The averaging theorem

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 $\Omega(\vecx)$ and the surface of this volume is $\partial\Omega$. The exterior surface has two parts $\partial\Omega= \partial E_0 + \partial E_Q$, with $\partial E_0$ being the part where the quantity of interest Q vanishes identically and $\partial E_Q$ being the part where $Q \ne 0$.In addition to the exterior surface, there is also an interior surface where Q changes abruptly to zero and we label this surface $\partial I_Q$,for interior. This interior surface is the bounding surface for the region we will label $\Omega_Q$, 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 $\hat{\bf n}_Q$ is the unit outward normal vector from the region containing nonzero Q. The main point of (aveofgrad) is just that $\partial E_Q + \partial I_Q$ is the entire bounding surface of Q in the volume $\Omega$.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 $\Omega(\vecx)$ and $\Omega(\vecx+\delta\vecx)$ 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 $V = \int_\Omega d^3x$ contained in $\Omega$ 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 $\left<Q\right\gt$ is an average over the whole volume of $\Omega$, while we will also want to consider the partial average $\bar{Q}$,related to the full volume average by

<Q> = v_Q Q,   where $\bar{v}_Q$ is the volume fraction of $\Omega$ 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 $\Omega(\vecx)$. In principle, $\Omega(\vecx)$ 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 $\Omega(\vecx)$.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.