Micromechanics of composites/Average stress power in a RVE

Average Stress Power in a RVE

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The equation for the balance of energy is

 

If the absence of heat flux or heat sources in the RVE, the equation reduces to

 

The quantity on the right is the stress power density and is a measure of the internal energy density of the material.

The average stress power in a RVE is defined as

 

Note that the quantities   and   need not be related in the general case.

The average velocity gradient   is defined as

 

To get an expression for the average stress power in terms of the boundary conditions, we use the identity

 

to get

 

Using the balance of linear momentum ( ), we get

 

Using the divergence theorem, we have

 

Now, the surface traction is given by  . Therefore,

 

{\scriptsize } In micromechanics, it is of interest to see how the average stress power of a RVE is related to the product of the average stress   and the average velocity gradient  . While homogenizing a RVE, we would ideally like to have

 

However, this is not true in general. We can show that if the gradient of the velocity is a symmetric tensor (i.e., there is no spin), then (see Appendix for proof)

 

We can arrive at   if either of the following conditions is met on the boundary  :

  1.   ~.
  2.   ~.

Linear boundary velocities

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If the prescribed velocities on   are a linear function of  , then we can write

 

where   is a constant second-order tensor.

From the divergence theorem

 

Therefore,

 

Hence, on the boundary

 

Using the identity (see Appendix)

 

and since   is constant, we get

 

From the divergence theorem,

 

Therefore,

 

Uniform boundary tractions

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If the prescribed tractions on the boundary   are uniform, they can be expressed in terms of a constant symmetric second-order tensor   through the relation

 

The tractions are related to the stresses at the boundary of the RVE by  .

The average stress in the RVE is given by

 

Using the identity   (see Appendix), we have

 

Since   is constant and symmetric, we have

 

Applying the divergence theorem,

 

Therefore,

 

Remark

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Recall that for small deformations, the displacement gradient   can be expressed as

 

For small deformations, the time derivative of   gives us the velocity gradient  , i.e.,

 

If  , we get

 

Hence, for small strains and in the absence of rigid body rotations, the stress power density is given by  . Then the average stress power is defined as

 

and the average strain rate is defined as

 

In terms of the surface tractions and the applied boundary velocities, we have

 

For small strains and no rotation, the stress-power difference relation becomes

 

We can arrive at   if either of the following conditions is met on the boundary  :

  1.   Linear boundary velocity field.
  2.   Uniform boundary tractions.

We can also show in an identical manner that

 

and that, when   is symmetric,

 

In this case, we can arrive at the relation   if either of the following conditions is met at the boundary:

  1.   Linear boundary displacement field.
  2.   Uniform boundary tractions.