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

Artina, Marco, Cagnetti, Filippo, Fornasier, Massimo and Solombrino, Francesco (2017) Linearly constrained evolutions of critical points and an application to cohesive fractures. Mathematical Models and Methods in Applied Sciences, 27 (2). pp. 231-290. ISSN 0218-2025

[img] PDF - Accepted Version
Restricted to SRO admin only until 3 February 2018.

Download (1MB)

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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Subjects: Q Science > QA Mathematics
Depositing User: Richard Chambers
Date Deposited: 22 Nov 2016 10:32
Last Modified: 03 Apr 2017 12:23
URI: http://sro.sussex.ac.uk/id/eprint/65562

View download statistics for this item

📧 Request an update