Module+41.key_points - ME ME 6201 Introduction to Continuum...

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Module 41: Balance of Angular Momentum ME 6201 Introduction to Continuum Mechanics D.L. McDowell D.L. McDowell Stress Outline of This Module: Balance of Angular Momentum Symmetry of Cauchy Stress Tensor
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Module 41: Balance of Angular Momentum ME 6201 Introduction to Continuum Mechanics D.L. McDowell D.L. McDowell Stress Balance of Angular Momentum (Moment of Linear Momentum, Nonpolar Case) For the instantaneous volume, V , currently occupied by a set of particles under consideration, P * * * * ~ ~ ~ ~ ~ ~ ~ ( ) ( ) ( ) * * * ~ ~ ~ ~ ~ ~ x V dm x bdm x t dS x d F V V S D x V dV x bdV x t dS Dt ρ ρ × × × = × × = × + × ± ²³´³µ ²´µ where the moment is taken with respect to the origin without loss of generality ( wolg) .
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Module 41: Balance of Angular Momentum ME 6201 Introduction to Continuum Mechanics D.L. McDowell ME 6201 D.L. McDowell Stress In the nonpolar case , we neglect the possibility of couple stresses, to be discussed later. Now, ( ) ~ ~ ~ * ~ ~ * * * ~ ~ ~ ~ ~ 0 0 V V V V V V D x V dV Dt D x D dV DV V dV x V x dV Dt Dt Dt ρ ρ ρ ρ ρ = × = × = × + × + ×
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