Theory — reading path
MeanFieldHomogenization 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 order below is the dependency chain, and it is worth reading in that order. It starts from the Eshelby problem and the tensors it produces, then the schemes assembled from them. Then come the three independent ways that one-inclusion picture is generalized — a richer morphological pattern in place of the ellipsoid, a different physics, a different time dependence — followed by periodic homogenization, which is a different construction rather than a generalization, and finally the N-body models, which drop the one-site assumption altogether.
Foundations: 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
The chain, in one picture
Everything the package does is one pipeline, read top to bottom. The gray boxes are what you supply — each entering at the stage that needs it — the blue ones what the package computes, and the green ones on the right the three places where you can plug in your own physics without touching the library.
The three green gates are the extension mechanism. A morphology MeanFieldHomogenization knows nothing about becomes a first-class citizen of every scheme by answering one of them — implement the lowest you can reach and the package derives the rest algebraically. Four routes already use them:
| Route | Gate | Where the answer comes from |
|---|---|---|
| ellipsoids, cylinders, cracks | A | closed forms, quadrature or residues |
| composite spheres and spheroids | B | the Hervé–Zaoui recurrences — a layered pattern has no Hill tensor |
| finite elements | B / COD | a Ferrite or Gridap solve of the Eshelby problem |
| neural surrogates | A / B | a trained network, differentiable in the morphology |
| your own | A, B or C | a formula, a solver, a table — anything callable |
Cracks and flat objects follow the same chain with
Homogenization schemes
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).
| Page | What it adds |
|---|---|
| Homogenization schemes | the scheme catalog, each as one assumption on the reference medium, and the bounds |
| Differential scheme | incorporation as an ODE in a fictitious time, and what a crack — which has no volume to replace — does to it |
The differential scheme is a scheme like the others, differing only in that its assembly is an integration rather than a closed form; it is grouped here rather than treated as a specialization of its own.
The generalized Eshelby problem: morphological patterns
A composite pattern has no Hill tensor — the strain inside it is not uniform, so Eshelby's result does not apply as such. What does exist, and is all any scheme needs, is its average concentration tensor, obtained by solving the generalized Eshelby problem for that pattern. Any morphology admitting that treatment belongs in this group; two are implemented.
| Page | Pattern |
|---|---|
| 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 |
Cracks
A crack is a degenerate ellipsoid rather than a composite pattern: it keeps the Eshelby framework and loses its volume, which is what forces a different descriptor.
| Page | Specialization |
|---|---|
| Crack opening displacement | the flat-inclusion limit: a crack has no volume, so it is described by |
| Thermal cracks | the same limit for scalar transport |
Extension to conductivity
| Page | What it adds |
|---|---|
| Extension to conductivity | the order-4 ↔ order-2 dictionary stated once — Hooke against Fourier, Fick, Darcy and Ohm — what transposes untouched, and the two things that genuinely differ |
Extension to viscoelasticity
Two distinct extensions, and the distinction matters: for a non-ageing material a transform maps the problem onto an elastic one, so nothing new has to be solved; for an ageing one no such map exists, and it is the Eshelby problem itself that must be generalized.
| Page | What it adds |
|---|---|
| The Laplace-Carson route | the correspondence principle: a non-ageing problem solved as an elastic one, transform by transform |
| Ageing linear viscoelasticity | moduli become Volterra operators; the algebra of the chain is unchanged |
Periodic homogenization
Not a morphological pattern, and not an inclusion problem at all: the laminate result is the closed form of a periodic homogenization problem, reached by interface algebra rather than by an Eshelby argument. It is grouped on its own for that reason.
| Page | What it adds |
|---|---|
| Laminate | a periodic stack: no inclusion at all, the interface algebra replaces the Hill tensor |
N-body models
These are the only pages that drop the one-site assumption: instead of averaging the interaction between inclusions, they resolve it pair by pair, and therefore need to know where the inclusions are.
| Page | What it adds |
|---|---|
| Interaction tensors | the two-inclusion tensor |
| The cluster model | the mean strain of every inclusion, resolved inside a cluster [1] |
| The equivalent inclusion method | the same physics as a variational Galerkin problem, with rigorous bounds [2] |
Appendices
| Page | Role |
|---|---|
| The finite Eshelby cell | a numerical device supporting the finite-element inclusions: the inclusion is solved on a finite cell and the truncation bias removed by its own dipole far field. Written in the language of localization and crack opening displacement, so it is read after both |
| Elliptic integrals | 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 with the convention
| Elasticity | Transport | |
|---|---|---|
| property | stiffness | conductivity |
| Hill tensor | ||
| Eshelby tensor | ||
| crack descriptor | COD tensor | COD scalar |
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 MeanFieldHomogenization are analytical far more often than the elastic ones.
Relation to the Echoes manual
MeanFieldHomogenization 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
and the compliance are the notable case. MeanFieldHomogenizationcomputesfirst and derives from it, where Echoes computes directly and never forms . The competing normalizations are named and compared in Crack opening displacement.