EN0175-20

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

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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. a Boundary conditions: Traction free @ a r = : = + = e e e r r rr r v v v , i.e. = rr , = r p a b Boundary conditions: p rr = , = r @ a r = = rr , = r @ b r = Governing equation: 2 2 = In Cartesian coordinates: 2 2 2 2 2 = + y x r r e v e v x e v y e v 1 EN0175 11 / 14 / 06 Proposition: use ( ) , r instead of , ( ) y x , ( ) , r = + = + = r e r e y e x e r y x v v v v + + = = 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 sin cos + = y x e e e v v v cos sin + = e e e e y x r v v v v = + == cos sin r y x e e e e v v v v = == sin cos It follows from above that 2 2 2 2 2 2 1 1 + + = r r r r Governing equation in polar coordinates: ( ) , r = 1 1 2 2 2 2 2 2 2 2 = + + = r r r r Stress components in Cartesian coordinates: 2 2 y xx = , y x xy = 2 , 2 2 x yy = 2 = + yy xx Stress components in polar coordinates: 2 2 r = , = r r r 1 , 2 2 2 1 1 + = r r r rr Equilibrium equations in polar coordinates: 1 = + + + r rr r rr f r r r 2 1 = +...
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EN0175-20 - EN0175 11 / 14 / 06 Linear elasticity solution...

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