# Distribution of Magnetic Flux Density under Stress and Its Application in Nondestructive Testing
## 2 Materials and Methods  
The equipment used in the experiment consists of a signal processing circuit, a carbon steel specimen, a power supply, a three-directional TMR sensor, and a loading system. The inspection setup block diagram is streamlined in Figure 1a. The I/O controller of the NI-USB device is employed to monitor the power supplied to the sensor, and the signal processing circuitry. The output signal from the TMR sensor is processed by high-pass filters, amplifiers, and root mean square (RMS) measurement circuits into a continuous signal that is digitized by the NI-USB A/D converter, and ultimately appears as software on a laptop. The XY stage is translated by a motor which is controlled by LabVIEW software via the USB port. The magnetic flux density (\(\overset{\rightarrow}{B}\)) in space resulting from an external magnetic field (\(\overset{\rightarrow}{H}\)) and the magnetization (\(\overset{\rightarrow}{M}\)) in the ferromagnetic material is outlined below:  
\[\overset{\rightarrow}{B}=\mu_{0}\overset{\rightarrow}{H}+\mu_{0}\overset{ \rightarrow}{M} \tag{1}\]  
where the magnetization vector can be defined as the total sum of the magnetic dipole moments expressed as [39]:  
\[\overset{\rightarrow}{M}=\underset{\Delta p\to 0}{lim}\frac{\sum_{k=1}^{n \Delta p}m_{k}}{\Delta p} \tag{2}\]  
Figure 1: Experiment setup. (**a**) The inspection setup block diagram; (**b**) experimental equipment. with \(m_{k}\) being the magnetic dipole moment. Even \(\overset{\rightarrow}{H}=0\), but \(\overset{\rightarrow}{M}\
eq 0\) with a stress strain condition, meaning that even without \(\overset{\rightarrow}{H}\), \(\overset{\rightarrow}{B}\) is not equal to zero [40]. To measure the \(\overset{\rightarrow}{B}\) in MMM, the flux gate is used in most cases, and the following expression is used:  
\[V_{emf}=-\frac{d\varnothing}{dt}=\int\limits_{S}\left(-\frac{\partial\overset{ \rightarrow}{B}}{dt}\right)\overset{\rightarrow}{ds} \tag{3}\]  
where \(\frac{\partial\overset{\rightarrow}{B}}{dt}\) is the frequency of excitation, and \(s\) is the area of the coil. In the rod type flux gate, which is used for the MMM usually, the lift-off is very small, the primary coil to detect \(\int_{s}\left(-\frac{\partial\overset{\rightarrow}{B}}{dt}\right)\overset{ \rightarrow}{ds}\) generates the magnetic field, as the lift-off is too close to the inspected material, so magnetic field occur; the effect is small, but it exists. In the case of the TMR sensor used in this paper, the output voltage is given by the Formula (4) [41]:  
\[V_{TMR}=2\frac{kV_{s}}{R}\Delta B \tag{4}\]  
where \(V_{s}\) is the applied voltage, \(R\) is the resistance of the TMR sensor, and \(k\) is a constant parameter of the sensor. The change in the TMR output is the consequence of stress. Changes in magnetization are caused by the applied mechanical stress, and can be inferred. Moreover, the TMR offers a very small measuring point. The experimental equipment is shown in Figure 1b, whereas the properties are given in Table 1, and the schematic of the four-point loading test, including the shape and the dimensions of the SA 106 plate, is illustrated in Figure 2.  
To figure out and estimate the 3D stress distribution on the target carbon steel plate, the latter was subjected to two load levels equivalent to the displacement (0 and 0.1 \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Young’s modulus & 210 GPa \\ Poisson ratio & 0.3 \\ Relative permeability & 260 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the SA 106 specimen. Figure 2: Schematic of the loading test used. (**a**) The dimensions of the plate; (**b**) top view of the specimen and the pins. mm), undertaken inside a laboratory at ambient temperature using a four-point stainless steel loading test so as not to influence the signal recovered by the TMR sensor. The plate was pushed in one side by two pins and two support pins: two pins in the same side and the other two in the other side, as shown in Figure 2b.