Tham khảo tài liệu 'the behavior of structures composed of composite materials part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 49 FIGURE . Shear Stresses and Strains In the above 10 is the displacement and ttj . i iLij iOxj. From elementary strength of materials the constant of proportionality between the shear stress O- I and the angle is í I . the shear modulus in the Vj .1 plane. From the theory of elasticity From Equation i a 6Ơ 1 or _ a66a21 _ a21 e12- 2 - 2G21 Hence a66 - G21 G12 2-19 Similarly a44 - G23 and a55 - G13 2-20 50 Thus all components have now been related to mechanical properties and it is seen that to characterize a three dimensional orthotropic body nine physical quantities are needed dial is Ep Ep Gp. G 3 G p Vp Vj . v. p Vp . V-p and . . and Hsing Equation . However because of only six separate tests are needed to obtain the nine physical quantities. The standardized tests used to obtain these anistropic elastic constants are given in ASTM standards and are described in a text by Carlsson and Pipes 5 . For convenience the compliance matrix is given explicitly as Methods to Obtain Composite Elastic Properties from Fiber and Matrix Properties There are several sets of equations for obtaining the composite elastic properties from those of the fiber and matrix materials. These include those of Halpin and Tsai 6 Hashin 7 and Christensen 8 . In 1980 Hahn 9 codified certain results for fibers of circular cross section which are randomly distributed in a plane normal to the unidirectionally oriented fibers. For that case the composite is macroscopically transversely isotropic that is p - i Ji p I Ip . i .p and I. -s where in the parentheses the quantity could be E G or v hence the elastic properties involve only five independent constants namely t jj p I p. . and If. . For several of the elastic constants Hahn states that they all have the same functional form p_ PfVf TlPmVm Vf nVm 51 where for the elastic constant P the I p l m and 11 are given in Table below and where v and m are the volume fractions of the fibers and matrix respectively .
