Extensions of the constitutive relations to describe the response of compressible nonlinear Kelvin–Voigt solids

This paper investigates the propagation of longitudinal waves in some possible models of compressible Kelvin-Voigt viscoelastic solids. With respect to the elastic part two extensions of the classical neo-Hookean material model are proposed: the A-model, which incorporates a logarithmic volumetric function, and the B-model, based on the deviatoric invariants and a power-law volumetric function. For both models, we assume the same dissipative part given by the classical Navier–Stokes constitutive equation. These models are analyzed for their ability to describe a recovery phenomenon, ensuring conditions for monotonicity, boundedness, and uniqueness of solutions. The propagation of longitudinal traveling waves is proved. We show that the equation governing such motions is indeed a special case of the viscous p-system and a weakly nonlinear analysis demonstrates the emergence of Burgers’ equations.