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Abstract

We investigate a 1-dimensional simple version of the Fried–Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau–Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified assumptions on the strain energy and data we prove the existence and uniqueness of a regular solution to the problem. The proof is based on the Leray–Schauder fixed point theorem.