## 7 Numerical Results
### Elliptic Cylinder  
The next application aims to evaluate the natural frequencies of a FC elliptic cylinder made of two FGM layers of equal thickness. The geometry under consideration is depicted in Figure 4b. The stacking sequence, as well as the volume fraction distribution of the constituents, are shown in Figure 5a for different values of the exponent \(p^{(1)}=p^{(2)}=p\). The middle surface of the shell is given by the following position vector  
\[\mathbf{r}(\alpha_{1},y)=a\cos\alpha_{1}\,\mathbf{e}_{1}-y\,\mathbf{e}_{2}+b \sin\alpha_{1}\,\mathbf{e}_{3} \tag{96}\]  
where \(a=1\) m and \(b=1.5\) m are the semi diameters of the ellipse. The coordinates \(\alpha_{1},y\) are bounded respectively by the limitations \(\alpha_{1}\in\left[\alpha_{1}^{0},\alpha_{1}^{1}\right]\) and \(y\in[0,L]\), with \(\alpha_{1}^{0}=-\pi/2\), \(\alpha_{1}^{1}=3\pi/2\), and \(L=4\) m. A sine-wave variation is applied along the first coordinate \(\alpha_{1}\) to define the thickness profile. Mathematically speaking, the thickness is described by the following expression  
\[h(\alpha_{1})=h_{0}\left(1+\frac{1}{2}\left(\sin\left(\pi\left(4\frac{\alpha_ {1}-\alpha_{1}^{0}}{\alpha_{1}^{1}-\alpha_{1}^{0}}+\frac{1}{2}\right)\right) \right)^{3}\right) \tag{97}\]  
with \(h_{0}=0.2\) m. The first 10 natural frequencies for the considered structure are shown in Table 3 for different values of the parameter \(p\in[0,\infty]\), varying the structural model. As in the previous example, the FSDT\({}_{RS}^{\chi=1.2}\), ED2\({}^{\chi=1.2}\), and ED3 are employed. The same considerations concerning the shear correction factor and the reduced elastic stiffnesses are still valid.  
It should be noted that the boundary values of \(p\) denote a structure completely made of Silicon nitride (\(p=0\)) and Stainless steel (\(p=\infty\)), respectively. In this cases, the comparison with the 3D-FEM solutions is performed with excellent results for all the structural theories. It should be noted that the finite element model is made of 69680 brick elements (924690 degrees of freedom). On the other hand, the LGDQ solutions is obtained by setting \(I_{N}=51\), \(I_{M}=31\), and \(I_{PN}=I_{PM}=16\). These values are chosen to describe accurately both the elliptical shape and the thickness variation. For completeness purposes, the variation of the first natural frequency is depicted graphically in Figure 6 as a function of the exponent \(p\). It is evident that for lower values of \(p\) the frequencies are highly affected by variation of the volume fraction distribution of the constituents. On the other hand, for \(p>4\div 6\) the variation in terms of natural frequencies is considerably restricted. The same behavior can be observed for each structural model. In the same plots, the black dashed lines represent the extreme cases of the isotropic homogeneous shell (\(p=0\) and \(p=\infty\)) and the corresponding frequencies are evaluated by the finite element software. For the sake of conciseness, Figure 6 shows only the variation of the first natural frequency, but it should be specified that higher modes exhibit the same trend. It is evident that the variation of the natural frequencies is bounded by these two lines. Finally, the first three mode shapes for a FC elliptic cylinder with variable thickness are shown in Figure 7 assuming \(p=1\), which corresponds to a linear distribution of the volume fraction. From these images, it can be noted that the boundary conditions are well-enforced. In this paper, the colormap used for the mode shapes stands for the intensity of the modal displacements. A zero displacement is specified by a dark blue color, whereas a red tone represents bigger movements.