## 2 Governing Equations
### FGM Layer  
The three-dimensional equilibrium equations are [36]:  
\[ \begin{array}{l}\sigma_{x,x}+\tau_{xy,y}+\tau_{xz,z}=0,\\ \tau_{xy,x}+\sigma_{y,y}+\tau_{yz,z}=0,\\ \tau_{xz,x}+\tau_{yz,y}+\sigma_{z,z}=0\end{array}  \tag{6}\]  
The strain-displacement relations are [36]:  
\[ \begin{array}{l}\varepsilon_{x}=\frac{\partial u}{\partial x},\;\varepsilon _{y}=\frac{\partial v}{\partial y},\;\varepsilon_{z}=\frac{\partial w}{ \partial z}\\ \gamma_{zx}=\frac{\partial u}{\partial x}+\frac{\partial w}{\partial x},\; \gamma_{zy}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial x},\; \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}  \tag{7}\]  
Since certain materials exhibit viscoelastic behavior at elevated temperatures, it is essential to account for viscoelasticity to predict the actual response of the materials in such environments. According to the Boltzmann superposition principle, constitutive relations of linear viscoelastic materials are introduced as follows [37]:  
\[\sigma_{i}=\int\limits_{-\infty}^{t}C_{ij}(t-\tau)\frac{d\left(\varepsilon_{j }(\tau)-\alpha_{j}T(\tau)\right)}{d\tau}d\tau \tag{8}\]  
in which \(C_{ij}(i,j=1,2,\ldots,6)\) are the relaxation moduli (see Equation (A1) in the Appendix A). Additionally, \(\sigma_{i}\)\((i=1,2,3)\) represent normal stresses \(\sigma_{x},\sigma_{y}\) and \(\sigma_{z}\) while \(\sigma_{i}\)\((i=4,5,6)\) are shear stresses \(\tau_{xy}\), \(\tau_{xz}\), and \(\tau_{yz}\). Likewise, \(\varepsilon_{i}\)\((i=1,2,3)\) represent normal strains, \(\varepsilon_{x},\varepsilon_{y}\), and \(\varepsilon_{z}\) while \(\varepsilon_{i}\)\((i=4,5,6)\) are the shear strains \(\gamma_{xy},\gamma_{xz}\), and \(\gamma_{yz}\). Before solving the equations, it is convenient to transform Equation (8) into the Laplace domain:  
\[\hat{\sigma_{i}}(s)=s\hat{\mathcal{C}_{ij}}(s)\big{(}\hat{\varepsilon}_{j}(s)- \alpha_{j}\hat{T}(s)\big{)} \tag{9}\]  
The Laplace transform of a variable is shown by the caret over it. Substitution of Equation (7) into Equation (9) results in the following stress-displacement relations for the FGM layer. \[ \begin{split}\partial_{fx}&=\frac{s\xi(z,s)}{(1+v)(1-2v)} \Big{[}(1-v)\dot{u}_{f,x}+v\partial_{f,y}+v\partial_{f,z}\Big{]}-\frac{a(z)s \xi(z,s)}{1-2v}\hat{T}_{f}\\ \partial_{fy}&=\frac{s\xi(z,s)}{(1+v)(1-2v)}\Big{[}v \dot{u}_{f,x}+(1-v)\partial_{f,y}+v\partial_{f,z}\Big{]}-\frac{a(z)s\xi(z,s)}{ 1-2v}\hat{T}_{f}\\ \partial_{fz}&=\frac{s\xi(z,s)}{(1+v)(1-2v)}\Big{[}v \dot{u}_{f,x}+v\partial_{f,y}+(1-v)\partial_{f,z}\Big{]}-\frac{a(z)s\xi(z,s)}{ 1-2v}\hat{T}_{f}\\ \hat{\tau}_{fxy}&=\frac{s\xi(z,s)}{2(1+v)}\Big{(} \dot{u}_{f,y}+\partial_{f,x}\Big{)},\ \hat{\tau}_{fxz}=\frac{s\xi(z,s)}{2(1+v)}\Big{(}\dot{u}_{f,z}+\dot{v}_{f,x} \Big{)}\,\hat{\tau}_{fyz}=\frac{s\xi(z,s)}{2(1+v)}\Big{(}\theta_{f,z}+\dot{v}_{f,y} \Big{)}\end{split}  \tag{10}\]  
The subscript \(f\) stands for the FGM layer. By assuming a Prony series for the time-dependence of relaxation modulus \(E\) and power law function for the thermal expansion \(\alpha\) and thermal conductivity coefficients \(k\), we can write for \(h_{p}\leq z\leq\Big{(}h_{p}+h_{f}\Big{)}\):  
\[ \begin{split}& E(z,t)=E_{1m}(\frac{z}{h_{p}})^{m_{1}}+E_{2m}( \frac{z}{h_{p}})^{m_{2}}e^{-(\frac{t}{t})},\alpha(z)=\alpha_{m}(\frac{z}{h_{p }})^{m_{3}},k(z)=k_{m}(\frac{z}{h_{p}})^{m_{4}}\\ & m_{1}=\frac{\ln\Big{(}\frac{E_{1m}}{h_{1m}}\Big{)}}{\ln\Big{(} \frac{h_{f}+h_{p}}{h_{p}}\Big{)}};m_{2}=\frac{\ln\Big{(}\frac{E_{2m}}{h_{2m}} \Big{)}}{\ln\Big{(}\frac{h_{f}+h_{p}}{h_{p}}\Big{)}};m_{3}=\frac{\ln\Big{(} \frac{E_{3m}}{h_{3}}\Big{)}}{\ln\Big{(}\frac{h_{f}+h_{p}}{h_{p}}\Big{)}};m_{4 }=\frac{\ln\big{(}\frac{\lambda_{c}}{\lambda_{c}}\big{)}}{\ln\Big{(}\frac{h_{f }+h_{p}}{h_{p}}\Big{)}}\end{split}  \tag{11}\]  
\(E_{1i}\) and \(E_{2i}(i=m,c)\) are the Prony series parameters for the relaxation modulus; the subscripts \(m\) and \(c\) denote the metal and ceramic constituents in the FGM layer.