  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, or equivalently, 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.
