Quasistatic crack growth in 2d-linearized elasticity

TL;DR

This paper establishes a rigorous mathematical framework for quasistatic crack growth in 2D linearized elasticity, proving existence of solutions using advanced variational methods and generalized function spaces.

Contribution

It introduces a new existence proof for crack evolution in 2D linearized elasticity using GSBD and a Jump Transfer Lemma, extending previous results to this setting.

Findings

Proved existence of quasistatic crack growth in 2D linearized elasticity.

Developed a compactness theorem for GSBD functions.

Established stability of static equilibrium without a-priori bounds.

Abstract

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.