The positive components of for an elemental parallelopiped are shown in Figure 4.08-1. It will be proved now that the nine components of stress are not independent.
If we consider that the parallelopiped is of very small dimensions , , and then the only important forces are the surface forces. Furthermore, the variations of each component of the surface stresses with x, y and z over the boundaries of the infinitesimal parallelopiped can be ignored.
Consider the projection of the parallelopiped on the x-y plane, Fig. 4.08-2.
The force couples shown in Fig. 4.08-2 will in general lead to a tendency to rotate the parallelopiped about an axis perpendicular to the x-y plane. If we let r represent the radius of gyration of the parallelopiped about such an axis through the center of the parallelopiped then
where m is the mass of the parallelopiped and is the angular turning in the clockwise sense. Now m is equivalent to so that
But r must be of the order of magnitude of and , therefore if the angular acceleration is to be finite, then in the limit as and approach zero we must have
It can be proved in a similar manner by examining the balance of moments in the y-z and x-z planes that, respectively,
These results can be represented with the concise statement
Any tensor which possesses this property is appropriately called a symmetrical tensor, since in writing out the components in a matrix array, it does not matter whether the subscript i refers to the rows or columns.
If we write the viscous stress tensor in such an array we have
It is evident therefore that there are actually only six different components of the viscous stress tensor, three normal components and three shear components.