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exam_1_eqn - Important relations and concepts in AE 322 1...

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Important relations and concepts in AE 322 1) Theory of elasticity 1.1) 3D theory Equilibrium equations : , 0 ij j i f σ + = ; σ ij = σ ji Cauchy’s relation : T i = σ ji n j Kinematic : ε ij = 1 2 u i , j + u j ,i ( ) Compatibility : ε ij, kl + ε kl,ij = ε ik, jl + ε jl, ik Constitutive (isotropic) : σ ij = λε mm δ ij + 2 με ij λ = ν E 1 + ν ( ) 1 2 ν ( ) , μ = G = E 2 1 + ν ( ) 1.2) 2D theory (plane stress) Equilibrium : ( ) , 0; , 1,2 ij j i ij ji f i σ σ σ + = = = j Kinematic : ( ) , , 1 2 ij i j j i u u ε = + Compatibility : 2 ε 12,12 = ε 11, 22 + ε 22,11 Constitutive (isotropic) : σ 11 = E 1 ν 2 ε 11 + νε 22 [ ] , σ 22 = E 1 ν 2 ε 22 + νε 11 [ ] , σ 12 = E 1 + ν ε 12 ε 33 = − ν 1 ν ε 11 + ε 22 ( ) , σ 33 = 0 For plane strain : replace E by E / 1 ν 2 ( ) ; ν by ν / 1 ν ( ) ; ε 33 = 0; σ 33 = ν σ 11 + σ 22 ( ) 1
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