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In this work we prove local existence of strong solutions to theinitial-value problem arising in one-dimensional strain-limitingviscoelasticity, which is based on a nonlinear constitutive relation betweenthe linearized strain, the rate of change of the linearized strain and thestress. The model is a generalization of the nonlinear Kelvin-Voigtviscoelastic solid under the assumption that the strain and the strain rate aresmall. We define an initial-value problem for the stress variable and then,under the assumption that the nonlinear constitutive function is strictlyincreasing, we convert the problem to a new form for the sum of the strain andthe strain rate. Using the theory of variable coefficient heat equationtogether with a fixed point argument we prove local existence of solutions.Finally, for several constitutive functions widely used in the literature weshow that the assumption on which the proof of existence is based is notviolated.

Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity