### Abstract

We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null-Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor when the phases are interchanged. It leads to optimal bounds on the effective conductivity when another effective constant, such as the effective diffusion coefficient, has been measured, or when one has information about ζ_{1} which is a parameter characteristic of the microgeometry, or when one knows the material is symmetric, i.e., invariant under phase interchange like a three-dimensional checkerboard.

Original language | English (US) |
---|---|

Pages (from-to) | 4989-5003 |

Number of pages | 15 |

Journal | Journal of Applied Physics |

Volume | 63 |

Issue number | 10 |

DOIs | |

State | Published - 1988 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Applied Physics*,

*63*(10), 4989-5003. https://doi.org/10.1063/1.340445

**On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality.** / Avellaneda, Marco; Cherkaev, A. V.; Lurie, K. A.; Milton, G. W.

Research output: Contribution to journal › Article

*Journal of Applied Physics*, vol. 63, no. 10, pp. 4989-5003. https://doi.org/10.1063/1.340445

}

TY - JOUR

T1 - On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality

AU - Avellaneda, Marco

AU - Cherkaev, A. V.

AU - Lurie, K. A.

AU - Milton, G. W.

PY - 1988

Y1 - 1988

N2 - We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null-Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor when the phases are interchanged. It leads to optimal bounds on the effective conductivity when another effective constant, such as the effective diffusion coefficient, has been measured, or when one has information about ζ1 which is a parameter characteristic of the microgeometry, or when one knows the material is symmetric, i.e., invariant under phase interchange like a three-dimensional checkerboard.

AB - We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null-Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor when the phases are interchanged. It leads to optimal bounds on the effective conductivity when another effective constant, such as the effective diffusion coefficient, has been measured, or when one has information about ζ1 which is a parameter characteristic of the microgeometry, or when one knows the material is symmetric, i.e., invariant under phase interchange like a three-dimensional checkerboard.

UR - http://www.scopus.com/inward/record.url?scp=0001582295&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001582295&partnerID=8YFLogxK

U2 - 10.1063/1.340445

DO - 10.1063/1.340445

M3 - Article

VL - 63

SP - 4989

EP - 5003

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 10

ER -