## 3 Higher-Order Equivalent Layer-Wise Approach
As a consequence, the relations shown below can be easily deduced  
\[U_{1}(\zeta=\zeta_{\tau}) =u_{1}^{(\tau)}+(-1)^{k}z_{k}(\zeta_{\tau})u_{1}^{(N+1)} \tag{20}\] \[U_{2}(\zeta=\zeta_{\tau}) =u_{2}^{(\tau)}+(-1)^{k}z_{k}(\zeta_{\tau})u_{2}^{(N+1)}\] \[U_{3}(\zeta=\zeta_{\tau}) =u_{3}^{(\tau)}+(-1)^{k}z_{k}(\zeta_{\tau})u_{3}^{(N+1)}\]  
This means that the three-dimensional displacement evaluated at the height coordinate linked to the \(\tau\)-th order of kinematic expansion coincides with the corresponding degree of freedom of the model, except for the Murakami's function. At this point it should be specified also that the kinematic expansion is not affected by either the lamination scheme or the placement of the various layers that compose the laminate, since this aspect is considered only when the elastic coefficients of the composite are evaluated as it will be shown in the following paragraphs. Finally, it should be noticed that the same kinematic model just illustrated is typically used for the LW approach to describe the displacement field of each lamina [79; 80; 81; 82; 83]. A peculiar notation can be introduced now to specify univocally the various higher-order ELW models that are employed in the following. Each theory is identified by the maximum order of kinematic expansion \(N\). In particular, the structural theories used in the current work are listed below for various orders of expansion  
\[ \begin{array}{ccc}N=1&\rightarrow&\text{ELD1}\\ N=2&\rightarrow&\text{ELD2}\\ N=3&\rightarrow&\text{ELD3}\\ N=4&\rightarrow&\text{ELD4}\end{array}  \tag{21}\]  
where \"EL\" specifies that an ELW approach is considered, whereas \"D\" means that the governing equations will be written in terms of generalized displacements. If the Murakami's function must be embedded in the model, the following theories are obtained for several orders of kinematic expansion  
\[ \begin{array}{ccc}N=1&\rightarrow&\text{ELDZ1}\\ N=2&\rightarrow&\text{ELDZ2}\\ N=3&\rightarrow&\text{ELDZ3}\\ N=4&\rightarrow&\text{ELDZ4}\end{array}  \tag{22}\]  
where \"Z\" stands clearly for the Murakami's function (zig-zag effect). Once the kinematic model is defined, both the constitutive relations and the motion equations are formally equal to the ones used for the ESL approach [61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74]. Thus, in the following only the main aspects of the model are presented for conciseness purposes. At this point, the vector \(\mathbf{u}^{(\tau)}=\mathbf{u}^{(\tau)}(\alpha_{1},\alpha_{2})\) that collects the degrees of freedom for each order \(\tau\) of kinematic expansion can be conveniently introduced  
\[\mathbf{u}^{(\tau)}=\left[\begin{array}{cc}u_{1}^{(\tau)}&u_{2}^{(\tau)}&u_ {3}^{(\tau)}\  
above, the interested reader can notice some similarities between the present approach and the aforementioned SaS formulation. In particular, the definitions of the first six generalized strains collected in (24) coincide exactly with the corresponding strain parameters of the SaS formulation [104; 105]. The stress resultants related to the \(\tau\)-th order of kinematic expansion can be collected in the algebraic vector \(\mathbf{S}^{(\tau)}=\mathbf{S}^{(\tau)}(\alpha_{1},\alpha_{2})\), whose extended vector form is shown below  
\[\mathbf{S}^{(\tau)}=\left[ \begin{array}{cccccc}N_{1}^{(\tau)}&N_{2}^{(\tau)} &N_{12}^{(\tau)}&N_{21}^{(\tau)}&T_{1}^{(\tau)}&T_{2}^{(\tau)}&P_{1}^{(\tau)}& P_{2}^{(\tau)}&S_{3}^{(\tau)}\end{array} \right]^{T} \tag{27}\]  
Due to the duality between strains and stresses, the generalized stresses (or stress resultants) can be related directly to the degrees of freedom of the model for \(\tau=0,1,\ldots,N,N+1\) according to the following equation  
\[\mathbf{S}^{(\tau)}=\sum\limits_{\eta=0}^{N+1}\mathbf{A}^{(\tau\eta)}\mathbf{ D}_{\Omega}\mathbf{u}^{(\eta)} \tag{28}\]  
where the constitutive operator \(\mathbf{A}^{(\tau\eta)}\) is introduced for each order of kinematic expansion (for \(\tau,\eta=0,1,2,\ldots,N,N+1\)).