## 2 Theoretical Model
### Shell Formulation  
The three-dimensional displacement field of the laminated composite shell can be expressed by the Carrera Unified Formulation (CUF), which was conceived to study the structural response of beams and plates. The displacement components, according to a general higher-order Equivalent Single Layer (ESL) approach, are given by  
\[ \begin{array}{l}\begin{array}{l}\begin{array}{l}U_{1}(\alpha_{1},\alpha_{2}, \zeta,t)=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta)u_{1}^{(\tau)}(\alpha_{1}, \alpha_{2},t)\\ U_{2}(\alpha_{1},\alpha_{2},\zeta,t)=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta )u_{2}^{(\tau)}(\alpha_{1},\alpha_{2},t)\\ U_{3}(\alpha_{1},\alpha_{2},\zeta,t)=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta )u_{3}^{(\tau)}(\alpha_{1},\alpha_{2},t)\end{array}\end{array}\end{array}  \tag{9}\]  
where \(U_{1}\), \(U_{2}\), \(U_{3}\) are the three-dimensional displacements; \(F_{\tau}\) represents the thickness or the shear functions and \(\tau=0,1,2,\ldots,N,N+1\) stands for the order of kinematic expansion; \(u_{1}^{(\tau)}\), \(u_{2}^{(\tau)}\), \(u_{3}^{(\tau)}\) are the generalized displacement components or degrees of freedom of the problem. The thickness functions can be chosen freely or according to the functions listed in the book [99]. In this paper, the thickness functions are given by the power function \(F_{\tau}=\zeta^{\tau}\) for \(\tau=0,1,2,\ldots,N\). The last order of expansion or the \((N+1)\)-th order is related to the [PERSON]'s function, by which the so-called zig-zag effect can be captured. More details about the zig-zag effect and the [PERSON]'s function can be found in [96; 97]. The [PERSON]'s function \(\mathrm{Z}=\mathrm{Z}(\zeta)\) is defined as  
\[\mathrm{Z}=(-1)^{k}\bigg{(}\frac{2}{\zeta_{k+1}-\zeta_{k}}\zeta-\frac{\zeta_{ k+1}+\zeta_{k}}{\zeta_{k+1}-\zeta_{k}}\bigg{)} \tag{10}\]  
where \(\zeta_{k}\) and \(\zeta_{k+1}\) represent, respectively, the bottom and the top coordinates of the \(k\)-th layer along the thickness direction, as shown in Figure 1. By varying the maximum order of kinematic expansion \(N\), one can obtain different shear deformation theories. For easier denotation of these theories, the acronyms \(\mathrm{ED}\,N\) and \(\mathrm{EDZ}\,N\) are introduced, where \"E\" stands for the ESL approach, \"D\" denotes the use of generalized displacement, and the letter \"Z\", if present, denotes the use of the Murakami's function. Once the displacement field (9) is set, the generalized strains evaluated on the shell middle surface can be computed as follows  
\[\mathbf{\epsilon}^{(\tau)}=\mathbf{D}_{\Omega}\mathbf{u}^{(\tau)} \tag{11}\]  
in which vector \(\mathbf{u}^{(\tau)}=\mathbf{u}^{(\tau)}(\alpha_{1},\alpha_{2},t)\) collects generalized displacement components as  
\[\mathbf{u}^{(\tau)}=\left[ \begin{array}{cc}u_{1}^{(\tau)}&u_{2}^{(\tau)}&u_ {3}^{(\tau)}\end{array} \right]^{T}\]  
All of the strains are collected in the algebraic vector \(\varepsilon^{(\tau)}=\varepsilon^{(\tau)}(\alpha_{1},\alpha_{2},t)\), defined as  
\[\varepsilon^{(\tau)}=\left[ \begin{array}{cccccccc}\varepsilon_{1}^{(\tau)}& \varepsilon_{2}^{(\tau)}&\gamma_{1}^{(\tau)}&\gamma_{2}^{(\tau)}&\gamma_{13}^{( \tau)}&\gamma_{23}^{(\tau)}&\omega_{13}^{(\tau)}&\omega_{23}^{(\tau)}& \varepsilon_{3}^{(\tau)}\end{array} \right]^{T} \tag{14}\]  
The relation between three-dimensional strain components collected in \(\varepsilon=\varepsilon(\alpha_{1},\alpha_{2},\zeta,t)\) and generalized strains (14) can be expressed as follows  
\[\varepsilon=\sum_{\tau=0}^{N+1}\mathbf{Z}^{(\tau)}\varepsilon^{(\tau)} \tag{15}\]  
where \(\varepsilon\) is described as  
\[\varepsilon=\left[ \begin{array}{cccccccc}\varepsilon_{1}&\varepsilon_{2}& \gamma_{12}&\gamma_{1n}&\gamma_{2n}&\varepsilon_{n}\end{array} \right]^{T} \tag{16}\]  
It should be specified that the subscript \(n\) in (16) denotes those strain components that involve the direction perpendicular to the shell middle surface. The matrix \(\mathbf{Z}^{(\tau)}\) is described, for each order of kinematic expansion \(\tau\), as follows  
\[\mathbf{Z}^{(\tau)}=\left[