9 Effective Elastic Constants of a Laminate 9.1 Basic Equations In this chapter,we introduce the concept of effective elastic constants for the lam- inate.These constants are the effective extensional modulus in the x direction Er, the effective extensional modulus in the y direction Ey,the effective Poisson's ratios Dy and Du,and the effective shear modulus in the r-y plane Gry The effective elastic constants are usually defined when considering the inplane loading of symmetric balanced laminates.In the following equations,we consider only symmetric balanced or symmetric cross-ply laminates.We therefore define the following three average laminate stresses [1]: 1 CH/2 二百J-HP (9.1) 1 H/2 y=百J-H2 Oudz (9.2 1 H/2 y=百J-H2 (9.3) where H is the thickness of the laminate.Comparing (9.1),(9.2),and (9.3)with (7.13),we obtain the following relations between the average stresses and the force resultants: -言N (9.4) ,=疗N (9.5) fy=五 (9.6) Solving (9.4),(9.5),and (9.6)for N:,Ny,and Nry,and substituting the results into (8.11)and(8.12)for symmetric balanced laminates,we obtain:
9 Effective Elastic Constants of a Laminate 9.1 Basic Equations In this chapter, we introduce the concept of effective elastic constants for the laminate. These constants are the effective extensional modulus in the x direction E¯x, the effective extensional modulus in the y direction E¯y, the effective Poisson’s ratios ν¯xy and ¯νyx, and the effective shear modulus in the x-y plane G¯xy. The effective elastic constants are usually defined when considering the inplane loading of symmetric balanced laminates. In the following equations, we consider only symmetric balanced or symmetric cross-ply laminates. We therefore define the following three average laminate stresses [1]: σ¯x = 1 H H/2 −H/2 σxdz (9.1) σ¯y = 1 H H/2 −H/2 σydz (9.2) τ¯xy = 1 H H/2 −H/2 τxydz (9.3) where H is the thickness of the laminate. Comparing (9.1), (9.2), and (9.3) with (7.13), we obtain the following relations between the average stresses and the force resultants: σ¯x = 1 H Nx (9.4) σ¯y = 1 H Ny (9.5) τ¯xy = 1 H Nxy (9.6) Solving (9.4), (9.5), and (9.6) for Nx, Ny, and Nxy, and substituting the results into (8.11) and (8.12) for symmetric balanced laminates, we obtain:
170 9 Effective Elastic Constants of a Laminate anH a12H 0 a12H a22H 0 (9.7) 0 0 a66 H The above 3 x 3 matrix is defined as the laminate compliance matrir for sym- metric balanced laminates.Therefore,by analogy with (4.5),we obtain the following effective elastic constants for the laminate: E:= 1 anH (9.8a) Ev= 1 a22H (9.8b) Gav= 1 (9.8c) a66H y=-012 (9.8d) a11 iyx=一 412 (9.8e) a22 It is clear from the above equations that Dry and Dyr are not independent and are related by the following reciprocity relation: 型=华 E Ev (9.9) Finally,we note that the expressions of the effective elastic constants of(9.8) can be re-written in terms of the components Aij of the matrix [A]as shown in Example 9.1. 9.2 MATLAB Functions Used The five MATLAB function used in this chapter to calculate the average laminate elastic constants are: Ebarr(A,H)-This function calculates the average laminate modulus in the r- direction Ez.There are two input arguments to this function-they are the thickness of the laminate H and the 3 x 3 stiffness matrix [A]for balanced symmetric lami- nates.The function returns a scalar quantity which the desired modulus. Ebary(A,H)-This function calculates the average laminate modulus in the y- direction Ey.There are two input arguments to this function-they are the thickness of the laminate H and the 3 x 3 stiffness matrix [A]for balanced symmetric lami- nates.The function returns a scalar quantity which the desired modulus. NUbarry(A,H)-This function calculates the average laminate Poisson's ratio Dy. There are two input arguments to this function-they are the thickness of the laminate H and the 3 x 3 stiffness matrix [A]for balanced symmetric laminates. The function returns a scalar quantity which the desired Poisson's ratio. NUbaryr(A,H)-This function calculates the average laminate Poisson's ratio uz. There are two input arguments to this function-they are the thickness of the
170 9 Effective Elastic Constants of a Laminate ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ a11H a12H 0 a12H a22H 0 0 0 a66H ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ σ¯x σ¯y τ¯xy ⎫ ⎪⎬ ⎪⎭ (9.7) The above 3 × 3 matrix is defined as the laminate compliance matrix for symmetric balanced laminates. Therefore, by analogy with (4.5), we obtain the following effective elastic constants for the laminate: E¯x = 1 a11H (9.8a) E¯y = 1 a22H (9.8b) G¯xy = 1 a66H (9.8c) ν¯xy = −a12 a11 (9.8d) ν¯yx = −a12 a22 (9.8e) It is clear from the above equations that ¯νxy and ¯νyx are not independent and are related by the following reciprocity relation: ν¯xy E¯x = ν¯yx E¯y (9.9) Finally, we note that the expressions of the effective elastic constants of (9.8) can be re-written in terms of the components Aij of the matrix [A] as shown in Example 9.1. 9.2 MATLAB Functions Used The five MATLAB function used in this chapter to calculate the average laminate elastic constants are: Ebarx(A, H) – This function calculates the average laminate modulus in the xdirection E¯x. There are two input arguments to this function – they are the thickness of the laminate H and the 3 × 3 stiffness matrix [A] for balanced symmetric laminates. The function returns a scalar quantity which the desired modulus. Ebary(A, H) – This function calculates the average laminate modulus in the ydirection E¯y. There are two input arguments to this function – they are the thickness of the laminate H and the 3 × 3 stiffness matrix [A] for balanced symmetric laminates. The function returns a scalar quantity which the desired modulus. NUbarxy(A, H) – This function calculates the average laminate Poisson’s ratio ¯νxy. There are two input arguments to this function – they are the thickness of the laminate H and the 3 × 3 stiffness matrix [A] for balanced symmetric laminates. The function returns a scalar quantity which the desired Poisson’s ratio. NUbaryx(A, H) – This function calculates the average laminate Poisson’s ratio ¯νyx. There are two input arguments to this function – they are the thickness of the
9.2 MATLAB Functions Used 171 laminate H and the 3 x 3 stiffness matrix [A]for balanced symmetric laminates. The function returns a scalar quantity which the desired Poisson's ratio. Gbarry(A,H)-This function calculates the average laminate shear modulus G There are two input arguments to this function-they are the thickness of the laminate H and the 3 x 3 stiffness matrix [A]for balanced symmetric laminates. The function returns a scalar quantity which the desired shear modulus. The following is a listing of the MATLAB source code for these functions: function y Ebarx(A,H) %Ebarx This function returns the average laminate modulus in the x-direction.Its input are two arguments: 名 A -3 x 3 stiffness matrix for balanced symmetric laminates. 名 H-thickness of laminate a inv(A) y=1/(H*a(1,1)); function y Ebary(A,H) %Ebary This function returns the average laminate modulus % in the y-direction.Its input are two arguments: % A -3 x 3 stiffness matrix for balanced symmetric % laminates. H-thickness of laminate a inv(A); y=1/(但*a(2,2); function y NUbarxy(A,H) %NUbarxy This function returns the average laminate Poisson's ratio NUxy.Its input are two arguments: % A-3 x 3 stiffness matrix for balanced symmetric % laminates. H-thickness of laminate a inv(A); y=-a(1,2)/a(1,1); function y NUbaryx(A,H) %NUbaryx This function returns the average laminate 名 Poisson's ratio NUyx.Its input are two arguments: % A-3 x 3 stiffness matrix for balanced symmetric laminates. 名 H-thickness of laminate a inv(A); y=-a(1,2)/a(2,2); function y Gbarxy(A,H) %Gbarxy This function returns the average laminate shear 名 modulus.Its input are two arguments: A -3 x 3 stiffness matrix for balanced symmetric
9.2 MATLAB Functions Used 171 laminate H and the 3 × 3 stiffness matrix [A] for balanced symmetric laminates. The function returns a scalar quantity which the desired Poisson’s ratio. Gbarxy(A, H) – This function calculates the average laminate shear modulus G¯xy. There are two input arguments to this function – they are the thickness of the laminate H and the 3 × 3 stiffness matrix [A] for balanced symmetric laminates. The function returns a scalar quantity which the desired shear modulus. The following is a listing of the MATLAB source code for these functions: function y = Ebarx(A,H) %Ebarx This function returns the average laminate modulus % in the x-direction. Its input are two arguments: % A -3x3 stiffness matrix for balanced symmetric % laminates. % H - thickness of laminate a = inv(A); y = 1/(H*a(1,1)); function y = Ebary(A,H) %Ebary This function returns the average laminate modulus % in the y-direction. Its input are two arguments: % A -3x3 stiffness matrix for balanced symmetric % laminates. % H - thickness of laminate a = inv(A); y = 1/(H*a(2,2)); function y = NUbarxy(A,H) %NUbarxy This function returns the average laminate % Poisson’s ratio NUxy. Its input are two arguments: % A -3x3 stiffness matrix for balanced symmetric % laminates. % H - thickness of laminate a = inv(A); y = -a(1,2)/a(1,1); function y = NUbaryx(A,H) %NUbaryx This function returns the average laminate % Poisson’s ratio NUyx. Its input are two arguments: % A -3x3 stiffness matrix for balanced symmetric % laminates. % H - thickness of laminate a = inv(A); y = -a(1,2)/a(2,2); function y = Gbarxy(A,H) %Gbarxy This function returns the average laminate shear % modulus. Its input are two arguments: % A -3x3 stiffness matrix for balanced symmetric
172 9 Effective Elastic Constants of a Laminate laminates. 名 H-thickness of laminate a inv(A); y=1/(日*a(3,3))； Example 9.1 Show that the effective elastic constants for the laminate can be written in terms of the components A;of the [A]matrix as follows: 瓦=AAA22-A经 A22H (9.10a) A11AA22-A2 ,= AuH (9.10b) 0y= A12 (9.10c A22 Dus A12 (9.10d) A11 Gv=Ace H (9.10e) Solution Starting with (8.11)and(8.12)as follows: N: A11 A12 A12 A22 (9.11) 0 A66 take the inverse of(9.11)to obtain: a11 a12 112 a22 (9.12) 0 0 66 N where A22 a11= (9.13a) A11A22-A72 Au 022= A11A22-A2 (9.13b) A12 a12=A11A2-A (9.13c 1 a66=A66 (9.13d)
172 9 Effective Elastic Constants of a Laminate % laminates. % H - thickness of laminate a = inv(A); y = 1/(H*a(3,3)); Example 9.1 Show that the effective elastic constants for the laminate can be written in terms of the components Aij of the [A] matrix as follows: E¯x = A11AA22 − A2 12 A22H (9.10a) E¯y = A11AA22 − A2 12 A11H (9.10b) ν¯xy = A12 A22 (9.10c) ν¯yx = A12 A11 (9.10d) G¯xy = A66 H (9.10e) Solution Starting with (8.11) and (8.12) as follows: ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ A11 A12 0 A12 A22 0 0 0 A66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ (9.11) take the inverse of (9.11) to obtain: ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ a11 a12 0 a12 a22 0 0 0 a66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ (9.12) where a11 = A22 A11A22 − A2 12 (9.13a) a22 = A11 A11A22 − A2 12 (9.13b) a12 = A12 A11A22 − A2 12 (9.13c) a66 = 1 A66 (9.13d)
9.2 MATLAB Functions Used 173 Next,substitute(9.13)into (9.8)to obtain the required expressions as follows: 瓦=A1AA22-A经 A22H (9.14a) E,= A11AA22-A12 (9.14b) Au A12 Dry=A22 (9.14c) Dyt A12 A11 (9.14d) ,= (9.14e) MATLAB Example 9.2 Consider a four-layer [0/90s graphite-reinforced polymer composite laminate with the elastic constants as given in Example 2.2.The laminate has total thickness of 0.800 mm.The four layers are of equal thickness.Use MATLAB to determine the five effective elastic constants for this laminate. Solution This example is solved using MATLAB.First,the reduced stiffness matrix [Q]for a typical layer using the MATLAB function ReducedStiffness as follows: EDU>>Q=ReducedStiffness(155.0,12.10,0.248,4.40) Q= 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000 Next,the transformed reduced stiffness matrix [is calculated for each layer using the MATLAB function Qbar as follows: EDU>>Qbar1 Qbar(Q,0) Obar1 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000
9.2 MATLAB Functions Used 173 Next, substitute (9.13) into (9.8) to obtain the required expressions as follows: E¯x = A11AA22 − A2 12 A22H (9.14a) E¯y = A11AA22 − A2 12 A11H (9.14b) ν¯xy = A12 A22 (9.14c) ν¯yx = A12 A11 (9.14d) G¯xy = A66 H (9.14e) MATLAB Example 9.2 Consider a four-layer [0/90]S graphite-reinforced polymer composite laminate with the elastic constants as given in Example 2.2. The laminate has total thickness of 0.800 mm. The four layers are of equal thickness. Use MATLAB to determine the five effective elastic constants for this laminate. Solution This example is solved using MATLAB. First, the reduced stiffness matrix [Q] for a typical layer using the MATLAB function ReducedStiffness as follows: EDU>> Q = ReducedStiffness(155.0, 12.10, 0.248, 4.40) Q = 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000 Next, the transformed reduced stiffness matrix ! Q¯" is calculated for each layer using the MATLAB function Qbar as follows: EDU>> Qbar1 = Qbar(Q, 0) Qbar1 = 155.7478 3.0153 0 3.0153 12.1584 0 0 0 4.4000