rIIÞ2þðrII rIIIÞ2þðrIII rIÞ2hirð7Þwhere rI, rII and rIII are the principal stresses. Fig. 7shows the equivalent von Mises stress in the compositeunit cell.Subsequently, let us consider a unit shear stress(szx ¼ 1) on the composite unit cell. Fig. 8 depicts thevariation of the stress component szx on the plane z ¼ c4of a unit cell. From this figure, the smaller hear stress szxat each point of the matrix region is predicted. On thecontrary, the larger shear stress szx occurs in the yarnregions. In addition, the equivalent von Mises stress is10 20 30 40 50 600.050.100.150.200.250.30poisson's ratiobraid angle ( o )γyxγyz,  γxzFig. 5. Influence of braid angle on the Poisson’s ratio of 3D braidedcomposites.Fig. 7. Prediction of the equivalent von Mises stress rVM on the planez ¼ c4of a unit cell under tensile stress rzUC ¼ 1. calculated for the unit cell. Fig. 9 shows the locationswhere high equivalent stresses occur. The four peakscorrespond to the yarn areas in braided composites.3.3. Mechanical properties of damaged 3D braidedcompositesIn this section, the BCAD program is applied foranalyzing the mechanical properties of damaged 3Dbraided composites. Fig. 10 shows the finite elementmesh used in this study. The length of crack is a0 and thenumber of total nodes and elements of mesh are 12439and 2400, respectively. Then, the stress and strain dis-tributions under a uniform tensile load along z-directioncan be calculated by using the BCAD program. For theelastic solid, the complementary energy can be written asPcðrijÞ¼ZVWcðrijÞdV ZSuTiu
 

idS ð8Þwhere Wc is the complementary energy density, Ti is theith component of traction vector, u

iis the ith compo-nent of the prescribed displacement vector on Su, and V ,Su denote the volume and the displacement-prescribedsurface part of the elastic solid, respectively. Here,WcðrijÞ¼ 12Sijrirj ð9ÞAccording to Hashin’s homogenization theorem [21]that the effective Young’s modulus of damaged 3Dbraided composites isEeffz¼ r20V02Pcð10Þwhere r0 is the uniform stress applied in the z-directionon the unit cell and V0 is the volume of a unit cell. Theeffective Poisson ratio ceffzx can be determined fromceffzx ¼   exx  ezzð11Þwhere   ezz ¼ r0=Eeffzand   exx is given by  exx ¼ 1V0ZV0exx dV ð12ÞTo illustrate the effect of crack on the mechanicalproperties of 3D braided composites, some sample casesare studied here. Three unidirectional composites sys-tems, i.e. glass/epoxy, graphite/epoxy and ceramic/ce-ramic, are used as yarn materials. The properties of thethree composites, treated as transversely isotropic ma-terials here, are listed in Table 1, in which subscripts  L’and  T’ denote axial and transverse directions, respec-tively and the braid angle is 45 .The numerical values of Eeffzand ceffzx are determinedfrom Eqs. (10) and (11) by using the BCAD program.The results are shown in Figs. 11 and 12. In these figures,Ez and czx denote Young’s modulus and Poisson’s ratioFig. 9. Prediction of the equivalent von Mises stress szx on the planez ¼ c4of a unit cell under shear stress szxUC ¼ 1.1.01.1 (Eeffz/Ez) Graphite/epoxy  Glass/epoxy of undamaged 3D braided composites, respectively.From11 and 12, it can be found that both the normalizedYoung’s modulus Eeffz=Ez and the normalized Poisson’sratio ceffzx =czx decreases monotonically with the growth ofthe crack: the larger a0, the smaller Eeffz=Ez and ceffzx =czxbecome. This has a good agreement with our previouswork [19]. The degree of father reduction in the nor-malized Young’s modulus due to crack depends on theratio ET=EL of yarn. For the ceramic/ceramic braidedcomposites with the larger value of ET=EL, the normal-ized Young’s modulus decreases slower as crack in-creases. In contrast, for the graphite/epoxy braidedcomposites having the smaller value of ET=EL, the nor-malized Young’s modulus decreases faster as crack in-creases. This indicates that the ceramic/ceramic braidedcomposite is safer to use in cracking susceptible envi-ronments.4. ConclusionsA simplified numerical model of 3D braided com-posites is presented. The unit cell is discretized into anumber of rectangular elements and this is different withconventional FEM method. By using this method, themechanical properties and the local stress within 3Dbraided composites are obtained.The present finite element method can account fordifferent 3D braided composites and geometrical di-mensions. Numerical results show the Young’s modulusand Poisson’s ratio of damaged 3D braided compositesdepend on the crack length and the yarn parameterET=EA.AcknowledgementsThis paper is supported by Trans-Century TrainingProgramme Foundation for the Talents by the Ministryof Education and the Foundation of Heilongjiangprovince for Outstanding Young Talents.References[1] Ishikawa T. Antisymmetric elastic properties of composite platesof satin weave cloth. Fiber Sci Technol 1981;15:127–45.[2] Ishikawa T, ChouTW. One dimensional micromechanical anal-ysis of woven hybrid composites. 三维编织复合材料力学性能的有限元分析英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_53607.html