## 3 Higher-Order Equivalent Layer-Wise Approach
If the strain along the shell thickness is neglected, the plane-stress-reduced elastic coefficients are needed (\(\overline{E}_{nm}^{(k)}=\overline{Q}_{nm}^{(k)}\)). On the contrary, if the stretching effect is taken into account the non-reduced elastic coefficients must be used (\(\overline{E}_{nm}^{(k)}=\overline{C}_{nm}^{(k)}\)). The trace over these symbols means that these quantities must be evaluated in the local reference system \(O^{\prime}\alpha_{1}\alpha_{2}\zeta\). In other words, the constitutive equations of each lamina must be transferred into the geometric coordinate system by means of the proper transformation laws which take into account the different orientation that can be given to an orthotropic medium. In fact, since each layer can be arbitrarily oriented with respect to the laminate reference system as shown in Figure 1, in which the symbols \(\alpha,\beta,\gamma,\delta\) are introduced for this purpose, the constitutive relations of each lamina must be converted into the local reference system \(O^{\prime}\alpha_{1}\alpha_{2}\zeta\). Further details concerning these relations can be found in the book by [PERSON] [1]. For the sake of completeness, it should be specified that in the present work the mechanical properties of each layer \(k\) are fully described by means of the engineering constants of the materials, which are \(E_{1}^{(k)}\), \(E_{2}^{(k)}\), \(E_{3}^{(k)}\), \(G_{12}^{(k)}\), \(G_{13}^{(k)}\), \(G_{23}^{(k)}\), \(v_{12}^{(k)}\), \(v_{13}^{(k)}\), \(v_{23}^{(k)}\) for an orthotropic medium, and \(E^{(k)}\), \(G^{(k)}\), \(v^{(k)}\) for an isotropic one. In any case, the lamination scheme is identified by the notation \((\alpha,\beta,\ldots,\gamma,\ldots,\delta)\), in which \(\alpha,\beta,\gamma,\delta\) represent the orientation of each ply. It should be recalled that in the current approach each layer is assumed to be linear elastic. In any circumstance, the use of both the shear correction factor and the plane stress-reduced elastic coefficients is specified by adding a proper notation to the acronym of the structural theory. In particular, the superscript \(\chi=1.2\) is added to specify the use of the constant value for the shear correction factor, whereas the subscript \(RS\) stands for \"Reduced Stiffness\" and means that the reduced elastic coefficients are employed. The motion equations are obtained by means of the well-known [PERSON]'s variational principle. For the sake of conciseness, only the equations at issue are shown. For each order of kinematic expansion \(\tau\), the free vibration problem is governed by the following matrix equation  
\[\mathbf{D}_{\Omega}^{*}\mathbf{S}^{(\tau)}=\sum_{\eta=0}^{N+1}\mathbf{M}^{( \tau\eta)}\ddot{\mathbf{u}}^{(\eta)} \tag{33}\]  
The equilibrium operator \(\mathbf{D}_{\Omega}^{*}\) takes the following aspect \[\mathbf{D}_{\Omega}^{*}=\left[ \begin{array}{ccc}\frac{1}{A_{1}}\frac{\partial}{ \partial\alpha_{1}}+\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\alpha_{1 }}&-\frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\alpha_{2}}&-\frac{1}{R _{1}}\\ -\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\alpha_{1}}&\frac{1}{A_{2}} \frac{\partial}{\partial\alpha_{2}}+\frac{1}{A_{1}A_{2}}\frac{\partial A_{1}} {\partial\alpha_{2}}&-\frac{1}{R_{2}}\\ \frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\alpha_{2}}&\frac{1}{A_{1}} \frac{\partial}{\partial\alpha_{1}}+\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}} {\partial\alpha_{1}}&0\\ \frac{1}{A_{2}}\frac{\partial}{\partial\alpha_{2}}+\frac{1}{A_{1}A_{2}}\frac{ \partial A_{1}}{\partial\alpha_{2}}&\frac{1}{A_{1}A_{2}}\frac{\partial A_{2}} {\partial\alpha_{1}}&0\\ \frac{1}{R_{1}}&0&\frac{1}{A_{1}}\frac{\partial}{\partial\alpha_{1}}+\frac{1 }{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\alpha_{1}}\\ 0&\frac{1}{R_{2}}&\frac{1}{A_{2}}\frac{\partial}{\partial\alpha_{2}}+\frac{1 }{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\alpha_{2}}\\ -1&0&0\\ 0&-1&0\\ 0&0&-1\end{array}