## 2 Effective Material Properties and Constitutive Relations  
Consider a curved nanobeam with length \(L\), thickness \(h\), and width \(b\) (Figure 1). The composite nanobeam is reinforced by GPLs, with \(N_{L}\) layers of thickness \(\Delta h\ =\ h/N_{L}\). The reinforcement features a uniform distribution within the polymer matrix at each layer. The GPL weight fraction changes linearly along the thickness direction to form a FG material structure. Four patterns are herein considered for functionalities, namely, a uniform distribution (UD) (labeled also as Pattern 1), two FG symmetric distributions (labeled as FG-O and FG-X Patterns, or equivalently Pattern 2 and 3), and an asymmetrical FG distribution (labeled as FG-A Pattern, or equivalently, Pattern 4), see Figure 2. The weight fraction of GPLs is, thus, varied along thickness direction, whereas the effective Young's modulus of each nanocomposite layer is computed based on the Halpin-Tsai algebraic formula, as follows [50]  
Figure 1: Mechanical model of a multi-layer graphene nanoplatelets (GPL)/polymer nanocomposite curved nanobeam. \[E_{C}^{(k)}\ =\ \frac{3}{8}\Bigg{[}\frac{1+\xi_{L}\eta_{L}V_{GPL}^{(k)}}{1-\eta_{L}V _{GPL}}\Bigg{]}\times E_{M}+\frac{5}{8}\Bigg{[}\frac{1+\xi_{w}\eta_{lw}V_{GPL}^{ (k)}}{1-\eta_{w}V_{GPL}}\Bigg{]}\times E_{M} \tag{1}\]  
where  
\[\eta_{L}\ =\ \frac{(E_{GPL}/E_{M})-1}{(E_{GPL}/E_{M})+\xi_{L}} \tag{2}\]  
\[\eta_{lw}\ =\ \frac{(E_{GPL}/E_{M})-1}{(E_{GPL}/E_{M})+\xi_{W}} \tag{3}\]  
and \(E_{M},E_{GPL}\) stand for the Young's modulus of the polymer and GPLs, respectively. In addition, in Equation (1), \(V_{GPL}^{(k)}\) is the volume fraction of GPLs for each layer, while \(\xi_{L},\xi_{w}\) denote two dimensionless length and weight parameters, depending on the dimension of GPLs, as follows  
\[\xi_{L}\ =\ 2\Bigg{(}\frac{l_{GPL}}{h_{GPL}}\Bigg{)} \tag{4}\]  
\[\xi_{w}\ =\ 2\Bigg{(}\frac{w_{GPL}}{h_{GPL}}\Bigg{)} \tag{5}\]  
With \(l_{GPL},w_{GPL},h_{GPL}\) being the length, width, and thickness of GPL, respectively. In addition, \(V_{GPL}^{(k)}\) is defined as follows  
\[V_{GPL}^{(k)}\ =\ \frac{g_{GPL}^{(k)}}{g_{GPL}^{(k)}+(\rho_{GPL}/\rho_{M})(1-g_{ GPL}^{(k)})} \tag{6}\]  
in which \(g_{GPL}^{(k)}\) is the weight fraction of GPL at each layer, \(\rho_{M}\) and \(\rho_{GPL}\) denote the density of the polymer and GPL, respectively. The effective density and Poisson's ratio of the nanocomposite material are defined analytically, according to the rule of mixtures, as follows  
\[ \begin{array}{ll}\rho_{C}^{(k)}&=\ \rho_{GPL}V_{GPL}^{(k)}+\rho_{M}V_{M}^{(k)}\\ \
u_{C}^{(k)}&=\ \
u_{GPL}V_{GPL}^{(k)}+\
u_{M}V_{M}^{(k)}\end{array}  \tag{7}\]  
with \(\
u_{M}\), \(\
u_{GPL}\) being the [PERSON]'s ratio of the polymer and GPL, respectively. Moreover, \(V_{M}^{(k)}\) is the volume fraction of the polymer at each layer. The weight fraction of the GPLs for each selected distribution reads as follows [16]  
\[g_{GPL}^{(k)}\ =\ \left\{\begin{array}{ll}\par\par\par\par\par\par\par \par\par  
where \(g^{*}_{GPL}\) denote the weight fraction of GPL.  
Now we introduce the constitutive relations of the nanocomposite curved nanobeam, based on the first order shear deformation theory. Thus, the displacement field is defined as [51; 52]  
\[u_{r}(r,\theta,t)\ =\ u_{r}(\theta,t) \tag{9}\]  
\[u_{\theta}(r,\theta,t)\ =\ u_{\theta}(\theta,t)+\zeta_{\chi}(\theta,t) \tag{10}\]  
where \(\zeta\ =\ r-R\) denotes the general distance from the reference surface. In addition, \(u_{r}(\theta,t)\) and \(u_{\theta}(\theta,t)\) refer to the radial and circumferential displacement components at the middle surface and \(\chi(\theta,t)\) is the rotation function.