inhomogeneous plastic slip distribution

and deformation patterning due to dislocation slip interactions.

The thesis concludes with a final chapter, summarizing the main achievements and

results obtained as well as an outlook to open challenges.

Chapter two

A finite strain BCC single crystalplasticity model and its experimental

identification1

Abstract / A crystal plasticity model for body-centered-cubic (BCC) single crystals, tak-ing into account the plastic anisotropy due to non-planar spreading of screw dislocationcores is presented. In view of the long-standing contradictory statements on the deforma-tion of BCC single crystals and their macroscopic slip planes, recent insights and devel-opments are reported and included in this model. The flow stress of BCC single crystalsshows a pronounced dependence on the crystal orientation and the temperature, mostlydue to non-planar spreading of a/2〈111〉 type screw dislocation cores. The main conse-quence here is the well-known violation of Schmid’s law in these materials, resulting inan intrinsic anisotropic effect which is not observed in e.g. FCC materials. Experimen-tal confrontations at the level of a single crystal are generally missing in the literature.To remedy this, uniaxial tension simulations are done at material point level for α-Fe,Mo and Nb single crystals and compared with reported experiments. Material param-eters, including non-Schmid parameters, are calibrated from experimental results usinga proper identification method. The model is validated for different crystal orientationsand temperatures, which was not attempted before in the open literature.

2.1 Introduction

In the present paper, attention is focused on the low temperature (room tempera-

ture and lower) properties of single BCC crystals includingα-Fe, metals of the group

VA (V, Nb, Ta) and of the group VIA (Mo, V, Cr) and certain alkali metals. These

materials show a peculiar mechanical behavior, mostly resulting from their screw

1This chapter is reproduced from Yalcinkaya et al. (2008)

5


identification

dislocation core configuration. They have a relatively high yield stress which is

strongly temperature, rate and orientation dependent. They exhibit complex slip

modes, dominated by the cross slip of a/2〈111〉 screw dislocations. Due to their dis-

location core structure, they show a severe glide direction sensitive behavior (slip

asymmetry) and the well-known Schmid law using the critical resolved shear stress

(CRSS) is violated. Another pronounced phenomenon is the anomalous slip (acti-

vation of an unexpected slip system at a certain orientation) observed in pure BCC

metals which, however, is only marginally dealt with in the present work.

Various discussions on the behavior of BCC crystals, reveal a number of contradic-

tions with respect to the slip plane activity. Even though there is no generally ac-

cepted explanation to the dislocation behavior of these materials recent studies pro-

vide a good basis for the constitutive model presented in the current paper. Seeger

(2001) states that the slip nature of BCC crystals depends highly on the tempera-

ture, upon which dislocations may accommodate either a straight 110 slip or a

wavy type 112 cross slip pattern. At room temperature a/2〈111〉 type of screw

dislocations move on 112 slip planes, which enables cross slip. The cross slip phe-

nomenon will not be modeled explicitly but the resulting effects are taken into ac-

count in the slip and hardening laws (for example (Pichl (2002)), showing how cross

slip can be included in a model).

The value of the critical resolved shear stress (CRSS) is independent from the slip sys-

tem and the sense of the slip of FCC metals. For these metals, it is generally accepted

that the only stress component affecting the glide is the Schmid stress. However,

BCC metals show an asymmetry in their slip behavior: the slip resistance in one

direction is different from the resistance in the opposite direction, indicated as the

twinning/anti-twinning asymmetry. Moreover, due to small edge fractional dislo-

cation components in the screw dislocation core, stress components other than the

resolved Schmid stress affect the glide or the CRSS of the material. Both of these ef-

fects result from the non-planar spreading of the dislocation cores. For these reasons

Schmid’s law is not applicable to BCC metals. In the presently proposed crystal plas-

ticity model these two types of intrinsic anisotropy effects will be taken into account.

The formulation of the constitutive crystal plasticity model departs from the papers

of Bronkhorst et al. (1992) and Kalidindi et al. (1992) related to FCC metals. Their de-

velopments are extended by including the intrinsic properties of BCC metals,using

a physical description of the slip law based on thermally activated dislocation ki-

netics. The barriers to dislocation movement are discriminated according to their

short-range or long-range nature. The short-range barrier can be overcome by ther-

mal activation, whereas the long-range barrier is affected slightly through changes

of the elastic moduli. Non-Schmid effects are included in the model by incorporat-

ing non-Schmid terms in the slip activation. The model is implemented at a material

2.1 Introduction 7

point in matlab and compared with experimental results.

The phenomena incorporated in this work are basically the orientation and the tem-

perature dependence of the flow stress and stress-strain behavior of single BCC crys-

tals including the non-Schmid behavior. Each of these characteristics has been inves-

tigated extensively in the literature, and especially the temperature dependence of

the flow stress was an active research area until the 90s. At present, atomistic com-

puter simulations studying screw dislocation cores are still an active area of research.

Many improvements have been achieved in this area and to our knowledge there is

no recent work combining these physical aspects of BCC structured materials with

crystal plasticity calculations. The objective of the present work is to exemplify this

combination.

The plan of this paper is as follows. Section 2 discusses the slip mechanisms in BCC

metals where the temperature dependence of the slip plane activation is strongly em-

phasized. Next, Schmid’s law and its violation in BCC crystals is handled in section

3, along with its connection to the non-planar spreading of screw dislocation cores.

In section 4, the crystal plasticity constitutive framework and its implementation is

outlined. Section 5 studies pronounced intrinsic properties, whereby examples are

presented and confronted with experimental results. Finally, concluding remarks are

given in section 6.

Cartesian tensors and associated tensor products will be used throughout this paper,

making use of a Cartesian vector basis e1e2e3. Using the Einstein summation rule

for repeated indices, the following conventions are used in the notations of vectors,

tensors, related products and crystallography:

• scalars a

• vectors a = aiei

• second-order tensors A = Ai jei ⊗ e j

• fourth-order tensors 4 A = Ai jklei ⊗ e j ⊗ ek ⊗ el

• C = a ⊗ b = aib jei ⊗ e j

• C = A · B = Ai jB jkei ⊗ ek

• C = 4 A : B = Ai jklBlkei ⊗ e j

• crystallographic direction, family [uvw], 〈uvw〉

• crystallographic plane, family (hkl),hkl

• slip system, family (hkl)[uvw], hkl 〈uvw〉


identification

2.2 Slip mechanisms in BCC metals

First, some of the long-standing contradictions in the identification of the slip planes,

and the active slip mechanisms of BCC crystals are highlighted. The first attempt

goes back to the introduction of the pencil glide mechanism by Taylor and Elam

(1926), where the slip was assumed to be oriented in the 〈111〉 crystallographic di-

rection while the mean plane of slip was the one having the maximal projected shear

stress. This plane might be a crystallographic but also a non-crystallographic plane.

After this pioneering research, there have been several of contradicting statements on

the active slip planes of BCC metals, for which an extended overview can be found

in Havner (1992). The different concepts will not be repeated here in detail, however,

a summary including the current developments will be presented in the following

paragraphs.

Gough (1928) and Barrett et al. (1937) state that the 110, 112 and 123 families

contain the crystallographic slip planes during the deformation of BCC metals, an

assumption that is still being used in many crystal plasticity works. Another fre-

quently used view is the participation of 110 and 112 slip planes only, whereby

it is assumed that 123 planes need a higher temperature for activation. Many re-

searchers state that only the 110 slip planes are active at room temperature, based

on the argument that apparent slip on both 112 and 123 planes is actually com-

posed of slip on two non-parallel 110 planes, e.g. Chen and Maddin (1954). Using

the latter argument, it would be physically more comprehensible to indeed model

only 110 planes in a crystal plasticity framework.

In this paper, attention is focused on an accurate description of the physical slip

mechanisms of BCC crystals, rather than re-advocating a discussion on the active

set of slip planes. As a result of their special slip mechanisms, BCC metals have

interesting intrinsic properties that are not observed in e.g. FCC metals. Many au-

thors (e.g. Vitek et al. (2004b), Vitek (2004), Duesbery and Vitek (1998)) related most

of the phenomena to the core structure of screw dislocations, e.g. by performing

atomistic simulations. Slip system activation in BCC metals is highly dependent on

the crystal orientation and especially on the temperature. Seeger (2001) and Seeger

and Wasserbach (2002) (see Fig. 2.1) provide a detailed explanation of temperature

dependent slip for high purity BCC single crystals with an orientation inducing a

Schmid factor µ = 0.500 for the slip system [111](101) and µ = 0.433 for the sys-

tems [111](112) and [111](211). The work of Seeger and co-workers is adopted here,

relying on the fact that BCC metals show different features and different slip mech-

anisms in different temperature ranges. The physical response below the so-called

knee temperature (which is around 0.2 times the melting temperature of the metal,

TK in Fig. 2.1) and above the knee temperature is thereby distinguished. Below the

knee temperature the slip is governed by the glide of a/2〈111〉 type screw disloca-

2.2 Slip mechanisms in BCC metals 9

0 100 200 300 400 5000

200

400

600

800

1000

Temperature T (K)

Flo

w S

tres

s (M

Pa)

T TK

cross slip of [111] screws; elementary steps on (211) and (112)(101)slip

wavy slip lines

slip linesstraight

T

Figure 2.1 / Flow stress vs. temperature curve for a pure Mo single crystal at aplastic shear strain rate 8.6 × 10−4 s−1, from Seeger (2001).

tions in kink pairs as the mobility of screw dislocations is lower than the mobility

of edge components. In this temperature range, the flow stress (the stress to main-

tain plastic deformation after yield) of the metal is highly temperature and strain

rate dependent. The flow stress decreases with increasing temperature and increases

with increasing strain rate. Above the knee temperature, the flow stress decreases

considerably due to self diffusion and recovery processes and the mobilities of screw

and non-screw dislocations are no longer substantially different. The high tempera-

ture range is out of the scope of the present work. Attention will be focused on the

behavior at lower temperatures, including the behavior at room temperature.

The flow stress dependence on the temperature and the slip mechanism in this tem-

perature range is visualized in Fig. 2.1, where the presented numerical values refer

to molybdenum single crystals. Below the lower limit T (70 K for Mo and 120 K for

α-Fe) the dislocation glide is confined to 110 planes. Screw dislocation glide pro-

duces straight step patterns. In this temperature range, dislocations are stated to be

in their ground state. They show a threefold symmetry and they are able to slip on

any of the three 110 slip planes. As a result of mirror symmetry and the absence of

cross slip, no plastic anisotropy is observed.

Above T, the dislocation core configuration changes and dislocations undergo a tran-

sition from their low temperature configuration (slipping on 110 planes) to their

high temperature configuration (slipping on 112 planes). The dislocations show

a wavy type of structure which results from the cross slip, a characteristic property


identification

associated with 112 slip. In BCC metals three 110 and three 112 slip planes in-

tersect on a common 〈111〉 direction and screw dislocations can therefore distribute

their core on these planes. This spreading is non-planar and nearly all peculiarities

in the mechanical properties of BCC metals can be attributed to this phenomenon.

The cross slip accommodated by 112 slip planes is the main source of the non-

Schmid behavior (orientation dependence of the flow stress). Similar explanations

of the change in the slip mechanism in BCC materials were presented by others. For

example Christian et al. (1990) focused on the mean jump distance of screw disloca-

tions in BCC metals, which decreases considerably above a critical temperature. The

explanation is again a change in the slip mechanism of BCC metals.

The above paragraphs emphasized some recent developments in the understanding

of the slip system activity of BCC crystals and the connection between the core struc-

ture and the intrinsic properties. The development of an adequate model will be

based on these considerations.

2.3 Violation of Schmid’s law in BCC metals

Schmid and co-workers (e.g. Schmid and Boas (1935)) first recognized that the yield

stress of a metal crystal is strongly depending on the crystal orientation with respect

to the load direction. Yield on the slip plane of a crystallographic family occurs at

a constant projected shear stress, which is called the critical resolved shear stress

(CRSS), for a particular material. Constant should be understood here as indepen-

dent from the slip system and the sense of slip. The resulting Schmid law assumes

that the only stress component triggering plastic flow of the material is the projected

shear stress on the slip system, in the direction of glide which is called the Schmid

stress. The other non-glide component, defined as the normal stress, does not have

any effect on the plastic deformation. These assertions are applicable on FCC metals,

however not on BCC metals. This violation becomes manifest through their plastic

anisotropy, revealing two distinct intrinsic non-Schmid effects in BCC metals.

The first intrinsic non-Schmid effect is the variation of the CRSS with the sense of the

shear. By definition shear on a certain plane of the family (e.g. 112) produces shear

in the twinning direction, whereas shear on another plane (e.g. 211) produces

shear in the anti-twinning direction (see Table 2.1 for the respective slip systems).

In BCC metals, twin and slip mechanisms share common slip systems and twinning

is only observed when the temperature is very low and the strain rate is extremely

high. In this context, the main source of the plastic deformation is just the glide of

dislocations, but the resistance to this movement in the twinning and anti-twining

directions is asymmetric. This is called the twinning/anti-twinning asymmetry of

BCC crystals in the literature and the source of this asymmetry is the strong cou-

2.3 Violation of Schmid’s law in BCC metals 11

pling of the screw dislocation to the BCC lattice, which constrains the core and its

properties to adopt the symmetry of the lattice (Duesbery and Vitek (1998)). From

the modeling point of view, this phenomenon is included in the models by taking a

different slip resistance for the twinning and anti-twinning planes.

Table 2.1 / 112 slip systems of BCC crystals, T and A referring to twinning/anti-twinning planes

(1) A (112)[111] (4) A (112)[111] (7) A (121)[111] (10)T (211)[111]

(2) A (112)[111] (5) A (121)[111] (8) A (121)[111] (11)T (211)[111]

(3) A (112)[111] (6) A (121)[111] (9) T (211)[111] (12)T (211)[111]

The second effect, which is sometimes denoted as an extrinsic non-Schmid effect,

e.g. Duesbery and Vitek (1998), is the sensitivity of the slip resistance to the non-

glide components of the applied stress. This effect originates from the non-planar

spreading of a/2〈111〉 type screw dislocation cores. Whereas this effect is often

called extrinsic in the literature in view of its relation to the applied stress, it es-

sentially remains a physically intrinsic non-Schmid effect owing its existence to the

dislocation core structure. The difference between intrinsic and extrinsic is therefore

not made further in this contribution. In BCC metals, the non-glide component of

the applied stress tensor (in a direction perpendicular to the Burgers vector) is cru-

cial. A Burgers vector in a BCC crystal can be decomposed into a screw component

and an edge component. The interaction of the applied stress field with the frac-

tional edge component explains this peculiarity. This component of the stress does

not contribute to the movement of dislocations, and thereby induces many interest-

ing features in BCC crystals. Especially in the last years many atomistic simulations

have been performed (e.g. Duesbery and Vitek (1998), Ito and Vitek (2001), Bas-

sani et al. (2001), Duesbery et al. (2002), Vitek et al. (2004a), Groger and Vitek (2005))

which support this effect. The hydrostatic pressure dependence of the flow stress (see

Spitzig (1979)), the strength differential effect or the tension-compression asymmetry

observed in BCC metals and intermetallic compounds (see Bassani (1994)) and crit-

ical conditions for the forming of shear bands and localization (e.g. Dao and Asaro

(1996)) typically result from this crystallographic non-Schmid effect, and its connec-

tion to non-associated plastic flow is well established by Racherla and Bassani (2007).

In the model presented next, the second intrinsic anisotropy effect will be included by

modifying the crystallographic flow rule which results in an update of the resolved

shear stress.


identification

2.4 A BCC crystal plasticity model at material point level

2.4.1 Kinematics in crystal plasticity

In the classical crystal plasticity theory as developed by Lee (1969), Rice (1971), Hill

and Rice (1972) and Asaro and Rice (1977), the deformation gradient tensor is de-

composed into an elastic part Fe and a plastic part F p according to:

F = Fe · F p (2.1)

The tensor F p defines the stress-free intermediate configuration. In this configura-

tion, resulting from plastic shearing along well-defined slip planes of the crystal lat-

tice, the orientation of the crystal lattice is identical to the orientation in the reference

state (see Fig. 2.2). The tensor Fe reflects the lattice deformation and local rigid body

rotations. The slip systems are labeled by a superscript α, with α = 1, 2..., ns where

nm

αα

n0α

mαp

e p

= F

= F

0 mα0

mα

nα

.

e

e

.

. n0α

mα0

−T

FeF = F F

Fn0

α

Figure 2.2 / Multiplicative decomposition of the deformation gradient.

ns is the total number of slip systems. The vectors mα0 and nα0 denote the slip direc-

tion and the slip plane normal in the reference and intermediate configurations. In

the current state they are represented by mα and nα, respectively.

The crystallographic split of the plastic flow rate is given by

Lp =ns

∑α=1

γαmα0 nα0 (2.2)

with γα the individual slip rates on the slip systems.

2.4 A BCC crystal plasticity model at material point level 13

2.4.2 Constitutive model

The deformation is composed of an elastic contribution and a plastic contribution.

The elastic part is related to the stress, based on a hyper-elastic formulation while

the plastic part is determined by a physically based flow rule.

Elastic contribution

The second Piola-Kirchhoff stress tensor S is expressed in terms of the elastic Green-

Lagrange strain tensor Ee, both relative to the intermediate state,

S = 4C : Ee and Ee =1

2(Ce − I) , Ce = FT

e · Fe (2.3)

with Ce the elastic right Cauchy-Green tensor and I the second order unity tensor.

The second Piola-Kirchhoff stress is the pull-back of the Kirchhoff stress tensor,

S = F−1e · τ · F−T

e (2.4)

where the Kirchhoff stress can be written in terms of the Cauchy stress using the

Jacobian according to

τ = Je ·σ with Je = det(Fe) (2.5)

The fourth order tensor 4C consists of the anisotropic elastic moduli.

The Schmid resolved shear stress is the projection of the Kirchhoff stress on the slip

systems, i.e.

τα = mα · τ · nα = mα0 · Ce · S · nα0 (2.6)

The slip systems of the 112 family in BCC crystals are given in Table 2.1, which is

the relevant set within the considered temperature range.

Plastic slip and hardening

In order to include the previously described crystallographic intrinsic properties, a

physical description of the slip law will be used instead of a classical phenomeno-

logical (power law) relation. Physically based slip laws have been formulated in

crystal plasticity models for materials with a symmetric planar slip dependency (e.g.

Kothari and Anand (1998)). The anisotropy in BCC crystals is included by adapting

the slip law accordingly.

The physical interpretation given hereafter, relies on the thermally activated dislo-

cation kinetics. Plasticity occurs by dislocation motion on certain slip planes in an


identification

energetically favorable direction. The actual flow stress is determined by the resis-

tance to this dislocation motion. The motion is obstructed by short-range and long-

range barriers. The short-range barriers in general are generated by the Peierls stress

(periodic resistance of the lattice) and the local forest of dislocations. The long-range

barriers originate from the elastic stress field due to grain boundaries, far field forests

of dislocations and other defects. The total resistance can be split accordingly

sα = sαt + sαa (2.7)

where the short-range barriers are responsible for the first part sαt , referred to as the

thermal part since thermal activation is sufficient to overcome this resistance. The

athermal part of the resistance sαa is related to the long-range barriers. Although

this contribution slightly decreases at higher temperatures (through a decrease of

the elastic moduli), this effect is negligible compared to the change of the thermal

resistance with varying temperature.

During their motion, dislocations are obstructed by a quasi-periodic short-range re-

sistance. The Helmholtz free energy required to isothermally cross a barrier is de-

noted by ∆F and the mechanical work of sαt can be written as ∆W. The energy differ-

ence between these two,

∆G = ∆F −∆W (2.8)

is the energy barrier to overcome by a dislocation through thermal activation. It is

well-known that the average dislocation velocity, vα on slip system α, can be esti-

mated by,

vα = lαω0 exp−∆G/kT (2.9)

with lα representing the distance between the barriers, ω0 the attempt frequency, k

the Boltzmann constant and T the absolute temperature. The relation between the

slip rate and the average velocity is given by the Orowan relation, γα = bρmvα where

ρm and b represent the mobile dislocation density and Burgers vector, respectively.

Substituting the velocity expression (2.9) into the Orowan relation leads to the slip

law according to,

γα =

0 if ταe f f ≤ 0

γα0 exp−∆GkT

sign(τα) if 0 < ταe f f

(2.10)

where ταe f f = |τα| − sαa is the driving force for the dislocation motion and γα0 =

bρmlαω0 is the reference strain rate, which is different for the different BCC slip plane

families since the distance between the barriers depends on the family. The energy

∆G to be supplied by the thermal fluctuations at constant temperature is calculated

as (see Kocks et al. (1975))

∆G = G0

[

1 −(ταe f f

sαt

)p]q

(2.11)

2.4 A BCC crystal plasticity model at material point level 15

where G0 is the activation free energy needed to overcome the obstacles without

the aid of an applied stress. The quantities p and q lie in the range 0 ≤ p ≤ 1

and 1 ≤ q ≤ 2, and in the numerical examples of this paper they are taken as 1.

The equations (2.10) and (2.11) constitute the slip law for materials that do not show

crystallographic asymmetry effects. The non-Schmid effects are included in equation

(2.10) by pursuing a similar strategy as introduced by Dao and Asaro (1993), where

the Schmid stress as defined by (2.6) is extended to account for the non-Schmid con-

tribution:

ταn = τα + ηα : τ (2.12)

with τ the Kirchhoff stress and ηα representing the tensor governing the non-Schmid

effects for slip system α aligned with mα, nα and zα = mα × nα defined as,

ηα = ηmm(m ⊗ m) + ηnn(n ⊗ n) + ηzz(z ⊗ z)+ηmz(m ⊗ z + z ⊗ m) + ηnz(n ⊗ z + z ⊗ n)

(2.13)

In this framework, ταn enters the equations (2.10) and (2.11) via ταe f f = |ταn | − sαa .

The definition of the non-Schmid stress tensor and incorporation in the slip law is

phenomenological, and the physical meaning is not immediately trivial. The non-

Schmid component is operative in the thermally activated process because of its ef-

fect on the fractional edge component of the screw dislocation cores.

For isothermal cases, the thermal part sαt of the slip resistance sα is taken constant

and the athermal part of the slip resistance is evolving such that

sα = sαa = ∑β

hαβ|γβ| (2.14)

The hardening moduli hαβ determine the rate of strain hardening on slip system α

due to slip on slip system β. This self and latent hardening are phenomenologically

described by (Asaro and Needleman (1985))

hαβ = qαβhβ (2.15)

where qαβ and hβ are further detailed. As explained in the discussions in section

2.2, BCC metals show a 112 dominated slip pattern at room temperature. For the

112〈111〉 slip system there are twelve different slip planes contributing to one slip

direction (see Table 2.1). This leads to the following definition of the q matrix (12 ×12),

qαβ =

1 qn . . . qn

qn 1 . . . qn...

.... . .

...qn qn . . . 1

(2.16)


identification

F

F e F p

S, τα, ταn γ

Fc = Fe · F p

R = F − Fc

Fe := Fe +∆F e

Figure 2.3 / Schematic overview of the BCC crystal plasticity model.

where qn represents the ratio of the latent hardening with respect to the self harden-

ing for non-coplanar slip systems.

Finally the specific form of the self hardening rate, which is motivated by Brown

et al. (1989), reads

hβ = hβ0

∣∣∣∣1 − sβa

sβs

∣∣∣∣

a

sign

(

1 − sβasβs

)

, (2.17)

where hβ0 , sβa and sβs are the initial hardening rate, the actual athermal slip resistance

and the saturation value of the slip resistance, respectively. The exponent a is con-

sidered as a constant material parameter.

Implementation of the constitutive model

The implementation of the above model follows an incremental-iterative solution

procedure. The first step in this iteration is the initial estimate for the elastic part Fe,

resulting in a plastic part F p through (2.1). With the kinematics defined, the stress,

the Schmid and non-Schmid stresses, are calculated. From these, the slip rate on each

slip system is calculated by using the slip law (2.10). As this slip law is non-linear

in terms of the slip rates, a sublevel Newton-Raphson iteration process is adopted

to solve for the slip rates.The plastic part of the deformation gradient is obtained

from the calculated slip rates through a time integration scheme. An updated F p

is determined, and the associated deformation gradient is calculated according to

2.5 Modeling some intrinsic properties of BCC single crystals 17

Fc = Fe · F p. Generally, the calculated Fc and the imposed F will be different, which

results in a residual R. The linearization of the residual by computing the sensitiv-

ity with respect to Fe leads to an update ∆F e of the elastic part of the deformation

gradient. The elastic deformation gradient is updated and the process is repeated

until convergence is achieved. The procedure is performed for all time steps, which

results in a full history of stress and slip evolution. The main steps of the procedure

are summarized in Fig. 2.3.

2.5 Modeling some intrinsic properties of BCC single crystals

In this section, orientation and temperature dependence of BCC materials are simu-

lated applying the presented crystal plasticity framework, and results are compared

with the single crystal experiments. To this purpose, material parameters are de-

termined using a proper identification procedure. This direct confrontation has, to

the best of our knowledge, not been done before for single crystals. Although, poly-

crystal BCC crystal plasticity simulations and validations have been conducted in

the literature (e.g. Kothari and Anand (1998), Liao et al. (1998), Lee et al. (1999), Xie

et al. (2004), Ganapathysubramanian and Zabaras (2005)) and non-Schmid effects

have been incorporated into constitutive models (e.g. Qin and Bassani (1992)), the

results cannot directly be exploited in the context of actual model.

2.5.1 Orientation dependence

The hardening curves, work hardening rate, temperature and rate sensitivity, activity

of slip planes, CRSS and the flow stress of BCC metals all depend on the orientation

of the crystal. As emphasized before, two crucial important physical aspects control-

ling this pronounced orientation dependence are the so-called slip asymmetry (or

twinning/anti-twinning asymmetry) and the non-planar spreading of screw dislo-

cation cores, which have been included in the model.

The twinning/anti-twinning asymmetry manifests itself on the 112 slip planes. On

these slip planes, the slip resistance in the anti-twinning direction is higher than in

the twinning direction, an effect which is difficult to quantify in experimental tests.

Guiu (1969) experimentally observed the asymmetry for Mo single crystals at dif-

ferent temperatures under direct shear. He concluded that the CRSS is roughly 1.5

times larger for the slip systems on the anti-twinning planes of 112〈111〉 compared

to the slip systems on the twining planes.

Due to the non-planar spreading of the screw dislocation cores, the non-glide com-

ponent of the applied stress affects the dislocation core and hence, the sense of the


identification

applied stress affects the yielding of a crystal. The associated non-Schmid parameters

will be identified together with the other parameters in the model.

2.5.2 Example: α-Fe single crystal

In the first example (Fig. 2.4) the orientation dependence ofα-Fe single crystals under

uniaxial tension is examined. Here, and in the following examples the lattice vector

was aligned with the tensile direction in the undeformed configuration and the pre-

sented framework automatically accounts for the lattice rotations. The experimental

curves (Keh (1964)) were reproduced from the reported shear stress-strain curves

which were initially determined from tensile data for the planes of (112), (211) and

(211) with the [001], [011] and [111] orientations respectively in the [111] direction.

The experiments and simulations are performed at a strain rate of 3.3 × 10−4s−1. The

0 1 2 3 4 5 60

20

40

60

80

100

120

140

160

180

Strain %

Str

ess

(MP

a)

[011]

[001]

[111]

Figure 2.4 / Tensile orientation dependence of [001], [011] and [111] oriented α-Fe single crystals. Solid lines are the simulation results and dashed lines representexperiments.

results show the pronounced influence of the orientation on the yielding and hard-

ening behavior of the crystal. The orientations selected in the example constitute the

corners of a unit triangle mapping the [001], [011] and [111] directions. It is a well-

established fact that at the corners and at the edges of the unit triangle multislip is

observed, while the deformation starts with single slip for directions mapped inside

the triangle. For these orientations the initial rate of work hardening is relatively

high and decreases rapidly with ongoing deformation.


Material parameters have been identified using a least-square optimization proce-

dure which minimizes an objective function that equals the sum of squares of the

differences between experimental and simulation results. Most of the parameters are

presented in Table 2.2. The remaining parameters are C11 = 236GPa, C12 = 134GPa,

Table 2.2 / Material parameters forα-Fe single crystals

Initial hardening rate h0 697.88 MPaSaturation value of slip resistance ss 132.10 MPaHardening rate exponent a 1.5Thermal slip resistance st 13.91 MPaAthermal slip resistance (atwin sense) sa0 9.59 MPaAthermal slip resistance (twin sense) sa0 5.75 MPaNon-Schmid parameter ηmm 0.0544Non-Schmid parameter ηnn -0.0293Non-Schmid parameter ηzz -0.0267Reference strain rate γ0 1.07 × 106s−1

Activation free energy G0 2.95 × 10−18J

C44 = 119GPa (Adams et al. (2006)), qn = 1.4, k = 1.3807 × 10−23J/K, while ηmz and

ηnz are taken zero..

2.5.3 Example: molybdenum single crystal

In the second example, the orientation dependence of molybdenum single crystals

under uniaxial tension is analyzed. The experimental curves for the [010], [101] and

[111] orientations are taken from the work of Irwin et al. (1974). They performed two

sets of experiments at 293K and 77K at a strain rate of 6 × 10−5s−1. The results are

compared for the 293K case and presented in Fig. 2.5. The material parameters that

have been identified using the same least-square minimization process are presented

in Table 2.3. The remaining parameters are C11 = 469GPa, C12 = 167.6GPa, C44 =

106.8GPa (Bolef and Klerk (1962)), qn = 1.4, k = 1.3807 × 10−23J/K.

2.5.4 Temperature dependence

BCC metals exhibit a different mechanical response compared to FCC metals, in par-

ticular in the presence of temperature changes. The dependence of the flow stress

on the temperature, has been studied thoroughly in the literature and illustrated for

many BCC single crystals such as Mo (e.g. Hollang et al. (1997)), Nb (e.g. Ackermann


identification

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

35

40

Strain %

Str

ess

(MP

a)

[101]

[111]

[010]

Figure 2.5 / Orientation dependence of [010], [101] and [111] oriented molybde-num single crystals. Solid lines are the simulation results and dashed lines representexperiments.

Table 2.3 / Material parameters for Mo single crystals

Initial hardening rate h0 251.37 MPaSaturation value of slip resistance ss 76.90 MPaHardening rate exponent a 1.05Thermal slip resistance st 11.89 MPaAthermal slip resistance (atwin sense) sa0 7.23 MPaAthermal slip resistance (twin sense) sa0 4.34 MPaNon-Schmid parameter ηmm -0.0528Non-Schmid parameter ηnn 0.0896Non-Schmid parameter ηzz -0.0369Reference strain rate γ0 1.4 × 107s−1

Activation free energy G0 0.1554 × 10−18J

et al. (1983)), Ta (e.g. Werner (1987)), α-Fe (e.g. Brunner and Diehl (1987), Brunner

and Diehl (1997)). Below the so-called knee temperature, where the flow stress is

controlled by the mobility of screw dislocations, the critical shear stress of BCC met-

als rises progressively with decreasing temperature (Seeger (1981)). The flow stress

in FCC metals on the contrary, only shows a moderate increase when the tempera-

ture is lowered below room temperature. Most of the work reported on BCC metals

concentrates on the behavior of pure single crystals in order to eliminate secondary


effects induced by interstitially dissolved foreign atoms. The purification process of

single crystals requires great effort, nevertheless, when tested usually impurities can

be identified.

The variation of the flow stress of a BCC metal under a temperature change has al-

ready been presented in Fig. 2.1 for Mo single crystals. For other BCC metals, the

behavior is qualitatively similar but quantitatively different. For the response below

the knee temperature TK, different regimes should be distinguished. Especially the

interval TK/2 < T < TK was analyzed by Seeger (1981) for which the rate and tem-

perature dependence was explained through the formation of kink pairs in screw

dislocations without invoking impurity effects.

Because of the non-uniformity of the temperature dependence of the flow stress,

Seeger (1981) proposed different formulations for different temperature regimes. A

high-temperature–low stress regime, an intermediate (diffusion controlled) regime

and a high-stress regime are distinguished. In the present paper, the flow is con-

trolled by the slip rate equation (2.10), which roughly corresponds to the third

regime, where only kink pairs controlling dislocation movement in the direction of

applied stress are taken into account.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30

35

40

45

50

Strain %

Str

ess

(MP

a)

77 K sim.77 K exp.113 K sim.113 K exp.175 K sim.175 K exp.

Figure 2.6 / Temperature dependence of [001] oriented niobium single crystals.

In Fig. 2.6, the tensile stress-strain curves for a [001] oriented Nb single crystal are

presented at three different temperature levels, at a strain rate of 1.3 × 10−4s−1. The

experimental data is taken from Duesbery and Foxall (1969). The true stress and true

strain curves are reproduced from the shear values reported for (112)[111] primary


identification

slip. The Schmid factor for the [001] orientation equals 0.471. The experiment and the

crystal plasticity simulations show an adequate agreement. The material parameters

for this material are: C11 = 250GPa, C12 = 135GPa, C44 = 30GPa (Carroll (1965)),

γ0 = 1.2 × 107s−1, qn = 1.4, G0 = 1.12 × 10−18J, k = 1.3807 × 10−23J/K, a = 1.2,

sa0 = 4MPa (atwin sense) and sa0 = 2.4MPa (twin sense). Due to the lack of experi-

mental evidence the identification of the non-Schmid parameters is disregarded and

the effect is excluded here. The variation of most parameters with the temperature is

negligible, with the exception of the parameters presented in Table 2.4.

Table 2.4 / Material parameters for Nb single crystals at three different temperatures

Temperature 77 K 113 K 175 Kh0 1500 MPa 922 MPa 800 MPass 21.7 MPa 19.9 MPa 13.42 MPast 15.3 MPa 8.06 MPa 3.18 MPa

2.6 Summary and Conclusion

In the present paper, a crystal plasticity model has been proposed and implemented,

revealing the unique characteristics of BCC single crystals. A comprehensive sum-

mary of the intrinsic properties of these materials has been presented, including re-

cent insights in the activation of different slip systems, violation of Schmid’s law,

temperature and orientation dependence of the flow stress and resulting stress-strain

curves. The parameters in the model are determined using a proper parameter iden-

tification process, relying on a least-square minimization procedure on the differ-

ences between the numerical and experimental uniaxial stress-strain responses.

Published results on the operational slip mechanisms in BCC crystals and the ex-

tended crystal plasticity models, reflect considerable contradictions. Most studies

take into account 110, 112 and 123 type of slip planes without confronting

the model with the expected temperature and orientation dependence of the crystal.

Following Seeger (2001), this work uses 112 planes at moderate temperatures and

110 planes at low temperatures. Contrary to most of the crystal plasticity elabora-

tions, 123 type of slip planes are not taken into account. Cross slipping phenomena

and non-planar spreading of screw dislocation cores are implicitly incorporated in a

phenomenological manner.

The applied slip law plays an essential role in the present model since all the intrinsic

2.6 Summary and Conclusion 23

characteristics result from the actual formulation of the slip rate equation. Additional

to the pre-mentioned existing crystal plasticity frameworks, in this contribution the

non-Schmid behavior is introduced in the slip law by modifying the effective shear

stress, where the non-Schmid contribution represents the dislocation cores spreading

in a non-planar manner. Actually, the emphasis was put on this intrinsic anisotropy

effect, even though other anisotropy effects such as texture development and dislo-

cation sub-structure evolution may be included in the model as well.

The necessity for this research results from the fact that there is a lack of published

works, that confront recent BCC single crystal plasticity models to single crystal ex-

perimental data that reveals the intrinsic orientation and temperature dependence of

these crystals.

Chapter three

A composite dislocation cell model todescribe strain path change effects in

BCC metals1

Abstract / Sheet metal forming processes are within the core of many modern man-ufacturing technologies, as applied in e.g. automotive and packaging industries. Ini-tially flat sheet material is forced to transform plastically into a three dimensional shapethrough complex loading modes. Deviation from a proportional strain path is associatedwith hardening or softening of the material due to the induced plastic anisotropy result-ing from the prior deformation. The main cause of these transient anisotropic effects atmoderate strains is attributed to the evolving underlying dislocation microstructures. Inthis paper, a composite dislocation cell model, which explicitly describes the dislocationstructure evolution, is combined with a BCC crystal plasticity framework to bridge themicrostructure evolution and its macroscopic anisotropic effects. Monotonic and multi-stage loading simulations are conducted for a single crystal and polycrystal BCC metal,and obtained macroscopic results and dislocation substructure evolution are comparedqualitatively with published experimental observations.

3.1 Introduction

For each car, the automotive industry manufactures more than 500 parts by multi-

stage forming operations, involving complex deformation paths. Deviation from

a proportional strain path is commonly associated with a change in the hardening

(or softening) behavior of the material. In order to achieve a first-time-right design,

modern predictive tools relying on the finite element method are commonly used

nowadays. The anisotropy induced by complex deformation paths, which may lead

1This chapter is reproduced from Yalcinkaya et al. (2009)

25

263 A composite dislocation cell model to describe strain path change effects

in BCC metals

to premature failure (e.g. Sang and Lloyd (1979)), is crucial in this sense and should

be included in the constitutive models used in the analysis.

The overall plastic anisotropy in BCC metals, induced by the imposed deformation,

originates from different sources at different length scales. Slip asymmetry and in-

trinsic anisotropy effects caused by the non-planar spreading of screw dislocation

cores are active at the micro level (e.g. Bassani et al. (2001), Duesbery and Vitek

(1998), Ito and Vitek (2001), Yalcinkaya et al. (2008)) whereas the development of dis-

location substructures is relevant at the meso level (e.g. Rauch and Schmitt (1989),

Wagoner and Laukonis (1983), Rao and Laukonis (1983), Wilson and Bate (1994),

Gardey et al. (2005)). At the macro level, the texture development of polycrystalline

metal is contributing dominantly (e.g. Bacroix et al. (1994), Bacroix and Hu (1995),

Nesterova et al. (2001)). Upon switching strain paths, the intrinsic properties ob-

viously have a substantial effect on the observed anisotropy due to changes in the

dislocation activity. However, the evolution of dislocation microstructures has been

recognized as a main driver triggering the observed anisotropic material behavior. In

a recent report Li et al. (2006) commented on the strong anisotropy, i.e., larger than

expected from the texture, induced by the dislocation structure in IF steel increasing

with the rolling prestrain. The prediction of dislocation microstructures within the

individual dislocation descriptions and continuum theories has been a challenging

subject in the last decades in the material science community (see Groma (1997) for an

overview). While transmission electron microscopy (TEM) observations have been a

powerful tool to understand their origin and to derive the physical parameters that

govern their evolution (e.g. Fernandes and Schmitt (1983)), discrete dislocation mod-

els and atomistic considerations improved the understanding of the formation and

the evolution of dislocation microstructures and the related plastic anisotropy. How-

ever, only a limited number of micromechanical modeling approaches have been ad-

dressing the anisotropy induced by evolving dislocation cells with a crystal plasticity

framework.

Among the attempts to develop plastic anisotropy models that incorporate the mi-

crostructure evolution for complex deformation histories, the most remarkable one is

the constitutive model proposed by Teodosiu and Hu (1995). This phenomenological

model uses the Hill criterion for the onset of yielding while the hardening is associ-

ated with the dislocation structures. The polarity of dislocation walls, the back-stress

and the strength of the dislocation structure are accounted for by internal variables.

Recently, Wang et al. (2008) presented an improvement of this model especially con-

centrating on continuous loading path changes from uniaxial tension to simple shear

without unloading the material.

Another attempt to describe the occurring phenomena is presented by Peeters et al.

(2000) dealing with a polycrystal plasticity model that incorporates more details of

3.1 Introduction 27

the microstructure evolution at the grain scale, where cell boundary dislocation den-

sities, cell block boundary dislocation densities, and directionally movable disloca-

tion densities are taken as internal variables. This model attributes a major part of the

strain path change effects to the evolution of cell block boundaries and polarization

of these structures. Additional to above mentioned models, Hoc and Forest (2001),

Mollica et al. (2001) and Tarigopula et al. (2008) presented some other approaches

dealing with the anisotropic strain path change effects. In the present paper the con-

r

ww

r

Figure 3.1 / Composite representation of the cell structure.

centration is focused on a crystal plasticity model that incorporates the evolution of

dislocation cell structures. As originally introduced by Mughrabi (1987), a cell struc-

ture can be idealized as a two-component material, distinguishing cell walls and cell

interiors. It is characterized by the wall thickness w, the cell size r (see Fig. 3.1), the

dislocation densities in the cell walls ρw and the cell interiors ρc. The macroscopic

anisotropy effects are obtained by the evolution of these internal variables during

monotonic deformation and multi-stage loading processes. Inside the cell structure,

a BCC crystal plasticity framework (Yalcinkaya et al. (2008)) is incorporated, which

goes beyond the developments of Viatkina et al. (2003) for FCC metals in which a

classical von-Mises plasticity model was used. From this perspective, it is the first

example that incorporates a physically motivated constitutive model into the evolu-

tion of dislocation substructures for BCC metals, in order to model the anisotropy

due to strain path changes.

The plan of this paper is as follows. Section 2 discusses the evolution of dislocation

substructures under monotonic and multi-stage deformations. Next, in section 3 the

formulation of the BCC crystal plasticity framework is summarized. Section 4 han-

dles the incorporation of the dislocation cell evolution model into the crystal plastic-

ity framework, along with a summary of the numerical implementation. Further, in

section 5 computational results of single crystal and polycrystal tests are presented

on the basis of which the crystal anisotropy is distinguished from the dislocation


in BCC metals

cell anisotropy. The accordance of the results with respect to published experimental

results is discussed. Finally, concluding remarks are given in section 6.

Cartesian tensors and associated tensor products will be used throughout this paper,

making use of a Cartesian vector basis e1e2e3. Using the Einstein summation rule

for repeated indices, the following conventions are used in the notations of vectors,

tensors, related products and crystallography:

• scalars a

• vectors a = aiei

• second-order tensors A = Ai jei ⊗ e j

• fourth-order tensors 4 A = Ai jklei ⊗ e j ⊗ ek ⊗ el

• C = a ⊗ b = aib jei ⊗ e j

• C = A · B = Ai jB jkei ⊗ ek

• C = 4 A : B = Ai jklBlkei ⊗ e j

• crystallographic direction, family [uvw], 〈uvw〉

• crystallographic plane, family (hkl),hkl

• slip system, family (hkl)[uvw], hkl 〈uvw〉

3.2 Dislocation substructure evolution

Dislocation substructuring is characterized by the clustering of dislocations after a

certain amount of plastic deformation, where an initially statistically homogeneous

distribution of dislocations develops towards a dislocation pattern with high density

dislocation walls enveloping low density dislocation areas. This self-organization of

the microstructure in the grains is often referred to as the low-energy, steady state

configuration of dislocations (Kuhlmann-Wilsdorf (1989)). TEM analyses (e.g. Keh

et al. (1963)) revealed that for deformations larger than 3-4 % a well-developed dis-

location cell structure forms in steel at ambient temperature. Further deformation

renders a polarized structure with dislocation sheets or cell block boundaries, which

envelope a number of dislocation cells. These structures are the result of the in-

teractions between dislocations gliding on the most active slip planes and the sec-

ondary dislocations (Teodosiu (1992)). However, the occurrence of the dislocation

3.2 Dislocation substructure evolution 29

sheets is not always manifest and sometimes the microstructure is partitioned by or-

dinary cell boundaries having no particular crystallographic or macroscopic orienta-

tion (Hansen and Huang (1997)). For that reason, depending on the grain orientation

and the strain direction, either parallel dislocation walls or more equiaxed closed

cells are observed (e.g. Rauch and Schmitt (1989)) in low carbon steels. Besides, dif-

ferent materials end up in different type of microstructures. It is neither experimen-

tally nor computationally an easy task to identify the type of evolving dislocation

microstructure, yet formation of dislocation cells is mostly observed. Thereof, this

paper concentrates on the formation and evolution of these dislocation cell struc-

tures.

CROSS TEST

MONOTONIC

STRESS REVERSAL

Figure 3.2 / Schematic evolution of a dislocation cell structure under strain pathchange.

As discussed above, dislocation cell structures develop upon plastic strain in most

metals, and evolve in a distinct way depending on the applied strain path. The main

features are visualized in Fig. 3.2. Under monotonic deformation a dislocation cell

structure appears and evolves towards a decreasing cell size r, and wall thickness w

accompanied by an increasing dislocation density in the cell walls ρw (e.g. Fernandes


in BCC metals

and Schmitt (1983)). After a strain path change, the developed cell structure adjusts

to the new loading and the dislocation microstructure induced by the prestrain be-

comes unstable. It is disrupted and dissolved, and a new dislocation structure typical

of the new strain path forms (Barlat et al. (2003)). The characteristic features of the

initial cell structure disappear as the deformation proceeds in the new direction. Un-

fortunately, there is no clear interpretation of what is occurring with the dislocation

microstructure during the adaptation nor is there a unique terminology to describe

this evolution. Here we distinguish between two different scenarios; dissolution of

cells as in the cross test and disruption as occurring under reversed loading. There

appears to be no consistency in the literature in the use of the dissolution, disruption

and disintegration of dislocation cells, and most of the time any cell evolution after

a strain path change is described as a dissolution process (e.g. Rauch and Schmitt

(1989), Rauch (1992), Rauch (1991) Gardey et al. (2005), Rao and Laukonis (1983)).

Indeed, both dissolved and disrupted structures appear as disorganized structures

with a higher degree of homogeneity compared to the state before the strain path

change. Nevertheless, there are indications that there is a morphological difference

between the two microstructure evolution scenarios mentioned above (e.g. Gardey

et al. (2005)) in correspondence with the difference between the driving forces and

their physical origins.

Figure 3.3 / Left: Experimental results for mild steel DC06 subjected to monotonicsimple shear and simple shear followed by load reversal (10 % and 30 %) [Bouvieret al. (2006)] (reversed loading) Right: Experimental results for IF-steel subjected tomonotonic simple shear and tensile tests (10 % and 20 %) followed by shear [Peeterset al. (2000)] (orthogonal loading).

In the example shown in Fig. 3.2, two different strain path changes have been consid-

ered where the two types of evolution phenomena can be distinguished. A cross test,

e.g. tension followed by simple shear or a tension test followed by tension in a dif-

3.3 Computational model 31

ferent direction, reveals progressive cell evolution (e.g. Rao and Laukonis (1983)). It

has been observed that after a strain path change cell walls become thicker while the

dislocation density in the walls becomes smaller (e.g. Schmitt et al. (1991)). Hence,

the dislocation distribution is more uniform and the cell structure is less organized.

This evolution process can cause partial or complete dissolution of the existing cell

structure, while concurrently a new cell structure develops with a morphology re-

lated to the new loading direction. The cell structure evolution resulting from a

stress reversal has received more attention in the context of the analysis of the well-

known Bauschinger effect (e.g. Rauch (1991)). The evolution of the cell structure

under stress reversal can be characterized by the disruption of cell walls (e.g. Vi-

atkina (2005), Christodoulou et al. (1986)). The thickness of the cell walls does not

change significantly, however, the walls tend to disconnect. Experimental observa-

tions (e.g. Christodoulou et al. (1986)) also report a strong flux of dislocations from

the walls to the cell interiors, decreasing the wall dislocation density and increasing

the density in the cell interiors. Accordingly, the descriptive modeling of the dislo-

cation distribution relies on an increase of cell size and a dislocation redistribution

(Viatkina (2005)). With ongoing deformation cell walls reappear and a new cell struc-

ture originates.

Upon sustained loading after a strain path change, the microstructure always evolves

such that transient effects disappear and the macroscopic stress-strain curve satu-

rates to the monotonic deformation curve (see Fig. 3.3).

3.3 Computational model

The constitutive behavior of each composite constituent (cell walls or cell interiors)

is modeled in a finite strain crystal plasticity framework with plastic slip governed

by the thermally activated motion of dislocations. The kinematics starts with the

multiplicative decomposition of the deformation gradient tensor into an elastic and

a plastic part of each component, as developed by Lee (1969), Rice (1971), Hill and

Rice (1972) in the classical plasticity theory,

F i = F ie · F i

p, (3.1)

where the superscript i indicates the specific component (w: wall, c: cell) and tensor

F ip defines the stress-free intermediate configuration. In this configuration, resulting

from plastic shearing along well-defined slip planes of the crystal lattice, the orien-

tation of the slip systems is unaltered. The tensor F ie reflects the lattice deformation

and local rigid body rotations. The slip systems are labeled by a superscript α, with

α = 1, 2..., ns where ns is the total number of slip systems. The vectors mα,i0 and nα,i

0


in BCC metals

denote the slip direction and the slip plane normal in the reference and intermediate

configurations. In the current state they are represented by mα,i and nα,i, respectively.

The crystallographic split of the plastic flow rate Lip = F

i

p · F ip−1 is given by

Lip =

ns

∑α=1

γα,imα,i0 ⊗ nα,i

0 , (3.2)

with γα,i the individual slip rate on the slip system α.

The second Piola-Kirchhoff stress tensor Si is expressed in terms of the elastic Green-

Lagrange strain tensor Eie, both relative to the intermediate state,

Si = 4C : Eie with Ei

e =1

2(F i

e

T · F ie − I), (3.3)

with I the second order unity tensor and 4C the fourth order tensor consisting of

elastic moduli.

From the second Piola-Kirchhoff stress the Kirchhoff stress in the current configura-

tion can be determined by a push-forward operation,

τ i = F ie · Si · F i

e

T. (3.4)

From the Kirchhoff stress the Cauchy stress can be derived according to,

σ i =1

J ie

τ i with J ie = det(F i

e). (3.5)

The Schmid resolved shear stress is the projection of the Kirchhoff stress on the slip

systems, i.e.

τα,i = mα,i · τ i · nα,i = mα,i0 · Ci

e · Si · nα,i0 with Ci

e = F ie

T · F ie, (3.6)

which is the driving force for the dislocation movement on a certain slip system α.

There has been various discussions and contradictions considering the active slip

systems of BCC crystals, and recent studies shows that the slip system activation

is highly temperature dependent (see Yalcinkaya et al. (2008)). At room tempera-

ture the 112 slip system family is dominantly active. The effect of the non-Schmid

stresses on the non-planar screw dislocation cores which contributes to the orien-

tation dependence of BCC crystals at single crystal level can be taken into account

as an additional contribution to the driving stress in equation (3.6) (Yalcinkaya et al.

(2008)), however, this contribution affects the initial anisotropy of these metals rather

than the transient effects observed during the strain path changes (see Fig. 3.3). In-

cluding this effect would increase the material parameters while it does not con-

tribute to the aim of this paper. Hence it was decided not to account for this effect

here.

3.3 Computational model 33

The motion of dislocations is obstructed by thermal and a-thermal barriers which

are caused by the dislocation interactions upon flow, the elastic stress field due to

other dislocations and grain boundaries. Hence the slip resistance distinguishably

originates from a thermal part sα,it and an a-thermal part sα,i

a . For the slip rates the

following slip law is adopted (see Yalcinkaya et al. (2008)),

γα,i = γα,i0 exp

−G0

kT

[

1 −(

τα,ie f f

sα,it

)]

sign(τα,i), (3.7)

where τα,ie f f = |τα,i| − sα,i

a is the effective driving stress on the slip systems, G0 is the ac-

tivation free energy, k is Boltzmann’s constant and T is the absolute temperature, γα,i0

is a reference strain rate. For isothermal cases, the thermal part sα,it of the slip resis-

tance is taken constant and the a-thermal slip resistance is related to the dislocation

densities on all slip systems through,

sα,ia = Gb

√ns

∑u=1

Aαu|ρu,i|, (3.8)

where G is the shear modulus, b is the magnitude of the Burgers vector, Aαu are

the interaction coefficients between the slip systems α and u, and ρu,i the dislocation

density on the slip system u of component i. The dislocation interaction coefficients

of the matrix Aαu depend on the type of interaction between dislocations on different

slip systems (e.g. Franciosi and Zaoui (1982), Queyreau et al. (2008)). Because of the

lack of data on 112 slip systems, only the interactions between the dislocations

belonging to the same slip system, i.e. α = u, and different slip systems, i.e. α 6= u,

will be distinguished for Aαu.

The macroscopic mechanical response of the composite model is obtained by ap-

plying a Taylor averaging assumption where the deformation in each component is

assumed to be equal to the macroscopic deformation and where the rule of mixtures

gives the macroscopic stress from the local stresses in each component according to,

σ = fσw + (1 − f )σ c. (3.9)

In this equation f represents the actual volume fraction of the cell walls, expressed

in terms of the microstructural morphology parameters w and r according to:

f =Vw

V= 3

w

r− 3

(w

r

)2

+(w

r

)3

, (3.10)

where V and Vw are the volumes of the entire composite and the wall component,

respectively.

Experimental studies (e.g. Fernandes and Schmitt (1983)) suggest that the wall thick-

ness w, the cell size r (see Fig. 3.1), the dislocation densities in the cells ρα,c, and


in BCC metals

the walls ρα,w evolve with increasing applied strain. Moreover, these quantities are

dependent on the deformation history, and therefore they are taken into account as

internal variables in this framework. Corresponding evolution equations are to be

formulated that describe the cell structure development during monotonic loading

as well as complex strain path histories. This is done in the following section, where

the incorporation of a dislocation cell structure evolution model into the crystal plas-

ticity framework is presented.

3.4 Modeling of microstructure evolution

In order to give a clear understanding of the model, three distinct types of loading

cases are considered, namely monotonic loading, orthogonal loading and reverse

loading. The purpose of the model consists in unifying these cases by capturing ef-

fects under continuous combinations of these deformations, through a single set of

evolution equations. It is assumed that the cell orientation is dictated by the loading,

and that there are always enough slip systems to accommodate that cell, indepen-

dently of the crystal orientation.

3.4.1 Monotonic deformation

The evolution of a two-phase dislocation cell structure has been schematized in Fig.

3.2. During monotonic deformation, the cell size r and the wall thickness w de-

crease, yielding a decrease of the length scales of the spatial dislocation patterns that

is inversely proportional to the flow stress, often referred to as the law of similitude

(Kuhlmann-Wilsdorf (1962)). Experimental observations (e.g. Mughrabi et al. (1986))

suggest that the dislocation density inside the cell interiors does not change signifi-

cantly. Accordingly, a constant dislocation density ρα,c in the cell interior component

is assumed here. The derivation of the evolution of the dislocation density in the

walls ρα,w departs from the frequently used balance between the multiplication of

mobile dislocations and annihilation events,

ρα,w =1

b

[I√ρα,w − Rρα,w

]

∑α

|γα,w|+ ρα,c − ρα,w

ff (3.11)

where R is the recovery length and I is a dislocation multiplication parameter. The

last term in the equation accounts for the change in volume occupied by the wall

component. The initial state of the composite is modeled as if the wall component

is occupying the entire volume; the initial value of its dislocation densities is de-

termined by the value ρ0 of the initial uniform distribution. This in fact correctly

represents the case when no dislocation pattern is present.

3.4 Modeling of microstructure evolution 35

The following empirical relation between the cell size r and the flow stress σ y is

suggested in the literature (e.g. Barker et al. (1989), Mughrabi (1987)),

σ y =CGb

rm, (3.12)

which is consistent with the experimental observations of Fernandes and Schmitt

(1983) and commonly used theoretical investigations (e.g. Mughrabi (1987)). The

parameter C, is a material constant and the exponent m is generally close to 1 for

cell structures. In the present framework, equation (3.12) is rewritten in terms of slip

variables at the slip system level instead of the continuum level yield stressσ y. This is

done by using evolving a-thermal slip resistances on the active slip systems, i.e. r ∼

CGb/σ y ∼ CGb/ ∑α sa, whereby the parameter C accounts for the scaling between

the two levels as well. The cell size evolution is approximated by incorporating the

rule of mixtures for the different components of the composite,

r =CGb

f ∑α sα,wa + (1 − f ) ∑α sα,c

a(3.13)

The evolution of the wall thickness w is adopted from Viatkina et al. (2003) and as-

sumed to be governed by an effective plastic strain rate measure ∑α |γα| according

to

w = km(win f − w) ∑α |γα| with

∑α |γα| = f ∑α |γα,w| + (1 − f ) ∑α |γα,c|.(3.14)

In this evolution law a decrease of the wall thickness with a saturation factor km is

incorporated, with a final saturation value equal to win f , which is consistent with

experimental observations (e.g. Fernandes and Schmitt (1983)).

The implementation of the model presented above follows an incremental-iterative

solution procedure, which is applied for each of the composite components with the

same imposed deformation (Taylor approach). The first step in this procedure is the

initial estimate for the elastic part F ie, resulting in an estimate for the plastic part F i

p

through (3.1). With the kinematics defined, both the stress and the Schmid stress is

calculated. These values together with the slip resistance (3.8) (which is calculated

from dislocation density evolution (3.11)) enter the slip law (3.7) resulting in the slip

rates on each slip system. The updated plastic part of the deformation gradient is

obtained from the calculated slip rates through a time integration scheme. Gener-

ally, the calculated and the imposed deformation will be different, which results in a

residual. Iteration on the residual leads to updated values of variables including F ip

and F ie. With the current values of r and w the volume fraction f is calculated with

(3.10), which is used to determine the macroscopic stress (3.9). The procedure is re-

peated for all time steps, which results in the entire history of stress, slip and internal

variable evolution.


in BCC metals

3.4.2 Orthogonal change of deformation

An orthogonal change of the deformation path leads to dissolution of dislocation

cell walls, which is captured through an increase of the wall thickness. In the limit

the wall occupies the whole material where w becomes equal to r, representing a

full recovery of a uniform dislocation configuration (i.e. no cells present). This limit

case is rarely observed in practice. After the dissolution process, new cells origi-

nate accommodating the new loading direction. Next, the cell size and dislocation

density are considered to evolve in the same way as for monotonic deformation, i.e.

the dislocation density increases in the walls and remains constant inside the cell.

More experimental evidence is needed to improve further on this phenomenological

relation.

The dissolution process is leading to a transient increase of the cell wall thickness

driven by the overall slip rate as given by

w = kd(r − w) ∑α

|γα| (3.15)

where kd is the dissolution factor. The saturation value of the wall thickness logically

equals the cell size r corresponding to a complete dissolution of the cell. When both

the dissolution and the redevelopment processes are taken into account, the wall

evolution becomes

w = pkd(r − w) ∑α

|γα| + (1 − p)km(win f − w) ∑α

|γα|, (3.16)

where the first contribution on the right-hand side reflects equation (3.15) accounting

for the effect of the loading in the new direction, i.e. the dissolution process. The

second contribution on the right-hand side of (3.16) represents the development of

the wall structure according to equation (3.14). The transition parameter p, to be

specified in the following, defines the relative contribution of the dissolution process

in the evolution of w. It is characterized by taking into account: (i) the dissolution

process depends only on the angle between successive deformation paths and (ii)

the dissolution effect disappears as the deformation proceeds in the new direction.

In this context, the following expression for p is taken:

p = (1 − |θ|) exp

−B

[

∑α

|γα| −∑α

|γαpre|]

, (3.17)

where θ is a scalar measure that identifies the strain path change and ∑α |γαpre| indi-

cates the accumulated plastic deformation prior to the strain path change and B is a

material parameter. To characterize the strain path change measure θ, a commonly

used definition is adopted here:

θ =Lp1 : Lp2

|Lp1||Lp2|, (3.18)

3.4 Modeling of microstructure evolution 37

where Lp1 and Lp2 are the macroscopic plastic velocity gradient tensors prior to and

after the strain path change. Here θ = 1 refers to monotonic deformation, θ = 0 to a

cross test, and θ = −1 to a reverse test.

Equations (3.16) and (3.17) describe the evolution of the wall thickness during the

whole deformation process. During monotonic deformation where θ = 1, equa-

tion (3.16) reduces to equation (3.14) describing cell wall thinning. After a strain

path change, the dissolution is initiated with an intensity proportional to (1 − |θ|).

The dislocation walls start widening, governed by the competition between the new

structure development and the old structure dissolution. As the deformation pro-

ceeds in the new direction, the dissolution process fades out and accordingly p ap-

proaches 0 due to (3.17) and the wall thickness tends to decrease again: a new dislo-

cation structure is developing.

3.4.3 Reverse deformation

The cell structure degeneration after a stress reversal, the so-called cell disruption,

is modeled by a temporary increase of the cell size. As already discussed in section

3.2, the thickness of the cell walls does not change significantly due to stress reversal.

Therefore, the other parameters, i.e. the wall thickness and the dislocation density,

are assumed to evolve in a similar way as under monotonic deformation. The dis-

ruption of cells is observed to be a rapid process in which the size of the cells rapidly

increases after a stress reversal and then slowly decreases (e.g. Viatkina et al. (2003),

Christodoulou et al. (1986)). As the deformation proceeds in the opposite direction

the cell size recovers to the level corresponding to the monotonic strain path (3.13).

In order to model this temporary increase of the cell size, an additional (transient)

term is incorporated in equation (3.13) for the cell size evolution:

r =CGb

f ∑α sα,wa + (1 − f ) ∑α sα,c

a+ A exp[−kc(∑

α

|γα| −∑α

|γαpre|)], (3.19)

where A defines the degree of disruption and kc is a constant reflecting the recovery

speed (governed by slip). The entire term in the right-hand side (added to (3.13))

determines the immediate increase of the cell size. This effect vanishes with ongoing

deformation depending on the value of the parameter kc. As soon as the disruption

contribution gradually disappears, the cell size again follows the evolution as given

for the monotonic deformation case. To reflect the fact that the cell disruption is

triggered by a stress reversal, the coefficient A depends on the strain path change

through

A =

a|θ| if θ < 00 if θ ≥ 0

(3.20)


in BCC metals

where a is a fitting parameter. Consequently, the Bauschinger test triggers the high-

est disruption. For more complex strain path changes with negative values of θ both

equations (3.16) and (3.19) have non-zero strain path change contributions (p > 0

and A > 0), describing a process with coexisting dissolution and disruption of cells.

When p = 0 and A = 0, the model describes the evolution under monotonic de-

formation. In this way the resulting system of equations effectively unifies different

types and combinations of strain path changes.

3.5 Numerical examples

Using reported experimental trends, this section presents and qualitatively validates

typical results: (i) the evolution of the internal variables during monotonic defor-

mation and the intrinsic orientation effect during strain path changes of BCC single

crystals and (ii) the macroscopic stress-strain behavior of BCC polycrystals during

multi-stage loading processes. Due to the lack of quantitative experimental data,

however, an adequate quantitative comparison is not possible, preventing a reliable

quantification of the material parameters. Within a physically acceptable range of

material parameters, it will be shown that the presented model is well capable of

capturing all experimental trends. Young’s modulus E, Poisson’s ratio ν, the refer-

ence strain rate γ0, the shear modulus G, the magnitude of the Burgers vector b, the

interaction coefficients Aαα and Aαu, the activation energy G0, thermal slip resistance

st, the dislocation multiplication parameter I and the recovery length R are the stan-

dard parameters in the constitutive model describing the BCC material, whereas ρ0,

ρc, kd, km, win f , C, B, a, kc are the additional parameters associated with the disloca-

tion cell structure evolution. The set of standard parameters are well documented in

the literature (see Table 3.1). The remaining parameters have a restricted range and

they are estimated to establish a qualitative agreement with experimental observa-

tions. The initial value of the dislocation densities in the walls ρ0 logically equals ρc

(no cells exist yet) and the initial value of the cell size r0 can be calculated by using

equation (3.13) where f = 1 and ραw = ρα0 .

3.5.1 Example 1: monotonic deformation of single crystals

Experimental observations of the microstructure evolution during monotonic defor-

mation have been reported in section 3.2. In most cases, the length scales of the spa-

tial patterns formed by the dislocations decrease with increasing strain. To analyze

this change in the microstructure, several studies have been conducted. For instance,

Sevillano et al. (1981) present several curves for the average cell size with respect to

the deformation during rolling and drawing processes for different materials, and

3.5 Numerical examples 39

Fernandes and Schmitt (1983) give data on the wall thickness and dislocation cell

size of low-carbon steel under various types of loading. In the current framework,

the parameters in the evolution equations are identified to retrieve this characteristic

behavior.

Figure 3.4 / Evolution of the cell structure variables during monotonic deformationforα-Fe. Left: wall thickness w and cell size r. Right: volume fraction of the walls.

In Fig. 3.4 the evolution of the microstructure of an α-Fe single crystal during a

uniaxial tension simulation at 298 K and a strain rate of 5 × 10−4s−1 is presented

with the material parameters given in Table 3.1 2. The cell size shows a decreasing

trend with increasing strain as expected. The wall thickness decreases quickly and

stabilizes at a constant value. The volume fraction approaches a value of around 0.1

after a sharp decrease. This trend can also be found in the literature, where it is stated

that the volume fraction of dislocation walls remains at a constant value (e.g. Peeters

(2002)).

3.5.2 Example 2: strain path change of single crystals

In this example the effect of the intrinsic anisotropy of single BCC crystals during

multi-stage loading is illustrated. In Fig. 3.5 the results of a cross loading simulation

is presented where the crystal was first loaded in the [001] direction and next in the

[011] direction after unloading at a strain value of 0.15. For comparison purposes

a reference calculation is performed in which the evolution of the microstructure

2Note that the number of papers focusing on the determination of the interaction coefficients inBCC single crystals are very limited. Moreover, the actual value strongly depends on the actual BCCcrystal considered and impurities present and the initial dislocation density. Values may thereforediffer considerably, where a difference of a factor 10 can be easily find throughout the literature.


in BCC metals

Young’s modulus E 139 GPaShear modulus G 64 GPaPoisson’s ratio ν 0.362Reference strain rate γ0 1.07 × 106s−1

Burgers vector length b 0.248 × 10−9mInteraction coefficient (self) Aαα 0.00072Interaction coefficient (latent) Aαu 0.001Initial dislocation density ρ0 0.18 × 1014m−2

Activation free energy G0 2.95 × 10−18JDislocation multiplication parameter I 0.228Dislocation annihilation rate parameter R 5.1 × 10−9

Boltzmann constant k 1.3807 × 10−23J/KThermal dislocation resistance st 15 MPaSaturation factor km 150Saturation value win f 0.18 × 10−6mMaterial constant C 20

Table 3.1 / Material parameters for monotonic loading case. Some of the param-eters, i.e. γ0, G0 and st have already been identified in previous work (Yalcinkayaet al. (2008)). The parameters b and G are taken from Frost and Ashby (1982). Thevalue of ρ0 is presented by Krejci and Lukas (1971). The latent interaction coefficientAαu is identified and the self interaction coefficient Aαα is obtained by assuming aratio of 1.4 between latent and self hardening. The remaining parameters are iden-tified by comparing the experimental trends with computational results. Additionalparameters needed for complex strain paths are commented in the text.

is not incorporated, i.e. transient hardening and softening effects due to the strain

path change (see solid line in Fig. 3.5) are absent. The dashed line presents the

outcome of the full microstructure evolution computation, in which both the crys-

tal slip anisotropy and the dislocation cell anisotropy are revealed. The purpose of

this example is to discriminate these two intrinsic sources of anisotropy at the single

crystal level. Obviously, both mechanisms here contribute to a larger yield stress af-

ter reloading. Whereas this increase is systematic for dislocation cell contribution, it

is obviously orientation dependent for the slip anisotropy contribution.

3.5.3 Example 3: strain path change of polycrystals

In this subsection, the performance of the model in the context of complex deforma-

tion histories is evaluated by determining the response of a BCC polycrystal with a

random texture under a sequence of: (i) two uniaxial tension tests in different direc-

tions to obtain the cross effect; (ii) simple shear and reversal in order to capture the

Bauschinger effect.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

10

20

30

40

50

60

Strain

Str

ess

(MP

a)

Figure 3.5 / Stress-strain curve for [001] uniaxial tension followed by [011] uniaxialtension with (dashed line) and without (solid lines) microstructure evolution effectforα-Fe single crystal.

As explained previously, at a single crystal level the effect of intrinsic crystallo-

graphic anisotropy during a strain path change is noticeably high. The initial

anisotropy, the so-called orientation dependence of BCC single crystals has been

studied before (e. g. Yalcinkaya et al. (2008)) and this effect adds up to anisotropy

due to the dislocation microstructure evolution. In this example the main interest

focuses on the anisotropy due to substructure evolution during multi-stage loading

processes for the case where the intrinsic orientation effect is known to contribute

less. To this purpose polycrystal simulations have been conducted, where 100 ran-

domly oriented crystals are considered interacting according to a Taylor averaging

scheme.

First, the cell dissolution process and its macroscopic cross effect are analyzed. The

characteristic feature of the stress-strain curve in Fig. 3.3 is the transient change in-

duced by a change of the deformation path. An initial increase in the yield stress is

followed by moderate softening. The cross effect vanishes gradually and the curve

saturates towards to the monotonic case. In order to measure this effect experimen-

tally either tension followed by simple shear or two successive orthogonal tensile

experiments need to be conducted. With respect to the latter approach Schmitt et al.

(1991) presented clear experimental results where various tensile sequences were ex-

amined, with different angles between the succeeding tensile directions equal to 15,

45 and 90 with different amounts of prestrain. In Schmitt et al. (1991), it was re-

ported that no evolution of cell-blocks was observed, supporting the case examined

here, where the cell structure development is assumed to be the main mechanism


in BCC metals

accompanying the strain path change. The sequence of two uniaxial tests with 45

between the tensile axes is, according to equation (3.18), characterized by θ = 0.25,

and it is rather close to a cross test exhibiting the highest cell dissolution. Additional

to the parameters used during monotonic deformation (see Table 3.1), kd and B are

identified as 300 [-] and 20 [-] respectively. The obtained typical cross effect is pre-

sented in Fig. 3.6. This transient effect is observed in many materials and the size

of the effect is determined by the amount of applied prestrain while the shape of the

hardening and softening zones depends on the material.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

10

20

30

40

50

60

Strain

Str

ess

(MP

a)

Figure 3.6 / Predicted stress-strain curve of a 45 cross tensile test of an α-Fe poly-crystal.

The second example concerns the Bauschinger effect, which yields a reduction of the

yield strength of the material after a load reversal. A simple shear and reversal sim-

ulation is presented in Fig. 3.7 , where the evolution of the cell size r is dominantly

contributing to the anisotropy at this continuum level. Additional to the parame-

ters used during monotonic deformation in Table 3.1, the parameters a and kc are

identified as 1 × 10−4 m and 14 [-] respectively, to validate this part of the model.

Even though both the cross effect and the Bauschinger effect are extensively docu-

mented in the literature, quantitative data on the evolution of dislocation cell struc-

ture during strain path changes remains hard to find. For this reason, a qualitative

analysis rather than a quantitative study has been conducted here.

3.6 Summary and Conclusion 43

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

5

10

15

20

25

30

35

40

45

50

Strain

Str

ess

(MP

a)

Figure 3.7 / Bauschinger effect for shear - reverse shear test of anα-Fe polycrystal.

3.6 Summary and Conclusion

This paper has presented a computational study on the anisotropy effects induced

by strain path changes for BCC structured metals. For this purpose a composite

dislocation cell model, which describes the dislocation substructure evolution, has

been combined with a BCC crystal plasticity framework to bridge the dislocation cell

structure evolution and its macroscopic anisotropic effects. The BCC crystal plastic-

ity framework was based on Yalcinkaya et al. (2008) and the composite cell model

was built upon the contribution of Viatkina et al. (2003), who analyzed strain path

dependency phenomena in a phenomenological plasticity framework at small strains

for FCC structured materials.

The presented computational framework assumed a composite aggregate, in which

the material with a cell structure was considered to consist of two components: a soft

cell interior component and hard cell wall components. The constitutive response of

each component has been obtained from crystal plasticity simulations, while a set of

phenomenological evolution equations for the cell size, the wall thickness and the

dislocation density inside the walls captured the evolution of the microstructure.

The numerical examples of this work have revealed an adequate qualitative agree-

ment between the simulations and the experimental trends for strain path change

tests, i.e. a cross test and a Bauschinger test. Further quantitative analyses call for

more extensive and more qualitative experimental results to compare with.

The paper clearly forwards a number of original contributions:

• A phenomenological cell structure evolution model embedded into a crystal


in BCC metals

plasticity framework is well able to reproduce all essential characteristics of

strain path changes reported, consistently with experimental observations at

two scales.

• The model proposed allows to study the interaction between different sources

of anisotropy, where a clear example at the single crystal and polycrystal has

been given.

• The level at which the enrichment of the crystal plasticity model was made,

enables its use in more complex microstructures as e.g. multi-phase steels.

Chapter four

Deformation patterning driven by ratedependent non-convex strain gradient

plasticity1

Abstract / A rate dependent strain gradient plasticity framework for the descriptionof plastic slip patterning in a system with non-convex energetic hardening is presented.Both the displacement and the plastic slip fields are considered as primary variables.These fields are determined on a global level by solving simultaneously the linear mo-mentum balance and the slip evolution equation which is postulated in a thermodynam-ically consistent manner. The slip law differs from classical ones in the sense that it in-cludes a non-convex free energy term, which enables patterning of the deformation field.The formulation of the computational framework is at least partially dual to a Ginzburg-Landau type of phase field modeling approach. The essential difference resides in thefact that a strong coupling exists between the deformation and the evolution of the plas-tic slip, whereas in the phase field type models the governing fields are only weaklycoupled. The derivations and implementations are done in a transparent 1D setting,which allows for a thorough mechanistic understanding, not excluding its extension tomultidimensional cases.

4.1 Introduction

During forming processes most metals develop cellular dislocation structures due to

dislocation slip patterning from moderate strains onwards. Typical examples of dis-

location microstructures are dislocation cells and dislocation walls (see e.g. Young

et al. (1986), Mughrabi (1987), Yalcinkaya et al. (2009)). Patterning typically refers

to the self organization of dislocations with formation of regions of high dislocation

1This chapter is reproduced from Yalcinkaya et al. (2011a)

45

464 Deformation patterning driven by rate dependent non-convex strain

gradient plasticity

density (dislocation walls) which envelop areas of low dislocation density (disloca-

tion cell interiors), also to be regarded as domains of high plastic slip and low plastic

slip activity. Due to the induced macroscopic anisotropic effects (see e.g. Peeters

et al. (2000), Li et al. (2006), Wang et al. (2008), Yalcinkaya et al. (2009)) the occurrence

of dislocation microstructures and their evolution have been an interesting topic

for the materials science community for decades. Starting with the studies on the

cold-worked sub-structure of polycrystals using transmission electron microscopy

in 1960s (e.g. Bailey and Hirsch (1960), Keh et al. (1963) and Swann (1963)), a vast

amount of experimental results have been collected, and several promising theoret-

ical models were presented dealing with dislocation (or slip) patterning. Neverthe-

less, a complete descriptive understanding of the occurring phenomena has never

been reached and the necessary input for computational models is still subject of

ongoing discussions.

In the context of the computational modeling of plastic slip pattering (or dislocation

sub-structure formation), different approaches have been pursued in the literature

that can be categorized into three main groups: (i) models using directly the me-

chanics of single dislocations or populations of dislocations, (ii) phase field modeling

of dislocation patterning, (iii) the incremental variational formulation of inelasticity

which has the advantage of applying the concept of relaxation resulting in fine scale

microstructure evolution.

In the first group, scientists deal either with the problem at a discrete dislocation level

by solving a system with only a limited number of dislocations (e.g. Lubarda et al.

(1993), Groma and Pavley (1993), Kubin and Canova (1992)), or they approach the

problem from a continuum point of view by using homogenized variables like dis-

location densities and internal stress fields in a system of coupled balance equations.

One of the fundamental studies in this class was introduced by Kuhlmann-Wilsdorf

(see e.g. Kuhlmann-Wilsdorf and Van der Merwe (1982)) within the so-called low-

energy dislocation structure (LEDS) approach, seeking for dislocation configurations

of minimal free energy under given constraints. In this framework, pattern formation

is driven by the reduction of the system energy. Holt (1980) improved this static de-

scription by introducing dynamics into the model, which is based on a conservation

law for the dislocation density by setting up a relation for the evolution of the dis-

location density in terms of elastic energy changes related to the dislocation density

fluctuations. Another approach in this class is the reaction-diffusion model (e.g. Wal-

graef and Aifantis (1985) and Aifantis (1987)), where mobile and immobile disloca-

tions are distinguished and the dynamics of the system is governed by diffusion and

reaction terms. The competition between the mobility and the non-linear interactions

(creation, annihilation and pinning) causes the instability of uniform dislocation dis-

tributions versus inhomogeneous ones and leads to the formation and persistence of

dislocation patterns. Even though there have been some improvements on the men-

4.1 Introduction 47

tioned models, they generally rely on assumptions that are difficult to validate at the

discrete dislocation level.

In the second group, there are the phase field modeling approaches of dislocation

patterning, which offer a valuable alternative for discrete dislocation dynamics sim-

ulations. These models have the advantage of making less small scale assumptions

on the dislocation interactions, such as multiplication/annihilation of dislocations.

Instead, elastic interactions of numerously interacting individual segments of dislo-

cations are considered, dealing with only a few density functions (see Wang et al.

(2001a), Wang et al. (2001b) and Wang et al. (2001c)). However, these models require

the use of large computational grids and time integration over large numbers of time

steps. These difficulties are overcome by the model of Koslowski et al. (2002) pre-

senting an analytically tractable theory which determines the value of phase field at

point-obstacle sites without using any grid. Even though the framework is successful

in predicting dislocation (slip) patterns, it is rather complicated to incorporate into

a finite-deformation formulation of single-crystal elastic-plasticity, as needed in the

context of large-scale finite element calculations of macroscopic samples.

The third category of methods aims to capture the phenomenological evolution of

dislocation microstructures by modeling a physically deformed crystal in finite plas-

ticity frameworks, which are the most relevant ones considering the present paper.

Ortiz and Repettto (1999) regard the dislocations as manifestations of the incompat-

ibility of the plastic deformation gradient field. Within this framework the incre-

mental displacements of inelastic solids follow as minimizers of a suitably defined

pseudoelastic energy function. Miehe et al. (2004) propose an approach based on

finite-step-sized incremental energy minimization. The boundary value problems

were recast into a principle of minimum incremental energy for standard dissipa-

tive solids. The general concepts are applied to analyze evolving deformation mi-

crostructures in single-slip plasticity. The derivations of the incremental variational

formulation of inelasticity in both frameworks are conceptionally parallel to each

other. Hackl and Kochmann (2008) employ energy principles as well to analyze the

microstructure formation and evolution as a result of energy minimization and re-

laxation via lamination. The idea is that, for non-quasiconvex energy potentials the

minimizers are no longer continuous deformation fields but small-scale fluctuations

related to probability distributions of deformation gradients to be calculated via en-

ergy relaxation.

The objective of the present paper consists in bridging models of the last two cat-

egories described above. To achieve this, a continuum plastic slip field model in a

non-convex strain gradient plasticity framework is proposed to simultaneously pre-

dict microstructure formation and the overall macroscopic elastic-plastic response in

a computationally low cost setting. A viscous relaxation scheme in a system with en-


gradient plasticity

ergetic hardening is used, which makes the method comparable to non-local phase

field models (e.g. Ubachs et al. (2004)) as used in diffusional evolving microstruc-

tures. Using the adequate formulations for the physics, thermodynamics and kine-

matics, phase field models are able to predict the evolution of microstructures and

morphologies without explicit tracking of the positions of the phases and interfaces.

Inspired by their success, the present paper extends this concept to the formation

and evolution of plastic slip patterns based on a consistent derivation of the ther-

modynamic relations and evolution equations. Plastic slip patterns in metals are

typically linked to the evolution of dislocation sub-structures, i.e. dislocation cell

walls and cell interiors. The computational framework yields a stable algorithm for

which the finite element formulation and implementation has been performed in a

fictitious 1D setting in order to preserve simplicity and transparency, and to easily

interpret the results. Moreover, in this simple presentation the theory stays general

and depending on the parameters and their physical interpretation, the model can

potentially describe different dislocation slip microstructures (from Luders bands

to dislocation cells) at different length scales. The formulation can be extended to

2D or 3D, where the multi-slip character of crystals is often considered as a natural

source of non-convexity due to latent hardening mechanism. In line with models of

the third category, a double-well free energy function is assumed here, which makes

the approach rather phenomenological for the considered 1D case, whereas the non-

convex nature in a single slip deformation state becomes more natural through the

interaction of multiple slip systems (e.g. through latent hardening). In the model, the

plastic slip and the displacement are taken as degrees of freedoms. These fields are

obtained on a global level by solving simultaneously the linear momentum balance

and the slip evolution equation. The thermodynamically consistent slip law is the

crucial part of the model. At first glance, the slip law (see equation (4.18)) is similar

to the one encountered in a classical rate dependent crystal plasticity approach (e.g.

Hutchinson (1976), Peirce et al. (1982), Yalcinkaya et al. (2008)), however the stress

expression in the numerator differs. Three participating stress contributions can be

distinguished: (1) the conventional resolved stress directly related to the external

loading, (2) a surface-like stress depending on the gradient of plastic slip which is

characteristic for strain gradient crystal plasticity models (e.g. Evers et al. (2004b))

and (3) the stress that emanates from a non-convex free energy, which is omitted

in classical convex theories and which triggers the patterning of the dislocation slip

field.

The paper is organized as follows. First, in section 2, the incremental variational ap-

proach for microstructure evolution is discussed shortly including some comments

on its connection with the present framework. Then, in section 3, the thermodynam-

ical consistency of the constitutive model and the derivation of the equations to be

solved in a finite element context are studied. Next, in section 4, the finite element

4.2 Macroscopic view: material instability and microstructure evolution in

inelastic materials 49

formulation for the 1D strain gradient plasticity framework and the incorporation of

the non-convexity into the problem are summarized. Further, section 5 addresses the

link between the non-convexity and the resulting patterning in phase field models in

general and in the presented model in particular. In section 6 numerical examples are

presented in order to demonstrate the performance of the proposed model. Finally,

some concluding remarks are given in section 7.

4.2 Macroscopic view: material instability and microstructure

evolution in inelastic materials

The mechanical response of many engineering components is often influenced by

an existing or an emerging microstructure (martensite, dislocation sub-structures,

voids, shear bands, etc.). To account for the actually relevant microstructure, there

are different approaches to model the formation and the development of the mi-

crostructure as additional fields in the material. The difficulty in the solution of

these conventional (local) multi-field problems is the localization of the correspond-

ing field and local strain hardening-softening elastic-plastic behavior, which yields

numerical instabilities. The boundary value problem becomes ill-posed and in the

context of finite element formulations it gives mesh dependent results (e.g. de Borst

(1987)). Regarding localized failure mechanisms and softening problems, a broad

range of regularizing approaches have been developed as viscoplasticity, non-local

continuum theories and Cosserat theories. Additional to these classical techniques,

another recent approach in the context of emerging and evolving microstructures for-

mulates the inelastic boundary value problems by using direct methods of calculus

of variations in an incremental setting (e.g. Ortiz and Repettto (1999), Miehe (2002),

Miehe et al. (2002), Lambrecht et al. (2003), Miehe et al. (2004), Svendsen (2004)).

Variational calculus principles are well-understood in the theory of elasticity. Mate-

rial stability conditions, and well-posedness of the problem specifications have been

studied extensively. The conclusion is that for a globally stable material, the free

energy function should be convex. However, due to its strong restrictions weak con-

vexity concepts (polyconvexity, quasiconvexity and rank-one convexity) have been

used in stability analyses and fine scale microstructure evolution descriptions in the

material. If the elastic energy function is non-convex the minimization problem does

not give a solution in the classical sense and fine scale microstructures are gener-

ated. Boundary conditions play also an important role in this procedure and the

variational problem might have infinitely many, a unique or no solution at all.

The deformation theory of plasticity, where all material points are assumed to follow

certain optimal deformation paths, can be elaborated by pursuing the well-defined


gradient plasticity

variational principles of elasticity as well. However, the flow theory of plasticity

and many other inelastic material descriptions cannot be handled in the same way

due to the intrinsic path dependence. Therefore, incremental variational principles

have been applied to remedy the numerical stability issues and to model the forma-

tion and evolution of microstructures in dissipative scale-invariant inelastic materials

(e.g. Miehe (2002), Miehe et al. (2002), Lambrecht et al. (2003), Miehe et al. (2004)).

The problem can be formulated as,

W(εn+1) = infI

∫ tn+1

tn

[ψ+ D

]dt (4.1)

where D = D(I , I) is the dissipation with I standing for the set of internal variables,

ε is the strain,ψ is the free energy functional and W is the minimum energy at the so-

lution point. The minimization problem is solved at each time increment (tn+1 − tn)

allowing for path dependent problems (e.g. plasticity) revealing microstructures.

The local minimization problem becomes non-local or scale variant when a length

scale is incorporated by supplying an additional energy component in the equation

(4.1) reflecting the size effect (e.g. Conti and Ortiz (2005)). The mathematical method

for the solution of the above problem is well known under the name relaxation. This

method produces a new well-posed relaxed problem in the sense of existence of so-

lutions and implicitly allows for the formation of microstructures. The relaxation is

associated with quasi-convexification of the non-convex function W.

The main disadvantages of the incremental variational formulation are its high com-

putational cost and the fact that it allows one to obtain only equilibrium states of

microstructures. Computational cost can be reduced by implementing analytical re-

laxation solutions to the problems (e.g. Conti et al. (2007)) which might be limited

for realistic calculations. The issue of tracking the actual non-equilibrium evolution

of microstructures is the aim of the framework that is presented in the following

sections. The viscous nature of the model allows the prediction of non-equilibrium

states of microstructure evolution depending on the rate of deformation.

4.3 Thermodynamics of strain gradient plasticity

In this section, the thermodynamical consistency and the derivation of the governing

system equations are discussed shortly. For conceptual simplicity, all formulations

are casted in a 1D setting (where a coordinate x is used to indicate the position), not

limiting their extension to 2D or 3D. In the geometrically linear small strain context

the time dependent displacement field is denoted by u = u(x, t), the strain is given

by ε = ∂u/∂x and the velocity is v = u. The strain is assumed to be decomposed

4.3 Thermodynamics of strain gradient plasticity 51

additively,

ε = εe +εp (4.2)

into an elastic part εe and a plastic part εp. In the single slip 1D case considered here,

the total amount of plastic slip γ and the plastic strain are identical. Note that in

general the plastic slip may be derived from a collection of internal variables, even

for a 1D problem (in particular if multiple slip systems are present). The free energy

ψ is assumed to be a function of the state variables,

state = εe,γ,∇γ (4.3)

where ∇γ = ∂γ/∂x is the gradient of the plastic slip. In the multi slip crystal plas-

ticity context the definition of plastic slip gradients is not trivial, and Cermelli and

Gurtin (2001) summarize the different options and requirements for such a formu-

lation. Following the arguments of Gurtin (e.g. Gurtin (2000), Gurtin (2002)), the

power expended by each independent rate-like kinematical descriptor is expressible

in terms of an associated force system consistent with its own balance. Yet, the basic

kinematical rate variables, namely εe, u and γ are not independent. It is therefore not

apparent what forms the associated force balances should take, and, for that reason,

these balances are established using the principle of virtual power.

Assuming that at a fixed time the fields u, εe and γ are known, we consider u, εe and

γ as virtual rates which are collected in the generalized virtual velocity V = (u, εe, γ).

The force systems are characterized through their work-conjugated nature with re-

spect to the state variables. Pext is the power expended on the domain P and Pint a

concomitant expenditure of power within P,

Pext(P, V) =∫

∂P t(n)udS +∫

∂P χ(n)γdS

Pint(P, V) =∫

PσεedV +

∫

P πγdV +∫

Pξ∇γdV(4.4)

whereσ , π and ξ are the thermodynamical forces conjugate to the internal state vari-

ables εe, γ and ∇γ respectively. In Pext, t(n) is the macroscopic surface traction while

χ represents the microscopic surface traction conjugate to γ at the boundary ∂P with

n indicating the normal direction.

Postulation of the principle of virtual power states that given any generalized virtual

velocity V the corresponding internal and external powers are balanced,

Pext(P, V) = Pint(P, V) (4.5)


gradient plasticity

Considering a generalized virtual velocity without slip, the virtual velocity field can

be chosen arbitrarily and this leads to the classical macroscopic force balance

∂σ∂x

= 0 (4.6)

and considering the virtual slip field arbitrarily without generalized virtual velocity

leads to the microscopic force balance,

∂ξ∂x

+σ − π = 0 (4.7)

In Gurtin (2000), Gurtin (2002), Gurtin et al. (2007) the previous derivations are fully

detailed in a generalized 3D setting while Del Piero (2009) presents an alternative

way to derive similar balance equations.

The local internal power expression can simply be written as,

Pi = σεe + πγ+ξ∇γ (4.8)

Hence, the local dissipation inequality can be expressed as

D = Pi − ψ ≥ 0 or D = σεe + πγ+ξ∇γ− ψ ≥ 0 (4.9)

The free energy is assumed to take the following form,

ψ = ψe +ψγ +ψ∇γ (4.10)

where ψe is the elastically stored energy, ψγ is the microstructurally stored energy

due to plastic slip andψ∇γ is the energy associated to the gradients of slip.

By exploiting (4.10), the dissipation (4.9) can be rewritten as,

D = σεe + πγ+ξ∇γ − ∂ψ∂εeεe − ∂ψ

∂γγ− ∂ψ

∂∇γ∇γ

=

(

σ − dψe

dεe

)

︸︷︷︸

0

εe +

(

π − dψγdγ

)

γ +

(

ξ − dψ∇γd∇γ

)

︸︷︷︸

0

∇γ ≥ 0 (4.11)

Additional to the stress σ , the microforce ξ conjugate to the slip gradient ∇γ is also

assumed to be energetic:

σ =dψe

dεe

ξ =dψ∇γd∇γ

(4.12)

The remaining term in the dissipation expression (4.11) reads,

4.3 Thermodynamics of strain gradient plasticity 53

D =

(

π − dψγdγ

)

︸︷︷︸

σdis

γ ≥ 0 (4.13)

where the term conjugated to the slip rate is identified as the dissipative stress σdis,

σdis = π − dψγdγ

(4.14)

The following constitutive equation which satisfies the inequality (4.13) is next pro-

posed,

σdis = sign(γ)ϕ (4.15)

whereϕ represents the actual slip resistance,

ϕ = s

( |γ|γ0

)m

(4.16)

In here, γ0 and m are the reference slip rate and the rate sensitivity exponent re-

spectively. Furthermore, s is the resistance to dislocation slip which is assumed to

be constant for simplicity. Substitution of equation (4.16) into equation (4.15) and

extracting the slip rate yields the slip law

γ = γ0

( |σdis|s

) 1m

sign(σdis) (4.17)

As a result, a thermodynamically consistent constitutive relation for the slip evo-

lution is obtained. By using the definition of σdis in equation (4.14) and using the

microforce balance according to equation (4.7), equation (4.17) can be written as,

γ = γ0

| ∂ξ∂x

+σ︸︷︷︸

π

−dψγdγ

|

s

1m

sign(π − dψγdγ

) (4.18)

Each of the stress contributions in the slip law is derived from the free energy, i.e.

σ = ∂ψ/∂εe, ξ = ∂ψ/∂∇γ and any change in the free energy directly affects the slip

law.


gradient plasticity

4.4 Particular choices of free energy functions

4.4.1 Slip based strain gradient plasticity

In this section the algorithm to solve equations (4.6) and (4.18) as a rate dependent

(convex) strain gradient plasticity problem over a domain 0 < x < L is outlined.

Treating this case provides a natural setting for the extension to the non-convex case

and its proper interpretation. The displacement field u and the plastic slip field γ

are selected as primary variables, which leads to the simultaneous solution of the

balance of linear momentum and the evolution of slip.

A convex free energy function,

ψ = ψe +ψ∇γ =1

2Eεe2 +

1

2A(∇γ)2 (4.19)

leads to classical strain gradient crystal plasticity frameworks, where E is Young’s

modulus and A (which includes an internal length parameter) is governing the ef-

fect of the slip gradient contribution to the internal stress field. Its expression in 3D

would be A = ER2/(16(1 − ν2)) as e.g. used in Evers et al. (2004b), Bayley et al.

(2006), Geers et al. (2007). In this expression R physically represents the radius of the

dislocation domain contributing to the internal stress field, ν is Poisson’s ratio. Note

thatψ in equation (4.19) does not include theψγ term. Adding a convex energy term

ψγ would result in a stress-like expression similar to the a-thermal slip resistance in

the numerator of the slip law (4.18) as used in some BCC crystal plasticity frame-

works (e.g. Yalcinkaya et al. (2008)). The governing system of equations is given by

the strong form of the linear momentum balance (equation (4.6)) and the plastic slip

evolution (equation (4.17)),

∂σ∂x

= 0

γ − γ0

( |σdis|s

) 1m

sign(σdis) = 0

(4.20)

For further elaborations it is more convenient to write the plastic slip evolution in the

following format,

|γ|msign(γ)− γm0

sσdis = 0 (4.21)

Implicit time integration gives,

∣∣∣∣

γ − γn

∆t

∣∣∣∣

m

sign

(γ − γn

∆t

)

− γm0

sσdis = 0 (4.22)

4.4 Particular choices of free energy functions 55

with ∆t representing the time step and γn is the plastic slip at the end of the previous

time step.

The dissipative stress in equation (4.22) can be written as

σdis = σ +∂ξ∂x

= σ +∂(∂ψ/∂∇γ)

∂x= σ + A

∂2γ

∂x2(4.23)

Substituting equation (4.23) into equation (4.22) gives,

∣∣∣∣

γ − γn

∆t

∣∣∣∣

m

sign

(γ − γn

∆t

)

− γm0

s

(

σ + A∂2γ

∂x2

)

= 0 (4.24)

The weak forms of the equations are obtained in a standard manner, using a Galerkin

procedure. Both the balance of linear momentum and the slip equation are tested

with virtual displacement δu and virtual slip δγ fields, respectively, and integrated

over the domain 0 < x < L. The obtained variational Galerkin functionals are solved

in a fully coupled manner (monolithic) by means of a Newton-Raphson scheme after

linearization and discretization procedures as outlined in the appendix section. Lin-

ear interpolation functions for the slip field γ and quadratic interpolation functions

for the displacement field u are used. The resulting system of equations is solved

for the increments of the displacement ∆u and the plastic slip ∆γ. The formulation is

numerically implemented and solutions for the nodal displacements and plastic slips

are obtained in a standard incremental iterative manner. Examples are presented in

section 4.6.

4.4.2 Slip based non-convex strain gradient plasticity

The non-convex case is recovered by adding a non-convex contribution to the free

energy (4.19). The additional term is a polynomial function of the plastic slip

ψ = ψe +ψ∇γ +ψγ

=1

2Eεe2 +

1

2A(∇γ)2 + (C1γ

4 + C2γ3 + C3γ

2 + C4γ+ C5)(4.25)

where non-convexity is obtained by specific values of the polynomial coefficients.

The above procedure is similar to the inclusion of non-convexity in the configura-

tional free energy of phase field models which have been studied extensively in the

literature. In many of the cases the non-convexity is approximated by a double well

potential (capturing the patterned field) where the wells correspond to the states

governing the minimum energy configuration of the mixture.

Considering the 3D evolution of the dislocation microstructures (e.g. labyrinth, mo-

saic, fence and carpet structures) driven by the imposed deformation, Ortiz and


gradient plasticity

Repettto (1999) present a comprehensive summary on the relation between the non-

convexity of the energy function and the dislocation microstructure evolution. In

crystals exhibiting latent hardening, the energy function is non-convex and has wells

corresponding to single-slip deformations. This favors microstructures consisting lo-

cally of single slip. Looking from thermodynamics perspective Kuhlmann-Wilsdorf

LEDS (low energy dislocation structure) theory (e.g. Kuhlmann-Wilsdorf (2001))

completes the explanation by stating that among all potentially accessible disloca-

tion structures, plastic deformation will generate the one with the lowest free energy.

This means that, limited only by dislocation mobility, availability of slip system and

insignificant entropy, dislocation structures always approach the lowest possible me-

chanical energy of the present dislocation population.

The present study clearly reveals the intrinsic role of the non-convexity in a 1D prob-

lem, even though it is a simplification of the underlying (more complex) 3D reality.

The non-convex energy which is coming from the accumulation of trapped disloca-

tions is introduced phenomenologically in terms of a dislocation slip potential rep-

resenting the microstructurally trapped energy. It is introduced in terms of plastic

slip representing the dislocation movement looking for the minimum energy config-

uration (in analogy to patterning in phase field models). The convex gradient term

represents the surface energy (penalizing spatial transitions from low to high values

of slip), which in fact regularizes the problem in the mathematical sense. Depending

on the value of the R (hidden in parameter A standing in front of the gradient term),

the presented theory is able to explain the formation and evolution of microstruc-

tures at different length scales.

The non-convex function does not have to be a polynomial one. The free energy func-

tion adopted is just a simple mathematical representation for a double-well function

motivated from phase field models. The parameters C1, C2 and C3 are chosen in a

way that they introduce a small modulation from a convex plastic slip potential (see

Fig. 4.3 (a)). Their exact values are therefore not essential if the non-convex modu-

lation is superimposed on a free energy function used for classical hardening laws.

It is clear that if a convex plastic slip potential enters equation (19) it would result in

only hardening (no softening branch) behavior and a homogeneous (constant) dis-

tribution of the plastic slip would be obtained.

The strong form of the system description according to the equations (4.20) remains

identical. The dissipative stress term in equation (4.23) can be rewritten as,

σdis = σ +∂ξ∂x

− ∂ψ∂γ

= σ + A∂2γ

∂x2− (4C1γ

3 + 3C2γ2 + 2C3γ+ C4)

(4.26)

4.5 Non-convexity and patterning in phase field modeling 57

Substituting equation (4.26) into equation (4.22) gives,∣∣∣∣

γ− γn

∆t

∣∣∣∣

m

sign

(γ− γn

∆t

)

− γm0

s

(

σ + A∂2γ

∂x2− (4C1γ

3 + 3C2γ2 + 2C3γ + C4)

)

= 0

(4.27)

The same procedure as in the previous section is followed to obtain the set of equa-

tions to be solved for the displacement and plastic slip increments (see the Appendix

for details).

4.5 Non-convexity and patterning in phase field modeling

In phase field models (e.g. Ubachs et al. (2004), Kuhl and Schmid (2007)), the con-

figurational energy of a two-phase material, is often represented by a (non-convex)

double-well potential (Fig. 4.1). It can be constructed from the configurational free

energy of the individual phases by assuming that at a certain composition only the

phase with the lowest energy will exist. Equilibrium is reached when the chemical

potential becomes homogeneous throughout the system. The configuration with the

lowest possible free energy is found when, due to phase separation, each material

point has the composition of either of the two phases, corresponding with the bin-

odal points. In the context of plastic slip patterning there are no physically distinct

phases but rather distinct regions in which the slip is either high (large dislocation

density) or low (small dislocation density). Two other points which are typically of

Figure 4.1 / Free energy curves of a two-phase material, and binodal and spinodalpoints.

interest are the so-called spinodal points. These points mathematically reflect the


gradient plasticity

state (e.g. composition) for which the second derivative of the configurational free

energy becomes zero. At these points the sign of the curvature (∂2ψc/∂c2) changes.

In the region where the curvature is positive the system is stable with respect to

small fluctuations in composition. However in the regions where the curvature is

negative (between the spinodal points) the system is locally unstable with respect to

small fluctuations and for that reason phase separation occurs (see e.g. Cahn (1961),

Langer (1971), Nauman and He (2001)). For the case considered here, this issue is

easily shown on the basis of a stability analysis of the steady state solution of the

plastic slip distribution, implying that γ = 0,

σdis = σ + A∂2γ

∂x2− ∂ψ

∂γ= 0 (4.28)

which can be considered as a nonlinear equation in terms of γ. A spatial wave per-

turbation δγ is applied to the γ distribution γ = γ0 + δγ = γ0 + gei(kx−ωt) with g the

complex amplitude, k the wave number andω the frequency. Linearization of (4.28)

around γ0 yields the behavior in terms of the perturbation

A∂2δγ

∂x2− ∂2ψ

∂γ2

∣∣∣γ=γ0

δγ = 0 (4.29)

Substitution of the spatial perturbation results in,

[

Ak2 +∂2ψ

∂γ2

∣∣∣γ=γ0

]

δγ = 0 (4.30)

Real values for the wave number k are found if

∂2ψ

∂γ2

∣∣∣γ=γ0

< 0 (4.31)

Indicating that physical solutions for such spatially non-homogenous perturbations

exist (corresponding to patterning of the plastic slip). For a double-well function

according to ψγ = C1γ4 + C2γ

3 + C3γ2 + C4γ + C5, the spinodal points are the roots

of

6C1γ2 + 3C2γ+ C3 = 0 (4.32)

and between these two spinodal points the system is unstable and any given pertur-

bation will give rise to patterning of the plastic slip.

4.6 Numerical examples

In this section, three different numerical cases of a fictitious 1D bar with length L

under tension are presented to study the behavior of the discussed strain gradient


models. The first example deals with a conventional convex free energy with slip

gradients (see section 4.4), where the influence of the internal length parameter de-

termining the value of the constant A (see equation (4.19)) is elucidated. The second

example studies the absence, onset and evolution of patterning of the plastic slip, de-

parting from a homogeneous distribution, depending on the deformation rate during

monotonic loading of a bar. The homogeneous (non-patterned) case is also compared

with the analytical solution. The third example studies plastic slip patterning in a re-

laxation test where the 1D bar is deformed to a certain state mapped between the

spinodal points. Constraining the deformation at this point leads to viscous relax-

ation of the microstructure evolution. The effect of the boundary conditions on the

slip patterning is analyzed in each example by considering hard (γ = 0) and soft

(∂γ/∂x = 0) cases.

4.6.1 Numerical example 1: convex case - monotonic loading

In this example, the influence of the internal length parameter R present in the ma-

terial constant A is briefly addressed, in terms of the plastic slip distribution and the

stress strain behavior. The (convex) strain gradient plasticity framework of section

4.4.1 is used to this purpose. The displacement at x = 0 is suppressed while displace-

ment at x = L is prescribed such that the average strain increases incrementally up

to 0.05. Hard boundary conditions (γ = 0) are applied at both ends of the bar. The

material parameters are: Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0.33,

slip resistance s = 15 MPa, reference slip rate γ0 = 5 × 10−2 s−1, the rate sensitivity

exponent m = 1 and the length of the bar is L = 1 mm. The bar is discretized into

100 finite elements and deformed with a rate of ε = 5 × 10−2 s−1. It is important

to note that, the material parameters do not represent a certain material, and taking

m=1 in the numerical examples may seem too simple at first sight. This choice was

of course made for simplicity only, yet with a large similarity to discrete dislocation

studies using linear drag relations. Simulations with other values of m do not change

the qualitative nature of the examples, yet they will affect the rate dependent (time

dependent) behavior. Considering the parameter s, it is remarked that its constant

value does result in redundancy, however, the introduction makes the stress term in

the slip evolution equation (4.22) dimensionless.

As pointed out in various strain gradient crystal plasticity models (e.g. Gurtin (2002),

Svendsen (2002), Bardella (2006), Bardella (2007), Bayley et al. (2006)), the effect of

the internal length scale enters the formulations via the (internal) back stress. In the

current framework, the general expression A∂2γ/∂x2 in equation (4.23) gives rise to

an internal back stress as occurring in a microstructure with dislocations.

The results presented in Fig. 4.2 show accordance with the solutions from the litera-


gradient plasticity

0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

35

40

Strain

Str

ess

(MP

a)

R = 0.01 mmR = 0.02 mmR = 0.05 mm

Figure 4.2 / 1D bar in tension with hard slip boundaries.

ture studying energetic size effects (e.g. Shu et al. (2001), Evers et al. (2004a), Dunstan

and Bushby (2004)). Because of the constrained slip at each end of the bar, plastic slip

gradients develop, resulting in inhomogeneous deformation with the occurrence of

boundary layers. The boundary layer thickness is typically increasing with R. As ex-

pected, Fig. 4.2 also reflects the size dependence of the stress vs. strain curve which

is consistent with the literature on this aspect.

4.6.2 Numerical example 2: non-convex case - monotonic loading

Theoretical homogenous solution

In this example we investigate homogeneous deformation (i.e. without patterning)

for the non-convex strain gradient plasticity framework and compare the results with

the analytical solution. The non-convexity is introduced through a small modulation

on top of a convex energy function (representing a classical hardening behavior in a

qualitative sense). The effect of this small non-convex modulation is discussed fur-

ther on. The rate dependent character of the model renders the possibility to stabilize

homogeneous deformation states at high rates of loading since microstructures do

not have enough time to evolve. A 1D bar is constrained at the ends (suppressed dis-

placement at x = 0 and prescribed displacement at x = L) where soft boundary con-

ditions (∂γ/∂x = 0) are applied. The results of the finite element computation will be

compared with the analytical solution by solving the slip equation without the non-

local effects in the steady state limit for a set of applied strain values. The material

parameters are: Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0.33, slip resis-

tance s = 35 MPa, reference slip rate γ0 = 5 s−1, internal length parameter R = 0.1


mm, the rate sensitivity exponent m = 1 and the length of the bar is L = 1 mm. The

overall deformation rate used in this calculation is ε = 2 s−1. This rather high rate

prevents microstructures from evolving and thereby favors a merely homogenous

solution even for the non-convex case considered here. The plastic slip dependent

free energy ψγ is taken, in analogy with phase field approaches, as a double-well

non-convex potential ψγ = 1.525 × 108γ4 − 5.2 × 106γ3 + 5 × 104γ2 MPa. Note,

however that with respect to this item there is insufficient experimental data to spec-

ify the precise form and nature of this non-convex contribution. The spinodal points

are identified by ∂2ψ/∂γ2 = 0 yielding γsp1 = 0.0042 and γsp2 = 0.0131. The binodal

points are obtained by extracting the points where the tangent line touches the free

energy curve, γbp1 = 0.001 and γbp2 = 0.0163 (see Fig. 4.3 (a)).

(a) Plastic slip dependent free energy.

0 0.005 0.01 0.015 0.020

50

100

150

200

250

300

350

Strain

Str

ess

(MP

a)

(b) Stress vs. strain.

Figure 4.3 / (a) Applied plastic slip dependent part of the free energy density func-tion for the non-convex case (solid line) with spinodal (stars) and binodal (polygons)points and for the convex case (dotted line). (b) Stress vs. strain response for a ho-mogeneous deformation (solid line: non-convex case, dotted line: convex case).

For the given free energy (ψγ shown in Fig. 4.3 (a)) and the other selected parameters

a homogeneous plastic slip distribution all along the bar is recovered. The resulting

stress vs. strain response for this particular case is shown in Fig. 4.3 (b). In Fig. 4.3

the effect of the convex plastic slip free energy density is also presented with material

parameters: C1 = 0.72 × 108 MPa, C2 = −2× 106 MPa and C3 = 2.1 × 104 MPa. This

set of parameters leads to a small change in the free energy curve, however results

in a significantly different behavior in the stress-strain response, i.e. the convex case

does not show a softening branch. The peculiar behavior described by Fig. 4.3 (b)

at this scale can be related to Luders bands and the Portevin-Le Chatelier effect (e.g.

Sun et al. (2003) and Halim et al. (2007)). However, note that this example represents


gradient plasticity

0 0.005 0.01 0.015 0.020

0.5

1

1.5

2

2.5

Strain

Fre

e en

ergy

den

sity

(M

Pa)

TotalPlasticElastic

Figure 4.4 / Energy plots of the homogeneous case (analytical solution).

the idealized homogeneous plastic slip distribution case which is not easy to obtain

in experiments due to the fact that plastic deformation always imposes evolution of

dislocation slip microstructures. Therefore, stress vs. strain curves generally include

a plateau corresponding to microstructure evolution (see following examples).

The analytical solution for this case is obtained assuming a steady state, γ = 0 reveal-

ing σdis = 0 and using the homogeneity condition A∂2γ/∂x2 = 0. The slip equation

then simplifies to:

E(ε− γ)− 4C1γ3 − 3C2γ

2 − 2C3γ = 0 (4.33)

Equation (4.33) is solved analytically for a set of strain values ranging from 0 to 0.02

which is a different way to recover the homogeneous solution. This solution is equal

to the homogeneous continuous loading case if the strain rates are not very high

(viscosity effects are minor). The contributing terms to the free energy densities are

plotted with respect to strain in Fig. 4.4. The obtained stress vs. strain response is

identical with the one from the finite element calculation (see Fig. 4.3 (b)). Within

the applied strain range, the total free energy also shows a double-well non-convex

behavior, which is an important prerequisite for patterning of the plastic slip.

Patterned solution and rate dependency

In this example, the rate dependent evolution of the plastic slip patterns during

monotonic loading is dealt with. The geometry and material parameters are iden-

tical to the ones in the previous example. First, results of hard and soft boundary

conditions are respectively presented in Figs. 4.5 and 4.6 at a strain rate of ε = 0.02

s−1 which is low compared to the value used in the previous subsection. The effect


of the loading rate on the stress vs. strain response and slip patterning is shown in

Figs. 4.7 and 4.8.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

400

Str

ess

(MP

a)

Strain

Patterned solutionHomogenous solution

(a) Stress vs. global strain. (b) Plastic slip evolution.

Figure 4.5 / Stress vs. strain response and plastic slip evolution (legend presentingthe global strain) for a low monotonic loading rate with hard boundary conditions.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

Strain

Str

ess

(MP

a)

Patterned solutionHomogenous solution

(a) Stress vs. global strain. (b) Plastic slip evolution.

Figure 4.6 / Stress vs. strain response and plastic slip evolution (legend presentingthe global strain) for a low monotonic loading rate with soft boundary conditions.

In Figs. 4.5 (a) and 4.6 (a), the stress vs. strain curve is presented for the patterned so-

lution together with the corresponding homogeneous solution. The stress response


gradient plasticity

shows a typical plateau for the values of strain where plastic slip patterning is ob-

served. Due to the low values of the elastic strain the total strain in the stress vs.

strain curves can roughly be linked to the plastic slip values. The plastic slip lim-

its of the plateau regime in the stress vs. strain curves coincide with the spinodal

points in the free energy and the distribution (decomposition) of the plastic slip dur-

ing the inhomogeneous plastic slip evolution matches with the binodal points (see

Fig. 4.3). The constant stress plateau corresponds to the convexified solution of the

non-convex stress potential, where a linear convex envelope yields constant stress

values (see e.g. Lambrecht et al. (2003) and Miehe et al. (2004)).

The patterned solutions in 4.5 (a) and 4.6 (a) converge in the rate independent limit

to Maxwell-lines which can be associated with a global convexification of rate in-

dependent problems, such as discussed in Lambrecht et al. (2003). However, while

a convexification approach is able to resolve full Maxwell-lines similar to classical

treatments in phase-decompositions of real gases, the presented non-convex gradient

theory catches the peaks which can be directly related to lower and upper yield phe-

nomena observed experimentally during the formation of microstructures in metals,

as in the case of Luders band formation and movement (e.g. Hahner (1994), Sun et al.

(2003), Halim et al. (2007), Yoshida et al. (2008)), considering the length scale in the

presented examples. The upper yield point (the peak in the stress-strain curve) can

be related to the formation of a microstructure (e.g. as an appearing Luders band)

and the plateau can be linked to evolution or the movement of the band. However,

the relaxed stress obtained in the convexification approach represents only the per-

fectly plastic response.

Regarding the distribution of the plastic slip, in case of both hard and soft bound-

ary conditions at low and at high strain levels (outside the binodal region) the same

stable behavior as in the case of convex strain gradient plasticity (see example 1) is

logically recovered. Hard boundary conditions typically trigger slip gradients at the

ends of the bar, resulting in characteristic boundary layers. Soft boundaries on the

other hand induce an initially homogeneous distribution of plastic slip. At strain

levels corresponding to the levels where the stress plateau is observed (between the

spinodal points) pronounced slip patterns develop depending on the boundary con-

ditions (see Fig. 4.5 (b) and 4.6 (b)). This is consistent with the expectations based on

phase field modeling on the one hand, and the incremental minimization procedure

on the other hand. What is new with respect to the latter, is the fact that transitory

regimes are obviously captured as well, highlighting the role of the rate dependent

character of the model and the spinodal characteristic of the free energy. The stress

drop in Fig. 4.6 (a) is essentially due to the sudden (rate dependent) loss of stability

in the spinodal regime. This drop is sharp for the considered idealized case, and it

is not expected to occur if a more gradual loss of stability is invoked through the

intrinsic statistically non-homogeneous nature of the material.


The analysis in the present paper concerns single laminates only. Multiple lami-

nates can be observed as in the energy-convexification methods, e.g. Lambrecht

et al. (2003) if a spatial fluctuation or random distribution is given to the material

parameters in the model. Departing from such fluctuations the obtained laminates

would coarsen in time, depending on the stabilizing gradient term in the free en-

ergy. Changes of the surface (gradient) term or introduction of higher gradient terms

(as in Cahn-Hilliard models) could give different response to a random fluctuations,

i.e. laminates may become more pronounced and/or stabilize at a particular size,

depending on the stabilizing (gradient) term in the slip evolution equation.

0 0.005 0.01 0.015 0.02 0.0250

500

1000

1500

2000

2500

3000

3500

4000

Strain

Str

ess

(MP

a)

(a) ε = 2000 s−1.

0 0.005 0.01 0.015 0.02 0.0250

500

1000

1500

Strain

Str

ess

(MP

a)

(b) ε = 200 s−1.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

400

450

SrainS

tres

s (M

Pa)

(c) ε = 20 s−1.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

400

Strain

Str

ess

(MP

a)

(d) ε = 2 s−1.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

Strain

Str

ess

(MP

a)

(e) ε = 0.2 s−1.

0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

350

Strain

Str

ess

(MP

a)

(f) ε = 0.02 s−1.

Figure 4.7 / Rate dependent stress vs. global strain response.

The examples in Figs. 4.7 and 4.8 deal with the dependence of stress vs. strain re-

sponse and the microstructure (plastic slip) evolution on the loading rate under soft

boundary conditions. A high loading rate (i.e. 2 s−1) tends to inhibit the microstruc-

ture evolution leading to a homogenous distribution of plastic slip. The stress vs.

strain response corresponds to the steady state analytical solution presented in sec-

tion 4.6.2. Higher loading rates (i.e. 20 s−1, 200 s−1, 2000 s−1) result in a homoge-

neous plastic slip distribution with a more stiff stress vs. strain response, approach-

ing the elastic limit behavior. Lower loading rates (i.e. 0.2 s−1, 0.02 s−1) typically

result in patterned plastic slip distributions, which implies that the patterns evolve

only close to the rate independent limit. The slip pattern development as a function


gradient plasticity

of the deformation rate is shown in Fig. 4.8. The deformation is initially homoge-

neous and the slip patterns after passing the first spinodal point. This slip pattern-

ing evolves further with time and finally vanishes after passing the second spinodal

point. The transition from the homogeneous to the inhomogeneous slip distribution

is clearly rate dependent. For low loading rates (Fig. 4.8(b)), the transition from

the homogeneous to the patterned state (after the first spinodal point) and from the

patterned to the homogeneous state (after the second spinodal point) appears con-

siderably faster compared to higher loading rates (Fig. 4.8(a)). This is most obvious

at an overall displacement of 7.2µm and a displacement equal to 14.8µm, where the

different loading rate cases clearly reveal different regimes.

0.2 0.4 0.6 0.8 10

u = 0 [t=0]

u = 4 m [t=0.02 s]μ

u = 7.2 0.036 s]μm [t=

u = 7.8 0.0392 s]μm [t=

u = 14.8 m [t=0.074 s]μ

u = 13.6 m [t=0.068 s]μ

u = 20 m [t=0.1 s]μ

(a) γ evolution at ε = 0.2 s−1.

0.2 0.4 0.6 0.8 10

u = 0 [t=0]

u = 4 m [t=0.2 s]μ

u = 7.2 0.36 s]μm [t=

u = 7.8 m [t=0.392 s]μ

u = 13.6 m [t=0.68 s]μ

u = 14.8 m [t=0.74 s]μ

u = 20 m [t=1 s]μ

(b) γ evolution at ε = 0.02 s−1.

Figure 4.8 / Rate dependent microstructure evolution for the same imposed dis-placement.

4.6.3 Numerical example 3: non-convex stress relaxation of a 1D bar

In the final example, we illustrate the evolution of slip patterning in a stress relax-

ation test, again including both hard and soft boundary conditions. The displace-

ment at x = 0 is suppressed and a displacement at x = L is applied with a rate of

ε = 0.02 s−1 until the homogeneous plastic slip reaches a value between the spinodal

points. Next, the displacement at x = L is kept constant, leading to relaxation of

the stress in the bar. In this way, one can observe the evolution of the plastic slip

microstructure at a constant (macroscopic) average strain level, accompanied by re-

laxation of the stress.

The material parameters and the plastic slip dependent non-convex free energy are


(a) Stress vs. global strain.

0.20 0.4 0.6 0.20.8 1

(b) Plastic slip evolution.

Figure 4.9 / Stress relaxation test: stress vs. strain response and plastic slip evolutionwith hard boundary conditions.

(a) Stress vs. global strain.

0.20 0.4 0.6 0.8 1

(b) Plastic slip evolution.

Figure 4.10 / Stress relaxation test: stress vs. strain response and plastic slip evolu-tion with soft boundary conditions.

the same as used in section 4.6.2. The relaxation starts immediately after the displace-

ment yielding an average overall strain equal to 0.0049 has been reached for the hard

boundary case and 0.0054 for the soft boundary case. The stress vs. strain response

and the evolution of the plastic slip are presented for hard and soft boundaries in

Figs. 4.9 and 4.10, respectively. Note that the soft boundary conditions lead to an ini-

tially homogeneous distribution, for which any perturbation may trigger patterning.

In order to stabilize this a small spatial fluctuation is applied to the Young’s modu-

lus E along the bar, which restores uniqueness and triggers a stable evolution of the


gradient plasticity

microstructure. The use of a small initial defect or a perturbation is also commonly

done within the analysis of non-local damage models in 1D systems.

In both relaxation and monotonic tension tests with soft boundaries the full binodal

decomposition of the plastic slip is observed, because of the fact that the slip is not

constrained contrary to the hard boundary case where the full decomposition of the

field is naturally prohibited.

4.7 Conclusion

Inspired by the efficiency of phase field models for microstructure formation and

evolution, we developed a non-convex strain gradient plasticity model in a sys-

tem with energetic hardening-softening, which takes a conceptually dual structure

to Ginzburg-Landau type of phase field models with high and low slipped regions

representing different phases. Focusing on the basics and simplicity, the derivation

and the implementation are conducted in a 1D setting in order to illustrate the ability

of the model to capture the patterning of plastic slip similar to a phase decomposi-

tion mechanism. The destabilizing non-convex term in the free energy is stabilized

through the gradient term in the free energy and the viscous nature of the thermo-

dynamically consistent slip law. Note that the resulting model typically yields a rate

dependent microstructure evolution. The framework can capture both homogeneous

and inhomogeneous deformations depending on the rate of the applied deformation.

The model is conceptually capable of covering the entire microstructure evolution

process, depending on the externally applied load and boundary conditions. Non-

equilibrium microstructures are thereby well at reach.

The non-convexity in the presented model is incorporated by a double-well function,

not addressing particular materials yet. We present a generic formulation in order to

demonstrate the microstructure evolution in a thermodynamical and mathematical

rigorous setting. A more practical emphasis applied to a particular material is ob-

viously needed in future work, provided reliable experimental data to recover the

non-convex term are available.

4.8 Appendix

4.8.1 Finite element implementation of slip based strain gradient plasticity

The weak forms of the equations are obtained in a standard manner, using a Galerkin

procedure. Firstly, the balance of linear momentum is tested with a field of virtual

4.8 Appendix 69

displacements δu and integrated over the domain 0 < x < L, which results in Gu,

Gu =∫ L

0δu

∂σ∂x

dx = 0 (4.34)

The slip equation (4.24) is tested with a field of virtual slips δγ and integrated towards

Gγ

Gγ =∫ L

0

[

δγ

∣∣∣∣

γ− γn

∆t

∣∣∣∣

m

sign

(γ− γn

∆t

)

− δγγm

0

sσ − δγ

γm0

sA

∂2γ

∂x2

]

dx = 0 (4.35)

Using integration by parts equation (4.34) can be written as

Gu = −∫ L

0

dδu

dxσ dx + [σδu]|L0 = 0 (4.36)

Similarly, equation (4.35) can be rewritten as

Gγ =∫ L

0

[

δγ

∣∣∣∣

γ − γn

∆t

∣∣∣∣

m

sign

(γ − γn

∆t

)

− δγCaσ + Cbd(δγ)

dx

∂γ∂x

]

dx

− Cb[δγ∂γ∂x

]|L0 = 0

(4.37)

with the abbreviations Ca = γm0 /s and Cb = γm

0 A/s.

The variational functionals Gu and Gγ are solved in a fully coupled manner (mono-

lithic) by means of a Newton-Raphson scheme. For this reason Gu and Gγ are lin-

earized with respect to the variations of the primary variables u and γ.

LinGu = ∆uGu + ∆γGu + G∗u = 0

LinGγ = ∆γGγ +∆uGγ + G∗γ = 0

(4.38)

where G∗u and G∗

γ stand for the values of the previous estimate and with

∆uGu =∫ L

0

dδu

dx

∂σ∂ε∆εdx

∆γGu =∫ L

0

dδu

dx

∂σ∂γ∆γdx

∆γGγ =∫ L

0 sign

(γ − γn

∆t

) ∣∣∣∣

γ − γn

∆t

∣∣∣∣

m (

mδγ

γ − γn

)

∆γ dx

− ∫ L0 δγCa

∂σ∂γ∆γ dx +

∫ L0 Cb

d(δγ)

dx

∂∆γ∂x

dx

∆uGγ = − ∫ L0 δγCa

∂σ∂ε∆ε dx

(4.39)


gradient plasticity

The equations (4.38) are discretized according to a finite element approach. Linear

interpolation functions Nγ are used for the slip field γ, and quadratic interpolation

functions Nu for the displacement field u.

δu = ∑J NuJ δ

Ju u = ∑K Nu

KuK

δγ = ∑L NγLδ

Lγ γ = ∑M Nγ

MγM

(4.40)

Substitution of the finite element interpolations into the linearized forms results in

the global system of coupled linear equations with element tangent matrices,

kuu =∫

Be ∑J ∑K

dNuJ

dx

∂σ∂ε

dNuK

dxdx

kuγ =∫

Be ∑J ∑M

dNuJ

dx

∂σ∂γ

NγM dx

kγγ =∫

Be ∑L ∑M

(

NγL sign

(γ− γn

∆t

) ∣∣∣∣

γ − γn

∆t

∣∣∣∣

m (m

γ− γn

)

NγM

)

dx

− ∫

Be ∑L ∑M

(

NγL Ca

∂σ∂γ

NγM − dNγ

L

dxCb

dNγM

dx

)

dx

kγu = − ∫Be ∑L ∑K NγL Ca

∂σ∂ε

dNuK

dxdx

(4.41)

where ∂σ/∂ε = E, ∂σ/∂γ = −E due to σ = E(ε − γ) and Be is the element do-

main. Element residual vectors are calculated by using the values of σ and γ from

the previous estimate

ru =∫

Be ∑ J

dNuJ

dxσ dx

rγ =∫

Be

[

∑L ∑M NγL Nγ

M

∣∣∣∣

γ− γn

∆t

∣∣∣∣

m

sign

(γ− γn

∆t

)]

dx

− ∫

Be

[

∑L NγL Caσ − ∑L ∑M

dNγL

dxCb

dNγM

dxγM

]

dx

(4.42)

The assembly operation gives the global tangent and residual i.e.,

Kuu = A kuu Kuγ = A kuγ Ru = A ru

Kγu = A kγu Kγγ = A kγγ Rγ = A rγ(4.43)

and the system of equations to be solved reads:[

Kuu Kuγ

Kγu Kγγ

] [

∆u

∆γ

]

=

[

−Ru + Rextu

−Rγ + Rextγ

]

(4.44)

where Rextu and Rext

γ originate from the boundary terms in the equilibrium (4.36) and

slip (4.37) weak forms.

4.8 Appendix 71

4.8.2 Finite element implementation of slip based non-convex strain gra-dient plasticity

Pursuing the same procedure as in the previous section, the weak form for equilib-

rium is written as

Gu =∫ L

0

dδu

dxσ dx − [σδu]|L0 = 0 (4.45)

and for the plastic slip evolution as

Gγ =∫ L

0

[

δγ

∣∣∣∣

γ − γn

∆t

∣∣∣∣

m

sign

(γ − γn

∆t

)

− δγCaσ + Cbd(δγ)

dx

∂γ∂x

]

dx

+∫ L

0 [δγCa(4C1γ3 + 3C2γ

2 + 2C3γ + C4)] dx − Cb[δγ∂γ∂x

]|L0 = 0

(4.46)

The variational functionals Gu and Gγ are again solved in a fully coupled manner

(monolithic) by means of a Newton-Raphson scheme. The corresponding element

tangent matrices are identical to (4.41), except the kγγ term which becomes,

kγγ =∫

Be ∑L ∑M

(

NγL sign

(γ− γn

∆t

) ∣∣∣∣

γ − γn

∆t

∣∣∣∣

m (m

γ− γn

)

NγM

)

dx

− ∫

Be ∑L ∑M

(

NγL Ca

∂σ∂γ

NγM − dNγ

L

dxCb

dNγM

dx

+ NγL Nγ

MCa (12C1γ2 + 6C2γ + 2C3)

)

dx

(4.47)

and rγ

rγ =∫

Be

[

∑L ∑M NγL Nγ

M

∣∣∣∣

γ− γn

∆t

∣∣∣∣

m

sign

(γ− γn

∆t

) ]

dx

− ∫

Be

[

∑L NγL Caσ − ∑L ∑M

dNγL

dxCb

dNγM

dxγM

]

dx

+∫

Be

[

∑L NγL Ca(4C1γ

3 + 3C2γ2 + 2C3γ+ C4)

]

dx

(4.48)

which are calculated for the values of γ and σ from the previous estimate. After as-

sembly the resulting system of equations in (4.44) is solved in a standard incremental

iterative manner.

Chapter five

Non-convex rate dependent straingradient crystal plasticity and

deformation patterning1

Abstract / A rate dependent strain gradient crystal plasticity framework is presentedwhere the displacement and the plastic slip fields are considered as primary variables.These coupled fields are determined on a global level by solving simultaneously the lin-ear momentum balance and the slip evolution equation, which is derived in a thermo-dynamically consistent manner. The formulation is based on the 1D theory presented inYalcinkaya et al. (2011a), where the patterning of plastic slip is obtained in a system withnon-convex energetic hardening through a phenomenological double-well plastic poten-tial. In the current multi-dimensional multi-slip analysis the non-convexity enters theframework through a latent hardening potential presented in Ortiz and Repettto (1999)where the microstructure evolution is obtained explicitly via lamination procedure. Thecurrent study aims the implicit evolution of deformation patterns due to the incorporatednon-convex potential.

5.1 Introduction

At the microscopic scale, deformed crystalline materials usually show heterogeneous

plastic deformation, where the amount of plastic strain varies spatially. At moderate

strain levels, regular cellular dislocation structures have been observed. Typical ex-

amples of dislocation microstructures are dislocation cells and dislocation walls (see

e.g. Rauch and Schmitt (1989), Gardey et al. (2005), Yalcinkaya et al. (2009)). Pattern-

ing typically refers to the self organization of dislocations, yielding regions with a

high dislocation density (dislocation walls) that envelop areas with a low dislocation

1This chapter is reproduced from Yalcinkaya et al. (2011b)

73

745 Non-convex rate dependent strain gradient crystal plasticity and

deformation patterning

density (dislocation cell interiors), also regarded as domains of high and low plastic

slip activity, respectively. In addition to the cellular microstructures at meso-scale,

clear band formation and related plastic flow localization in irradiated materials (see

e.g. Sauzay et al. (2010)) at lower scales and macroscopic plastic slip bands such as

Luders bands (see e.g. Shaw and Kyriakides (1998)) are also commonly observed

structures due to plastic deformation. These microstructures macroscopically e.g.

manifest themselves through softening of the material or through plastic anisotropy

under strain path changes.

The softening of the material and macroscopic anisotropic effects under strain path

changes (see e.g. Peeters et al. (2000), Yalcinkaya et al. (2009)), resulting from evolv-

ing dislocation microstructures, have been an interesting topic for the materials sci-

ence community and the metal forming industry for decades. Starting with the stud-

ies on the cold-worked sub-structure of polycrystals using transmission electron mi-

croscopy in 1960s (e.g. Bailey and Hirsch (1960), Keh et al. (1963) and Swann (1963)),

a vast amount of experimental results have been collected, and several promising

theoretical models were presented dealing with dislocation (or slip) patterning. Nev-

ertheless, a complete descriptive understanding of the occurring phenomena has not

been reached and the necessary input for computational models is still subject of

ongoing discussions.

In the context of the computational modeling of plastic slip pattering (or disloca-

tion sub-structure formation), different approaches have been pursued in the liter-

ature which can be categorized into three main groups: (i) models using directly

the mechanics of single dislocations or populations of dislocations, (ii) phase field

modeling of dislocation patterning, (iii) the incremental variational formulation of

inelasticity by applying relaxation concepts for fine scale microstructure evolution.

See Yalcinkaya et al. (2011a) for a global overview of the available approaches.

The objective of the present paper is to develop a rate dependent strain gradient crys-

tal plasticity finite element framework for the simulation of dislocation microstruc-

ture evolution, where the non-convexity is treated as an intrinsic property of the plas-

tic free energy of the material. The constitutive model aims to simulate deformation

patterning and the macroscopic material behavior in a thermodynamically consistent

manner. Hence, the influence of latent hardening on the dislocation microstructure

evolution is studied through a physically based latent hardening potential proposed

by Ortiz and Repettto (1999). The physically based non-convex formulation is rele-

vant in multi-slip deformation states, accounting for the interaction of slip systems.

In the model, the plastic slip and the displacement are taken as degrees of freedoms.

These fields are determined on a global level by solving simultaneously the linear

momentum balance and the slip evolution equation. The latter, expressed through a

thermodynamically consistent slip law is a crucial part of the model. At first glance,

5.2 Strain gradient crystal plasticity and finite element implementation 75

the slip law (see equation (5.18)) is similar to the one encountered in classical rate

dependent crystal plasticity approaches (e.g. Hutchinson (1976), Peirce et al. (1982),

Yalcinkaya et al. (2008)), however the contribution of the actual stress state differs.

Three stress contributions can be distinguished: (1) the conventional resolved shear

stress directly related to the external loading, (2) an internal stress depending on the

gradient of the plastic slip, which is characteristic for strain gradient crystal plastic-

ity models (e.g. Evers et al. (2004b), Yefimov et al. (2004), Bayley et al. (2006)) and

(3) the stress that emanates from the non-convex part in the free energy. The latter

contribution eventually triggers the patterning of the dislocation slip field.

The paper is organized as follows. First, in section 2, the rate dependent strain gra-

dient crystal plasticity and its finite element solution is briefly discussed. Then, in

section 3, the incorporation of non-convexity into the model is presented, using a

physically based latent hardening potential. A detailed analysis of the latent hard-

ening based non-convex function is performed in this section in order to clarify the

conditions enabling microstructure evolution. In section 4, numerical examples are

presented in order to demonstrate the capability of the proposed model. First, the

size effect related to plastic slip gradients and rate dependent deformation evolution

in the context of convex strain gradient crystal plasticity is studied. Then, the rate

dependent microstructure evolution via the physically motivated latent hardening

non-convex potential is addressed in this section. Finally, some concluding remarks

are given in section 5.

5.2 Strain gradient crystal plasticity and finite element im-

plementation

In this section, the theoretical framework of the slip based strain gradient crystal

plasticity is presented and its incorporation into a finite element formulation is ad-

dressed briefly. First, the thermodynamical consistency and the derivation of the

governing system equations are discussed. In a geometrically linear context, with

small displacements, strains and rotations, the time dependent displacement field

is denoted by u = u(x, t), where the vector x indicates the position of a material

point. The strain tensor ε is defined as ε = 12(∇u + (∇u)T), and the velocity vector

is represented as v = u. The strain is decomposed additively as

ε = εe +εp (5.1)

into an elastic part εe and a plastic part εp. The plastic strain rate can be written as

a summation of plastic slip rates on the individual slip systems, εp = ∑α γαPα with

Pα = 12(sα ⊗ nα + nα ⊗ sα) the symmetrized Schmid tensor, where sα and nα are the



unit slip direction vector and unit normal vector on slip system α, respectively. The

state variables are chosen to be given by the set

state = εe,γα, ∇γα (5.2)

where γα contains the plastic slips on the different slip systemsα and ∇γα represents

the gradient of the slips on these slip systems. Following the arguments of Gurtin

(e.g. Gurtin (2000), Gurtin (2002)), the power expended by each independent rate-

like kinematical descriptor is expressible in terms of an associated force consistent

with its own balance. However, the basic kinematical fields of rate variables, namely

εe, u and γα are not spatially independent. It is therefore not immediately clear how

the associated force balances are to be formulated, and, for that reason, these balances

are established using the principle of virtual power.

Assuming that at a fixed time the fields u, εe and γα are known, we consider û, ˆεe

and ˆγα as virtual rates, which are collected in the generalized virtual velocity V =

( û, ˆεe, ˆγα). Pext is the power expended on the domain Ω and Pint a concomitant

expenditure of power within Ω, given by

Pext(Ω, V) =∫

S t(n) · û dS +∫

S ∑α(χα(n) ˆγα) dS

Pint(Ω, V) =∫

Ωσ : ˆεe dΩ+∫

Ω ∑α(πα ˆγα)dΩ+

∫

Ω ∑α(ξα ·∇ ˆγα)dΩ

(5.3)

where the stress tensor σ , the scalar internal forces πα and the microstress vectors

ξα are the thermodynamical forces conjugate to the internal state variables εe, γα

and ∇γα, respectively. In Pext, t(n) is the macroscopic surface traction while χα

represents the microscopic surface traction conjugate to γα at the boundary S with n

indicating the boundary normal.

The principle of virtual power states that for any generalized virtual velocity V the

corresponding internal and external power are balanced, i.e.

Pext(Ω, V) = Pint(Ω, V) ∀ V (5.4)

Considering a generalized virtual velocity without slip, the virtual velocity field can

be chosen arbitrarily and this leads to the classical macroscopic force balance,

∇ ·σ = 0 (5.5)

and considering arbitrary virtual slip fields without a generalized virtual velocity

leads to the microscopic force balances,

∇ ·ξα + τα − πα = 0 (5.6)


on each slip system α, where τα is the resolved Schmid stress given by τα = σ : Pα.

The local internal power expression can be written as

Pi = σ : εe + ∑α

(παγα +ξα · ∇γα) (5.7)

The local dissipation inequality results in

D = Pi − ψ = σ : εe + ∑α

(παγα +ξα · ∇γα)− ψ ≥ 0 (5.8)

The material is assumed to be endowed with a free energy with different contribu-

tions according to

ψ = ψe +ψγ +ψ∇γ (5.9)

The time derivative of the free energy is expanded and equation (5.8) is elaborated

to

D = σ : εe + ∑α(παγα +ξα ·∇γα − ∂ψ∂εe

: εe − ∂ψ∂γα

γα − ∂ψ∂∇γα

· ∇γα)

= (σ − dψe

dεe)

︸︷︷︸

0

: εe + ∑α(πα −∂ψγ∂γα

)γα + ∑α (ξα − ∂ψ∇γ∂∇γα

)︸︷︷︸

0

·∇γα ≥ 0(5.10)

The stress σ and the microstress vectors ξα are regarded as energetic quantities hav-

ing no contribution to the dissipation

σ =dψe

dεe

ξα =∂ψ∇γ∂∇γα

(5.11)

whereas πα does have a dissipative contribution,

D = ∑α

(πα − ∂ψγ∂γα

)γα ≥ 0 (5.12)

The multipliers of the plastic slip rates are identified as the set of dissipative stresses

σαdis

σαdis = πα − ∂ψγ

∂γα(5.13)

In order to satisfy the reduced dissipation inequality at the slip system level the fol-

lowing constitutive equation is proposed

σαdis =ϕαsign(γα) (5.14)



whereϕα represents the mobilized slip resistance of the slip system under consider-

ation

ϕα =sα

γα0|γα| (5.15)

with sα is the resistance to dislocation slip which is assumed to be constant and γ0 is

the reference slip rate. Substituting (5.15) into (5.14) gives

γα =γα0sασαdis (5.16)

Substitution of σαdis according to (5.13) into (5.16) reveals,

γα =γα0sα

(

πα − ∂ψγ∂γα

)

(5.17)

Using the microforce balance (5.6) results in the plastic slip equation

γα =γα0sα

(

τα + ∇ ·ξα − ∂ψγ∂γα

)

(5.18)

In addition to the explicit contribution of ψγ, other contributions of the free energies

defined in (5.9) enter the slip equation via (5.11) with τα = dψe/dεe : Pα and ξα =

∂ψ∇γ/∂∇γα. Quadratic forms are used for the elastic free energy ψe and the plastic

slip gradients free energy contribution ψ∇γ, i.e.

ψe =1

2εe : 4C : εe

ψ∇γ = ∑α

1

2A∇γα ·∇γα

(5.19)

where A is a scalar quantity, which includes an internal length scale parameter, gov-

erning the effect of the plastic slip gradients on the internal stress field. It may be

expressed as A = ER2/(16(1 − ν2)) as e.g. used in Bayley et al. (2006) and Geers

et al. (2007), where R physically represents the radius of the dislocation domain con-

tributing to the internal stress field, ν is Poisson’s ratio and E is Young’s modulus.

The plastic slip dependent free energy ψγα will be defined in the following section.

In order to solve the initial boundary value problem for this rate dependent strain

gradient crystal plasticity framework, a fully coupled finite element solution algo-

rithm is used in which both the displacement u and plastic slips γα are considered

as primary variables. These fields are determined in the solution domain by solving

simultaneously the linear momentum balance (5.5) and the slip evolution equation

(5.18), which constitute the local strong form of the balance equations:

∇ ·σ = 0

γα − γα0sατα − γα0

sα∇ ·ξα +

γα0sα

∂ψγ∂γα

= 0(5.20)


In order to obtain variational expressions representing the weak forms of the govern-

ing equations given above, these equations are multiplied by weighting functions δu

and δαγ and integrated over the domainΩ. Using the Gauss theorem (S is the bound-

ary of Ω) results in

Gu =∫

Ω

∇δu : σdΩ−∫

Sδu · tdS

Gαγ =

∫

Ω

δαγ γα dΩ−

∫

Ω

δαγγα0sατα dΩ+

∫

Ω

∇δαγ ·γα0sα

A ∇γαdΩ

+∫

Ω

δαγγα0sα

∂ψγ∂γα

dΩ−∫

Sδαγ χ

α dS

(5.21)

where t is the external traction vector on the boundary S, and χα =γα0sα

A∇γα · n.

The domain Ω is subdivided into finite elements, where the unknown fields of the

displacement and slips and the associated weighting functions within each element

are approximated by their nodal values multiplied with the interpolation shape func-

tions stored in the Nu and Nγ matrices, using a standard Galerkin approach.

δu = Nuδu u = Nuu

δαγ = Nγδαγ γα = Nγγα(5.22)

with u, δu, γα and δαγ are columns containing the nodal variables. Bilinear interpo-

lation functions for the slip field and quadratic interpolation functions for the dis-

placement field are used. An implicit backward Euler time integration scheme is

used for γα in a typical time increment [tn, tn+1] which gives γα = [γαn+1 − γαn ]/∆t.

The discretized element weak forms read

Geu = δT

u

[∫

ΩeBu σ dΩe −

∫

SeNu t dSe

]

Gαeγ = δαγ

[∫

ΩeNγT Nγ

[γα

n+1− γα

n

∆t

]

dΩe −∫

Ωe

γα0sα

NγTτα dΩe

]

+δαγ

[∫

Ωe

γα0sα

A BγT Bγ γα dΩe +∫

Ωe

γα0sα

NγT∂ψ

γα

∂γαdΩe −

∫

SeNγT

χα dSe

]

(5.23)

The weak form of the balance equations (5.23) are linearized with respect to the vari-

ations of the primary variables u and γα and solved by means of a Newton-Raphson

solution scheme for the increments of the displacement field ∆u and the plastic slips

∆γα. The procedure results in a system of linear equations which can be written in

the following matrix format,[

Kuu Kuγ

Kγu Kγγ

] [

∆u

∆γα

]

=

[

−Ru + Rextu

−Rγ + Rextγ

]

(5.24)



where Kuu, Kuγ, Kγu and Kγγ represent the global tangent matrices while Ru and Rγ

are the global residual columns. The contributions Rextu and Rext

γ originate from the

boundary terms.

5.3 Latent hardening based non-convex plastic potential

In this section, a latent hardening based non-convex plastic potential proposed by

Ortiz and Repettto (1999) is examined, whereby the conditions for the occurrence of

plastic slip patterning is studied.

In physically deformed crystals one of the main presumed reasons for dislocation

microstructure formation, is latent hardening accompanied with non-convexity of

the energy function due to slip system interactions. Such a function is proposed by

Ortiz and Repettto (1999) which gives parabolic-like hardening in single slip and

latent (off-diagonal) hardening in multi-slip (see Fig. 5.1 for two slip systems)

ψγ =2

3τ0 γ0

[

∑α

∑β

aαβ|γα|γ0

|γβ|γ0

]3/4

(5.25)

Here τ0 and γ0 are a reference resolved shear stress and a reference slip value, respec-

tively, and aαβ are interaction coefficients. For the values of the matrix aαβ, a simple

geometrical model is used proposed by Cuitino and Ortiz (1993),

aαβ =2

π

√

1 − (nα · nβ)2 (5.26)

where nα and nβ are normals on the slip planes of the systems considered. Using

(5.26), slip systems do not self-harden in a multi-slip context. The reasoning leading

to (5.26) is based on the fact that the typical resolved shear stress required to de-

form a well-annealed crystal in single slip tends to be small compared to the stress

required for multi-slip. For the purpose of understanding the morphology of dislo-

cation structures, self-hardening can be neglected at first instance. Ortiz and Repettto

(1999) and Ortiz et al. (2000) employ (5.25) and (5.26) in the context of crystal plas-

ticity to obtain lamellar dislocation structures via a sequential lamination procedure.

The purpose here is to study patterning driven by this latent hardening based non-

convex potential.

5.3.1 Conditions for plastic slip patterning

Plastic deformation tends to generate dislocation microstructures that minimize the

free energy (see e.g. Kuhlmann-Wilsdorf (2001)). From a thermodynamics point of

5.3 Latent hardening based non-convex plastic potential 81

0

0.02

0.04

0.06

0

0.02

0.04

0.060

0.002

0.004

0.006

0.008

0.01

0.012

γ2γ1

ψγ

00.02

0.040.06

00.010.020.030.040.050.06

0

0.002

0.004

0.006

0.008

0.01

0.012

γ1γ2

ψγ

Figure 5.1 / Two different views of ψγ (MPa) for different amounts of slip on twoslip systems oriented with respect to x axis as 60 and 120 where γ0 = 1 and τ0 = 1MPa.

view, a patterned microstructure will develop if it has a lower energy than a state

with a homogeneous plastic deformation distribution. Yalcinkaya et al. (2011a) stud-

ied this aspect on a double-well potential and showed that plastic slip patterning

occurs at relatively low strain rates due to the existence of an unstable regime in

the double-well potential, where the microstructure evolves in a patterned way to

lower its free energy. Convex energy potentials preserve stability and do not trigger

patterning. Therefore, the presence of non-convexity is the first condition for a het-

erogeneous microstructure evolution to develop, yielding a lower energy than the

homogeneous state. In what follows, the latent hardening function (5.25), which is

assumed to be the driving force for the evolution of dislocation microstructures, is

examined in this sense.

For this purpose, a single crystal in 2D having 2 slip systems oriented θ1 and θ2

with respect to x axis is considered. A pure shear deformation is applied with a

macroscopic strain field εM, whose components are written in Cartesian coordinates

as

εM =

[

0 γ

γ 0

]

(5.27)

which satisfies the plastic incompressibility condition tr (εM) = 0. Assume that the

plastic deformation is patterned in the solution domain and that there exist two states

with volume fraction f and 1 − f . The weighted average strain of the two phases

should be equal to the macroscopic strain, i.e.

f ε1 + (1 − f )ε2 = εM (5.28)



The contributing strains ε1 and ε2 are calculated according to,

ε1 = ∑2α=1γ

α1 Pα

ε2 = ∑2α=1γ

α2 Pα

(5.29)

with both patterned regions having the same crystal orientation and hence the same

Pα. The strain tensors, ε1 = ε1(γ11 ,γ2

1 ,θ1,θ2) and ε2 = ε2(γ12 ,γ2

2 ,θ1,θ2) are symmetric

and incompressible. Using (5.29), equation (5.28) reduces to a set of 2 linear equa-

tions with 5 unknowns γ11 ,γ2

1 ,γ12 ,γ2

2 , f . For a fixed value of f and given two of the

unknown slips γ11 and γ2

1 the value of the other plastic slips γ12 and γ2

2 can be calcu-

lated according to

γ12 = − f

1 − fγ1

1 +P21

P22P11 − P12P21

1

1 − fγ

γ22 = − f

1 − fγ2

1 +P11

P22P11 − P12P21

1

1 − fγ

(5.30)

with P11 = P1(1, 1), P12 = P1(1, 2), P21 = P2(1, 1), P22 = P2(1, 2). The energy of this

assumed patterned state can be calculated as,

ψγ = fψγ1(γ1

1 ,γ21 ,θ1,θ2) + (1 − f )ψγ2

(γ12 ,γ2

2 ,θ1,θ2) (5.31)

which is plotted in Fig. 5.2. The latent hardening energy corresponding to the macro-

scopic shear strain εM is calculated via (5.25) in terms of the plastic slips γ1M and γ2

M

on given slip systems as,

ψγ = ψγ(γ1M,γ2

M,θ1,θ2) (5.32)

The purpose of this analysis is to investigate if there exists a domain of a lower latent

hardening based energy for the assumed patterned case ψγ than the macroscopic

latent hardening based energy ψγ (which reflects the homogeneous non-patterned

state). To this end, ψγ is calculated for different values of γ11 and γ2

1, and for specific

(chosen) orientations and volume fractions.

If a macroscopic shear deformation tensor εM with γ = 0.02 is applied on a com-

binations of slip systems with orientations θ1 = 60 and θ1 = 120, the amount of

slips on the two slip systems for the local homogeneous state equals 0.04. Using

these values for the slips and the orientations, the locally homogeneous latent hard-

ening plastic potential is calculated via Equation (5.25). Assuming τ0 = 1 MPa and

γ0 = 1 the characterizing latent hardening energy for the non-patterned state equals

ψγ = 0.0057 MPa. Presuming the existence of patterned states for different values of

f and (γ11, γ2

1), the values of γ12 and γ2

2 are calculated via (5.30) and the latent hard-

ening energy follows from (5.31). In order to obtain a patterned microstructure the

5.3 Latent hardening based non-convex plastic potential 83

00.05

0.1

00.020.040.060.080.1

4.8

5

5.2

5.4

5.6

5.8

6

6.2

6.4

x 10−3

γ2

1

f = 0.1

γ1

1

ψγ

5

5.2

5.4

5.6

5.8

6

x 10−3

00.05

0.100.050.1

3.5

4

4.5

5

5.5

6

6.5

7

x 10−3

γ2

1

f = 0.2

γ1

1

ψγ

4

4.5

5

5.5

6

6.5

x 10−3

0 0.05 0.10

0.050.1

2

3

4

5

6

7

8

x 10−3

γ2

1

f = 0.3

γ1

1

ψγ

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

x 10−3

0 0.02 0.04 0.06 0.08 0.10

0.050.1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

γ2

1

f = 0.5

γ1

1

ψγ

0

2

4

6

8

10

12

x 10−3

0 0.05 0.10

0.050.1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

γ2

1

f = 0.6

γ1

1

ψγ

2

4

6

8

10

12

14

16

x 10−3

0 0.05 0.10

0.050.1

0

0.005

0.01

0.015

0.02

0.025

γ2

1

f = 0.7

γ1

1

ψγ

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0 0.05 0.10

0.050.1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

γ2

1

f = 0.8

γ1

1

ψγ

0.005

0.01

0.015

0.02

0.025

0.03

0 0.05 0.10

0.050.1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

γ2

1

f = 0.9

γ1

1

ψγ

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Figure 5.2 / ψγ (MPa) in terms of a set of given γ11 and γ2

1 for different values of fwith θ1 = 60 and θ1 = 120 .



energy of the patterned state should be smaller than the energy of the local homoge-

neous state ψγ < ψγ. As seen in Fig. 5.2 there are many combinations satisfying this

inequality for different values of the volume fraction f and the plastic slip values γ11

and γ21.

5.4 Numerical analysis

In this section, two different numerical examples are presented to study the behavior

of the proposed rate dependent strain gradient crystal plasticity models. The first

example deals with a conventional convex free energy in terms of plastic slip gra-

dients and elastic strains, where the effect of the applied shear rate and the internal

length parameter R are analyzed. Then, the influence of the physically based non-

convex latent hardening plastic potential on the mechanical behavior of the material

is discussed.

5.4.1 Convex strain gradient crystal plasticity

In this subsection, the plastic potential ψγ is assumed to be zero therefore having no

influence on patterning and the hardening of the material. The incorporated hard-

ening is only due to the gradients of the plastic slip. This is referred to as convex

strain gradient crystal plasticity because there is no plastic potential inducing a lack

of convexity.

To reveal the main characteristics of this convex strain gradient viscous crystal plas-

ticity model, a constrained plane strain shear problem of an infinite strip, induced by

periodic boundary conditions, is studied in the first example. A strip, with height H

(in y direction) is bounded by rigid walls that are impenetrable for dislocations, i.e.

a no slip condition applies (γα = 0) at top and bottom edges. These so-called micro

clamped boundary conditions for the plastic slips invoke an inhomogeneous plastic

deformation state (e.g. as present near grain boundaries in polycrystals or at the sur-

face of a thin film). The displacements at y = 0 are suppressed (ux = 0 and uy = 0)

and prescribed at y = H as ux = u(t) and uy = 0. In addition, in x direction all field

quantities are taken to be independent of x. Consequently, the field quantities on the

left side are assumed to be identical to those on the right side ul = ur and γαl = γαr .

Locally, two slip systems with orientations 60 and 120 with respect to the hori-

zontal axis are considered. The material is assumed to be elastically isotropic with

Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0.33, slip resistance for both slip

systems s = 35 MPa, and a reference slip rate γ0 = 0.15 s−1. Results presented in

the following correspond to a discretization with 1x100 rectangular elements. One

5.4 Numerical analysis 85

element in the x-direction is sufficient, given the uniformity in this direction.

0 0.005 0.01 0.015 0.020

100

200

300

400

500

600

700

800

900

Applied shear Γ = u/H

Shea

rst

ressτ

[MP

a]

Γ = 2.5s−1

0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

140


Shea

rst

ressτ

[MP

a]

Γ = 0.25s−1

0 0.005 0.01 0.015 0.020

2

4

6

8

10

12

14

16


Shea

rst

ressτ

[MP

a]

Γ = 0.025s−1

0 0.005 0.01 0.015 0.020

0.5

1

1.5

2

2.5

3

3.5

4


Shea

rst

ressτ

[MP

a]

Γ = 0.0025s−1

Figure 5.3 / Rate dependent shear stress vs. the applied shear for the plane strainshear problem of an infinite strip.

The examples addressed in this subsection are carried out for different applied shear

rates and R/H ratios for varying R and constant height H, where the relative effect

of the internal length scale parameter R determining A, acting on the higher order

microstresses ξα (see equations (5.11) and (5.19)), is analyzed.

First, the resulting shear stress versus the applied macroscopic shear Γ is presented

in Fig. 5.3 for different overall shear rates Γ , using R = 0.35 µm, and H = 20 µm.

Γ is the macroscopic shear defined as Γ = u/H. In this case, the average of the local

strain ε12 should be half of the macroscopic shear Γ . If the elastic deformation is

small, the average value of the local plastic shear strain εp12 will be roughly equal to

half of the macroscopic shear Γ as well. When the applied macroscopic shear rate

increases, the material shows a stiffer (dominantly elastic) response. In Fig. 5.4, the

local rate dependent plastic shear strain evolution is presented at a macroscopic shear

level of Γ = 0.02. There is a clear boundary layer width dependence on the applied

shear rate. For high values of the applied shear rate, e.g. Γ = 2.5s−1 we observe

a sharp boundary layer, while for low values of the shear rate, e.g. Γ = 0.0025s−1



the boundary layer thickness is more diffuse. This type of rate dependent dislocation

slip profile is also observed in Yalcinkaya et al. (2011a) where a high strain rate causes

that the plastic slip has not enough time to evolve.

Γ = 2.5s−1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Γ = 0.25s−1

0

1

2

3

4

5

6

7

8

9

x 10−3

Γ = 0.025s−1

0

2

4

6

8

10

12x 10

−3

Γ = 0.0025s−1

0

2

4

6

8

10

12

14

x 10−3

Figure 5.4 / Rate dependent plastic shear strain εp12 distribution for the plane strain

shear problem of an infinite strip at a global shear level of Γ = 0.02.

The next example in this subsection concerns the influence of the internal length scale

parameter R on the mechanical behavior for a fixed height H = 20µm. The value of R

is taken as 0.35µm, 0.7µm and 1.75µm corresponding to a R/H ratio equal to 0.0175,

0.035 and 0.0875 respectively. The shear stress versus the applied macroscopic shear

response is plotted in Fig. 5.5 for different values of R at Γ = 0.025s−1. Note that

the formulation is intrinsically viscous, corresponding to models using a linear drag

law for dislocation motion to determine the slip. Fig. 5.5 illustrates a significant

effect of the internal length parameter where the strip is exhibiting a stiffer response

for larger values of R. The results are consistent with many strain gradient models

(e.g. Shu et al. (2001), Evers et al. (2004b), Yalcinkaya et al. (2011a)) in which the

influence of the length scale was shown. Physically, this effect results from the fact

that large R values induce a large internal stress and hence penalize high plastic slip


gradients (see Fig. 5.6) spreading the geometrically necessary dislocation densities.

In this convex example, the slip gradient is the dominating factor in the plastic and

hardening behavior of the material and therefore a clear size effect is observed. In

Fig. 5.6 a clear dependence of the boundary layer evolution on R is presented at

Γ = 0.02. With increasing R an increased influence of the boundary conditions is

observed with a more diffuse boundary layer.

0 0.005 0.01 0.015 0.020

10

20

30

40

50

60

70


Shea

rst

ressτ[M

Pa]

R = 0.35 µm

R = 0.7 µm

R = 1.75 µm

Figure 5.5 / The effect of the internal length scale parameter R on the stress vs.applied shear response for the plane strain shear problem of an infinite strip.

R = 0.35µm

2

4

6

8

10

12x 10

−3

R = 0.7µm

2

4

6

8

10

12

x 10−3

R = 1.75µm

2

4

6

8

10

12

14

x 10−3

Figure 5.6 / The effect of the internal length scale parameter R on the plastic shearstrainε

p12 distribution for the plane strain shear problem of an infinite strip at a global

shear level of Γ = 0.02.

The distribution of the plastic slips on each of the slip systems is illustrated in Fig. 5.7

for the same example with R = 0.35µm and Γ = 0.025s−1 at Γ = 0.02. The amount



θ = 60

−0.0242

−0.0121

0

θ = 120

−0.0242

−0.0121

0

Figure 5.7 / The distribution of the plastic slip on both slip systems at a global shearlevel of Γ = 0.02.

of the slip on both slip systems is identical and the evolution is similar to the plastic

shear strain evolution, exhibiting a boundary layer.

5.4.2 Non-convex strain gradient crystal plasticity

In this subsection the influence of the latent hardening plastic potential ψγ (5.25) on

the mechanical behavior and the deformation patterning is illustrated in the present

rate dependent strain gradient crystal plasticity framework. In addition to ψγ, an

elastic strain energy potentialψe and a plastic slip gradient free energy potentialψ∇γα

are incorporated as well. The viscous formulation of the problem and the gradient

free energy potential regularize the problem.

The paper of Ortiz and Repettto (1999) shows that in crystals exhibiting latent hard-

ening the energy function is non-convex, which favors the development of mi-

crostructures. Therefore, uniform deformation fields are not the minimizers of the in-

cremental work of deformation. In other words, in the context of classical variational

formulations the crystals exhibiting latent hardening might not reach the expected

solution, i.e. the minimum free energy is not achieved. It is possible to construct

deformation mappings to recover the minimum value. Such deformation mappings

make use of the existence of fine microstructures. Using a sequential lamination

method, Ortiz and Repettto (1999) were able to characterize analytically several dis-

location structures. The same procedure is followed later in Ortiz et al. (2000) where

microstructures are regarded as instances of sequential lamination during deforma-

tion. The microstructures are explicitly constructed by recursive lamination and their

subsequent equilibration.

The numerical study in section 5.3 proves that the latent hardening potential (5.25)

is non-convex and satisfies the energetic conditions for plastic slip phase separation


in the context of the present small strain rate dependent crystal plasticity formula-

tion. Compared to the previous work of Ortiz and Repettto (1999) and Ortiz et al.

(2000) relying on the explicit construction of cellular dislocation microstructures via

a lamination procedure, the present framework is based on the implicit evolution of

microstructures driven by the deformation. Throughout the incremental deforma-

tion process the state of the plastic slip enters energetically favorable regimes, (as

illustrated in section 5.3) which might eventually result in deformation heterogene-

ity.

The numerical study in this subsection concerns a plane strain pure shear problem of

a square representative volume element (RVE) in order to have a direct link with the

study in section 5.3. Locally, two slip systems with orientations 60 and 120 with

respect to the x axis are considered. The material is assumed to be elastically isotropic

with Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0.33, slip resistance for both

slip systems s = 35 MPa, and a reference slip rate γ0 = 0.15 s−1. The reference slip

strain and resolved shear stress in (5.25) are assumed to be γ0 = 0.015 and τ0 = 50

MPa, respectively.

In the square RVE the displacements and the plastic slips on the left edge are tied to

the ones on the right edge and the ones on the bottom edge are tied to the ones on the

top edge which makes it a fully periodic configuration. The vertical displacement at

the right bottom corner and the horizontal displacement at the left top corner are pre-

scribed, both equal to u(t). The displacements at left bottom corner are suppressed

together with the horizontal displacement at the right bottom corner and vertical dis-

placement at left top corner. These boundary and loading conditions result in a pure

shear deformation mode. The length of each edge of the square is H = 20µm. Re-

sults presented in the following correspond to a discretization with 20x20 rectangular

elements.

In Fig. 5.8 the shear stress versus applied macroscopic shear, defined as Γ = 2u/H,

is plotted for an applied shear rate Γ = 0.005s−1 and an internal length scale param-

eter R = 0.01µm. In this case, the average of the local strain ε12 should be half of

the macroscopic shear Γ . If the elastic deformation is small, the average value of the

local plastic shear strain εp12 will be roughly equal to half of the macroscopic shear

Γ as well. The corresponding shear strain, plastic shear strain and plastic slips of

the slip systems are plotted in Fig. 5.9. All plotted fields exhibit a strong patterned

response due to the incorporated latent hardening potential. Note that the applied

periodic boundary conditions lead to an initially homogeneous distribution of the

strain and plastic slip fields. In order to properly trigger a small perturbation a small

spatial fluctuation is applied to the Young’s modulus E in the RVE. In order to il-

lustrate the influence of the selected spatial fluctuation of E on the evolution of the

plastic microstructure, two different types of fluctuations are applied in the next two



0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

10

20

30

40

50

60

70

80

90

100

Applied shear Γ = 2u/H

Shea

rst

ressτ

[MP

a]

Figure 5.8 / Latent hardening based non-convex shear stress vs. applied macro-scopic shear for a plane strain pure shear problem of a fully periodic RVE.

ε12

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021

0.022

εp

12

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021

θ = 60

−0.055

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

θ = 120

−0.05

−0.04

−0.03

−0.02

−0.01

Figure 5.9 / Shear strain (first figure), plastic shear strain (second figure), and plasticslip (last two figures) distributions at Γ = 0.034 for a plane strain pure shear problemof a fully periodic RVE.

examples. In Fig. 5.10 the field quantities are plotted for two additional cases with

different fluctuations applied in Young’s modulus. First, the fluctuation is given only


ε12

0.014

0.015

0.016

0.017

0.018

0.019

0.02

εp

12

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

θ = 60

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

θ = 120

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

ε12

0.014

0.015

0.016

0.017

0.018

0.019

0.02

εp

12

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

θ = 60

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

θ = 120

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

Figure 5.10 / Shear strain, plastic shear strain, and plastic slip distributions at Γ =0.034 for a plane strain pure shear problem of a fully periodic RVE with a fluctuationapplied to the central element (first four figures) and to the middle element on theright edge (last four figures).



to an element in the center of the RVE and in the second case the fluctuation is given

to an element in the middle of the right edge. The applied fluctuation has certainly

an effect on the distribution of the deformation patterns, however, the spacing of

high and low strain areas and the amplitude do not depend on the applied fluctua-

tion. The macroscopic stress versus strain diagram is not plotted in these two cases

because it is identical to the previous case plotted in Fig. 5.8.

It is remarked that the obtained deformation patterns depend considerably on the

applied rate of deformation. Increasing the strain rate results in a stiffer stress versus

strain response while the amplitude of the obtained patterns decreases. Another

important parameter affecting the patterning of deformation fields is the internal

length scale R which should be small enough to obtain a stable numerical solution

and pronounced patterns. These two factors and other material parameters play

an important role in the convergence of the numerical solution as well. Due to the

non-convexity of the latent hardening potential at each increment of the deformation

some combinations of the parameters might give convergence problems and it is not

always possible to reach the same state of deformation with different combinations

of the rate of deformation and material parameters.

As stated before the non-convex potential has been used to recover a cellular kind of

dislocation microstructures by Ortiz and Repettto (1999) and Ortiz et al. (2000) via

an external lamination procedure, while we try to obtain deformation patterns via

a non-convex evolution problem. In the current examples we observe deformation

patterning at low loading rates and a small internal length scale parameter due to

the latent hardening based non-convexity. This shows agreement with the study in

section 5.3 illustrating the capability of the latent hardening function for deformation

patterning.

5.5 Summary and conclusion

A plastic slip based rate dependent non-convex strain gradient crystal plasticity

model is proposed and embedded in a FEM solution framework using displacements

and plastic slips as degree of freedoms. A physically based latent hardening non-

convex plastic potential (Ortiz and Repettto (1999)) is incorporated into the thermo-

dynamically consistent viscous strain gradient crystal plasticity model based on the

1D formulation as presented in Yalcinkaya et al. (2011a) in order to obtain deforma-

tion and plastic slip patterns due to slip system interactions in physically deforming

crystals. The presented approach models the implicit evolution of deformation pat-

terns through the intrinsic non-convexity of the free energy function. This kinetics

driven method offers an alternative to the explicit construction of the microstructure

evolution (e.g. in Ortiz and Repettto (1999)).

5.5 Summary and conclusion 93

Selected examples demonstrate the ability of the model to obtain a deformation

driven plastic slip microstructure evolution. While a convex theory explicitly il-

lustrates the size dependent and rate dependent boundary layer development, the

non-convex formulation, originating from the slip interaction phenomena in crys-

tals, allows for deformation and plastic slip patterning.

In the model the destabilizing non-convex term in the free energy is mathematically

stabilized through the gradient term in the free energy and the viscous nature of the

thermodynamically consistent slip law. The obtained microstructure evolution is rate

dependent, where the homogeneity or inhomogeneity of the deformation basically

depends on the applied rate. Due to the non-convexity at each increment of the

applied deformation, the convergence and the patterning of the field are sensitive to

many parameters, where we have shown only the effect of the applied fluctuation.

A more detailed investigation of the dependence of the results on the loading type,

loading rates, slip system orientations, material parameters, the mesh and boundary

conditions would help for a deeper understanding of the observed microstructure

evolution phenomena.

Chapter six

Discussion and conclusions

Abstract / The main conclusions drawn from the thesis are summarized in this conclud-ing chapter and an outlook to further work related to microstructure evolution in crystalplasticity models is presented.

The objective of this thesis was the investigation of the macroscopic transient harden-

ing and softening effects due to the dislocation microstructure evolution in BCC met-

als. The project started with the development of a large strain BCC crystal plasticity

framework where the intrinsic anisotropy due to non-planar spreading of screw dis-

location cores was addressed. Next, the evolution of an existing dislocation cell struc-

ture has been investigated in a composite cell model which was incorporated into

the BCC crystal plasticity model. This framework starts with an assumed dislocation

structure and studies the evolution of this microstructure via phenomenological dis-

location evolution equations. However, the formation of dislocation microstructures

or deformation patterning cannot be simulated by these kind of models. Therefore,

in order to model the formation and evolution of inhomogeneous deformation pat-

terns, a non-convex rate dependent strain gradient crystal plasticity model has been

developed in a small strain context in the last part of the thesis.

The BCC crystal plasticity framework, presented in Yalcinkaya et al. (2008) (chapter

2), reveals some unique characteristics of BCC crystals. A comprehensive summary

of intrinsic properties of these materials are presented, including recent insights in

the activation of different slip systems, violation of Schmid’s law, temperature and

orientation dependence of the flow stress and resulting stress-strain curves. With

respect to the BCC crystal plasticity model the following conclusions are drawn:

• The applied slip evolution equation which is based on the thermally activated

dislocation kinetics plays an essential role in the model since all the intrinsic

characteristics result from the actual formulation of this equation.

95

96 6 Discussion and conclusions

• Contrary to the common assumption of using the 110, 112 and 123 type

of slip planes in all conditions, it is explained that the activation of slip systems

depends on the temperature. At moderate temperatures only 112 slip planes

are activated which affect considerably the mechanical behavior of BCC metals.

• The non-Schmid behavior is introduced in the slip law by modifying the effec-

tive shear stress, where the non-Schmid contribution represents the dislocation

cores spreading in a non-planar manner.

Deviation from a proportional strain path is associated with hardening or softening

of the material due to the induced plastic anisotropy. At moderate strains the dom-

inating effect is attributed to the evolving underlying dislocation microstructures.

Chapter 3 (Yalcinkaya et al. (2009)) deals with a combination of a composite disloca-

tion cell model, which explicitly describes the dislocation structure evolution, with

the BCC crystal plasticity framework from chapter 2 to bridge the microstructure

evolution and its macroscopic anisotropic effects. The main conclusions from this

part are:

• A phenomenological cell structure evolution model embedded into a crystal

plasticity framework is well able to reproduce all essential characteristics of

strain path changes reported, consistently with experimental observations at

two scales.

• The model proposed allows to study the interaction between different sources

of anisotropy, where a clear example at the single crystal and polycrystal has

been given.

• The level at which the enrichment of the crystal plasticity model was made,

enables its use in more complex microstructures as e.g. multi-phase steels.

In order to complete the missing link between the formation of the microstructure

and its evolution in crystal plasticity frameworks, the second part of the thesis

concentrated on the development of a non-convex rate dependent crystal plastic-

ity model, which may eventually simulate rate dependent dislocation microstruc-

ture formation and evolution together with macroscopic hardening-softening stress-

strain responses. In chapter 4 (Yalcinkaya et al. (2011a)), inspired by the efficiency

of phase field models for microstructure formation and evolution, a 1D non-convex

strain gradient plasticity model has been developed yielding combined hardening-

softening. The resulting formulation takes a conceptually dual structure to the

Ginzburg-Landau type of phase field models where high and low slipped regions

represent different phases. The main conclusions related to this part are:

97

• The thermodynamically consistent constitutive model is able to capture pat-

terning of plastic slip, qualitatively similar to a phase decomposition mecha-

nism.

• The mathematical structure of the employed double-well plastic potential al-

lows to control the amount and the timing of deformation patterning.

• The framework can capture both homogeneous and inhomogeneous deforma-

tions depending on the rate of the applied deformation.

• The model is conceptually capable of covering many aspects of microstructure

evolution processes, depending on the externally applied load and boundary

conditions. Non-equilibrium microstructures are thereby well at reach.

• The model could be used for various materials. Especially the clear band for-

mation observed in irradiated materials or Luders band formation and motion

in low carbon steels exhibit one-to-one correspondence with the obtained re-

sults from the numerical examples.

Chapter 5 (Yalcinkaya et al. (2011b)) extends the 1D non-convex strain gradient crys-

tal plasticity formulation of the previous chapter to 2D crystals with multi-slip sys-

tems. In this chapter a more physically based non-convex plastic potential is con-

sidered which originates from the slip interactions in crystals. The main conclusions

are:

• Studying the effect of the latent hardening based non-convexity (Ortiz and

Repettto (1999)) on the deformation patterning revealed that the latent hard-

ening potential satisfies the energetic conditions for patterning.

• In the numerical examples, the convex theory explicitly illustrates the effect of

the internal length scale parameter and the deformation rate on the microstruc-

ture evolution due to the hard boundary conditions. The results show agree-

ment with chapter 4 and the literature.

• The non-convex formulation presents the possibility of deformation pattern-

ing depending on the loading rate. Strong deformation patterns are obtained

during monotonic pure shear deformation.

The non-convex strain gradient plasticity frameworks in the last two chapters of the

thesis present an energetic approach to deformation patterning and related soften-

ing of the materials. While in many crystal plasticity models the softening effect is

98 6 Discussion and conclusions

introduced externally, in the current approach it becomes a natural product. More-

over, compared to incremental variational principles and relaxation theories for mi-

crostructure evolution it is simpler to implement and computationally less expen-

sive. Capturing both equilibrium and non-equilibrium states of the microstructures

makes it a unique framework in computational plasticity. The crucial aspect in the

present models is the plastic potential entering the formulation in a thermodynami-

cally consistent manner.

In this thesis two different plastic potentials have been formulated. The double-well

based non-convexity in the fourth chapter presents a phenomenological description

of plastic slip patterning. The coefficients of the potential and the rate of deforma-

tion control the whole patterning mechanism. Even though the material parame-

ters are not identified for real materials it offers a broad range of application. A

more practical emphasis applied to a particular material is obviously needed in fu-

ture work, provided reliable experimental data to recover the non-convex term are

available. On the other hand, the latent-hardening based non-convex potential used

in the last chapter addresses a physical phenomenon, i.e. slip interactions in crystals.

It induces a clear implicit deformation patterning in the present non-convex strain

gradient crystal plasticity framework, compared to its previous usage for external

microstructure evolution via lamination procedure. A more detailed study on the

available functions and possible other descriptions of the non-convexity for specific

materials is obviously needed as a future work.

Bibliography

ACKERMANN, F., MUGHRABI, H., and SEEGER, A. (1983). Temperature and strain-

rate dependence of the flow stress of ultrapure niobium single crystals in cyclic

deformation. Acta Metall., 31, 1353–1366.

ADAMS, J. J., AGOSTA, D. S., LEISURE, R. G., and LEDBETTER, H. (2006). Elastic

constant of monocrystal iron from 3 to 500 K. J. Appl. Phys., 100, 113530.

AIFANTIS, E. C. (1987). The Physics of plastic deformation. Int. J. Plast., 3, 211–247.

ARGON, A. S. (2008). Strengthening mechanisms in crystal plasticity. Oxford Univer-

sity Press.

ASARO, R. and NEEDLEMAN, A. (1985). Texture Development and Strain Hardening

in Rate Dependent Polycrystals. Acta Metall., 33, 923–953.

ASARO, R. J. and RICE, J. R. (1977). Strain localization in ductile single crystals. J.

Mech. Phys. Solids, 25, 309–338.

BACROIX, B. and HU, Z. (1995). Texture evolution induced by strain path changes

in low carbon steel sheets . Metall. Mater. Trans. A, 26, 601–613.

BACROIX, B., GENEVOIS, P., and C., TEODOSIU (1994). Plastic anisotropy in low

carbon steels subjected to simple shear with strain path changes. Eur. J. Mech.

A/Solids, 13, 661–675.

BAILEY, J. E. and HIRSCH, P. B. (1960). The dislocation distribution, flow stress, and

stored energy in cold-worked polycrystalline silver. Philos. Mag., 5, 485.

BARDELLA, L. (2006). A deformation theory of strain gradient crystal plasticity that

accounts for geometrically necessary dislocations. J. Mech. Phys. Solids, 54, 128–160.

BARDELLA, L. (2007). Some remarks on the strain gradient crystal plasticity mod-

elling, with particular reference to the material length scales involved. Int. J. Plast.,

23, 296–322.

BARKER, I., HANSEN, N., and RALPH, B. (1989). The development of deformation

substructures in face-centered cubic metals. Mat. Sci. Eng. A, 113, 449–454.

99

100 Bibliography

BARLAT, F., FERREIRA DUARTE, J. M., GRACIO, J. J., LOPES, A. B., and RAUCH,

E. F. (2003). Plastic flow for non-monotonic loading conditions of an aluminum

alloy sheet sample. Int. J. Plast, 19, 1215–1244.

BARRETT, C. S., ANSEL, G., and MEHL, R F (1937). Slip, twinning and cleavage in

iron and silicon ferrite. Trans. Am. Soc. Met., 25, 702–733.

BASSANI, J. L. (1994). Plastic flow of crystals. Adv. Appl. Mech., 30, 191–257.

BASSANI, J. L., ITO, K., and VITEK, V. (2001). Complex macroscopic plastic flow

arising from non-planar dislocation core structures. Mat. Sci. Eng. A, 319-321, 97–

101.

BAYLEY, C. J., BREKELMANS, W. A. M., and GEERS, M. G. D. (2006). A comparison

of dislocation induced back stress formulations in strain gradient crystal plasticity.

Int. J. Solids Struct., 43, 7268–7286.

BOLEF, D. I. and KLERK, J. D. (1962). Elastic Constants of Single-Crystal Mo and W

between 77 and 500 K. J. Appl. Phys., 33, 2311–2314.

BOUVIER, S., GARDEY, B., HADDADI, H., and C., TEODOSIU (2006). Characteriza-

tion of the strain-induced plastic anisotropy of rolled sheets by using sequences of

simple shear and uniaxial tensile tests. J. Mater. Process. Tech., 174, 115–126.

BRONKHORST, C. A., KALIDINDI, S. R., and ANAND, L. (1992). Polycrystalline

plasticity and the evolution of crystallographic texture in FCC metals. Philos. T.

Roy. Soc. A, 341, 443–477.

BROWN, S. B., KIM, K. H., and ANAND, L. (1989). An internal variable constitutive

model for hot working of metals. Int. J. Plast., 5, 95–130.

BRUNNER, D. and DIEHL, J. (1987). The use of stress-relaxation measurements for

investigations on the flow stress of alpha-iron. Phys. Stat. Sol. A, 104, 145–155.

BRUNNER, D. and DIEHL, J. (1997). The effect of atomic lattice defects on the soften-

ing phenomena of high-purity alpha-iron. Phys. Stat. Sol. A, 160, 355–372.

CAHN, J. W. (1961). On spinodal decomposition. Acta Metall., 9, 795–801.

CARROLL, K. J. (1965). Elastic Constants of Niobium from 4.2 to 300 K. J. Appl. Phys.,

36, 3689–3690.

CERMELLI, P. and GURTIN, M. E. (2001). On the characterization of geometrically

necessary dislocations in finite plasticity. J. Mech. Phys. Solids, 49, 1539–1568.

CHEN, N. K. and MADDIN, R. (1954). Slip planes and energy of dislocations in a

body centered cubic structure. Acta Metall., 2, 49–51.

CHRISTIAN, A., KANERT, O., and DE HOSSON, J. TH. M. (1990). Dislocation dy-

namics in Vanadium: A nuclear magnetic and transmission electron microscopic

study. Acta Metall. Mater., 38, 2479–2484.

Bibliography 101

CHRISTODOULOU, N., WOO, O. T., and MACEWEN, S. R. (1986). Effect of stress

reversals on the work hardening behaviour of polycrystalline copper. Acta Metall.,

34, 1553–1562.

CONTI, S. and ORTIZ, M. (2005). Dislocation microstructures and the effective be-

havior of single crystals. Arch. Ration. Mech. Anal., 176, 103–147.

CONTI, S., HAURET, P., and ORTIZ, M. (2007). Concurrent multiscale computing of

deformation microstructure by relaxation and local enrichment with application to

single-crystal plasticity. Multiscale Model. Simul., 6, 135–157.

CUITINO, A. M. and ORTIZ, M. (1993). Constitutive modeling of L12 intermetallic

crystals. Mater. Sci. Eng. A, 170, 111–123.

DAO, M. and ASARO, R, J. (1993). Non-Schmid effects and localized plastic flow in

intermetallic alloys. Mater. Sci. Eng., 170, 143–160.

DAO, M. and ASARO, R. J. (1996). Localized deformation modes and non-Schmid

effects in crystalline solids, Part I and II. Adv. Appl. Mech., 23, 71–132.

DEL PIERO, G. (2009). On the method of virtual power in continuum mechanics.

Journal of Mechanics of Materials and Structures, 4, 281–292.

DUESBERY, M. S. and FOXALL, R. A. (1969). A detailed study of the deformation of

high purity niobium single crystals. Philos. Mag., 20, 719–751.

DUESBERY, M. S. and VITEK, V. (1998). Plastic anisotropy in BCC transition metals.

Acta Mater., 46, 1481–1492.

DUESBERY, M. S., VITEK, V., and CSERTI, J. (2002). Understanding Materials, 165.

DUNSTAN, D. J. and BUSHBY, A. J. (2004). Theory of deformation in small volumes

of material. Proc. R. Soc. Lond. A, 460, 2781–2796.

EVERS, L. P., BREKELMANS, W. A. M., and GEERS, M. G. D. (2004a). Non-local

crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids,

52, 2379–2401.

EVERS, L. P., BREKELMANS, W. A. M., and GEERS, M. G. D. (2004b). Scale de-

pendent crystal plasticity framework with dislocation density and grain boundary

effects. Int. J. Solids Struct., 41, 5209–5230.

FERNANDES, J. V. and SCHMITT, J. H. (1983). Dislocation Microstructures in steel

during deep drawing. Philos. Mag. A, 48, 841–870.

FRANCIOSI, P. and ZAOUI, A. (1982). Multislip in FCC crystals; a theoretical ap-

proach compared with experimental data. Acta Metall., 30, 1627–1637.

FROST, H. J. and ASHBY, M. F. (1982). Deformation-Mechanism Maps: The plasticity

and creep of metals and ceramics. Pergamon Pr.

GANAPATHYSUBRAMANIAN, S. and ZABARAS, N. (2005). Modeling the

thermoelastic-viscoplastic response of polycrystals using a continuum represen-

tation over the orientation space. Int. J. Plast., 21, 119–144.

102 Bibliography

GARDEY, B., BOUVIER, S., RICHARD, V., and BACROIX, B. (2005). Texture and

dislocation structures observation in a dual-phase steel under strain-path changes

at large deformations. Mater. Sci. Eng., A, 400-401, 136–141.

GEERS, M. G. D., BREKELMANS, W. A. M., and BAYLEY, C. J. (2007). Second-order

crystal plasticity: internal stress effects and cyclic loading. Model. Simul. Mater. Sci.

Eng., 15, 133–145.

GOUGH, H. J. (1928). The behavior of a single crystal alpha-iron subjected to alter-

nating torsional stresses. P. Roy. Soc. Lond. A, 118, 498–534.

GROGER, R. and VITEK, V. (2005). Breakdown of the Schmid law in BCC molybde-

num related to the effect of shear stress perpendicular to the slip direction. Mater.

Sci. Forum, 482, 123.

GROMA, I. (1997). Link between the microscopic and mesoscopic length scale de-

scription of the collective behavior of dislocations. Phys. Rev. B, 56, 5807–5813.

GROMA, I. and PAVLEY, G. S. (1993). Role of the secondary slip system in a computer

simulation model of the plastic behaviour of single crystals. Mater. Sci. Eng. A, 164,

306–311.

GUIU, F. (1969). Slip asymmetry in molybdenum single crystals deformed in direct

shear. Scr. Metall., 3, 449–454.

GURTIN, M. E. (2000). On the plasticity of single crystals: free energy, microforces,

plastic-strain gradients. J. Mech. Phys. Solids, 48, 989–1036.

GURTIN, M. E. (2002). A gradient theory of single-crystal viscoplasticity that ac-

counts for geometrically necessary dislocations. J. Mech. Phys. Solids, 50, 5–32.

GURTIN, M. E., ANAND, L., and LELE, S. P. (2007). Gradient single-crystal plasticity

with free energy dependent on dislocation densities. J. Mech. Phys. Solids, 55, 1853–

1878.

HACKL, K. and KOCHMANN, D. M. (2008). Relaxed potentials and evolution equa-

tions for inelastic microstructures. IUTAM Symposium on Theoretical, Computational

and Modelling Aspects of Inelastic Media, 27–41.

HAHNER, P. (1994). Theory of solitary plastic waves. Part I: Luders bands in poly-

crystals. Appl. Phys. A, 58, 41–48.

HALIM, H., WILKINSON, D. S., and NIEWCZAS, M. (2007). The Portevin–Le Chate-

lier (PLC) effect and shear band formation in an AA5754 alloy. Acta Metall., 55,

4151–4160.

HANSEN, N. and HUANG, X. (1997). Dislocation Structures and Flow Stress. Mat.

Sci. Eng. A, 234, 602–605.

HAVNER, K. S. (1992). Finite Plastic Deformation of Crystalline Solids. Cambridge

Monographs on Mechanics and Applied Mathematics.

Bibliography 103

HILL, R. and RICE, J. R. (1972). Constitutive analysis of elastic-plastic crystals at

arbitrary strains. J. Mech. Phys. Solids, 20, 401–413.

HOC, T. and FOREST, S. (2001). Polycrystal modelling of IF-Ti steel under complex

loading path. Int. J. Plast, 17, 65–85.

HOLLANG, L., HOMMEL, M., and SEEGER, A. (1997). The flow stress of ultra-high-

purity molybdenum single crystals. Phys. Stat. Sol. A, 160, 329–354.

HOLT, D. L. (1980). Dislocation cell formation in metals. J. Appl. Phys., 41, 3197–3201.

HUTCHINSON, J. W. (1976). Bounds and self-consistent estimates for creep of poly-

crystalline materials. Proc. R. Soc. London, A, 348, 101–127.

IRWIN, G. J., GUIU, F., and PRATT, P. L. (1974). The influence of orientation on slip

and strain hardening of molybdenum single crystals. Phys. Stat. Sol. A, 22, 685–698.

ITO, K. and VITEK, V. (2001). Atomistic study of non-Schmid effects in the plastic

yielding of BCC metals. Philos. Mag., 81, 1387–1407.

KALIDINDI, S. R., BRONKHORST, C. A., and ANAND, L. (1992). Crystallographic

texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys.

Solids, 40, 537–569.

KEH, A. S. (1964). Work hardening and deformation sub-structure in iron single

crystals deformed in tension at 298 K. Philos. Mag., 12, 9–30.

KEH, A. S., WEISSMAN, S., THOMAS, G., and WASHBURN, J. (1963). Electron Mi-

croscopy and Strength of Crystals. Interscience.

KOCKS, U. F., ARGON, A. S., and ASHBY, M. F. (1975). Thermodynamics and

Kinetics of Slip. Prog. Mater Sci., 19, 1–291.

KOSLOWSKI, M., CUITINO, A. M., and ORTIZ, M. (2002). A phase field theory of

dislocation dynamics, strain hardening and hysteresis in ductile single crystals. J.


KOTHARI, M. and ANAND, L. (1998). Elasto-viscoplastic constitutive equations for

polycrystalline metals: Application to tantalum. J. Mech. Phys. Solids, 46, 51–83.

KREJCI, J. and LUKAS, P. (1971). Dislocation substructure in fatigued α-iron single

crystals. Phys. Status Solidi A, 5, 315–325.

KUBIN, L. P. and CANOVA, G. (1992). The modelling of dislocation patterns. Scr.

Metall. Mater., 27, 957–962.

KUHL, E. and SCHMID, D. W. (2007). Computational modeling of mineral unmixing

and growth. Comput. Mech., 39, 439–451.

KUHLMANN-WILSDORF, D. (1962). A new theory of work hardening. Trans. Am.

Inst. Min. Metall. Eng., 224, 1047–1061.

KUHLMANN-WILSDORF, D. (1989). Theory of Plastic Deformation: properties of low

energy dislocation structures. Mat. Sci. Eng. A, 113, 1–41.

104 Bibliography

KUHLMANN-WILSDORF, D. (2001). Q: Dislocation structures - how far from equilib-

rium? A: Very close indeed. Mater. Sci. Eng. A, 315, 211–216.

KUHLMANN-WILSDORF, D. and VAN DER MERWE, J. H. (1982). Theory of disloca-

tion cell sizes in deformed metals. Mater. Sci. Eng., 55, 79–83.

LAMBRECHT, M., MIEHE, C., and DETTMAR, J. (2003). Energy relaxation of non-

convex incremental stress potentials in a strain-softening elastic-plastic bar. Int. J.

Solids Struct., 40, 1369–1391.

LANGER, J. S. (1971). Theory of spinodal decomposition in alloys. Ann. Phys., 65,

53–86.

LEE, E. H. (1969). Elastic-plastic deformation at finite strains. J. Appl. Mech., 36, 1–6.

LEE, Y. J., SUBHASH, G., and RAVICHANDRAN, G. (1999). Constitutive modeling of

textured body-centered-cubic (BCC) polycrystals. Int. J. Plast., 15, 625–645.

LI, Z. J., WINTHER, G., and HANSEN, N. (2006). Anisotropy in rolled metals in-

duced by dislocation structure. Acta Mater., 54, 401–410.

LIAO, K. C., FRIEDMAN, P. A., and PAN, J. (1998). Texture development and plastic

anisotropy of BCC strain hardening sheet metals. Int. J. Solids Struct., 35, 5205–

5236.

LUBARDA, V. A., BLUME, J. A., and NEEDLEMAN, A. (1993). An Analysis of Equi-

librium Dislocations Distributions. Acta Metall. Mater., 41, 625–642.

MIEHE, C. (2002). Strain-driven homogenization of inelastic microstructures and

composites based on an incremental variational formulation. Int. J. Numer. Methods

Eng., 55, 1285–1322.

MIEHE, C., SCHOTTE, J., and LAMBRECHT, M. (2002). Homogenization of inelastic

solid materials at finite strains based on incremental minimization principles. Ap-

plication to the texture analysis of polycrystals. J. Mech. Phys. Solids, 50, 2123–2167.

MIEHE, C., LAMBRECHT, M., and GURSES, E. (2004). Analysis of material instabili-

ties in inelastic solids by incremental energy minimization and relaxation methods:

evolving deformation microstructures in finite plasticity. J. Mech. Phys. Solids, 52,

2725–2769.

MOLLICA, F., RAJAGOPAL, K. R., and SRINIVASA, A. R. (2001). The inelastic behav-

iors of metals subject to loading reversal. Int. J. Plast, 17, 1119–1146.

MUGHRABI, H. (1987). A Two-parameter Description of heterogeneous dislocation

Distributions in Deformed Metal Crystals. Mater. Sci. Eng., 85, 15–31.

MUGHRABI, H., UNGAR, T., and WILKENS, W. (1986). Long range internal stresses

and asymmetric X-ray line-broadening in tensile-deformed [001]-oriented copper

single crystals. Philos. Mag. A, 53, 793–813.

NAUMAN, E. B. and HE, D. Q. (2001). Nonlinear diffusion and phase separation.

Chem. Eng. Sci., 56, 1999–2018.

Bibliography 105

NESTEROVA, E. V., BACROIX, B., and TEODOSIU, C. (2001). Microstructure and

texture evolution under strain-path changes in low-carbon interstitial-free steel.

Metall. Mater. Trans. A, 32, 2527–2538.

ORTIZ, M. and REPETTTO, E. A. (1999). Nonconvex energy minimization and dislo-

cation structures in ductile single crystals. J. Mech. Phys. Solids, 47, 397–462.

ORTIZ, M., REPETTO, E. A., and STAINER, L. (2000). A theory of subgrain disloca-

tion structures. J. Mech. Phys. Solids, 48, 2077–2114.

PEETERS, B. (2002). Multiscale modelling of the induced plastic anisotropy in IF steel

during sheet forming. Ph.D. thesis, Katholieke Universiteit Leuven.

PEETERS, B., KALIDINDI, S. R., VAN HOUTTE, P., and AERNOUDT, E. (2000). A

Crystal Plasticity Based Work-Hardening/Softening Model for B.C.C. Metals Un-

der Changing Strain Paths. Acta Mater., 48, 2123–2133.

PEIRCE, D., ASARO, R. J., and NEEDLEMAN, A. (1982). An analysis of non-uniform

and localized deformation in ductile single crystals. Acta Metall., 30, 1087–1119.

PICHL, W. (2002). Slip geometry and plastic anisotropy of body-centered cubic met-

als. Phys. Stat. Sol. A, 189, 5–25.

QIN, Q. and BASSANI, J. L. (1992). Non-Schmid yield behavior in single crystals. J.


QUEYREAU, S., MONNET, G., and DEVINCRE, B. (2008). Slip system interactions

in alpha-iron determined by dislocation dynamics simulations. Int. J. Plast, 25,

361–377.

RACHERLA, V. and BASSANI, J. L. (2007). Strain burst phenomena in the necking

of a sheet that deforms by non-associated plastic flow. Modelling Simul. Mater. Sci.

Eng., 15, 297–311.

RAO, B. V. N. and LAUKONIS, J. V. (1983). Microstructural mechanism for the

anomalous tensile behavior of aluminum-killed steel prestrained in plane tension.

Mater. Sci. Eng., 60, 125–135.

RAUCH, E. F. (1991). Stress reversal tests imposed by shear on mild steel. n: Strength

of Metals and Alloys, (D.G. Brandon, R. Shaim, A. Rosen, Eds.), Haifa, Israel, July 14-19,

1991, (Proc. of ICSMA 9, Freund Publ. House, Ltd., London), 1, 187–194.

RAUCH, E. F. (1992). The flow law of mild steel under monotonic or complex strain

path. Solid State Phenom., 23-24, 317–333.

RAUCH, E. F. and SCHMITT, J. H. (1989). Dislocation Substructures in Mild Steel

Deformed in Simple Shear. Mat. Sci. Eng. A, 113, 441–448.

RICE, J. R. (1971). Inelastic constitutive relations for solids: an internal variable

theory and its application to metal plasticity. J. Mech. Phys. Solids, 19, 433–455.

SANG, H. and LLOYD, D. J. (1979). The influence of biaxial prestrain on the tensile

properties of three aluminum alloys. Metal. Trans. A, 10, 1773–1776.

106 Bibliography

SAUZAY, M., BAVARD, K., and KARLSEN, W. (2010). TEM observations and finite

element modelling of channel deformation in pre-irradiated austenitic stainless

steels Interactions with free surfaces and grain boundaries. J. Nucl. Mater., 406,

152–165.

SCHMID, E. and BOAS, W. (1935). Kristallplastizitat. Springer.

SCHMITT, J. H., FERNANDES, J. V., GRACIO, J. J., and F., VIEIRA M. (1991). Plastic

behavior of copper sheets during sequential tension tests. Mat. Sci. Eng. A, 147,

143–154.

SEEGER, A. (1981). Temperature and strain-rate dependence of the flow stress of

body-centered cubic metals. A theory based on kink-kink interactions. Z. Metallkd.,

72, 369–380.

SEEGER, A. (2001). Why anomalous slip in body-centered cubic metals? Mat. Sci.

Eng. A, 319-321, 254–260.

SEEGER, A. and WASSERBACH, W. (2002). Anomalous slip - A feature of high-purity

body-centered cubic metals. Phys. Stat. Sol. A, 189, 27–50.

SEVILLANO, J. G., VAN HOUTTE, P., and AERNOUDT, E. (1981). Large strain work

hardening and textures. Prog. Mater Sci., 25.

SHAW, J. A. and KYRIAKIDES, S. (1998). Initiation and propagation of localized

deformation in elasto-plastic strips under uniaxial tension. Int. J. Plast., 13, 837–

871.

SHU, J. Y., FLECK, N. A., VAN DER GIESSEN, E., and NEEDLEMAN, A. (2001).

Boundary layers in constrained plastic flow: comparison of nonlocal and discrete

dislocation plasticity. J. Mech. Phys. Solids, 49, 1361–1395.

SPITZIG, W. A. (1979). Effect of hydrostatic pressure on plastic properties - flow

properties of iron single crystals. Acta Metall., 27, 523–534.

SUN, H. B., YOSHIDA, F., MA, X., KAMEI, T., and OHMORI, M. (2003). Finite ele-

ment simulation on the propagation of Luders band and effect of stress concentra-

tion. Matter. Lett., 57, 3206–3210.

SVENDSEN, B. (2002). Continuum thermodynamic models for crystal plasticity in-

cluding the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids, 50,

1297–1329.

SVENDSEN, B. (2004). On thermodynamic- and variational-based formulations of

models for inelastic continua with internal length scales. Comput. Meth. Appl. Mech.

Eng., 193, 5429–5452.

SWANN, P. R. (1963). Electron Microscopy and Strength of Crystals. G. Thomas and J.

Washburn, eds. Interscience, New York, 131.

TARIGOPULA, V., HOPPERSTAD, O. S., LANGSETH, M., and H., CLAUSEN A. (2008).

Elastic-plastic behavior of dual-phase, high-strength steel under strain path

changes. Eur. J. Mech. A/Solids, 27, 764–782.

Bibliography 107

TAYLOR, G. I. and ELAM, C. F. (1926). The distortion of iron crystals. P. Roy. Soc.

Lond. A, 112, 337–361.

TEODOSIU, C. (1992). Materials Science Input to Engineering Models. Modeling of

Plastic Deformation and its Engineering Applications (Proc. 13th Riso), 125–146.

TEODOSIU, C. and HU, Z. (1995). Simulation of Materials Processing: Theory and Appli-

cations (Proc. of NUMIFORM ’95), 173–182.

UBACHS, R. L. J. M., SCHREURS, P. J. G., and GEERS, M. G. D. (2004). A non-

local phase field model for microstructure evolution of Sn-Pb solder. J. Mech. Phys.

Solids, 52, 1763–1792.

VIATKINA, E. (2005). Micromechanical modelling of strain path dependency in FCC

metals. Ph.D. thesis, Eindhoven University of Technology.

VIATKINA, E. M., BREKELMANS, W. A. M., and GEERS, M. G. D. (2003). Strain path

dependency in metal plasticity. J. Phys. IV, 105, 355–362.

VITEK, V. (2004). Core structure of screw dislocations in body-centred cubic metals:

Relation to symmetry and interatomic bonding. Philos. Mag., 84, 415–428.

VITEK, V., MROVEC, M., GROGER, R., BASSANI, J. L, RACHERLA, V., and YIN,

L. (2004a). Effects of non-glide stresses on plastic flow of single and polycrystals

of molybdenum. Mater. Sci. Eng. A, 387, 138–142.

VITEK, V., MROVEC, M., and BASSANI, J. L. (2004b). Influence of non-glide stresses

on plastic flow: from atomistic to continuum modeling. Mater. Sci. Eng. A, 365,

31–37.

WAGONER, R. H. and LAUKONIS, J. V. (1983). Plastic behavior of aluminum-killed

steel following plane-strain deformation. Metal. Trans. A, 14, 1487–1495.

WALGRAEF, D. and AIFANTIS, E. C. (1985). Dislocation patterning in fatigued metals

as a result of dynamical instabilities. J. Appl. Phys., 58, 688–691.

WANG, J., LEVKOVITCH, V., REUSCH, F., SVENDSEN, B., HUETINK, J., and

VAN RIEL, M. (2008). On the Modeling of Hardening in Metals During Non-

Proportional Loading. Int. J. Plast., 24, 1039–1070.

WANG, Y. U., JIN, Y. M., CUITINO, A. M., and KHACHATURYAN, A. G. (2001a).

Application of phase field microelasticity theory of phase transformations to dis-

location dynamics: Model and three dimensional simulations in a single crystal.

Philos. Mag. Lett., 81, 385–393.

WANG, Y. U., JIN, Y. M., CUITINO, A. M., and KHACHATURYAN, A. G. (2001b).

Nano scale phase field microelasticity theory of dislocations: Model and 3D simu-

lations. Acta Mater., 49, 1847–1857.

WANG, Y. U., JIN, Y. M., CUITINO, A. M., and KHACHATURYAN, A. G. (2001c).

Phase field microelasticity theory and modeling of multiple dislocation dynamics.

Appl. Phys. Lett., 78, 2324–2326.

108 Bibliography

WERNER, M. (1987). Temperature and strain-rate dependence of the flow stress of

ultrapure tantalum single crystals. Phys. Stat. Sol. A, 104, 63–78.

WILSON, D. V. and BATE, P. S. (1994). Influence of cell walls and grain boundaries

on transient response of an IF steel to changes in strain path. Acta Metall. Mater.,

42, 1099–1111.

XIE, C. L., GHOSH, S., and GROEBER, M. (2004). Modeling Cyclic Deformation of

HSLA Steels Using Crystal Plasticity. J. Eng. Mater. Technol., 126, 339–352.

YALCINKAYA, T., BREKELMANS, W. A. M., and GEERS, M. G. D. (2008). BCC single

crystal plasticity modeling and its experimental identification. Model. Simul. Mater.

Sci. Eng., 16, 085007.

YALCINKAYA, T., BREKELMANS, W. A. M., and GEERS, M. G. D. (2009). A com-

posite dislocation cell model to describe strain path change effects in BCC metals.

Model. Simul. Mater. Sci. Eng., 17, 064008.

YALCINKAYA, T., BREKELMANS, W. A. M., and GEERS, M. G. D. (2011a). Deforma-

tion patterning driven by rate dependent nonconvex strain gradient plasticity. J.

Mech. Phys. Solids., 59, 1–17.

YALCINKAYA, T., BREKELMANS, W. A. M., and GEERS, M. G. D. (2011b). Non-

convex rate dependent strain gradient crystal plasticity and deformation pattern-

ing. In Prep.

YEFIMOV, S., GROMA, I., and VAN DER GIESSEN, E. (2004). A comparison of a

statistical-mechanics based plasticity model with discrete dislocation plasticity cal-

culations. J. Mech. Phys. Solids, 52, 279–300.

YOSHIDA, F., KANEDA, Y., and YAMAMOTO, S. (2008). A plasticity model describing

yield-point phenomena of steels and its application to FE simulation of temper

rolling. Int. J. Plast., 24, 1792–1818.

YOUNG, C. T., HEADLEY, T. J., and LYTTON, J. L. (1986). Dislocation substructures

formed during the flow stress recovery of high purity aluminum. Mater. Sci. Eng.,

81, 391–407.

DE BORST, R. (1987). Computation of post-bifurcation and post-failure behaviour of

strain-softening solids. Comput. Struct., 25, 211–224.

Acknowledgements

Firstly, I would like to express my deep gratitude to my promoter Marc Geers, who

gave me the opportunity to start my PhD research in his group and who has been

continuously supporting me scientifically and socially throughout the whole PhD

process. He was not only the person to ask questions about the convergence prob-

lems in the models or the thermodynamical relations but also he has been the exam-

ple researcher widening my horizon considerably. His positive approach during the

stressful times has always ended up with high motivation. Marc, thank you.

I would like to thank my co-promoter Marcel Brekelmans for his valuable input in the

process of developing each chapter and his fruitful discussions and criticism during

the development phase of the models and also on writing articles. Even though he

has been partially stopped with the university he never gave up supporting me.

I sincerely thank to the committee members; Prof.Dr. Klaus Hackl, Prof.Dr. Fionn

Dunne, Prof.Dr. Bob Svendsen, Prof.Dr. Marc Peletier and Dr. Henk Vegter for their

interest and constructive comments on the thesis.

A special thanks goes to Dr. Izzet Ozdemir who has always been there for discussions

and input. He has taught me how to be stubborn on debugging process and how to

overcome numerical problems during the implementation of some of the models.

I am grateful to Prof.Dr. Jeff de Hosson whom I visited in Groningen to discuss on

BCC crystals and who supported me via emails when I needed physical insight for

my computational models. Even though I have never met him, Prof.Dr. Ali Argon

from MIT has supported me remotely by answering my emails on the difficulties I

have faced in crystal plasticity modeling. I would like to thank him for his valuable

comments.

My sincere thanks goes to the IT administrators Patrick and Leo for their never-

ending support related to computer issues. And I would like to thank Alice van

Litsenburg in Mate and Monika Hoekstra from M2i for the support on solving prac-

tical matters faced throughout the PhD years in the Netherlands.

109

110 Acknowledgements

I would like to thank Karl Fredrik Nilsson, Peter Haehner and Vesselina Rugnelova

at the Institute of Energy in Joint Research Centre of European Commission, where I

have already started my post-doctoral research, for their understanding and support

on wrapping up my thesis. Considering the fact that chapter 5 of the thesis has

been completed at the Institute of Energy, the support of the institute is gratefully

acknowledged.

Finally, I am extremely grateful to my parents Duran and Tazegul and my sister

Derya. I am such a lucky person to have such lovely and open-minded family. They

have always supported me for all the decisions I have taken.

Tuncay Yalcınkaya,

Petten, August 2011.

Curriculum Vitae

Tuncay Yalcınkaya was born on January 31, 1980 in Ankara, Turkey. After completing

his high school education in Batıkent High School in 1998, he studied Aerospace

Engineering at Middle East Technical University, Ankara and got his B.Sc. degree in

2003. Directly thereafter, he continued his studies with a full DAAD scholarship at

the University of Stuttgart and got his M.Sc. degree on computational mechanics of

materials and structures (COMMAS) in 2005. After his graduation he was employed

by the Materials Innovation Institute (M2i) in Delft and has been working on his PhD

thesis on strain path change effects and crystal plasticity modeling of BCC metals at

the Mechanical Engineering Department of the Eindhoven University of Technology

under the supervision of Prof.Dr.Ir. M.G.D. Geers and Dr.Ir. W.A.M. Brekelmans.

He is currently a post-doctoral researcher at the Institute for Energy - Joint Research

Centre of European Commission.

111

Microstructure evolution in crystal plasticity : strain ...

Documents

Transcript of Microstructure evolution in crystal plasticity : strain ...