Calculus of Variations and Geometric Measure Theory

S. Kholmatov - P. Piovano

A unified model for stress-driven rearrangement instabilities

created by piovano on 18 Feb 2019
modified on 15 May 2020


Accepted Paper

Inserted: 18 feb 2019
Last Updated: 15 may 2020

Journal: Arch. Ration. Mech. Anal.
Year: 2020


A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. Existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. Compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number $m$ of connected components, of sets of finite perimeter. Finally, it is shown that, as $m\to\infty$, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface.

Keywords: Lower Semicontinuity, Thin films, surface energy, fracture, SDRI, interface instabilities, crystal cavities, elastic energy, existence of minimal configurations