## 2 Magnetic Domain Models for Stress and Fatigue Cracks
The equivalent magnetization \(\overset{\rightarrow}{\mathrm{M}}_{\mathrm{eqv}}\) can be expressed as the product of magnetic susceptibility \(\mathrm{x}_{\mathrm{m}}\), which is an intrinsic property of a material, and the external magnetic field \(\overset{\rightarrow}{\mathrm{H}}\), which is expressed in Equation (4). \[\overset{\rightarrow}{\mathrm{M}}_{\mathrm{eqv}}=\chi_{\mathrm{m}}\overset{ \rightarrow}{\mathrm{H}},\ \left(\mathrm{A}/\mathrm{m}\right) \tag{4}\]  
The magnetic susceptibility is not a constant entity, but corresponds to a parameter for an external magnetic field, which causes magnetic hysteresis. In other words, even after the external magnetic field is removed, the residual magnetization phenomenon of \(\overset{\rightarrow}{\mathrm{M}}_{\mathrm{eqv}}\
eq 0\) appears. In this state, for \(\overset{\rightarrow}{\mathrm{M}}_{\mathrm{eqv}}=0\), an external magnetic field in the reverse direction, i.e., a coercive force, must be applied. Figure 1: Magnetic domain models; (**a**) rotational model without and (**b**) with external magnetic field, (**c**) concentration model without and (**d**) with external magnetic field, (**e**) vertical model without and (**f**) with external magnetic field. The MFD \(\overset{\rightarrow}{\text{B}}\) outside the ferromagnetic metal can be expressed as the superposition of the external magnetic field \(\overset{\rightarrow}{\text{H}}\) and the equivalent magnetization, as in Equation (5). Here, \(\mu_{0}\), \(\mu_{r}\), and \(\mu\) are the permeability in free space, relative permeability, and permeability, respectively. According to Equation (5), when \(\overset{\rightarrow}{\text{H}}=0\), \(\overset{\rightarrow}{\text{M}}_{\text{eqv}}=0\). \[\overset{\rightarrow}{\text{B}}=\mu_{0}\overset{\rightarrow}{\text{H}}+\mu_ {0}\overset{\rightarrow}{\text{M}}_{\text{eqv}}=\mu_{0}(1+\chi_{\text{m}}) \overset{\rightarrow}{\text{H}}=\mu_{0}\mu_{r}\overset{\rightarrow}{\text{H}} =\mu\overset{\rightarrow}{\text{H}},\ (\text{Wb}/\text{m}^{2}) \tag{5}\]  
\[\mu_{0}=4\pi\times 10^{-7},\ (\text{H}/\text{m}) \tag{6}\]  
Although it is not a magnetic hysteresis phenomenon, as shown in Figure 2, when a mechanical load is applied and deformation occurs, a phenomenon in which \(\overset{\rightarrow}{\text{M}}_{\text{eqv}}\
eq 0\) may occur at \(\overset{\rightarrow}{\text{H}}=0\). Figure 2a-c show conceptual diagrams applying RDM, CDM, and VDM, respectively, with n = 8 to a cantilever beam. Figure 2d-f show the deformation that occurred when a load \(\overset{\rightarrow}{\text{F}}\) was applied to the tip of the cantilever beam. The area S\({}_{k}\) of each magnetic domain was deformed into S\({}_{k}\)\({}^{\prime}\) by the load \(\overset{\rightarrow}{\text{F}}\). The strain \(\varepsilon_{\alpha\beta}\) in each area can be expressed using Equation (7). In addition, the stress \(\sigma_{\alpha\beta}\) in each area can be expressed using Equation (8). In this study, both the tensile and shear stresses are expressed as \(\sigma_{\alpha\beta}\), (Equation (8)), where E is the elastic modulus and \(\alpha\) or \(\beta\) refers to the x or y direction  
\[\text{S}_{k}^{\prime}=\varepsilon_{\alpha\beta,k}\text{S}_{k},\ (\text{m}^{2}) \tag{7}\]  
\[\sigma_{\alpha\beta,k}=\text{E}\ \varepsilon_{\alpha\beta,k},\ (\text{N}/\text{m}^{2}) \tag{8}\]  
Figure 2: Distributions of magnetization; (**a**) on the rotational domain model (RDM) without and (**b**) with external force, (**c**) on the concentration domain model (CDM) without and (**d**) with external force, and (**e**) on the vertical domain model (VDM) without and (**f**) with external force. In the state of Figure 2d,e, when \(\overset{\rightarrow}{\mathrm{H}}=0\), the equivalent magnetization \(\overset{\rightarrow}{\mathrm{M}}^{\prime}_{\mathrm{eqv}}\) and the magnetic flux density \(\overset{\rightarrow}{\mathrm{B}}^{\prime}\) can be expressed by Equations (9) and (10), respectively. S is the sum of the total area. In addition, Equation (10) can be expressed as Equations (11) and (12) by substituting in Equations (7) and (8).