Classification of linear viscoelastic solids based on a failure criterion

Abstract

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.