Effective conductivity ( $⟨ a \nabla u ⟩ = a^{*} ⟨ \nabla u ⟩$ )

Effective conductivity describes the averaged response of a microscopic conductivity field to a macroscopic temperature or electric-potential gradient. Here the positive conductivity flux is

a \nabla u

; physical heat flux has the opposite sign. For periodic media it is computed by cell correctors, and can be a tensor even for isotropic constituents.

Table of contents

Self-consistent conductivity approximation ( $C_{SC} = H (C_{SC})$ )

 0  0

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities

a, b > 0

, the comparison formula gives

C_{SC}^{2} = ab

, so its positive solution is

ab

. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.

Hashin-Shtrikman conductivity variational principle

 0  0

For a scalar reference

b

, choose a trial polarization

q

and solve

b Δ v + \nabla \cdot q = 0

with zero Dirichlet data. The trial functional is

F_{b} (q) = b ∣Ω∣∣ λ ∣^{2} + 2 λ \cdot \int_{Ω} q + \int_{Ω} q \cdot \nabla v - \int_{Ω} q \cdot (a - b I)^{- 1} q .

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A reference above all phase conductivities gives

J (u) \leq F_{b} (q)

; one below all of them gives

J (u) \geq F_{b} (q)

. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

Two-dimensional isotropic polycrystal conductivity ( $H (c) = \frac{2 ab + c ( a + b )}{a + b + 2 c}$ )

 0  0

For uniformly distributed rotations of a two-dimensional conductivity tensor with positive principal values

a \leq b

, the Hashin-Shtrikman conductivity variational principle with isotropic comparison

c I

yields

H (c) I

. Averaging a rotated diagonal tensor replaces it by half its trace times the identity. Since

H^{'} (c) = (a - b)^{2} / (a + b + 2 c)^{2}

, the admissible comparison endpoints give

a (a + 3 b) / (3 a + b) \leq C_{eff} \leq b (3 a + b) / (a + 3 b)

within the statistically isotropic polycrystal model.

Hashin-Shtrikman bounds for conductivity ( $a_{1} + \frac{3 a _{1} ( a _{2} - a _{1} ) p _{2}}{3 a _{1} + p _{1} ( a _{2} - a _{1} )} \leq a^{*} \leq a_{2} + \frac{3 a _{2} ( a _{1} - a _{2} ) p _{1}}{3 a _{2} + p _{2} ( a _{1} - a _{2} )}$ )

 0  0

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with

0 < a_{1} < a_{2}

. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.

Voigt bound ( $a^{*} ⪯ ⟨ a ⟩$ )

 0  0

An affine trial temperature in the minimum-energy principle gives the upper tensor bound by the volume-averaged local conductivity. For an isotropic two-phase composite,

a^{*} \leq p_{1} a_{1} + p_{2} a_{2}

. The same uniform-field variational idea supplies the familiar corresponding elastic upper bound.

Laminate conductivity ( $a^{*} = a_{H} n \otimes n + a_{A} (I - n \otimes n)$ )

 0  0

For isotropic phases with planar interfaces normal to

n

, normal conductivity is the weighted harmonic mean

a_{H} = (\sum p_{i} / a_{i})^{- 1}

and tangential conductivity is the weighted arithmetic mean

a_{A} = \sum p_{i} a_{i}

. Continuity of normal flux and tangential temperature gradient gives the two averages directly.

Conductivity homogenization

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Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

Locally periodic one-dimensional harmonic-mean coefficient ( $\overset{a}{^} (x) = (\int_{0}^{1} a (x, X)^{- 1} d X)^{- 1}$ )

 0  0

For

(a (x, x / ϵ) y^{'})^{'} = f (x, x / ϵ)

with positive one-periodic fast coefficient, a periodic two-scale expansion has leading term independent of the fast variable. The first corrector satisfies

y_{1 X} = C (x) / a - y_{0}^{'}

. Its zero-mean derivative requires

C = \overset{a}{^} y_{0}^{'}

. Averaging the next equation gives

C^{'} = \int_{0}^{1} f d X

. Thus the leading equation has the local harmonic-mean transport coefficient and the arithmetic-mean source. Positivity is what makes the secular drift incompatible with periodicity.

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

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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.

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

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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

a_{0} + pβ + p^{2} β^{2} / (3 a_{0}) + O (p^{3})

. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.

Dipole interaction correction to effective conductivity ( $a^{*} \approx a_{0} I + pβ I - β^{2} \sum_{i} Λ_{i}$ )

 0  0

If the first exciting-field correction at inclusion

i

- β Λ_{i} E / p_{i}

, its revised polarization is

β χ_{i} (E - β Λ_{i} E / p_{i})

. Averaging weights it by

p_{i}

and gives the displayed second-reflection approximation. Further dipole reflections and finite-size multipoles are neglected.

Periodic conductivity cell problem ( $\nabla_{y} \cdot [a (y) (e_{j} + \nabla_{y} w_{j})] = 0$ )

 0  0

The periodic zero-mean corrector

w_{j}

adjusts a unit imposed gradient so microscopic flux is divergence-free. For bounded uniformly elliptic coefficients, the weak problem has a unique solution modulo constants. The averaged flux is

a^{*} e_{j} = ⟨ a (e_{j} + \nabla w_{j})⟩

. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.

Two-scale expansion for periodic conductivity ( $\nabla = \nabla_{x} + ϵ^{- 1} \nabla_{y}$ )

 0  0

A periodic expansion with

y = x / ϵ

separates the macroscopic variable from the cell variable. Its leading cell equation forces

u_{0}

to be independent of

y

, the next equation gives

u_{1} = w_{j} (y) \partial_{x_{j}} u_{0}

, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.

Boundary corrector in periodic homogenization ( $b^{ϵ} ∣_{\partial Ω} = - ϵ w_{j} (x / ϵ) \partial_{j} u_{0}$ )

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A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.

Effective conductivity ( $⟨ a \nabla u ⟩ = a^{*} ⟨ \nabla u ⟩$ )

Self-consistent conductivity approximation ( $C_{SC} = H (C_{SC})$ )

Hashin-Shtrikman conductivity variational principle

Two-dimensional isotropic polycrystal conductivity ( $H (c) = \frac{2 ab + c ( a + b )}{a + b + 2 c}$ )

Hashin-Shtrikman bounds for conductivity ( $a_{1} + \frac{3 a _{1} ( a _{2} - a _{1} ) p _{2}}{3 a _{1} + p _{1} ( a _{2} - a _{1} )} \leq a^{*} \leq a_{2} + \frac{3 a _{2} ( a _{1} - a _{2} ) p _{1}}{3 a _{2} + p _{2} ( a _{1} - a _{2} )}$ )

Voigt bound ( $a^{*} ⪯ ⟨ a ⟩$ )

Laminate conductivity ( $a^{*} = a_{H} n \otimes n + a_{A} (I - n \otimes n)$ )

Conductivity homogenization

Locally periodic one-dimensional harmonic-mean coefficient ( $\overset{a}{^} (x) = (\int_{0}^{1} a (x, X)^{- 1} d X)^{- 1}$ )

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

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

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

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

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

Dipole interaction correction to effective conductivity ( $a^{*} \approx a_{0} I + pβ I - β^{2} \sum_{i} Λ_{i}$ )

Periodic conductivity cell problem ( $\nabla_{y} \cdot [a (y) (e_{j} + \nabla_{y} w_{j})] = 0$ )

Two-scale expansion for periodic conductivity ( $\nabla = \nabla_{x} + ϵ^{- 1} \nabla_{y}$ )

Boundary corrector in periodic homogenization ( $b^{ϵ} ∣_{\partial Ω} = - ϵ w_{j} (x / ϵ) \partial_{j} u_{0}$ )

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