Microstructural model of the behavior of a ferroalloy with shape memory in a magnetic field

Abstract

In the article, within the framework of the theory of micromagnetism, a magneto-elastic microstructural model of the behavior of a ferromagnetic material with the shape memory (Heusler alloy) in a magnetic field is constructed. The dynamics of the magnetic process is described by the Landau-Lifshitz-Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out. A martensitic structure of the type “herringbone” (a twinned variant of martensite) with magnetic domains located at an angle of ${180}^{°}$ is considered. 90-degree magnetic domain walls are the boundaries of the twins. The finite element method models the formation of these walls and the distribution of the magnetization vector in them. The evolution of this magnetic structure is investigated, namely the movement and interaction of 180-degree magnetic domain walls when an external magnetic field is applied in various directions. The problems of the formation of twins from the martensite variants (the twinning problem) and their disappearance (the detwinning problem) in an external magnetic field are considered. On the plane separating the two variants of martensite, the Hadamard compatibility condition (or twinning equation) is written out. The solution of this equation is given and the processes, that this equation describes, are considered in detail. The detwinning condition for a ferroalloy with the shape memory in a magnetic field is proposed and the processes related to the reorientation (detwinning) of the martensitic variants forming a twin are discussed. The problems considered earlier without taking into account the detwinning process are supplemented by this process, the magnetization curves are constructed and the components of the strain tensor are determined. The results obtained are in good agreement with the known experimental data.

Keywords:

• Heusler alloy
• micromagnetism
• magnetic domains
• variational formulation
• finite element method
• twinning and detwinning processes
• detwinning condition

Acknowledgements

Authors appreciate greatly financial support of the Russian Foundation for Basic Research (RFBR), project 20-01-00031.

Notes

1 The N-degree domain boundary is a transition layer between neighboring domains A and B with opposite or coinciding directions of the magnetization vectors ${\mathit{m}}_A$ and ${\mathit{m}}_B,$ in which the magnetic moment gradually turns from the direction ${\mathit{m}}_A$ to the direction ${\mathit{m}}_B.$ If in this case the rotation of the magnetization vector occurs in a plane coinciding with the wall plane, the boundary is called the Bloch boundary, if it occurs in a plane perpendicular to the wall plane, then the boundary is called the Neel boundary [35].

2 The tensor $\mathbf{Q}$ for which ${\mathbf{Q}}^T={\mathbf{Q}}^{-1}$ is called the orthogonal tensor. This tensor, when scalar multiplied by a vector, rotates the latter in space, preserving its modulus. With a similar action on two vectors, it, by rotating them in space, also preserves the angle between them. The determinant of the orthogonal tensor is equal to ±1. The determinant of the orthogonal tensor proper is equal to + 1.

3 In the work of Mennerich et al. [10] it is stated that the short axes of two tetragonal martensite cells forming a twin in Ni₂MnGa alloy are located at an angle of ${86.5}^{°}$ to each other, and reference is made to the experimental work of Solomon et al. [29]. But in the latter, another alloy is considered, namely Ni₅₁Mn₂₉Ga₂₀. Probably the difference in the angles is related to this.