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Modelling the effective conductivity function of an arbitrary two–dimensional polycrystal using sequential laminates

Published online by Cambridge University Press:  14 November 2011

Karen E. Clark  and
Graeme W. Milton
Karen E. Clark
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 U.S.A
Graeme W. Milton
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 U.S.A
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Abstract

The effective conductivity tensor σ* of a two-dimensional polycrystalline material depends on the conductivity tensor σ0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation field R(x) giving the orientation of the crystal at each point x. Here it is established that the dependence of σ* on σ0 in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 757 - 783
Copyright
Copyright © Royal Society of Edinburgh 1994

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