4. Numerical results transversely isotropic. The elastic properties of these materials are presented in Table 1. Depending on the constitutive materials 4. 1. The material input nd directions, the composite can be transversely isotropic to com pletely anisotropic. The algorithm will determine the degree of our composite materials chosen consist of four sets of constitu- isotropy by counting the independent number of constants result- tive materials. The composite #1 consists of an epoxy matrix rein- ing from the analysis. It should be noted that the accuracy of the forced with glass fibers, composite #2 is made of ceramic fibers method is dependent on the accuracy of the FEM procedure and embedded in glass matrix, composite #3 contains a ceramic matrix therefore, the solutions are mesh-dependent. However, in the re- reinforced by another ic fiber, and finally composite #4 is sults that are presented here, the solution convergence has been made of an epoxy matrix reinforced with carbon fibers. All the reached with the mesh employed as shown in Fig. 1d materials chosen are isotropic except the carbon fibers which are 4.2 Stress distributions in ruts For each of the composites and at fiber cross angles between 0 Elastic properties of the constituents of the matrix and 90, a periodic unit cell of the assumed fiber packing(shown in Constituent Figs. 1 and 2) was analyzed under six loading conditions using ABAQUS 37. Results are stored for all six load cases at each cross Glass fiber [39 E=72.9 v=022 angle Using the volume averaging utine program interfaced Epoxy matrix [35 E=4.5 v=0.45 with ABAQUS, the volume-averaged responses of the RUCs are stored and used to obtain the resultant composite compliance Textron SCS-6 SiC E=423 and stiffness properties. In Figs. 5 and 6. the cross-sectional view fiber [40] v=0.2 stress distribution(over half of the cell) in the off-axis fibers due to load cases1.and2 for angles of 0°.45°and90° are plotted. iC matrix [411 E=251 In see in Fig. 5 that for load case 1, stresses Sn1 induced in iC fiber [411 he off-axis fibers decreases with increase in fiber angles wherea or axial fibers stresses Snl increase with increase in fiber angle. AS4 Carbon fiber E=2010, E2=E3=13.5. v12=v13=0.22and Opposite behavior is seen in Fig. 6 for load case 2, where S22 in- G12=G13=95,G23=49 v23=025 creases for off-axis fiber with increase in fiber angle and the same Epoxy matrix [35] E=4.5 v=45 decreases for axial fibers with increase in fiber angle Also looking at Fig. 7, it can be observed that for shear cases, the composite Fig. 5. Stress distribution contours for the cross-section of the off-axis fiber inside the ruc under load case 1, for fiber cross angles of o=0. 45. 90 and with the constitutive aterials of the composite#1.(a)φ=0°,(b)=45°(c)φ=90°4. Numerical results 4.1. The material input Four composite materials chosen consist of four sets of constitu￾tive materials. The composite #1 consists of an epoxy matrix rein￾forced with glass fibers, composite #2 is made of ceramic fibers embedded in glass matrix, composite #3 contains a ceramic matrix reinforced by another ceramic fiber, and finally composite #4 is made of an epoxy matrix reinforced with carbon fibers. All the materials chosen are isotropic except the carbon fibers which are transversely isotropic. The elastic properties of these materials are presented in Table 1. Depending on the constitutive materials and directions, the composite can be transversely isotropic to com￾pletely anisotropic. The algorithm will determine the degree of isotropy by counting the independent number of constants result￾ing from the analysis. It should be noted that the accuracy of the method is dependent on the accuracy of the FEM procedure and therefore, the solutions are mesh-dependent. However, in the re￾sults that are presented here, the solution convergence has been reached with the mesh employed as shown in Fig. 1d. 4.2. Stress distributions in RUCs For each of the composites and at fiber cross angles between 0 and 90, a periodic unit cell of the assumed fiber packing (shown in Figs. 1 and 2) was analyzed under six loading conditions using ABAQUS [37]. Results are stored for all six load cases at each cross angle. Using the volume averaging subroutine program interfaced with ABAQUS, the volume-averaged responses of the RUCs are stored and used to obtain the resultant composite compliance and stiffness properties. In Figs. 5 and 6, the cross-sectional view stress distribution (over half of the cell) in the off-axis fibers due to load cases 1, and 2 for angles of 0, 45 and 90 are plotted. One can see in Fig. 5 that for load case 1, stresses S11 induced in the off-axis fibers decreases with increase in fiber angles whereas, for axial fibers stresses S11 increase with increase in fiber angle. Opposite behavior is seen in Fig. 6 for load case 2, where S22 in￾creases for off-axis fiber with increase in fiber angle and the same decreases for axial fibers with increase in fiber angle. Also looking at Fig. 7, it can be observed that for shear cases, the composite Fig. 5. Stress distribution contours for the cross-section of the off-axis fiber inside the RUC under load case 1, for fiber cross angles of u = 0, 45, 90 and with the constitutive materials of the composite #1. (a) u = 0, (b) u = 45, (c) u = 90. Table 1 Elastic properties of the constituents of the matrix Constituent E or G (GPa) v Composite # 1 E-Glass fiber [39] E = 72.9 v = 0.22 Epoxy matrix [35] E = 4.5 v = 0.45 Composite # 2 Textron SCS-6 SiC fiber [40] E = 423 v = 0.15 F-glass matrix [40] E = 59 v = 0.2 Composite # 3 SiC matrix [41] E = 251 v = 0.16 SiC fiber [41] E = 200 v = 0.25 Composite # 4 AS4 Carbon fiber [35] E1 = 201.0, E2 = E3 = 13.5, G12 = G13 = 95, G23 = 4.9 v12 = v13 = 0.22 and v23 = 0.25 Epoxy matrix [35] E = 4.5 v = .45 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1199