Why are bounds useful?

Very efficient numerical algorithms are currently available for calculating the effective tensors of quite complicated two-dimensional microgeometries. The numerical evaluation of effective tensors for three-dimensional microgeometries is also progressing rapidly. In light of these advances one might ask: Why is there a need for developing bounds on effective tensors? One reason is that they often provide quick and simple estimates for the effective tensors.

Another reason for favoring bounds is that in most experimental situations we do not have a complete knowledge of the composite geometry. Even when an accurate determination of the three-dimensional composite microgeometry is possible, obtaining this information and numerically parameterizing it (which may involve the triangulation of boundaries between phases) can be a very time-consuming process. Cross-sectional photographs give only limited information. For example, in a two-phase microgeometry it can be difficult to judge whether a phase is connected if a cross-sectional photograph shows only islands of that phase surrounded by the second phase. In the three-dimensional microgeometry, does the first phase consist of connected wire-like filaments, or does it consist of isolated elongated inclusions? The answer could have a large influence on one's estimates for, say, the effective conductivity when both phases have widely different conductivities. The problem of reconstructing the three-dimensional microstructure from a cross-sectional photograph is the subject of active research; see Yeong and Torquato (1998) and references therein.