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)
![{\displaystyle \int _{\partial {\Omega }}\mathbf {v} \otimes ({\boldsymbol {S}}^{T}\cdot \mathbf {n} )~{\text{dA}}=\int _{\Omega }[{\boldsymbol {\nabla }}\mathbf {v} \cdot {\boldsymbol {S}}+\mathbf {v} \otimes ({\boldsymbol {\nabla }}\bullet {\boldsymbol {S}}^{T})]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c526aaf5e74863755e2cc1709c003642c21894e7)
In this case, we have
,
,
,
,
, and
. Then
![{\displaystyle \int _{\partial {\Omega }}{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}=\int _{\Omega }[{\boldsymbol {\nabla }}_{0}~{\dot {\mathbf {x} }}\cdot {\boldsymbol {P}}+{\dot {\mathbf {x} }}\otimes ({\boldsymbol {\nabla }}_{0}\bullet {\boldsymbol {P}}^{T})]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8c328c31c0f60cc0b4d43fe87d088325563393)
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)
![{\displaystyle {\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e391e8782b16ba0291b257e0fa3f733ec1542284)
Using similar arguments, if we assume that
is a deformation that is compatible with an applied boundary displacement
,we can show that
![{\displaystyle {\begin{aligned}\langle {\boldsymbol {F}}\cdot {\boldsymbol {P}}\rangle -\langle {\boldsymbol {F}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\mathbf {x} -\langle {\boldsymbol {F}}\rangle \cdot \mathbf {X} ]\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\mathbf {x} -\langle {\boldsymbol {F}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\mathbf {x} \otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942ce4eae345ada758edfe58c4fd8186773f0794)
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,
![{\displaystyle {\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-{\dot {\boldsymbol {G}}}\cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}=\mathbf {0} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63f2b49a3c0d48e5469b9ed70a4b990be83557a)
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,
![{\displaystyle \langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/966ca46809534424457f51b9e940b67140551862)
or,

Similarly,

Hence, using the same argument as for the previous case, we have
