  We consider shear wave propagation in soft viscoelastic solids of rate type.
Based on objective stress rates, the constitutive model accounts for finite
strain, incompressibility, as well as stress- and strain-rate viscoelasticity.
The theory generalises the standard linear solid model to three-dimensional
volume-preserving motions of large amplitude in a physically-consistent way.
The nonlinear equations governing shear motion take the form of a
one-dimensional hyperbolic system with relaxation. For specific objective rates
of Cauchy stress (lower- and upper-convected derivatives), we study the
propagation of acceleration waves and shock waves. Then we show that both
smooth and discontinuous travelling wave solutions can be obtained
analytically. We observe that the amplitude and velocity of steady shocks are
very sensitive to variations of the stress relaxation time. Furthermore, the
existence of steady shocks is conditional. Extension of these results to the
case of multiple relaxation mechanisms and of the Jaumann stress rate is
attempted. The analysis of simple shearing motions is more involved in these
cases.
