On the transient elastic field for an expanding spherical inclusion in 3-D solid

The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.