Polarization field of a conductivity inclusion ( $P = (a - a_{0} I) \nabla u$ )

Relative to a uniform reference conductivity

a_{0}

, this polarization is the excess positive conductivity flux. The equation becomes

a_{0} Δ u + \nabla \cdot P = 0

. For a prescribed mean gradient

E

, averaging gives

a^{*} E = a_{0} E + ⟨ P ⟩

. This sign convention differs from the physical heat-flux sign.

Table of contents
- Exciting field of a conductivity inclusion Polarization field of a conductivity inclusion
- Spherical conductivity inclusion Polarization field of a conductivity inclusion
  - Dilute spherical-inclusion conductivity Spherical conductivity inclusion
    - Maxwell approximation for conductivity Dilute spherical-inclusion conductivity
    - Dipole interaction correction to effective conductivity Dilute spherical-inclusion conductivity

Exciting field of a conductivity inclusion ( $E_{exc} = E + e_{regular} (y)$ )

 0  0

The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre

y

. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.

Spherical conductivity inclusion ( $β = 3 a_{0} (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

 0  0

A sphere of conductivity

a_{1}

in a host

a_{0}

under a uniform incident gradient

E

has interior gradient

3 a_{0} E / (a_{1} + 2 a_{0})

. Outside a sphere of radius

R

, its potential correction is

- R^{3} (a_{1} - a_{0}) (E \cdot x) / [(a_{1} + 2 a_{0}) ∣ x ∣^{3}]

. Its polarization is

βE

inside and zero outside. Temperature and normal-flux continuity determine these coefficients.

Dilute spherical-inclusion conductivity ( $a^{*} = a_{0} I + pβ I + o (p)$ )

 0  0

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

Maxwell approximation for conductivity ( $a^{*} = a_{0} (1 + 2 p k) / (1 - p k), k = (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

 0  0

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is