On the steady-state propagation of an anti-plane shear crack in an infinite general linearly viscoelastic body

The steady-state propagation of a semi-infinite anti-plane shear crack is considered for a general infinite homogeneous and isotropic linearly viscoelastic body. Inertial terms are retained and the only restrictions placed on the shear modulus are that it be positive, continuous, decreasing and convex. For a given integrable distribution of shearing tractions travelling with the crack, a simple closed-form solution is obtained for the stress intensity factor and for the entire stress field ahead of and in the plane of the advancing crack. As was observed previously for the standard linear solid, the separate considerations of two distinct cases, defined by parameters $c$ and $c*$, arises naturally in the analysis. Specifically, $c$ and $c*$ denote the elastic shear wave speeds corresponding to zero and infinite time, and the two cases are (1) $0 < \upsilon < c*$ and (2) $c* < \upsilon < c$, where $\upsilon$ is the speed of propagation of the crack. For case (1) it is shown that the stress field is the same as in the corresponding elastic problem and is hence independent of $\upsilon$ and all material properties, whereas, for case (2) the stress field depends on both $\upsilon$ and material properties. This dependence is shown to be of a very elementary form even for a general viscoelastic shear modulus.