Classification of linear viscoelastic solids based on a failure criterion

An isotropic, incompressible linear viscoelastic solid subjected to a step shear displacement fails if the relaxation function G(s) is such that 0<G(0)< and –<G(0)0. In this case, the discontinuity in displacement propagates into the interior of the body. The discontinuity will not propagate however if G(0)= or G(0)=–. In the former case there is a diffusion-like smoothening of discontinuous data characteristic of parabolic equations. The case G(0)= may be achieved by composing the kernel as a sum of a smooth kernel and a delta function at the origin times a viscosity coefficient. If the viscosity is small, the smoothing will take place in a propagating layer which scales with the small viscosity. The case of G(0)=– is interesting in the sense that the solution is C smooth but the boundary of the support of the solution propagates at a constant wave spped. If 0<G(0)< and –<G(0)<0, then the material accomodates stress waves under step traction leading to an elastic steady state.