Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the calculation of bulk properties of heterogeneous materials

Author: John J. McCoy
Journal: Quart. Appl. Math. 37 (1979), 137-149
DOI: https://doi.org/10.1090/qam/99634
MathSciNet review: QAM99634
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Abstract | References | Additional Information

Abstract: Models for calculating the effective, or bulk, properties of heterogeneous materials are considered in the light of a Dyson equation formalism. It is seen that convergence difficulties, of the same nature as those which necessitated the renormalization effected by the Dyson equation in the first place, are frequently reintroduced, but now in calculating the parameters to use in the equation. Thus the validity of the models must be suspect. Fortunately, in most instances the validity of the models, if properly interpreted, can be established. The proper interpretation to be given is considered.

References [Enhancements On Off] (What's this?)

  • [1] Statistical homogeneity is obtained in a limit in which the test specimen is unboundedly large.
  • [2] In an alternate, but equivalent, definition one equates the energy stored in a fictitious, homogeneous effective medium and the averaged energy in the randomly heterogeneous test specimen.
  • [3] The appropriateness of such an appellation might be questioned by some. I use it here since its use is not uncommon; it is a convenient shorthand term, and the precise meaning is made clear by the equation that describes the formalism.
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  • [16] The outlined proof is clearly only a formal proof. To make the proof rigorous would require a demonstration that all the series involved converge, a task that would have to be accomplished on a problem-by-problem basis. This author knows of no attempt to prove convergence for the effective property problem.
  • [17] All that has been demonstrated here, of course, is that a proper cancellation is achieved in the term of fourth order in $ {L_1}$ . In the Appendix we consider terms of higher orders in $ {L_1}$ .
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  • [23] There is no relationship between $ G$ here used to denote the temperature gradient and $ {G_0}$ used previously to denote a Green's function operator.
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Additional Information

DOI: https://doi.org/10.1090/qam/99634
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society