## 2 Material Statistics
### Meso-Scale Statistics  
Statistics of meso-scale properties of unidirectional (UD) E-glass/epoxy are derived from micro-scale properties by a combination of Monte-Carlo simulation and bridging micromechanical model [28]. For E-glass/epoxy composite, the bridging micromechanical model is expressed as:  
\[E_{1}=E_{f}V_{f}+E_{m}V_{m} \tag{1}\]  
\[E_{22}=\frac{(V_{f}+V_{m}a_{11})(V_{f}+V_{m}a_{22})}{(V_{f}+V_{m}a_{11})(V_{f}S _{f22}+a_{22}V_{m}S_{m22})+V_{f}V_{m}(S_{m12}-S_{f12})a_{12}} \tag{2}\]  
\[\
u_{12}=\
u_{f}V_{f}+\
u_{m}V_{m} \tag{3}\]  
\[G_{12}=\frac{(G_{f}+G_{m})+V_{f}(G_{f}-G_{m})}{(G_{f}+G_{m})-V_{f}(G_{f}-G_{m})} \tag{4}\]  
where \(a_{11}\), \(a_{22}\), \(a_{12}\), \(S_{f11}\),..., \(S_{m12}\) are functions of the elastic properties of fibre and matrix as given in Huang [28]; \(V_{m}\) denotes matrix volume ratio (\(V_{m}=1-V_{f}\)). \(G_{m}(G_{f})\) is the shear modulus of matrix (fibre) which is dependent on \(E_{m}\) and \(\
u_{m}\) (\(E_{f}\) and \(\
u_{f}\)) due to isotropy of matrix (fibre). \(E_{1}\) denotes normal modulus in the fibre direction; \(E_{2}\) denotes the normal modulus in the transverse direction; \(\
u_{12}\) denotes the in-plane major Poisson's ratio; \(G_{12}\) denotes the in-plane shear modulus. In a composite material coordinate, 1 denotes fibre direction and 2 denotes transverse direction (perpendicular to fibre direction). Lamina density is derived by the rule of mixture which is written as:  
\[\rho=\rho_{f}V_{f}+\rho_{m}V_{m} \tag{5}\]  
In Monte-Carlo method, the accuracy (\(\zeta\)) on the estimation of a parameter \(P\) is expressed as [29]:  
\[\varsigma\approx\frac{s/\sqrt{N}}{\hat{P}} \tag{6}\]  
where \(\hat{P}\) is estimation of \(P\), \(N\) is sampling number, and \(s\) is standard deviation of \(P\) which is estimated as:  
\[s=\sqrt{\frac{1}{N}\sum_{k=1}^{N}g(\mathbf{X}_{k})^{2}-(\frac{1}{N}\sum_{k=1} ^{N}g(\mathbf{X}_{k}))^{2}} \tag{7}\]  
where \(\mathbf{X}_{k}\) is the \(k\)-th sample of random variables, and \(g(\mathbf{X}_{k})\) is a function on \(\mathbf{X}_{k}\) such as Equations (1)-(5). \(\zeta\) represents the relative difference between the mathematical expectation of a group of samples and the true mean value of corresponding random variable. Equation (6) indicates that a large enough number of samples would provide very accurate estimation on random parameters in Monte-Carlo simulation. In this work, \(1\times 10^{5}\) group of samples were drawn for the micro-scale random variables listed in Table 1 and they were substituted into Equations (1)-(5) to derive statistics of meso-scale properties. According to Equation (6), the sampling number of \(1\times 10^{5}\) provides estimation accuracy on meso-scale random variables better than 0.02%. The derived mean values, coefficient of variation (CoV) and linear correlation coefficients of UD E-glass/epoxy are shown in Table 2. The probabilistic distribution of derived py mechanical properties was determined by Kolmogorov-Smirnov and Anderson-Darling tests, and it was shown that \(E_{11}\), \(E_{22}\), \(\
u_{12}\), \(G_{12}\) and \(\rho\) all follow normal distribution with a confidence level at 95%. Table 2 shows that significant statistical correlation exists between py mechanical properties (\(E_{1}\), \(E_{2}\), \(G_{12}\), \(v_{12}\)), which agrees with observation in [19; 20; 22]. A new observation is that statistical correlation between py density (\(\rho\)) and ply elastic properties (\(E_{1}\), \(E_{2}\), \(G_{12}\)) was also significant, and this correlation seems to be rarely reported or noticed in previous studies. Actually, experimental data in Huang [28] provided a clear positive correlation between \(V_{f}\) (fibre volume ratio) and \(E_{1}\), \(E_{2}\), \(G_{12}\). Since it is known that ply density is positively dependent on the \(V_{f}\), the theoretically predicted positive correlation between ply density and \(E_{1}\), \(E_{2}\), \(G_{12}\) shown Table 2 is reasonable.