Statistical Continuum Mechanics

In these sections we outline some basic ideas underlying the solution of statistical problems in conductivity, acoustics and elasticity. The problems we consider are both static and dynamic and the statistical nature of the problems sterns from the fact that coefficients like the heat conductivity or sound speed are random functions of position. We first point out that simple averaging procedures are inadequate to determine effective properties of a medium except in the limit of small perturbations. In general they lead to an infinite hierarchy of statistical equations. For large variations of the random coefficients different techniques are required to obtain useful information. In the static problem variational principles are used extensively to find bounds on effective constants. However, we also discuss very approximate techniques like the self-consistent scheme. In the dynamic case we consider three techniques used in different-type problems. We show how to treat the problem of scattering by a dilute collection of discrete scatterers, propagation of acoustic waves through a medium like the ocean and forward and backward scattering of acoustic waves by a one-dimensional random medium.