Micromechanics of composites/Average stress power in a RVE with finite strain

Average stress power in a RVE

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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)

 

Remark

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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:

  1.   or  .
  2.  .

Linear boundary velocities/displacements

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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,

 

Uniform boundary tractions

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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