## 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
When \(M\) is represented by the ordered parameter \(e\), the external field is represented as a magnetic field \(H\). When \(\varepsilon\) is represented by the ordered parameter \(e\), the external field is represented as stress \(\sigma\). \(\psi\) (\(e,T\)) is used to express the general relationship of [PERSON]'s free energy. When no external field is added, the condition of thermodynamic equilibrium is that \(e\) is at a minimum, which is:  
\[\phi(e,T)\ =\ \frac{\partial\psi(e,T)}{\partial e}\ =\ 0. \tag{3}\]  
The energy response of the system can be expressed as \(\phi\). When the system is acted on by the external field \(\vec{\phi}\), total free energy can be expressed as Equation (4):  
\[\psi_{\vec{\phi}}(e,T)\ =\ \psi(e,T)-\vec{\phi}e. \tag{4}\]  
The equilibrium conditions applied by an external field are expressed by Equation (5):  
\[\phi(e,T)\ =\ \vec{\phi}. \tag{5}\]  
Figure 4: The magnetization process of plane \(\left(\overline{110}\right)\) of the Terfenol-D single crystal under the magnetic field in the [112] direction. (**a**) Non-zero spontaneous magnetization state; (**b**) Magnetization caused by the reversible domain wall motion when the input field is low; (**c**) Two irreversible magnetization mechanisms appearanced when the input field increases gradually; (**d**) The magnetization saturation state of the material. In the case where the pre-pressure is sufficiently large, two magnetization directions in [111] and [111] in the giant magnetostrictive material will be easily produced. It is necessary to consider both the internal energy and the magnetic moment energy of the material in these two magnetization directions. It is assumed that the interaction between magnetic moments is adiabatic, and the temperature is stable below the Curie temperature, the Helmholtz potential energy \(\psi\) can be expressed by Equation (6). \[\psi(M,T)\ =\ \frac{H_{h}M_{\rm s}}{2}[1-(M/M_{\rm s})^{2}]+\frac{H_{h}T}{2T_{c}}[ M\ln(\frac{M+M_{R}}{M_{\rm s}-M})+M_{\rm s}\ln(1-(M/M_{\rm s})^{2})], \tag{6}\]  
where \(H_{h}\) is the biasing magnetic field, \(M_{\rm s}\) is saturation magnetization, and \(T_{c}\) is the Curie temperature. According to statistical mechanics, Helmholtz free energy can be represented by piecewise quadratic relations under isothermal conditions. \[\psi(M)\ =\ \left\{ \begin{array}{l}\frac{1}{2}\eta(M+M_{R})^{2},M\leq-M_{I}\\ \frac{1}{2}\eta(M-M_{R})^{2},\ \ M\geq M_{I}\\ \frac{1}{2}\eta(M_{I}-M_{R})^{2}(\frac{M^{2}}{M_{I}}-M_{R}),|M|<M_{I}\end{array}  \right., \tag{7}\]  
where \(M_{I}\) is the magnetization generated at the inflection point, \(M_{R}\) is residual magnetization, \(\eta\) is the converted slope, and \(\eta\ =\ \frac{dH}{dM}\). By Equation (6), Gibbs energy can be written by Equation (8):  
\[G(H,M,T)\ =\ \psi(M,T)-HM. \tag{8}\]  
(3) Hysteresis model of homogeneous material  
For homogeneous materials, the effective field \(H_{e}\) is equal to the application field \(H\), and the average magnetization can be written by Equation (9). \[\overline{M}\ =\ x_{+}\langle M_{+}\rangle+x_{-}\langle M_{-}\rangle, \tag{9}\]  
where \(x_{+}/\ x_{-}\) is the probability of positive or negative magnetic moment, and \(M_{+}/\ M_{-}\) is the expected value of magnetization. \(\langle M_{+}\rangle\) and \(\langle M_{-}\rangle\) can be obtained by Equation (10):  
\[\langle M_{+}\rangle =\ \int_{M_{I}}^{\infty}M\mu(G)dM \tag{10}\] \[\langle M_{-}\rangle =\ \int_{-\infty}^{-M_{I}}M\mu(G)dM\]  
where:  
\[\mu(G)\ =\ Ce^{-GV/kT}. \tag{11}\]  
The probability of Gibbs energy obtained is expressed quantitatively in the above equation, where \(k\) is the Boltzmann constant, \(C\) is a constant, and this constant can be selected according to the magnetization integral value \(1\), \(V\) is lattice volume. Considering the Boltzmann energy balance, the approximate relaxation process characteristics of the material can be obtained in the lattice volume \(V\), and then the constant \(C\) can be estimated to obtain the average magnetization value.