Stress-Strain Relations for On-Axis and Off-Axis Composite Elements A.1 On-Axis System

Abstract

Consider an element of a unidirectional on-axis composite, i.e., a composite where the principal material axes (1, 2, 3) are aligned with the coordinate system, see Figure A.1. Figure A.1 also shows the definition of stress components associated with the material coordinate system, 1, 2, 3, where the stresses are volume averages over the fiber and matrix domains. The normal stresses are σ 1 , σ 2 , and σ 3 , while the shear stresses are τ 12 , τ 13 , and τ 23. Corresponding normal strains are ε 1 , ε 2 , and ε 3 , and the engineering shear strains are γ 12 , γ 13 , and γ 23 , see Hyer (1998) for additional discussion. In thin, sheet-like structures such as a ply in a laminate, it is common to assume a state of plane stress by setting σ 3 = τ 13 = τ 23 = 0. (A.1) It may be shown that such a state of stress leads to vanishing of the out-of-plane shear strains, i.e. γ 13 = γ 23 = 0. (A.2) The out-of-plane extensional strain, ε 3 , does not vanish but becomes coupled to the in-plane stresses σ 1 and σ 2 and does not remain an independent quantity. The stress-strain relation for plane stress becomes ⎡ ⎢ ⎣ σ 1 σ 2 τ 12 ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ Q 11 Q 12 0 Q 12 Q 22 0 0 0 Q 66