## 3 Construction of a Hysteresis Model of a Giant Magnetostrictive Microactuator Based on Free Energy
### Theoretical Basis for the Establishment of the Free Energy Hysteresis Model
To assume that the parameter \(H_{c}\ =\ \eta(M_{R}-M_{I})\) is a normal distribution, with respect to the average coercivity \(\overline{H}_{c}\), the total magnetization can be written by Equation (21). \[M(H)\ =\ \int_{0}^{\infty}\overline{M}(H;H_{c},\xi)f(H_{c})dH_{c}, \tag{21}\]  
where \(H_{c}\) is the density distribution function and can be expressed by Equation (22). \[f(H_{c})\ =\ C_{1}e^{-\left(H_{c}-\overline{H}_{c}\right)^{2}/b}, \tag{22}\]  
where \(C_{1}\) and \(b\) are coefficients, and \(\overline{M}\) can be obtained from Equation (10) or (17). In Equation (21), its lower limit of integration is 0, which indicates that the hysteresis core width must be non-negative. In addition, it is defined as a logarithmic function to be also used to reflect the non-negative nature of the hysteresis kernel width. In the Jiles-Atherton model, the Weiss field effect caused by the coupling between domains can be represented by Equation (23). \[H_{\epsilon}\ =\ H+\alpha M, \tag{23}\]  
where \(\alpha\) is a constant mean field parameter. Weiss fields quantitatively describe the interaction between atoms in giant magnetostrictive material. The effective field \(H_{\epsilon}\) exhibits changes caused by uneven magnetic moment distribution, and it is assumed that the effective field \(H_{\epsilon}\) is a normal distribution of the applied field. The magnetization can be expressed by Equation (24). \[M(H)\ =\ \int_{-\infty}^{\infty}C_{2}\overline{M}(;H_{c},\xi)e^{-\left(H- \right)^{2}/\overline{b}}d. \tag{24}\]  
Thus, under the effective field of the low-frequency range without considering the eddy current loss, the complete magnetization model of the heterogeneous polycrystalline material can be expressed as:  
\[[M(H)](t)\ =\ C\int_{0}^{\infty}\int_{-\infty}^{\infty}[\overline{M}(+H;H_{c}, \xi)](t)e^{-2/\overline{b}}e^{-\left(H_{c}-\overline{H}_{c}\right)^{2}/b}ddH_ {c}. \tag{25}\]  
(5) Construction of a magnetic-mechanical coupling eigen model of giant magnetostrictive material The Gibbs energy expression of Equation (8) combines the internal energy and magnetic moment energy of the isotropic material on the magnitude of the magnetic domain, but the magnetic coupling effect is ignored, and the magnetostrictive ability of the material cannot be exhibited. When anisotropy is exhibited for the lattice dominated by the actuator's pre-stress, the Helmholtz free energy relation embodying magnetoelasticity can be expressed by Equation (26). \[\psi_{e}(M,\varepsilon)\ =\ \psi(M)+\frac{1}{2}\gamma^{M}\varepsilon^{2}-\gamma^{ M}\gamma\varepsilon M^{2}. \tag{26}\]  
The corresponding Gibbs energy is expressed by Equation (27). \[G(H,M,\varepsilon)\ =\ \psi(M)+\frac{1}{2}Y^{M}\varepsilon^{2}-Y^{M}\gamma \varepsilon M^{2}-HM-\sigma\varepsilon, \tag{27}\]  
where \(\psi\) can be obtained from Equation (7). \(Y^{M}\) is the Young's modulus at a certain magnetization. \(\gamma\) is the magnetic coupling coefficient. In the case of intense thermal motion, the magnetization \(\overline{M}\) can be obtained by Equation (9), and the Gibbs energy can be obtained by Equation (27). Magnetization is expressed by Equation (28) when the thermal motion is ignored. \[[\overline{M}(H,\varepsilon;H_{c},\xi)](t)\ =\ \left\{ \begin{array}{l}[ \overline{M}(H,\varepsilon;H_{c},\xi)](0),\,\tau(t)\ =\ \phi\\ \frac{H}{\eta-2Y^{M}\gamma\varepsilon}-\frac{M_{R}\eta}{\eta-2Y^{M}\gamma \varepsilon},\tau(t)\
eq\phi\,and\,H[\max_{\Upsilon}(t)]\ =\ -H_{c}\\ \frac{H}{\eta-2Y^{M}\gamma\varepsilon}+\frac{M_{R}\eta}{\eta-2Y^{M}\gamma \varepsilon},\tau(t)\
eq\phi\,and\,H[\max_{\Upsilon}(t)]\ =\ H_{c}\end{array} \right., \tag{28}\]  
where \(H_{c}\ =\ \eta(M_{R}-M_{l})\), \(\tau(t)\) can be obtained by Equation (18), and Equation (29) is obtained. \[[\overline{M}(H,\varepsilon;H_{c},\xi)](0)\ =\ \left\{ \begin{array}{l}\frac{H}{ \eta-2Y^{M}\gamma\varepsilon}-\frac{M_{R}\eta}{\eta-2Y^{M}\gamma\varepsilon},\ H(0)\leq-H_{c}\\ \xi,-H_{c}<H(0)<H_{c}\\ \frac{H}{\eta-2Y^{M}\gamma\varepsilon}+\frac{M_{R}\eta}{\eta-2Y^{M}\gamma \varepsilon},H(0)\geq H_{c}\end{array}