  We consider the equation of motion for one-dimensional nonlinear
viscoelasticity of strain-rate type under the assumption that the stored-energy
function is $\lambda$-convex, which allows for solid phase transformations. We
formulate this problem as a gradient flow, leading to existence and uniqueness
of solutions. By approximating general initial data by those in which the
deformation gradient takes only finitely many values, we show that under
suitable hypotheses on the stored-energy function the deformation gradient is
instantaneously bounded and bounded away from zero. Finally, we discuss the
open problem of showing that every solution converges to an equilibrium state
as time $t \to \infty$ and prove convergence to equilibrium under a
nondegeneracy condition. We show that this condition is satisfied in particular
for any real analytic cubic-like stress-strain function.
