A quasistatic model of the evolution of an interface inside a deformed solid

A one-dimensional integral equation, the solution of which enables one to follow the (small and continuous) change in the form of the interface as a function of a time-like loading parameter, is derived by constructing a formal trinomial asymptotic form of the elastic fields. The operator and other data of the equation are expressed in terms of the Steklov-Poincaré operators for separated phases at the initial instant and the solutions of the problem with a fixed interface. An investigation of the equation establishes the stability of the development and the possibility of bifurcations or the need to take dynamic effects into account. A well-known thermodynamic condition at the interface and a new condition of its classical stability are obtained as a special case.