## 2 Theoretical Basics
The constitutive equations for stresses (\(\sigma\)), electric displacements (\(D\)) and magnetic displacement (\(B\)) for the piezoelectric facesheets obtain the following form:  
\[\sigma_{r}\ =\ Q_{11}\varepsilon_{r}+Q_{12}\varepsilon_{\theta}-e_{31}E_{z}-d_{31 }H_{z}-\lambda_{11}\Delta T-\xi_{11}\Delta C, \tag{8}\]  
\[\sigma_{\theta}\ =\ Q_{12}\varepsilon_{r}+Q_{22}\varepsilon_{\theta}-e_{31}E_{ z}-d_{31}H_{z}-\lambda_{22}\Delta T-\xi_{22}\Delta C, \tag{9}\]  
\[\tau_{rz}\ =\ Q_{55}\varepsilon_{rz}+\epsilon_{15}E_{r}+d_{15}H_{r}, \tag{10}\]  
\[D_{r}\ =\ e_{15}\varepsilon_{rz}+\ \varepsilon_{11}\ E_{r}+g_{11}H_{r}+\gamma_{1 }\Delta T+\chi_{1}\Delta C, \tag{11}\]  
\[D_{z}\ =\ e_{31}(\varepsilon_{rz}+\varepsilon_{\theta})+\ \varepsilon_{33}\ E_{z}+g_{33}H_{z}+\gamma_{3}\Delta T+\chi_{3}\Delta C, \tag{12}\]  
\[B_{r}\ =\ d_{15}\varepsilon_{rz}+g_{11}E_{r}+\mu_{11}H_{r}+\tau_{1}\Delta T+ \
u_{1}\Delta C, \tag{13}\]  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]0  
where \(Q_{11}\) and \(Q_{12}\) describe the elastic stiffness; \(Q_{55}\) is the shear moduli of the piezoelectric layers; \(\varepsilon_{11}\) and \(\varepsilon_{33}\) are the dielectric constants; \(e_{15}\) and \(e_{31}\) are the piezoelectric coefficient; \(d_{15}\) and \(d_{31}\) describe the piezomagnetic coefficients; \(g_{11}\) and \(g_{33}\) are the electromagnetic coefficients; \(\mu_{11}\) and \(\mu_{33}\) are the magnetic permeability coefficients; \(\tau_{1}\) and \(\tau_{3}\) are the pyromagnetic coefficients; \(\
u_{1}\) and \(\
u_{3}\) are the hygromagnetic coefficients; \(\xi_{11}\) and \(\lambda_{11}\) represent the hygroscopic stress and thermal stress coefficients; and \(p_{q}\) and \(\chi_{q}\ (q\ =\ 1,2)\) are the pyroelectric and hygroelectric coefficients, respectively. It should be noted that the thermal stress and hygroscopic stress coefficients are related to the elastic coefficient, thermal expansion coefficient \(\alpha\) and moisture expansion coefficient \(\beta\) as follows:  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]1  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]2  
Meanwhile, the electric and magnetic fields \(E_{j},\ H_{j}(j\ =\ r,z)\) in terms of electric potential (\(\Phi\)) and magnetic potential (\(\Psi\)) can be written as follows [6]:  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]3  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]4  
The electric and magnetic potential distributions \(\Phi(r,\theta,t),\Psi(r,\theta,t)\) in the thickness direction of the piezoelectric layer, at time, \(t\), are defined as a combination of a half-cosine and linear variation, which satisfies the Maxwell equation [31]:  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]5  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]6  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]7\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]8  
where \(\Phi^{b}\) and \(\Phi^{b}\) represent the electric potential at the top and bottom layers of the sandwich structure, respectively, whereas \(\Psi^{t}\) and \(\Psi^{b}\) refer to the magnetic potential at the top and bottom layers, respectively. The same considerations can be repeated for the magnetic potential. Additionally, \(V_{0}\) and \(\lambda_{0}\) represent the external electric and magnetic potentials, respectively. The stress components in the core layer are assumed to be graded as follows:  
\[V_{c}\ =\ \left(\frac{z}{h_{c}}+\frac{1}{2}\right)^{g},\ \ g\geq 0 \tag{1}\]9  
\[P(z)\ =\ P_{m}+P_{cm}\bigg{(}\frac{z}{h_{c}}+\frac{1}{2}\bigg{)}^{g},\ \ P_{cm}\ =\ P_{c}-P_{m} \tag{2}\]0  
\[P(z)\ =\ P_{m}+P_{cm}\bigg{(}\frac{z}{h_{c}}+\frac{1}{2}\bigg{)}^{g},\ \ P_{cm}\ =\ P_{c}-P_{m} \tag{2}\]1