Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T02:29:47.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Which electric fields are realizable in conducting materials?

Published online by Cambridge University Press:  20 January 2014

Marc Briane ,
Graeme W. Milton  and
Andrejs Treibergs
Marc Briane
Affiliation:
Institut de Recherche Mathématique de Rennes, INSA de Rennes, France. mbriane@insa-rennes.fr
Graeme W. Milton
Affiliation:
Department of Mathematics, University of Utah, USA; milton@math.utah.edu
Andrejs Treibergs
Affiliation:
Department of Mathematics, University of Utah, USA; treiberg@math.utah.edu
Get access

Abstract

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

Keywords

Isotropic conductivityelectric fieldgradient systemlaminate
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 307 - 323
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G. and Nesi, V., Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal. 158 (2001) 155–171. Google Scholar
G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Appl. Math. Sci. Springer-Verlag, New-York (2002) 456.
Ancona, A., Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J. 165 (2002) 123–158. Google Scholar
V.I. Arnold, Ordinary differential equations, translated from the third Russian edition by R. Cooke, Springer Textbook. Springer-Verlag, Berlin (1992) 334.
N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Mathematical Problems in the Mechanics of Composite Materials, translated from the Russian by D. Leĭtes, vol. 36 of Math. Appl. (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1989) 366.
Bauman, P., Marini, A. and Nesi, V., Univalent solutions of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50 (2001) 747–757. Google Scholar
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in vol. 5 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam-New York (1978) 700.
Briane, M., Correctors for the homogenization of a laminate. Adv. Math. Sci. Appl. 4 (1994) 357–379. Google Scholar
Briane, M., Milton, G.W. and Nesi, V., Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal. 173 (2004) 133–150. Google Scholar
Briane, M., and Nesi, V., Is it wise to keep laminating? ESAIM: COCV 10 (2004) 452–477. Google Scholar
Cherkaev, A. and Zhang, Y., Optimal anisotropic three-phase conducting composites: Plane problem. Int. J. Solids Struct. 48 (2011) 2800–2813. Google Scholar
B. Dacorogna, Direct Methods in the Calculus of Variations, in vol. 78 of Appl. Math. Sci. Springer-Verlag, Berlin-Heidelberg (1989) 308.
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, translated from the Russian by G.A. Yosifian. Springer-Verlag, Berlin (1994) 570.
G.W. Milton, Modelling the properties of composites by laminates, Homogenization and Effective Moduli of Materials and Media, in vol. 1 of IMA Math. Appl. Springer-Verlag, New York (1986) 150–174.
G.W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002) 719.
F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, in vol. 31 of Progr. Nonlinear Differ. Equ. Appl., edited by L. Cherkaev and R.V. Kohn. Birkhaüser, Boston (1997) 21–43.
Nesi, V., Bounds on the effective conductivity of two-dimensional composites made of n ≥ 3 isotropic phases in prescribed volume fraction: the weighted translation method. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1219–1239. Google Scholar
Raitums, U., On the local representation of G-closure. Arch. Rational Mech. Anal. 158 (2001) 213–234. Google Scholar