## 5 Conclusions  
In this study, theoretical models of magnetic metal memory (MMM) have been proposed. The idea was to utilize these models to estimate the depth direction of a fatigue crack. Three models were used: the rotation domain model (RDM), concentration domain model (CDM), and vertical domain model (VDM). In addition, case studies on the deformation and non-deformation conditions when cracks occur were also studied theoretically. In general, the results found that using the magnetic domain model, the self-magnetic flux density (SMFD) changes owing to the stress-strain state (SSS) and fatigue cracking, even when no external magnetic field is applied to the ferromagnetic specimen. This indicates that MMM can be used to detect internal defects by measuring the self-magnetic flux leakage (SMFL) on the specimen. This result was observed on both the theoretical and experimental approaches, which supports the model validity.  
In the theoretical model, if the magnetic domain is not deformed when the fatigue cracking occurs, RDM explains that the SMFL will appear on the longitudinal component of the crack (y-direction). Meanwhile, the SMFL of the longitudinal component (y-direction) of the crack occurs according to the deformed RDM, accompanied by fatigue cracking. On the other hand, CDM explains that if the magnetic domain is not deformed when fatigue cracks occur, the SMFL components perpendicular to the crack length (x-and z-directions) will exist, while the longitudinal component of the crack (y-direction) does not occur. Compared to the CDM, VDM pursues a similar tendency on each SMFL component. However, VDM found that the larger the depth direction of crack angle, the larger the SMFD intensity of each component.  
Figure 20: Relationship between the angle of fatigue crack and angle estimation functions.  
Furthermore, to reinforce the results from theoretical models, the magnetic flux density was measured using a 3-axis TMR sensor in a state where no external magnetic field was applied to the specimen with fatigue cracks. It was found that although no external magnetic field was applied, the SMFD of the x-, y-, and z-components changed. Furthermore, in the identical tendency with the VDM, as the inclination angle of the crack increased, the magnitude of the magnetic flux density of each component increased. Therefore, it is justifiable to explain the change in the self-magnetic flux density due to the occurrence of fatigue cracks using a magnetic domain model in which the undeformed RDM and VDM are superimposed. Therefore, by using the integral value of the z-component reflecting the peak position of the x-component, the depth angle of the cracks can be quantitatively evaluated.  
Finally, it can be concluded that both the theoretical models and experimental results have similar trends. This supports the model validity, and therefore, the model can be used to estimate the depth direction of a fatigue crack.  
Conceptualization, J.L.; methodology, J.L., D.W. and I.D.M.O.D.; software, J.L. and I.D.M.O.D.; validation, J.L. and I.D.M.O.D.; formal analysis, J.L.; investigation, J.L., D.W. and I.D.M.O.D.; resources, J.L. and D.W.; data curation, J.L., D.W. and I.D.M.O.D.; writing--original draft preparation, J.L.; writing--review and editing, J.L. and I.D.M.O.D.; visualization, J.L., D.W. and I.D.M.O.D.; supervision, J.L. and I.D.M.O.D.; visualization project administration, J.L.; funding acquisition, J.L. This paper was prepared with the contributions of all authors. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A2C2006064). The code and dataset will be made available on request. The authors declare no conflict of interest.