International Journal of Engineering Science 48 (2010) 1401–1412

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International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

Instability origin of subgrain formation in plastically deformed materials

Jan Kratochvíl a,b,⇑, Martin Kruzík c,a, Radan Sedlácek d,1

a Czech Technical University, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague, Czech Republicb Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague, Czech Republicc Institute of Information Theory and Automation of the ASCR, Pod vodárenskou vezí 4, 182 08 Prague, Czech Republicd Technische Universität München, Fakultät für Maschinenwesen, Boltzmannstr. 15, 85747 Garching, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Available online 28 October 2010Communicated by: K.R. Rajagopal

Keywords:Crystal plasticityEnergetic solutionsGradient plasticitySubgrain formation

0020-7225/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijengsci.2010.09.017

⇑ Corresponding author at: Czech Technical UniveE-mail address: kratochvil@fsv.cvut.cz (J. Kratoch

1 Present address: AREVA NP, Erlangen, Germany.

The review is focused on two methods of formulation and solution of the subgrain forma-tion problem: an energetic approach and a model of incremental deformations. Both meth-ods are based on a reduced single slip version of crystal plasticity. The mathematicalanalysis of the energetic approach is done for a single slip model only; in the incrementalapproach the deformation are assumed small, hence, multi slip can be treated as a sum ofsingle slips. The energetic approach has been employed in analysis of the crystal plasticitymodel of shear and kink bands. The incremental higher strain gradient model provides aninsight into an initial stage of the subgrain formation and the mechanism controlling thesubgrain size.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Formation of misoriented structural elements, typically in a form of subgrains or misoriented dislocation cells (in the fol-lowing text we use the short term ‘‘subgrains”), is a fundamental process of dislocation patterning accompanying plasticdeformation [1]. Subgrains can be found in plastically deformed metals on very different scales, from sub-micron subgrainsinduced by severe plastic deformation to mm-size subgrains in metals deformed near the melting temperature. The impor-tance of the phenomenon is recognized by considering the process of work hardening, which is clearly correlated with for-mation of subgrains.

Conventional explanations of formation of subgrains assume either a pre-existing structure of obstacles in the crystal [2]or use statistical arguments [3,4]. In the statistical approach misoriented dislocation structures are considered as a randomaccumulation process of excess dislocations in dislocation boundaries. Two types of boundaries are distinguished: ordinaryboundaries are assumed to be caused by a statistical mutual trapping of dislocations and excess dislocations are accumulatedby stochastic reasons only, whereas a different activation of slip systems is expected on both sides of planar dislocationboundaries termed geometrically necessary boundaries. As noted in [4], such imbalance in the activation of slip systems be-tween different regions can arise from an intrinsic instability of the deformation process.

As shown in the present paper, the reason for the plastic deformation being non-homogeneous is a possible instability ofhomogeneous plastic flow driven by energy minimization. The long-range internal stresses which would be set up by defor-mation of a single volume element are reduced by deformation and rotation of neighboring volume elements [5,6]. Accord-ing this alternative approach, dislocations are forced by the laws of non-linear continuum mechanics to arrange themselvesinto patterns of varying density. From this point of view, the intrinsically developed inhomogeneity of plastic deformation is

. All rights reserved.

rsity, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague, Czech Republic.víl).

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the basic reason for subgrain formation. In addition to the macroscopic instabilities of plastic deformation in the form ofnecking or shear band formation, there is an instability in the form of internal buckling [7–9]. This instability is made pos-sible by the inherent anisotropy of plastic deformation. Under certain circumstances, more energy is needed in the uniformplastic deformation than is required to initiate an internal mode of buckling. In such a case, the internal instability makes itsappearance. It has been suggested to interpret the internal instability of homogeneous plastic flow in terms of the formationof subgrains in [10–13]. It has been shown that the internal buckling leads to the build up of lattice misorientations betweenneighboring volume elements. The periodic patterns of excess dislocations necessary to accommodate the lattice misorien-tations were interpreted as the beginning of subgrain formation.

The aim of this work is to review shortly the latter approach, focusing attention on two methods of formulation and solu-tion of the subgrain formation problem: an energetic approach, Section 3, and a model of incremental deformations, Section4. The methods are of a different origin and remain disjointed at present using even different scientific language; hopefullythey will converge in the future. Both methods are based on a reduced single slip version of crystal plasticity. The detailmathematical analysis of the energetic approach is done for a single slip model only; in another energetic attempt [14] multislip is treated as a succession of single slips. In the incremental approach the deformation are assumed small, hence, multislip can be treated as a sum of single slips.

The energetic method employed in crystal plasticity has been inspired by the very successful mathematical theory of rate-independent processes [15]. The energetic approach in crystal plasticity leads to a problem of a minimization of an energyfunctional subjected to boundary conditions and dissipation inequality. The minimization may result in a spontaneous struc-tural inhomogeneity (subgrains can be treated as a composition of lamellar deformation modes, Sections 3 and 4). The exactmathematical proof of the existence of lamellar structures was given by Conti and Theil [16] for a single slip rigid-plasticmodel with zero hardening (elastic deformation is reduced to lattice rotations and the hardening coefficient h = 0). Moreover,they predicted existence of a boundary layer which accommodates the lamellar structure to displacement boundary condi-tions. Their results indicate that the dominant effect, which causes formation of the lamellar structure, is the minimization ofthe dissipative energy (the rigidity excludes the elastic energy and h = 0 causes no change of the dislocation stored energy).The energetic approach has been employed in analysis of the crystal plasticity model of shear and kink bands [17] summa-rized in Section 3.

The other method presented in Section 4 is based on mechanics of incremental deformations proposed by Biot [9]. Thetheory provides rigorous and completely general equations governing the dynamics and stability of solids and fluids underinitial stress in the context of small perturbations. It is applicable to anisotropic, viscoelastic, or plastic media. In Section 4Biot’s approach is employed in an analysis of a strain gradient rigid-plastic model of crystalline solids [13]. It provides aninsight into an initiation of the subgrain formation and into the mechanism controlling the subgrain size. In order to intro-duce a physically relevant scale to our problem we assume, following earlier works of Dillon and Kratochvíl [18], that theenergy in the system depends also on the gradient of the plastic variables. The gradient terms represent non-local effectscaused by short-range interactions among dislocations. It is not clear, however, which function of the gradient should beused. We refer to Kratochvíl and Sedlácek [19], and to Groma and Bakó [20] for attempts to derive it from statistics of dis-crete dislocations which reveal the complexity of the problem. Constitutive relations of gradient continua are advocated e.g.by Gurtin [21], Mainik and Mielke [22], or Conti and Ortiz [14] and also investigated in [23] mostly in relation to the so-calledsize effect. Mathematical theory of rate-independent isothermal evolution with gradients of plastic variables is developed in[24]. An interesting recent contribution by Gurtin and Anand [25] discusses the flow rules for rate-independent gradientplasticity proposed by Fleck and Hutchinson. As a key result, they showed that the flow rule of Fleck and Hutchinson [26]is incompatible with thermodynamics unless its nonlocal term is skipped. A physically sound gradient plasticity theory with-in the framework of small deformations is developed in [27]. A survey of non-local models in plasticity appeared in Bazantand Jirásek [28]. Numerical approaches are surveyed in [29,30].

2. Crystal plasticity

The energetic and incremental methods are based on the crystal plasticity framework introduced in classical papers, e.g.[31,32] and recalled e.g. in [21]. In the present paper the rigid-plastic, rate independent approximation to crystal plasticity isconsidered; this framework seems to be sufficient to catch the essence of the subgrain formation problem.

Each material point of a crystal can be identified by its position in a reference configuration. The point which was at posi-tion X in the reference configuration is in the current configuration in time t in the position x(X, t). The difference u = x � X isthe displacement of the material point X. The deformation of the material is described by the transformation F of an infin-itesimal material fiber from the reference to the current configuration,

dx ¼ FdX: ð1Þ

Assuming that x(X, t) is a continuous and differentiable vector field, this transformation can be introduced as the deformationgradient F = @x/@X = I + @u/@X, where I is the second order identity tensor. In the rigid-plastic approximation the crystal lat-tice can (rigidly) rotate but it is not (elastically) strained. The plastic deformation of a crystal can be decomposed in twosteps. First, the material flows through the crystal lattice by shearing along the active slip systems to reach an intermediateconfiguration. This step is described by the plastic deformation gradient Fp, detFp = 1. Second, the plastic deformation Fp is

J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412 1403

followed by a rigid rotation R of the lattice representing the elastic part of the deformation gradient.2 The correspondingdecomposition reads

2 In t3 In d

distorti4 In a

F ¼ RFp ð2Þ

hence, detF = 1. The plastic deformation gradient Fp transforms the reference configuration into the lattice (intermediate)configuration, R transforms the lattice configuration to the current configuration, and F transforms the reference configura-tion into the current configuration.

Unlike F, the tensors Fp and R do not generally correspond to a gradient of a vector field, i.e. they may be individuallyincompatible. In case of inhomogeneous plastic deformation Fp, the lattice rotation R can reestablish the compatibility ofthe overall material deformation. The density of excess dislocations, usually called geometrically necessary dislocations(GNDs), required for the material to be compatible can be characterized by the GND density tensor K = RTcurl(RT) (an over-view and analysis of various measures of GND density is given in [34]).

The velocity v of a material point is given by the material time derivative of its position, vðx; tÞ ¼ _xðX; tÞ. Now we performthe material time derivative of Eq. (1),

d _x ¼ _FdX ¼ @vðx; tÞ@X

dX ¼ @v@x

FF�1dx ¼ Ldx; ð3Þ

where Lðx; tÞ ¼ _FF�1 ¼ @v=@x is the velocity gradient. Using Eq. (2), the latter can be decomposed3 as

L ¼ Lp þ _RRT ; ð4Þ

where Lp is the plastic flow represented by the rate of plastic distortion in the current configuration and _RRT is the latticespin.

The motion of glide dislocations carrying plastic flow takes place on slip systems (i), i = 1, 2, . . ., N. The (i) slip system isdefined by the unit vector s(i) in the direction of slip and by the unit normal to the glide plane m(i). In the lattice configurationthe vectors s(i) and m(i) are constant, given by the crystallographic structure. The plastic flow is governed by slip rates m(i)(x, t)on the individual slip systems via the flow rule

Lp ¼XN

i¼1

mðiÞsðiÞ �mðiÞ: ð5Þ

The slip rate m(i) in the current configuration is driven by the resolved shear stress s(i),

sðiÞ ¼ RsðiÞ � ðTRmðiÞÞ; ð6Þ

where T is the Cauchy stress tensor and Rs(i) and Rm(i) represent the slip direction and the normal to the slip plane in thecurrent configuration, which rotates rigidly with the lattice. In a quasi static process with no body forces the stress T hasto satisfy the equilibrium equation

divT ¼ 0: ð7Þ

Within the present mechanical framework the second law of thermodynamics is reduced to the requirement that the rate ofplastic dissipation be nonnegative

T � D ¼XN

i¼1

sðiÞmðiÞ P 0; ð8Þ

where D = 1/2(rv + (rv)T) and the equality follows from (4)–(6), see also [21, p.998]. The constitutive relations of the rate-independent rigid-plastic material are represented by the yield condition: the slip system remains active, i.e. m(i) may be non-zero, if and only if

sðiÞ ¼ sðiÞy signmðiÞ ð9Þ

if jsðiÞj < sðiÞy , the rate m(i) = 0. The critical resolved shear stress sðiÞy ðx; tÞP 0 represents local dissipative internal forces thatoppose slip. In a rate-independent material, sðiÞ 6 sðiÞy . The requirement sðiÞy P 0 and the yield condition (9) guarantee thevalidity of the dissipation inequality (8).4

Gurtin [21] has shown that as a consequence of rate-independence the evolution equation for sðiÞy has the form

_sðiÞy ¼X2

j¼1

HijjmðjÞj: ð10Þ

erms of Rajagopal and Srinivasa [33] Re represents the rotation of the crystal lattice from its natural configuration.etail: L ¼ _FF�1 ¼ R _FpðFpÞ�1RT þ _RRT , where _FpðFpÞ�1 is the rate of plastic distortion in the reference lattice, R _FpðFpÞ�1RT ¼ Lp is the rate of plastic

on rotated with the lattice into the current configuration, and _RRT is the lattice spin.n alternative approach [21] the yield condition (9) is treated as a consequence of the rate-independence and (8).

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The hardening matrix components Hij are generally functions of variables, which may be themselves controlled by evolutionequations. In the present context such variables are sðiÞy and K. A higher level of the critical resolved shear stress sðiÞy promotesdislocation annihilation, hence, it may decrease some components of hardening matrix Hij and clusters of dislocations rep-resented by the density tensor K serve as favorable centers of generation and annihilation of dislocations.

A special case, important for the energetic and incremental methods reviewed in the present paper, occurs when only oneslip system is active, i.e. the case of single slip. Assuming that Fp = I initially, the time integration of the flow rule (5) adjustedfor single slip, and the identities (s �m) � (m � s) = 1 and (s �m) � (s �m) = 0 yield

Fp ¼ I þ cs�m; m ¼ _c; c ¼ 0 initially: ð11Þ

Hence, the admissible deformation gradient F has to be of the form

F ¼ RðI þ cs�mÞ: ð12Þ

From the last equation it follows that

c ¼ ðFmÞ � ðFsÞ: ð13Þ

As emphasized by Gurtin [21], the existence of a physically meaningful kinematical variable c whose material time deriva-tive is the slip rate m is unique to the single slip only.

3. Energetic approach to rate-independent plasticity

Mathematical models describing rate-independent elasto-plastic behavior of materials typically lead to partial differen-tial equations and inclusions which are not analytically tractable without suitable regularizing terms, e.g., plastic strain gra-dients. This is often because the models predict formation of infinitely fine microstructures. Based on the timesemidiscretization of these models a so-called energetic solution was developed by Mielke and his collaborators (see e.g.[22] for gradient plasticity) following earlier work of Ortiz and Repetto [35]. This concept of solution is based on two require-ments. First, as a consequence of the conservation law for linear momentum, all work put into the system by external forcesor boundary conditions is spent on increasing the stored energy or it is dissipated. Secondly, the formulation must satisfy the2nd law of thermodynamics, which has in the present mechanical framework the form of a dissipation inequality. The lastrequirement enters the framework as the assumption of the existence of a nonnegative convex potential of dissipative forces.As a consequence the imposed deformation evolves in such a way that the sum of stored and dissipated energies is alwaysminimized. In case of a rigid-plastic material the elastic deformation gradient is always a rotation, hence, the stored energy isconstant. It means that all work put to the system by the external loading must be dissipated. If [0,T] is a process-time inter-val, let us define cðiÞðtÞ :¼

R t0 mðiÞðsÞds. Then _cðiÞ ¼ mðiÞ and the dissipation rate (8) reads Rð _cÞ :¼

PNi¼1s

ðiÞy j _cðiÞj. Following Mainik

and Mielke [22] we define the dissipation metric g as follows: fixing x 2X and having two states of an N-slip system, i.e., twoN-tuples of accumulated slip c0ðxÞ :¼ ðcð1Þ0 ðxÞ; . . . ; cðNÞ0 ðxÞÞ and c1 :¼ ðcð1Þ1 ðxÞ; . . . ; cðNÞ1 ðxÞÞ we calculate

gðc0ðxÞ; c1ðxÞÞ :¼ infZ 1

0Rð _zðsÞÞds; zð0Þ ¼ c0ðxÞzð1Þ ¼ c1ðxÞ

� �; ð14Þ

where the infimum is taken over all smooth paths z joining c0(x) and c1(x). The overall dissipation is then the integral of gover the domain

Dðc0; c1Þ ¼Z

Xgðc0ðxÞ; c1ðxÞÞdx: ð15Þ

Hence, the dissipation metric measures how much energy is dissipated if a unit volume of the material transforms from onestate to the other. The overall dissipation then evaluates the dissipated energy released during the transformation of thewhole specimen X.

If C1 � @X and Lðt; xÞ :¼R

X f ðt; xÞ � xðxÞdxþRC1

gðt; xÞ � xðxÞdS denotes the work done on the body by external volumeforces, f, and surface forces, g, we have the following energy functional

Iðt; x; Fp; cÞ ¼Z

XWðx;rx; Fp; cÞdx� Lðt; xÞ;

where W is the stored energy density and the energy balance in the time interval [t1, t2]

Iðt2; xðt2Þ; Fpðt2Þ; cðt2ÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}stored energy at time t2

þ VarðD; ½t1; t2�Þ|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}energy dissipated during½t1 ;t2 �

¼ �Z t2

t1

_Lðn; xðnÞÞdn|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}work done by mechanical load

þ Iðt1; xðt1Þ; Fpðt1Þ; cðt1ÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}stored energy at time t1

: ð16Þ

We say that the process ðx1; Fp1; ~cÞ is stable if for all t 2 [0,T]:

Iðt; x; Fp; cÞ 6 Iðt; x1; Fp1; ~cÞ þ Dðc; ~cÞ ð17Þ

for every admissible state ðx1; Fp1; ~cÞ.

J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412 1405

The stability inequality just says that the state x, Fp, c is optimal at the time t from the following point of view: if there isx1; F

p1; ~c with smaller stored energy then the amount of dissipation needed to move the system to this state is too high to

undergo this process.Given the data f, g, and an initial state of the body we say that the process ðx1; F

p1; ~cÞ is a solution, if the energy balance (16)

is satisfied, the process is stable, and Fp and sy evolve, so that _Fp ¼ LpFp, and (10) holds. It is also possible to include theseevolutionary laws into the definition of the dissipation. A solution is usually constructed as a limit of time-incremental pro-cesses for a vanishing time step. Specifically, for a given division of [0,T] by a time step and given an initial conditionðcðiÞ0 ; F

pÞjt¼0 we minimize for k = 1, . . ., N:

Iðt; x; Fp; cÞ þ Dðc; ck�1Þ;

where ck�1 is the value of c in the previous time step.The main advantage of this approach is, firstly, that the formulation does not assume smoothness of evolving variables in

time, and, secondly, that the proof suggests a numerical scheme for a solution. To our knowledge, the first attempt to use theenergetic approach is due to Francfort and Marigo [36]. Further there are many papers on this topic related to evolution ofshape-memory materials see e.g. [37–39] and references therein.

Let us consider an example of a single slip. Here we treat the case of no plastic stored energy, i.e., all work put to the sys-tem must be dissipated. For the present setting of the energetic method, it is convenient to formulate a power balance in thereference configuration. For rigid-plastic model the integration over the body X in the reference configuration with theboundary oX provides the power balance, see [21],

Z

XT ref � _FdV ¼

Z@X

T ref n � vdA; ð18Þ

where n is the unit outer normal to oX and Tref = TF�T denotes the stress in the reference configuration. We assume that thevelocity v = v0 is prescribed on a part oXv of the boundary and the stress Tref on the complementary part @XT. In the case ofsingle slip the kinematical relations (4), (5) and the resolved shear stress (6) yield T ref � _F ¼ s _c. The corresponding weak for-mulation of the boundary value problem reads

Z

Xsdj _cjdV �

Z@XT

T ref n � dvdA ¼ 0 ð19Þ

for all virtual slip rates d _c and virtual velocities dv on X that satisfy d _c ¼ 0 and dv = 0 on @Xv.The relation (19) can be understood as a condition for an extremum of an energy functional. If a displacement is pre-

scribed on the whole boundary oX or if the complementary part oXT is stress free, the surface integral in (19) disappears.Such boundary conditions are typical of standard strain controlled tensile and cycling experiments.

For the considered model the problem of a specification of the energy functional and a method of determination of itsminimum represented by a lamellar structure has been analyzed in [16] under the restrictive assumption that there is nohardening, hence, the critical resolved shear stress sy > 0 is constant. For no hardening and a zero surface integral the bound-ary value problem (19), where the yield condition (9) is incorporated, is reduced to

Z

Xsydj _cjdV ¼ 0; ð20Þ

which means the condition of minimum of dissipation. Consider the case of no hardening, i.e., _sy ¼ 0. Then it is easy to cal-culate g in (14), namely g(c0(x),c1(x)) :¼ syjc0(x) � c1(x)j. Conti and Theil [16] demonstrated rigorously that the model pre-dicts the formation of a lamellar structure as the lowest energy microstructure formed. The mathematical proofs and therelevant references can be found in their paper. Due to (11) the time integration in (20) can be evaluated and slip c canbe expressed through F using (13), hence, the problem can be implicitly formulated as minimization in terms of the positionvector field x, i.e. we are looking for x(X,t) which satisfies: (i) the kinematical rigid-plastic restriction rx = R(I + cs �m), (ii)the displacement boundary conditions at oXv, (iii) it minimizes the incremental functional J(x):

JðxÞ ¼Z

XWepðF; F0Þ;dV ; ð21Þ

where Wep is the energy density

WepðF; F0Þ ¼syjcðFÞ � cðF0Þj if F ¼ RðI þ cs�mÞ;þ1 otherwise;

�ð22Þ

where knowing x at time t0 we denote its gradient F0 =rx(t0) = R0(I + c0s �m) for the rotation R0 = R(t0) and slip c0 = c(t0).The infinity in formula (22) guarantees in a formal way that the lattice strain is negligible. The latter corresponds to theassumption that the elastic energy is infinite whenever the elastic part of the deformation gradient F is not a lattice rotation.Note that the restriction to single slip which guarantees the existence of c is a key ingredient of the presented energeticmethod.

2

F1

F2

F1

F2

1F

a/

μ

(1−μ)

d

d

b/

(1−λ) d λd

G1 G2 G1G

Fig. 1. Horizontal shear band lamellae a and vertical kink band lamellae b with finite periodicity. Each construction matches the displacement boundaryconditions on two opposite sides. However, refining this construction leads to lower energies. In the limit for d ? 0, the periodicity of lamellar structuretends to zero.

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The main aim of the paper [16] was to study of minimization for (21) and (22). Strong anisotropy of the crystal caused byshearing just along a single active slip system makes the energy density Wep not convex, which in turn favors a spontaneousformation of a microstructure. However, the minimization of the energy functional J(x) in (21) leads to highly oscillatorybehavior of minimizing sequences and its minimum can not be reached due to finer and finer oscillations of F.

The mathematical study of the problem and its boundary conditions is based on the concept of quasiconvexity. In theconstitutive assumption (22) the density Wep is replaced by quasiconvex envelope Wqc; the envelope Wqc is defined as thelargest quasiconvex function which is less than or equal to Wep. We recall that a function w : R3�3 ! R is quasiconvex if

5 Theconvex

6 A fuconvex

wðAÞjXj 6Z

XwðAþruðxÞÞdx; ð23Þ

for all A and all smooth u : X! R3 vanishing at the boundary of an open bounded regular domain X. Minimization of J(x)corresponds to the microstructure formed, and minimizer of the functional Jqc(x) related to Wqc characterizes the averagemacroscopic properties of the microstructure. The relation between J and Jqc is demonstrated by a recovery sequence[14]: for each x, there is a sequence xh converging to x such that

JqcðxÞ ¼ limh!1

JðxhÞ: ð24Þ

The energy density Wep is non-quasiconvex in F; its quasiconvex envelope Wqc and the recovery sequence can be constructedin the following way. Roughly speaking, an imposed deformation gradient F such that Wep(F,Fk) = +1 can be obtained as the‘‘average” of two rigid-plastic deformations F1 and F2 with finite energy and rank (F1 � F2) = 1, i.e., F = lF1 + (1 � l)F2 forsome volume fraction 0 < l < 1. The position vector field xh(X) of the recovery sequence is represented by a lamellar structure[16]

xdðXÞ ¼ ðlF1 þ ð1� lÞF2ÞX þ advln � X þ c

d

� �; ð25Þ

where d sets the scale of the lamellar structure and serves as the sequence number h = 1/d, where d ? 0. The vector a deter-mines the direction of the structure amplitude and n is the unit normal to the lamellae; they are related throughF1 � F2 = a � n, c represents the phase relation between successive lamellae, and vl(n) is a continuous, periodic, piece-wise-linear function of the argument n = (n � x + c)/d such that @vl/@X = 1 � l for n 2 (0,l), and @vl/@X = �l for n 2 (l,1).Hence, to accommodate the deformation imposed by the boundary conditions the system oscillates between two rigid-plas-tic deformations F1 and F2. However, replacing in formula (21) Wep by the quasiconvex envelope Wqc sets up a minimizationproblem which possesses a minimizer and describes a limiting behavior of the oscillations if their length scale d tends tozero. However, explicit evaluations of Wqc are hardly possible in practice. The main difficulty that in order to check quasicon-vexity of a particular function one needs to solve a global minimization problem verifying (23). Nevertheless, there are gen-eric ways how to obtain bounds on the quasiconvex envelope. The upper bound is typically received by subsequentlamination which leads in the limit to the so-called rank-one convex envelope of Wep

5 We refer e.g. to [40,41] for this ap-proach. The lower bound can be calculated by estimating the polyconvex envelope of Wep.6 We refer to [42] for an interestingcomputational strategy. Nevertheless, both bounds are strict, in general.

rank-one convex envelope of Wep is a pointwise supremum of all rank-one convex functions not greater than Wep. A function is rank-one convex if it isalong segments whose endpoints differ by a rank-one matrix.nction is polyconvex if it is a convex function of all subdeterminants of its argument. The polyconvex envelope is defined similarly as the rank-oneone.

Fig. 2. Composition of two basic constructions from Fig. 1. The dashed line denotes the macroscopic deformation.

J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412 1407

In [17] there has been demonstrated that two basic lamellar structures may be formed: shear bands and kink bands sche-matically shown in Fig. 1. For the shear band lamellae the amplitude vector a is parallel to the slip direction s and the unitnormal to the shear band lamellae n = m. For the kink band lamellae the directions are reversed, a is parallel to the slip planenormal m and n = s. These two basic constructions can be composed to a structure depicted in Fig. 2. In the present termi-nology the structure in Fig. 2 can be understood as a pattern of subgrains. There, the shear stress is carried by the shear bandsand the kink bands adjust the crystal lattice orientation in average to the applied shear. The observed structures which canbe interpreted as a combinations of the shear bands and kink bands have been reviewed and analyzed in [43–45].

The model does not introduce any length scale into the problem. Namely, as already mentioned before, finer and finerlamellae decrease the bulk energy. Nevertheless, as demonstrated by Conti and Theil [16] the boundary conditions bringan additional energy contribution which vanishes with the thickness of the lamellae, d ? 0. However, the observations showthat the lamella patterns have a well-defined finite wavelength. The reason is that the present model neglects the energyneeded to build interface boundaries or an inner structure of the pattern. The effects controlling the refinement of the micro-structure is discussed for the incremental model in the following Section 4.

4. Symmetric incremental double slip

Here we present the analysis of an incremental double slip. This method applies to more general situations [9], too, butwe restrict ourselves to the double slip for the sake of simplicity. The crystal plasticity model of symmetric double slip wasproposed by Asaro [46] and considered e.g. by Peirce et al. [47,48]. It falls into the class of orthotropic-symmetric modelsinvestigated by Biot [7–9] and Hill and Hutchinson [49]. The incremental rigid-plastic version of the model which includeshigher plastic strain gradients [13] is reviewed in this Section.

Within the framework of crystal plasticity summarized in Section 2 the incremental deformation of a prestressed crystaldeformed in plane strain by symmetric double slip, Fig. 3, is represented by increments ru, x* and bp such that

F ¼ I þru; Fp ¼ I þ bp; R ¼ I þx�: ð26Þ

bp is a plastic distortion which transforms the pre-stressed homogeneous (reference) configuration into the lattice (interme-diate) configuration, x* measures the lattice rotation from the lattice configuration into the incremental (current) configu-ration, andru transforms the reference configuration into the current configuration. With higher order terms neglected thedecomposition (2) in the incremental form reads

ru ¼ bp þx�: ð27Þ

From (5) for i = 1, 2, and from R _FpðFpÞ�1RT ¼ Lp in footnote 4 the rate of plastic distortion _bp correct in the first order resultsin the form _bp ¼ mð1Þsð1Þ �mð1Þ þ mð2Þsð2Þ �mð2Þ. Assuming that bp = 0 initially, the time integration yields

bp ¼X2

i¼1

cðiÞsðiÞ �mðiÞ; mðiÞ ¼ _cðiÞ; cðiÞ ¼ 0; initially: ð28Þ

Hence, the incremental plastic distortion is expressed as the sum of two incremental single slips. The slip increments c(1) andc(2) have the physical meaning only in this limited sense.

In the pre-stressed state the slip systems are characterized by the slip planes with unit normals m(1) = (�cos/, sin/),m(2) = (cos/, sin/) and slip directions s(1) = (sin/, cos/), s(2) = (�sin/, cos /); / is the orientation angle of the slip planes withrespect to the symmetry axis that coincides with axis y of x–y coordinate system, Fig. 3. The lattice rotation x* is related tothe material rotation x = xxy = (@yux � @xuy)/2 and plastic rotation xp = (c(1) + c(2))/2

x� ¼ x�xp: ð29Þ

x

σ

φ φ

s(1)

m(1)

y

s(2)

m(2)

x

y

σ

Fig. 3. Symmetric double slip.

1408 J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412

The density of the geometrically necessary dislocations (GNDs) K = RT curl(RT) in the incremental form required for thematerial deformation to be compatible is K = curlbp = �curl x*. K can be resolved in the densities K(i), i = 1, 2,

7 Theaix.upo

K ¼ Kð1Þ þ Kð2Þ ¼X2

i¼1

curl cðiÞsðiÞ � nðiÞ� �

: ð30Þ

If misoriented cell boundaries are present in the crystal, the tensors K(i) represent their GND composition.The slips c(i) are driven by the resolved shear stress s(i) given by (6)

sðiÞ ¼ RsðiÞ � T RmðiÞ

¼ sðiÞ � S mðiÞ; ð31Þ

where S = RTTR is the stress in the lattice configuration. Assuming the incremental stress r0 and rotations to be first orderquantities the lattice stress S consists of three contributions,

S ¼ �rþxp �rþ �rðxpÞT þ r0; ð32Þ

where we suppose that the pre-stress tensor �r has only diagonal components �rxx and �ryy, Fig. 3; we denote D�r ¼ �ryy � �rxx.The term xp �rþ �rðxpÞT is of the geometrical origin caused by rotation of the lattice with respect to the pre-stress configu-ration; it accounts for so-called ‘‘geometrical hardening/softening”. The incremental stress r0 is of the physical nature in-duced by incremental strain.7 In the pre-stress configuration the projection of the pre-stress tensor �r into each slip systemhas to be equal at yield to the initial value of critical yield stress ± s0, i.e. D�r sin /=2 ¼ s0.

The constitutive relations (9) adjusted to the double slip model are represented by the yield condition: the slip systemremains active, i.e. _cðiÞ, i = 1, 2, may be non-zero, if and only if

sðiÞ ¼ sðiÞy sign _cðiÞ: ð33Þ

Due to the assumed existence of the incremental slips c(i) the evolution Eq. (10) for sðiÞy may be integrated in time and beexpressed as (see Eqs. (45) and (46) in [13])

sð1Þy ¼ hcð1Þ þ qcð2Þ þ ~hðsð1Þ � rÞ2cð1Þ þ ~qðsð1Þ � rÞðsð2Þ � rÞcð2Þ þ s0; ð34Þ

sð2Þy ¼ qcð1Þ þ hcð2Þ þ ~hðsð2Þ � rÞ2cð2Þ þ ~qðsð2Þ � rÞðsð1Þ � rÞcð1Þ þ s0: ð35Þ

where a unidirectional incremental loading is considered; h = H11 = H22 and q = H12 = H21 are components of the symmetricmatrix of the local hardening, ~h and ~q represent nonlocal hardening coefficients. In [13] the non-local effects have been ana-lyzed using a model of an infinite crystal deformed by symmetric double slip, where plastic strain is carried by straight, par-allel, edge dislocations. The constitutive Eqs. (34) and (35) have been derived from the statistical mechanics description ofcollective behavior of discrete dislocations proposed by Groma et al. [52,20,53]. The non-local effects are caused by the short-range interactions among dislocations.

In analogy to (32) the Cauchy stress T in the current configuration can be expressed in the incremental form

T ¼ �rþx�rþ �rxT þ r0; ð36Þ

pre-stressed (reference, laboratory), current (material) and lattice configurations relevant to crystal plasticity are analyzed in [50,51] and thesis http://l.cz/furst.

J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412 1409

hence, the incremental stress tensor r0 has to satisfy the condition of the quasi-static stress equilibrium (7) for the pre-stressed sample [9]

@xr0xx þ @yr0xy þ D�r@yx ¼ 0 @xr0xy þ @yr0yy þ D�r@xx ¼ 0: ð37Þ

A variational formulation of the quasi-static stress equilibrium for a pre-stressed sample was found in [54,55] using the prin-ciple of virtual displacements (weak formulation of the problem). The formulation requires that an energy functional A at-tains a minimum

dA ¼ 0: ð38Þ

As has been shown in [13], the functional A in domain X can be expressed in the form

A ¼Z

Xfhðcð1Þ þ cð2ÞÞ2 þ 2qcð1Þcð2Þ þ D�r cos 2/ cð1Þ � cð2Þ

� �cð1Þ � cð2Þ þx� �

=2þ ~h ðsð1Þ � rcð1ÞÞ2 þ ðsð2Þ � rcð2ÞÞ2h i

þ 2~q sð1Þ � rcð1Þ� �

sð2Þ � rcð2Þ� �

gdx: ð39Þ

In general, the variational formulation leads to a difficult optimization problem deciding which of kinematically admis-sible instability modes takes place in the material under load. The admissible modes are formed by linear combinations ofstream functions u(x + ny). The stream function u is related to the displacement u, so that ux = @yu, uy = �@xu. The modesrepresent deformation bands perpendicular to the direction n = tanh, where h is the angle between the y-axis and the normalto the band.

To illustrate the proposed incremental variational method a reduced problem was analyzed in [13]. The set of deforma-tion modes to be optimized is restricted to subgrain structures arising from superposition of two periodically arranged setsof parallel deformation bands in a form of lamellae perpendicular to directions n and �n. In each lamella, and hence, in eachsubgrain the increments c(1), c(2) and x are supposed to be uniform. Slip is adjusted such that in the direction perpendicularto the lamellae the incremental slips average to zero. The width L of the lamellae, which specifies the size of the subgrains, istaken to be the same for all the lamellae; for physical reasons it is required L > 0. In this case, the gradient terms s(i) � rc(i) arereplaced by the difference in slips c(i) in the neighboring subgrains divided by the width of the boundary d. The non-localterms led by ~h and ~q in (39) thus represent the energy of the subgrain boundaries. The optimization is then reduced tothe problem to find out the subgrain structure of a preferred subgrain size L and a preferred orientations ± n which minimizethe functional A. With s(i) � rc(i) replaced by the jump in c(i) divided by d and multiplied by the subgrain boundary surface,the functional (39) becomes the function of L and n

AredðL; nÞ ¼16Vp2g2

L2

hþ q

sin2 2/n2 þ h� qþ D�r cos 2/

4 cos2 2/ð1� n2Þ2 � D�r

4ð1� n4Þ

!

þ 64Vp2g2

L3

~hþ ~q

sin2 2/n2 þ

~h� ~q4 cos2 2/

ð1� n2Þ2 !

; ð40Þ

where V is the volume of the domain X and g is the amplitude of the incremental displacement in the subgrain boundaries.The subgrain structure will form if the difference between the function Ared given by (40) and the corresponding function

for the equivalent homogeneous deformation is negative. In that case the non-homogeneous deformation is energeticallypreferred. As the average of the incremental deformation is zero, the function for the equivalent homogeneous deformationvanishes and the condition for appearance of the subgrain structure considered is Ared(L,n) < 0. For a given orientation of theslip systems / and a pre-stress D�r the condition Ared < 0 delimits instability regions of the material parameters under whichthe considered deformation substructure may occur. This finding is an extension of the result for symmetric double slip inplane strain with the non-local effect excluded that was discussed in detail by means of bifurcation maps in [50,51].

The necessary conditions Ared < 0 for appearance of subgrains has been expressed in [13] in terms of the hardeningcoefficients

~h > 0 � ~h < ~q < ~h q 6 hþ 2D�r cos 2/ cos2 /: ð41Þ

To specify sufficient conditions we are looking for L > 0, so that Ared < 0 is valid. The sufficient conditions expressed in termsof hardening coefficients are cumbersome to write; they are derived and stated in [13] together with the bifurcation maps.Let us note that there exist three types of instability regions where both necessary and sufficient conditions are satisfied: A, Band C. The regions A and C correspond to the existence of shear and kink bands (Biot’s instability mode of the second kind [9]somewhat modified by nonlocal hardening coefficients), the region B corresponds to Biot’s instability mode of the first kind,which represents an internal buckling.

The minimum of Ared with respect to the subgrain size L and the orientation n indicates the most favorable structureformed in a bulk of a crystal. The conditions @Ared/@(n2) = 0 and @Ared/@L = 0 yield the formulas for L and n derived and ana-lyzed in [13]. Here the results are represented by the sketch of a preferred subgrain structure shown in Fig. 4. The content ofthe geometrical necessary dislocations in the subgrain boundaries is evaluated from (30).

The subgrain size L can be expressed as L ¼ Rgðq=h; ~q=~h;D�r=h;/Þ, where g is the function of the parameters of the modelspecified in [13] assuming that h – 0 and ~h–0. R ¼ 4~h=dh is the leading factor controlling the size L. The factor R

Fig. 4. Sketch of the distribution of geometrically necessary dislocations in subgrain boundaries.

1410 J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412

demonstrates the characteristic feature of the model: bulk strain and dissipative energy tends to decrease the size, whileshort-range dislocation interactions restrict that tendency. The size L grows with non-local effect represented by ~h (if thenon-local effect is neglected, ~h! 0 the size L ? 0). On the other hand, increasing hardening, represented by h, decreasesthe size. As shown in [13], the pre-stress has the similar effect; the size L decreases with the absolute value of the pre-stress.

5. Summary and outlook

We reviewed two models of plastic rate-independent evolution. The first one, the energetic approach, is more general,relies on modern methods of the calculus of variations, and the incremental minimization problems suggest a numerical pro-cedure for a solution. On the other hand, in many cases it is rather difficult/impossible to obtain an analytical solution. Thesecond one, the incremental deformation concept, can be seen as a one step in the energetic formulation. More precisely,having constitutive laws, the static equilibrium equations are solved to get a deformed configuration. These equations formfirst-order optimality condition for a minimum of an energy functional. As shown for the case of a symmetric double slip in[13] this strategy can give us a fairly deep insight into the structure of a solution. Any substantial progress in the problem ofdeformation substructure formation has to overcome at least three obstacles:

To formulate crystal plasticity for multi slip. The reason is that the current models reviewed here are based on single slip;as mentioned at the end of Introduction the energetic approach is done for a single slip model only or multi slip is treatedas a succession of single slips. In the incremental approach the deformation are assumed small, hence, multi slip can betreated as a sum of single slips. As noted by Gurtin [21] the existence of a physically meaningful kinematical variable cwhose material time derivative is the slip rate m is unique to single slip. A multi slip model has to overcome this problem. To specify mechanisms controlling the scale represented in the present paper by the subgrain size. It seems that the size is

governed by the short range interactions among dislocations. Unfortunately, it is not clear which plastic variable woulddescribe properly this non-local effect. The current custom is to employ gradients of plastic strain. However, the attemptsto describe the short range dislocation interactions by averaging an assembly of discrete dislocations [19,20] reveal thecomplexity of the problem and demonstrate that the gradients represent rather rough approximation. To combine the presented crystal plasticity approach with the statistical treatment of subgrain boundaries [3,4]. The

approaches seem to express just two aspects of the same reality. As noted by Pantleon [4], the imbalance in the activationof slip systems between different regions, which serves as the background of the statistics, can arise from an intrinsicinstability of the deformation process. Therefore, it is feasible to propose a joint model combining the deterministic originof structuralization presented here with the statistical nature of the boundaries.

Acknowledgment

This work was supported by the grants P201/10/0357 (GACR�), A 100750802 (GA AVCR), VZ6840770021, andVZ6840770003 (MŠMTCR). The authors are indebted to anonymous referees for helpful comments.

J. Kratochvíl et al. / International Journal of Engineering Science 48 (2010) 1401–1412 1411

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