MeanFieldHomogenization.jl

Appearance

MeanFieldHomogenization.jlEffective properties of heterogeneous materials

One Eshelby chain — Hill tensor, localization, scheme — generalized one direction at a time, in pure Julia and differentiable throughout.

Get started
Theory
API
View on GitHub
A representative volume element of ellipsoidal inclusions in a matrix
📐

Theory

The Eshelby/Hill chain in the order it is built, then the ways it is generalized — richer patterns, conductivity, viscoelasticity, periodicity, N-body.

🧰

Manual

Every inclusion family, cell and scheme, with the call that produces it — from an ellipsoid to a neural surrogate.

🎓

Tutorials

The API by worked example, topic by topic, each page runnable end to end.

🧪

Applications

Full micromechanical models of real materials — cement paste, concrete, bituminous mixtures, clays.

🔁

Tools and migration

Build a model in the browser with MFH Studio, or port one from the Echoes C++/Python codebase.

🏗️

Finite-element coupling

One microstructure per Gauss point, handing a structural code a stress, a consistent tangent and Biot coefficients.

📖

API reference

Every exported function, grouped by sub-module.

🛠️

Developer guide

How the package is put together, and how to add an inclusion, an algorithm or a scheme without touching the rest.

What it does ​

Given a reference medium and an inclusion shape, MeanFieldHomogenization builds the Hill polarization tensor (Eshelby's result), derives the localization tensor for each phase, and assembles a homogenization scheme — dilute, Mori–Tanaka, self-consistent, differential, PCW, or the classical bounds — into an effective stiffness or conductivity.

That chain is the whole library. Everything else is the chain generalized in one direction at a time: a richer morphological pattern in place of the ellipsoid, a degenerate one for cracks, a different physics for transport, a different time dependence for viscoelasticity, a periodic problem for the multilayer, and no one-site assumption at all for the N-body schemes. The reading path takes them in that order.

MeanFieldHomogenization is a pure-Julia reimplementation of the Eshelby/Hill machinery of the Echoes C++/Python codebase; see From Echoes to MeanFieldHomogenization for the translation guide.

A first calculation ​

A porous solid, and every scheme the package ships, against the bounds that frame them. The void is given a small but non-zero stiffness, which is what lets the self-consistent branch stay on the physical solution:

julia
using MeanFieldHomogenization, TensND, Plots
gr()  # headless backend; GKSwstype is set to "100" in make.jl

C_solid = iso_stiffness(90.0, 30.0)     # GPa
C_void  = iso_stiffness(0.01, 0.005)

function bulk(f, scheme)
    r = RVE()
    add_phase!(r, :M, Ellipsoid(1.0), Dict(:C => C_solid); fraction = :rest)
    f > 0 && add_phase!(r, :V, Ellipsoid(1.0), Dict(:C => C_void); fraction = f)
    return first(k_mu(homogenize(r, scheme, :C)))
end

fs = range(0.0, 0.5; length = 61)
schemes = ("Voigt" => Voigt(),
           "Reuss" => Reuss(),
           "Mori-Tanaka" => MoriTanaka(),
           "Dilute (dual)" => DiluteDual(),
           "Self-consistent" => AsymmetricSelfConsistent(; abstol = 1.0e-10,
                                                           maxiters = 200,
                                                           select_best = true),
           "Differential" => DifferentialScheme(; nsteps = 100))

plt = plot(; xlabel = "porosity f", ylabel = "effective bulk modulus k [GPa]",
             legend = :topright, framestyle = :box, size = (760, 470))
for (name, s) in schemes
    plot!(plt, fs, [bulk(f, s) for f in fs]; label = name, lw = 2)
end
plt

Voigt and Reuss bracket the others; the three estimates between them differ by how much of the load each void is assumed to see. Porous materials works through why the standard self-consistent scheme fails on this problem and what replaces it.