Macromechanical analysis of 3-D textile reinforced composites 115 Table 4.7.Properties to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Theory of Ex,Ey,Gxy and Vxy X,X',Y,Y and Sxy Criterion 3 laminated plates Shell Irons's Ex,Ey Gxy,Gxz X,X,Y,Y,So Criterion 4 theory Gyz and Vxy S and Sv Shell First-order Ex,Ey.GxY,Gxz X,X',Y,Y,Sx Criterion 4 shear Gyz and Vxy S and Sy theory Shell Higher-order Ex,Ey GxY,Gxz X,X',Y,Y',Sx Criterion 4 shear Gyz and Vxy Sx and Sv theory Solid Elasticity Ex,Ey Ez GxY Gxz X,X',Y,Y,ZZ, Criterion 1 theory Gvz,Vxy,Vxz and Vyz Sxy Sx and Sy the type of theory used for the 2-D and 3-D warp knitting technologies respectively. Multilayer knitting 4/5-D This technology generates textile plane layers,as is the case for 3-D weaving and warp and weft knitted plain techniques.For all these cases,the numeri- cal models to be implemented from multilayer knitting 4/5-D preforms may 8 be constituted by shell or solid elements.There are a number of theories available,with various degrees of accuracy [27]. The 3-D Tsai-Wu criterion can be applied for analysing multilayer knit- ting 4/5-D technologies.The interaction factors between normal and shear stresses must be implemented to obtain more accurate results.If a theory other than the one based on elasticity is applied,the number of stress com- ponents is drastically reduced,and therefore the expression of the failure criterion becomes more simple (criteria 2,3 and 4). Table 4.7 shows the type of finite element,needed stiffness and strength properties and failure criterion to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology. Stitching The stitching technology can be differentiated from the other techniques above described,since this is a joint technology and not a process to obtain
Macromechanical analysis of 3-D textile reinforced composites 115 the type of theory used for the 2-D and 3-D warp knitting technologies respectively. Multilayer knitting 4/5-D This technology generates textile plane layers, as is the case for 3-D weaving and warp and weft knitted plain techniques. For all these cases, the numerical models to be implemented from multilayer knitting 4/5-D preforms may be constituted by shell or solid elements. There are a number of theories available, with various degrees of accuracy [27]. The 3-D Tsai–Wu criterion can be applied for analysing multilayer knitting 4/5-D technologies. The interaction factors between normal and shear stresses must be implemented to obtain more accurate results. If a theory other than the one based on elasticity is applied, the number of stress components is drastically reduced, and therefore the expression of the failure criterion becomes more simple (criteria 2, 3 and 4). Table 4.7 shows the type of finite element, needed stiffness and strength properties and failure criterion to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology. Stitching The stitching technology can be differentiated from the other techniques above described, since this is a joint technology and not a process to obtain Table 4.7. Properties to be applied as a function of the type of theory used for the multilayer knitting 4/5-D technology Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Theory of EX, EY, GXY and vXY X, X¢, Y, Y¢ and Sxy Criterion 3 laminated plates Shell Irons’s EX, EY, GXY, GXZ, X, X¢, Y, Y¢, Sxy Criterion 4 theory GYZ and vXY Sxz and Syz Shell First-order EX, EY, GXY, GXZ, X, X¢, Y, Y¢, Sxy Criterion 4 shear GYZ and vXY Sxz and Syz theory Shell Higher-order EX, EY, GXY, GXZ, X, X¢, Y, Y¢, Sxy Criterion 4 shear GYZ and vXY Sxz and Syz theory Solid Elasticity EX, EY, EZ, GXY, GXZ, X, X¢, Y, Y¢, Z, Z¢, Criterion 1 theory GYZ, vXY, vXZ and vYZ Sxy Sxz and Syz RIC4 7/10/99 7:43 PM Page 115 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:21 AM IP Address: 158.132.122.9
116 3-D textile reinforcements in composite materials textile preforms.However,this technique is essential for joining,from a highly structural point of view,sub-structures manufactured by different preform technologies.Therefore,the analysis of the stitching technique is critical when a macromechanical analysis of a complete structure composed of a number of preformed substructures is carried out [28]. Since the joint area between two preforms by means of a stitching tech- nology is constituted by stitched fibres,this material system can be consid- ered unidirectional.For this case,the Hashin criterion seems to be the most appropriate (criteria 6 and 7): Fibre failure:the Hashin criterion takes into account the interaction between compression and shear.The failure of the fibre occurs when one of the following conditions is met: Ox=Oata for 01>0 t12+T13 ，2 =1 for o1<0 (criterion 6)[4.15] T12sa Matrix failure:by means of this criterion,the matrix failure occurs when the stresses exceed an interactive combination of normal and maximum shear stresses (criterion 7): t232+t A 2 =1 for on<0 Tsa T2+t1 =1 for o <0 (criterion 7)[4.16] where:oia=allowable compression stress in i-direction, Onta allowable tension stress in i-direction, =allowable shear stress in the 12-plane, ta=allowable shear stress in the plane perpendicular to the fibres. In order to assess the stress components in the joint area,it is not neces- sary to build a finite element mesh of this area,but the stress values in the joint surfaces must be obtained from the finite element mesh of the pre- formed substructures. 4.3 Determination of the stiffness and strength properties of braided composite materials An analytical model for the prediction of stiffness and strength properties of 2-D triaxial braided composite materials [29]is presented in this section
116 3-D textile reinforcements in composite materials textile preforms. However, this technique is essential for joining, from a highly structural point of view, sub-structures manufactured by different preform technologies. Therefore, the analysis of the stitching technique is critical when a macromechanical analysis of a complete structure composed of a number of preformed substructures is carried out [28]. Since the joint area between two preforms by means of a stitching technology is constituted by stitched fibres, this material system can be considered unidirectional. For this case, the Hashin criterion seems to be the most appropriate (criteria 6 and 7): • Fibre failure: the Hashin criterion takes into account the interaction between compression and shear. The failure of the fibre occurs when one of the following conditions is met: sx = sita for s1 > 0 (criterion 6) [4.15] • Matrix failure: by means of this criterion, the matrix failure occurs when the stresses exceed an interactive combination of normal and maximum shear stresses (criterion 7): (criterion 7) [4.16] where: sica = allowable compression stress in i-direction, sita = allowable tension stress in i-direction, t12sa = allowable shear stress in the 12-plane, tsa = allowable shear stress in the plane perpendicular to the fibres. In order to assess the stress components in the joint area, it is not necessary to build a finite element mesh of this area, but the stress values in the joint surfaces must be obtained from the finite element mesh of the preformed substructures. 4.3 Determination of the stiffness and strength properties of braided composite materials An analytical model for the prediction of stiffness and strength properties of 2-D triaxial braided composite materials [29] is presented in this section. t t t s 23 2 13 2 2 2 1 0 Ê + Ë Á ˆ ¯ ˜ = < sa for n s s t t t s n n 2 2 23 2 13 2 2 2 1 0 ta sa for Ê Ë ˆ ¯ + Ê + Ë Á ˆ ¯ ˜ = < s s t t t s 1 2 12 2 13 2 12 2 2 1 0 ica sa for 1 Ê Ë ˆ ¯ + Ê + Ë Á ˆ ¯ ˜ = < RIC4 7/10/99 7:43 PM Page 116 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:21 AM IP Address: 158.132.122.9