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 follows the Conventions and the Nomenclature, 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 with the problems of a single heterogeneity in an infinite medium: the Eshelby problem and the tensors it produces, the interaction of two inclusions, the crack as the flat limit of an ellipsoid, and the layered patterns that generalize the Eshelby problem. The homogenization schemes come next, one family per page, and turn those solutions into effective properties. The chain is then extended to another physics, conduction, and to another time dependence, viscoelasticity.

The symbols

Every symbol of these pages is in the Nomenclature, with its meaning and its unit. Hovering an equation shows the symbols it holds, with their meaning on the page.

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 — this is Eshelby's result, and depends only on the inclusion shape and the reference moduli (Hill polarization tensors). From follows the concentration tensor, which says how much of the remote load each phase actually sees, and hence each phase's contribution to the effective stiffness (Concentration).

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 in place of and a density in place of a volume fraction; transport is the same picture at tensor order 2, and aging viscoelasticity the same picture with Volterra products in place of tensor products. The contract is written up in Adding a new inclusion.

The Eshelby problem

Page	What it adds
The Eshelby inclusion problem	the uniform strain inside an ellipsoid, and the three tensors , ,
Hill polarization tensors	the closed forms of , shape by shape and symmetry class by symmetry class
Concentration and contribution tensors	the response of an inhomogeneity to a remote load, and its contribution to the effective stiffness
Two-inclusion interaction tensors	the field one inclusion induces in another, through the Green operator — the ingredient of the N-body schemes

Cracks and degenerate shapes

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	What it adds
Crack opening displacement	the flat-inclusion limit: a crack has no volume, so it is described by and instead of a volume fraction
Thermal cracks	the same limit for scalar transport
Degenerate shapes	the closed forms of the Hill, Eshelby and crack opening tensors at the shapes where they reach limits: sphere, disc, needle

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
The elastic confocal spheroid	the axisymmetric elastic problem on the same chart

Homogenization schemes

Assembling the solutions of the previous chapters under an assumption about what surrounds each inclusion gives a homogenization scheme. The overview states what every scheme estimates and gathers them in one table; each family then has its page, with its figure, its assumption and its equations.

Page	What it adds
Homogenization schemes	the framework common to all schemes, the identities between contribution tensors, how a crack family enters, the concentration tensors of an estimate, and the table of schemes
Voigt and Reuss bounds	a uniform strain or a uniform stress, and the energy principles that make them bounds
Dilute estimates	each inclusion alone in the matrix, under the remote strain or the remote stress
Mori–Tanaka	each inclusion under the mean strain of the matrix
Maxwell	a domain holding the inclusions, filled with the effective medium, disturbs the matrix far away as they do
Ponte Castañeda–Willis	the Hashin–Shtrikman energy principle with the two-point statistics of the inclusions, one distribution ellipsoid per pair of phases
Self-consistent schemes	each phase in the effective medium, with or without a distinguished matrix
Differential scheme	incorporation as an ODE in a fictitious time, and what a crack — which has no volume to replace — does to it
Laminate	a periodic stack solved exactly: no inclusion at all, the interface algebra replaces the Hill tensor
The cluster model	the mean strain of every inclusion, resolved inside a cluster (Molinari and El Mouden, 1996)
The equivalent inclusion method	the same physics as a variational Galerkin problem, with rigorous bounds (Brisard et al., 2014)
Levin's theorem	the response to free strains and prestresses of the phases — thermal expansion, Biot tensor — from the concentration tensors of any scheme

The cluster model and the equivalent inclusion method 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. Every scheme also estimates the concentration tensors of the phases, and the last page draws from them what the effective stiffness alone does not give.

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-aging material a transform maps the problem onto an elastic one, so nothing new has to be solved; for an aging 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-aging problem solved as an elastic one, transform by transform
Aging linear Volterra behavior	moduli and conductivities become Volterra operators; the algebra of the chain is unchanged, the order of its factors is not free, and every scheme is restated in that algebra

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 concentration 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    fixed in Elasticity and transport: one set of formulas the algebra is not merely analogous but identical, symbol for symbol:

	Elasticity	Transport
property	stiffness	conductivity
Hill tensor	(order 4)	(order 2)
Eshelby tensor
crack descriptor	COD tensor , compliance	COD scalar , resistivity

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. MeanFieldHomogenization computes first and derives from it, where Echoes computes directly and never forms . The competing normalizations are named and compared in Crack opening displacement.