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It canbe expressed as½K ¼X Mm¼1½KY þX Nn¼1½KM þX Ll½KMix ð3Þwhere ½KY , ½KM and ½KMix represent the stiffness matrixof yarn element, matrix element and mixed element,respectively. The stiffness matrix of each kind of ele-ments can be written as½Ki ¼ZVi½B T½Di ½B dV¼Z 1 1Z 1 1Z 1 1½B T½Di ½B det½J dndgdfi ¼ Y;M;Mix ð4Þwhere ½Di is the material property matrix and ½B is thestrain displacement matrix. In order to compute the el-ement stiffness matrix, the numerical integration will beused. Guass quadrature formulae are utilized to com-pute all element integrals in this paper. Therefore, Eq.(4) can be expressed as½Ki ¼X 3i¼1X 3j¼1X 3k¼1WiW je Wk ½B T½Di ½B det½J n¼ni ;g¼gj ;f¼fkð5Þwhere ðni; gj; fk Þ are integration points.Fig. 2. Discretization of a unit cell: (a) unit cell and (b) three kindsof subcells. Matrix elements are isotropic and ½DM denotes thematerial property matrix. Although yarn elements con-tain fiber and resin, they are assumed to be homoge-neous and transversely isotropic. The correspondingmaterial property matrix ½DY can be written as½DY ¼½T T½D0Y ½T ð6Þwhere ½D0Y is the stiffness matrix in the material coor-dinate system and ½T is the transformation matrix. Sincemixed elements are nonhomogeneous, the materialproperty matrix ½DMix varies with the coordinates ofGauss intergration points. If the intergration point is inthe yarn volume, the material property matrix ½DY istaken; otherwise the material property matrix ½DM istaken.According to the above strategy, a Fortran computercode, labeled BCAD, is developed for computing thestiffness and stress field of 3D braided composites.3. Results and discussionsTo confirm the validity of the BCAD program, thestiffness properties, stress distribution of undamaged 3Dbraided composites are first calculated. The calculatedstiffness properties are compared with [22]. Then, themechanical properties of damaged 3D braided compos-ites are studied by using averaging method. All theanalyses in the present study are done for 3D braidedcomposites by the 4-step 1 · 1 procedure. In Sections 3.1and 3.2, we use graphite as yarn material and themechanical properties of yarns and resin are given inTable 1.3.1. Effective properties of undamaged 3D braided com-positesFig. 3 shows the variation of the predicted Young’smoduli of undamaged 3D braided composites with thebraid angle. From Fig. 3, it can be seen that Young’smodulus Ez decreases monotonically with increasing ofthe braid angle and has a difference between the presentresult and one given by Chen et al. [22]. However,Young’s moduli Ex and Ey almost uniform and areconsistent with results given by Chen et al. [22].Fig. 4 illustrates the variation of shear moduli withthe braid angle. From Fig. 4, it can be found that theshear modulus Gxy increases monotonically with the in-creasing of the braid angle and Gzx, Gyz first increaseand then decrease. When the braid angle is equal to 45 ,Gzx,Gyz has the maximum values.Fig. 5 shows the variation of Possion’s ratio with thebraid angle. With increasing the braid angle, cyx first10 20 30 40 50 600102030405060708090Tensile modulus (GPa)braid angle ( o ) Ex,Ey Ez Ex,Ey [22] Ez [22]Fig. 3. Influence of braid angle on the Young’s modulus of 3D braidedcomposites.10 20 30 40 50 606810121416shear modulus (GPa)braid angle ( o ) Gzx, Gyz GxyFig. 4. Influence of braid angle on the shear modulus of 3D braidedcomposites. decreases and then increases; cxz and cyz first increase andthen decrease.3.2. Stress field within a unit cellThe stress distribution of the unit cell is very impor-tant because it is used to analyze the failure of 3Dbraided composites. Without loss of generally, a uni-form longitudinal stress (rzUC ¼ 1) is considered whenthe mechanical behavior of materials is analyzed. Fig. 6depicts the variation of the stress rz on the plane z ¼ c4ofa unit cell in the z-direction. It is seen that the stress rz inthe yarn regions is much higher than that in the matrixregion and the largest tensile stress rz in the yarn regionsgets to 239% of the applied tress. This indicates that theyarns in 3D braided composites share most of the tensileload.For the isotropic matrix, it is well known that thedirection of principal stress or strain has no significance.The stress field of matrix can be characterized by the vonMises equivalent stressrVM 2ðrI