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We consider a quasistatic system involving a Volterra kernel modellingan 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 possiblyanisotropic material law.We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the materiallaw and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneousfunctions of degree $\frac12$ 
or have a more complicated dependence onthe distance variable r to the crack tips. In the latter situation,we observe a novel behavior of the singular functions, incompatible withthe usual fracture criteria, involving super polynomial functions of ln r growing in time.
