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 LLC2 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