Theory — reading path
MeanFieldHom computes effective properties of heterogeneous materials by mean-field homogenization. This section states the theory it implements, in the order in which it is built. Every page is self-contained on notation (Notation and conventions) and every formula is either cited or derived on the page.
The chain, in one paragraph
An inclusion embedded in an infinite reference medium responds to a remote load in a way entirely captured by one object, the Hill polarization tensor $\mathbb{P}$ — this is Eshelby's result, and $\mathbb{P}$ depends only on the inclusion shape and the reference moduli (Hill polarization tensors). From $\mathbb{P}$ follows the localization tensor, which says how much of the remote load each phase actually sees, and hence each phase's contribution to the effective stiffness (Localization). Assembling those contributions under an assumption about how phases interact gives a homogenization scheme — dilute, Mori–Tanaka, self-consistent, differential, and the bounds (Homogenization schemes).
Everything else in this section is a specialization of that chain:
| Page | Specialization |
|---|---|
| The finite Eshelby cell | when no closed form exists: the inclusion is solved on a finite cell and the truncation bias removed by its own dipole far field |
| Differential scheme | incorporation as an ODE in a fictitious time, and what a crack — which has no volume to replace — does to it |
| Crack opening displacement | the flat-inclusion limit: a crack has no volume, so it is described by $\boldsymbol{B}$ and $\mathbb{H}$ instead of a volume fraction |
| Thermal cracks | the same limit for scalar transport |
| Layered sphere | a composite inclusion: no Hill tensor exists, the response is assembled by a radial recurrence |
| Layered spheroid | the same for confocal spheroids, where imperfect interfaces couple harmonic degrees |
| Ageing linear viscoelasticity | moduli become Volterra operators; the algebra of the chain is unchanged |
| Elliptic integrals | mathematical appendix: the special functions the closed forms need |
Where the two physics meet
The Eshelby framework applies verbatim to elasticity (order-4 tensors) and to scalar transport — heat conduction, diffusion, electric conduction, Darcy flow (order-2 tensors). The documentation treats them in parallel rather than in separate sections, because the algebra is identical:
| Elasticity | Transport | |
|---|---|---|
| property | stiffness $\mathbb{C}$ | conductivity $\boldsymbol{K}$ |
| Hill tensor | $\mathbb{P}(\boldsymbol{A},\mathbb{C})$ (order 4) | $\boldsymbol{P}(\boldsymbol{A},\boldsymbol{K})$ (order 2) |
| Eshelby tensor | $\mathbb{S} = \mathbb{P}:\mathbb{C}$ | $\boldsymbol{s} = \boldsymbol{P}\cdot\boldsymbol{K}$ |
| crack descriptor | COD tensor $\boldsymbol{B}$, compliance $\mathbb{H}$ | COD scalar $b$, resistivity $\boldsymbol{R}$ |
One asymmetry is worth knowing in advance: for an arbitrarily anisotropic matrix the order-2 Hill tensor has a closed form (via a square-root transformation), whereas the order-4 one requires numerical cubature. This is why the conductivity paths in MeanFieldHom are analytical far more often than the elastic ones.
Relation to the Echoes manual
MeanFieldHom is a pure-Julia reimplementation of the Eshelby/Hill machinery of the Echoes manual, and the Hill-tensor pages follow its appendix closely — same expressions, same conventions, same bibliography. Two families of difference are flagged explicitly wherever they occur:
- extensions: infinite cylinders and 2-D plane strain as first-class inclusion types, an analytical transversely isotropic path, automatic differentiation and symbolic number types throughout;
- conventions that genuinely differ: the crack opening displacement tensor $\boldsymbol{B}$ and the compliance $\mathbb{H}$ are the notable case.
MeanFieldHomcomputes $\boldsymbol{B}$ first and derives $\mathbb{H}$ from it, where Echoes computes $\mathbb{H}$ directly and never forms $\boldsymbol{B}$. The competing normalizations are named and compared in Crack opening displacement.