  In this work we prove local existence of strong solutions to the
initial-value problem arising in one-dimensional strain-limiting
viscoelasticity, which is based on a nonlinear constitutive relation between
the linearized strain, the rate of change of the linearized strain and the
stress. The model is a generalization of the nonlinear Kelvin-Voigt
viscoelastic solid under the assumption that the strain and the strain rate are
small. We define an initial-value problem for the stress variable and then,
under the assumption that the nonlinear constitutive function is strictly
increasing, we convert the problem to a new form for the sum of the strain and
the strain rate. Using the theory of variable coefficient heat equation
together with a fixed point argument we prove local existence of solutions.
Finally, for several constitutive functions widely used in the literature we
show that the assumption on which the proof of existence is based is not
violated.
