Strain-limiting viscoelasticity with stress rate dependence via Edelen’s theory of primitive thermodynamics

Abstract The overarching aim of this paper is to investigate the viscoelastic material response within the context of strain-limiting theory of elasticity by following Edelen’s approach of primitive thermodynamics. To achieve this, first, a thermodynamic derivation of strain-limiting viscoelasticity is given, where the constitutive relation is expressed in terms of the strain, stress, and stress rate, based on Edeelen’s representation theory. This also shows that it is possible to obtain a geometrically linear and physically nonlinear viscoelastic model including a non-dissipative contribution. Secondly, a parallelism of the model under consideration with Maxwell-Cattaneo heat conduction is shown. This is the first time in the literature that a viscoelastic material model without the presence of the rate of the strain is derived in connection with the well-known linear models of viscoelasticity. Third, the equation of motion representing the dynamics of the viscoelastic material is derived where the elastic part of the stress tensor is assumed to be linear and basic solutions to this equation are given. Finally, we find that, in the planar case under quasi-static traction boundary conditions, a shift in material properties is possible such that the stress and stress rate fields remain unchanged.