4 Macromechanical analysis of 3-D textile reinforced composites A.MIRAVETE,R.CLEMENTE AND L.CASTEJON 4.1 Introduction Both laminated composite materials [1]and 3-D textile reinforced com- posite materials [2]are characterized by being composed of biphasic ma- terials:fibres and matrices [3].The macromechanical analysis of 3-D textile reinforced composite materials is defined by the strain-stress relations,the 2 WV IZOE failure modes and the degradation properties from first failure(FF)to last failure (LF)of the material system.Fibre properties and geometry of the fibres [4,5]inside the matrix are considered in the micromechanical analy- 5 sis,the result being the following macromechanical parameters: pua aic the strain-stress relations; 。the failure modes; 、 degradation properties from first failure to last failure 豆 The 3-D textile reinforced composite material [6]is no longer considered a biphasic material,but as a system with the properties listed above,as a result of the micromechanical study. To carry out the macromechanical analysis of a certain complex structure made of 3-D textile reinforced composite materials,the following informa- tion is required: 1 The definition of the material models,which will govern the behaviour of the material system in terms of stiffness and strength at a macro- mechanical level. 2 The introduction of the stiffness and strength properties necessary for the total implementation of the material models of each of the composite systems manufactured by the textile technologies available nowadays. 3 The definition of the geometry of the structure to be analysed,includ- ing the geometry of the borders between the different substructures, which can be made by different manufacturing textile technologies. 100
4.1 Introduction Both laminated composite materials [1] and 3-D textile reinforced composite materials [2] are characterized by being composed of biphasic materials: fibres and matrices [3]. The macromechanical analysis of 3-D textile reinforced composite materials is defined by the strain–stress relations, the failure modes and the degradation properties from first failure (FF) to last failure (LF) of the material system. Fibre properties and geometry of the fibres [4,5] inside the matrix are considered in the micromechanical analysis, the result being the following macromechanical parameters: • the strain–stress relations; • the failure modes; • degradation properties from first failure to last failure. The 3-D textile reinforced composite material [6] is no longer considered a biphasic material, but as a system with the properties listed above, as a result of the micromechanical study. To carry out the macromechanical analysis of a certain complex structure made of 3-D textile reinforced composite materials, the following information is required: 1 The definition of the material models, which will govern the behaviour of the material system in terms of stiffness and strength at a macromechanical level. 2 The introduction of the stiffness and strength properties necessary for the total implementation of the material models of each of the composite systems manufactured by the textile technologies available nowadays. 3 The definition of the geometry of the structure to be analysed, including the geometry of the borders between the different substructures, which can be made by different manufacturing textile technologies. 4 Macromechanical analysis of 3-D textile reinforced composites A. MIRAVETE, R. CLEMENTE AND L. CASTEJON 100 RIC4 7/10/99 7:43 PM Page 100 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 101 4 The definition of the boundary of loading conditions,which can be con- stant or variable with time,owing to the existence of contact or friction conditions from dynamic loads. 5 The design requirements of the structure,object of study. This information is processed by means of a mathematical model.In most cases,the finite element technique seems to be the most appropriate to solve the numerical problem of obtaining the response of the structure to the loading conditions. Usually,1,3,4 and 5 are known and 2,which refers to the strength and stress properties,must be either obtained from testing or estimated by means of micromechanical studies.Testing is recommended,when possible, since the accuracy of the results is extremely high when a proper statistical analysis is made.However,in those cases when testing cannot be carried out owing to the high complexity of the characterization,as is the case with some through-thickness normal and transverse properties,or for other reasons,the estimation of properties by means of micromechanical analy- ses and the finite element technique or analytical procedures constitutes an alternative method,although it is much less accurate. op 4.2 Determination of the stiffness and 号 strength properties of 3-D textile reinforced composite materials The object of the macromechanical analysis is the mechanical prediction of structures made of 3-D textile reinforced composite materials,under given 8 working conditions.The implementation of appropriate material models, simulating the behaviour of the material system under given working con- ditions,is necessary to carry out this type of analysis. In order to define the material model properly,simulating the behaviour of the material system,the introduction of a number of stiffness and strength properties is required.However,the material model varies as a function of the following issues: type of macromechanical analysis; ·.type of theory; type of 3-D textile technology. Owing to the fact that the textile technologies present differences in terms of the type of construction [7],the stiffness and strength properties will also vary in terms of the textile technology used. The type of macromechanical analysis also affects the material model to be used,and therefore the stiffness and strength properties needed to carry
Macromechanical analysis of 3-D textile reinforced composites 101 4 The definition of the boundary of loading conditions, which can be constant or variable with time, owing to the existence of contact or friction conditions from dynamic loads. 5 The design requirements of the structure, object of study. This information is processed by means of a mathematical model. In most cases, the finite element technique seems to be the most appropriate to solve the numerical problem of obtaining the response of the structure to the loading conditions. Usually, 1, 3, 4 and 5 are known and 2, which refers to the strength and stress properties, must be either obtained from testing or estimated by means of micromechanical studies. Testing is recommended, when possible, since the accuracy of the results is extremely high when a proper statistical analysis is made. However, in those cases when testing cannot be carried out owing to the high complexity of the characterization, as is the case with some through-thickness normal and transverse properties, or for other reasons, the estimation of properties by means of micromechanical analyses and the finite element technique or analytical procedures constitutes an alternative method, although it is much less accurate. 4.2 Determination of the stiffness and strength properties of 3-D textile reinforced composite materials The object of the macromechanical analysis is the mechanical prediction of structures made of 3-D textile reinforced composite materials, under given working conditions. The implementation of appropriate material models, simulating the behaviour of the material system under given working conditions, is necessary to carry out this type of analysis. In order to define the material model properly, simulating the behaviour of the material system, the introduction of a number of stiffness and strength properties is required. However, the material model varies as a function of the following issues: • type of macromechanical analysis; • type of theory; • type of 3-D textile technology. Owing to the fact that the textile technologies present differences in terms of the type of construction [7], the stiffness and strength properties will also vary in terms of the textile technology used. The type of macromechanical analysis also affects the material model to be used, and therefore the stiffness and strength properties needed to carry RIC4 7/10/99 7:43 PM Page 101 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
102 3-D textile reinforcements in composite materials out the analysis.These parameters will be a function of the type of analy- sis;linear/non-linear,static or dynamic [8,9],stress or displacement-based, hygrothermal,buckling,modal,crash analysis [10],etc. Finally,the elastic and strength properties needed also depend on the theory on which the analysis is based. 4.2.1 Theories used for the macromechanical analysis of 3-D textile reinforced composite materials In this section,several theories applicable to the macromechanical analysis of 3-D textile reinforced composite materials are described.The textile technologies studied correspond to those composite materials constituted by preforms generated from 3-D textiles and those joined by means of the stitching technology.The stiffness properties needed are also analysed. The most appropriate theories for analysing every textile technology will poo be selected according to the material typology and the desired degree of accuracy [11]. The classical beam theory The classical beam theory [11,12 is based on the fourth-order differential pua equations used in the Euler-Bernoulli bending theory,the torsion and the axial tension-compression theories.The Euler-Bernoulli theory assumes that the transverse section perpendicular to the beam axis remains plane 豆 and perpendicular to this axis after deformation.The transverse deflection w is governed by a fourth-order differential equation: [4.1 where fx)is the transverse distributed load,Ex is the elasticity modules in the beam axis direction (x),and I(x)is the inertia momentum as a function of the x-direction.A scheme of these variables is represented in Figs.4.1 and 4.2. The following stiffness properties are needed when using the classical beam theory:E Gy and vy.The following parameters must also be imple- mented:I(x),lo(x),A(x),Ac(x),ki(x,y,and k2(x,y,z )where:I(x)is inertia momentum,lo(x)is torsion inertia momentum,A(x)is cross- sectional area,Ac(x)is shear cross-sectional area,ki(x,y,z)is the function dependent on the cross-sectional shape in position x,used to determine the strain component Yo and k2(,y,z)is the function dependent on the cross- sectional shape in position x,used to determine the strain component Y
102 3-D textile reinforcements in composite materials out the analysis. These parameters will be a function of the type of analysis; linear/non-linear, static or dynamic [8,9], stress or displacement-based, hygrothermal, buckling, modal, crash analysis [10], etc. Finally, the elastic and strength properties needed also depend on the theory on which the analysis is based. 4.2.1 Theories used for the macromechanical analysis of 3-D textile reinforced composite materials In this section, several theories applicable to the macromechanical analysis of 3-D textile reinforced composite materials are described. The textile technologies studied correspond to those composite materials constituted by preforms generated from 3-D textiles and those joined by means of the stitching technology. The stiffness properties needed are also analysed. The most appropriate theories for analysing every textile technology will be selected according to the material typology and the desired degree of accuracy [11]. The classical beam theory The classical beam theory [11,12] is based on the fourth-order differential equations used in the Euler–Bernoulli bending theory, the torsion and the axial tension–compression theories. The Euler–Bernoulli theory assumes that the transverse section perpendicular to the beam axis remains plane and perpendicular to this axis after deformation. The transverse deflection w is governed by a fourth-order differential equation: [4.1] where f(x) is the transverse distributed load, Ex is the elasticity modules in the beam axis direction (x), and I(x) is the inertia momentum as a function of the x-direction. A scheme of these variables is represented in Figs. 4.1 and 4.2. The following stiffness properties are needed when using the classical beam theory: Ex, Gxy and nxy. The following parameters must also be implemented: I(x), IO(x), A(x), AC(x), k1(x, y, z ) and k2(x, y, z ), where: I(x) is inertia momentum, IO(x) is torsion inertia momentum, A(x) is crosssectional area, AC(x) is shear cross-sectional area, k1(x, y, z) is the function dependent on the cross-sectional shape in position x, used to determine the strain component gxy and k2(x, y, z) is the function dependent on the crosssectional shape in position x, used to determine the strain component gxz. d d d d for 2 2 2 2 0 x EIx w x x ( ) fx x L Ê Ë Á ˆ ¯ ˜ = ( ) < < RIC4 7/10/99 7:43 PM Page 102 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 103 M 4.1 Scheme of a beam subjected to bending,torsion and tension. (x,y,Z) Z 4.2 Scheme of a general cross-section. WV IZOE Classical laminated plate theory Nonisotropic and lamination aspects of composite materials are introduced 9 by means of the classical laminated plate theory [2,11,13,14].This theory takes into account in-plane and bending stresses.Interlaminar stresses are not considered,and therefore the application field of this theory is limited to thin plates with small displacements subject to uniform loads.Those structures subject to impact,free edge effects,stress concentrations,point loads,mechanical and bonded joint,or thick structures are beyond the scope of the classical laminated plate theory. In terms of order of magnitude,a plate is considered thin when: plate thickness characteristic length <10 [4.2] This theory is based on the following assumptions: linear variations of strains; the perpendicular line to the mid-surface remains perpendicular after deformation;i.e.the strains generated by the shear forces are neglected. The local axes(x,y,z),the mid-plane x-y and the displacements associated with these axes are represented in Fig.4.3.The displacement fields are: ux(x,y)=u(x,y) u,(x,y)=v(x,y) [4.3] u.(x,y)=w(x,y)
Macromechanical analysis of 3-D textile reinforced composites 103 Classical laminated plate theory Nonisotropic and lamination aspects of composite materials are introduced by means of the classical laminated plate theory [2,11,13,14]. This theory takes into account in-plane and bending stresses. Interlaminar stresses are not considered, and therefore the application field of this theory is limited to thin plates with small displacements subject to uniform loads. Those structures subject to impact, free edge effects, stress concentrations, point loads, mechanical and bonded joint, or thick structures are beyond the scope of the classical laminated plate theory. In terms of order of magnitude, a plate is considered thin when: [4.2] This theory is based on the following assumptions: • linear variations of strains; • the perpendicular line to the mid-surface remains perpendicular after deformation; i.e. the strains generated by the shear forces are neglected. The local axes (x, y, z), the mid-plane x–y and the displacements associated with these axes are represented in Fig. 4.3. The displacement fields are: [4.3] u xy wxy z ( ) , , = ( ) u xy vxy y ( ) , , = ( ) u xy uxy x ( ) , , = ( ) plate thickness characteristic length < 10 4.1 Scheme of a beam subjected to bending, torsion and tension. 4.2 Scheme of a general cross-section. RIC4 7/10/99 7:43 PM Page 103 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
104 3-D textile reinforcements in composite materials 27 4.3 Definition of a local axis. z(w) z(w) y(v) wo'ssaudmau'peaypoow/:dny x (u) ZEL'8SI 4.4 Definition of plate displacements. When this theory is applied,the following stiffness properties are needed: Ex,Ey,Gxy and Vxy. Irons theory This theory is based on the following assumptions: The perpendicular line to the mid-surface of the laminated plate remains straight after deformation. The strain energy corresponding to the stresses perpendicular to the mid-surface is neglected. However,the assumption that the perpendicular line to the mid-surface remains perpendicular after deformation is not imposed.Therefore,inter- laminar shear stresses are accounted for in this case.Irons theory consid- ers in-plane and shear stresses for each ply of the laminate. The relationship between stresses and strains proceeds from a three- dimensional approach.The local axis and the definition of the displace- ments of the plate are represented in Fig.4.4.The displacement fields are:
104 3-D textile reinforcements in composite materials When this theory is applied, the following stiffness properties are needed: EX, EY, GXY and nXY. Irons theory This theory is based on the following assumptions: • The perpendicular line to the mid-surface of the laminated plate remains straight after deformation. • The strain energy corresponding to the stresses perpendicular to the mid-surface is neglected. However, the assumption that the perpendicular line to the mid-surface remains perpendicular after deformation is not imposed. Therefore, interlaminar shear stresses are accounted for in this case. Irons theory considers in-plane and shear stresses for each ply of the laminate. The relationship between stresses and strains proceeds from a threedimensional approach. The local axis and the definition of the displacements of the plate are represented in Fig. 4.4. The displacement fields are: 4.3 Definition of a local axis. 4.4 Definition of plate displacements. RIC4 7/10/99 7:43 PM Page 104 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 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