Elastic radiation from a propagating phase boundary

Abstract

The elastic effects which occur when a solid undergoes a phase change from a metastable to a stable structure are considered. In such a change, stored energy is released, and some of this energy may appear in elastic form. Equations describing the elastic effects of the process are derived from an examination of internal strain and an assumption of linearity for frequencies in the seismic range. According to concepts presented in this paper, a solid can be strained internally by the presence of dislocations (real internal strain), and also by possessing a potential to deform at every point through a change in crystal structure (frozen internal strain). Equations for elastic effects are derived by using these concepts, and they are solved for plane phase boundaries in steady motion. These results are valid only for times sufficiently short that the linear approximation still applies. The speed of boundary motion cannot be determined from macroscopic classical theory, but the equations have solutions that correspond to two different mechanisms of boundary propagation. The first occurs if a stress pulse is able to induce the transformation. The plane phase boundary then propagates at the speed of sound, with a growing elastic wave tied to it, using up some of the energy released in the transformation. The second propagation mechanism occurs when externally applied stress exerts a force on the real strain field associated with the phase boundary. In this case, the speed of propagation is different from the sound velocity, but the process again involves an increase in elastic energy at the boundary due to the release of internal strain under phase transition. The amount of internal strain release depends on the thermodynamic nature of the process. If the reaction is sudden, strain release is small, and the transformation essentially diffusionless, the initial process is considered to occur at constant entropy. If the reaction continues, it becomes non-linear and the entropy is expected to increase. The non-linear problem is not treated in this paper. An estimate of energy release for the linear problem is given for the purpose of considering the propagating phase boundary as a possible earthquake source mechanism.