## 3 Solution Procedure
### Piezoelectric Layer
Inserting \(\overline{\psi}\Big{(}\left(h_{p}+h_{f}\right)/h\Big{)}=0\) Equation (40), into Equation (38) yields:  
\[\overline{\delta}^{m}(h)=T_{a}\overline{\delta}^{m}\Big{(}\left(h_{p}+h_{f} \right)/h\Big{)}+C_{a}+B_{a}N^{a} \tag{41}\]  
where  
\[T_{a}=\left[\left[m_{\overline{g}j}^{a}\right]-\left\{m_{\overline{g}j}^{a} \right\}\frac{1}{m_{\overline{g}k}^{m}}\left[m_{\overline{g}j}^{a}\right] \right],\ C_{a}=\left\{m_{\overline{g}j}^{a}\right\}\frac{\overline{\psi}(h)}{m _{\overline{g}j}^{a}},\]  
\[B_{a}=\left[m_{ik}^{\prime a}\right]-\left\{m_{\overline{g}j}^{a}\right\}\frac {1}{m_{\overline{g}k}^{m}}\left[m_{\overline{g}k}^{a}\right],\ i=1,2,\ldots,6, \ j=1,2,\ldots,6,\ k=1,2,\ldots,8.\]  
Using Equation (39), the electric potential \(\psi\) at the lower surface of the sensor can be derived as:  
\[\overline{\delta}_{a}(\overline{z})=e^{\overline{C}_{a}z}\bigg{[}e^{-\overline {C}_{a}(\frac{h_{p}+h_{f}}{h})}\overline{\delta}_{a}\left(\frac{h_{p}+h_{f}}{h} \right)+\int_{\frac{\overline{z}}{h}+h_{f}}^{\overline{z}}e^{-\overline{C}_{a} \eta}\overline{B}_{p}(\eta)\overline{T}_{a}(\eta)d\eta\bigg{]}, \tag{37}\]0  
Substitution of \(D_{z}=0\) and inserting Equation (42) into Equation (39) results in  
\[\overline{\delta}_{a}(\overline{z})=e^{\overline{C}_{a}z}\bigg{[}e^{-\overline {C}_{a}(\frac{h_{p}+h_{f}}{h})}\overline{\delta}_{a}\left(\frac{h_{p}+h_{f}}{h} \right)+\int_{\frac{\overline{z}}{h}+h_{f}}^{\overline{z}}e^{-\overline{C}_{a} \eta}\overline{B}_{p}(\eta)\overline{T}_{a}(\eta)d\eta\bigg{]}, \tag{37}\]1  
where  
\[\overline{\delta}_{a}(\overline{z})=e^{\overline{C}_{a}z}\bigg{[}e^{-\overline {C}_{a}(\frac{h_{p}+h_{f}}{h})}\overline{\delta}_{a}\left(\frac{h_{p}+h_{f}}{h} \right)+\int_{\frac{\overline{z}}{h}+h_{f}}^{\overline{z}}e^{-\overline{C}_{a} \eta}\overline{B}_{p}(\eta)\overline{T}_{a}(\eta)d\eta\bigg{]}, \tag{37}\]2 \[\overline{\delta}_{a}(\overline{z})=e^{\overline{C}_{a}z}\bigg{[}e^{-\overline {C}_{a}(\frac{h_{p}+h_{f}}{h})}\overline{\delta}_{a}\left(\frac{h_{p}+h_{f}}{h} \right)+\int_{\frac{\overline{z}}{h}+h_{f}}^{\overline{z}}e^{-\overline{C}_{a} \eta}\overline{B}_{p}(\eta)\overline{T}_{a}(\eta)d\eta\bigg{]}, \tag{37}\]3Since the transverse normal and shear stresses and displacements at the interfaces of the actuator, FGM, and sensor are continuous, from Equations (25), (41) and (43), the mechanical state variables at the top surface relate to those at the bottom surface with the following matrix equation  
\[\overline{\delta}_{a}(\overline{z})=e^{\overline{C}_{a}z}\bigg{[}e^{-\overline {C}_{a}(\frac{h_{p}+h_{f}}{h})}\overline{\delta}_{a}\left(\frac{h_{p}+h_{f}}{h} \right)+\int_{\frac{\overline{z}}{h}+h_{f}}^{\overline{z}}e^{-\overline{C}_{a} \eta}\overline{B}_{p}(\eta)\overline{T}_{a}(\eta)d\eta\bigg{]}, \tag{37}\]4  
where \(S=T_{a}MT_{s}\). Applying surface traction boundary conditions at the bottom and top surfaces to Equation (44) leads to the following equations  
\[ \begin{bmatrix}s_{12}&s_{13}&s_{14}\\ s_{52}&s_{53}&s_{54}\\ s_{62}&s_{63}&s_{64}\end{bmatrix}\begin{Bmatrix}\overline{u}_{0}\\ \overline{v}_{0}\\ \overline{w}_{0}\end{Bmatrix} = \begin{Bmatrix}Y(1)\\ Y(5)\\ Y(6)\end{Bmatrix}  \tag{45}\]  
By solving Equation (45), the displacement components at the bottom surface of the sensor layer \( \begin{bmatrix}\overline{u}_{0}&\overline{v}_{0}&\overline{w}_{0}\end{bmatrix} \) are obtained; then, using Equations (23), (26), and (37), mechanical and electrical state variables can be computed. Finally, modified Dubner's and Abats's numerical techniques are employed to transform the displacement and stress fields from Laplace to the time domain [39].