The equilibrium of material forces in a 1D phase transition problem

This paper aims at the use of material forces in estimating the interface location in an elastic, one-dimensional, two-phase problem. A variational formulation based on variations of both the dependent and the independent variables results in the weak form of the momentum and the canonical momentum equations, thus providing an appropriate framework for the use of the finite element method. Using this, one can compute, apart from the finite element solution to the equilibrium equation for the physical forces, the corresponding one to the equilibrium equation for the material forces which provides the location of the interface. In this way the total energy is minimized with respect to the deformations as well as to the interface location. Our theoretical expectations are confirmed by a numerical example concerning a one-dimensional, deformation-induced phase transition problem with known analytical solution.