Phenomenological modeling for magneto-mechanical couplings of martensitic variant reorientation in ferromagnetic shape memory alloys

Abstract

The paper presents a 2D phenomenological model to characterize the magneto-mechanical coupled behavior and its temperature dependences for martensitic variant reorientation in ferromagnetic shape memory alloys. A set of state variables are chosen to describe different microscopic mechanisms, and the evolution rules for the state variables are determined by minimizing the thermodynamic potential. The modified Landau free energy function is employed to account for the hysteretic switching between martensitic variants. It is assumed that the local switching thresholds are distributed due to the non-uniform internal stress throughout the specimen. The governing equation for martensite reorientation is then homogenized by taking this distribution property into account. The concept of density reassignment is also deployed to improve the modeling accuracy for partial reorientations. To validate the model capability, the model formulation is numerically implemented for the typical loading condition (i.e., uniaxial stress and a perpendicular magnetic field). As one of the superiorities, the temperature-dependent properties are remarked for the proposed model as well. Comparisons between the model predictions and the experimental results demonstrate the model capability in addressing the magneto-mechanical couplings and the temperature dependences of martensite reorientation.

Appendix

1.1 Model calibration

In this work, the experimental data in [8, 26] are employed to validate the model capabilities, in which the single crystalline Ni_49.7Mn_29.1Ga_21.2 (at %) rectangular prism $9\times 5\times 5 {\mathrm{mm}}^{3}$ in size was used for the simultaneous measurement of the strain and magnetization responses. Besides, several reported data in [25] are employed to characterize the temperature-dependent properties of the Ni–Mn–Ga sample. The model calibration is discussed in this section, providing the details on the identification of the material constants and the model parameters given in Table 1.

Fig. 18

Calibration of several material constants: a Young’s moduli fitted from the measured stress–strain data in [8], b temperature-dependent magnetic properties fitted using the reported data in [25], and c temperature-dependent martensite tetragonality fitted using the reported data in [25]

The Curie temperature ${T}_{\mathrm{c}}$, and the austenitic transformation temperature ${T}_{0}$ are taken from [25], while the mass density $\rho$ are taken from [34]. Other material constants are determined from the experimental data by curve fitting. As shown in Fig. 18a, Young’s moduli ${E}_{1}$ and ${E}_{2}$ can be obtained using the stress–strain data under zero bias field in [8]. The reported data in [25] are exploited to illustrate the temperature dependences of the magnetic properties and the martensite tetragonality in Fig. 18b, c, from which the constants ${M}_{0}$, $p$, ${K}_{0}$, $q$, ${k}_{\varepsilon }$, $r$, and ${\varepsilon }_{0}$ are fitted. Apart from these material constants, there are several model parameters to be identified in the proposed model. Given the specimen size, the diagonal entries of the demagnetization tensor, i.e., ${D}_{1}$ and ${D}_{2}$, can be simply computed using the analytic formula in [35]. In this work, the identification of the remaining parameters, i.e., ${\kappa }_{1}$, ${k}_{\phi }$, and ${\phi }_{0}$, is treated as a least-square optimization problem, which is to minimize the error between the modeling results and experimental counterparts. These parameters are utilized to compute the predicted strain and magnetization responses and then updated by optimizing the prediction error. Herein, the optimization problem is given by

$$\mathop {{\text{min}}}\limits_{{\kappa _{1} ,\,k_{\phi } ,\, \phi _{0} }} \,\,\sum _{{{\text{j}} = 1}}^{{\text{N}}} \left( {\frac{{\hat{\varepsilon }_{{\text{j}}} - \varepsilon _{{\text{j}}} }}{{\varepsilon _{{{\text{max}}}} }}} \right)^{2} + \left( {\frac{{\hat{M}_{{\text{j}}} - M_{{\text{j}}} }}{{M_{{\text{s}}} }}} \right)^{2} ,$$

(36)

where ${\widehat{\varepsilon }}_{\mathrm{j}}$ and ${\widehat{M}}_{\mathrm{j}}$ denote the measured values of strain and relative magnetization while ${\varepsilon }_{\mathrm{j}}$ and ${M}_{\mathrm{j}}$ denote the predicted ones, and $\text{N}$ is the number of the sampled points. ${\varepsilon }_{\mathrm{max}}$ is the maximum strain that is used to obtain the relative strain error. The optimization of the prediction error is carried out using the MATLAB function fminsearch which is based on the Nelder-Mead algorithm, and three parameters are calibrated with the measured data in Figs. 7, 10, and 13.