## Abstract

We investigated the effect of spontaneous formation of microstructures on the macroscopic material behaviour in a class of models in finite single-crystal plasticity without hardening. In particular, we show that, even in the presence of a single active slip system, the formation of slip bands leads to a very soft behaviour of the sample in response to a large class of applied loads. This contrasts with results obtained under the assumption of rigid elasticity.

## 1. Introduction and main results

Multiscale analysis has become an indispensable tool for a reliable prediction of complex material behaviour in the presence of microstructures. In many situations, a variational modelling is possible, which permits us to explain in a natural way the spontaneous formation of fine structures across a wide range of length scales as a consequence of the lack of convexity of the relevant energy density *W*. This approach has proven successful in the investigation of the behaviour of a large class of materials, for example in the study of magnetic materials (DeSimone *et al*. 2001) or shape memory alloys (Ball & James 1987, 1992). In the case of crystal plasticity, the fundamental modelling aspects of the multiscale analysis have been investigated in Ortiz & Repetto (1999), Carstensen *et al*. (2002), Miehe *et al*. (2002) and Aubry & Ortiz (2003), but far-reaching studies, despite a significant technological relevance, are still at an early stage of development. In this paper, we contribute to this line of thought by predicting surprising material behaviour for a large class of models in finite plasticity.

Crystal plasticity can be put in a variational framework by resorting to a time-discretized setting, where, in each time increment, a variational problem is solved, whose parameters depend on the value of the internal variables at the end of the previous time step (Ortiz & Repetto 1999; Carstensen *et al*. 2002; Miehe *et al*. 2002). One of the successes of the approach is that it allows us to explain the formation of microstructures via a natural principle of minimization of energy and dissipation. General theory shows that the macroscopic effects of microstructures can be captured by analysing the relaxed (or macroscopic) energy; in mathematical terms this is called the quasiconvex envelope of the microscopic energy density (Morrey 1966; Dacorogna 1989; Müller 1999; Dolzmann 2003). Despite the fact that the abstract framework provides a simple definition of the relaxed energy, an explicit analytical or numerical computation is often a challenging problem, and only partial results in special cases are available in the literature.

The starting point of the models investigated in this paper is the assumption of finite plasticity with a multiplicative decomposition of the deformation gradient (Lee 1969) in a rate-independent setting. We focus on the fundamental building block of time-discretized models, namely on a single time step in a monotone loading path (Ortiz & Repetto 1999; Carstensen *et al*. 2002; Miehe *et al*. 2002). For simplicity, we shall discuss here the case that only one slip system is active and that the model is two dimensional; the main result holds in more general situations. Our main finding is that there is a crucial difference between the situation without self-hardening and the one with linear hardening. More precisely, in the absence of self-hardening, the material response to a large class of deformations is characterized by the formation of microstructures and a macroscopically soft behaviour, with macroscopic stress equal to zero. The situation is, however, different if linear hardening is included, at least with a single slip system.

The variational model for one time step is given by(1.1)subject to suitable boundary conditions, where *u* is the deformation field and ∇*u*=*F*_{e}*F*_{p} is the usual multiplicative decomposition of the deformation gradient into an elastic part and a plastic part. Moreover, *W*_{e} denotes the elastic stored energy and *W*_{p} contains the plastic stored energy and the dissipative terms. Minimizing locally in the internal variables, assuming that a single slip system is active and that the model does not include hardening, the integrand can be written as *W*(∇*u*), where *W* is the condensed energy,where *I*+γ*s*⊗*m* denotes the simple shear along the slip system with slip direction *s* and slip plane normal *m*; *s* and *m* are orthogonal unit vectors; and *τ*>0 is a parameter representing the yield stress.

It was demonstrated by Ortiz & Repetto (1999) that even the simplified model with rigid elasticity, namely with , where is given by for all *F*∈SO(2) and otherwise, predicts the formation of microstructures and explains geometric softening. Here, SO(2) is the set of all proper rotations in the plane. The relaxation of the corresponding condensed energy was determined in Conti & Theil (2005) and is given bywhere *λ*_{min}(*F*)≤*λ*_{max}(*F*) are the two (non-negative) singular values of *F* andThe case of two slip systems was investigated in Albin *et al*. (in press).

In this paper, we investigate a version of the model with a realistic elastic energy *W*_{e}. This is closely related to the models (which include hardening) studied numerically in Aubry *et al*. (2003), Miehe *et al*. (2004) and Carstensen *et al*. (2008). Our main result is the following.

*Suppose that the elastic energy* *satisfies the following hypotheses*:

*W*_{e}(*RF*)=*W*_{e}(*F*),*for all R*∈SO(2), (*frame indifference*);*W*_{e}(*R*)=0,*for all R*∈SO(2) (*rigid body motions are minimizers*);*there exist N*,*M*≥0,*b*>0*and c*_{1},*c*_{2}≥0,*such that**for all F*∈^{2×2},*with*det*F*>*M and*|*F*|≥*N*(*growth condition*);*and**W*_{e}*is continuous at the identity*.

*Then*, *W*^{qc}(*F*)=0, *for all F*∈*N*^{(2)}.

The result means that every macroscopic deformation gradient

*F*∈*N*^{(2)}can be realized as an average of a microstructure with arbitrarily small energy. That is, for every*F*∈*N*^{(2)}and*ϵ*>0, and for every bounded and smooth domain , there exists a Lipschitz-continuous map , such that*u*(*x*)=*Fx*on the boundary of*Ω*and(1.2)An explicit formula for the relaxation on

^{2×2}is not known. It is tempting to use the relaxation of the elastically rigid case as an approximation for the relaxation of the case with finite elastic energy. Our result shows that this leads to qualitatively different results (outside SO(2)), independent of the value of the elastic constants. Indeed, in*N*^{(2)}\SO(2), the relaxed energy*W*^{qc}vanishes, whereas is strictly positive. Outside of the set*N*^{(2)}, the macroscopic energy is identically +∞, whereas*W*^{qc}is finite on all deformation gradients on which*W*_{e}is finite (in a typical model, these are all deformation gradients with a positive determinant).Note that the hypotheses include densities that are infinite on deformation gradients with a negative determinant.

The key observation is that there exist curves in the space of all deformation gradients along which the energy has sublinear growth. In order to illustrate this effect, consider for

*t*,*α*>1 the matrixFor this decomposition of*F*, for*t*≫1, we have |*F*_{e}|∼|*F*_{t}|^{1/α}and |*F*_{p}|∼|*F*_{t}|^{(α−1)/α}. Hence,If we choose*α*=*b*+1, then we find thatThis observation allows one to find a rank-one direction along which*W*has a sublinear growth. From general theory, it then follows that the rank-one convex envelope, and hence the quasiconvex one, vanish along that line (details are given in §2). See Dacorogna (1989) for the various notions of convexity and their properties.

If linear hardening is included, a rather different behaviour is obtained. In both cases, spontaneous formation of microstructures leads to a separation into phases with different slip and softening of the macroscopic material response. The deviatoric macroscopic stress equals zero in the case without hardening, but it remains non-zero in the model with linear hardening.

*Consider the condensed energy**with τ*≥0, *μ*>0, *and assume that* *satisfies**for some c*>0. *Then*,

It is a natural question to ask what happens if the dependence of the plastic energy dissipation term *W*_{p} on *γ* is intermediate between linear and quadratic, as, for example, in the condensed energyEssentially, the same proof shows that the result of theorem 1.1 holds for all *q* between 1 and 2. For *q*=2 or larger, instead, the conclusion of theorem 1.3 holds.

## 2. Proofs

First, suppose that *W*_{e} and hence *W* are finite valued. Without loss of generality, we assume that *s*=*e*_{1}, *m*=*e*_{2} and *τ*=1. Let *M*^{(2)} be defined byNote that *N*^{(2)} is the rank-one convex hull of *M*^{(2)} (see Conti & Theil 2005). Therefore, it suffices to prove the assertion for all matrices in *M*^{(2)}. Since the rank-one convex relaxation of the energy is again frame indifferent, we may consider matrices of the formwith *σ*∈{−1,1} and *γ*_{0}>0. First, we construct a rank-one line close to *F*_{*}. Let , defineand set . For , we findwith . We finally letThen, *F*_{t}(0)=*H*_{t} and *F*_{t}(γ_{0}*t*^{−b−1})=*F*_{*}. Moreover,i.e. *F*_{t}(0) is close to the identity matrix andBy (H4), we deduce by writing *F*_{e}=*F*_{t}(0) and *F*_{p}=*I* that *W*_{e}(*F*_{t}(0))→0 as *t*→∞. Moreover, for sufficiently large *t*, we find, with *γ*=*σt*^{b} in view ofthat(2.1)Since the rank-one convex relaxation is convex along the rank-one line *s*↦*F*_{t}(*s*), we obtainThis implies that, for *t*→∞, *W*^{rc}(*F*_{*})=0, and hence *W*^{qc}(*F*_{*})=0.

In the case that is extended valued, we cannot conclude based on the fact that rank-one convexity implies quasiconvexity, and a slightly more refined construction is needed. Let *ϵ*>0. By (H4), there is *ρ*>0, such that *W*_{e}<*ϵ* on *B*_{ρ}(*I*). Furthermore, for sufficiently large *t*, we have |*F*_{t}(0)−*I*|<*ρ*/2. Analogously, a simple modification of the above computations shows that there is *δ*=*δ*(*t*)∈(0,*ρ*/2), such that *W*(*F*)≤*c*(*t*^{b}+1), for all *F*∈*B*_{δ}(*F*_{t}(1)). By lemma 2.1 of Müller & Šverák (1999), there is a Lipschitz map , such that ∇*u*∈*B*_{δ}(*F*_{t}(0))∪*B*_{δ}(*F*_{t}(1)) almost everywhere and *u*(*x*)=*F*_{*}*x*, for all *x*∈∂(0,1)^{2}. Furthermore, from the proof of the mentioned lemma, it is clear that the area of the set on which ∇*u*∉*B*_{δ}(*F*_{t}(0)) is no larger than *λ*_{t}. Therefore, , for all *ϵ*>0 and all *t* sufficiently large. This implies *W*^{qc}(*F*_{*})≤0 and concludes the proof. ▪

Without loss of generality, we assume that *s*=*e*_{1}, *m*=*e*_{2}, *τ*=0, *μ*=1 and *W*_{e}(*F*)=dist^{2}(*F*, SO(2)). The condensed energy *W*_{LH} is then a continuous function, which is strictly positive away from SO(2). We first show that *W*_{LH} has (at least) linear growth at infinity. Fromwe deduce that, for any *F*,Applying *x*^{2}≥2*x*−1 to the first two terms, we obtainThis, in particular, implies *W*_{LH}(*F*)≥(1/8)|*F*|, for all *F*, with |*F*|≥8.

Now, suppose that is given with . By the definition of the quasiconvex envelope, this means that there is a sequence of Lipschitz functions , such that *u*_{k}(*x*)=*F*_{*}*x*, for all *x*∈∂(0,1)^{2}, and(2.2)We truncate *u*_{k}, using a result from Evans & Gariepy (1992, §6.6.2) and Friesecke *et al*. (2002, proposition A.1), to obtain , such that |∇*v*_{k}|≤*c* almost everywhere andHere, *c* is a universal constant. From (2.2), we obtain that dist(∇*u*_{k}, SO(2)) converges to zero in measure, and the above equation shows that the same is true for ∇*v*_{k}. Since ∇*v*_{k} is uniformly bounded, a standard rigidity result (Müller 1999, theorem 2.4) shows that *v*_{k} converges to a rigid-body motion. Hence, the same is true for *u*_{k}. By the boundary condition, this implies that *F*_{*}∈SO(2). ▪

## 3. Conclusions

We have investigated a class of models in crystal plasticity which is used for multiscale analysis of material behaviour. Our results show a critical (discontinuous) dependence of the effective material response on some parameters in the model. We illustrate this with a specific example. Letwithbe a family of elastic energies with bulk modulus *κ* and elastic modulus *μ* (as used, for example, in Carstensen *et al*. 2002). Let *W*^{(κ,μ)} denote the corresponding condensed energies without hardening. It is easy to see that, for each deformation gradient *F*, the elastic energy converges, as *κ* and *μ* tend to ∞, to the rigid energy . Fix now any *F*∈*N*^{(2)}\SO(2). Then, the relaxation obtained from the rigid energy equals the positive real number *λ*_{max}(*F*)−*λ*_{min}(*F*). However, the relaxation of *W*^{(κ,μ)} vanishes at *F*, for all *κ* and *μ*. In particular, as *κ* and *μ* tend to infinity, the relaxation remains zero and does not converge to . This shows that is not a good approximation for the relaxation. In particular, this result illustrates the fact that the relaxation of extended-valued energy densities is a subtle issue that often leads to surprising effects.

## Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 ‘Analysis and computation of microstructure in finite plasticity’, projects CO 304/4-1 and DO 633/2-1. The work of G.D. was also supported by the National Science Foundation through grant no. DMS0405853.

## Footnotes

- Received September 29, 2008.
- Accepted February 4, 2009.

- © 2009 The Royal Society