## 2 Functionally Graded Materials  
It is well-known that FGMs allows to define a continuous gradual variation of the mechanical properties of two isotropic constituents, in general one ceramic and one metal, from the bottom to the top surfaces of the layer in which they are employed. In the present paper, the rule of mixture is used to evaluate the mechanical properties of these composites. For the sake of simplicity, let us assume that the \(k\)-th layer of the structure is made of FGM and the two constituents are ceramic and metallic materials, respectively. Each mechanical quantity related to the first constituent is denoted by the subscript \"C\", whereas the subscript \"\(M\)\" is used to specify the properties of the metal. Since the two constituents are assumed isotropic and linear elastic, they are fully characterized through the Young's modulus (\(E_{C},E_{M}\)), Poisson's ratio (\(\
u_{C},\
u_{M}\)), and the density (\(\rho_{C},\rho_{M}\)). By hypothesis, these composites are isotropic and non-homogenous along the thickness direction, which is denoted by \(\zeta\). Thus, the rule of mixture provides the following relations  
\[E^{(k)}(\zeta) =E_{C}V_{C}^{(k)}(\zeta)+E_{M}V_{M}^{(k)}(\zeta) \tag{1}\] \[\
u^{(k)}(\zeta) =\
u_{C}V_{C}^{(k)}(\zeta)+\
u_{M}V_{M}^{(k)}(\zeta)\] (2) \[\rho^{(k)}(\zeta) =\rho_{C}V_{C}^{(k)}(\zeta)+\rho_{M}V_{M}^{(k)}(\zeta) \tag{3}\]  
where \(E^{(k)}(\zeta)\), \(\
u^{(k)}(\zeta)\), and \(\rho^{(k)}(\zeta)\) stand for the overall mechanical properties of the \(k\)-th FGM layer. On the other hand, \(V_{C}^{(k)}(\zeta)\) and \(V_{M}^{(k)}(\zeta)\) represent the volume fraction distributions of the ceramic and the metal, respectively. These two quantities are related as follows  
\[V_{C}^{(k)}(\zeta)+V_{M}^{(k)}(\zeta)=1 \tag{4}\]  
Thus, Equations (1)-(3) could be rewritten also as a function only of the volume fraction of one constituent. Several expressions are available in the literature to define the continuous and gradual variation of the mechanical properties along the thickness of the FGM layer [67]. In the current paper, a four-parameter power law is introduced for this purpose. Let us assume that the volume fraction is strictly related to the materials located to the external surfaces of the \(k\)-th FGM layer. With reference to Figure 1, in which a generic stacking sequence made of FGM and isotropic layers is depicted, the \(k\)-thply is identified by the thickness coordinates \(\zeta_{k}\) and \(\zeta_{k+1}\), which denote respectively the coordinates of its bottom and top surfaces in each point of the reference domain. On the other hand, \(h_{k}\) represents the thickness of the \(k\)-th layer in a generic point of the domain. If the constituent at the top surface is the ceramic material, the volume fraction \(V_{C}^{(k)}(\zeta)\) is defined as follows  
\[V_{C}^{(k)}(\zeta)=\left(1-a^{(k)}\left(\frac{\zeta_{k+1}-\zeta}{h_{k}}\right) +b^{(k)}\left(\frac{\zeta_{k+1}-\zeta}{h_{k}}\right)^{c^{(k)}}\right)^{p^{(k)}} \tag{5}\]  
where \(a^{(k)}\), \(b^{(k)}\), \(c^{(k)}\) and \(p^{(k)}\) are the four parameters that characterize the distribution at issue. As specified in the paper by [PERSON] [60], their value must be chosen accurately so that Equation (4) is satisfied for any value of \(p^{(k)}\). On the other hand, if the constituent at the top surface is the metallic material the volume fraction \(V_{C}^{(k)}(\zeta)\) assumes the following aspect  
\[V_{C}^{(k)}(\zeta)=\left(1-a^{(k)}\left(\frac{\zeta-\zeta_{k}}{h_{k}}\right)+ b^{(k)}\left(\frac{\zeta-\zeta_{k}}{h_{k}}\right)^{c^{(k)}}\right)^{p^{(k)}} \tag{6}\]It should be noted that homogeneous and isotropic material can be obtained as a particular case of the FGM, by setting properly the value of the exponent \(p^{(k)}\). In other words, by setting \(p^{(k)}=0\) or \(p^{(k)}=\infty\) it is possible to model an isotropic layer. In addition, it should be specified that sandwich structures can be obtained by choosing properly the volume fraction distributions of the constituents in the various plies.