Dimension reduction and homogenization for thin plates with disconnected rigid inclusions

This paper addresses the simultaneous homogenization and dimension reduction of thin composite plates reinforced with rigid substructures, within the framework of non-linear elasticity. The reference (undeformed) configuration consists of a periodically perforated elastic matrix, where the holes are occupied by rigid inclusions. We first establish a decomposition of the deformation for such composite structures. Under the assumption that the thickness parameter delta δ is asymptotically smaller than the periodicity epsilon ϵ, we derive a reduced asymptotic model as both parameters tend to zero. Using rescaled unfolding operators, we characterize the limiting behaviour of the Green-St. Venant strain tensor. Through Γ-convergence, we obtain the homogenized limit energy and establish the existence of a minimizer. The resulting limit model is of constrained Kármán type, where the limiting displacement satisfies a first-order infinitesimal isometry constraint.