N. Abolfathi et aL/ Computational Material Science 43(2008)1193-1206 C1111一C2222 0.12 s2222 S3333 E8n9 006 8 10 Fiber Cross Angle(deg) Fig 8. The variation of the compliance and the stiffness components of the composite #l in longitudinal directions as a function of the fiber cross angle (ol C2323 0.20 8生苏 Fiber Cross Angle(deg) Fig 9. The variation of the compliance and the stiffness components of the composite #1 in shear directions as a function of the fber cross angle (ol composite then becomes completely anisotropic at all other angles. in direction 2(S2222)decreases with an increase in angle, while the The compliance coefficients presented in Table 8 for a cross angle stiffness(C2222)increases. Such changes show the natural behavior of 45 show this. In the Tables 3, 5 and 7 where the data for Youngs of a composite material when it transforms from a unidirectional moduli and Poissons ratios were presented where E11, E22, E33, V12,(0)to a completely bidirectional(90)composite For composite V21, V13 are the elastic constants of the composite, and Em, vm, Ef, Vr #1 the fiber is much stiffer than the matrix and therefore, the stiff are the elastic constants of the matrix and fiber Vf and Vm are the ness contribution in direction 2 should increase with an increase in volume fractions As shown in Figs. 8-11, the variations in the stiff- cross fiber angle. In addition, as the results show the contribution ness and compliance coefficients of composites considered with of stiffness in direction 1(Cull) for unidirectional (0 angle)com cross fiber angles are obviously functions of the constituent mate- posite is higher than in 2-direction(C2222)of 90 bidirectional. This rial properties of the composite as one set of fibers change their is obvious as the percentage of fiber in direction 1, in case of uni- orientations within the nd therefore contribute to the directional composite is 100%, whereas in case of 90 bidirectional l directions. For example, in directo rameters in different composite it is less than 100% Since the fiber will change direction 1(Cu111)for the com- only in the plane of directions 1 and 2, the coefficients in direction posite #1 as plotted in Fig 8 decreases with an increase in fiber 3 should not be affected much by the fiber angles. This has been cross angle, while the compliance(S1111)increases The compliance observed from the presented data in the tables 2, 4 and 6composite then becomes completely anisotropic at all other angles. The compliance coefficients presented in Table 8 for a cross angle of 45 show this. In the Tables 3, 5 and 7 where the data for Young’s moduli and Poisson’s ratios were presented where E11, E22, E33, v12, v21, v13 are the elastic constants of the composite, and Em, vm, Ef, vf are the elastic constants of the matrix and fiber. Vf and Vm are the volume fractions. As shown in Figs. 8–11, the variations in the stiff￾ness and compliance coefficients of composites considered with cross fiber angles are obviously functions of the constituent mate￾rial properties of the composite as one set of fibers change their orientations within the matrix, and therefore contribute to the changes in the stiffness and compliance parameters in different directions. For example, stiffness in direction 1 (C1111) for the com￾posite #1 as plotted in Fig. 8 decreases with an increase in fiber cross angle, while the compliance (S1111) increases. The compliance in direction 2 (S2222) decreases with an increase in angle, while the stiffness (C2222) increases. Such changes show the natural behavior of a composite material when it transforms from a unidirectional (0) to a completely bidirectional (90) composite. For composite #1 the fiber is much stiffer than the matrix and therefore, the stiff￾ness contribution in direction 2 should increase with an increase in cross fiber angle. In addition, as the results show the contribution of stiffness in direction 1 (C1111) for unidirectional (0 angle) com￾posite is higher than in 2-direction (C2222) of 90 bidirectional. This is obvious as the percentage of fiber in direction 1, in case of uni￾directional composite is 100%, whereas in case of 90 bidirectional composite it is less than 100%. Since the fiber will change direction only in the plane of directions 1 and 2, the coefficients in direction 3 should not be affected much by the fiber angles. This has been observed from the presented data in the Tables 2, 4 and 6. 0.00 0.03 0.06 0.09 0.12 0 10 20 30 40 0 15 30 45 60 75 90 Compliance Coefficients (1/GPa) Stiffness Coefficients (GPa) Fiber Cross Angle (deg) C1 111 C2222 C3 333 S1111 S2222 S3333 Fig. 8. The variation of the compliance and the stiffness components of the composite #1 in longitudinal directions as a function of the fiber cross angle (u). 0.00 0.10 0.20 0.30 0.40 0 2 4 6 8 0 15 30 45 60 75 90 Compliance Coefficients (1/GPa) Stiffness Coefficients (GPa) Fiber Cross Angle (deg) C1212 C1313 C2323 S1212 S1313 S2323 Fig. 9. The variation of the compliance and the stiffness components of the composite #1 in shear directions as a function of the fiber cross angle (u). 1202 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206