## 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  
(1) Magnetization process of the giant magnetostrictive material  
In order to establish the energy model of the material, it is necessary to analyze the lattice structure of the material by means of vector representation. In the paper, the symbol '()' represents a plane, the symbol '()' represents a vector, and the symbol '()' represents the general name of the series represented by the mark. Therefore, the various planes of the cube are represented as: \((100)\), \((010)\), \((001)\), \((001)\), \((001)\), \((001)\), and \((001)\), and the direction vector is expressed as: \([100]\), \([010]\), \([001]\), \([\overline{100}]\), and \([00\overline{1}]\), among them,\(\overline{1}\) indicates a negative direction. In the current processing conditions, the lattice of Terfenol-D in the \([1\overline{2}]\) direction grows in the form of a tree (as shown in Figure 3). In room temperature, the easy magnetization axis of Terfenol-D is generally in the \(\langle 111\rangle\) direction series, and the maximum strain occurs when the magnetization \(M\) is rotated from direction \([111]\) to \([11\overline{1}]\) or from direction \([1\overline{1}1]\) to \([1\overline{1}1]\). To take a sample that is not magnetized (as shown in Figure 4). Figure 4a shows that all regions in the specimen been not magnetized have nonzero spontaneous magnetization \(M_{0}\). Figure 4b shows that the change in magnetization is mainly caused by the reversible domain wall motion when the input field is low. Figure 4c shows that two irreversible mechanisms are appearanceal when the input field gradually increases, which a magnetic domain is increased caused by domain wall movement (consistent with the direction of the magnetic field), and the magnetic moment is rotated to the direction of easy magnetization \([11\overline{1}]\). In this phase, inputting a small change in the driving magnetic field produces a large change in magnetization and output mechanical strain. Figure 4d shows that the magnetization saturation state of the material is reached. At this time, the sample can  
Figure 3: Lattice direction representation of Terfenol-D.  
Figure 2: Free energy hysteresis model construction process. be approximated as a single magnetic domain, and the magnetic moment is completely rotated from the easy magnetization direction to be consistent with the magnetic field direction. In this study, it is assumed that the pre-stress of the material perpendicular to the direction of the magnetic moment is large enough to dominate the anisotropy of the lattice, then, the magnetization direction of the magnetic domain is easily changed from the original direction \(\langle 111\rangle\) to the [111] and \(\langle\overline{111}\rangle\) directions perpendicular to the axis [112]. This can be considered that the output strain is mainly caused by the rotation of the magnetic moment:  
\[\lambda\ =\ \frac{3}{2}\lambda_{s}(\frac{M}{M_{s}})^{2}, \tag{2}\]  
where \(\lambda_{s}\) is the saturation magnetostriction rate. \(M_{s}\) is the saturation magnetization. \(M\) is magnetization. \(\lambda\) is the magnetostriction rate. In order to simulate the hysteresis behavior of a giant magnetostrictive actuator, the nonlinear relationship between the input magnetic field \(H\) and the magnetization \(M\) needs to be described quantitatively. Then, Equation (2) is introduced to derive the relationship between the input magnetic field \(H\) and the output strain \(\varepsilon\). (2) Description function of Helmholtz and Gibbs energy  
To describe the relationship between magnetostatic and magnetoelastic properties of a giant magnetostrictive actuator, it is assumed that free energy is a function of temperature \(T\) and other ordered parameters, where \(e\) represents magnetization \(M\) in magnetostatics and strain \(\varepsilon\) in magnetoelasticity. The external field is represented by \(\vec{\phi}\) and is conjugated with \(e\) in thermodynamics.