N. Abo b ig. rials es the composite ono ur) for te oss-s4stioc the or- xis nibe inside the RUc under load case 2. for nber cross angles of p-0. 45, 90 and with the constitutive attends highest stress values at an angle of 45 with minimum at 0 sumed to be 40%. In addition, to study the impact due to fiber vol- and 90% ume fractions, the analysis are extended to another two values Although the magnitude of the stress is a function of the load- 20%, and 60% The results at a 45 fiber angle for the different fiber ing, but the constitutive parameters are evaluated based on the lin- volume fractions are plotted in Figs. 12a and 12b for composite# ear stress-strain relations, therefore the deformation is linearly dependent on the load. For convenience, the amount of applied 43. Material property change with fiber cross angle loads will be calculated based on an assumption that the ruc will be under a uniform stress distribution of magnitude 1(S11= 1 for In the following, unless otherwise specified, the fiber volume load case 1, S22=1 for load case 2, S33=1 for load case 3, S12=1 fraction(Vf/) is 40% Tables 2 and 3 show the compliance coeffi- for load case 4, S13=1 for load case 4 and S23=1 for load case 6) cients for the composite #1. Although the constituent materials if the ruc is made of a purely uniform elastic material. Therefore, for this case are isotropic, the composite becomes transversely iso- isotropic at all other angles. This conclusion is made foo monoclinic the average stress in each of the contour plots in Figs 5-7 should tropic for fiber cross angle of o orthotropic for 90 an We have calculated the compliance and stiffness coefficients for ber of symmetric planes observed and from the nonzero indepen each case which are tabulated in Tables 2, 4, 6 and 8. In these ta- dent coefficient values evaluated from the methodology as bles, the numerical solutions are compared with the solutions from presented in Table 2. The material properties of this composite in the lamination theory solution procedure developed in previous terms of Youngs modulus and Poisson s ratio are also presented section. In general the agreement is good especially for the cases in Table 3. As expected, the Youngs modulus in direction 1 at 0 when the material approaches orthotropic material symmetry(o is higher than that of 90 fiber cross angle. For the direction 2 and 90). This is expected as the lamination theory will yield more opposite behavior to direction 1 is observed. Similar data and con- accurate solutions as the material symmetry is observed. Also, clusions were made for the composite #2 as only the numerical when the material properties of the two constituents become clo- values for the constitutive material properties have been changed. ser and closer the agreement between the two solution procedures The resulted data for composite #2 are shown in Tables 4 and 5. mproves. Again this is expected from the lamination theory to The stiffness coefficients for the composite #3 are tabulated ld more accurate results for closer constituents'materials In for the three cross angles of 0, 45 and 90 in Table 6. In addition these tables the solutions for the lamination theory at 0 cross an the material properties in terms of Young s modulus and Poissons gle are taken from the micromechanics solutions. This is because ratios are presented in Table 7. Due to close material properties of he needs the homogenized material properties of the layers at a the constituent matrix and fiber and the 40% fiber volume fraction, cross angle (say 0 here)to go ahead with the lamination theory a small change in material properties of the composite due to and the transformation materials rules involved. The microme- change in cross angle is resulted. The material is orthotropic at chanics solutions at 0 cross angle agree with the solutions of the 0 and 90, and remains anisotropic at other angles. For composite #4, the stiffness properties are shown in tables 8 and The compliance and stiffness parameters are also plotted in the carbon fibers are transversely isotropic and the epe Figs. 8-11. The fiber volume fractions for all these figure are as- isotropic, the composite is monolithic isotropic at 0oattends highest stress values at an angle of 45 with minimum at 0 and 90. Although the magnitude of the stress is a function of the load￾ing, but the constitutive parameters are evaluated based on the lin￾ear stress–strain relations, therefore the deformation is linearly dependent on the load. For convenience, the amount of applied loads will be calculated based on an assumption that the RUC will be under a uniform stress distribution of magnitude 1 (S11 = 1 for load case 1, S22 = 1 for load case 2, S33 = 1 for load case 3, S12 = 1 for load case 4, S13 = 1 for load case 4 and S23 = 1 for load case 6) if the RUC is made of a purely uniform elastic material. Therefore, the average stress in each of the contour plots in Figs. 5–7 should be 1. We have calculated the compliance and stiffness coefficients for each case which are tabulated in Tables 2, 4, 6 and 8. In these ta￾bles, the numerical solutions are compared with the solutions from the lamination theory solution procedure developed in previous section. In general the agreement is good especially for the cases when the material approaches orthotropic material symmetry (0 and 90). This is expected as the lamination theory will yield more accurate solutions as the material symmetry is observed. Also, when the material properties of the two constituents become clo￾ser and closer the agreement between the two solution procedures improves. Again this is expected from the lamination theory to yield more accurate results for closer constituents’ materials. In these tables the solutions for the lamination theory at 0 cross an￾gle are taken from the micromechanics solutions. This is because one needs the homogenized material properties of the layers at a cross angle (say 0 here) to go ahead with the lamination theory and the transformation materials rules involved. The microme￾chanics solutions at 0 cross angle agree with the solutions of the mixture rules very well. The compliance and stiffness parameters are also plotted in Figs. 8–11. The fiber volume fractions for all these figure are as￾sumed to be 40%. In addition, to study the impact due to fiber vol￾ume fractions, the analysis are extended to another two values of 20%, and 60%. The results at a 45 fiber angle for the different fiber volume fractions are plotted in Figs. 12a and 12b for composite #1. 4.3. Material property change with fiber cross angle In the following, unless otherwise specified, the fiber volume fraction (Vf/V) is 40%. Tables 2 and 3 show the compliance coeffi- cients for the composite #1. Although the constituent materials for this case are isotropic, the composite becomes transversely iso￾tropic for fiber cross angle of 0, orthotropic for 90 and monoclinic isotropic at all other angles. This conclusion is made from the num￾ber of symmetric planes observed and from the nonzero indepen￾dent coefficient values evaluated from the methodology as presented in Table 2. The material properties of this composite in terms of Young’s modulus and Poisson’s ratio are also presented in Table 3. As expected, the Young’s modulus in direction 1 at 0 is higher than that of 90 fiber cross angle. For the direction 2 opposite behavior to direction 1 is observed. Similar data and con￾clusions were made for the composite #2 as only the numerical values for the constitutive material properties have been changed. The resulted data for composite #2 are shown in Tables 4 and 5. The stiffness coefficients for the composite #3 are tabulated for the three cross angles of 0, 45 and 90 in Table 6. In addition, the material properties in terms of Young’s modulus and Poisson’s ratios are presented in Table 7. Due to close material properties of the constituent matrix and fiber and the 40% fiber volume fraction, a small change in material properties of the composite due to change in cross angle is resulted. The material is orthotropic at 0 and 90, and remains anisotropic at other angles. For composite #4, the stiffness properties are shown in Tables 8 and 9. Although the carbon fibers are transversely isotropic and the epoxy matrix is isotropic, the composite is monolithic isotropic at 0 and 90. The Fig. 6. Stress distribution contours for the cross-section of the off-axis fiber inside the RUC under load case 2, 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. 1200 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206