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# INTERIOR BOUNDARY CONDITIONS

The medium can be totally heterogeneous internally on any scale. The shapes of these heterogeneities define material discontinuity surfaces on which we specify interior boundary conditions. On each point on these interior boundaries, the normal stress components and the tangential strain components are continuous. We can define a local transformation of stress and strain and stiffness matrix at such an interior boundary into normal and tangential components. In order to get this decomposition for any arbitrary surface, we rotate the coordinate system in order to align the local surface normal with the z axis of the coordinate frame. The determination of the tangential stress and strain components is then identical to the one given in Schoenberg & Muir (1989) for the layered case.

Using this local mapping we can cast the constitutive relation into a form that indicates the boundary conditions in effect. Thereby we create the hybrid matrix (consisting of stiffnesses and compliances) . The left hand side consists of quantities () which are not preserved when crossing the interior boundary. These quantities however are a linear combination of quantities that are preserved (). We end up with the constitutive relation
 (4)
The quantity will be a function of the position on the discontinuity surface and thus implicitly depend on the normal vector at that point. Thus the mapping of the stiffness at those points into the group matrix as described before, will be a function of spatial coordinates.

The mapping is not defined when the normal onto the surface is not defined. This requires that the discontinuity surfaces cannot have points with infinite curvature. Thus the discontinuity surface has itself to be continuous on the scale where we carry out the mapping and summation. With these restrictions in mind, the static total energy H in the medium is given by
 (5)

Next: EVALUATION OF VOLUME INTEGRALS Up: Karrenbach: Equivalent Medium Previous: EQUIVALENT QUANTITIES
Stanford Exploration Project
1/13/1998