ElasticityPolars_02_PolarsDifferential

ElasticityPolars_02_PolarsDifferential - Section 4.2 Solid...

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Unformatted text preview: Section 4.2 Solid Mechanics Part II Kelly 60 4.2 Differential Equations in Polar Coordinates Here, the two-dimensional Cartesian relations of Chapter 1 are re-cast in polar coordinates. 4.2.1 Equilibrium equations in Polar Coordinates One way of expressing the equations of equilibrium in polar coordinates is to apply a change of coordinates directly to the 2D Cartesian version, Eqns. 1.1.8, as outlined in the Appendix to this section, 4.2.6. Alternatively, the equations can be derived from first principles by considering an element of material subjected to stresses , rr and r , as shown in Fig. 4.2.1. The dimensions of the element are r in the radial direction, and r (inner surface) and ( ) + r r (outer surface) in the tangential direction. Figure 4.2.1: an element of material Summing the forces in the radial direction leads to ( ) ( ) ( ) 2 cos 2 cos 2 sin 2 sin + + + + + = r r r r r r r r r F r r r rr rr rr r (4.2.1) For a small element, 1 cos , sin and so, dividing through by r , ( ) 2 + + + r rr rr r r r (4.2.2) A similar calculation can be carried out for forces in the tangential direction { Problem 1}. In the limit as , r , one then has the two-dimensional equilibrium equations in polar coordinates: rr r r rr rr + r r + r r r r + + r r Section 4.2 Solid Mechanics Part II Kelly 61 ( ) 2 1 1 1 = + + = + + r r r r r r r r rr r rr Equilibrium Equations (4.2.3) 4.2.2 Strain Displacement Relations and Hookes Law The two-dimensional strain-displacement relations can be derived from first principles by...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.

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ElasticityPolars_02_PolarsDifferential - Section 4.2 Solid...

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