## 4 Higher-Order Equivalent Single Layer Approach  
A unified formulation is employed in the present paper to define several HSDTs in a compact manner. In particular, an Equivalent Single Layer (ESL) approach is taken into account. The three-dimensional displacements \(U_{1}(\alpha_{1},\alpha_{2},\zeta,t)\), \(U_{2}(\alpha_{1},\alpha_{2},\zeta,t)\), \(U_{3}(\alpha_{1},\alpha_{2},\zeta,t)\) for a generic composite shell made of several layers can be written as a function of a generic order of kinematic expansion \(\tau\). \[ \begin{array}{l}U_{1}=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta)u_{1}^{(\tau )}(\alpha_{1},\alpha_{2},t)\\ U_{2}=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta)u_{2}^{(\tau)}(\alpha_{1}, \alpha_{2},t)\\ U_{3}=\sum\limits_{\tau=0}^{N+1}F_{\tau}(\zeta)u_{3}^{(\tau)}(\alpha_{1}, \alpha_{2},t)\end{array}  \tag{27}\]  
Where the generalized displacements \(u_{1}^{(\tau)}(\alpha_{1},\alpha_{2},t)\), \(u_{2}^{(\tau)}=u_{2}^{(\tau)}(\alpha_{1},\alpha_{2},t)\), \(u_{3}^{(\tau)}=u_{3}^{(\tau)}(\alpha_{1},\alpha_{2},t)\) represent the degrees of freedom of the problem and are defined on the shell middle surface. They can be conveniently collected in the algebraic vector \(\mathbf{u}^{(\tau)}(\alpha_{1},\alpha_{2},t)\) for each order \(\tau\) of kinematic expansion. \[\mathbf{u}^{(\tau)}=\left[ \begin{array}{cc}u_{1}^{(\tau)}&u_{2}^{(\tau)}&u_ {3}^{(\tau)}\end{array} \right]^{T} \tag{28}\]  
On the other hand, \(F_{\tau}(\zeta)\) denotes the shear function (or thickness functions) linked to the \(\tau\)-th order of kinematic expansion and can assumes different meanings. In the current work, the power function \(\zeta^{\tau}\), for \(\tau=0,1,2,...,N\), is chosen for this purpose. The \((N+1)\)-st order of kinematic expansion denotes the Murakami's function that is employed to model the zig-zag effect [94; 95; 96]. For the laminated shell element shown in Figure 1, the [PERSON]'s function is defined as follows:  
\[\mathrm{Z}=(-1)^{k}\bigg{(}\frac{2}{\xi_{k+1}-\xi_{k}}\zeta-\frac{\xi_{k+1}+ \xi_{k}}{\xi_{k+1}-\xi_{k}}\bigg{)} \tag{29}\]  
in which the index \(k\) is introduced to specify the \(k\)-th ply. Thus, the \((N+1)\)-st shear function is assumed equal to Equation (29) if the zig-zag effect is included in the structural theory (in other words, one gets \(F_{N+1}=\mathrm{Z}\)). It should be noted that each HSDT is identified by the maximum order of kinematic expansion \(N\). If the power function \(\zeta^{\tau}\) is employed to define all the thickness functions of the model, the following HSDT can be introduced. \[ \begin{array}{lcl}N=1&\rightarrow&\mathrm{ED1}\\ N=2&\rightarrow&\mathrm{ED2}\\ N=3&\rightarrow&\mathrm{ED3}\\ N=4&\rightarrow&\mathrm{ED4}\end{array}  \tag{30}\]  
When the zig-zag effect is contemplated, the same order theories are specified as follows:  
\[ \begin{array}{lcl}N=1&\rightarrow&\mathrm{EDZ1}\\ N=2&\rightarrow&\mathrm{EDZ2}\\ N=3&\rightarrow&\mathrm{EDZ3}\\ N=4&\rightarrow&\mathrm{EDZ4}\end{array}  \tag{31}\]  
With reference to the Equations (30) and (31), it should be noted that the letter \"E\" specifies an ESL model, \"D\" means that the fundamental equations will be written in terms of generalized displacements, and \"Z\" stands for the Murakami's function. At this point it is important to underline that the displacement field of the well-known first-order shear deformation theory can be obtained from Equation (27) by choosing properly the shear functions and the degrees of freedom. In the following, this theory will be indicated by the notation FSDT (First-order Shear Deformation  
where  
\[