Derivation of Kirchhoff-type plate theories for elastic materials with voids

Abstract

We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of Santili and Schmidt (2022), by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.