## 2 Theory  
### Magnetostrictive Equation  
In ferromagnetic materials, adjacent atoms construct magnetic domains spontaneously because of the forces caused by electron spins [18]. As shown in Figure 2, without external magnetic fields, the magnetic domains distribute randomly, which results in the material having a total magnetic intensity of zero. When the external magnetic field is nonzero, the magnetic domains will move and spin to be parallel to it, changing the length of the object along the same direction. The magnetic domain theory gives a qualitative explanation of the magnetostrictive effect. The constitutive equations of magnetic and elastic properties in ferromagnetic materials are provided as follows [35]:  
\[S=\mathbf{s}\sigma+\mathbf{d}\mathbf{H} \tag{1}\]  
\[\sigma=\mathbf{c}\mathbf{S}-\mathbf{e}\mathbf{H} \tag{2}\]  
\[B=\mathbf{d}^{T}\sigma+\mathbf{\mu}\mathbf{H} \tag{3}\]  
where the superscript \(T\) denotes the transposition; \(\mathbf{S}\) and \(\sigma\) are the strain and stress, respectively; \(\mathbf{H}\) is the magnetic field intensity; \(\mathbf{B}\) is the magnetic flux density; \(\mathbf{s}\) and \(\mathbf{c}\) are the elastic compliance matrix and stiffness matrix, respectively; \(\mathbf{d}\) is the piezomagnetic matrix; \(\mathbf{e}\) is the inverse matrix; and \(\mathbf{\mu}\) is the permeability. Equations (1) and (2) account for the magnetostrictive effect while Equation (3) contributes to the inverse magnetostrictive effect. Figure 3 shows the diagram of a MPT with planar coils. The static bias magnetic field in the magnetostrictive patch was provided by pre-magnetization, making the structure of the MPT thin and flexible without permanent magnets. The dynamic magnetic field was supplied by the coils sealed in the FPC, which sensed induced dynamic magnetic fields during the wave detection. In this paper, the model of the MPT was considered for the generation and detection of wave modes with shear vibrations in plates (shear horizontal waves) and pipes (torsional waves), which had the advantage of non-dispersion at low frequencies for inspection. The models for the longitudinal wave MPTs can be found in [36]. In shear mode generation, the dynamic magnetic  
Figure 1: Coils commonly used in the magnetostrictive transducers.  
field is perpendicular to the static magnetic field, of which the intensity is much smaller than that of the static magnetic field. The total strain is caused by the field superposition:  
\[\mathbf{S_{t}}=\mathbf{s}\mathbf{\sigma_{t}}+\mathbf{d}(\mathbf{H_{d}}+\mathbf{H_{s}}) \tag{4}\]  
where  
\[\mathbf{d}=\left[\begin{array}{cccccc}0&0&0&0&0&\frac{3\varepsilon_{M}}{\mathbf{H_{s }}}\\ -\frac{\varphi}{2}&\varphi&-\frac{\varphi}{2}&0&0&0\\ 0&0&0&\frac{3\varepsilon_{M}}{\mathbf{H_{s}}}&0&0\end{array}\right]^{T} \tag{5}\]  
\(\mathbf{\sigma_{t}}\) is the applied stress; \(\mathbf{H_{d}}\) and \(\mathbf{H_{s}}\) denote the intensities of the dynamic and static magnetic fields, respectively; \(\varphi\) is the gradient of the magnetostrictive curve of the material at \(\mathbf{H_{s}}\); and \(\varepsilon_{M}\) is the total magnetostrictive strain, which mostly depends on the static magnetic field since the dynamic field is much smaller. Considering that the wave amplitude is correlated with the dynamic shear strain \(S_{d^{\prime}}^{s}\). Equation (4) can be simplified as:  
\[S_{d}^{s}=\frac{3\varepsilon_{M}}{|\mathbf{H_{s}}|}|\mathbf{H_{d}}| \tag{6}\]  
It can be concluded from Equation (6) that the dynamic shear strain is proportional to the amplitude of the dynamic magnetic field. Thus, it may be an effective way to raise the wave energy by enhancing the dynamic magnetic field.