Take a material volume with no body couples acting on it

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

Take a material volume

V(t)

with no body couples acting on it, so the only sources of torque about its centroid are the surface traction

t_i=\sigma_{ij}n_j

and gravity. By writing the balance of angular momentum

\int_S \epsilon_{ijk}x_j t_k\,\mathrm{d}S+\int_V \epsilon_{ijk}x_j\rho g_k\,\mathrm{d}V=\frac{\mathrm{d}}{\mathrm{d}t}\int_V \epsilon_{ijk}x_j\rho u_k\,\mathrm{d}V,

where

x_i

is measured from the centroid, and using the divergence theorem, Cauchy's equation of motion and the identity of the previous problem, show that

\epsilon_{ijk}\sigma_{kj}=0

for every material volume, however small, and hence that

\sigma_{ij}=\sigma_{ji}

Browse the problem library