From derivatives to a strength criterion

This capstone tutorial combines the last three pages — porous materials, scheme comparison, and automatic differentiation — into a single applied result: a macroscopic strength criterion for a porous solid, derived without ever writing a closed-form strength formula by hand. It ports the porous strength-criterion benchmark of the Echoes book [57].

Setup

The solid and pore are both oblate spheroids (aspect ratio $\omega = 0.1$) with a uniform spatial distribution of orientations (IsoSymmetrize, as in the porous benchmark tutorial), so the homogenized stiffness is isotropic. The solid is much stiffer than the pore, at a fixed porosity $\varphi = 0.15$:

using MeanFieldHom
using TensND
using ForwardDiff
using LinearAlgebra
using Plots
gr()  # headless backend; GKSwstype is set to "100" in make.jl

const k_s, μ_s0 = 1.0e6, 1.0
const TINY = 1.0e-12
const ω = 0.1
const φ = 0.15

The criterion

Under a macroscopic stress split into a mean part $\Sigma_m$ and a deviatoric magnitude $\Sigma_d$, a porous solid governed by a von Mises-type local strength $\sigma_o$ in the solid phase admits, to second order in the contrast, an elliptical macroscopic strength domain:

\[\frac{(\Sigma_m/\sigma_o)^2}{2A} + \frac{(\Sigma_d/\sigma_o)^2}{B} = \frac{1}{1-\varphi}, \qquad A = \Big(\frac{\mu_s}{k_{\text{hom}}}\Big)^{\!2}\frac{\partial k_{\text{hom}}}{\partial\mu_s}, \qquad B = \Big(\frac{\mu_s}{\mu_{\text{hom}}}\Big)^{\!2}\frac{\partial \mu_{\text{hom}}}{\partial\mu_s}.\]

$A$ and $B$ are not independent material data — they are the sensitivities of the two homogenized moduli to the solid's own shear modulus, exactly the kind of derivative built up over the previous tutorial. Differentiating a closure that homogenizes and reads back (k_hom, μ_hom) gives both at once:

function k_mu_hom(μ_s::T, scheme) where {T}
    r = RVE(:SOLID; T = T)
    add_matrix!(r, Spheroid(ω), Dict(:C => iso_stiffness(convert(T, k_s), μ_s)); symmetrize = IsoSymmetrize())
    add_phase!(r, :PORE, Spheroid(ω), Dict(:C => iso_stiffness(convert(T, TINY), convert(T, TINY)));
               fraction = convert(T, φ), symmetrize = IsoSymmetrize())
    C = homogenize(r, scheme, :C)
    return [k_mu(best_fit_iso(C))...]
end

function ellipse_radii(scheme)
    k_hom, μ_hom = k_mu_hom(μ_s0, scheme)
    dk_dμs, dμ_dμs = ForwardDiff.derivative(μ_s -> k_mu_hom(μ_s, scheme), μ_s0)
    A = (μ_s0 / k_hom)^2 * dk_dμs
    B = (μ_s0 / μ_hom)^2 * dμ_dμs
    a = sqrt((1 - φ) / (2A))   # semi-axis along Σ_m/σ_o
    b = sqrt((1 - φ) / B)      # semi-axis along Σ_d/σ_o
    return a, b
end

k_mu_hom uses only public accessors — iso_stiffness, homogenize, best_fit_iso, k_mu — the same building blocks as every earlier tutorial; there is no need to reach into the package's internals to differentiate through a homogenization call, whatever scheme is used.

Strength ellipses for six schemes

SCHEMES = [
    (MoriTanaka(), "MT"),
    (DiluteDual(), "DiluteDual"),
    (SelfConsistent(; abstol = 1.0e-8, maxiters = 300, select_best = true), "SelfConsistent"),
    (AsymmetricSelfConsistent(; abstol = 1.0e-8, maxiters = 300, select_best = true), "AsymmetricSelfConsistent"),
    (PonteCastanedaWillis(), "PCW"),
    (Maxwell(), "Maxwell"),
]

θ = range(0, π; length = 100)
plt = plot(;
    xlabel = "Σ_m / σ_o", ylabel = "Σ_d / σ_o",
    aspect_ratio = :equal, legend = :outerright, framestyle = :box, size = (760, 480),
)
for (scheme, label) in SCHEMES
    a, b = ellipse_radii(scheme)
    plot!(plt, a .* cos.(θ), b .* sin.(θ); lw = 2, label = label)
end
hline!(plt, [0.0]; color = :black, lw = 0.5, label = "")
vline!(plt, [0.0]; color = :black, lw = 0.5, label = "")
plt

The ellipses differ noticeably between schemes — SelfConsistent and AsymmetricSelfConsistent predict a markedly smaller admissible domain than Mori-Tanaka or Maxwell at this porosity, the same qualitative ranking (percolating vs. isolated-pore topology) seen throughout the porous benchmark tutorial. Mean-field homogenization gives the moduli; automatic differentiation gives their sensitivities; together they yield a macroscopic strength criterion — with no closed-form derivative ever written by hand, and no assumption beyond the choice of scheme. For the same construction at three nested scales, see Quasi-brittle strength of cement paste and mortar.