Analysis of crack singularities in an aging elastic material

Costabel, Martin; Dauge, Monique; Nazarov, Sergeï A.; Sokolowski, Jan

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 3, pp. 553-595.

We consider a quasistatic system involving a Volterra kernel modelling an hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possibly anisotropic material law. We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the material law and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneous functions of degree $\frac{1}{2}$ or have a more complicated dependence on the distance variable $r$ to the crack tips. In the latter situation, we observe a novel behavior of the singular functions, incompatible with the usual fracture criteria, involving super polynomial functions of $lnr$ growing in time.

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