## 2 Theoretical Model
### Shell Formulation
\begin{split} I_{f1}^{(\tau\eta)a_{1}}&=\frac{1}{3}\rho_{f}^{(-) }h_{f}^{(-)}F_{\eta}^{a_{1}(-)}F_{\tau}^{a_{1}(-)}H_{1}^{(-)}H_{2}^{(-)}+\frac{ 1}{3}\rho_{f}^{(+)}h_{f}^{(+)}F_{\eta}^{a_{1}(+)}F_{\tau}^{a_{1}(+)}H_{1}^{(+ )}H_{2}^{(+)}\\ I_{f2}^{(\tau\eta)a_{2}}&=\frac{1}{3}\rho_{f}^{(-)}h_{f}^{(-)}F_{\eta}^{a _{2}(-)}F_{\tau}^{a_{2}(-)}H_{1}^{(-)}H_{2}^{(-)}+\frac{1}{3}\rho_{f}^{(+)}h_{f }^{a_{2}(+)}F_{\eta}^{a_{2}(+)}F_{\tau}^{a_{2}(+)}H_{1}^{(+)}H_{2}^{(+)}\\ I_{f3}^{(\tau\eta)a_{3}}&=\frac{1}{3}\rho_{f}^{(-)}h_{f}^{(-)}F_{\eta}^{a _{3}(-)}F_{\tau}^{a_{3}(-)}H_{1}^{(-)}H_{2}^{(-)}+\frac{1}{3}\rho_{f}^{(+)}h_{ f}^{(+)}F_{\eta}^{a_{3}(+)}F_{\tau}^{a_{3}(+)}H_{1}^{(+)}H_{2}^{(+)}\end{split}  \tag{31}\]  
where \(\rho_{f}^{(\pm)}\) is the density of the elastic foundation. Finally, the equations of motion and the corresponding boundary conditions are deduced [99]. For each order \(\tau\) of kinematic expansion, the three equations of motion can be written in matrix form as follows  
\[\mathbf{D}_{\Omega}^{*}\mathbf{S}^{(\tau)}+\mathbf{q}_{f}^{(\tau)}=\sum_{\eta =0}^{N+1}\mathbf{M}^{(\tau\eta)}\ddot{\mathbf{u}}^{(\eta)} \tag{32}\]  
in which \(\mathbf{D}_{\Omega}^{*}\) represents the equilibrium differential operator defined as  
\[\mathbf{D}_{\Omega}^{*}=\left[ \begin{array}{ccc}\frac{1}{A_{1}}\frac{ \partial}{\partial\mathbf{a}_{1}}+\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{ \partial\mathbf{a}_{1}}&-\frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial \mathbf{a}_{2}}&-\frac{1}{R_{1}}\\ \quad-\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\mathbf{a}_{1}}&\frac{ 1}{A_{2}}\frac{\partial}{\partial\mathbf{a}_{2}}+\frac{1}{A_{1}A_{2}}\frac{ \partial A_{1}}{\partial\mathbf{a}_{2}}&-\frac{1}{R_{2}}\\ \quad\frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\mathbf{a}_{2}}&\frac{ 1}{A_{1}}\frac{\partial}{\partial\mathbf{a}_{1}}+\frac{1}{A_{1}A_{2}}\frac{ \partial A_{2}}{\partial\mathbf{a}_{1}}&0\\ \quad\frac{1}{A_{2}}\frac{\partial}{\partial\mathbf{a}_{2}}+\frac{1}{A_{1}A_{2}} \frac{\partial A_{1}}{\partial\mathbf{a}_{2}}&\frac{1}{A_{1}A_{2}}\frac{ \partial A_{2}}{\partial\mathbf{a}_{1}}&0\\ \quad\frac{1}{R_{1}}&0&\frac{1}{A_{1}}\frac{\partial}{\partial\mathbf{a}_{1}}+ \frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\mathbf{a}_{1}}\\ \quad 0&\frac{1}{R_{2}}&\frac{1}{A_{2}}\frac{\partial}{\partial\mathbf{a}_{2}}+ \frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\mathbf{a}_{2}}\\ \quad-1&0&0\\ \quad 0&-1&0\\ \quad 0&0&-1\end{array} \right] \tag{33}\]  
The inertia matrix \(\mathbf{M}^{(\tau\eta)}\) is defined as follows, for \(\tau,\eta=0,1,2,\ldots,N,N+1\)  
\[\mathbf{M}^{(\tau\eta)}=\left[ \begin{array}{ccc}I^{(\tau\eta)}&0&0\\ 0&I^{(\tau\eta)}&0\\ 0&0&I^{(\tau\eta)}\end{array} \right] \tag{34}\]where the inertia terms \(I^{(\tau\eta)}\) are evaluated for \(\tau,\eta=0,1,2,\ldots,N,N+1\) once the mass density \(\rho^{(k)}\) of the \(k\)-th layer is introduced  
\[I^{(\tau\eta)}=\sum_{k=1}^{l}\int\limits_{\xi_{k}}^{\xi_{k+1}}\rho^{(k)}F_{\tau} F_{\eta}H_{1}H_{2}d\zeta \tag{35}\]  
All the results are summarized in the fundamental system of equations with the size of \(3\times(N+2)\) equilibrium equations, in which, for each order \(\tau=0,1,2,\ldots,N,N+1\) of kinematic expansion, the compact form is written as follows  
\[\sum_{\eta=0}^{N+1}\Big{(}\mathbf{L}^{(\tau\eta)}-\mathbf{L}_{f}^{(\tau\eta)} \Big{)}\mathbf{u}^{(\eta)}=\sum_{\eta=0}^{N+1}\Big{(}\mathbf{M}^{(\tau\eta)}+ \mathbf{M}_{f}^{(\tau\eta)}\Big{)}\tilde{\mathbf{u}}^{(\eta)} \tag{36}\]  
where the fundamental operator \(\mathbf{L}^{(\tau\eta)}=\mathbf{D}_{\Omega}^{*}\mathbf{A}^{(\tau\eta)} \mathbf{D}_{\Omega}\) is a \(3\times 3\) matrix defined as shown below  
\[\mathbf{L}^{(\tau\eta)}=\left[ \begin{array}{cccc}L_{11}^{(\tau\eta)}&L_{12}^ {(\tau\eta)}&L_{13}^{(\tau\eta)}\\ L_{21}^{(\tau\eta)}&L_{22}^{(\tau\eta)}&L_{23}^{(\tau\eta)}\\ L_{31}^{(\tau\eta)}&L_{32}^{(\tau\eta)}&L_{33}^{(\tau\eta)}\end{array} \right] \tag{37}\]  
The explicit definitions of the term \(L_{fg}^{(\tau\eta)}\), for \(f,g=1,2,3\) can be found in the book by [PERSON] et al. [99].