Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys

Abstract

The magnetically induced martensitic variant reorientation process under applied mechanical load in magnetic shape memory alloys (MSMAs) is considered. Of particular interest is the associated nonlinear and hysteretic macroscopic strain response under variable applied magnetic field in the presence of stress, also known as the magnetic shape memory effect (MSME). A thermodynamically consistent phenomenological constitutive model is derived which captures the magnetic shape memory effect caused by the martensitic variant reorientation process, using internal state variables, which are chosen in consideration of the crystallographic and magnetic microstructure. The magnetic contributions to the free energy function considered in this work are the Zeeman energy and the magnetocrystalline anisotropy energy. Activation functions for the onset and termination of the reorientation process are formulated and evolution equations for the internal state variables are derived. The model is applied to a two-dimensional special case in which the application of a transverse magnetic field produces axial reorientation strain in a NiMnGa single-crystal specimen under a constant compressive axial stress. It is explicitly shown how the model parameters are obtained from experimental data. Model predictions of magnetic field-reorientation strain hysteresis loops under different applied stresses are discussed.

†Dedicated to Professor Gerard Maugin on the occasion of his receiving of the 2003 SES A.C. Eringen Medal.

Acknowledgments

This work was supported by the Army Research Office, contract no. DAAD 19-02-1-0261 and the National Science Foundation, award no. CMS-0324537. The authors would also like to acknowledge the impact of the experimental MSMA work performed by Dr. Ibrahim Karaman and his group at the Mechanical Engineering Department at Texas A&M University on the modelling efforts presented in this paper. An extensive part of this manuscript was written during a visit to Prof. Hans Jürgen Maier's group at the Lehrstuhl für Werkstoffkunde, Universität Paderborn, as part of an ongoing research collaboration.

Notes

†Dedicated to Professor Gerard Maugin on the occasion of his receiving of the 2003 SES A.C. Eringen Medal.

† To be precise, the austenite in Ni₂MnGa is of L2₁ Heusler type structure, which is based on a fcc lattice. The martensite phase exhibits a bct structure, where the unit cell is rotated by 45° against the austenite lattice (cf. 47). However, in most of the literature the more convenient description of the martensite structure depicted here has been adopted.

† An initial single variant configuration can actually be obtained in experiments at stress levels below σ ^sv by first applying a higher stress to completely detwin the martensitic specimen and then lower the applied stress to the desired level.

† This series expansion is valid for an unstrained crystal. However, because of the high magnetocrystalline anisotropy observed in MSMA materials, the influence of stress application on the anisotropy through ordinary magnetostriction is negligible, as was indicated by experiments performed by Tickle 53.

† Recall that the experimentally measured strain vs. applied magnetic field curves cannot be interpreted as the pure constitutive response of the MSMA material, since the applied magnetic field depends on the specimen geometry through the demagnetization factor. In order to extract the constitutive response from experimental data, one has to compute the corresponding strain vs. internal magnetic field curves by subtracting the demagnetization field from the applied field. However, since the demagnetizing field depends on the magnetization, the corresponding magnetization curves need to be considered simultaneously. The matter is complicated by the fact that the measured magnetization curves are themselves given in terms of the applied magnetic field, not the internal magnetic at a specific material point. Since not all of the necessary data needed to extract these curves is provided by Tickle 53, the parameters for this numerical application example are determined based on the measured strain vs. applied magnetic field curves, which in this case are just interpreted as the constitutive response. This assumption only limits the predictability of quantitative data for this specific experiment; the procedure is, however, sufficient to illustrate the applicability and validity of the proposed constitutive model as well as the presented method of determining model parameters.

‡ Throughout this paper SI units are used. However, the CGS system is more commonly used in the literature on magnetic materials. For the magnetic field strength the conversion factor applies 55. Despite being the same unit as the magnetic field strength in the SI system, emu cm⁻³ is the commonly used unit for the magnetization in the CGS system. The corresponding conversion factors are given by: and . For example, the following material constants, as specified in table 2, are converted to CGS units as: H ^s(1,2) = 3460 Oe and M ^sat = 75 emu g⁻¹.