Transformation front kinetics in deformable ferromagnets

Materials such as magnetic shape-memory alloys possess an intrinsic coupling between material’s magnetisation and mechanical deformation. These materials also undergo structural phase transitions, with phase boundaries separating different phases and the kinetics of the phase boundaries governed by the magnetic field and the mechanical stresses. There is a multiplicity of other materials revealing similar phenomena, e.g. magnetic perovskites. To model the propagation of the phase boundaries in deformable magnetic materials at the continuum scale, three ingredients are required: a set of governing equations for the bulk behaviour with coupled magnetic and mechanical degrees of freedom, a dependency of the phase boundary velocity on the governing factors, and a reliable computational method. The expression for the phase boundary velocity is usually obtained within the continuum thermodynamics setting, where the entropy production due to phase boundary propagation is derived, which gives a thermodynamic driving force for the phase boundary kinetics. For deformable ferromagnets, all three elements (bulk behaviour, interface kinetics, and computational approaches) have been explored, but under a number of limitations. The present paper focuses on the derivation of the thermodynamic driving force for transformation fronts in a general magneto-mechanical setting, adapts the cut-finite-element method for transformation fronts in magneto-mechanics, which allows for an exceptionally efficient handling of the propagating interfaces, without modifying the finite-element mesh, and applies the developments to qualitative modelling of magneto-mechanics of magnetic shape-memory alloys.

💡 Research Summary

This paper addresses the challenging problem of modeling and numerically simulating the propagation of phase‑transformation fronts in deformable ferromagnets, with a particular focus on magnetic shape‑memory alloys (MSMAs). The authors first develop a fully general thermodynamic framework for magneto‑mechanical continua in which the lattice (mechanical) and spin (magnetic) fields coexist in the same material volume. By assuming saturated magnetisation of constant magnitude, they derive the kinematic relation (\dot{\mathbf m}= \mathbf m\times\mathbf a) and connect magnetisation to angular momentum through the gyromagnetic relation (\mathbf l = \gamma_0^{-1}\mathbf m).

Using Reynolds’ transport theorem and Gauss’ theorem, the authors systematically treat a moving interface (\Sigma(t)) that cuts through the reference domain (\Omega). They introduce jump (\llbracket\cdot\rrbracket) and average (\langle\cdot\rangle) operators to handle discontinuities of stress, traction, magnetic field, and other field quantities across the front. By evaluating the entropy production associated with front motion, they obtain a universal expression for the thermodynamic driving force (\mathcal F) that depends on the jump in mechanical traction, magnetic body forces, and the local magnetic induction. The second law of thermodynamics imposes (\mathcal F,W^\ast \ge 0), where (W^\ast) is the normal velocity of the front. Consequently, a linear kinetic law (W^\ast = M,\mathcal F) (with mobility (M)) is proposed, which does not rely on any specific constitutive model. For non‑dissipative solids and quasistatic conditions, the authors show that solving the governing equations is equivalent to minimizing an appropriate energy functional, thereby linking front kinetics to an energy‑gradient flow.

The second major contribution is the adaptation of the Cut‑Finite‑Element Method (CutFEM) to this magneto‑mechanical setting. In CutFEM the computational mesh is independent of the moving front; the front is represented as an arbitrary cut through the background mesh. The authors enforce continuity of traction and magnetic field across the cut using Nitsche’s method and add ghost‑penalty stabilization to guarantee numerical robustness. This approach eliminates the need for remeshing as the front propagates, dramatically reducing computational cost while preserving high accuracy even for complex three‑dimensional geometries. The method is extended to solve the coupled Maxwell equations and nonlinear elasticity equations simultaneously, allowing full magneto‑mechanical coupling at the interface.

Finally, the framework is applied to a qualitative simulation of twin‑boundary motion in MSMAs. The high magnetic anisotropy of these alloys causes structural domains to reorient under an external magnetic field, which in turn drives the motion of twin boundaries. The authors demonstrate that their model captures key phenomena: (i) acceleration of the front under increasing magnetic field strength, (ii) retardation or arrest of the front when mechanical stresses oppose the magnetic driving force, and (iii) the existence of critical field‑stress combinations that trigger rapid front propagation followed by stabilization in a new phase configuration. The simulated behavior aligns with experimental observations reported in the literature and showcases the advantages of a mesh‑independent CutFEM implementation over traditional sharp‑interface or phase‑field methods, which typically require frequent mesh updates.

In summary, the paper delivers (1) a rigorous, constitutive‑independent thermodynamic derivation of the driving force for transformation fronts in magneto‑mechanical continua, (2) a robust, efficient CutFEM‑based numerical scheme for solving the resulting coupled PDE system without remeshing, and (3) a proof‑of‑concept application to magnetic shape‑memory alloys that reproduces essential twin‑boundary kinetics. The work opens pathways for future extensions, such as incorporating temperature gradients, non‑saturated magnetisation dynamics, multi‑front interactions, and quantitative calibration against experimental data.