110 3-D textile reinforcements in composite materials When Irons's first-or higher-order shear theories are applied,the stress components are o,and ty and thus the failure criterion is rep- resented by the following expression: FuGx2+F2Cy2+Futo2+Fsstx2+F66tyz2 +F10x+F2Oy+2F1200y+2F140xtxy +2F1sOxte +2F160xtyz +2F240ytoy (criterion 4)[4.13] +2F2s0,tx+2F26O,ty:=1 Table 4.2 shows the type of finite element,theory,needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. 3-D weaving The models to be carried out for simulations of 3-D weaving preforms can be constituted by shell or solid finite elements.The most appropriate failure criterion seems to be the 3-D Tsai-Wu criterion,implementing interaction WV IZOE factors between normal and shear stresses.This criterion can be simplified (criteria 2,3 and 4)should theories different from the one based on elas- ticity be applied. Table 4.3 shows the type of finite element,theory,needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. Weft knitting 豆 PPV The analysis for the weft knitting technology coincides with the study for 3-D weaving.The models to be carried out for simulations of the weft knit- ting can be constituted by shell or solid finite elements.The most ap- propriate failure criterion seems to be the 3-D Tsai-Wu criterion, implementing interaction factors between normal and shear stresses.This criterion can be simplified(criteria 2,3 and 4)should theories different from the one based on elasticity be applied. Table 4.4 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. Warp knitting Several options exist in terms of type of finite element and theory to be applied,depending on the kind of warp knitted composite material studied (Tables 4.5 and 4.6).When using plain warp knitting [24-26 the treatment is similar to the one described for weft knitting.However,if a
110 3-D textile reinforcements in composite materials • When Irons’s first- or higher-order shear theories are applied, the stress components are sx, sy, txy, txz and tyz and thus the failure criterion is represented by the following expression: F11sx 2 + F22sy 2 + F44txy2 + F55txz2 + F66tyz2 + F1sx + F2sy + 2F12sxsy + 2F14sxtxy + 2F15sxtxz + 2F16sxtyz + 2F24sytxy + 2F25sytxz + 2F26sytyz = 1 (criterion 4) [4.13] Table 4.2 shows the type of finite element, theory, needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. 3-D weaving The models to be carried out for simulations of 3-D weaving preforms can be constituted by shell or solid finite elements.The most appropriate failure criterion seems to be the 3-D Tsai–Wu criterion, implementing interaction factors between normal and shear stresses. This criterion can be simplified (criteria 2, 3 and 4) should theories different from the one based on elasticity be applied. Table 4.3 shows the type of finite element, theory, needed stiffness and strength properties and failure criterion to be applied depending on the type of braiding technology used. Weft knitting The analysis for the weft knitting technology coincides with the study for 3-D weaving. The models to be carried out for simulations of the weft knitting can be constituted by shell or solid finite elements. The most appropriate failure criterion seems to be the 3-D Tsai–Wu criterion, implementing interaction factors between normal and shear stresses. This criterion can be simplified (criteria 2, 3 and 4) should theories different from the one based on elasticity be applied. Table 4.4 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. Warp knitting Several options exist in terms of type of finite element and theory to be applied, depending on the kind of warp knitted composite material studied (Tables 4.5 and 4.6). When using plain warp knitting [24–26] the treatment is similar to the one described for weft knitting. However, if a RIC4 7/10/99 7:43 PM Page 110 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
Macromechanical analysis of 3-D textile reinforced composites 111 Table 4.3.Properties to be applied as a function of the type of braiding technology Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Laminated Ex,Ey,Gxy and VxY X,X',Y,Y and Sxy Criterion 3 plates theory Shell Irons's Ex,Ey Gxy,Gxz X,X,Y,Y,So Criterion 4 theory Gyz and Vxy S and Svz 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,Z Criterion 1 theory Gvz,Vxy,Vxz and Vyz Z',Sxy Sx and Sv dmau'pea Table 4.4.Properties to be applied as a function of the theory used 0 Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Laminated Ex,Ey Gxy and VxY X,X',Y,Y and Sy Criterion 3 plates theory Shell Irons's Ex Ey,GxY,Gxz X,X,Y,Y,So Criterion 4 theory Gyz and Vxy S and Syz Shell First-order Ex,Ey,GxY,Gxz, X,X',Y,Y',So Criterion 4 shear Gyz and Vxy Sxz and Syz theory Shell Higher-order Ex,Ey GxY,Gxz X,X',Y,Y',So Criterion 4 shear Gyz and Vxy S and Syz theory Solid Elasticity Ex,Ey Ez Gxy,Gxz X,X,Y,Y,Z Criterion 1 theory Gyz,Vxy,Vxz and vyz Z',Sxy S and Sye 3-D warp knitted material is studied,there are a number of possibilities, since the sandwich skins can be modelled as a 2-D warp knitted material but the core can be analysed by means of several methods. The 3-D Tsai-Wu criterion can be applied for analysing 2-D warp knitted materials or the sandwich skins of a 2-D warp knitted system.The interac- tion factors between normal and shear stresses must be implemented to
Macromechanical analysis of 3-D textile reinforced composites 111 Table 4.3. Properties to be applied as a function of the type of braiding technology Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Laminated EX, EY, GXY and vXY X, X¢, Y, Y¢ and Sxy Criterion 3 plates theory 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, Criterion 1 theory GYZ, vXY, vXZ and vYZ Z¢, Sxy Sxz and Syz Table 4.4. Properties to be applied as a function of the theory used Type of Theory Requested Requested Strength element stiffness strength criterion properties properties Shell Laminated EX, EY, GXY and vXY X, X¢, Y, Y¢ and Sxy Criterion 3 plates theory 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, Criterion 1 theory GYZ, vXY, vXZ and vYZ Z¢, Sxy Sxz and Syz 3-D warp knitted material is studied, there are a number of possibilities, since the sandwich skins can be modelled as a 2-D warp knitted material but the core can be analysed by means of several methods. The 3-D Tsai–Wu criterion can be applied for analysing 2-D warp knitted materials or the sandwich skins of a 2-D warp knitted system. The interaction factors between normal and shear stresses must be implemented to RIC4 7/10/99 7:43 PM Page 111 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
112 3-D textile reinforcements in composite materials Table 4.5 Properties to be applied as a function of the type of theory used for the 2-D warp knitting 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.S Criterion 4 theory Gyz and Vxy Sxz and Sv Shell First-order Ex Ey,Gxy,Gxz X,X',Y,Y,So Criterion 4 shear Gyz and Vxy Sz and Sy theory Shell Higher-order Ex,Ev GxY,Gxz X X',Y,Y,Sx Criterion 4 shear Gyz and Vxy Sx and Sy theory Solid Elasticity Ex Ey,Ez Gxy,Gxz, X,X',Y,Y,Z Criterion 1 poo theory Gyz,Vxy,Vxz and Vyz Z',SxY S and Syt 山 obtain more accurate results.If a theory different from the one based on the 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). The sandwich core is constituted by a pile structure and optionally by a foam.The failure criterion number 5 is considered the most appropriate since shear and peeling stresses are accounted for.Since the pile structure may be different in the two principal directions,the core strength will also vary in both directions. The expressions of criterion 5 are represented in expression 4.14.There is no peeling failure if: Sc"≤o:≤San There is no shear failure if: 2 2 Tu T (criterion 5)[4.14] where:S."=core compression strength in the z-direction, S"=core tension strength in the z-direction, S"=core shear strength in the x-z plane, S"=core shear strength in the y-z plane. Tables 4.5 and 4.6 represent the type of finite element,needed stiffness and strength properties and failure criterion to be applied as a function of
112 3-D textile reinforcements in composite materials obtain more accurate results. If a theory different from the one based on the 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). The sandwich core is constituted by a pile structure and optionally by a foam. The failure criterion number 5 is considered the most appropriate since shear and peeling stresses are accounted for. Since the pile structure may be different in the two principal directions, the core strength will also vary in both directions. The expressions of criterion 5 are represented in expression 4.14. There is no peeling failure if: Szcn £ sz £ Szt n There is no shear failure if: (criterion 5) [4.14] where: Szcn = core compression strength in the z-direction, Szt n = core tension strength in the z-direction, Sxzn = core shear strength in the x–z plane, Syzn = core shear strength in the y–z plane. Tables 4.5 and 4.6 represent the type of finite element, needed stiffness and strength properties and failure criterion to be applied as a function of t xz t xzn yz yzn 2 2 2 2 1 S S + £ Table 4.5 Properties to be applied as a function of the type of theory used for the 2-D warp knitting 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, Criterion 1 theory GYZ, vXY, vXZ and vYZ Z¢, SXY Sxz and Syz RIC4 7/10/99 7:43 PM Page 112 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
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Table 4.6. Properties to be applied as a function of the type of theory used for the 3-D warp knitting sandwich technology Type of element Theory Stiffness properties requested1 Strength properties requested1,2,3 Strength criterion Shell including Theory of laminated Skins: EX, EY, GXY and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 3 skins and core plates Core: EXn, EYn, GXYn and vXYn Core: Sxzn and Syzn Core: criterion 5 Shell including Irons’s theory Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 skins and core Core: EXn, EYn, GXYn, GXZn, GYZn and Core: Sxzn and Syzn Core: criterion 5 vXYn Shell including First-order shear theory Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 skins and core Core: EXn, EYn, GXYn, GXZn, GYZn and Core: Sxzn and Syzn Core: criterion 5 vXYn Shell including Higher-order shear theory Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 skins and core Core: EXn, EYn, GXYn, GXZn, GYZn Core: Sxzn and Syzn Core: criterion 5 and vXYn Skins: Shell Skins: theory of laminated Skins: EX, EY, GXY and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 3 Core: Solid plates Core: EXn, EYn, EZn, GXYn, GXZn, Core: Sxzn and Syzn, SzCn and SzTn Core: criterion 5 Core: elasticity theory GYZn, vXYn, vXZn and vYZn Skins: shell Skins: Irons’s theory Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 Core: solid Core: elasticity theory Core: EXn, EYn, EZn, GXYn, GXZn, Core: Sxzn and Syzn, SzCn and SzTn Core: criterion 5 GYZn, vXYn, vXZn and vYZn Skins: shell Skins: First-order shear Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 Core: solid theory Core: EXn, EYn, EZn, GXYn, GXZn, Core: Sxzn and Syzn, SzCn and SzTn Core: criterion 5 Core: Elasticity theory GYZn, vXYn, vXZn and vYZn 113 RIC4 7/10/99 7:43 PM Page 113 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
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Table 4.6. (cont.) Type of element Theory Stiffness properties requested1 Strength properties requested1,2,3 Strength criterion Skins: shell Skins: higher-order shear Skins: EX, EY, GXY, GXZ, GYZ and vXY Skins: X, X¢, Y, Y¢ and Sxy Skins: criterion 4 Core: solid theory Core: EXn, EYn, EZn, GXYn, GXZn, Core: Sxzn and Syzn, SzCn and SzTn Core: criterion 5 Core: elasticity theory GYZn, vXYn, vXZn and vYZn Skins: solid Elasticity theory Skins: EX, EY, EZ, GXY, GXZ, GYZ, Skins: X, X¢, Y, Y¢, Z, Z¢, Sxy Sxz Skins: criterion 1 Core: solid vXY, vXZ and vYZ and Syz Core: criterion 5 Core: EXn, EYn, EZn, GXYn, GXZn, Core: Sxzn and Syzn, SzCn and SzTn GYZn, vXYn, vXZn and vYZn Skins: solid Skins and core: Skins: EX, EY, EZ, GXY, GXZ, GYZ, Skins: X, X¢, Y, Y¢, Z, Z¢, Sxy Sxz Skins: criterion 1 elasticity theory vXY, vXZ and vYZ and Syz Core: solid Piles: unidimensional Core: model of Krieg and Key Core: ST, SC, tmax Core: criterion of Piles: beams Piles: EX p, GXYp and vXYp Piles: smaxp and tmaxp hydrostatic pressure failure Piles: criterion 2 1The superscript(n) indicates that the stiffness and strength properties refer to the sandwich core, and the superscript(p) indicates that the properties refer to the piles. 2ST, SC, tmax are the tension, compression and shear strengths of the core, respectively. 3smaxp and tmaxp are the maximum tension and shear strengths that the pile fibres can bear. 114 RIC4 7/10/99 7:43 PM Page 114 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