7 Laminate Analysis -Part I 7.1 Basic Equations Fiber-reinforced materials consist usually of multiple layers of material to form a laminate.Each layer is thin and may have a different fiber orientation-see Fig.7.1. Two laminates may have the same number of layers and the same fiber angles but the two laminates may be different because of the arrangement of the layers. In this chapter,we will evaluate the influence of fiber directions,stacking arrange- ments and material properties on laminate and structural response.We will study a simplified theory called classical lamination theory for this purpose(see [1]). Figure 7.2 shows a global Cartesian coordinate system and a general laminate consisting of N layers.The laminate thickness is denoted by HI and the thickness of an individual layer by h.Not all layers necessarily have the same thickness,so the thickness of the kth layer is denoted by hk. The origin of the through-thickness coordinate,designated z,is located at the laminate geometric midplane.The geometric midplane may be within a particular layer or at an interface between layers.We consider the +z axis to be downward and the laminate extends in the z direction from-H/2 to +H/2.We refer to the layer at the most negative location as layer 1,the next layer in as layer 2,the layer at an arbitrary location as layer k,and the layer at the most positive z position as layer N.The locations of the layer interfaces are denoted by a subscripted z;the 0000000000000000 0000000000000000 Fig.7.1.Schematic illustration of a laminate with four layers
7 Laminate Analysis – Part I 7.1 Basic Equations Fiber-reinforced materials consist usually of multiple layers of material to form a laminate. Each layer is thin and may have a different fiber orientation – see Fig. 7.1. Two laminates may have the same number of layers and the same fiber angles but the two laminates may be different because of the arrangement of the layers. In this chapter, we will evaluate the influence of fiber directions, stacking arrangements and material properties on laminate and structural response. We will study a simplified theory called classical lamination theory for this purpose (see [1]). Figure 7.2 shows a global Cartesian coordinate system and a general laminate consisting of N layers. The laminate thickness is denoted by H and the thickness of an individual layer by h. Not all layers necessarily have the same thickness, so the thickness of the kth layer is denoted by hk. The origin of the through-thickness coordinate, designated z, is located at the laminate geometric midplane. The geometric midplane may be within a particular layer or at an interface between layers. We consider the +z axis to be downward and the laminate extends in the z direction from −H/2 to +H/2. We refer to the layer at the most negative location as layer 1, the next layer in as layer 2, the layer at an arbitrary location as layer k, and the layer at the most positive z position as layer N. The locations of the layer interfaces are denoted by a subscripted z; the Fig. 7.1. Schematic illustration of a laminate with four layers
116 7 Laminate Analysis-Part I y N (a) layer N 5 C (b) Fig.7.2.Schematic illustration showing a cross-section and a plan view first layer is bounded by locations zo and z1,the second layer by z1 and z2,the kth layer by zk-1 and zk,and the Nth layer by zN-1 and zN [1]. Let us examine the deformation of an r-z cross-section [1].Figure 7.3 shows in detail the deformation of a cross-section,and in particular the displacements of point P,a point located at an arbitrary distance z below point po,a point on the reference surface,with points P and po being on line AA'.The superscript 0 will be reserved to denote the kinematics of point po on the reference surface.In particular,the horizontal translation of point po in the x direction will be denoted by u.The vertical translation will be denoted by w.The rotation of the reference surface about the y axis at point po is ow/Or.An important part of the Kirchhoff hypothesis is the assumption that line AA'remains perpendicular to the reference surface.Because of this,the rotation of line AA'is the same as the rotation of the reference surface,and thus the rotation of line AA',as viewed in the r-z plane,is 0w/Or.It is assumed that [1]: 0w0 Ox <1 (7.1) By less than unity is meant that sines and tangents of angles of rotation are replaced by the rotations themselves,and cosines of the angles of rotation are replace by 1.With this approximation,then,the rotation of point po causes point P to translate horizontally in the minus z direction by an amount equal to: 8w (7.2)
116 7 Laminate Analysis – Part I Fig. 7.2. Schematic illustration showing a cross-section and a plan view first layer is bounded by locations z0 and z1, the second layer by z1 and z2, the kth layer by zk−1 and zk, and the Nth layer by zN−1 and zN [1]. Let us examine the deformation of an x-z cross-section [1]. Figure 7.3 shows in detail the deformation of a cross-section, and in particular the displacements of point P, a point located at an arbitrary distance z below point P0, a point on the reference surface, with points P and P0 being on line AA . The superscript 0 will be reserved to denote the kinematics of point P0 on the reference surface. In particular, the horizontal translation of point P0 in the x direction will be denoted by u0. The vertical translation will be denoted by w0. The rotation of the reference surface about the y axis at point P0 is ∂w0/∂x. An important part of the Kirchhoff hypothesis is the assumption that line AA remains perpendicular to the reference surface. Because of this, the rotation of line AA is the same as the rotation of the reference surface, and thus the rotation of line AA , as viewed in the x-z plane, is ∂w0/∂x. It is assumed that [1]: ∂w0 ∂x < 1 (7.1) By less than unity is meant that sines and tangents of angles of rotation are replaced by the rotations themselves, and cosines of the angles of rotation are replace by 1. With this approximation, then, the rotation of point P0 causes point P to translate horizontally in the minus x direction by an amount equal to: z = ∂w0 ∂x (7.2)
7.1 Basic Equations 117 e r Fig.7.3.Schematic illustration showing the kinematics of deformation of a laminate Therefore,the horizontal translation of a point P with coordinates (z,y,z)in the direction of the r-axis is then given by: u,,习=ue,0-00e, (7.3) Also,the vertical translation of point P in the direction of the z-axis is given by: w(x,y,z)=w(,y) (7.4) The horizontal translation of point P in the direction of the y-axis is similar to that in the direction of the z-axis and is given by: u(z,y,z)=v°(x,)-z9 0u°(x,y） (7.5) oy Therefore,we now have the following relations: u(x,y,z)=u°(x,)）-z 0w°(x, (7.6a)
7.1 Basic Equations 117 Fig. 7.3. Schematic illustration showing the kinematics of deformation of a laminate Therefore, the horizontal translation of a point P with coordinates (x, y, z) in the direction of the x-axis is then given by: u(x, y, z) = u0 (x, y) − z ∂w0(x, y) ∂x (7.3) Also, the vertical translation of point P in the direction of the z-axis is given by: w(x, y, z) = w0 (x, y) (7.4) The horizontal translation of point P in the direction of the y-axis is similar to that in the direction of the x-axis and is given by: v(x, y, z) = v0 (x, y) − z ∂w0(x, y) ∂y (7.5) Therefore, we now have the following relations: u(x, y, z) = u0 (x, y) − z ∂w0(x, y) ∂x (7.6a)
118 7 Laminate Analysis-Part I v(e,y,z）=v(x,y)-z 0u°(t,y) (7.6b) w(x,y,z)=w°(x,) (7.6c) Next,we investigate the strains that result from the displacements according to the Kirchhoff hypothesis.This can be done by using the strain-displacement relations from the theory of elasticity.Using these relations and(7.6a,b,c),we can compute the strains at any point within the laminate,and by using these laminate strains in the stress-strain relations,we can compute the stresses at any point within the laminate. From the strain-displacement relations and (7.6a),the extensional strain in the z direction,Er,is given by: ez,,动=0u红y,=0ue,边-0uz,边 (7.7) Ox Ox 0x2 Equation (7.7)may be re-written as follows: er(x,,z)=e(x,)+z9(x,) (7.8) where the following notation is used: (红，)=0ue (7.9a) (红，0= 82w°(x,y） (7.9b) 0x2 The quantity is referred to as the extensional strain of the reference surface in the r direction,and is referred to as the curvature of the reference surface in the z direction.The other five strain components are given by: E(x,y,z)三 0u(x,2=8(红，）)+z(x,) (7.10a) 则 e:e,)=0咖=0ug业=0 (7.10b) :l,斯，=加e8+m色 0z -0mE边_0wz边=0 (7.10c) ay oy x:(x,y,2）≡ 0(,y,+0u(x,,】 0x 8z 0m(E,_u(x,型=0 (7.10d) Ox Ox Yxy(x,,2) 0u(c,,+0ue,42=,+z, (7.10e) Ox 8y where the following notation is used: (z,到= 0u(x,) (7.11a) by
118 7 Laminate Analysis – Part I v(x, y, z) = v0 (x, y) − z ∂w0(x, y) ∂y (7.6b) w(x, y, z) = w0 (x, y) (7.6c) Next, we investigate the strains that result from the displacements according to the Kirchhoff hypothesis. This can be done by using the strain-displacement relations from the theory of elasticity. Using these relations and (7.6a,b,c), we can compute the strains at any point within the laminate, and by using these laminate strains in the stress-strain relations, we can compute the stresses at any point within the laminate. From the strain-displacement relations and (7.6a), the extensional strain in the x direction, εx, is given by: εx(x, y, z) ≡ ∂u(x, y, z) ∂x = ∂u0(x, y) ∂x − z ∂2w0(x, y) ∂x2 (7.7) Equation (7.7) may be re-written as follows: εx(x, y, z) = ε 0 x(x, y) + zκ0 x(x, y) (7.8) where the following notation is used: ε 0 x(x, y) = ∂u0(x, y) ∂x (7.9a) κ0 x(x, y) = −∂2w0(x, y) ∂x2 (7.9b) The quantity ε0 x is referred to as the extensional strain of the reference surface in the x direction, and κ0 x is referred to as the curvature of the reference surface in the x direction. The other five strain components are given by: εy(x, y, z) ≡ ∂v(x, y, z) ∂y = ε 0 y(x, y) + zκ0 y(x, y) (7.10a) εz(x, y, z) ≡ ∂w(x, y, z) ∂z = ∂w0(x, y) ∂z = 0 (7.10b) γyz(x, y, z) ≡ ∂w(x, y, z) ∂y + ∂v(x, y, z) ∂z = ∂w0(x, y) ∂y − ∂w0(x, y) ∂y = 0 (7.10c) γxz(x, y, z) ≡ ∂w(x, y, z) ∂x + ∂u(x, y, z) ∂z = ∂w0(x, y) ∂x − ∂w0(x, y) ∂x = 0 (7.10d) γxy(x, y, z) ≡ ∂v(x, y, z) ∂x + ∂u(x, y, z) ∂y = γ0 xy + zκ0 xy (7.10e) where the following notation is used: ε 0 y(x, y) = ∂v0(x, y) ∂y (7.11a)
7.2 MATLAB Functions Used 119 9(x,)=- 82w°(x, 0y2 (7.11b) 0,e,=0m色+ 0u°(x, (7.11c) Ox by k(x,)=-2 0u°(x, 8xoy (7.11d) The quantitiesandare referred to as the reference surface extensional strain in the y direction,the reference surface curvature in the y direction,the reference surface inplane shear strain,and the reference surface twisting curvature, respectively. The second important assumption of classical lamination theory is that each point within the volume of a laminate is in a state of plane stress.Therefore,we can compute the stresses if we know the strains and curvatures of the reference surface. Accordingly,using the strains that result from the Kirchhoff hypothesis,(7.8)and (7.10a,e),we find that the stress-strain relations for a laminate become: 012 Q16 (7.12) Finally the force and moment resultants in the laminate can be computed using the stresses as follows: H/2 Nr= ordz (7.13a) -H/2 H/2 Ny= (7.13b) -H/2 H/2 (7.13c -H/2 H/2 M:= 0x2d2 (7.13d) -H/2 H/2 My= (7.13e) -H/2 H/2 My= Tryzdz (7.13f) -H/2 7.2 MATLAB Functions Used The only MATLAB function used in this chapter to calculate the strains is:
7.2 MATLAB Functions Used 119 κ0 y(x, y) = −∂2w0(x, y) ∂y2 (7.11b) γ0 xy(x, y) = ∂v0(x, y) ∂x + ∂u0(x, y) ∂y (7.11c) κ0 xy(x, y) = −2∂2w0(x, y) ∂x∂y (7.11d) The quantities ε0 y, κ0 y, γ0 xy, and κ0 xy are referred to as the reference surface extensional strain in the y direction, the reference surface curvature in the y direction, the reference surface inplane shear strain, and the reference surface twisting curvature, respectively. The second important assumption of classical lamination theory is that each point within the volume of a laminate is in a state of plane stress. Therefore, we can compute the stresses if we know the strains and curvatures of the reference surface. Accordingly, using the strains that result from the Kirchhoff hypothesis, (7.8) and (7.10a, e), we find that the stress-strain relations for a laminate become: ⎧ ⎨ ⎩ σx σy τxy ⎫ ⎬ ⎭ = ⎡ ⎢ ⎣ Q¯11 Q¯12 Q¯16 Q¯12 Q¯22 Q¯26 Q¯16 Q¯26 Q¯66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x + zκ0 x ε0 y + zκ0 y γ0 xy + zκ0 xy ⎫ ⎪⎬ ⎪⎭ (7.12) Finally the force and moment resultants in the laminate can be computed using the stresses as follows: Nx = H/ 2 −H/2 σxdz (7.13a) Ny = H/ 2 −H/2 σydz (7.13b) Nxy = H/ 2 −H/2 τxydz (7.13c) Mx = H/ 2 −H/2 σxzdz (7.13d) My = H/ 2 −H/2 σyzdz (7.13e) Mxy = H/ 2 −H/2 τxyzdz (7.13f) 7.2 MATLAB Functions Used The only MATLAB function used in this chapter to calculate the strains is: