Linearly constrained evolutions of critical points and an application to cohesive fractures

Marco Artina, Filippo Cagnetti, Massimo Fornasier, Francesco Solombrino

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)231-290
Number of pages60
JournalMathematical Models and Methods in Applied Sciences
Volume27
Issue number2
DOIs
StatePublished - 1 Feb 2017

Keywords

  • Quasistatic evolution
  • cohesive fracture
  • numerical approximation

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