Page 468 - Introduction to Continuum Mechanics

452 integral Formulation of General Principles

Therefore, Eq. (7.9.1) becomes

where V is the axial vector of the antisymmetric part of the stress tensor T. Now the first
4
term in Eq. (7.9.2) vanishes because of Eq. (7.6.5), therefore, t = 0 and the symmetry of the
stress tensor

is obtained.
On the other hand, if we use the Reynold's transport theorem, Eq. (7.4.1), for the left side
of Eq. (7.9.1), we obtain

That is, the total moment about a fixed point due to surface and body forces acting on the
material instantaneously inside a control volume = total rate of change of moment of
momenta inside the control volume + total net rate of outflow of moment of momenta across
the control surface
If the control volume is fixed in a moving frame, then the following terms should be added
to the left side of Eq. (7.9.4)

where to and <» are absolute angular velocity and acceleration of the moving frame (and of the
control volume), the vector x of (dm) is measured from the arbitrary chosen point O in the
control volume, a 0 is the absolute acceleration of point O and v is the velocity of (dm) relative
to the control volume.

Example 7.9.1

Each sprinkler arm in Fig. 7.8 discharges a constant volume of water Q and is free to rotate
around the vertical center axis. Determine its constant speed of rotation.