2016/04/28 by Friedrich, Manuel, Solombrino, Francesco · 2 citations
#49J45 #70G75 #74R10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci)
paper · doi:10.48550/arxiv.1604.08338
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.