## 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
\[\langle M_{+}\rangle =\ \frac{\int_{M_{I}}^{\infty}Me^{-G(HM)V/kT}dM}{\int_{M_{e}}^{ \infty}e^{-G(HM)V/kT}dM} \tag{12}\] \[\langle M_{-}\rangle =\ \frac{\int_{-\infty}^{-M_{I}}Me^{-G(HM)V/kT}dM}{\int_{-\infty}^{-M_{I}}e^ {-G(HM)V/kT}dM}\]  
According to the normalized equation \(x_{+}+x_{-}\ =\ 1\), the matrix component can be quantitatively described by the differential Equation (13). \[\dot{x}_{+}\ =\ -p_{+-}x_{+}+p_{-+}(1-x_{+}). \tag{13}\]The likelihood that the magnetic moment changes from the positive direction to the negative direction is expressed by Equation (14). \[ \begin{array}{l}p_{+-}\ =\ \frac{1}{\Gamma(T)}\frac{\int_{M_{t}-e}^{M_{t}}e^{-C( HAM)V/kTdM}}{\int_{M_{t}-e}^{M_{t}}e^{-C(HAM)V/kTdM}}\\ p_{-+}\ =\ \frac{1}{\Gamma(T)}\frac{\int_{-M_{t}}^{M_{t}+e}e^{-C(HAM)V/kTdM}}{\int_{- \infty}^{M_{t}+e}e^{-C(HAM)V/kTdM}}\end{array}  \tag{14}\]  
where \(\in\) is constant, \(\Gamma\) is the relaxation time of the material, and \(\omega\ =\ \frac{1}{\Gamma}\), which indicates the frequency of magnetic moment conversion. In order to qualitatively analyze the model and quantitatively analyze the thermal motion region, it is necessary to simplify the complex behavior of the average magnetization \(\overline{M}\) in the region where the thermal motion is negligible, according to the equilibrium conditions by Equation (15). \[\frac{\partial G}{\partial M}\ =\ 0. \tag{15}\]  
Combined with (8), Equation (16) is obtained. \[\frac{\partial H}{\partial M}\ =\ \frac{\partial^{2}\psi}{\partial M^{2}}. \tag{16}\]  
It can be seen from the above equation that the slope of the hysteresis kernel in this linear domain is \(1/\eta\), and the inflection point \(M_{I}\) is the key point. At this time, the resilience is the largest, and the original estimated parameter value of the free magnetic hysteresis model of the giant magnetostrictive actuator can be obtained according to the above equilibrium conditions. In the case where magnetization occurs for a short time and the thermal motion is negligible, the jump will be completed in an instant, so the magnetization can be simplified by an asymptotic relationship (as shown in Figure 5). \(M_{\min}\) can be obtained by solving Equation (15), and \(\overline{M}\ =\ M_{\min}\), then, \(\overline{M}\) is calculated. Combining with the Helmholtz free energy expression (7), the magnetization can be obtained. \[[\overline{M}(H;H_{c},\xi)](t)\ =\ \left\{ \begin{array}{l}[\overline{M}(H;H_{c},\xi)](0 ),\tau(t)\ =\ \phi\\ \frac{H}{\eta}-M_{R},\tau(t)\
eq\phi\,and\,H[\max\tau(t)]\ =\ -H_{c}\\ \frac{H}{\eta}+M_{R},\tau(t)\
eq\phi\,and\,H[\max\tau(t)]\ =\ H_{c}\end{array} \right., \tag{17}\]  
Figure 5: Relationship between magnetization \(\overline{M}\) and magnetic field strength \(H\). When the transition occurs, the set of times \(\tau(t)\) can be expressed by Equation (18). \[\tau(t)\ =\ \Big{\{}t\in(0,T_{f}]\Big{|}H(t)\ =\ -H_{c}\,\text{or}\,H(t)\ =\ H_{c}\Big{\}}. \tag{18}\]  
The initial value of the magnetic moment \([\overline{M}(H;H_{c},\xi)](0)\) can be expressed by Equation (19). \[[\overline{M}(H;H_{c},\xi)](0)\ =\ \left\{ \begin{array}{l}\frac{H}{\eta}-M_{R},H(0 )\leq-H_{c}\\ \xi,-H_{c}<H(0)<H_{c}\\ \frac{H}{\eta}+M_{R},H(0)\geq H_{c}\end{array} \right.. \tag{19}\]  
Coercivity \(H_{c}\) can be expressed by Equation (20). \[H_{c}\ =\ \eta(M_{R}-M_{I}). \tag{20}\]  
(4) Hysteresis model of heterogeneous materials  
The magnetization model (Equation (17)) is established on the premise that the internal lattice and magnetic domain structure of the material are completely uniform. In fact, the non-uniformity of the lattice structure of the material is led by the defects of the material itself and the non-uniform free energy profile of its different regions, according to the hypothesis that the effective field \(H_{\epsilon}\) is equal to the application field \(H\) in the homogeneous material, the magnetic interaction and Weiss mean field effect are ignored.