三 FIGURE 3.11 Composite shear stiffness,G12/Gm 的品品 Materials,1,4-17.With Permission from Technomic Publishing Co.) fibers in a square array.(From Adams,D.E.and Doner,D.R.1967.Journal of Composite Normalized composite shear stiffness,G/Gvs.shear modulus ratio,G/G for circular 810 nts in a square array Shear modulus ratio GGm 060100 Filament volume(v Typical results are shown in figure 3.11,where the ratio of the composite problem for shear along y=b yields the associated shear modulus G where tz is the average shear stress along x=a.Asimilar boundary value C71104/元 (3.54),and the effective shear modulus was then determined from substituting these displacements in the finite difference forms of equation w(xy)at each node of the finite difference grid.Stresses were found by difference scheme.The solution yielded the values of the displacements and continuity conditions at the fiber/matrix interface by using a finite 心/g which was solved subject to the displacement boundary conditions 1006001000 4(0.89 55039 7086(0.115 75(004 789(0.012 Principles of Composite Material Mechanics B.5 FIGURE 3.12 Normalized composite transverse stiffness,E2/Em 3 6 的 & [19]and Caruso [23]is another example of a numerical elasticity solution. The previously mentioned finite element analysis of Caruso and Chamis as squares and ellipses in a rectangular array [21,22]. factors were also obtained for a variety of fiber cross-sectional shapes such of complex geometries.For example,stiffness and stress concentration numerical solutions such as finite differences is the capability for analysis calculation of stress concentration factors is possible.One advantage of plete stress and strain distributions in the RVE are generated,and the [21,22].One of the advantages of the elasticity approach is that the com- high fiber volume fractions can significantly increase Gi2 and E2.Unfor- tunately,these same combinations also generate very high stress concen- tration factors at the fiber/matrix interfaces,as shown in the same papers effect for both Gi2 and E2 only becomes significant for fiber volume frac- tions above about 50%,but that combinations of high fiber stiffness and determine the transverse modulus E2 and typical results are shown in figure 3.12.It is seen in figure 3.11 and figure 3.12 that the reinforcement In a separate paper Adams and Doner [22]used a similar approach to modulus ratio G/Gm for various fiber volume fractions. shear modulus to the matrix shear modulus is plotted versus the shear 1,152-164.With permission from Technomic Publishing Co.) Normalized composite transverse stiffness,E/E versus modulus ratio,E/E for circular fibers in a square array.(Adams,D.R.and Doner,D.R.1967.Journal of Composite Materials 6810 Circular filaments In a square array Constituent stiffness ratio EEm Effective Moduli of a Continuous Fiber-Reinforced Lamina 0006001000 #8(6.88- 08(0.80)- 1500.390 700(0118) 75%001 780(04012 [Filament spacing( Filament volume(