Calculus of Variations and Geometric Measure Theory

M. Kruzik - P. M. Mariano - D. Mucci

Crack occurrence in bodies with gradient polyconvex energies

created by mucci on 18 Feb 2021
modified on 30 Dec 2021


Published Paper

Inserted: 18 feb 2021
Last Updated: 30 dec 2021

Journal: Journal of Nonlinear Science
Volume: 32
Pages: 26
Year: 2022
Links: Link to Arxiv


In a set of infinitely many reference configurations differing from a chosen fit region $\mathcal{B}$ in the three-dimensional space and from each other only by possible crack paths, a set parameterized by special measures, namely curvature varifolds, energy minimality selects among possible configurations of a continuous body those that are compatible with assigned boundary conditions of Dirichlet-type. The use of varifolds allows us to consider both "material phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove existence of minimizers for such an energy. They are pairs of deformations and curvature varifolds. The former ones are taken to be $SBV$ maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.

Keywords: calculus of variations, microstructures, varifolds, shells, fracture, ground states