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Optimal anisotropic three-phase conducting composites: Plane problem

Andrej Cherkaev, Yuan Zhang

Abstract

The paper establishes tight lower bound for effective conductivity tensor $K_*$ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin-Shtrikman and Translation bounds to multiphase structures, it is derived using the technique of {\em localized polyconvexity} that is a combination of Translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi (1995) and Cherkaev (2009) for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of $K_*$, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin Cherkaev and Nesi (2007) and Cherkaev (2009). The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.

Optimal anisotropic three-phase conducting composites: Plane problem

Abstract

The paper establishes tight lower bound for effective conductivity tensor of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin-Shtrikman and Translation bounds to multiphase structures, it is derived using the technique of {\em localized polyconvexity} that is a combination of Translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi (1995) and Cherkaev (2009) for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of , that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin Cherkaev and Nesi (2007) and Cherkaev (2009). The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.

Paper Structure

This paper contains 57 sections, 127 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The problem
  3. Bounds
  4. Optimal structures
  5. The problem
  6. Equations and notations
  7. Energy
  8. Basis
  9. Bounds
  10. Energy Bounds
  11. G-closure boundary
  12. Methods for bounds and optimal structures
  13. Formulation and procedure
  14. Calculation of the translated energy $Y$ in the materials
  15. Structure of minimizers
  16. ...and 42 more sections

Figures (7)

  • Figure 1: Scheme of fields in a structure that realizes the translation bound
  • Figure 2: Regions A-E of multifaceted boundary for the energy in an anisotropic field in the $r, m_1$ plane.
  • Figure 3: The eigenvalues of the gradient field- minimizers, according to the bounds. The equilibrium condition is not assumed.
  • Figure 4: Cartoon of optimal structures in regions A -D1 and the presumed optimal structure in region E.
  • Figure 5: The eigenvalues of effective tensor $K_*$ at the G-closure boundary in dependence of $r$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark 1.1
  • Remark 1.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 5.1
  • Remark 5.2
  • Remark 6.1
  • Remark 6.2