*3.2. Discussion and Analysis*

In order to explain the experimental results, the magneto-mechanical effect and the finite depth slit leakage magnetic field theory are employed. It is generally known that after demagnetization of magnetic material, the orientation of magnetic domain is random, so when stress is 0 MPa, the amplitude of *H*p(*y*) signal is very low in experimental results. When the stress is lower than the yield strength of material, as stress increases, the orientation of the magnetic domain becomes orderly gradually. For that process, the change in orientation of the magnetic domain is described as three stages, including magnetic domain movement, magnetic domain merger and magnetic domain rotation; after that, the orientation is parallel to the stress direction. It indicates that as stress increases, the magnetic characteristic of material becomes stronger, so the amplitude of *H*p(*y*) signal becomes higher gradually as shown in Figure 4. When the stress reaches yield strength, as the stress increases further, lots of dislocations are generated and gathered together in material, and an internal stress field is formed simultaneously. Because there is a pinning effect of dislocation on the magnetic domain, which has been reported by many studies, the orientation change in the magnetic domain is restricted as stress increases. As we all know, the magnetization degree of material largely depends on the order degree of magnetic domain orientation, thus the magnetization degree is decreased as stress increases in the plastic deformation stage. Thist means that the amplitude of *H*p(*y*) signal decreases, as shown in Figure 4f. However, there were some studies showing that when the material was in the state of plastic deformation, the magnetization degree still increased gradually as stress increased, because that pinning effect could be broken. However, some other scholars had different opinions, and they thought that for the material with good plastic deformation capability, that pinning effect was hardly broken [23]. For the material in this study, its plastic deformation capability is very good, so a larger number of dislocations are generated in the plastic deformation stage, and the pinning effect of dislocation cannot be broken. To verify the theoretical analysis, the fracture morphology of experimental material was observed and shown in Figure 9.

**Figure 9.** The fracture morphology of the experimental sample.

From Figure 9, it can be seen that many dimples are generated, which can be seen in the material fracture. The fracture theory of metal states clearly that more dimples represent better deformation performance, thus the experimental result agrees well with the theoretical analysis.

To explain the change in *H*p(*y*) signal, corresponding to grooves, the leakage magnetic field model of finite depth slit, which was built by Förster [24], was employed, and it was shown in Figure 10.

**Figure 10.** The leakage magnetic field model of finite depth slit.

As the definition of the coordinate (*X*,*Y*), shown in Figure 10, the leakage magnetic field intensity can be written as:

$$H\_y = \frac{(l+W)\mu H\_0 W}{\pi (l+\mu W)} [\frac{\text{x}}{\text{x}^2 + \text{y}^2} - \frac{\text{x}}{\text{x}^2 + (y+D)^2}] \tag{2}$$

where *Hy* is the normal component of leakage magnetic field, *x* is the horizontal distance between testing point and the center of slit, *y* is the vertical distance between testing point and the center of slit, *W* and *D* are the width and depth of the slit, *H*<sup>0</sup> is internal magnetic field intensity of material, *l* is the length of sample, and μ is the magnetic permeability.

Generally speaking, the *H*0, which is determined by the earth's magnetic field and the magnetic field induced by stress, can be written as:

$$H\_0 = H\_E + \frac{H\_E M\_S (a\mu\_0 + 3\gamma \sigma)}{3\alpha \mu\_0 - M\_S (a\mu\_0 + 3\gamma \sigma)}\tag{3}$$

where *H*<sup>E</sup> is the earth's magnetic field intensity, *M*<sup>S</sup> is the saturation magnetization of sample, α is the parameter of molecular field determined by material, μ<sup>0</sup> is the vacuum permeability, and γ is determined by the magnetostrictive coefficient.

Taking Equation (3) into Equation (2), the *Hy* can be written as:

$$H\_y = [H\_E + \frac{H\_E M\_S (a\mu\_0 + 3\gamma \sigma)}{3a\mu\_0 - M\_S (a\mu\_0 + 3\gamma \sigma)}] \frac{(l + \mathcal{W})\mu \mathcal{W}}{\pi (l + \mu \mathcal{W})} [\frac{\mathbf{x}}{\mathbf{x}^2 + \mathbf{y}^2} - \frac{\mathbf{x}}{\mathbf{x}^2 + (\mathbf{y} + \mathcal{D})^2}] \tag{4}$$

Then, taking Equation (4) into Equation (1), the magnetic intensity gradient *K*, which is defined in Equation (1), can be expressed as:

$$K = \begin{array}{c} K = \frac{H\_{q1} - H\_{q2}}{x\_1 - x\_2} \\ \frac{H \lnot M \lnot (a \mu \emptyset + 3 \nu \alpha)}{3 \kappa \mu\_0 - M\_3 (a \mu\_0 + 3 \nu \alpha)} \cdot \frac{(l + \mathcal{W}) \mu \mathcal{W}}{\pi (l + \mu \mathcal{W})} \cdot \left[ \frac{x\_1 (y + D)}{(x\_1^2 + y^2) (x\_1^2 + (y + D)^2)} - \frac{x\_2 (y + D)}{(x\_2^2 + y^2) (x\_2^2 + (y + D)^2)} \right] \cdot \frac{1}{x\_1 - x\_2} \end{array} \tag{5}$$

To discuss the relationship of *K* and groove width, the partial derivation of *K* respect to *W* is calculated and reduced as:

$$\mathcal{K}' = \left[H\mathbb{E} + \frac{H\_{\mathbb{E}}M\_{\mathbb{S}}(a\mu\_{0} + \mathfrak{H}\gamma\sigma)}{3a\mu\_{0} - M\_{\mathbb{S}}(a\mu\_{0} + \mathfrak{H}\gamma\sigma)}\right] \cdot \frac{1}{\text{x}\_{1} - \text{x}\_{2}} \cdot \frac{\mu\mathcal{W}^{2} + 2\mathcal{W}\mathcal{L} + \mathcal{L}^{2}}{(\text{L} + \mu\mathcal{W})^{2}} \cdot \left[\frac{\text{x}\_{1}(y+D)}{(\text{x}\_{1}^{2} + y^{2})[\text{x}\_{1}^{2} + (y+D)^{2}]} - \frac{\text{x}\_{2}(y+D)}{(\text{x}\_{2}^{2} + y^{2})[\text{x}\_{2}^{2} + (y+D)^{2}]}\right] \tag{6}$$

where *K* is defined as partial derivation of *K* respect to *W*.

Comparing the value of *L* and μ, it can be known that the value of *L* >> μ can be accepted, so Equation (6) is simplified to:

$$K'=\left[H\_{\mathbb{E}}+\frac{H\_{\mathbb{E}}M\_{\mathbb{S}}(a\mu\_{0}+3\gamma\sigma)}{3a\mu\_{0}-M\_{\mathbb{S}}(a\mu\_{0}+3\gamma\sigma)}\right]\cdot\frac{1}{x\_{1}-x\_{2}}\cdot\frac{2W+L}{L}\cdot\left[\frac{x\_{1}(y+D)}{(x\_{1}^{2}+y^{2})[x\_{1}^{2}+(y+D)^{2}]}-\frac{x\_{2}(y+D)}{(x\_{2}^{2}+y^{2})[x\_{2}^{2}+(y+D)^{2}]}\right] \tag{7}$$

In this study, the groove depth is constant in the same one sample, meaning that the value of *D* is a constant, so the relation of *K* and *W* is linear in Equation (7). Based on the derivative theory, it can be known that *K* is a quadratic polynomial function of *D*, so it agrees well with the experimental result, as shown in Figure 6.

Similarly, the relation of *K* and *D* is discussed. The partial derivation of *K* respect to *D* is calculated, so the Equation (5) is reduced to:

$$K'' = \left[H\_E + \frac{H\_E M\_S (a\mu\_0 + 3\gamma\sigma)}{3a\mu\_0 - M\_S (a\mu\_0 + 3\gamma\sigma)}\right] \cdot \frac{1}{x\_1 - x\_2} \cdot \frac{\mu (L + \mathcal{W}) \mathcal{W}}{(L + \mu\mathcal{W})\pi} \cdot \text{s} \tag{8}$$
  $\left[\frac{1}{(x\_1^2 + y^2)[x\_1^2 + (y + D)^2]} \cdot \left[\frac{x\_1^3}{x\_1^2 + (y + D)^2} - x\_1\right] - \frac{1}{(x\_2^2 + y^2)[x\_2^2 + (y + D)^2]} \cdot \left[\frac{x\_2^3}{x\_2^2 + (y + D)^2} - x\_2\right]\right] \tag{8}$ 

where *K*" is defined as the partial derivation of *K* respect to *D*.

Simplifying Equation (8), it can be written as:

$$\begin{aligned} K'' &= \left[ H\_E + \frac{H\_E M\_S (a\mu\_0 + 3\gamma\sigma)}{3a\mu\mu - M\_S (a\mu\mu + 3\gamma\sigma)} \right] \cdot \frac{1}{\mathbf{x}\_1 - \mathbf{x}\_2} \cdot \frac{\mu (L + W) W}{(L + \mu W) \pi} \cdot \\\ \left[ \frac{\mathbf{x}\_1^3}{(\mathbf{x}\_1^2 + y^2)(\mathbf{x}\_1^2 + (y+D)^2)} - \frac{\mathbf{x}\_2^3}{(\mathbf{x}\_1^2 + y^2)[\mathbf{x}\_2^2 + (y+D)^2]^2} - \frac{\mathbf{x}\_1}{(\mathbf{x}\_1^2 + y^2)[\mathbf{x}\_1^2 + (y+D)^2]} + \frac{\mathbf{x}\_2}{(\mathbf{x}\_2^2 + y^2)[\mathbf{x}\_2^2 + (y+D)^2]} \right] \end{aligned} \tag{9}$$

Because the value of *x* is far bigger than the value of *D* in this study, the influence of change in *D* on *K*" can be ignored. It means that the relationship of *K* and *D* can be seen as linear, so it agrees well with results as shown in Figure 7. In a word, the experimental result agrees well with theoretical discussion.
