# Random Fiber Array Generation Considering Actual Noncircular Fibers with a Particle-Shape Library
## Appendix A Computational Homogenization Schemes  
The stress and strain relationship in a unidirectional lamina is typically expressed as  
\[\sigma_{i}(\mathbf{x})=D_{ij}(\mathbf{x})\varepsilon_{j}(\mathbf{x}),\;(i,\;j =\;1,\;2,\;\ldots,\;6) \tag{1}\]  
where \(\sigma_{i}\), \(\varepsilon_{j}\), and \(D_{ij}\) are stress, strain, and stiffness matrices in the form of the vector notation, respectively, and \(\mathbf{x}\) is the position vector at an arbitrary point in an RVE. Due to the transversely isotropic symmetry of the stiffness \(D_{ij}\), it is well known that there are only five independent elastic moduli: \(E_{1}(=E_{L})\), \(E_{2}=E_{3}(=E_{T})\), \(G_{12}=G_{13}(=G_{L})\), \(G_{23}(=G_{T})\), and \(v_{12}=v_{13}(=v_{L})\). In the case of random fiber arrays, due to an anisotropic ratio, one more parameter, \(v_{23}=v_{32}(=v_{T})\), is needed. To determine the elastic properties of unidirectional composites in a theoretical manner, computational homogenization in relation to the present study is briefly summarized. To predict the equivalent elastic properties of a unidirectional lamina in the computational homogenization scheme, an RVE comprising fibers and a matrix with a predefined fiber-volume ratio is first modeled with a set of FE meshes. Proper periodic boundary conditions in Equation (1) are then applied to satisfy Hill-Mandel or macro-homogeneity condition [33; 34]:  
\[u_{p}^{r+}-u_{p}^{r-}=\overline{\varepsilon}_{pq}\Delta x_{q}^{r},\;(p,\;q,\;r =\;1,\;2,\;3) \tag{2}\]  
where the superscripts \(r+\) and \(r-\) indicate the positive and negative sides of the RVE in the \(r\)-direction, respectively. That is, the displacements \(u_{p}\) for each periodic pair are constrained by Equation (1). In addition, \(\Delta x_{q}^{r}\) indicates the difference between the coordinates \(x_{q}\) for the periodic pair in the \(r\)-direction. Only in the case of \(r=q\), \(\Delta x_{q}^{r}\) corresponds to the length of the RVE with respect to the \(r\)-direction, i.e., \(\Delta x_{1}^{1}=X_{1}\), \(\Delta x_{2}^{2}=X_{2}\), \(\Delta x_{3}^{3}=X_{3}\), respectively; and the others are zero when the periodic boundary surfaces of the RVE are assumed to be perpendicular to the Cartesian coordinates axes. The prescribed strain imposed on the RVE is denoted by \(\overline{\varepsilon}_{pq}\). We consider six independent macroscopic periodic deformations of the RVE, which correspond to six strain components in the form of vector notation, respectively. First, one of the six strain components, corresponding to the prescribed \(\overline{\varepsilon}_{pq}\), is non-zero, while the others are zero. For example, when \(\overline{\varepsilon}_{23}(=\overline{\varepsilon}_{32})\) is prescribed, the corresponding constraints are given as  
\[ \begin{array}{cccc}u_{1}^{1+}-u_{1}^{1-}=0,&u_{2}^{1+}-u_{2}^{1-}=0,&u_{3}^{ 1+}-u_{3}^{1-}=0,\\ u_{2}^{2+}-u_{1}^{2-}=0,&u_{2}^{2+}-u_{2}^{2-}=0,&u_{3}^{2+}-u_{3}^{2-}= \overline{\varepsilon}_{32}X_{2},\\ u_{1}^{3+}-u_{1}^{3-}=0,&u_{2}^{3+}-u_{2}^{3-}=\overline{\varepsilon}_{23}X_ {3},&u_{3}^{3+}-u_{3}^{3-}=0\end{array}  \tag{3}\]  
Then, the FE analysis was conducted imposing the above constraints at all the nodes on boundary surfaces of the RVE. Similarly, the FE analyses for the other five strain components were carried out. For each of the macroscopic periodic deformations, the average stresses can be obtained by  
\[<\sigma_{i}>=\frac{1}{\Omega}\int_{\Omega}\sigma_{i}(\mathbf{x})d\Omega, \hskip 28.452756 pt(i\;=\;1,\;2,\;\ldots,\;6) \tag{4}\]where \(\Omega\) is the RVE volume. The macroscopic strain-stress relationship in the form of the vector notation is given with the compliance matrix \(S_{ij}\), as follows:  
\[<\varepsilon_{i}>=S_{ij}<\sigma_{j}>,\ \ \ \ (i,\ j\ =\ 1,\ 2,\ \ldots,\ 6) \tag{10}\]  
where the macroscopic averaged strain \(<\varepsilon_{i}>\) corresponds to the prescribed strain \(\overline{\varepsilon}_{pq}\). The compliance matrix \(S_{ij}\) can be thus computed from Equations (10) and (11).