Modeling of materials capable of solid-solid phase transformation. Application to the analytical solution of the semi-infinite mode III crack problem in a phase-changing solid

The paper proposes two frameworks for the derivation of constitutive models for solids undergoing phase transformations. Two distinct approaches are considered: 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 different phase fragments and uses the homogenization theory to derive constitutive relations at the macroscale. For both approaches, the mathematical representation of the material behavior is intentionally kept simple and the derivations are fully developed for ease of future modification and use by the readership. It is shown that in the case of reversible phase transformation, the energy of the material can be obtained by convexifiying the energy functions of the constituent phases. It is further shown that for dissipative phase transformation the material behavior can be made stable by deriving the evolution equations of the state variables from adequately chosen dissipation potentials. Some new results of existence and uniqueness of the solution of boundary value problems of structures undergoing phase change are also given. As an application, a schematic model is derived and used to obtain analytical solutions for the problem of semi-infinite mode III crack in a solid capable of phase transformation.