We show that a morphological instability driven by deviatoric applied stresses can generate elastically induced particle splitting during phase transformations. The splitting instability occurs when the elastic fields are above some critical value. For elastic fields below critical, one observes a small perturbation of the particle shape consistent with splitting, but this perturbation is stabilized by surface tension. Both the onset of the splitting instability and the nonlinear evolution of the particle towards splitting depend on the precise form of the applied stress, the elastic constants of the precipitate and matrix, and the initial shape of the precipitate. We also investigate whether non-dilatational mistif strains can generate splitting instabilities in the absence of applied stress; however the results are inconclusive.