EN0175-18

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

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Unformatted text preview: EN0175 11 / 07 / 06 Chap. 6 Boundary value problems in linear elasticity Equilibrium equations: i i j ij u f & & = + , ( 1 ) Kinematic equations: ( i j j i ij u u , , 2 1 + = ) ( 2 ) Hookes law: ij kk ij ij E E + = 1 or ij kk ij ij + = 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 , , , + + = ( 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 & & & & = + + + = + + + , , , , , ( 5 ) Equation (5) is called Navier displacement equation. The vector form of this equation is ( ) u f u u & & v v v v = + + + 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, v v + = u where is the dilatational term and v is the distortional term....
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This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.

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

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