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.

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Abstract.Bounds are here derived for the effective bulk modulus in heterogeneous media, denoted by k*, using the two standard variational principles of elasticity. As trial functions for the stress and strain fields we use perturbation expansions that have been modified by the inclusion of a set of multiplicative constants. The first order perturbation effect is explicitly calculated and bounds for k* are derived in terms of the correlation functions (/(0)k'(i)lc'(s)) and ([fc'(r)A/(s)/|u(0)]) where n' and k' are the fluctuating parts of the shear modulus n and the bulk modulus, fc, respectively. Explicit calculations are given for two phase media when fi'(x) = 0 and when the media are symmetric in the two phases. Results are also included for the dielectric problem when the media are composed of two symmetric phases.

In this note a thought experiment will be used to show that relativistic contraction can introduce stress effects in a moving body.