## 2 Multi-Scale Micromechanical Model of Nanocomposites Reinforced by GPLs  
The GPL-reinforced nanocomposite circular arch with small width \(b\) and total thickness \(H\) is shown in Figure 1, where \(\theta_{0}\) denotes the subtended angle, \(R_{0}\) is the radius of the mid-surface, and \(L_{0}=\theta_{0}R_{0}\) is the length of the centerline of the arch. The coordinate system \(r-\theta\) is established with its origin being at the centerline endpoint of the arch, such that \(R_{0}-H/2\leq r\leq R_{0}+H/2\), and \(0\leq\theta\leq\theta_{0}\).  
The GPLs are dispersed with the weight fraction varying functionally graded along the thickness direction as symmetric and non-symmetric types. Typical linear patterns of GPL distributions, namely UD, FG-X, FG-O, and FG-A, as shown in Figure 2, are expressed according to the following relations:  
\[\begin{array}{ll}\text{UD}:&V_{\text{GPL}}(r)=V_{\text{GPL}}^{*}\\ \text{FG-X}:&V_{\text{GPL}}(r)=4V_{\text{GPL}}^{*}|r-R_{0}|/H\\ \text{FG-O}:&V_{\text{GPL}}(r)=2V_{\text{GPL}}^{*}(1-2|r-R_{0}|/H)\\ \text{FG-A}:&V_{\text{GPL}}(r)=V_{\text{GPL}}^{*}[1-2(r-R_{0})/H].\end{array} \tag{1}\]  
where \(V_{\text{GPL}}^{*}\) is the volume fraction of GPLs, given as:  
\[V_{\text{GPL}}^{*}=\frac{W_{\text{GPL}}}{W_{\text{GPL}}+(\rho_{\text{GPL}}/ \rho_{\text{M}})(1-W_{\text{GPL}})}. \tag{2}\]  
in which \(\rho_{\text{GPL}}\) and \(\rho_{\text{M}}\) are the mass densities of GPLs and polymer matrix, respectively, and \(W_{\text{GPL}}\) represents the total weight fraction of GPLs. It can be noted that the content of GPLs decreases monotonically from the bottom surface to the top surface for FG-A, and the pattern FG-X are GPLs rich near the top and bottom surfaces, while FG-O is GPLs rich at mid-plane of the arch. Specially, the GPLs for pattern UD are distributed uniformly through the thickness, which corresponds to a homogeneous arch.  
The effective Young's modulus of nanocomposites with randomly oriented nanofillers can be approximated by the modified Halpin-Tsai model [11], which is:  
\[E=\frac{3}{8}\frac{1+\xi_{l}\eta_{l}V_{\text{GPL}}}{1-\eta_{l}V_{\text{GPL}}} \times E_{\text{M}}+\frac{5}{8}\frac{1+\xi_{w}\eta_{w}V_{\text{GPL}}}{1-\eta_ {w}V_{\text{GPL}}}\times E_{\text{M}}. \tag{3}\]  
Figure 1: Sketch of the FG-GPLRC circular arch and the polar coordinate system.  
where the parameters \(\eta_{l}\) and \(\eta_{w}\) are:  
\[\eta_{l}=\frac{(E_{\text{GPL}}/E_{\text{M}})-1}{(E_{\text{GPL}}/E_{\text{M}})+ \xi_{l}},\ \eta_{w}=\frac{(E_{\text{GPL}}/E_{\text{M}})-1}{(E_{\text{GPL}}/E_{\text{M}})+ \xi_{w}}. \tag{4}\]  
in which \(E_{\text{GPL}}\) and \(E_{\text{M}}\) denote the [PERSON]'s moduli of GPLs and matrix, respectively, and \(\xi_{l}\) and \(\xi_{w}\) indicate both the geometry and size parameters of GPLs, and can be determined by:  
\[\xi_{l}=2l_{\text{GPL}}/h_{\text{GPL}},\ \xi_{w}=2w_{\text{GPL}}/h_{\text{GPL}}. \tag{5}\]  
with \(l_{\text{GPL}}\), \(w_{\text{GPL}}\), and \(h_{\text{GPL}}\) being the length, width and thickness of the GPL nanofiller, respectively. The effective [PERSON]'s modulus of GPL/polymer nanocomposites predicted by the modified Halpin-Tsai model (Equation (3)) is only 2.7% higher than the experimental result [31], which validated the accuracy of the present micromechanical model.  
By employing the rule of mixtures, the effective mass density \(\rho\) and Poisson's ratio \(\
u\) of the GPL-reinforced nanocomposites can be determined as:  
\[\begin{array}{l}\rho=\rho_{\text{GPL}}V_{\text{GPL}}+\rho_{\text{M}}(1-V_{ \text{GPL}})\\ \
u=v_{\text{GPL}}V_{\text{GPL}}+\
u_{\text{M}}(1-V_{\text{GPL}}).\end{array} \tag{6}\]  
where \(v_{\text{GPL}}\) and \(v_{\text{M}}\) are the Poisson's ratios of GPLs and matrix, respectively.  
It can be found that the material properties of FG-GPLRCs (see Equations (3) and (6)), vary smoothly through the thickness of the arch. Hence, the patterns FG-X, FG-O, and FG-A appear the functionally-graded behaviors in the thickness direction.