The averaging theorem used is due to Slattery (1967) and is based on the idea that volume averages of derivatives are closely related to derivatives of volume averages, but care must be taken to account properly for behavior of the averaged quantities at points or surfaces where abrupt changes occur. In particular, when the quantity to be averaged exists on one side of an interface and does not exist on the other side, an interior interface term will contribute to the volume average of the derivative, but not to the derivative of the volume average.

Suppose that *Q* is a quantity to be averaged; *Q* can be a scalar,
vector, or tensor.
For convenience of the discussion, we will assume that the averaging volume
is a finite sphere centered at position , although other choices are also possible.
We label this volume and
the surface of this volume is . The exterior surface has two parts
, with being the part where the quantity of interest *Q*
vanishes identically and being the part where .In addition to the exterior surface, there is also an interior surface
where *Q* changes abruptly to zero and we label this surface ,for interior.
This interior surface is the bounding surface for the region we will label
, *i.e.*, the region wherein the quantity to be averaged *Q* is nonzero.
With these definitions, it is straightforward to show that

_Qd^3x= __Q Qd^3x = _E_Q B<>n_Q QdS +
_I_Q B<>n_Q QdS,
where *dS* is the infinitesimal of the surface volume element,
and is the unit outward normal vector from the region
containing nonzero *Q*.
The main point of (aveofgrad) is just that is the entire bounding
surface of *Q* in the volume .As an example of the meaning of this result, consider *Q* to be a vector
quantity, take the trace of (aveofgrad), and the result is just a statement of the
well-known divergence theorem for vectors.

The second result is that

_Qd^3x = __Q Qd^3x = _E_Q B<>n_Q QdS. The result (gradofave) follows from the fact that the volumes and contain virtually the same internal surfaces and so these do not contribute to the gradient.

Combining these results finally gives

_E_Q B<>n_Q QdS = _Qd^3x = _Qd^3x - _I_Q B<>n_Q QdS. Then, dividing by the volume contained in gives the averaging theorem:

<Q> = <Q> - 1V_I_Q B<>n_Q QdS.

One further definition is required to understand the notation to be used for the single solid analysis. The average is an average over the whole volume of , while we will also want to consider the partial average ,related to the full volume average by

<Q> = v_Q Q,
where is the volume fraction of in which *Q* is nonzero.

Note that, although we generally neglect to show this dependence, all the
average quantities are in fact functions of the particular choice of
averaging volume . In principle, can be as large as
the sample being studied, or as small as desired. The legitimacy of the
averaging theorem does not depend on the size of the averaging volume.
However, some intermediate choice will generally be made for .Too small of an averaging volume implies rapid fluctuations in the
quantities of interest (like the fluid and solid dilatations), while a
very large averaging volume implies all the coefficients in the equations are
universal constants and therefore prevents us from studying the effect of
local inhomogeneities. Also, if the averaging volume is too large, then oscillatory changes in
particle displacement must average to zero over the averaging volume, which
is clearly an undesirable result when studying wave propagation.
Pride *et al.* (1992) provide further discussion of criteria
for choosing the size of the averaging volume.

11/12/1997