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EN0175-18 - EN0175 11 07 06 Chap 6 Boundary value problems...

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EN0175 11 / 07 / 06 Chap. 6 Boundary value problems in linear elasticity Equilibrium equations: i i j ij u f & & ρ σ = + , (1) Kinematic equations: ( i j j i ij u u , , 2 1 + = ε ) (2) Hooke’s law: ij kk ij ij E E δ σ ν σ ν ε + = 1 or ij kk ij ij δ λε με σ + = 2 (3) For the most general problems of linear elasticity, you have to solve a system of 15 independent equations with 15 unknown variables. There are two kinds of solution techniques: 1. Displacement based solution methods 2. Stress based solution methods 1. Displacement based method (Navier displacement equation) Eliminate strain by inserting (2) into (3): ( ) ij k k i j j i ij u u u δ λ μ σ , , , + + = (4) Eliminate stress by inserting (4) into (1): ( ) ( ) i i ki k jj i i i ij kj k ij j jj i u f u u u f u u u & & & & ρ λ μ μ ρ δ λ μ = + + + = + + + , , , , , (5) Equation (5) is called Navier displacement equation. The vector form of this equation is ( ) u f u u & & v v v v ρ λ μ μ = + ∇∇ + + 2 Similar to the concept of splitting stress/strain into a volumetric part and a deviatoric part, the displacement field can also be decomposed into a dilatational part plus a distortional part, ψ ϕ v v × + = u where ϕ is the dilatational term and ψ v × is the distortional term. The volume change u u V V k k kk v = = = Δ , ε . If ψ v v × = u , 0 = × = ψ u v , which means the distortional part causes no volume change.
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