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

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Average stress power in a RVE[edit | edit source]

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[edit | edit source]

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[edit | edit source]

If a linear velocity field is prescribed on the boundary , we can express this field as


Recall that





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


Uniform boundary tractions[edit | edit source]

A uniform boundary traction field in the reference configuration can be represented as


Since the surface tractions are related to the nominal stress by , we must have




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