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Fine Properties of Solutions to Conductivity Equations With Applications to Composites

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Mathematics of Multiscale Materials

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 99))

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Abstract

The simplest possible approach in bounding effective moduli enjoying a variational definition, consists in plugging into the variational principle an “admissible test field”. If only the volume fraction of the phases is known, the traditional approach is to use a field which is constant throughout the media. In the context of conductivity, this very simple choice leads to the so called Wiener (or harmonic and arithmetic mean) bounds. On the other hand rank one laminates provide us with composites which saturate these elementary bounds.

If one considers “sum of energies” and restricts attention to isotropic composites, Wiener bounds are no longer attainable. This is because the constant test field no longer matches the actual solution for any microgeometry.

We explain how to use recent results from the theory of quasiconformal mappings, to make a more flexible and in general more appropriate choice of test field when a sum of energies is considered. Our method only requires information about the volume fraction and, if used in combination with the translation method, is at least as successful as any other known method.

The method, as developed so far, works only in two dimensional conductivity. However it applies to non linear problems as well. In the non linear context, when applied to a polycrystalline material, the method delivers a new lower bound when the energy is subquadratic. The only other known method to do so has been recently developed by D. Talbot and J. R. Willis. A brief review of recently established very fine properties of solutions to the conductivity equations in the linear context will be given.

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Author information

Authors and Affiliations

  1. Universita’ di Roma ‘La Sapienza’, Dipartimento di Matematica, Rome, 00185, Italy

    V. Nesi

  2. Dipartimento di Matematica Pura & Applicata, Université degli Studi di L’Aquila, L’Aquila, 67010, Italy

    V. Nesi

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  1. V. Nesi
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Kenneth M. Golden  & Graeme W. Milton  & 

  2. Statistical Laboratory, University of Cambridge, Cambridge, CB2 1SB, England

    Geoffrey R. Grimmett

  3. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Richard D. James

  4. Schlumberger-Doll Research Center, Old Quarry Road, Ridgefield, CT, 06877, USA

    Pabitra N. Sen

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Nesi, V. (1998). Fine Properties of Solutions to Conductivity Equations With Applications to Composites. In: Golden, K.M., Grimmett, G.R., James, R.D., Milton, G.W., Sen, P.N. (eds) Mathematics of Multiscale Materials. The IMA Volumes in Mathematics and its Applications, vol 99. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1728-2_11

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  • DOI: https://doi.org/10.1007/978-1-4612-1728-2_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7256-4

  • Online ISBN: 978-1-4612-1728-2

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