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 param￾eters 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