The interfacial transition zone in concrete

Around every aggregate in a concrete there is a thin shell of cement paste that is more porous than the bulk paste: packing of cement grains against a rigid surface leaves more room for water, and the resulting interfacial transition zone (ITZ) is softer and weaker than the material around it. How much stiffness does that cost? Microstructural model of [68].

The three scales

The paste scale is the one used in Quasi-brittle strength — the hydrate foam and cement paste model of [67], shared as scripts/common/pichler_model.jl. The new ingredient is the fourth phase geometry: coated aggregates.

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

# Densities (relative to water) and phase moduli of pichler2011.
const d_clin, d_hyd, d_san = 3.15, 2.073, 2.648
const K_clin, μ_clin = 116.7, 53.8
const K_hyd, μ_hyd = 18.7, 11.8
const K_san, μ_san = 37.8, 44.3

# Powers hydration model: volume fractions from w/c and hydration degree α.
f_clin(wc, α) = (1 - α) / (1 + d_clin * wc)
f_w(wc, α) = d_clin * (wc - 0.42 * α) / (1 + d_clin * wc)
f_hyd(wc, α) = 1.42 * d_clin / d_hyd * α / (1 + d_clin * wc)
fh_san(wc, sc) = sc / d_san / (1 / d_clin + wc + sc / d_san)

const WC, ALPHA, SC_RATIO = 0.5, 1.0, 3.0

@printf("w/c = %.2f, α = %.1f, s/c = %.1f → aggregate volume fraction %.4f\n",
        WC, ALPHA, SC_RATIO, fh_san(WC, SC_RATIO))

w/c = 0.50, α = 1.0, s/c = 3.0 → aggregate volume fraction 0.5809

Scale 1–2: the bulk cement paste

The hydrate foam is a self-consistent polycrystal of needle-shaped hydrates ($\omega = 10^4$) discretized into $N$ polar orientation bins, plus water and air; clinker grains are then embedded by Mori-Tanaka. Derivation and cross-check against Echoes: Quasi-brittle strength.

const N_BINS = 20
const ω_needle = 1.0e4
const TINY = 1.0e-3      # regularizes the exactly-zero water/air stiffness

function hydrate_foam(wc, α; N = N_BINS, ω = ω_needle)
    fclin = f_clin(wc, α)
    fw = f_w(wc, α)
    fhyd = f_hyd(wc, α)
    fair = max(0.0, 1 - fclin - fw - fhyd)
    ## Fractions renormalized to the foam, from which clinker is excluded.
    scale = 1 / (1 - fclin)
    C_hyd = TensISO{3}(3K_hyd, 2μ_hyd)
    C_fluid = TensISO{3}(3TINY, 2TINY)
    ez = (0.0, 0.0, 1.0)

    rve = RVE(:M)
    ## Zero-volume matrix phase: a self-consistent seed only, since the
    ## inclusion fractions already sum to 1.
    add_matrix!(rve, Ellipsoid(1.0), Dict(:C => C_hyd); symmetrize = :iso)
    for (i, bin) in enumerate(polar_orientation_bins(N))
        add_phase!(
            rve, Symbol(:HYD, i), Spheroid(ω; euler_angles = (bin.θ, 0.0, 0.0)),
            Dict(:C => C_hyd); fraction = fhyd * scale * bin.weight,
            symmetrize = TISymmetrize(ez),
        )
    end
    add_phase!(rve, :W, Ellipsoid(1.0), Dict(:C => C_fluid);
        fraction = fw * scale, symmetrize = :iso)
    add_phase!(rve, :AIR, Ellipsoid(1.0), Dict(:C => C_fluid);
        fraction = fair * scale, symmetrize = :iso)
    return homogenize(
        rve, SelfConsistent(; abstol = 1.0e-8, maxiters = 1000, damping = 0.5),
        :C; select_best = true,
    )
end

The converged foam estimate carries a residual transverse isotropy about the binning axis — an artifact of discretizing a uniform orientation distribution, not a physical texture. It is projected back to isotropy before the aggregate scale, which is also what the composite-sphere recurrence of the next section requires.

C_hf = best_fit_ti(hydrate_foam(WC, ALPHA), (0.0, 0.0, 1.0))

rve_cp = RVE(:HF)
add_matrix!(rve_cp, Ellipsoid(1.0), Dict(:C => C_hf))
add_phase!(
    rve_cp, :CLIN, Ellipsoid(1.0),
    Dict(:C => TensISO{3}(3K_clin, 2μ_clin)); fraction = f_clin(WC, ALPHA),
)
k_cp, μ_cp = k_mu(TensND.proj_tens(Val(:ISO), get_array(homogenize(rve_cp, MoriTanaka(), :C)))[1])
C_cp = TensISO{3}(3k_cp, 2μ_cp)

@printf("bulk cement paste : E = %.2f GPa, ν = %.3f\n", E_nu(C_cp)...)

bulk cement paste : E = 19.83 GPa, ν = 0.231

Scale 3: what an ITZ is made of

The ITZ is modelled as the same paste carrying additional capillary porosity $\varphi_{\rm ITZ}$, introduced as spherical voids by Mori-Tanaka. No empirical stiffness–porosity relation is involved: the softening comes out of the same scheme used everywhere else on this page.

function itz_stiffness(C_cp, φ)
    φ ≤ 0 && return C_cp
    rve = RVE(:CP)
    add_matrix!(rve, Ellipsoid(1.0), Dict(:C => C_cp))
    add_phase!(rve, :PORE, Ellipsoid(1.0), Dict(:C => TensISO{3}(0.0, 0.0)); fraction = φ)
    return homogenize(rve, MoriTanaka(), :C)
end

for φ in (0.1, 0.3, 0.5)
    @printf("φ_ITZ = %.1f → E_ITZ = %5.2f GPa  (bulk paste %.2f GPa)\n",
            φ, E_nu(itz_stiffness(C_cp, φ))[1], E_nu(C_cp)[1])
end

φ_ITZ = 0.1 → E_ITZ = 16.22 GPa  (bulk paste 19.83 GPa)
φ_ITZ = 0.3 → E_ITZ = 10.67 GPa  (bulk paste 19.83 GPa)
φ_ITZ = 0.5 → E_ITZ =  6.60 GPa  (bulk paste 19.83 GPa)

Scale 4: coated aggregates

An aggregate of radius $R$ surrounded by an ITZ of thickness $t$ is a two-layer LayeredSphere — a composite sphere, which MeanFieldHom accepts as an RVE phase directly: it enters the scheme through its concentration tensors, with no Hill tensor of its own.

Two bookkeeping points, both easy to get wrong:

• the volume fraction to declare is that of the whole coated particle, $f_{\rm agg}\,(1+t/R)^3$, not of the aggregate core alone;
• the matrix around it stays the bulk paste — the ITZ is already inside the inclusion, and counting it twice would double the softening.

const C_SAN = TensISO{3}(3K_san, 2μ_san)
const F_AGG = fh_san(WC, SC_RATIO)

function E_concrete(φ_itz, t_over_R)
    coated = LayeredSphere((1.0, 1.0 + t_over_R), (C_SAN, itz_stiffness(C_cp, φ_itz)))
    rve = RVE(:CP)
    add_matrix!(rve, Ellipsoid(1.0), Dict(:C => C_cp))
    add_phase!(
        rve, :AGG, coated, Dict(:C => C_SAN);
        fraction = F_AGG * (1 + t_over_R)^3,
    )
    return E_nu(homogenize(rve, MoriTanaka(), :C))[1]
end

E_ref = E_concrete(0.0, 0.05)   # ITZ as stiff as the bulk paste ⇒ no ITZ at all
@printf("concrete without ITZ contrast : E = %.2f GPa\n", E_ref)

concrete without ITZ contrast : E = 44.36 GPa

How much stiffness does the ITZ cost?

φs = range(0.0, 0.5; length = 21)
plt = plot(
    xlabel = "additional ITZ porosity φ_ITZ", ylabel = "E_concrete / E (no ITZ)",
    legend = :bottomleft, title = "Elastic cost of a porous ITZ (w/c = $WC, α = $ALPHA)",
)
for t in (0.02, 0.05, 0.10)
    plot!(plt, φs, [E_concrete(φ, t) / E_ref for φ in φs];
          lw = 2, marker = :circle, ms = 2, label = "t/R = $t")
end
plt

Three readings of the figure:

• the loss is roughly linear in $\varphi_{\rm ITZ}$ and grows with thickness, as one would expect of a thin compliant shell in series with the load path;
• it is modest for realistic ITZs: at $t/R = 0.05$ and $\varphi_{\rm ITZ} = 0.3$ — a 5 % shell with a third of it porous — concrete stiffness falls by about 11 %. A porous ITZ does not, on its own, explain a concrete that is half as stiff as expected;
• the sensitivity to thickness is what makes the ITZ hard to identify from stiffness measurements alone: $t/R$ and $\varphi_{\rm ITZ}$ trade off against each other along the curves.

That last point is the reason [68] turn to the elastic limit rather than the elastic modulus: the ITZ is a far more conspicuous feature of where concrete starts to fail than of how stiff it is. Reproducing that argument would need the strength machinery listed in the scope note above.

@printf("t/R = 0.05, φ_ITZ = 0.30 → E/E₀ = %.3f\n", E_concrete(0.3, 0.05) / E_ref)

t/R = 0.05, φ_ITZ = 0.30 → E/E₀ = 0.894