Extension to conductivity

Everything in the preceding chapters was written for elasticity, where the unknown is a displacement and the constitutive tensor has order four. The same machinery solves a second family of problems, in which the unknown is a scalar potential and the constitutive tensor has order two. Nothing has to be rederived: the Eshelby problem, the Hill tensor, the localization tensors and every scheme carry over term by term.

This chapter states the correspondence once, so that the remaining chapters can give their order-2 counterparts in a line instead of arguing for them.

One problem, several physics

The order-2 problem is a linear elliptic equation for a scalar potential whose gradient drives a flux. Four classical theories are that equation with different names on the symbols:

Theory	Potential	Flux
Fourier	temperature	heat flux	thermal conductivity
Fick	concentration	species flux	diffusivity
Darcy	pressure head	filtration velocity	permeability
Ohm	electric potential	current density	electrical conductivity

MeanFieldHomogenization implements the mathematics, not the physics, so a single set of routines serves all four: what the caller supplies is a symmetric positive-definite , and what the caller reads back is an effective in the same units. The chemical route of a hydrating paste and the permeability of a fractured rock use the same conductivity_contribution as a heat-conduction estimate.

The one asymmetry, and how it is removed

Hooke's law carries no minus sign; Fourier's and Fick's do. Left alone, that single difference makes order-2 and order-4 formulas look as though they disagreed on a sign. The convention fixed in Elasticity and transport: one set of formulas takes as the stress analog minus the flux,

so that   is, in both theories, what the exterior transmits to the interior across a surface. With that substitution the dictionary below is literal: every entry on the right is the entry on the left with the symbols renamed, and no sign is ever flipped.

The dictionary

Elasticity — order 4	Conductivity — order 2
displacement — vector	potential — scalar
strain — 2-tensor	gradient — vector
stress — 2-tensor	— vector
stiffness — 21 components	conductivity — 6 components
Hill tensor — 4-tensor	Hill tensor — 2-tensor
Eshelby tensor
localization , contribution	,

The routine names follow the same rule, and the correspondence is listed in full under Conductivity (2nd-order transport): each strain/stress becomes a gradient/flux, each stiffness a conductivity, each compliance a resistivity.

What transposes untouched, and what does not

The schemes transpose entirely. Dilute, Mori-Tanaka, self-consistent, differential, the bounds, the N-body models: all of them are written on localization and contribution tensors, never on the order of the constitutive tensor. Dispatching on a 2-tensor is enough — see Homogenization schemes.

Two things genuinely differ, and both are in the order-2 problem's favor.

The acoustic tensor of the order-4 problem is a sextic polynomial in the wave direction, which is why an arbitrarily anisotropic needs the residue algorithm. Its order-2 counterpart is the quadratic form   , so has a closed form at any matrix anisotropy [8] — no quadrature, no residues. The derivation is in Hill polarization tensors.

The order-2 problem also carries fewer modes. A crack opens in three modes in elasticity, hence a 6-component opening tensor; in conduction only the normal flux jumps, so a single scalar suffices, and it carries all the anisotropy of the matrix. That reduction is worked through in Thermal cracks.

Where this is used

• Hill polarization tensors — the closed form of and its dispatch.

• Localization and contribution tensors — the full table of order-2 routine names.

• Thermal cracks — the flat-inclusion limit in conduction, and its intensity factor.

• Conductivity — how to call all of this.