We determine the effective behavior of a class of composites in finite-strain crystal plasticity, based on a variational model for materials made of fine parallel layers of two types. While one component is completely rigid in the sense that it admits only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, we obtain explicit homogenization formulas by means of -convergence. Due to the anisotropic nature of the problem, the findings depend critically on the orientation of the slip direction relative to the layers, leading to three qualitatively different regimes that involve macroscopic shearing and blocking effects. The technical difficulties in the proofs are rooted in the intrinsic rigidity of the model, which translates into a non-standard variational problem constraint by non-convex partial differential inclusions. The proof of the lower bound requires a careful analysis of the admissible microstructures and a new asymptotic rigidity result, whereas the construction of recovery sequences relies on nested laminates. Mathematics Subject Classification 49J45 (primary) · 74Q05, 74C15

In the context of elasticity theory, rigidity theorems allow to derive global properties of a deformation from local ones. This paper presents a new asymptotic version of rigidity, applicable to elastic bodies with sufficiently stiff components arranged into fine parallel layers. We show that strict global constraints of anisotropic nature occur in the limit of vanishing layer thickness, and give a characterization of the class of effective deformations. The optimality of the scaling relation between layer thickness and stiffness is confirmed by suitable bending constructions. Beyond its theoretical interest, this result constitutes a key ingredient for the homogenization of variational problems modeling high-contrast bilayered composite materials, where the common assumption of strict inclusion of one phase in the other is clearly not satisfied. We study a model inspired by hyperelasticity via Γ-convergence, for which we are able to give an explicit representation of the homogenized limit problem. It turns out to be of integral form with its density corresponding to a cell formula.MSC (2010): 49J45 (primary); 74Q05, 74B20

A variational model for composite materials with rigid horizontal layers is introduced within the framework of geometrically nonlinear single-slip plasticity. Considering the orientation of the slip system, we discuss the effective material response in the case of vanishing layer thickness and give explicit examples of possible macroscopic deformations which are realized by simple laminate constructions. The intention of this contribution is to give some new insight into the effective behavior of heterogeneous models in finite crystal plasticity. In particular, we consider a variational model for bilayered crystalline solids built from fine, periodic oscillations of a rigid and a softer component.Let Ω ⊂ R 2 be a bounded Lipschitz domain modeling the reference configuration of an elastoplastic body, and denote by u : Ω → R 2 its deformation. As common in finite plasticity, we assume the multiplicative splitting of the deformation gradient ∇u = F e F p with F e and F p the elastic and the plastic part, see [1]. In the following, we take F e ∈ SO(2), which corresponds to excluding elasticity. In our model, plastic deformation can take place along one active slip system with slip direction s and slip plane normal m, precisely s ∈ R 2 with |s| = 1 and m = s ⊥ , so that integration of the plastic flow rule yields F p = I + γs ⊗ m with γ the amount of slip. To model the layered nature of the material under consideration, we introduce sets Y soft = ∪ i∈Z R × (i, i + λ) with λ ∈ (0, 1) and Y rig = R 2 \ Y soft , as well as a parameter > 0, which stands for the thickness of two neighboring layers and hence captures the length scale of the oscillations. In particular, Y rig ∩ Ω is the union of all rigid layers of thickness , where only local rotations are allowed, meaning that no plastic deformation can occur.Inspired by the time-discrete variational approach initiated in [2, 3], we consider the energy to be minimized in the first incremental problem as given by the functional E : L 2 (Ω; R 2 ) → R ∪ {∞},

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