MeanFieldHomogenization.jl

Julia framework for mean-field homogenization: Hill tensors, cracks, layered inclusions, aging viscoelasticity and hereditary conduction, all classical schemes, ForwardDiff sensitivities; shapes with no closed form enter by finite elements or a surrogate.

Julia
Micromechanics
Homogenization
Machine Learning

MeanFieldHomogenization.jl predicts the effective elastic, transport and viscoelastic properties of a heterogeneous material from the properties, shapes, orientations and volume fractions of the phases of its microstructure. All the ingredients — inclusion morphologies, localization tensors and homogenization schemes — live under a common abstraction hierarchy with a shared numerical core and a central dispatch mechanism. A microstructure can also serve as the constitutive law at each Gauss point of a finite-element computation. Part of the MicroPoroChemoMechanics ecosystem, built on TensND.jl.

Key features

  • Hill polarization tensors — ellipsoidal inclusions and infinite cylinders, 2D and 3D, isotropic, anisotropic or transversely isotropic coaxial matrices
  • Cracks — crack-opening-displacement tensor, compliance contribution, stress and displacement intensity factors for elliptic and ribbon cracks
  • Transport — second-order Hill tensor with a closed form for any matrix anisotropy
  • Layered inclusions — n-layer composite spheres and confocal spheroids with imperfect interfaces (perfect, spring, membrane, Kapitza, surface-conductive)
  • Homogenization schemes — bounds, dilute, Mori–Tanaka, Maxwell, Ponte Castañeda–Willis, self-consistent and asymmetric self-consistent, differential; the cluster model and the equivalent inclusion method for interacting particles
  • Aging linear Volterra behavior — aging viscoelasticity and hereditary conduction as time-domain Volterra operators with structured ISO/TI/orthotropic kernel storage, in every scheme; and, for non-aging behavior, the Laplace–Carson route with autodiff-capable numerical inversion
  • Open inclusion contract — any morphology joins every scheme through one of three entry gates (Hill tensor, localization tensors, contribution tensors), whatever computed the answer; the two below are its current tenants, not its limits
  • Finite-element inclusions — Eshelby problem solved numerically (3-D elliptical crack, sphere with an off-center core) behind a two-backend contract
  • Neural-surrogate inclusions — a trained multilayer perceptron standing in for the Eshelby response, differentiable with respect to the morphology itself and orders of magnitude cheaper than the finite-element solve it replaces
  • Prototyping-oriented — ForwardDiff sensitivities with respect to fractions, moduli and inclusion geometry, plus symbolic simplification via SymPy and Symbolics

Try it in the browser

The simplest case the package handles: an isotropic matrix holding isotropic spherical inclusions. The chart gives the effective Young’s modulus as the inclusions take up more of the volume, on a logarithmic or a linear scale; drag across it to read the values at a given fraction. The boxes under it choose the curves: the Voigt–Reuss bounds and the tighter Hashin–Shtrikman ones, which no isotropic mixture of these two phases can leave; the dilute estimates, on the stiffness or on the compliance, which hold at small fractions only; and the Mori–Tanaka, three-phase (generalized self-consistent), differential and self-consistent estimates. For spheres the Maxwell and Ponte Castañeda–Willis schemes coincide with Mori–Tanaka, and the asymmetric self-consistent scheme with the self-consistent one. Every curve is a closed form, a fixed point or a short differential equation, computed in your browser rather than by the package, and checked against the package to ten significant digits.

The interactive chart needs JavaScript.

Installation

MeanFieldHomogenization.jl is registered in Julia’s General registry:

using Pkg
Pkg.add("MeanFieldHomogenization")

Optional package extensions activate on weak dependencies: DECUHR.jl with Integrals.jl (adaptive cubature backend), NonlinearSolve.jl (SciML fixed-point algorithms for the self-consistent schemes), SymPy.jl and Symbolics.jl (symbolic computation), MittagLeffler.jl (closed-form time values of the fractional rheological models), two finite-element backends — Ferrite.jl + Gmsh.jl or Gridap.jl + GridapGmsh.jl — and Lux.jl with Optimisers.jl and Zygote.jl to train your own surrogates. Loading a shipped surrogate needs none of them: the trained network carries no machine-learning dependency.

Documentation

Full documentation is available at MicroPoroChemoMechanics.github.io/MeanFieldHomogenization.jl, including a gallery of complete micromechanical models in the Applications section — hydrating cement paste, chloride diffusivity, the interfacial transition zone in concrete, quasi-brittle strength, the elastic limit of concrete, the friction of a granular medium, the strength of a sandstone, bituminous mixtures and aging creep.