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Effective moduli of plane polycrystals with a rank-one local compliance tensor

Published online by Cambridge University Press:  14 November 2011

L. V. Gibiansky
L. V. Gibiansky
Affiliation:
Department of Civil Engineering and Operations Research and the Princeton Materials Institute, Princeton University, Princeton, NJ 08544, U.S.A.
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Extract

Recently, Grabovsky and Milton have studied effective properties of plane polycrystals made of a crystal with a rank-one local compliance tensor and a positive definite ‘weak’ direction. This paper continues their investigation. By using several complementary approaches, it is proved that the effective compliance tensor of such a polycrystal has exactly one weak direction. The new proofs clarify the link of the degenerate polycrystal problem with previously obtained results on the polycrystal properties. Investigation of the phase boundary conditions has allowed us to find the L-closure set of the effective properties of all polycrystals that can be constructed by sequential laminations of the original degenerate crystals. Bounds on the G-closure, i.e. a set of the effective properties of all the polycrystals that can be built from the given set of degenerate crystals, are found. They are given by a polygonal set in the plane of two linear invariants of the effective compliance tensors. The effective moduli of the polycrystals made from crystals with non-definite ‘weak’ directions are also studied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1325 - 1354
Copyright
Copyright © Royal Society of Edinburgh 1998

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