## 2 Functionally Graded GPLRC Plate with a Central Crack  
Figure 1a shows a uniform-thickness rectangular plate of GPLs mixed according to a specific distribution through thickness, where black dots indicate GPL particles. In addition, an inclined central crack exists within the plate. The plate mid-surface is denoted by \(\omega\) with two side lengths \(a\) and \(b\) and uniform thickness \(h\); then the material domain \(\Omega\in\Re^{3}\) of the plate can be expressed by \(\Omega=\omega\times[-h/2,h/2]\). Figure 1b shows different FG distribution patterns of GPLs indicated by black dots: FG-U, FG-O, FG-\(\Lambda\), and FG-X. GPLs are uniform for FG-U, biased at the mid-surface for FG-O, biased at the bottom side for FG-\(\Lambda\), and biased at the top and bottom sides for FG-X, respectively. The neutral surface \(\varpi\) is away from the mid-surface when the material composition is not symmetric with respect to the mid-surface, and the distance \(e\) between the two surfaces is calculated using the formulae given later. Figure 1: Graphene platelet-reinforced rectangular plate: (**a**) Geometry and dimensions; (**b**) Primitive GPL distribution patterns. The underlying matrix material and GPLs are mixed in the GPLRC plate, and the volume fractions of matrix and GPL are denoted by \(V_{m}(z)\) and \(V_{GPL}(z)\) in the function of thickness co-ordinate \(z\). Both volume fractions satisfy the following physical relation [14]:  
\[V_{GPL}(z)+V_{m}(z)=1 \tag{1}\]  
So the choice of either one is sufficient to represent both volume fractions. The present study chooses the thickness-wise GPL volume fraction \(V_{GPL}(z)\), which is expressed as  
\[V_{GPL}(z)= \begin{cases}V_{GPL}^{*},&\text{FG }-\text{ U}\\ 2(1-2|z|/h)V_{GPL}^{*},&\text{FG}-\text{O}\\ (1-2z/h)V_{GPL}^{*},&\text{FG}-\Lambda\\ 2(2|z|/h)V_{GPL}^{*},&\text{FG}-X\end{cases}  \tag{2}\]  
for the above-mentioned four GPL distribution patterns shown in Figure 1b. \(V_{GPL}^{*}\) indicates the total GPL volume fraction calculated as [19]  
\[V_{GPL}^{*}=\frac{g_{GPL}}{g_{GPL}+\rho_{GPL}(1-g_{GPL})/\rho_{m}} \tag{3}\]  
in terms of the total mass fraction \(g_{GPL}^{*}\) and density \(\rho_{GPL}\) of the GPLs and the density \(\rho_{m}\) of the matrix material (hereafter, the subscript \(m\) stands for the matrix material). Graphene platelets are typically modeled as rectangular solids of length \(l_{GPL}\), width \(w_{GPL}\), and thickness \(t_{GPL}\), and are assumed to be uniformly mixed into a polymer matrix _in a local sense_. Thus, the GPLRC structures are treated as isotropic with the effective elastic modulus \(E\) given by [19; 44]  
\[E=\frac{3}{8}\cdot\frac{1+\xi_{L}\eta_{L}V_{GPL}}{1-\eta_{L}V_{GPL}}E_{m}+ \frac{5}{8}\cdot\frac{1+\xi_{T}\eta_{T}V_{GPL}}{1-\eta_{T}V_{GPL}}E_{m} \tag{4}\]  
according to the Halphin-Tsai micromechanics approach. Two parameters \(\eta_{L}\) and \(\eta_{T}\) are defined by [19]  
\[\eta_{L}=\frac{E_{GPL}-E_{m}}{E_{GPL}+\xi_{L}E_{m}},\ \eta_{T}=\frac{E_{GPL}-E_{m} }{E_{GPL}+\xi_{T}E_{m}} \tag{5}\]  
in terms of the elastic moduli \(E_{GPL}\) and \(E_{m}\) of GPL and matrix material and the geometric parameters \(\xi_{L}=2l_{GPL}/t_{GPL}\) and \(\xi_{T}=2w_{GPL}/t_{GPL}\). The effective [PERSON]'s ratio \(\
u\) and thermal expansion coefficient \(\alpha_{t}\) are calculated by  
\[\
u=V_{GPL}\
u_{GPL}+V_{m}\
u_{m} \tag{6}\]  
\[\alpha_{T}=V_{GPL}\alpha_{t,GPL}+V_{m}\alpha_{t,m} \tag{7}\]  
using the simplest linear rule of mixtures. Since the volume fractions are in the function of co-ordinate \(z\), the effective material properties in Equations (4), (6), and (7) are also in the function of \(z\). The neutral surface may not coincide with the mid-surface, as mentioned above, but the present work will be formulated based on the neutral surface. The vertical distance \(e\) of the neutral surface from the mid-surface is determined by [45]  
\[e=\int_{-h/2}^{h/2}zE(z)dz/\int_{-h/2}^{h/2}E(z)dz \tag{8}\]  
Figure 2 depicts a central crack within the GPLRC plate, where the crack line is represented by phase-field formulation.