Crack Nucleation in Shells with Through-the-Thickness Microstructure

We refer to the common low-dimensional description of shells and thin films: surfaces endowed with out-of-surface directors satisfying a nondegeneracy condition under large strain. We consider in addition through-the-thickness material microstructure described by elements of a complete and intrinsic Riemannian manifold. We look at brittle materials. Among all possible cracked and uncracked admissible configurations, the one obtained under Dirichlet-type boundary conditions realizes the minimum of a regularized Griffith's energy that includes curvature terms. For it, we prove the existence of minimizers for different constitutive functional choices and geometric structures with or without active through-the-thickness microstructure. Deformations are taken as SBV maps with a jump set included in the support of a curvature varifold with boundary. Through-the-thickness descriptors of the material microstructure are taken first as manifold-valued Sobolev maps. Then we consider pertinent SBV versions.