Phase-field theory for martensitic phase transformations at large strains

Abstract

General phase-field theory for multivariant martensitic phase transformations and explicit models are formulated at large strains. Each order parameter is unambiguously related to the transformation strain of the corresponding variant. Thermodynamic potential includes energy related to the gradient of the order parameters that mimics the interface energy. Application of the global form of the second law of thermodynamics resulted in the determination of the driving force for change of the order parameters and the boundary conditions for the order parameters. Kinetic relationships between the rate of change of the order parameters and the conjugate driving force lead to the Ginzburg–Landau equations. For homogeneous fields, conditions for instabilities of the equilibrium states (which represent criteria for the phase transformation between austenite and martensitic variants and between martensitic variants) are found for the prescribed Piola–Kirchoff stress tensor. It was proved that these criteria are invariant with respect to change in the prescribed stresses. The expression for the rigid-body rotation tensor is derived for the prescribed Piola–Kirchoff stress. The explicit expressions for the Helmholtz free energy and for transformation strain in terms of order parameters are derived for the most general case of large elastic and transformational strains, rotations, as well as nonlinear, anisotropic, and different elastic properties of phases. For negligible elastic strains, explicit expression for the Gibbs potential is formulated. Results are obtained for fifth- and sixth-degree potentials in Cartesian order parameters and for similar potentials in hyperspherical order parameters. Geometric interpretation of transformation conditions in the stress space and similarity with plasticity theory are discussed. All material parameters are obtained for cubic to tetragonal transformation in NiAl. Phase transformations in NiAl, boron nitride, and graphite to diamond under uniaxial loading are described explicitly, and the importance of geometrically nonlinear terms is demonstrated. A similar approach can be applied for twinning, dislocations, reconstructive transformations, and fracture.