Calculus of Variations and Geometric Measure Theory

M. Friedrich - F. Solombrino

Quasistatic crack growth in 2d-linearized elasticity

created by friedrich on 06 May 2016
modified by solombrin on 04 Jan 2018


Published Paper

Inserted: 6 may 2016
Last Updated: 4 jan 2018

Journal: Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire
Volume: 35
Number: 1
Pages: 28--64
Year: 2018
Doi: 10.1016/j.anihpc.2017.03.002


In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ($GSBD$). As the time-discretization step tends to $0$, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of Francfort and Larsen (Comm. Pure Appl. Math., 56 (2003), 1465--1500) to the $GSBD$ setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without the necessity of a-priori bounds on the displacements or applied body forces.