EN0175-20

EN0175-20 - EN0175 11 14 06 Linear elasticity solution in...

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Unformatted text preview: EN0175 11 / 14 / 06 Linear elasticity solution in polar coordinates Typical problems: Stress around a circular hole in an elastic solid. σ σ a Boundary conditions: Traction free @ a r = : = + = θ θ σ σ σ e e e r r rr r v v v , i.e. = rr σ , = θ σ r p a b Boundary conditions: p rr − = σ , = θ σ r @ a r = = rr σ , = θ σ r @ b r = Governing equation: 2 2 = ∇ ∇ φ In Cartesian coordinates: 2 2 2 2 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ φ y x r θ r e v θ e v x e v y e v 1 EN0175 11 / 14 / 06 Proposition: use ( ) θ , r instead of , ( ) y x , ( ) θ φ φ , r = θ θ ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∇ r e r e y e x e r y x v v v v ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ∇ ⋅ ∇ = ∇ θ θ θ θ r e r e r e r e r r 1 1 2 v v v v y x r e e e v v v θ θ sin cos + = y x e e e v v v θ θ θ cos sin + − = θ θ θ θ e e e e y x r v v v v = + − == ∂ ∂ cos sin r y x e e e e v v v v − = − − == ∂ ∂ θ θ θ θ sin cos It follows from above that 2 2 2 2 2 2 1 1 θ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ r r r r Governing equation in polar coordinates: ( ) θ φ φ , r = 1 1 2 2 2 2 2 2 2 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ ∇ φ θ φ r r r r Stress components in Cartesian coordinates: 2 2 y xx ∂ ∂ = φ σ , y x xy ∂ ∂ ∂ − = φ σ 2 , 2 2 x yy ∂ ∂ = φ σ φ σ σ 2 ∇ = + yy xx Stress components in polar coordinates: 2 2 r ∂ ∂ = φ σ θθ , ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − = θ φ σ θ r r r 1 , 2 2 2 1 1 θ φ φ σ ∂ ∂ + ∂ ∂ = r r r rr Equilibrium equations in polar coordinates: 1 = + − + ∂ ∂ + ∂ ∂ r rr r rr f r r r θθ θ σ σ θ σ σ 2 1 = +...
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EN0175-20 - EN0175 11 14 06 Linear elasticity solution in...

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