A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity

We introduce a one-dimensional stress-rate type nonlinear viscoelastic model for solids that obey the assumptions of the strain-limiting theory. Unlike the classical viscoelasticity theory, the critical hypothesis in the present strain-limiting theory is that the linearized strain depends nonlinearly on the stress and the stress rate. We show the thermodynamic consistency of the model using the complementary free energy and then using the Gibbs free energy. This allows us to take the stress and the stress rate as primitive variables instead of kinematical quantities such as deformation or strain. We also show that the non-dissipative part of the materials in consideration have a stored energy. We compare the new stress-rate type model with the strain-rate type viscoelastic model due to Rajagopal from the points of view of energy decay, the nonlinear differential equations of motion and Fourier analysis of the corresponding linear models.