## 3 Higher-Order Equivalent Layer-Wise Approach  
As already illustrated in the introduction, the well-known UF represents an extremely valid and efficient tool to analyze a huge variety of HSDTs, which are classified in general as Equivalent Single Layer (ESL) theories [61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74] and Layer-Wise (LW) models [79; 80; 81; 82; 83]. The former approach establishes that all the mechanical and geometric parameters are evaluated on the shell middle surface, whereas each layer is analyzed independently from the others in the latter method. The main novelty introduced by the present work is a new higher-order formulation which keeps the main features of the ESL approaches but uses the kinematic expansion of the displacement field commonly employed in the LW models. For this purpose, the term Equivalent Layer-Wise (ELW) is introduced. Let us consider the three-dimensional displacements \(U_{1}=U_{1}(\alpha_{1},\alpha_{2},\zeta)\), \(U_{2}=U_{2}(\alpha_{1},\alpha_{2},\zeta)\), and \(U_{3}=U_{3}(\alpha_{1},\alpha_{2},\zeta)\). They can be defined in each point of the shell structure according to the following expressions  
\[ \begin{split} U_{1}&=F_{0}u_{1}^{(0)}+F_{1}u_{1}^{ (1)}+F_{2}u_{1}^{(2)}+F_{3}u_{1}^{(3)}+\cdots+F_{N}u_{1}^{(N)}+F_{N+1}u_{1}^{(N +1)}\\ U_{2}&=F_{0}u_{2}^{(0)}+F_{1}u_{2}^{(1)}+F_{2}u_{2}^{ (2)}+F_{3}u_{2}^{(3)}+\cdots+F_{N}u_{2}^{(N)}+F_{N+1}u_{2}^{(N+1)}\\ U_{3}&=F_{0}u_{3}^{(0)}+F_{1}u_{3}^{(1)}+F_{2}u_{3}^{ (2)}+F_{3}u_{3}^{(3)}+\cdots+F_{N}u_{3}^{(N)}+F_{N+1}u_{3}^{(N+1)}\end{split}  \tag{9}\]  
in which \(F_{\tau}=F_{\tau}(\zeta)\), for \(\tau=0,1,\ldots,N,N+1\), specify the thickness functions related to the \(\tau\)-th order of kinematic expansion. On the other hand, \(u_{1}^{(\tau)}=u_{1}^{(\tau)}(\alpha_{1},\alpha_{2})\), \(u_{2}^{(\tau)}=u_{2}^{(\tau)}(\alpha_{1},\alpha_{2})\), and \(u_{3}^{(\tau)}=u_{3}^{(\tau)}(\alpha_{1},\alpha_{2})\), for \(\tau=0,1,\ldots,N,N+1\), are the degrees of freedom of the current model and denote the displacements of the section placed at the height \(\zeta_{\tau}\) of the shell, as it can be noticed from Figure 2. This aspect represents the main difference in comparison with the ESL approach, where each degree of freedom is always evaluated on the shell middle surface. In other words, the \(\tau\)-th order of kinematic expansion is strictly related to the number of points along the shell thickness (Figure 2). For the sake of clarity, it should be noticed that the displacements \(u_{1}^{(0)},u_{2}^{(0)},u_{3}^{(0)}\) linked to the order of expansion \(\tau=0\) are the displacements of the points laying on the shell bottom surface (for \(\zeta=-h/2\)). On the contrary, the displacements \(u_{1}^{(N)},u_{2}^{(N)},u_{3}^{(N)}\) are related to the \(\tau=N\) order of kinematic expansion and represent the displacements of the top surface of the shell. Figure 2: Kinematic expansion and degrees of freedom related to the order for the Equivalent Layer-Wise (ELW) approach. Similarly to the ESL approach, the \((N+1)\)-th order of kinematic expansion is always related to the well-known Murakami's function, which can be introduced to capture the zig-zag effect that can be noticed especially if a sandwich structure is analyzed. Having in mind Figure 1, in which a general lamination scheme is depicted, the [PERSON]'s function \(Z=Z(\xi)\) can be defined as follows  
\[Z=(-1)^{k}z_{k} \tag{10}\]  
The non-dimensional parameter \(z_{k}\in[-1,1]\) is assumed equal to the following expression  
\[z_{k}=\frac{2}{\xi_{k+1}-\xi_{k}}\zeta-\frac{\zeta_{k+1}+\zeta_{k}}{\xi_{k+1}- \xi_{k}} \tag{11}\]  
in which \(\zeta_{k}\) represents the boundary coordinate of the \(k\)-th layer along the normal direction. Thus, if a soft-core effect must be studied, one gets \(F_{N+1}=Z\). The reader can find further details concerning the use and the peculiar features of the [PERSON]'s function in the works [57; 58; 59]. Let us consider the thickness functions related to the other orders of kinematic expansions.