Mathematical analysis of fractional and nonlocal models from nonlinear Solid Mechanics.

Seminario de Ecuaciones

Speaker: Javier Cueto, Universidad de Castilla La Mancha

Abstract: This study is the result of a search for finding a suitable model of hyperelasticity that remains valid for discontinuous deformations. After noticing that nonlinear bond-based peridynamics does not fit into Solid Mechanics we decided to change our initial approach to one based on the fractional gradient. In particular, we show existence of minimizers of nonlinear vector fractional functionals, for what is needed a proper study of functional spaces (related to the fractional gradient) and fractional vector calculus. This is completed with a recovering of the classical model when the fractional index s goes to 1. Since the operators in the last approach are defined over the whole domain, we introduce and study a similar framework but with operators (integral nonlocal gradients) acting over bounded domains (relevant in applications), inspired by state-based peridynamics. In this framework more tools had to be developed, including nonlocal versions of the fundamental theorem of calculus and Poincaré inequality. Finally, we manage to determine the existence of minimizers of nonlocal vector polyconvex energy functionals under (nonlocal) Dirichlet conditions in a functional space admitting functions exhibiting some singularity phenomena. This also leads to the development of a framework for nonlocal vectorial calculus.