Extension to conductivity
Before this page
The Eshelby inclusion problem §3, where the transport form of the inclusion problem is stated, and Concentration and contribution tensors §4 for the order-2 routines named below.
The foundations and the schemes were written above 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 concentration tensors and every scheme carry over term by term.
This chapter states the correspondence once, so that the other chapters can give their order-2 counterparts in a line instead of arguing for them.
1. 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 property_contribution as a heat-conduction estimate.
2. 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
3. The dictionary
| Elasticity — order 4 | Conductivity — order 2 |
|---|---|
| displacement | potential |
| strain | gradient |
| stress | |
| stiffness | conductivity |
| Hill tensor | Hill tensor |
| Eshelby tensor | |
| concentration |
The routine names follow the same rule, and the correspondence is listed in full in Concentration and contribution tensors §4: each strain/stress becomes a gradient/flux, each stiffness a conductivity, each compliance a resistivity.
4. 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 concentration 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
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 to go next
The next chapter changes the time dependence rather than the physics: The Laplace-Carson route reduces a non-aging viscoelastic problem to an elastic one at each value of a transform variable. The order-2 counterparts stated on this page are developed on the following pages.
Hill polarization tensors — the closed form of
and its dispatch. Concentration 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.