Modeling framework for materials capable of solid-solid phase transformation: Application to the analytical solution of the semi-infinite mode III crack problem in an idealized shape memory alloy

We propose two frameworks for the derivation of constitutive models for solids undergoing phase transformations. The first is based on the assumption that solid phases within the material are finely mixed whereas the second considers the material as a heterogeneous solution of phase fragments and uses the homogenization theory to derive equilibrium conditions for displacement fields and phase distributions. It is shown that in the case of reversible phase transformation, the energy of the material can be obtained by taking the convex envelope of the energy functions of the constituent phases. As an application, a schematic model is derived for an idealized shape memory alloy and used to obtain a novel analytical solution for the problem of semi-infinite mode III crack in this material. The derivation of the analytical solution uses the hodograph method to map Cartesian coordinates into the hodograph plane. The resulting boundary-value problem for the mode III crack considered becomes analytically tractable for the idealized shape memory alloy considered and leads to closed-form expressions for the displacement and phase volume fraction fields near the crack tip as well as for the boundaries between different phase regions.