###### Abstract  
In this study, theoretical models were proposed to explain the changes in self-magnetic flux density (SMFD) due to fatigue cracks in the presence and absence of external magnetic fields. Three theoretical models were proposed: rotation domain model (RDM), concentration domain model (CDM), and vertical domain model (VDM), considering the deformation and non-deformation possibilities. To prove the theoretical model, fatigue cracks with different depth angles were fabricated through fatigue testing and EDM processing on the CT specimens. In addition, tunnel magnetoresistance (TMR) sensors were used to evaluate the 3-axis distribution of SMFD. Comparing the simulation and experimental results, similar tendencies of the occurrence and depth angle of fatigue cracks and their effect on the distribution of SMFD were observed. According to the RDM, the distribution of SMFD occurs in the direction of the crack length (y-direction), while the CDM explains that the SMFD does not occur if the fatigue crack is in a direction perpendicular to the surface. In addition, the VDM shows that SMFDs occur in a direction perpendicular to the crack length (x-direction) and the specimen surface (z-direction). Interestingly, these trends agree with the experimental results, which confirms the validity of the theoretical model and thus can be used to estimate the depth direction of a fatigue crack. magnetic domain model; self magnetization; magnetic flux density; stress; strain; fatigue crack; depth direction; nondestructive testing +
Footnote †: journal: Applied  
consequently, the allowable tolerance sizes are different [6; 7; 8]. Therefore, it is essential to quantitatively evaluate the size and shape of the defects detected by NDT. One of the challenges of NDT is detecting fatigue cracks that consist of a very thin crack layer. The initiation of fatigue crack on the metal and alloy is a convergence phenomenon that is influenced by many parameter interactions, physically and chemically. Widely known as damage accumulation, crack nucleation and growth occur when the local stress in the atomic displacement exceeds the yield strength of the material [9; 10]. The accumulation of this displacement leads to continuous faults, such as deformation, dislocation motions, and evolution, in each atomic layer and initiates micro voids which transform into micro defects with high tendencies to grow with the applied loads. The fatigue crack can be initiated on the material surface or at an internal layer [11]. If fatigue cracks develop in a direction close to the normal to the surface, the structure will fracture rapidly. Furthermore, if fatigue cracks develop in a direction inclined parallel to the surface, the probability of delaminating or shelling increases. Therefore, it is important to evaluate the angle between the fatigue crack and the surface of the structure because this phenomenon can significantly reduce the life span of components and increase the probability of unexpected in-service failures [12]. Many approaches to NDT have been used to detect fatigue defects, one of which is surface defect detection. In the case of a surface defect in NDT, large mechanical structures are detected using several methods such as visual, penetration, and magnetic particle testing, which can effectively determine the presence and length of a defect. However, it is difficult to implement these methods to evaluate the depth of the material defect [13]. A study on magnetic metal memory (MMM) has been reported in leakage magnetic flux testing which is able to detect an internal defect by measuring the self-magnetic flux leakage (SMFL) [14; 15; 16; 17]. Many related studies have reported the development of this technology for solid ferromagnetic materials and assemblies of ferromagnetic materials, through experimental data models [18; 19; 20], the probability evaluation model [21], and the magnetic theoretical model.