Each carbon nanotube crossing a nanoscale crack in the cement matrix behaves like a shear-lag pullout fiber. As the crack opening displacement δ grows, a bridging tube slides a distance s = δ / cos θ along its own axis (θ = inclination to the crack normal):
Elastic (bonded) stage, s ≤ s_deb:
F(s) = k·s, F(s_deb) = F_bond = τ·π·d·L_e·e^(f·θ)
Frictional pullout stage, s > s_deb:
L_e(s) = L_e0 − (s − s_deb)
F(s) = 0.7τ·π·d·L_e(s)·e^(f·θ)
Rupture criterion (checked at debonding):
F_bond > F_rupture = σ_CNT·π·d²/4 → tube snaps instead of sliding
τ is the CNT–cement interfacial shear strength, d the tube diameter, L_e the embedded length on the shorter side, e^(f·θ) a snubbing factor for inclined tubes (f ≈ 0.7), and σ_CNT ≈ 30 GPa a fixed single-wall tensile strength. Summing the axial force's opening-direction component over every tube per unit crack area gives the crack-bridging stress:
σ_bridge(δ) = Σᵢ F_i(δ)·cos θᵢ / A_crack
G(δ) = ∫₀^δ σ_bridge(δ') dδ' (toughening energy)
- Crack opening displacement — widens the crack; tubes stretch elastically, then debond and slide, then either rupture or fully pull out.
- Areal CNT density — how many nanotubes cross a unit area of the crack face; more tubes raise bridging stress but each carries the same load.
- Interfacial shear strength τ — bond quality between tube and cement paste; higher τ raises pullout capacity but also raises the odds of rupture instead of sliding.
- CNT diameter — bigger tubes have more bonded surface (∝ d) but the rupture load grows faster (∝ d²), so thick tubes tend to pull out rather than snap.
This crack-bridging mechanism — not just stiffer bulk material — is why adding a small mass fraction of multiwall carbon nanotubes to cement paste ("nanoconcrete") measurably increases fracture toughness and delays crack growth in real structural tests. The top panel below is a cross-section through the crack face: tubes span the gap, embedded stubs (dimmer) reach into the solid cement on each side, and colour tracks each tube's state. The lower-left panel plots the bridging-stress curve against COD with the shaded area under it equal to the toughening energy already absorbed; the lower-right panel tallies how many of the active tubes are in each state right now.