## 2 Magnetstrictive Mechanism of the Giant Magnetostrictive Material and Its Modeling Method
### Factors Affecting Magnetic Coupling Characteristics of Giant Magnetostrictive Material  
The giant magnetostrictive material is magnetic anisotropy, and its saturation magnetostriction coefficient is changed along with the changing of the direction of the magnetic field. The magnetic domain distribution in the process of technical magnetization has dual effects of wall shift and domain rotation [21; 22], which makes it extremely difficult to study the intrinsic properties of giant magnetostrictive material. According to references [23; 24], a relationship between the magnetic field and magnetostriction coefficient of different crystal orientations for Tb\({}_{0.3}\)Dy\({}_{0.7}\)Fe\({}_{1.95}\) has been obtained at 20 \({}^{\circ}\)C and 11 Mp.  
When magnetostriction is saturated, there is a certain hysteresis between the magnetostriction and the magnetic field strength, and its relationship is nonlinear. Because of the magnetostrictive effect and magnetostrictive inverse effect inside the giant magnetostrictive material, the two subsystems independent of each other between the magnetic system and the mechanical system are coupled [25; 26]. At this time, if the giant magnetostrictive material is operated at a constant temperature, and only the strain in the longitudinal direction is considered, the magnetic field variables (magnetic field strength \(H\), magnetic induction \(B\)) and mechanical field variables (stress \(\sigma\), strain \(\varepsilon\)) are correlated with each other. Assuming that the magnetic field strength \(H\) and stress \(\sigma\) are independent variables, the coupling relationship between the magnetic field and the mechanical field is expressed as a linear piezomagnetic equation [12].  
\[\left\{\begin{array}{l}\varepsilon\ =\ E^{H}\sigma+d_{33}H\\ B\ =\ d_{33}\sigma+\mu H\end{array}\right. \tag{1}\]  
where \(\varepsilon\) is the total strain in the length direction of the GMM material. \(E^{H}\) is [PERSON]'s modulus when the magnetic field strength \(H\) is a constant. \(d_{33}\) is pressure magnetic coefficient. \(\sigma\) indicates the magnitude of stress. \(\mu\) indicates the magnetic permeability when the stress is a constant.  
According to Equation (1), the magnetostrictive effect is to make the magnetic system of the giant magnetostrictive material transfer energy to the elastic system. and the magnetostrictive inverse effect is to make the elastic system of the giant magnetostrictive material transfer energy to the magnetic system., simultaneously, energy exchange is achieved between the magnetic system and the elastic system of the material [27; 28].  
In the actual alternating magnetic field, the magnetic induction intensity \(B\) of the giant magnetostrictive material is closer to one phase than the applied alternating magnetic field \(H\) due to factors such as domain wall resonance, the hysteresis effect, and natural resonance. Thus, as the giant magnetostrictive material is continuously magnetized in the alternating magnetic field, the applied energy is continuously consumed, which causes hysteresis loss.  
In addition, the giant magnetostrictive material also has ohmic loss, compressive stress characteristics [29], temperature characteristics [30], \(\Delta E\) effects [31], electromechanical equivalent characteristics [32], and so on, these characteristics affect the hysteresis characteristics of giant magnetostrictive material and construction of the microatuor magnetic coupling model.