When two solid constituents are present, the averaging theorem gives some significantly different results that we will present and discuss here. For simplicity, we will limit the discussion to averages of divergence of displacement and of the displacement itself. We make one assumption implicitly here, that the averaging volume is large enough so that statistical differences between the bulk porosity in the volume and the outcrop of porosity at the surface of the averaging volume are negligible.
We restrict discussion to a problem studied previously by Berryman and Milton (1991): two porous constituents in fully welded contact. Welded contact between porous constituents implies that no cracks/fractures can open up between these constitutents due to applied temperature or stress. Welded contact is somewhat easier to analyze than nonwelded or partially welded contact. Our intention is to treat these more general situations in a later publication, but the main ideas will be presented here.
As a means of simplifying the algebra in the following analysis, we introduce in the second subsection two new quantities that we call and , the divergences of which are just the dilatations of the corresponding solid constituents. This step helps to avoid introducing various terms that would ultimately cancel in the final formulas.