## 2 Theoretical Model
### Shell Formulation
Vector \(\bar{\mathbf{u}}^{(\tau)}=\bar{\mathbf{u}}^{(\tau)}(\alpha_{1},\alpha_{2},t)\) collects the generalized acceleration components that correspond to the generalized displacements  
\[\bar{\mathbf{u}}^{(\tau)}=\left[ \begin{array}{cc}\bar{u}_{1}^{(\tau)}&\bar{u }_{2}^{(\tau)}&\bar{u}_{3}^{(\tau)}\end{array} \right]^{T} \tag{25}\]  
The operator \(\mathbf{L}_{f}^{(\tau\eta)}\) takes the following form  
\[\mathbf{L}_{f}^{(\tau\eta)}=\left[ \begin{array}{ccc}L_{f1}^{(\tau\eta)\alpha _{1}}&0&0\\ 0&L_{f2}^{(\tau\eta)\alpha_{2}}&0\\ 0&0&L_{f3}^{(\tau\eta)\alpha_{3}}\end{array} \right] \tag{26}\]  
with the quantities \(L_{f1}^{(\tau\eta)\alpha_{1}}\), \(L_{f2}^{(\tau\eta)\alpha_{2}}\), \(L_{f3}^{(\tau\eta)\alpha_{3}}\) defined as follows  
\[ \begin{split}& L_{f1}^{(\tau\eta)\alpha_{1}}=k_{1f}^{(-)}F_{ \eta}^{\alpha_{1}(-)}F_{\tau}^{\alpha_{1}(-)}H_{1}^{(-)}H_{2}^{(-)}+k_{1f}^{(+ )}F_{\eta}^{\alpha_{1}(+)}F_{\tau}^{\alpha_{1}(+)}H_{1}^{(+)}H_{2}^{(+)}\\ & L_{f2}^{(\tau\eta)\alpha_{2}}=k_{2f}^{(-)}F_{\eta}^{\alpha_{2}(-)}F_{\tau} ^{\alpha_{2}(-)}H_{1}^{(-)}H_{2}^{(-)}+k_{2f}^{(+)}F_{\eta}^{\alpha_{2}(+)}F_{ \tau}^{\alpha_{2}(+)}H_{1}^{(+)}H_{2}^{(+)}\\ & L_{f3}^{(\tau\eta)\alpha_{3}}=\left(k_{3f}^{(-)}-G_{f}^{(-)} \
abla_{(-)}^{2}\right)F_{\eta}^{\alpha_{3}(-)}F_{\tau}^{\alpha_{3}(-)}H_{1}^ {(-)}H_{2}^{(-)}+\left(k_{3f}^{(+)}-G_{f}^{(+)}\
abla_{(+)}^{2}\right)F_{\eta }^{\alpha_{3}(+)}F_{\tau}^{\alpha_{3}(+)}H_{1}^{(+)}H_{2}^{(+)}\end{split}  \tag{27}\]  
where \(\
abla_{(\pm)}^{2}\) represents the Laplacian operator in curvilinear orthogonal coordinates applied at the top or the bottom surfaces of the shell defined below  
\[ \begin{split}&\
abla_{(\pm)}^{2}=\left(\frac{1}{A_{1}^{2}\left(H_{ 1}^{(\pm)}\right)^{2}}\frac{\partial^{2}}{\partial\alpha_{1}^{2}}+\frac{1}{A_{ 2}^{2}\left([PERSON]_{2}^{(\pm)}\right)^{2}}\frac{\partial^{2}}{\partial\alpha_{2}^{2 }}+\left(\frac{1}{A_{1}^{2}A_{2}\left([PERSON]_{1}^{(\pm)}\right)^{2}}\frac{\partial A _{2}}{\partial\alpha_{1}}-\frac{h}{2A_{1}^{2}R_{2}^{2}\left([PERSON]_{1}^{(\pm)} \right)^{2}H_{2}^{(\pm)}}\frac{\partial R_{2}}{\partial\alpha_{1}}+\right.\\ &\left.-\frac{1}{A_{1}^{3}\left(H_{1}^{(\pm)}\right)^{2}}\frac{ \partial A_{1}}{\partial\alpha_{1}}+\frac{h}{2A_{1}^{2}R_{1}^{2}\left(H_{1}^{( \pm)}\right)^{3}}\frac{\partial R_{1}}{\partial\alpha_{1}}\right)\frac{ \partial}{\partial\alpha_{1}}+\left(\frac{1}{A_{1}A_{2}^{2}\left(H_{2}^{(\pm)} \right)^{2}}\frac{\partial A_{1}}{\partial\alpha_{2}}-\frac{h}{2A_{2}^{2}R_{1 }^{2}\left(H_{2}^{(\pm)}\right)^{2}H_{1}^{(\pm)}}\frac{\partial R_{1}}{ \partial\alpha_{2}}+\right.\\ &\left.-\frac{1}{A_{2}^{2}\left(H_{2}^{(\pm)}\right)^{2}}\frac{ \partial A_{2}}{\partial\alpha_{2}}+\frac{h}{2A_{2}^{2}R_{2}^{2}\left(H_{2}^{( \pm)}\right)^{3}}\frac{\partial R_{2}}{\partial\alpha_{2}}\right)\frac{ \partial}{\partial\alpha_{2}}\right)\end{split}  \tag{28}\]  
where  
\[H_{1}^{(\pm)}=1\pm\frac{h}{2R_{1}},\qquad H_{2}^{(\pm)}=1+\frac{h}{2R_{2}} \tag{29}\]The quantity \(G_{f}^{(\pm)}\) is the shear modulus of the elastic foundation, according to the Pasternak model. On the other hand, the operator \(\mathbf{M}_{f}^{(\tau\eta)}\) is defined as  
\[\mathbf{M}_{f}^{(\tau\eta)}=\left[ \begin{array}{ccc}I_{f1}^{(\tau\eta)a_{1}}& 0&0\\ 0&I_{f2}^{(\tau\eta)a_{2}}&0\\ 0&0&I_{f3}^{(\tau\eta)a_{3}}\end{array} \right] \tag{30}\]  
with the foundation inertial components \(I_{f1}^{(\tau\eta)a_{1}}\), \(I_{f2}^{(\tau\eta)a_{2}}\), \(I_{f3}^{(\tau\eta)a_{3}}\) defined as follows  
\[