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 $\frac12$ 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 ln r growing in time.