Homogenisation in finite elasticity for composites with a high contrast in the vicinity of rigid-body motions

We present a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic two-material composites with high contrast in a finite-strain setting. Our derivation starts with the geometrically nonlinear description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions, where the contrast between the components is coupled with the period ε. We assume that the deformation field is a rigid-body motion on some part of the boundary of the composite, and consider various loading regimes, which are controlled by an additional parameter λε. We show that for particular regimes the elastic displacements are small throughout the material as ε→ 0 and that the deformation gradients in the stiff material are situated in the vicinity of the rigid-body motion specified by the boundary condition. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component by its linearised version. The behaviour of the deformation gradients on the soft component depends on the asymptotics of λε. In particular, in the case λε ∼ 1 they are shown to be of order one, and the related contribution to the homogenised energy is expressed via the quasiconvex envelope of the original stored-energy density defined on suitably rescaled gradient fields. As a main result we derive, in the spirit of Γ-convergence, a limit energy functional and establish a precise two-scale expansion for minimising sequences.