## 3 Functionally Graded Materials  
The material considered in this study is a combined metal and ceramic, whose volume fractions are assumed to be variable along the \(\zeta\)-direction. In a context where even or uneven distributions of porosity are commonly assumed in the literature, the present work considers an even distribution of porosity, equally distributed within the material.  
The [PERSON]'s Modulus \(E\), the Poisson's ratio \(\
u\), and the density of the shell \(\rho\), are assumed to vary through the thickness according to a power law distribution and even porous distribution, as defined in the following:  
\[E(\zeta)=[E_{c}-E_{m}]V_{c}+E_{m}-\frac{\beta}{2}[E_{c}-E_{m}] \tag{9}\]  
\[\
u(\zeta)=[v_{c}-\
u_{m}]V_{c}+\
u_{m}-\frac{\beta}{2}[v_{c}-\
u_{m}] \tag{10}\]  
\[\rho(\zeta)=[\rho_{c}-\rho_{m}]V_{c}+\rho_{m}-\frac{\beta}{2}[\rho_{c}  
where the subscripts \(m\) and \(c\) refer to the metal and ceramic, respectively, \(\beta\) (with \(\beta\ll 1\)) is the porosity volume fraction, and \(V_{c}\) and \(V_{m}\) refer to the volume fraction of the ceramic and metal, respectively. As can be noted from Equations (9)-(11), the overall mechanical properties of the composite are computed through the well-known rule of mixture, which has been properly modified to take into account the porosity of the medium [31]. The same empirical approach has been recently employed and validated by many authors to describe the porous phase of FGMs [30; 31; 32; 33; 34; 35]. To the best of the authors' knowledge, the rule of mixture represents a simple homogenization technique that allows us to evaluate in an efficient manner the elastic properties of a graded composite. Nevertheless, it can be recalled that more refined approaches are available in the literature, as highlighted in the paper by [PERSON] et al. [39]. For instance, the Mori-Tanaka scheme could be used as an alternative to the theory of mixture, as illustrated in [3], where the classic theory of mixtures is compared to the more refined scheme. Nevertheless, the numerical analyses shown in the paper by [PERSON] et al. [3] proved that similar results can be obtained by using these two micromechanical approaches. Thus, a proper set of experimental analyses should be performed to understand which is the best methodology for evaluating the mechanical properties of porous FGMs. Nevertheless, the Mori-Tanaka scheme could provide a better physical interpretation of the mechanical behavior of FGMs [3]. As specified in the introduction, the paper does not aim to discuss the micromechanics of porous media, but is focused on the effect that porosities could have on the structural response of FGM shells by using existing approaches.  
Based on the rule of mixture, the following relation can be deduced:  
\[V_{m}+V_{c}=1. \tag{12}\]  
In this study, the volume fraction of ceramic \(V_{c}\) is modeled according to the following power-law distributions, namely  
\[FGM-type\ 1:V_{c}=\left(\frac{2\zeta+h}{2h}\right)^{p} \tag{13}\]  
\[FGM-type\ 2:V_{c}=\left(\frac{2\zeta-h}{2h}\right)^{p}, \tag{14}\]  
where \(p\in[0,\infty)\) is the FG power-law index. Figure 2 illustrates the variation of the ceramic volume fraction vs. the dimensionless thickness coordinates for different values of \(p\).  
Figure 2: Different volume fraction of ceramic along the thickness for different values of power-law index: (**a**) FGM (Functionally Graded Material)-type 2; (**b**) FGM-type 1 [38].