Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.

► A non-linear rate-type constitutive equation generalizes the Maxwell model.
► The constitutive equations are framed within the scheme of materials with internal variables.
► The thermodynamic restrictions on the evolution equation are determined.
► The propagation condition of shock and acceleration waves is determined.
► The effective acoustic tensor is the sum of a thermoelastic tensor and of a tensor arising from the rate-type equation.