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Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation

Published online by Cambridge University Press:  14 November 2011

Yury Grabovsky  and
Graeme W. Milton
Yury Grabovsky
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84102, U.S.A.
Graeme W. Milton
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84102, U.S.A.
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Abstract

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 2 , 1998 , pp. 283 - 299
Copyright
Copyright © Royal Society of Edinburgh 1998

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