Statistical Continuum Theories

Abstract

There are presently a large number of problems in continuum physics that may be treated from a statistical point of view. In this paper we shall discuss what is meant by the statistical solution to a continuum problem. We shall initially use the nature of the stress and strain fields in a heterogeneous material as an example. We shall then emphasize the essential similarity of all statistical continuum theories. The more developed statistical continuum theories like the statistical theory of turbulence and the theory of partial coherence will be compared to the statistical theory of heterogeneous materials. For the heterogeneous material problem we shall outline a method for deriving a hierarchy of statistical moment equations from the equations of elasticity for a medium with variable elastic properties. We conclude with a discussion of the use of perturbation procedures in determining bounds for the effective elastic constants.