A real-space and constraint-free phase field model for the microstructure of ferromagnetic shape memory alloys

Abstract

A continuum real-space and constraint-free phase field model is proposed for simulating the microstructure of ferromagnetic shape memory alloys. For the ferroelastic orderings, a new set of internal variables \(\lambda _\text {I}\) motivated by the multi-rank laminated microstructure of martensitic variants are used as the order parameters. For the ferromagnetic orderings, the model takes the polar and azimuthal angles \((\vartheta _1,\vartheta _2)\) as the order parameters. In this way, the model is free from the constraint on the volume fraction of martensitic variants and the magnetization magnitude. The phase field model is developed from a thermodynamic framework which involves a configurational or micro force system for the order parameters, thermodynamically consistent constitutive relations, and the generalized evolution equations. The 3D finite element implementation of the model in real space is straightforward. Numerical examples show that the model can capture the correct micrographs of the ferroelastic and ferromagnetic microstructures and their evolution under external mechanical or magnetic loading. The distribution of ferromagnetic and ferroelastic domains in the equilibrium state is found to be dependent on both the boundary constraints and the sample geometry. The 3D ferroelastic variants distribution and the ferromagnetic vortices can also be resolved.