MeanFieldHomogenization.jl

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Conductivity ​

Before this page

Extension to conductivity, where the order-2 counterparts of the elastic tensors are derived, and Hill polarization tensors, whose section on conductivity gives the closed form used below.

Everything the elastic API does, the conduction API does with order-2 tensors instead of order-4. Heat conduction, mass diffusion, electric conduction and Darcy flow are the same mathematical problem, so the same functions serve all four — the dispatch is on the order of the property tensor you pass, not on a keyword.

julia
using MeanFieldHomogenization, TensND

C₀ = TensISO{3}(3 * 70.0, 2 * 30.0)   # order 4 → elasticity
K₀ = TensISO{3}(5.0)                  # order 2 → conduction

hill_tensor(Ellipsoid(1.0), C₀)       # order-4 Hill tensor ℙ
hill_tensor(Ellipsoid(1.0), K₀)       # order-2 Hill tensor 𝐏

The stress analog is minus the flux

Fourier's and Fick's laws carry a minus sign that Hooke's does not, so the package takes      as the stress analog: on a surface of normal ,   is then in both theories what the exterior transmits to the interior. That is why one implementation serves both physics with no sign anywhere, and why the tensors named flux_* below carry , not . Full statement: Elasticity and transport: one set of formulas.

Building a conductivity tensor ​

Isotropic, transversely isotropic and fully anisotropic conductivities are TensND order-2 tensors:

julia
using MeanFieldHomogenization, TensND, LinearAlgebra

K_iso   = TensISO{3}(2.5)                              # isotropic
K_ti    = TensND.TensTI{2}(1.0, 4.0, (0., 0., 1.))     # transverse, axial, axis
K_aniso = TensND.Tens(Matrix(Diagonal([3.2, 0.5, 0.6]))) # orthotropic

The Hill and Eshelby tensors ​

julia
ell = Ellipsoid(3.0, 1.0, 1.0)        # prolate spheroid

P = hill_tensor(ell, K_iso)           # 𝐏(𝐀, 𝐊)
S = eshelby_tensor(ell, K_iso)        # Eshelby tensor, P ⋅ K

For a sphere in an isotropic matrix   these are   and  , independent of .

Anisotropy is cheap here

Unlike the order-4 case, the order-2 Hill tensor has a closed form for any matrix anisotropy, via the square-root transformation     (Giraud et al., 2019). Passing an anisotropic costs no more than an isotropic one — no cubature is involved, and the result stays ForwardDiff-compatible.

Homogenization ​

An RVE carries its conduction properties under the :K key, and every scheme works unchanged:

julia
rve = RVE()
add_phase!(rve, :M, Ellipsoid(1.0), Dict(:K => TensISO{3}(1.0)); fraction = :rest)
add_phase!(rve, :I, Ellipsoid(1.0), Dict(:K => TensISO{3}(20.0)); fraction = 0.3)

K_MT = homogenize(rve, MoriTanaka(), :K)
K_SC = homogenize(rve, SelfConsistent(), :K)
K_D  = homogenize(rve, DifferentialScheme(), :K)

The third argument selects the physics: :C for elasticity, :K for conduction. A single RVE may carry both, and the two are homogenized independently.

Cracks ​

The transport analog of the crack compliance is the crack resistivity contribution , a rank-1 order-2 tensor: only the normal component of the flux can jump across a crack. It is assembled from a scalar COD , with the same geometric factors (elliptic) and (ribbon) as in elasticity.

julia
pc = PennyCrack(1.0)
K₀ = TensISO{3}(2.5)

b  = cod_tensor(pc, K₀)                    # scalar COD, = 2/(π²K) for a penny
R  = inverse_property_contribution(pc, K₀)       # 𝐑 = (3/4)·b·(ŵ⊗ŵ)
ΔR = delta_inverse_property(pc, R, 0.05)        # dilute correction at density ε

Theory: Thermal cracks.

Imperfect interfaces ​

In conduction, composite particles with imperfect interfaces are provided by:

julia
s = LayeredSpheroid(
    (1.0,), (2.0,), (TensISO{3}(1e-6),);            # insulating oblate core
    interfaces = (MeanFieldHomogenization.SurfaceConductiveInterface(2.0),),
    Nseries = 5,
)

A = gradient_gradient_concentration(s, TensISO{3}(1e-6), TensISO{3}(1.0))  # ⟨∇T⟩ = 𝐀_Ω·𝐇
B = flux_gradient_concentration(s, TensISO{3}(1e-6), TensISO{3}(1.0))      # ⟨𝐊∇T⟩ = 𝐁_Ω·𝐇
k_eq = B[1, 1] / A[1, 1]                                          # equivalent particle

Theory: Layered sphere and Layered spheroid. Worked examples: geometry and effective conductivity (Kapitza sweep and equivalent particle), what an interface does to the local fields, and highly conducting interfaces.

Conduction and elasticity are not independent — a microstructure that stiffens a material also changes how it conducts. Explicit cross-property correlations for two-phase composites are given in Sevostianov and Kachanov (2002); the Transport properties application page works through one.

Where to go next ​

Viscoelastic homogenization passes from a second physics to time dependence, on the same RVE machinery. The tutorial Transport properties applies the calls of this page to effective conductivities, and Cement paste: chloride diffusivity and elasticity computes both properties on one microstructure.