An elastically deformable solid separated into two regions consisting of atoms of different types and having different elastic properties is considered. To describe the motion of the interface separating the two regions (or phases), a phase-field model is introduced and studied in a one-dimensional setting. It describes the space-time evolution of the displacement \((u,0,0)\in {\mathbb R}^3\), the Cauchy stress tensor \(T\) which is a symmetric \(3\times 3\) matrix, and an order parameter \(S\) describing the location of the two regions in the solid, and couples an elliptic equation for the deformation with a fourth-order nonlinear degenerate parabolic equation for \(S\). The main difficulty lies in the fact that the equation for \(S\) is not uniformly parabolic where \(\partial_x S\) vanishes.