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MIT2_094S11_lec9 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 9 - u / p formulation Prof. K.J. Bathe MIT OpenCourseWare We want to solve Reading: Sec. 4.4.3 I. Equilibrium τ ij,j + f i B = 0 in Volume (9.1) S f τ ij n j = f i on S f II. Compatibility III. Stress-strain law Use the principle of virtual displacements T C� dV = R (9.2) V We recognize that if ν 0 . 5 V 0 ( V = xx + yy + zz ) (9.3) E κ = 3(1 2 ν ) → ∞ (9.4) p = κ� V must be accurately computed (9.5) Solution τ ij = κ� V δ ij + 2 G� ij (9.6) where 1 i = j δ ij = Kronecker delta = (9.7) 0 i = j Deviatoric strains: V ij = ij 3 δ ij (9.8) τ kk τ ij = ij + 2 G� ij p = 3 (9.9) (9.2) becomes T C dV + V κ� V dV = R (9.10) V V T C dV T V p dV = R (9.11) V V 37
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= MIT 2.094 9. u / p formulation We need another equation because we now have another unknown p .
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