Macromechanical analysis of 3-D textile reinforced composites 105 4.5 Displacement field in a thin plate according to YNS theory. 山(x,y,z）=(x,y,z uz(x,y,z)=v(x,y,z) [4.4 u3(x,y,z)=w(x,y,z) The strain vector is represented in expression 4.5: e=[exEYoYx.Yy]= du dv du.dvdu∂wdv,dw Lox'ax'dy ax'oz ax'dy dy [4.5] The following stiffness properties are needed for this theory:Ex,Ey,Gxy, Gxz.Gyz and vxy. First-order shear theory This theory [11]is based on the work from Yang-Norris-Stavsky (YNS), which is a generalization of Mindlin theory to laminated non-isotropic materials.In-plane,bending and shear stresses are accounted for.This theory is applicable to both thin and thick laminated plates,by using an appropriate correction factor. Figure 4.5 represents a plate with constant thickness h and the param- eters needed to define the displacement field.The following equations govern the displacement field by applying YNS theory: u(x,y,z)=uo(x,y,z)+zPy(x,y,z) v(x,y,z)=vo(x,y,z)+zPx(x,y,z) [4.6 w(x,y,z)=wo(x,y,z) where:u,v,w displacement components in the x,y,z directions, uo,vo,wo=mid-plane linear displacements, Yx,Yy angular displacements around the x,y axes. The following stiffness properties are needed:Ex,Ey,Gxy,Gxz,Gyz and Vxy
Macromechanical analysis of 3-D textile reinforced composites 105 [4.4] The strain vector is represented in expression 4.5: [4.5] The following stiffness properties are needed for this theory: EX, EY, GXY, GXZ, GYZ and vXY. First-order shear theory This theory [11] is based on the work from Yang–Norris–Stavsky (YNS), which is a generalization of Mindlin theory to laminated non-isotropic materials. In-plane, bending and shear stresses are accounted for. This theory is applicable to both thin and thick laminated plates, by using an appropriate correction factor. Figure 4.5 represents a plate with constant thickness h and the parameters needed to define the displacement field. The following equations govern the displacement field by applying YNS theory: [4.6] where: u, v, w = displacement components in the x,y,z directions, uO, vO, wO = mid-plane linear displacements, YX, YY = angular displacements around the x,y axes. The following stiffness properties are needed: EX, EY, GXY, GXZ, GYZ and vXY. wxyz w xyz ( ) ,, ,, = O( ) vxyz v xyz z xyz ( ) ,, ,, ,, = O( ) + YX ( ) uxyz u xyz z xyz ( ) ,, ,, ,, = O( ) + YY ( ) e eeg g g ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = [ ] = + È Î Í ˘ ˚ ˙ x y xy xz yz u x v x u y v x u z w x v y w y ,, , , , , + , + , u xyz wxyz 3 ( ) ,, ,, = ( ) u xyz vxyz 2 ( ) ,, ,, = ( ) u xyz uxyz 1 ( ) ,, ,, = ( ) 4.5 Displacement field in a thin plate according to YNS theory. RIC4 7/10/99 7:43 PM Page 105 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
106 3-D textile reinforcements in composite materials Higher-order shear theory According to the first-order shear theories,shear strains are constant through the laminate thickness,and therefore they do not satisfy the equi- librium equation at the top and bottom surfaces,where shear strain must be zero if no external force is applied. For thick laminate plates,an accurate shear strain distribution through the laminate thickness is essential.To satisfy the equilibrium equation above mentioned,a higher-order shear theory must be applied [11,15,16]. In this section,a theory developed by Reddy will be described.In-plane, bending and shear stresses are taken into account,the number of variables being the same as in the first-order shear theories.A parabolic shear strain distribution through the laminate thickness is implemented,the shear strains being zero at both top and bottom surfaces. The displacement field according to Reddy theory is: u(x,y,z)=(x,y)+zΨ，(x,y)+z2ξx(x,y)+zpx(x,y) x,y,z=(x,y)+zΨ(x,y)+z2ξ，(x,y)+zp(x,y) [4.7] 宇 w(x,y,z)=w。(x,y) 2-0 where:uo,vo,wo=linear displacements of a point (x,y)at the laminate mid-plane, x,Yy angular displacements around the x and y axes, ξx,5xPx, Py functions to be determined by applying the condition ont that interlaminar shear stresses must be zero at top and bottom surfaces: ox(x,y,±h/2)=0 [4.8] o(x,y,±h/2)=0 The following stiffness properties are needed:Ex,Ey,Gxy,Gxz,Gyz and vxy. Elasticity theory The elasticity theory [17]is applicable to both isotropic and non-isotropic materials,owing to the fact that all the effects related to the elasticity are taken into account.This theory is very efficient in those analyses where the whole stress tensor must be considered,including the interlaminar normal or peeling stress.The displacement field is shown in Fig.4.6 The strain tensor is given by: E=[ex,Ey,E:,Yo,Yn,Yn] du dv ow du dv du ow dv ow [4.9] Lax'ay'azyox'zox'yy
106 3-D textile reinforcements in composite materials Higher-order shear theory According to the first-order shear theories, shear strains are constant through the laminate thickness, and therefore they do not satisfy the equilibrium equation at the top and bottom surfaces, where shear strain must be zero if no external force is applied. For thick laminate plates, an accurate shear strain distribution through the laminate thickness is essential. To satisfy the equilibrium equation above mentioned, a higher-order shear theory must be applied [11,15,16]. In this section, a theory developed by Reddy will be described. In-plane, bending and shear stresses are taken into account, the number of variables being the same as in the first-order shear theories. A parabolic shear strain distribution through the laminate thickness is implemented, the shear strains being zero at both top and bottom surfaces. The displacement field according to Reddy theory is: [4.7] where: uO, vO, wO = linear displacements of a point (x,y) at the laminate mid-plane, YX, YY = angular displacements around the x and y axes, xX, xY, rX, rY = functions to be determined by applying the condition that interlaminar shear stresses must be zero at top and bottom surfaces: [4.8] The following stiffness properties are needed: EX, EY, GXY, GXZ, GYZ and vXY. Elasticity theory The elasticity theory [17] is applicable to both isotropic and non-isotropic materials, owing to the fact that all the effects related to the elasticity are taken into account. This theory is very efficient in those analyses where the whole stress tensor must be considered, including the interlaminar normal or peeling stress. The displacement field is shown in Fig. 4.6. The strain tensor is given by: [4.9] e eeeg g g ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = [ ] = ++ È Î Í ˘ ˚ ˙ x y z xy xz yz u x v y w z u y v x u z w x v y w y ,,, , , ,, ,+, , syz ( ) xy h , , ± 2 0 = sxz ( ) xy h , , ± 2 0 = wxyz w xy ( ) ,, , = o ( ) vxyz v xy z xy z xy z xy ( ) ,, , , , , = o ( ) + Yxyy ( ) + ( ) + ( ) 2 3 x r uxyz u xy z xy z xy z xy ( ) ,, , , , , = o ( ) + Yyxx ( ) + ( ) + ( ) 2 3 x r RIC4 7/10/99 7:43 PM Page 106 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 107 4.6 Displacement field in a thick plate. Table 4.1.Stiffness properties as a function of the theory used Theory Needed stiffness properties Beams theory Ex Gxy and Vxy Laminated plates theory Ex Ey,Gxy and Vxy Irons's theory Ex Ey,Gxy Gxz Gyz and VxY First-order shear theory Ex Ey,Gxy,Gxz Gyz and Vxy Higher-order shear theory Ex Ey,Gxy,Gxz Gyz and Vxy Elasticity theory Ex Ey,Ez,Gxy,Gxz Gvz,VxY,Vxz and Vyz The following stiffness properties are needed:Ex,Ey,Ez,Gxy,Gxz.Gyz, Vxy,Vxz and vyz. Table 4.1 represents the stiffness properties needed as a function of the theory applied. 4.2.2 Stiffness and strength properties as a function of the 3-D textile preform The 3-D textile preform used as a reinforcement for the composite ma- terial affects the needed stiffness and strength properties for two reasons [18-23: On the one hand,every 3-D textile technology is associated with one or more theories among the ones described in Section 4.2.1.Each of these theories requires a specific list of stiffness properties for an appropriate implementation. On the other hand,every 3-D textile technology requires one or more specific strength criteria and,therefore,a number of strength properties
Macromechanical analysis of 3-D textile reinforced composites 107 The following stiffness properties are needed: EX, EY, EZ, GXY, GXZ, GYZ, vXY, vXZ and vYZ. Table 4.1 represents the stiffness properties needed as a function of the theory applied. 4.2.2 Stiffness and strength properties as a function of the 3-D textile preform The 3-D textile preform used as a reinforcement for the composite material affects the needed stiffness and strength properties for two reasons [18–23]: • On the one hand, every 3-D textile technology is associated with one or more theories among the ones described in Section 4.2.1. Each of these theories requires a specific list of stiffness properties for an appropriate implementation. • On the other hand, every 3-D textile technology requires one or more specific strength criteria and, therefore, a number of strength properties. Table 4.1. Stiffness properties as a function of the theory used Theory Needed stiffness properties Beams theory EX, GXY and vXY Laminated plates theory EX, EY, GXY and vXY Irons’s theory EX, EY, GXY, GXZ, GYZ and vXY First-order shear theory EX, EY, GXY, GXZ, GYZ and vXY Higher-order shear theory EX, EY, GXY, GXZ, GYZ and vXY Elasticity theory EX, EY, EZ, GXY, GXZ, GYZ, nXY, nXZ and vYZ 4.6 Displacement field in a thick plate. RIC4 7/10/99 7:43 PM Page 107 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
108 3-D textile reinforcements in composite materials In the following sections,the most important 3-D textile technologies will be analysed.Special attention will be paid to the strength criterion for each case. Braiding Depending on the type of braiding technology considered(2-D or 3-D), there are several options in terms of type of finite element used and type of theory applied.This issue is analysed in Table 4.2.The most appropriate failure criterion for static analyses of braided preforms is the 3-D Tsai-Wu criterion [1].For dynamic studies,the maximum strain criterion gives very interesting results until the final failure occurs. For those studies where out-of-plane stresses must be considered,the introduction of interaction factors between normal and shear stress com- ponents in the 3-D Tsai-Wu criterion generates more accurate results.In this case,the general quadratic criterion to be applied is governed by the following equations: F0g+Foa=0i,j=1÷6 (criterion 1) where: :E=E=E=0 1 1 %m1 s、1 F8= ZZ [4.10] 1 1 F44=1 1 F5= F66= Fs=F6=Fs6=0F=0.5 VFaFn,ij=1÷6andi≠j When other theories on general elasticity are applied,the stress tensor is considerably reduced: For beam theory,the stress tensor is composed of o,and t,and there- fore the criterion can be simplified to the following expression: Fn02+Futo2+F1Gx+2FuGxtry=1 (criterion 2)[4.11] For the classical laminated plate theory,the stress components are ox, oy and tx,and the failure criterion corresponds to: FuGx+F2Cy2+Futo2+F10x F2Cy +2F120xOy +2F140xtoy 2F240ytry =1 (criterion 3)[4.12]
108 3-D textile reinforcements in composite materials In the following sections, the most important 3-D textile technologies will be analysed. Special attention will be paid to the strength criterion for each case. Braiding Depending on the type of braiding technology considered (2-D or 3-D), there are several options in terms of type of finite element used and type of theory applied. This issue is analysed in Table 4.2. The most appropriate failure criterion for static analyses of braided preforms is the 3-D Tsai–Wu criterion [1]. For dynamic studies, the maximum strain criterion gives very interesting results until the final failure occurs. For those studies where out-of-plane stresses must be considered, the introduction of interaction factors between normal and shear stress components in the 3-D Tsai–Wu criterion generates more accurate results. In this case, the general quadratic criterion to be applied is governed by the following equations: Fijsij + Fisii = 0 i, j = 1 ∏ 6 (criterion 1) where: When other theories on general elasticity are applied, the stress tensor is considerably reduced: • For beam theory, the stress tensor is composed of sx and txy, and therefore the criterion can be simplified to the following expression: F11sx 2 + F44txy2 + F1sx + 2F14sxtxy = 1 (criterion 2) [4.11] • For the classical laminated plate theory, the stress components are sx, sy and txy, and the failure criterion corresponds to: F11sx 2 + F22sy 2 + F44txy2 + F1sx + F2sy + 2F12sxsy + 2F14sxtxy + 2F24sytxy = 1 (criterion 3) [4.12] F F F F FF i j i j 45 46 56 = = = =- = ∏ π 0 05 1 6 ij ii jj . , and F S F S F xy xz yz S 44 2 55 2 66 2 111 === F XX F YY F ZZ 11 22 33 1 11 = ¢ = ¢ = ¢ F X X F Y Y F Z Z 1 2 3 456 FFF 1 1 11 11 = - 0 ¢ = - ¢ = - ¢ === [4.10] RIC4 7/10/99 7:43 PM Page 108 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 4.2. Properties to be applied as a function of the type of braiding technology Type of Type of Theory implemented Requested stiffness Requested strength properties Strength criterion braiding element properties 2-D Beam Unidimensional EX, GXY and vXY X, X¢ and Sxy Criterion 2 2-D Shell Laminated plates theory EX, EY, GXY and vXY X, X¢, Y, Y¢ and SXY Criterion 3 2-D Shell Irons’s theory EX, EY, GXY, GXZ, GYZ and vXY X, X¢, Y, Y¢, Sxy Sxz and Syz Criterion 4 2-D Shell First-order shear theory EX, EY, GXY, GXZ, GYZ and vXY X, X¢, Y, Y¢, Sxy Sxz and Syz Criterion 4 2-D Shell Higher-order shear theory EX, EY, GXY, GXZ, GYZ and vXY X, X¢, Y, Y¢, Sxy Sxz and Syz Criterion 4 2-D Solid Elasticity theory EX, EY, EZ, GXY, GXZ, GYZ, X, X¢, Y, Y¢, Z, Z¢, Sxy Sxz and Syz Criterion 1 vXY, vXZ and vYZ 3-D Beam Unidimensional EX, GXY and vXY X, X¢ and Sxy Criterion 2 3-D Solid Elasticity theory EX, EY, EZ, GXY, GXZ, GYZ, X, X¢, Y, Y¢, Z, Z¢, Sxy Sxz and Syz Criterion 1 vXY, vXZ and vYZ 109 RIC4 7/10/99 7:43 PM Page 109 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