  The formation of shear shock waves in the brain has been proposed as one of
the plausible explanations for deep intracranial injuries. In fact, such
singular solutions emerge naturally in soft viscoelastic tissues under dynamic
loading conditions. To improve our understanding of the mechanical processes at
hand, the development of dedicated computational models is needed. The present
study concerns three-dimensional numerical models of incompressible
viscoelastic solids whose motion is analysed by means of shock-capturing finite
volume methods. More specifically, we focus on the use of the artificial
compressibility method, a technique that has been frequently employed in
computational fluid dynamics. The material behaviour is deduced from the
Fung--Simo quasi-linear viscoelasiticity theory (QLV) where the elastic
response is of Yeoh type. We analyse the accuracy of the method and demonstrate
its applicability for the study of nonlinear wave propagation in soft solids.
The numerical results cover accuracy tests, shock formation and wave focusing.
