Recall the equation for the balance of energy (with respect to the reference
configuration)
The quantity is the stress power.
The average stress power is defined as
Here is an arbitrary self-equilibrating nominal stress field
that satisfies the balance of momentum (without any body forces or
inertial forces) and is the time rate of change of .
The reference configuration can be arbitrary. Also, the nominal stress
and the rate need not be related.
Note that in that case
We can express the stress power in terms of boundary tractions and
boundary velocities using the relation (see Appendix)
In this case, we have , , , ,
, and . Then
Using the balance of linear momentum (in the absence of body and inertial
forces), we get
Recalling that
we then have
If is a self equilibrating traction applied on the boundary
that leads to the stress field , i.e., ,
then we have
Note that the fields and need not be related and hence the velocities and the tractions are not related.
If the boundary velocity field leads to the rate , using the identity (see Appendix)
we can show that (see Appendix)
Using similar arguments, if we assume that is a deformation that is compatible with an applied boundary displacement ,we can show that
We can arrive at or
if either of
the following conditions is satisfied at the boundary:
- or .
- .
If a linear velocity field is prescribed on the boundary ,
we can express this field as
Now,
Recall that
Therefore,
Hence,
Then,
Hence,
Similarly, if a linear displacement field is prescribed on the
boundary such that
we can show that
This leads to the equality
Recall that, the average Kirchhoff stress is given by
.
Therefore, if a uniform boundary displacement is prescribed, we
have
or,
A uniform boundary traction field in the reference configuration can be represented as
Now,
Since the surface tractions are related to the nominal stress by , we must have
Therefore,
or,
Similarly,
Hence, using the same argument as for the previous case, we have