Localization in heterogeneous materials: A variational approach and its application to polycrystalline solids

International Journal of Plasticity 48 (2013) 92–110

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Localization in heterogeneous materials: A variational approach and its application to polycrystalline solids J.L. Dequiedt CEA, DAM, DIF, F-91297 Arpajon, France

a r t i c l e

i n f o

Article history: Received 18 September 2012 Received in final revised form 14 February 2013 Available online 27 February 2013 Keywords: Stability and bifurcation (C) Microstructures (A) Crystal plasticity (B) Variational calculus (C)

a b s t r a c t In the case when macroscopic instability in a heterogeneous material is defined as the loss of rank-one convexity of the homogenized tangent modulus (namely, the propensity to develop bands of localization), the stability domain is the one of positive definiteness of a potential function on a representative volume element defining the unit cell of the microsctructure. It is possible to bound this domain by minimizing the functional on a set of kinematically admissible velocity fields which intend to catch the localization of deformation in the microstructure. The approach is applied to polycrystal plasticity in a bidimensional geometry: the plane double-slip model of Asaro is retained for the local behavior. The stability domain of a polycrystalline aggregate, schematized by a bidimensional pavement, is displayed as a function of the heterogeneity of crystal orientations, latent hardening and stress distributions. The localization velocity field is estimated in each case. In particular, the destabilizing effect of the coupling between shear strain and rotation at the microscopic level is exhibited. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Localization phenomena usually develop during plastic deformation of metals in both quasi-static and dynamic loading conditions: they tend to limit the ductility of the material since they quickly lead to the failure of the device. In thick metal specimens, they appear as shear bands which concentrate the deformation and are sites for the initiation of damage and fracture. In thin shells, necking is also observed which leads to local thinning of the shell giving possible edges of fragments. Localization is usually explained at the macroscopic scale by instability and bifurcation of the homogeneous deformation of the material or structure. The conditions for such phenomena are linked with the evolution of the elasto-plastic behavior during deformation: the saturation of the flow stress resulting from the competition between work hardening and any softening mechanism (such as damage or thermal softening in adiabatic loading conditions) is the prominent factor. Geometrical effects such as the concentration of stresses in stretched regions also influence the phenomenon. The case of thin shells in biaxial stretching has been widely studied for application in forming processes; the question of localization is addressed in a plane stress framework: the initiation of the phenomenon is characterized either by the existence of instability modes quickly growing in time such as in Dudzinski and Molinari (1991), or by the propensity of the shell to concentrate deformation in a pre-existing defect such as in Marciniak et al. (1973). Forming limit diagrams are then displayed giving the point of initiation as a function of the ratio of the strain rates in the two stretching axes. The case of thick shells was analyzed by tridimensional linear stability analyses in plane strain conditions for different geometries, such as in Shenoy and Freund (1999) or Mercier and Molinari (2003). It has been extended to biaxial stretching by Jouve (2010).

E-mail address: [email protected] 0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2013.02.007

J.L. Dequiedt / International Journal of Plasticity 48 (2013) 92–110

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The instability of bulk materials is usually associated with the development of bands of localization. In quasi-static loading, the criterion for the activation of a band is given by the loss of positive-definiteness of the so-called acoustic tensor such as in Rice (1976). This gives an orientation of the first band appearing as the flow stress saturates. However, all these approaches enlightened the sensitivity of the different criteria to the precise description of the elastoplastic behavior of the material. In multiaxial loading, the sensitivity to the shape of the plasticity yield surface and its evolution has been emphasized by different studies. Barlat (1987) compares the forming limit diagrams given by the Marciniak approach for different shapes of yield surfaces and notices a strong influence on the limit in biaxial stretching. Needleman and Rice (1978) analyze the influence on the quasi-static bifurcation criterion of various deviations from the Prandtl–Reuss plasticity model (i.e. a Von Mises yield surface and a flow rule obeying the normality law): they consider a slight sensitivity of the yield surface to pressure and the existence of a vertex in proportional loading. A fine description of tridimensional elasto-plasticity lies on the modeling of phenomena acting at the microscopic scale, namely the one of the polycrystalline microstructure of metals: the shape change of the macroscopic plasticity yield surfaces and the development of anisotropy are caused by the deformation and rotation of grains and the activation of favorably oriented slip systems in each of them. One way to incorporate these physics in instability analyses is to keep a macroscopic point of view but replace phenomenological constitutive models by ones drawn from the average response of a polycrystalline aggregate. Such models are obtained by different homogenization techniques on a representative distribution of crystal orientations. Among them, the Taylor model (1938) assumes that the strain rate is uniform on all orientations. Toth et al. (1996) characterized polycrystal yield surfaces and their evolution during a proportional uniaxial or biaxial tension with this model: then, they identified an anisotropic Hill surface fitting the yield surface and applied it to the linear perturbation analysis of Dudzinski and Molinari (1991). Inal et al. (2002) applied the Taylor model to the simulation of localization during the plane strain tension of an aluminum sheet: they analyzed the influence of different parameters and showed that the model was able to catch the onset of necking and its triggering to shear banding. Yang and Bacroix (1996) used the Taylor model in a shear band analysis performed for application to localization in rolling processes. Other works retain the self-consistent model applied to polycrystals (Berveiller and Zaoui, 1979) rather than the Taylor one in order to take inter-granular heterogeneity into account. Franz et al. (2009) analyzed the limit to ductility of steel with the Rice bifurcation criterion and a self consistent visco-plastic model; they found quite different results than the ones given by classical forming limit diagrams obtained with a plane stress formalism. The drawback of such approaches, although constituting a clear improvement compared with the ones carried on with phenomenological models, is that homogenized models do not catch the concentration of deformation at the microscopic scale which may noticeably modify the overall response of the representative volume element (R.V.E.) which characterize the microstructure and thus, the conditions at which macroscopic instability criteria are fulfilled. More precisely, before macroscopic instability is reached, it is likely that instability conditions are satisfied in some points of the microstructure, namely, in the case of polycrystals, in a few favorably oriented grains and deformed zones. The question is whether micro-localization may initiate in each of these points and affect the whole grain and the neighboring grains: obviously, it depends on the orientation of bifurcation modes in the instable points and the many parameters which characterize the behavior in the surroundings such as neighboring grains misorientation, stress fluctuations or strain hardening of the different slip systems. For these reasons, an accurate definition of a macroscopic instability criterion implies re-considering the problem of the response of the R.V.E. and identifying the equilibrium strain field in the former conditions. Bifurcation criteria at the microscopic scale is the topic of several papers which characterize the onset of localization in monocrystals; these papers evaluate the influence of different micromechanical and microstructural parameters such as the critical shear stress on the different glide systems, the orientation of these systems from the principal directions of stress, the self and latent strain hardening. The case of one slip system per grain is treated in Asaro and Rice (1977) who analyzed the influence of a deviation from the Schmid law (slip rate function of the resolved shear stress): they showed that this deviation allows localization for positive work hardening of the slip system as was observed by Rudnicki and Rice (1975) for any nonnormality flow rule. The approach was later extended to double slip in the next papers (Asaro, 1979; Pierce et al., 1982; Dao and Asaro, 1996). They showed that the multiplicity of slip systems destabilizes the crystal and displayed the influence of geometrical effects such as slip plane rotation with deformation and of latent hardening. Simultaneously, the development of polycrystal simulation tools gives a way to better understand how localization may organize at the scale of a crystalline aggregate. In Barbe et al. (2001) simulations for instance, areas of intense plastic deformation crossing several grains were exhibited. More recently, Cheong and Busso (2006) investigated the onset of a macroscopic shear band in a few grains assembly and showed that it strongly depends on the lattice misorientation of neighboring grains. Similar observations had been displayed in the work of Harren and Asaro (1989) for a bidimensional geometry, where the crossing of a grain boundary by a microscopic shear band was precisely analyzed: a pavement of hexagonal grains was simulated using idealized plane slip systems. The influence of texture and grain boundary lattice misorientation was confirmed by the simulations of Anand and Kalidindi (1994) on copper with the actual tridimensional FCC slip systems. Polycrystal simulations are validated by experimental results obtained with different techniques such as digital image correlation for strain fields or electron back-scattering diffraction (EBSD) for crystal orientations: an overview of the comparison methods is presented in Héripré et al. (2007). In the present work, we study the onset of a macroscopic localization band in a heterogeneous medium representing a polycrystalline aggregate by a variational approach. The loss of rank-one convexity of the macroscopic tangent modulus

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is evaluated by minimizing a bilinear functional on the R.V.E. among strain rates the average of which is a simple glide. The minimization is then operated by considering particular families of kinematically admissible velocity fields in the R.V.E. and the localization criterion is then bounded. For the sake of simplicity, the approach is developed in plane strain for a bidimensional periodic microstructure. In Section 2, the variational method to estimate the macroscopic band criterion is presented in a general framework: it applies to any heterogeneous medium so long as the microscopic tangent modulus possesses the major symmetry. In Section 3, the constitutive model of a monocrystal undergoing multi-slip is recalled; the incremental stress–strain relation is then particularized to the idealized plane double slip model of Asaro (1979). Then, the heterogeneous medium localization criterion is applied to a bidimensional polycrystal in plane strain loading conditions. Families of strain rate fields are defined in the R.V.E. in such a way as to reproduce heterogeneities of strain rates between the different grains; the estimated criterion is compared to the one obtained with a homogeneous strain rate on the R.V.E. like in the Taylor hypothesis. 2. Macroscopic stability of a heterogeneous material At the macroscopic scale in quasistatic loading conditions, the onset of a band of localization is possible when there exists a bifurcation of the homogeneous solution of the problem: the difference between the two possible incremental solutions consists in a simple glide of direction M in a narrow band of normal N (N and M are supposed to be normalized vectors). In other words, the velocity field associated with the bifurcation, i.e. the difference between the two solutions, writes in the form:

DV ¼ DVðX N ÞM

ð1Þ

With XN = X  N the macroscopic coordinate in the direction of N. The velocity gradient of the bifurcation is thus:

DG ¼ gradðDVÞ ¼

dDV M  N ¼ DGðX N ÞM  N dX N

ð2Þ

The equation of equilibrium applied to both incremental solutions leads to the following relation:

N  DT_ ¼ 0

ð3Þ

T_ is the time derivative of the nominal stress tensor as defined by Rice (1976) (the current configuration is taken as the reference one). Let us suppose that the material behavior allows us to write a linear relation between T_ and G:

T_ ¼ L : G

ð4Þ

The tangent modulus L is a fourth order tensor and the above tensorial form is taken as equivalent to the following form with indices (convention is adopted to consider products of matrices to operate on the nearest indices):

T_ ij ¼ Lijkl Glk

ð5Þ

So long as L possesses the major symmetry (namely Lijkl = Lklij), T_ is derived from a macroscopic potential W which is a quadratic function of G (as introduced by Stolz (1982)):

  dW dW 1 with WðGÞ ¼ G : L : G T_ ¼ i:e:T_ ij ¼ dG dGji 2

ð6Þ

In this case, the criterion for the activation of a band of normal N is the loss of positive definiteness of the bilinear form: M ´ W(M  N) and the associated eigenvector M is the direction of glide in the band. When the material is heterogeneous at a microscopic scale, the macroscopic variables are mean values which are evaluated on a representative volume element (R.V.E.):

G ¼ hgðxÞiRVE

_ and T_ ¼ htðxÞi RVE

ð7Þ

_ g(x) and tðxÞ are respectively the gradient of the microscopic velocity field v(x) and the rate of microscopic nominal stress; they are functions of the position x in the R.V.E. In each point x, the incremental constitutive relation writes:

_ tðxÞ ¼ lðx; kÞ : gðxÞ

ð8Þ

lðx; kÞ is the microscopic tangent modulus which depends on the position in the R.V.E. and on all the characteristics of the material at the time at which the incremental constitutive relation is written (distribution of phases, fields of stresses and of all the internal state variables associated with the plasticity). These characteristics depend on the whole loading history of the material up to the current time and are summarized formally in a k parameter. In the case when lðx; kÞ also possesses the major symmetry, it is possible to define a microscopic potential wðx; k; gÞ ¼ 12 g : lðx; kÞ : g such that:

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@w _ tðxÞ ¼ ðxÞ @g

ð9Þ

Let us remark that the condition for local bifurcation in point x is fulfilled when the bilinear form M # wðx; k; M  NÞ loses positive-definiteness. However, bifurcation in a few points of the R.V.E. does not automatically imply macroscopic localization. The relation between microscopic and macroscopic quantities is established in the following way. Let us suppose that the macroscopic velocity gradient G is given, that either conditions of periodicity or conditions of homogeneous G on the boundary are assumed (in the first case, the R.V.E. needs to have such a shape that periodicity is possible). For any kinematically admissible velocity field v⁄(x), let us define the functional:

Jðk; g ðxÞÞ ¼ hwðx; k; g ðxÞÞiRVE

ð10Þ

The unicity of the true velocity field v(x) is guaranteed when the functional J is convex, namely when:

@2J ½dg ; dg  ¼ @g @g

*

@2w dg ðxÞ :   ðx; kÞ : dg ðxÞ @g @g

+



¼ hdg ðxÞ : lðx; kÞ : dg ðxÞiRVE > 0

ð11Þ

RVE

for any kinematically admissible dv⁄ such that hdg⁄(x)iRVE = 0 (or such that dv⁄(x) = 0 on the boundary if conditions of homogeneous velocity gradient are retained). Then, the true velocity field v minimizes J among all kinematically admissible velocity fields v⁄; this mean value is moreover equal to the macroscopic potential:

Jðk; gðxÞÞ ¼ MinfJðk; g ðxÞÞ=hg ðxÞiERV ¼ Gg ¼ WðGÞ

ð12Þ

~  N, the difference Dv of the two actual Identically, for two macroscopic velocity gradients differing by a simple glide M microscopic velocity fields satisfy:

~  Ng ¼ WðM ~  NÞ Jðk; DgðxÞÞ ¼ MinfJðk; Dg ðxÞÞ=hDg ðxÞiRVE ¼ M

ð13Þ

Let C(N) be the set of velocity fields compatible with a macroscopic glide in a band of normal N, namely:

~  N; kMk ~ ¼ 1g CðNÞ ¼ fv  ðxÞ=hg iRVE ¼ M

ð14Þ

So long as condition (11) is fulfilled, the existence of a macroscopic band of normal N is equivalent to the condition:

~  NÞ=kMk ~ ¼ 1g ¼ 0 bðk; NÞ ¼ MinfJðk; g ðxÞÞ=v  2 CðNÞg ¼ MinfWðM

ð15Þ

So, for a given N, the stability domain Sðk; NÞ in the space of loading parameter k, i.e. the domain in which no band may develop, is the set of k for which bðk; NÞ exists (which ensures that condition (11) is satisfied since J is quadratic) and is strictly positive. ^ In the following, approximations of the stability domain are built by minimizing J on restricted sets CðNÞ  CðNÞ of velocity fields:

^ ^ NÞ ¼ MinfJðk; g ðxÞÞ=v  2 CðNÞg bðk; > bðk; NÞ

ð16Þ

The approximated stability domain ^ Sðk; NÞ contains the actual stability domain Sðk; NÞ. The aim of the following sections is to develop this approach in the case of an idealized polycrystal. The parameter k describes either microstructural characteristics of the R.V.E. such as the scattering of grain orientations or mechanical characteristics such as the strain hardening parameters on the different slip systems and the stress fields. The estimated stability domain is all the more precise that the set of velocity fields is able to approach the actual velocity field, namely to reproduce the concentrations of strains in the R.V.E. at the onset of localization. We would be tempted to think that these highly strained zones appear in the areas where the local instability criterion is fulfilled, but as we will see further, this is not always the case. 3. Application of the variational approach to an idealized polycrystal 3.1. General formulation of the monocrystal behavior In this section, the incremental relationship between the rate of nominal stress and the velocity gradient is written for a monocrystal in the form established by Asaro and Rice (1977) for single slip and extended to multi-slip in Asaro (1979). In finite deformations, the deformation gradient F decomposes into a plastic part Fp which contains the plastic shearing on the different crystallographic slip systems but keeps the crystallographic orientations unchanged and an elastic part Fe which contains all the elastic deformations and rotations of the monocrystal:

F ¼ Fe  Fp

ð17Þ

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The velocity gradient is thus a sum of an elastic and a plastic part:

g ¼ F_  F1 ¼ F_ e  Fe1 þ Fe  ðF_ p  Fp1 Þ  Fe1 ¼ ge þ gp

ð18Þ

Each of these terms can be written as the sum of a symmetric part and an anti-symmetric part, i.e. a strain rate and a spin rate:

g ¼ D þ X;

ge ¼ De þ Omegae

and gp ¼ Dp þ Xp

ð19Þ

The plastic velocity gradient is the sum of shearing rates on all the active slip systems:

gp ¼

X

c_ ðaÞ ðmðaÞ  nðaÞ Þ

ð20Þ

a

with n(a) the normal to the slip plane (a) and m(a) the direction of slip. The symmetric and anti-symmetric part of the tensor (m(a)  n(a)) are noted P(a) and W(a) respectively. The plastic strain rate and plastic spin rate are thus:

Dp ¼

X

X

a

a

c_ ðaÞ PðaÞ and Xp ¼

c_ ðaÞ WðaÞ

ð21Þ

The slip systems are rotated with the deformation such that m(a) and n(a) do not remain constant. Their time evolution is given by one of the formulations introduced by Asaro and Rice (1977) (m(a) is convected with the lattice motion and n(a) is convected as a vector of the reciprocal base):

_ ðaÞ ¼ ge  mðaÞ m

and n_ ðaÞ ¼ nðaÞ  ge

ð22Þ

It is assumed that the material is hypo-elastic, namely that elasticity is ruled by a linear relation between the elastic strain rate and an objective derivative of the Cauchy stress: we chose the Jaumann derivative of r/q (q is the mass density) formed on axes which rotate with the lattice as proposed in Stolz (1982):

DeJ

 

 

 

 

r d r r r 1 ¼  Xe  þ  Xe ¼ C : De q q q dt q q

ð23Þ

The activation of slip systems is ruled by the Schmid law. Non-Schmid effects, which are supposed to be linked with cross slip and which are considered in Asaro and Rice (1977) and Dao and Asaro (1996), are neglected in this study; let us recall that they remove the normality of the flow rule and so the major symmetry of the tangent modulus without which the variational approach is no more valid. In the Schmid case, the deformation on system (a) is possible provided that the resolved shear stress reaches a critical value

sðcaÞ :

sðaÞ ¼ mðaÞ  r  nðaÞ ¼ PðaÞ : r ¼ sðcaÞ

ð24Þ

The critical shear stress increases with both the deformation on system (a) (self hardening) and the deformations on the other systems (latent hardening):

s_ ðcaÞ ¼

X hab c_ ðbÞ

ð25Þ

b

Thus, for all active slip systems (a) and provided that unloading happens for none of them:

s_ ðaÞ ¼ P_ ðaÞ : r þ PðaÞ : r_ ¼

X hab c_ ðbÞ

ð26Þ

b

Let us remark that, at constant stress (namely when r_ ¼ 0), the resolved shear stress may increase or decrease due to the rotation of the slip system. This is one of the ‘‘geometrical effects’’ which influences the stability. After some straightforward calculations which use relations (22) on the evolution of lattice vectors, elasticity relation (23) and the expression for plastic deformation (21), the slip rate on each active system is given as a function of the total strain rate:

c_ ðaÞ ¼

X 1 ð@ Þab ðPðbÞ : C þ bðbÞ Þ : D

ð27Þ

b

(@1)ab is the inverse of matrix @ab whose components are:

@ab ¼ hab þ PðaÞ : C : PðbÞ þ bðaÞ : PðbÞ

ð28Þ

b(a) are symmetric tensors given by :

bðaÞ ¼ ðWðaÞ  r  r  WðaÞ Þ

ð29Þ

Expressions for the Jaumann derivative of the Cauchy stress and thus for the rate of nominal stress are deduced as functions of the strain and spin rates (the difference between both stress rates consists of other ‘‘geometrical’’ terms representing the convection of stress with the material motion):

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qD$J

 

X X r ¼C:D c_ ðaÞ ðPðaÞ : CÞ ¼ C : D  ð@1 Þab ððPðbÞ : C þ bðbÞ Þ : DÞðPðaÞ : CÞ q a a;b

t_ ¼ C : D 

X 1 ð@ Þab ððPðbÞ : C þ bðbÞ Þ : DÞðPðaÞ : C þ bðaÞ Þ  r  X  D  r

ð30Þ

ð31Þ

a;b

The summation applies on all active slip systems (a) and (b). Let us remark that the tangent modulus depends on the slip systems which are actually active. Thus, the approach of Section 2 only applies when the active slip systems are the same for the fundamental and bifurcated solutions, which excludes bifurcations with unloading on one of them. A potential w (cf. Eq. (9)) exists so long as the (@1)ab and @ab matrices are symmetric. This condition is not satisfied when the strain hardening matrix hab is symmetric; thus we have to assume, as in Pierce et al. (1982), that the hardening matrix satisfies the following relation in which the matrix Hab is symmetric:

1 hab ¼ Hab þ ðbðbÞ : PðaÞ  bðaÞ : PðbÞ Þ 2

ð32Þ

In this case, w is:

( ) X 1 1 1 1 1 ðaÞ ðbÞ ða Þ ðbÞ w¼ ð@ Þab ððP : C þ b Þ : DÞððP : C þ b Þ : DÞ  D  r  D  X  r  X þ ðX  r  D  D  r  XÞ ð33Þ D: C: D 2 2 2 2 a;b 3.2. Plane double-slip model and local bifurcation In the following, the approach of Section 2 is particularized to a bidimensional geometry in plane strain loading conditions. Bands of localization are considered with normal N in the loading plane. For the sake of simplicity, the material is supposed to be isotropic and incompressible and thus the direction of glide is necessarily the direction T tangent to the band in the plane. The idealized polycrystal which is considered here is constituted of several phases of infinite extent in the direction normal to the plane, each phase being characterized by a uniform tangent modulus i.e. a uniform crystal orientation, hardening matrix and stress state. Let us remark that in the initial unloaded configuration, phases are superimposed with grains, but as soon as the polycrystal has deformed, this is no longer the case since the stress and plastic strain distributions become heterogeneous inside grains (and subsequently the orientation of slip systems which spin during plastic deformation and the hardening moduli which are functions of the cumulated slip c(a) on the different systems). The material is supposed to deform by plastic slip on two systems and the plane double-slip model of Asaro (1979) is taken. The two slip planes are separated by the angle 2U. The crystal orientation of each phase is referenced by the angle h between the plane of symmetry of the two systems and the plane of the band (Fig. 1). The slip planes normals and directions of slip thus write in the (N, T) basis:

nð1Þ ¼ sinðh þ UÞT  cosðh þ UÞN mð1Þ ¼ cosðh þ UÞT þ sinðh þ UÞN nð2Þ ¼  sinðh  UÞT þ cosðh  UÞN mð2Þ ¼ cosðh  UÞT þ sinðh  UÞN

ð34Þ

N m (1)

σ II n (1)

σ I > σ II

θs

Φ n (2 )

m (2 )

θ T

Fig. 1. Slip systems and principal directions of stress in the band orientation referential (Asaro (1979) plane double-slip monocrystal).

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Tensors P(1), P(2) and W(1), W(2) are respectively:

Pð1Þ ¼

1 ½sinð2ðh þ UÞÞðT  T  N  NÞ  cosð2ðh þ UÞÞðT  N þ N  TÞ 2

Pð2Þ ¼

1 ½ sinð2ðh  UÞÞðT  T  N  NÞ þ cosð2ðh  UÞÞðT  N þ N  TÞ 2

1 Wð1Þ ¼  ðT  N  N  TÞ ¼ Wð2Þ 2

ð35Þ

The principal components of the stress tensor in the plane, rI and rII such that rI P rII , are supposed to form an angle hs with the plane of symmetry. So:

r ¼ ðrI cos2 ðh þ hs Þ þ rII sin2 ðh þ hs ÞÞT  T þ ðrI sin2 ðh þ hs Þ þ rII cos2 ðh þ hs ÞÞN  N þ

ðrI  rII Þ sinð2ðh þ hs ÞÞ 2

 ðT  N þ N  TÞ

ð36Þ

The hardening matrix is taken with identical self strain hardening coefficients for the two systems and a latent hardening coefficient which is proportional to them but higher as is usually admitted in crystal plasticity:

H11 ¼ H22 ¼ H

and H12 ¼ qH with q > 1

ð37Þ

In this framework, the coefficients of matrix @ab are:

1 2

@11 ¼ H þ G  ðrI  rII Þ cosð2ðU  hs ÞÞ 1 4

@12 ¼ qH  G cosð4UÞ þ ðrI  rII Þðcosð2ðU  hs ÞÞ þ cosð2ðU þ hs ÞÞÞ 1 2

@22 ¼ H þ G  ðrI  rII Þ cosð2ðU þ hs ÞÞ

ð38Þ

The saturation of strain hardening is characterized by the parameter m such that:

H ¼ H0 expðmÞ

ð39Þ

The ‘‘local’’ stability domain, i.e. the domain inside which no local bifurcation of normal N exists, is the one of positivity of the potential w(m, q, h, hs, rI, rII, T  N), (m, q, h, hs, rI, rII) being the set of loading parameters (U, H0 and G are considered as constant material parameters and we take U = 30° and H0/G = 0, 05). This stability domain is plotted on Fig. 2 in the (m, h) plane with U = 30°, H0/G = 0, 05, q = 1, 4, rI/G = 0, 05, rII = 0, hs = 0. Fig. 2 displays two distinct instability zones. For slip system orientation h near 37°, the monocrystal is unstable for high m; in other words, localization tends to appear when strain hardening saturates; the bifurcation is a sum of positive slips in both slip systems. One can guess that, in this situation, geometrical effects such as slip system rotations are destabilizing and lead to localization when the stabilizing effect of strain hardening has decreased. For h near 0°, the monocrystal is unstable for low m, i.e. for high strain hardening. Moreover, the bifurcation is a sum of slips of opposite sign in the two slip systems.

Fig. 2. Stability domain of the plane double slip monocrystal in the (m, h) plane.

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Namely, the bifurcated solution would be the initiation of a ‘‘patchy slip’’ phenomenon: it concentrates slip on one system rather than activate equal slip on the two systems. This effect has been documented in the literature (as for example in Pierce et al. (1982)): when latent hardening is high, the monocrystal tends not to deform homogeneously on several systems but to divide into subgrains each one deforming mainly on one of the systems. One can guess that this effect is all the more significant that strain hardening is high. Let us now analyze the influence of other loading parameters on the local stability domain. Even in the simplest crystal strain hardening models (the model of Asaro and Needleman (1985) for instance), q depends on the pair of slip systems involved in the double slip model; thus, there is no reason why it should be taken as a material constant. The sensitivity of the instability to latent hardening is evaluated by plotting the stability domain in the (m, q) plane (Fig. 3). For h = 37°, latent hardening has a slight stabilizing effect (i.e. the stability domain increases with q) which can be explained by the fact that it globally increases strain hardening. For h = 0, the destabilizing effect of latent hardening is displayed since the stability domain diminishes while q increases; moreover, no instability zone exists when q is beyond a critical value around 1.25. At last, the influence of the stress state on the instability (linked with ‘‘geometrical effects’’ as described in section 3.1) is displayed by plotting the stability domain as a function of the principal stress orientation hs (with rII = 0) and of the biaxiality rate rII/rI (Fig. 4). For h = 37°, instability is favored by negative hs and rII/rI and for h = 0, instability is favored by high values of rII/rI and any value of hs deviating from 0. 3.3. Macroscopic stability domain An idealized two dimensional polycrystal is now considered and the variational approach of Section 2 is applied to it in order to get an estimation of the stability domain, as a function of the inhomogeneity of its micromechanical properties. The polycrystal is supposed to be periodic and the unit cell is a 2D square area of dimension 1 made of a K ⁄ K pavement as schematized on Fig. 5(a) (the dimension of the unit cell can be chosen arbitrarily since the monocrystal plasticity model does not introduce any characteristic length). The set of loading parameters and thus potential w are supposed to be constant in each of the K ⁄ K square areas. More precisely, let us suppose that the unit cell is constituted of two phases A and B, each square area belonging to one of the two. The set of kinematically admissible velocity fields which is considered is the one of piecewise linear fields of the following form:

Dv  ðxT ; xN Þ ¼ xN T þ Dv T ðxN ÞT þ Dv N ðxT ÞN

ð40Þ

Each field is a sum of the average macroscopic glide plus fluctuating glides in both directions T and N. Dv N ðxT Þ and Dv T ðxN Þ obey the periodicity conditions Dv T ð0Þ ¼ Dv T ð1Þ and Dv N ð0Þ ¼ Dv N ð1Þ; they are supposed to be piecewise linear on each interval [(i  1)/K, i/K] and [(j  1)/K, j/K] respectively. The set of velocity gradient is defined by:

Dg ðxT ; xN Þ ¼ ð1 þ ðDg TN Þj ÞT  N þ ðDg NT Þi N  T With:

PK

 j¼1 ðDg TN Þj

¼ 0 and

PK

 j¼1 ðDg TN Þj

¼ 0 such that hDg⁄iVER = T  N

Fig. 3. Stability domain of the plane double slip monocrystal in the (m, q) plane for the two crystal orientations h = 37° and h = 0°.

ð41Þ

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Fig. 4. Stability domain of the plane double slip monocrystal as a function of the principal stress orientation hs and the biaxiality of stresses rII/rI for the two crystal orientations h = 37° and h = 0°.

(a)

(b)

Fig. 5. Unit cell of the idealized polycrystal (a) and set of velocity fields (b) (strain and spin rates are uniform on each square area the deformation of the unit cell is visualized).

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We only keep velocity gradients satisfying ðDg NT Þi þ ðDg TN Þj P 1 for all (i, j) such that in any square area the strain rate DDTN is positive. Such a velocity field is schematized on Fig. 5(b) which reproduces the associated deformation of the unit cell. One can observe that this family of velocity fields is able to catch heterogeneities of deformation and spin rates between the different phases. In the following, we try to obtain estimations ^ S of the stability domain for three different distributions of phase A and B in a 4 ⁄ 4 pavement representing the unit cell (the three distributions are noted I, II and III respectively and represented on Fig. 6). Phase A is supposed to be characterized by the loading parameters q = 1, 4, rI/G = 0, 05, rII = 0 and hs = 0 and one of the two slip system orientations identified as favorable to localization for the monocrystal (h = 37° or h = 0). Phase B is defined by a deviation from phase A of successively the slip system orientation h, the latent hardening parameter q, the principal stress orientation hs and the biaxiality rate rII/rI (saturation of strain hardening m is supposed to be homogeneous in the unit cell). Functional J is minimized on the former family of velocity fields for each phase distribution. This estimation is compared with the one obtained for a uniform velocity gradient in the R.V.E. (the latter assumption is referred to as Taylor hypothesis (1938) and the associated stability domain is denoted by ^ ST ). Moreover, in a few cases for which the deviation in phase B is taken in the sense of stabilization of phase B alone (according to paragraph 3.2), the velocity field at the approximated onset of localization (i.e. on ^ S boundary) is represented. In a first time, the crystal orientation h = 0 is considered for phase A. On Fig. 7, a deviation dh of the slip system orientation in phase B is considered: the ^ S and ^ ST stability domains are plotted in the (m, dh) plane. The two domains are symmetric by the line dh = 0 which is coherent with the symmetry of the problem; they differ and the difference increases when |dh| increases. Moreover, such as the monocrystal stabilizes when the slip systems move from orientation h = 0 whether in negative or positive sense, the polycrystal also stabilizes when |dh| increases. For dh = 12°, the shape of the velocity field which defines the limit of ^ S is plotted for the three phase distributions defined on Fig. 6: it gives an approximation of the bifurcation velocity field in the unit cell. The square areas where wij is strongly negative are indicated on this picture with a minus sign and the ones where wij is strongly positive with a plus sign: this gives an idea of the zones in the unit cell which enhance localization and of the zones which penalize it. In this first case, the three velocity fields consist of pure glide (namely strain plus clockwise spin) in lines j which contains the higher density of phase A, the other lines remaining undeformed and unrotated: phase distribution III, which is characterized by a high heterogeneity of the density of phase A and B between the different lines also exhibits the higher difference between the stability domains ^ S and ^ ST . In the sheared lines, phase A favors localization (wij < 0) whereas phase B penalizes it (wij > 0): in other words, the phase B square areas have to deform due to geometrical compatibility even though they would tend to remain unstrained. On Fig. 8, phase B differs from phase A by the latent hardening parameter q which is varied in B from q = 1 to q = 2. The stability domains ^ S and ^ ST are plotted in the (m, q) plane: the polycrystal such as the monocrystal stabilizes with decreasing q. The localization velocity fields, plotted for q = 1.2 in B, are comparable with the one exhibited on Fig. 7: they nearly consist of pure glide in lines with a majority of phase A, with very slight deformations and rotations in the other lines. The sign of wij in the different square areas indicates that these rotations play a second order role since the strongly negative values of wij always correspond to the sheared phase A square areas. On Fig. 9, phase B is characterized by a non-zero biaxiality rate varying from rII/rI = 1 to rII/rI = 1. The stability domains are plotted in the (m, rII/rI) plane exhibiting an increasing stability with a decreasing rII/rI. The localization velocity fields

xN

Distribution I

xN

xT

Phase A

Distribution II

xN

xT

Phase B Fig. 6. Distributions of phases A and B in the unit cell.

Distribution III

xT

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Fig. 7. Stability domains of the two phase 4 ⁄ 4 polycrystal as a function of the crystal misorientations dh between the two phases (phase A is oriented h = 0° and phase B is oriented h =  dh) and estimated localization velocity field for dh = 12°.

plotted for rII/rI = -0.5 also consist of intense glide in lines with a majority of phase A but also glide in other lines and additive spin rates more significant than in the latent hardening case. According to the sign of wij, localization is still enhanced by shear in phase A and penalized by deformation in phase B or rotation in phase A (in several lines). The additive spin in the intense shear line is always clockwise in phase B.

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Fig. 8. Stability domains of the two phase 4 ⁄ 4 polycrystal as a function of the heterogeneity of the latent hardening coefficient q between the two phases of crystal orientation h = 0° (q = 1.4 in phase A and q = qB in phase B) and estimated localization velocity field for qB = 1.2.

In a second time, the crystal orientation h = 37° for phase A is studied. As we shall see, the results are somewhat different. On Fig. 10, the influence of crystal orientation is analyzed (in the same way as on Fig. 6 for h = 0): the stability domains ^ S and ^ ST are plotted in the (m, dh) plane for the three phase distributions. Both positive and negative dh are stabilizing as was observed for the monocrystal but the domains are not symmetric by the line dh = 0. Moreover, ^ S and ^ ST differ significantly espe-

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Fig. 9. Stability domains of the two phase 4⁄4 polycrystal as a function of the heterogeneity of the biaxiality rate rII/rI between the two phases of crystal orientation h = 0° (rII/rI= 0 in phase A and rII/rI = (rII/rI)B in phase B) and estimated localization velocity field for (rII/rI)B= 0.5.

cially for high dh which means that the heterogeneity of deformation at the mesoscopic scale is influent. Localization velocity fields are represented for both dh = 12° and dh = 10° on Figs. 10 and 11 respectively. For dh = 12°, they consist of glide in the lines with the higher density of phase A plus slight rotations; in the former lines, the rotation is always clockwise in phase B. The sign of wij indicates that shearing instability in phase A is the prominent factor. For dh = -10° on the contrary, the sign of wij tends to prove that the most important destabilizing effect is linked with glide plus counter-clockwise spin in phase B; however, the lines submitted to the higher level of glide still contain the highest density of phase A.

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Fig. 10. Stability domains of the two phase polycrystal as a function of the crystal misorientation dh between the two phases (phase A is oriented h = 37° and phase B is oriented h = 37°  dh) and estimated localization velocity field for dh = 12°.

On Fig. 12, the influence of the principal stress orientation hs in phase B is analyzed (hs = 0 in phase A and hs = dhs in phase B) for the three distributions. ^ S and ^ ST are plotted in the (m, dhs) plane: whereas the Taylor hypothesis predicts an increase of stability with increasing dhs (as for the monocrystal), the domain ^ S predicts a stability which increases first and decreases for sufficiently high positive dhs. Localization velocity fields are given for dhs = 15° for which ^ S and ^ ST differ significantly. The destabilizing zones (wij < 0) still consist of glide plus counter-clockwise spin in phase B. In all the distributions, the velocity

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Fig. 11. Stability domains of the two phase polycrystal as a function of the crystal misorientation dh between the two phases (phase A is oriented h = 37° and phase B is oriented h = 37°  dh) and estimated localization velocity field for dh =  10°.

field can also be seen as glide in lines j with the highest density of phase A plus orthogonal glide in columns i with maximum density of phase B. Fig. 13 is devoted to the influence of the biaxiality rate rII/rI in phase B. Whereas the Taylor hypothesis exhibits an increasing stability with rII/rI, the former family of velocity fields predicts a destabilizing effect of both positive and negative

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Fig. 12. Stability domains of the two phase polycrystal as a function of the principal stress misorientation between phases of crystal orientation h = 37° (hs = 0 in phase A and hs = dhs in phase B) and estimated localization velocity field for dhs = 15°.

rII/rI (except in a narrow interval around rII/rI = 0). Localization velocity fields are represented for the biaxiality rate rII/ rI = 0.5. By comparing the three distributions, it seems that these fields consist of a superposition of glide in lines j with maximum density of A and orthogonal glide in columns i with maximum density of phase B, the second being the most influent on instability. The destabilizing zones (wij < 0) approximately consist of pure strain in phase B.

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Fig. 13. Stability domains of the two phase polycrystal as a function of the heterogeneity of the biaxiality rate rII/rI between phases of crystal orientation h = 37° (rII/rI = 0 in phase A and rII/rI = (rII/rI)B in phase B) and estimated localization velocity field for (rII/rI)B = 0.5.

4. Concluding remarks In this work, we tried to identify the stability domain of a heterogeneous material in the sense of the absence of macroscopic bands of localization. In the case when, at the microscopic scale, the incremental stress–strain relation writes in such a manner that the tangent modulus is symmetric, we developed a variational approach which defines stability as the positivity of a potential function on the unit cell defining the microstructure (it is assumed to be periodic). For a given orientation of band, the latter condition is bounded by minimizing the potential on a family of kinematically admissible velocity fields

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which allow heterogeneity of deformation in the unit cell and so strain concentrations in the weakest phases. This condition is compared to the one which would be obtained with an assumption of a homogeneous strain rate on the unit cell (namely the Taylor hypothesis). Moreover, the minimizing velocity field gives an estimation of the shape of the localization pattern at the microscopic scale. The approach is applied to polycrystal plasticity in a bidimensional geometry. The polycrystal is schematized by a pavement of two phases, each phase being characterized by a homogeneous incremental stress–strain relation (i.e. homogeneous crystalline orientation, stress distribution, strain hardening parameters). The first phase named A has such fixed characteristics that a monocrystal analysis would identify as favorable to localization and the second named B has varying ones. Common sense would say that the best bound and so the most accurate estimation of the localization pattern is obtained for a velocity field which concentrates shear in a plane parallel to the band crossing the highest density of the phase which is the most favorable to localization when taken alone. Actually, this is only verified in some cases whereas in others, the localization pattern concentrates shear deformation in zones of the other phase. It seems that in these zones, the geometrical effects linked with spin rate and coupling of strain and spin rate are at least as important as pure strain rate effects. Moreover, there are some cases for which an increase of stability of phase B alone results in a decrease of stability of the polycrystal. The present study gives a first idea of how localization may organize at a microscopic level on a bidimensional microstructure representing (very schematically) a polycrystalline material. It also gives an estimation of the error made by neglecting the microscopic heterogeneity of deformation (when a macroscopic instability criterion is derived from the Taylor model). One question which is not addressed in the present work is the transition between microscopic and macroscopic localization. It has been studied in different papers mainly for periodic media: different kinds of approaches are proposed. Some of these works still consider macroscopic instability but they assume that the macroscopic to microscopic scale size ratio is not infinite and the macroscopic bifurcation criterion is derived from a potential including higher order gradients in strain rate (the formalism is laid in Luscher et al. (2010) for instance). Other papers show that, in the periodic case, macroscopic instability can be preceded by conditions when solutions exist for equilibrium having wavelengths of several unit cells (this phenomenon is referred to as ‘‘microscopic buckling’’, see for example Triantafyllidis et al. (2006)). The mathematical bases of the former approach are set in Geymonat et al. (1993). The comparison of both phenomena was later developed for different materials and behaviors (in hyperelasticity for porous elastomers in Michel et al. (2007) among others). However, in the case of polycrystals, whether such considerations are relevant is not obvious since there is no actual periodicity in the microstructure. Concerning macroscopic localization, further developments should consider more accurate monocrystal constitutive models, more sophisticated microstructures (which would be more representative of the space distributions of crystal orientations, stresses and slip systems saturation) and more elaborate velocity fields. Among others, one of the limits of the present work is the absence of a physical characteristic length for the localization pattern since it is fixed by the pavement representing the unit cell (the strain and spin rates are homogeneous on each square area). One could guess that some of the phase distributions would give localization in infinitely fine shear bands (at the mesoscopic scale) if this was permitted by the chosen family of velocity fields. However, this has probably no physical sense and it is due to the absence of a characteristic length in the constitutive behavior of the monocrystal. Such lengths would be introduced by gradient plasticity models including the resistance of monocrystals to lattice distorsions (see Mandel (1973) for instance) More generally, in the topic of localization in polycrystalline materials, analytical and numerical studies are complementary. Especially, polycrystal plasticity simulations may help to characterize the microstructure of the initial and pre-deformed polycrystal (when macroscopic shear banding occurs after some amount of deformation); it also may give an idea of the shape of the localization pattern in order to define accurate velocity fields in the variational approach.

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