# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
## 3 Results, Analysis, and Discussion
### Analysis Area and Comparison
The error range is 0.0316 to 0.1353 (3.16% to 13.53%); however, not all errors are acceptable; cases 7 and 10 are optimal. There is, indeed, a good constancy in cases 8, 9, and 12. Furthermore, a good linearity is also observed in cases 7, 13, and 16. As a result of the applied load, the magnetic data measurable by the TMR sensor change in response to the domain motion. The variations of the magnetic signals in the ferromagnetic plate under the effect of the applied load were used as a means for the MMM to be performed, i.e., the changes in the magnetic properties of the ferromagnetic specimen under the effect of the applied load. Thus, these affect the direction and structure of the domains, and produce a magnetic moment on the specimen. According to existing knowledge in the literature, in the ferromagnetic system, the domains can be aligned or randomly aligned according to the presence or absence or of an external magnetic field, respectively; magnetic alterations in ferromagnetic materials can be assigned to the change in the structure of the magnetic domains. The central intent of this investigation was to seek the relationship between the measured triaxial magnetic flux density and the simulated stress and shear stress by the Ansys software, allowing one to assess the state of the tested SA 106 plate. It was found that \(\sigma_{Z}\) is smaller than \(\sigma_{X}\) and \(\sigma_{Y}\), and maintains a nearly constant trend and value, as there is little or no residual magnetization on the plate along the Z directions. On the other hand, \(\sigma_{X}\), \(\sigma_{Y}\), and \(\tau_{XY}\) are affected, and show a visible change in attention in such a loading configuration. Figure 11: Error and features of the transformations coefficients. (**a**) Error, (**b**) linearity, (**c**) constancy. It was tough to explain the relationship between each stress and shear stress as a function of the magnetic flux density presented in Figures 8-10, so it was important to find an alternative that guaranteed clear relationships; hence, the use of a third-order polynomial equation type to obtain clear relationships that help understand the effect of the stress, and to achieve the goal set in the beginning. By the time the material is first subjected to a modest load, it adopts an elastic behavior by stretching its atomic boundaries, where stiffness is considered as the ability of the material to resist deformation when acted upon by external stress. Elastic behavior essentially implies that regardless of the movement of the material when stressed, it returns to its initial stance when the load is released. The ratio between the stress and the elastic deformation is known as Young's modulus. However, if this elastic deformation is exceeded and the yield strength is reached, the material undergoes permanent deformation. To safeguard security, it is very important that the working range of our materials used in various constructions and transportations in everyday life remain in the elastic regime, so the material does not undergo any permanent deformation. An important inference is the need to search for the relationship using the shear stress, and not just with each direction of stress to obtain the latter. As pointed out above, the last figure shows an obvious correlation and identification of the relationships obtained. These relationships are pivotal in identifying the state of the stressed plate for the purposes of establishing a preventive approach method for possible use in several applications in the upcoming timeframe.