Calculus of Variations and Geometric Measure Theory

P. M. Mariano - D. Mucci

Crack nucleation in shells with through-the-thickness microstructure

created by mucci on 17 Dec 2020
modified on 28 Sep 2023


Published Paper

Inserted: 17 dec 2020
Last Updated: 28 sep 2023

Journal: SIAM J. Math. Anal.
Volume: 55
Number: 5
Pages: 4977--4997
Year: 2023


We refer to the common low-imensional description of shells and thin films: surfaces endowed with directors satisfying a non-degeneracy 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 reached under Dirichlet-type boundary conditions realizes the minimum of a regularized Griffith's energy that includes curvature terms. For it we prove 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 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.

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