Calculus of Variations and Geometric Measure Theory

M. Artina - F. Cagnetti - M. Fornasier - F. Solombrino

Linearly Constrained Evolutions of Critical Points and an Application to Cohesive Fractures

created by cagnetti on 12 Aug 2015
modified by solombrin on 09 Feb 2017


Published Paper

Inserted: 12 aug 2015
Last Updated: 9 feb 2017

Journal: Mathematical Models and Methods in Applied Sciences
Volume: 27
Number: 02
Pages: 231-290
Year: 2017
Doi: 10.1142/S0218202517500014


We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.