Stable and unstable growth of crack tip precipitates

A model is established that describes stress driven diffusion, resulting in formation and growth of an expanded precipitate at the tip of a crack. The new phase is transversely isotropic. A finite element method is used and the results are compared with a simplified analytical theory. A stress criterium for formation of the precipitate is derived by direct integration of the Einstein-Smoluchowski law for stress driven diffusion. Thus, the conventional critical concentration criterium for precipitate growth can be replaced with a critical hydrostatic stress. The problem has only one length scale and as a consequence the precipitate grows under self-similar conditions. The length scale is given by the stress intensity factor, the diffusion coefficient and critical stress versus remote ambient concentrations. The free parameters involved are the expansion strain, the degree of anisotropy and Poisson's ratio. Solutions are obtained for a variation of the first two. The key result is that there is a critical phase expansion strain below which the growth of the new phase is stable and controlled by the stress intensity factor. For supercritical expansion strains, the precipitate grows even without remote load. The anisotropy of the expansion strongly affects the shape of the precipitate, but does not have a large effect on the crack tip shielding.