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We introduce a one-dimensional stress-rate type nonlinear viscoelastic modelfor solids that obey the assumptions of the strain-limiting theory. Unlike theclassical viscoelasticity theory, the critical hypothesis in the presentstrain-limiting theory is that the linearized strain depends nonlinearly on thestress and the stress rate. We show the thermodynamic consistency of the modelusing the complementary free energy, or equivalently, the Gibbs free energy.This allows us to take the stress and the stress rate as primitive variablesinstead of kinematical quantities such as deformation or strain. We also showthat the non-dissipative part of the materials in consideration have a storedenergy. We compare the new stress-rate type model with the strain-rate typeviscoelastic model due to Rajagopal from the points of view of energy decay,the nonlinear differential equations of motion and Fourier analysis of thecorresponding linear models.

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