fraction. FIGURE 3.8 后 Division of RVE into subregions based on square fiber having and from equation (3.10),the size of the RVE is must then have the dimension area as the round fiber.The equivalent square fiber shown in figure 3.8 for more detailed analysis if we convert to a square fiber having the same array is shown in figure 3.8.The RVE is easily divided into subregions A square array of fibers is shown in figure 3.1,and the RVE for such an derivation is adapted from ref.[12]. equivalent fiber volume array and a method of dividing the RVE into subregions.The following model for transverse and shear properties based on a square fiber-packing volume fraction.Hopkins and Chamis [12]have developed a refined tion allows us to use simple relations among fiber size,spacing,and have any hope of developing simple design equations.Such an assump- the assumption of a regular array is a logical simplification if we are to of the models.Although real composites have random-packing arrays, the assumption of a specific fiber-packing array is one possible refinement that E2 and Gi2 may be more sensitive to fiber-packing geometry.Thus, independent of fiber-packing geometry.By the same reasoning,it appears so favorable,we can conclude that those properties must be essentially any particular fiber-packing geometry.Since the results for E and v12 were of materials approach(fig.3.5),the resulting equations were not tied to Due to the simplified RVE that was used for the elementary mechanics mechanics of materials models. generally poor.We will now discuss several refinements of the elementary Principles of Composite Material Mechanics .4) equations for effective moduili of the three-nhase model isr121 equation for E2 in ref.[12]reduces to equation(3.50).The complete set of composites.When the fiber diameter is equal to the interphase diameter,the interphase regions exist in many metal matrix [12]and polymer matrix [13] which is assumed to be an annular volume surrounding the fiber.Such also includes the effect of a third phase,a fiber/matrix interphase material A similar result may be found for G2.The detailed derivation in ref.[121 gives the final result, Substitution of equation (3.47)and equation(3.48)in equation (3.49)then course,is the rule of mixtures analogous to equation(3.21) order to find the effective transverse modulus of the RVE.The result,of transverse normal stress and the procedure of section 3.2.1 is followed in The parallel combination of subregions A and B is now loaded by a so that equation (3.46)now becomes tion (3.45),it is seen that where the matrix dimension is s=s-s From equation(3.44)and equa- 1 Em s 一当 verse modulus for this subregion,EB,is found to be normal stress.Following the procedure of section 3.2.2,the effective trans- the series arrangement of fiber and matrix in subregion B to a transverse order to find the effective transverse modulus for the RVE,we first subject The RVE is divided into subregions A and B,as shown in figure 3.8.In Effective Moduli of a Continuous Fiber-Reinforced Lamina 8.50 3.8) 8.5 86) 云