Variational Analysis of Sharp Folds and Cusps in Shells: A Generalized Functional Setting with Atomic Measures

We characterize the functional structure of deformations that describe the occurrence of possibly irrecoverable sharp folds and cusps in shell-like bodies, while they avoid self-penetration of matter. The combination of special bounded variation maps with one-dimensional jump set and atomic measures provides a functional environment in which it is possible to represent configurations not otherwise described by special bounded Hessian maps, often used for energy minimization of shells. The introduction of atomic measures provides necessary control over the singular component of the generalized gradient measure, ensuring the robust compactness required by the direct method. In the proposed framework, we prove the existence of minimizers under Dirichlet boundary conditions for a class of energies that are appropriate for an ambient setting foreseeing large strains. Although we treat a traditional topic, the analysis goes beyond the classical format of nonlinear elasticity.