# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
## 3 Results, Analysis, and Discussion
### Analysis Area and Comparison  
The focus of this part was to compare the distribution of the stress simulated by the finite element analysis and the variations of the measured magnetic flux density, in order to determine the differential magnetic flux density and the stress distributions on the specimen when applying the load. In the comparison between the two datasets, the same target area was selected. The center of the selected area corresponds to the center of the plate for the simulation and experiment, and 10 mm before and after the center line were selected to conduct the study by matching the two areas, as shown by the red dashed line, and to avoid the pinning effect and edge effect, as shown in Figure 6.  
Figure 5: FEM analysis of the stresses (\(\sigma_{X}\), \(\sigma_{Y}\), and \(\sigma_{Z}\)) and shear stresses (\(\tau_{XY}\), \(\tau_{YZ}\), and \(\tau_{XZ}\) ) on the SA 106 plate. Figure 6: Analysis region for experiment and FEM analysis. Figure 7 illustrates the variation of stress and shear stress obtained by the software, as well as the magnetic flux density and differential magnetic flux density obtained by the experiment. It can be observed that for the FEM results, the principal stress distribution follows the x direction; in other words, the range of \(\sigma_{X}\) is the more significant stress than of \(\sigma_{Y}\) and \(\sigma_{Z}\), where \(\sigma_{Z}\) is the least significant stress. An almost identical feature is observed for the three stresses. The shear stresses, \(\tau_{XY}\), \(\tau_{YZ}\), and \(\tau_{XZ}\), show a variant distribution: upward and leftward variation, upward and rightward variation, and downward and leftward variation, respectively. They appear to be small, but larger than the least significant stress, especially \(\tau_{XY}\). \(B_{X}\) varies from left to right, and \(B_{Y}\) varies from bottom to top; \(B_{Z}\) is a mixture of the two trends of \(B_{X}\) and \(B_{Y}\) (upward and rightward variation). Furthermore, the obtained results show that the behavior of the differential magnetic flux density (\(\Delta B_{X}\), \(\Delta B_{Y}\), and \(\Delta B_{Z}\)) is not uniform, and coincides with the simulated stress and shear stress. Figure 7: _Cont._  
It turns out that the simulated data have, more or less, a resemblance to the measured data; although, there are some differences. However, a graded variation tendency can be seen, and a good correlation is also observed, mainly with the differential magnetic flux density. This implies that there is a good match between the results. This first step of comparison can be used as a criterion to judge the efficiency of the simulation. These results will be used to evaluate the stresses depending on the differential flux density. In several cases, the simulated and measured data have the same trend, but it is rather difficult to make an analysis by focusing only on one of them or their behavior. In the following, an attempt is made to find relationships between the two data (Figures 8-10). Since the \(\Delta B_{X}\) data are small, on the order of 0.04 and 0.15 for both \(\Delta B_{Y}\) and \(\Delta B_{Z}\), the Tylor series can be used to approximate these data in polynomial form. On the other hand, the Taylor series computation merely involves the function's acquaintance with the proximity of a point. Therefore, working with the Taylor series results in a considerably small error in the proximity of the point where it is calculated. The Taylor series of a function is an infinite sum of its derivatives at a single point, as given below [42, 43]:  
\[T\{(f(x))\}=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n} \tag{6}\]  
where \(T\{(f(x))\}\) denotes the Taylor series of a real valuated function, \(f(x)\); the right-hand side is called the \(n\)th Taylor polynomial, and \(n\) takes on non-negative integer values; \(n!\) is the factorial of \(n\); and \(f^{(n)}(a)\) is the \(n\)th derivative of the function, \(f(x)\), evaluated at the point \(x=a\).