2 Analysis of Composite Materials 2.1 Constitutive Relations Laminated composites are typically constructed from orthotropic plies (laminae)containing unidirectional fibers or woven fabric.Generally,in a macroscopic sense,the lamina is assumed to behave as a homogeneous orthotropic material.The constitutive relation for a linear elastic orthotropic material in the material coordinate system (Figure 2.1)is [1-6] Su S12 S13 0 0 07 01 E2 S S23 0 0 0 02 E3 9 S23 S3 0 0 0 63 0 0 Sa 0 (2.1) 0 0 个9 Y13 0 0 0 0 Ss5 0 0 0 0 0 0 2 where the stress components(o,t)are defined in Figure 2.1 and the S are elements of the compliance matrix.The engineering strain components(Y) are defined as implied in Figure 2.2. In a thin lamina,a state of plane stress is commonly assumed by setting 03=t23=t13=0 (2.2) For Equation(2.1)this assumption leads to E3=S1301+S2302 (2.3a) Y2s=h3=0 (2.3b) Thus,for plane stress the through-the-thickness strain E3 is not an independent quantity and does not need to be included in the constitutive relationship. Equation (2.1)becomes ©2003 by CRC Press LLC
2 Analysis of Composite Materials 2.1 Constitutive Relations Laminated composites are typically constructed from orthotropic plies (laminae) containing unidirectional fibers or woven fabric. Generally, in a macroscopic sense, the lamina is assumed to behave as a homogeneous orthotropic material. The constitutive relation for a linear elastic orthotropic material in the material coordinate system (Figure 2.1) is [1–6] (2.1) where the stress components (σi, τij) are defined in Figure 2.1 and the Sij are elements of the compliance matrix. The engineering strain components (εi , γij) are defined as implied in Figure 2.2. In a thin lamina, a state of plane stress is commonly assumed by setting σ3 = τ23 = τ13 = 0 (2.2) For Equation (2.1) this assumption leads to ε3 = S13σ1 + S23σ2 (2.3a) γ23 = γ13 = 0 (2.3b) Thus, for plane stress the through-the-thickness strain ε3 is not an independent quantity and does not need to be included in the constitutive relationship. Equation (2.1) becomes ε ε ε γ γ γ σ σ σ τ τ τ 1 2 3 23 13 12 11 12 13 12 22 23 13 23 33 44 55 66 1 2 3 23 13 12 000 000 000 000 00 0000 0 00000 = SSS SSS SSS S S S TX001_ch02_Frame Page 11 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
◆3 8..8888。 T23 02 FIGURE 2.1 Definitions of principal material directions for an orthotropic lamina and stress components. E1 Su S12 0 E2 S12 0 62 (2.4 Y12 0 0 The compliance elements Si may be related to the engineering constants (E1,E2,G12V12V2 S11=1/E1,S2=-V12/E1=-V2:/E2 (2.5a) S2=1/E2,S6=1/G12 (2.5b) The engineering constants are average properties of the composite ply.The quantities E and vi2 are the Young's modulus and Poisson's ratio,respectively, corresponding to stress o1(Figure 2.2a) E1=o1/e1 (2.6a) V12=-e2/e (2.6b) E2 and v2 correspond to stress 02(Figure 2.2b) E2=o2/e2 (2.7a V21=-e1/e2 (2.7b) For a unidirectional composite E2 is much less than E,and va is much less than v12.For a balanced fabric composite E=E2 and vi2=V2.The Poisson's ratios vz and vz are not independent ©2003 by CRC Press LLC
(2.4) The compliance elements Sij may be related to the engineering constants (E1, E2, G12, ν12, ν21), S11 = 1/E1, S12 = –ν12/E1 = –ν21/E2 (2.5a) S22 = 1/E2, S66 = 1/G12 (2.5b) The engineering constants are average properties of the composite ply. The quantities E1 and ν12 are the Young’s modulus and Poisson’s ratio, respectively, corresponding to stress σ1 (Figure 2.2a) E1 = σ1/ε1 (2.6a) ν12 = –ε2/ε1 (2.6b) E2 and ν21 correspond to stress σ2 (Figure 2.2b) E2 = σ2/ε2 (2.7a) ν21 = –ε1/ε2 (2.7b) For a unidirectional composite E2 is much less than E1, and ν21 is much less than ν12. For a balanced fabric composite E1 ≈ E2 and ν12 ≈ ν21. The Poisson’s ratios ν12 and ν21 are not independent FIGURE 2.1 Definitions of principal material directions for an orthotropic lamina and stress components. 1 2 12 11 12 12 22 66 S S S S S ε ε γ σ σ τ = 0 0 0 0 1 2 12 TX001_ch02_Frame Page 12 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
3 *2 3D VIEW TOP VIEW () ↑3 2 2 61 3D VIEW 6) TOP VIEW 13 12 2 2 T12 F12 -12 T12 3D VIEW ( TOP VIEW FIGURE 2.2 Illustration of deformations of an orthotropic material due to (a)stress o,(b)stress o2,and(c) stress t2. V21=V12E2/E (2.8) The in-plane shear modulus,G2,is defined as(Figure 2.2c) G12=t12/M12 (2.9) It is often convenient to express stresses as functions of strains.This is accomplished by inversion of Equation(2.4) 1 Qi Q12 0 E1 Q12 Q22 0 (2.10) 0 0 Q66 Y12 ©2003 by CRC Press LLC
ν21 = ν12E2/E1 (2.8) The in-plane shear modulus, G12, is defined as (Figure 2.2c) G12 = τ12/γ12 (2.9) It is often convenient to express stresses as functions of strains. This is accomplished by inversion of Equation (2.4) (2.10) FIGURE 2.2 Illustration of deformations of an orthotropic material due to (a) stress σ1, (b) stress σ2, and (c) stress τ12. 1 2 12 11 12 12 22 66 1 2 12 = Q Q Q Q Q σ σ τ ε ε γ 0 0 0 0 TX001_ch02_Frame Page 13 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
where the reduced stiffnesses,Qcan be expressed in terms of the engineering constants Q1=E1/(1-V12V2) (2.11a) Q12=V12E2/(1-V12V2i)=V2E1/(1-V12V2) (2.11b) Q22=E2/(1-v12V2) (2.11c) Q66=G12 2.11d) 2.1.1 Transformation of Stresses and Strains For a lamina whose principal material axes(1,2)are oriented at an angle,0, with respect to the x,y coordinate system(Figure 2.3),the stresses and strains can be transformed.It may be shown [1-6]that both the stresses and strains transform according to 01 Ox 02 =[T] y (2.12) Ty】 and E2 (2.13) Y2/2 Y/2 z,3 2 8o888 FIGURE 2.3 Positive (counterclockwise)rotation of principal material axes(1,2)from arbitrary x,y axes. ©2003 by CRC Press LLC
where the reduced stiffnesses, Qij, can be expressed in terms of the engineering constants Q11 = E1/(1 – ν12ν21) (2.11a) Q12 = ν12E2/(1 – ν12ν21) = ν21E1/(1 – ν12ν21) (2.11b) Q22 = E2/(1 – ν12ν21) (2.11c) Q66 = G12 (2.11d) 2.1.1 Transformation of Stresses and Strains For a lamina whose principal material axes (1,2) are oriented at an angle, θ, with respect to the x,y coordinate system (Figure 2.3), the stresses and strains can be transformed. It may be shown [1–6] that both the stresses and strains transform according to (2.12) and (2.13) FIGURE 2.3 Positive (counterclockwise) rotation of principal material axes (1,2) from arbitrary x,y axes. 1 2 12 x y xy T σ σ τ σ σ τ = [ ] ε ε γ ε ε γ 1 2 12 / / 2 2 = [ ] T x y xy TX001_ch02_Frame Page 14 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
where the transformation matrix is [1-6] 3 n2 2mn [T]= n2 m2 -2mn (2.14) -mn mn m2-n2 and m cos 0 (2.15a) n=sin 0 (2.15b) From Equations (2.12)and (2.13)it is possible to establish the lamina strain-stress relations in the (x,y)coordinate system [1-6] Ex 516 6 Ey 69 (2.16) 56 Sao The S terms are the transformed compliances defined in Appendix A. Similarly,the lamina stress-strain relations become x Q16 y Qu2 y (2.17)） txy] Q16 Q where the overbars denote transformed reduced stiffness elements,defined in Appendix A. 2.1.2 Hygrothermal Strains If fibrous composite materials are processed at elevated temperatures,ther- mal strains are introduced during cooling to room temperature,leading to residual stresses and dimensional changes.Figure 2.4 illustrates dimensional changes of a composite subjected to a temperature increase of AT from the reference temperature T.Furthermore,polymer matrices are commonly hygroscopic,and absorbing moisture leads to swelling of the material.The analysis of moisture expansion strains in composites is mathematically ©2003 by CRC Press LLC
where the transformation matrix is [1–6] (2.14) and m = cos θ (2.15a) n = sin θ (2.15b) From Equations (2.12) and (2.13) it is possible to establish the lamina strain–stress relations in the (x,y) coordinate system [1–6] (2.16) The Sij terms are the transformed compliances defined in Appendix A. Similarly, the lamina stress–strain relations become (2.17) where the overbars denote transformed reduced stiffness elements, defined in Appendix A. 2.1.2 Hygrothermal Strains If fibrous composite materials are processed at elevated temperatures, thermal strains are introduced during cooling to room temperature, leading to residual stresses and dimensional changes. Figure 2.4 illustrates dimensional changes of a composite subjected to a temperature increase of ∆T from the reference temperature T. Furthermore, polymer matrices are commonly hygroscopic, and absorbing moisture leads to swelling of the material. The analysis of moisture expansion strains in composites is mathematically T = m n mn n m mn mn mn m n [ ] − − − 2 2 2 2 2 2 2 2 x y xy y xy S S S SSS SSS ε ε γ σ σ τ = 11 12 16 12 22 26 16 26 66 x x y xy x y xy Q Q Q QQQ QQQ σ σ τ ε ε γ = 11 12 16 12 22 26 16 26 66 TX001_ch02_Frame Page 15 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC