## 2 Magnetstrictive Mechanism of the Giant Magnetostrictive Material and Its Modeling Method  
### Ferromagnetic Properties of Material and Their Magnetstrictive Mechanisms  
(1) Ferromagnetic principle of material In an atom of matter, electrons orbit around a nucleus to produce orbital magnetic moments, at the same time, electrons also produce a spin magnetic moment by rotating around their own fixed axis [17], which produces ferromagnetic properties of the material.  
Generally, the electrons of the ferromagnetic material atoms are not arranged in an ideal order. There are unfilled electron layers inside the atoms, the spin magnetic moment of the electrons is not neutralized, the atoms have a \"permanent magnetic moment\", and the spin magnetic moment in the ferromagnetic material can be fixed in a small local area, this process is called spontaneous magnetization. The spontaneous magnetization zone is called a magnetic domain, and the boundary layer of adjacent magnetic domains is called a magnetic domain wall. If an external magnetic field is applied to the ferromagnetic material, the arrangement of some of the magnetic domains changes along the direction of the external magnetic field, which shows magnetic properties. In addition, the degree of magnetization of ferromagnetic materials is also related to temperature: when the temperature exceeds a certain critical value (Curie point), the ferromagnetism of the ferromagnetic material disappears, and the ferromagnetic material is changed to paramagnetic.  
(2) Magnetostrictive mechanism  
Specifically, there are three reasons to change the magnetization state of the ferromagnetic material by changing the shape and volume of ferromagnetic materials [18; 19]. The main effect of magnetostrictive strain on giant magnetostrictive material is field-induced deformation [20].  
1 Spontaneous deformation caused by spontaneous magnetization  
Spontaneous magnetization is caused by an exchange force independent of the external magnetic field. It is imaginary that there is a single domain crystal in the giant magnetostrictive material, which is spherical above the Curie temperature (380 \({}^{\circ}\)C). When it is cooled from the Curie temperature, the exchange force causes the crystal to spontaneously magnetize, and the shape of the crystal changes (shown in Figure 1). However, since the directions of spontaneous deformation of the magnetic domains inside the giant magnetostrictive material are random, there is no obvious elongation or shortening in the macroscopic direction.  
2 Field-induced deformation caused by technical magnetization  
When the strength of the external magnetic field gradually increases from zero, the magnetization of the giant magnetostrictive material gradually rises to a saturated state, then a magnetostriction phenomenon occurs, this process is called field-induced deformation. The field-induced deformation process can be divided into four stages: the reversible magnetostriction stage, the irreversible magnetostriction stage, the stage where the rotation of the magnetic moment of the magnetic domain plays a major role, and the magnetization asymptotic saturation phase.  
After saturation of magnetostriction is reached, and the magnetic field strength \(H\) of the external magnetic field is lowered, even if the applied magnetic field strength \(H\) is reduced to zero, the magnetic domain cannot be completely restored to the initial state, and the magnetostriction rate \(\lambda\) cannot be completely zeroed, this results in a certain residual magnetization inside the giant magnetostrictive material, which is called hysteresis.  
Figure 1: Spontaneous deformation diagram.