Propagation of Weak Shock Waves in Materials with Memory

The propagation of longitudinal shock waves is studied for Green-Rivlin viscoelastic materials. Significant results can be obtained only for a class of materials, almost-elastic materials. Using the integral-expansion method along with the derivative-expansion method, well-known Burgers' equation is derived as a far field equation for solid-like materials with a normal memory. For both solid-like and fluid-like materials with an anomalous memory, a far field equation becomes a new type of integro-partially differential equation. In practice, the mechanical models consisting of spring and dashpot elements are considered as concrete and manifest examples. The structure of steady shocks is examined by solving the far field equations thus obtained.