microstructure evolution in crystal plasticity : strain
TRANSCRIPT
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
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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
viii
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
62 A finite strain BCC single crystal plasticity model and its experimental
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〉
82 A finite strain BCC single crystal plasticity model and its experimental
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
102 A finite strain BCC single crystal plasticity model and its experimental
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.
122 A finite strain BCC single crystal plasticity model and its experimental
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
142 A finite strain BCC single crystal plasticity model and its experimental
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)
162 A finite strain BCC single crystal plasticity model and its experimental
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
182 A finite strain BCC single crystal plasticity model and its experimental
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.
2.5 Modeling some intrinsic properties of BCC single crystals 19
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
202 A finite strain BCC single crystal plasticity model and its experimental
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
2.5 Modeling some intrinsic properties of BCC single crystals 21
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
222 A finite strain BCC single crystal plasticity model and its experimental
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.
24
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
283 A composite dislocation cell model to describe strain path change effects
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
303 A composite dislocation cell model to describe strain path change effects
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
323 A composite dislocation cell model to describe strain path change effects
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
343 A composite dislocation cell model to describe strain path change effects
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.
363 A composite dislocation cell model to describe strain path change effects
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)
383 A composite dislocation cell model to describe strain path change effects
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.
403 A composite dislocation cell model to describe strain path change effects
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.
3.5 Numerical examples 41
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
423 A composite dislocation cell model to describe strain path change effects
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
443 A composite dislocation cell model to describe strain path change effects
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-
484 Deformation patterning driven by rate dependent non-convex strain
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
504 Deformation patterning driven by rate dependent non-convex strain
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)
524 Deformation patterning driven by rate dependent non-convex strain
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.
544 Deformation patterning driven by rate dependent non-convex strain
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
564 Deformation patterning driven by rate dependent non-convex strain
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
584 Deformation patterning driven by rate dependent non-convex strain
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
4.6 Numerical examples 59
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-
604 Deformation patterning driven by rate dependent non-convex strain
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
4.6 Numerical examples 61
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
624 Deformation patterning driven by rate dependent non-convex strain
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
4.6 Numerical examples 63
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
644 Deformation patterning driven by rate dependent non-convex strain
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.
4.6 Numerical examples 65
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
664 Deformation patterning driven by rate dependent non-convex strain
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
4.6 Numerical examples 67
(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
684 Deformation patterning driven by rate dependent non-convex strain
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)
704 Deformation patterning driven by rate dependent non-convex strain
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.
72
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
765 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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 ˆu, ˆεe
and ˆγα as virtual rates, which are collected in the generalized virtual velocity V =
( ˆu, 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) · ˆu 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)
5.2 Strain gradient crystal plasticity and finite element implementation 77
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)
785 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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)
5.2 Strain gradient crystal plasticity and finite element implementation 79
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)
805 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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)
825 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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 .
845 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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
Applied shear Γ = u/H
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
Applied shear Γ = u/H
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
Applied shear Γ = u/H
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
865 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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
5.4 Numerical analysis 87
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
Applied shear Γ = u/H
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
885 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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
5.4 Numerical analysis 89
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
905 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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
5.4 Numerical analysis 91
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).
925 Non-convex rate dependent strain gradient crystal plasticity and
deformation patterning
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.
94
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.
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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