Microstructure Evolution in Crystal Plasticity

8/19/2019 Microstructure Evolution in Crystal Plasticity

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

patterning


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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 Yalçınkaya

Microstructure evolution in crystal plasticity: strain path effects and dislocationslip patterning /

by T. Yalçınkaya – 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 c2011 by Tuncay Yalçınkaya, all rights reserved.

This thesis was prepared with the LATEX 2ε documentation system.

Reproduction: Universiteitsdrukkerij TU Eindhoven, Eindhoven, TheNetherlands.


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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 Yalçınkaya

geboren te Ankara, Turkije


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Dit proefschrift is goedgekeurd door de promotor:

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

Copromotor:

dr.ir. W.A.M. Brekelmans


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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 . . . . . . . . . . . . . . . . . . 102.4 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


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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 . . . . . . . . . 393.5.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.8.1 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 . . 755.3 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


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Contents vii

Dankwoord / Acknowledgements 109

Curriculum Vitae 111


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viii


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Summary

During deformation polycrystalline metals tend to develop heterogeneous plasticdeformation 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 (Lüders 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). Thesemicrostructures 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 theanisotropy 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


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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 areconsidered 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-equilibriumstates 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.


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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 plasticityA 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


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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 thesisemploys 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-


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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 historieswhich 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 screwdislocation 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


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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 componentand 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 plasticityframework 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.


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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/2111 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 orientations

and 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 including α -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


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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/2111 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 awavy type {112} cross slip pattern. At room temperature a/2111 type of screwdislocations 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 acceptedthat 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


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

tensors, related products and crystallography:

• scalars a• vectors a = a iei• second-order tensors A = Ai jei ⊗ e j

• fourth-order tensors 4 A = Aijklei ⊗ e j ⊗ ek ⊗ el• C = a ⊗ b = a ib jei ⊗ e j• C = A · B = Ai jB jk ei ⊗ ek • C = 4 A : B = Aijkl Blk ei ⊗ e j• crystallographic direction, family [uvw], uvw• crystallographic plane, family (hkl),{hkl}• slip system, family (hkl)[uvw], {hkl} uvw


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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 followingparagraphs.

Gough (1928) and Barrett et al. (1937) state that the {110}, {112} and {123} familiescontain 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, wherebyit 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, basedon 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). Usingthe 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 onthe 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](1̄01) and µ = 0.433 for the sys-

tems [111](1̄1̄2) and [111](2̄11). 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,

T K in Fig. 2.1) and above the knee temperature is thereby distinguished. Below theknee temperature the slip is governed by the glide of a/2111 type screw disloca-


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2.2 Slip mechanisms in BCC metals 9

0 100 200 300 400 5000

200

400

600

800

1000

Temperature T (K)

F l o w

S t r e s s ( M P a )

T TK

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

slip

wavy slip lines

slip lines

straight

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 strainrate 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 Ť (70 K for Mo and 120 K forα -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 Ť , the dislocation core configuration changes and dislocations undergo a tran-

sition from their low temperature configuration (slipping on {110} planes) to theirhigh temperature configuration (slipping on {112} planes). The dislocations showa wavy type of structure which results from the cross slip, a characteristic property


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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 peculiaritiesin 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 ata 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 shearin the twinning direction, whereas shear on another plane (e.g. {2̄11}) producesshear 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-


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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 adifferent 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)[111̄] (4) A (112̄)[111] (7) A (12̄1)[111] (10)T (2̄11)[111]

(2) A (1̄12)[11̄1] (5) A (121)[11̄1] (8) A (121̄)[1̄11] (11)T (21̄1)[111̄]

(3) A (11̄2)[1̄11] (6) A (1̄21)[111̄] (9) T (211)[1̄11] (12)T (211̄)[11̄1]

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/2111 type screw dislocation cores. Whereas this effect is oftencalled 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), Gröger and Vitek (2005))

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


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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 F e and a plastic part F p according to:

F = F e · 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 F e reflects the lattice deformation and local rigid body

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

n

m α

α

n0α

mα

p

e p

= F

= F

0 mα0

mα

nα

.

e

e

.

. n0α

mα0

−T

Fe F = F F

F n0

α

Figure 2.2 / Multiplicative decomposition of the deformation gradient.

ns is the total number of slip systems. The vectors mα 0 and n

α 0 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

L p =ns

∑α =1

γ̇ α mα 0nα 0 (2.2)

with γ̇ α the individual slip rates on the slip systems.


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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 = 4

C : E e and Ee =

1

2 (C e − I ) , C e = F T e · F e (2.3)

with C e 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 = J e·σ with J e = det( F e) (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.

τ α = mα ·τ ·nα = mα 0 ·C e · S · nα 0 (2.6)The slip systems of the {112} family in BCC crystals are given in Table 2.1, which isthe 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


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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 = sα t + sα a (2.7)

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

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

athermal part of the resistance sα a 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 sα t 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, v̄α on slip system α , can be esti-

mated by,

v̄α = lα ω0 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, γ̇ α = bρm v̄α 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,

γ̇ α =

0 if τ α e f f ≤ 0γ̇ α 0 exp{−∆GkT } sign(τ α ) if 0 < τ α e f f

(2.10)

where τ α e f f = |τ α | − sα a is the driving force for the dislocation motion and γ̇ α 0 =bρml

α ω0 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 −τ α e f f sα t pq (2.11)


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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:

τ α n = τ α + 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) + η zz (z⊗ z)+

ηmz (m⊗ z + z ⊗m) + ηnz(n⊗ z + z ⊗ n) (2.13)

In this framework, τ α n enters the equations (2.10) and (2.11) via τ α e f f = |τ α n | − sα a .

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 sα t of the slip resistance sα is taken constant

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

ṡα = ṡα a = ∑β

hαβ| γ̇ β| (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

= qαβ

hβ

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

qαβ =

1 qn . . . qnqn 1 . . . qn...

...

.. .

...

qn qn . . . 1

(2.16)


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identification

F

F e F p

S, τ α , τ α n γ̇

F c = F e · F p

R = F − F c

F e := F e + ∆ 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 Brownet al. (1989), reads

hβ = hβ0

1 − sβasβsa sign

1 − s

βa

sβs

, (2.17)

where hβ0 , sβa and s

βs 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 F e,

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


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2.5 Modeling some intrinsic properties of BCC single crystals 17

F c = F e · F p. Generally, the calculated F c and the imposed F will be different, whichresults in a residual R. The linearization of the residual by computing the sensitiv-

ity with respect to F e leads to an update ∆ F e of the elastic part of the deformationgradient. 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. Onthese 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 comparedto 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


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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 of α -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 (1̄1̄2), (2̄11) and

(2̄11) with the [001], [011] and [1̄11] 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 %


[011]

[001]

[1̄11]

Figure 2.4 / Tensile orientation dependence of [001], [011] and [1̄11] 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 [1̄11] 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.


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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 arepresented 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 η zz -0.0267Reference strain rate γ̇ 0 1.07 × 106s−1Activation free energy G0 2.95 × 10−18 J

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


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identification

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

35

40

Strain %


[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 η zz -0.0369Reference strain rate γ̇ 0 1.4 × 107s−1Activation free energy G0 0.1554 × 10−18 J

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


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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 T K , different regimes should be distinguished. Especially the

interval T K /2 < T < T K 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 %


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

strain curves are reproduced from the shear values reported for (1̄12)[11̄1] primary


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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)),γ̇ 0 = 1.2 × 107s−1, qn = 1.4, G0 = 1.12 × 10−18 J, k = 1.3807 × 10−23 J/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 confrontingthe 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 phenomenaand 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


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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 shearstress, 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.


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24


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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 shape

through 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


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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 apowerful 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 isthe 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


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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 thestrain 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

w

w

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 firstexample 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 presentedon the basis of which the crystal anisotropy is distinguished from the dislocation


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

tensors, related products and crystallography:

• scalars a

• vectors a = a iei

• second-order tensors A = Ai jei

⊗e j

• fourth-order tensors 4 A = Aijklei ⊗ e j ⊗ ek ⊗ el• C = a ⊗ b = a ib jei ⊗ e j• C = A · B = Ai jB jk ei ⊗ ek • C = 4 A : B = Aijkl Blk ei ⊗ 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


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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 orientationand 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 wallsρw (e.g. Fernandes


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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 typicalof 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-


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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 i p 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 superscriptα , withα = 1, 2..., ns where ns is the total number of slip systems. The vectors m

α ,i0 and n

α ,i0


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Microstructure Evolution in Crystal Plasticity

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