# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
## 3 Results, Analysis, and Discussion
### Analysis Area and Comparison
Note that the partial sums of the series are broadly used to approximate the function. The accuracy of the approximation increases with the number of terms included. However, underneath, the third-order polynomial type is used, which is accurate and sufficient to handle the presented data. By taking advantage of this type of polynomial, the stresses and shear stresses (\(\sigma_{a}\) and \(\tau_{\beta}\)) are approximated by the differential magnetic flux density, \(\Delta B_{\alpha}\), in the form of a third-order polynomial functional relation as [44]:  
\[\sigma_{a}.\tau_{\beta}=p_{0}+p_{1}\;\Delta B_{\alpha}+p_{2}\;\Delta B_{a}{}^{2 }+p_{3}\;\Delta B_{a}{}^{3} \tag{7}\]  
Figure 10: FEM analysis versus experimental results of the differential magnetic flux density (\(\Delta B_{Z}\)) on the area of interest: (**a**) FEM analysis of \(\sigma_{X}\) vs. \(\Delta B_{Z}\); (**b**) FEM analysis of \(\sigma_{Y}\) vs. \(\Delta B_{Z}\); (**c**) FEM analysis of \(\sigma_{Z}\) vs. \(\Delta B_{Z}\); (**d**) FEM analysis of \(\tau_{XY}\) vs. \(\Delta B_{Z}\). The approximated data by the Equation (7) study the relationship between the stress and shear stress versus the differential magnetic flux density. The error between the approximated data and the experimented data is expressed by the following formula:  
\[E_{\alpha}=\frac{\sum\left(\left(\sigma_{\alpha}\cdot\tau_{\beta}\right)_{approximated}- \left(\sigma_{\alpha}\cdot\tau_{\beta}\right)_{experimented}\right)^{2}}{\sum \left(\sigma_{\alpha}\cdot\tau_{\beta}\right)_{experimented}} \tag{8}\]  
The purpose of calculating the error is to determine whether the marriage Equation (7) for estimating the data is accurate; in other words, whether the behavior of the data is accurately reflected by the equation applied. As a guideline, Figure 8a-f, Figure 9a-f, and Figure 10a-f correspond to Cases 1-6, 7-12, and 13-18, respectively, as portrayed in Table 2.  
Figures 8-10 reveal two distribution patterns: a gradient distribution between each data, and a gradient distribution of the whole. \(\sigma_{X}\) versus the differential magnetic flux density describes a strong relationship, as \(\sigma_{X}\) represents the principal stress with more significant magnitude than \(\sigma_{Y}\) and \(\sigma_{Z}\). In Figure 8a, a perfect third polynomial trend is observed, whereas \(\sigma_{X}\) exhibits an approximately linear change, with indirect and direct slopes in Figures 9a and 10a, respectively. Both \(\sigma_{Y}\) and \(\sigma_{Z}\) plot an almost horizontal trend in Figure 8b,c, although \(\sigma_{Y}\) is higher than \(\sigma_{Z}\), and the same trend is also seen in Figure 9b,c, without significant variation, and an approximately constant trend in Figure 10b,c. In all cases, \(\sigma_{X}\) yields a better response. \(\tau_{XY}\) shows a curve variation with a downward peak, whereas \(\tau_{YZ}\) depicts a perfect polynomial trend with a small amplitude variation. \(\tau_{ZX}\) exhibits an approximately horizontal trend, with small amplitude variation in Figures 8d and 9d, and keeps a nearly constant variation in Figure 10d. There is a good correlation between the quantities represented. This ascertainment leads to the assumption that these quantities are complementary to each other. It is obvious that these findings may differ; however, the fundamental tendency of the variations is basically the same. Incidentally, similar trends appear, in certain cases, to be well correlated; therefore, it is reasonable to conclude that the relationship can be well characterized for both the stresses and shear stresses.