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'yy106 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: [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