Two Methods for Analyzing Waves in Composites with Random Microstructure

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TWO METHODS FOR ANALYZING WAVES IN COMPOSITES WITH RANDOM MICROSTRUCTURE JOHN R. WILLIS School of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

ABSTRACT The problem of calculating the mean wave in a composite with random microstructure is addressed. Exact characterizations of the problem can be given, in the form of stochastic variational principles. Substitution of simple configuration-dependent trial fields into these generates approximations which are, in a sense, 'optimal'. It is necessary in practice to employ only trial fields which will generate, in the variational principle, no more statistical information than is actually available. Trial fields that require knowledge of two-point statistics generate equations that can also be obtained directly, through use of the QCA. The same fields can be substituted into an alternative variational principle to yield an approximation that makes use of three-point statistics - this approximation is less easy to obtain by direct reasoning. When not even two-point information is available, some more elementary approximation is needed. One such approximation, which is simple and direct in its application, is an extension to dynamics of a "self-consistent embedding" scheme which is widely used in static problems. This is also discussed, together with some illustrative results for a matrix containing inclusions and for a polycrystal.

INTRODUCTION The theory of the effective, or mean, behaviour of a composite is well-developed for the case of static deformations. When the response of the composite is linear, methods of analysis include analytic continuation [1], the "translation method" [21 and the use of variational principles [3,4]. All of these yield bounds for effective properties. The use of variational principles has also been extended to provide bounds for the effective energy density of a wide class of nonlinear composites [5,6,7]. These bounds, while they have the virtue of precision, may in some situations be too far apart to be useful, or they may not make use of all of the information that is available. A case in point is that of a composite, containing a known volume fraction of inclusions, of known shapes and sizes, but whose pairwise correlations are unknown: the only known bounds allow only for the volume fractions. Methods of approximate analysis can be devised, however. Rigour is sacrificed, but answers can be obtained which may be practically useful. One such method is based on "self-consistent embedding" [8,9]. For dynamic problems, it is more usual to apply "multiple scattering theory" in some form [10,11] and to close a statistical hierarchy by making some plausible closure assumption. Such approaches have a very different appearance to those employed for statics and yet they frequently lead to approximate dispersion relations which predict (in the long-wavelength limit) exactly the same static response as the direct static methods. The purpose of this paper is to provide a brief review of some methods for dynamic problems, whi