When a polymer matrix is strained, load reaches a stiff, short nanofiller (a carbon nanotube, graphene flake or clay platelet) only indirectly — through interfacial shear at the filler surface, not through a direct end-grip. This is the Cox/Kelly–Tyson shear-lag model:
σ_f(x) = E_f·ε·[1 − cosh(βx)/cosh(βL/2)]
τ_i(x) = (r·β/2)·E_f·ε·sinh(βx)/cosh(βL/2)
β² = 2G_m / [E_f·r²·ln(R/r)], ln(R/r) ≈ −½ln(V_f)
x runs from −L/2 to +L/2 along the filler's axis. Axial stress σ_f is zero at the tips (nothing to grip onto) and rises to a plateau near E_f·ε at the center once the filler is long enough — interfacial shear τ_i does the opposite, peaking at the tips and vanishing at the center.
A filler shorter than the critical length l_c = r·E_f·ε_matrix,max /τ_interface,max never reaches its full load-bearing stress before it runs out of length — this is exactly why aspect ratio (L/d) is the single biggest lever nanofillers have over microscale fibers: nanoscale diameters make huge aspect ratios achievable even at short absolute lengths.
The reinforcement this delivers to the whole composite is captured by the Krenchel/Cox modified rule of mixtures used for the E꜀ readout:
η_l = 1 − tanh(βL/2)/(βL/2) (length efficiency)
E꜀ = η_l·η_o·E_f·V_f + E_m·(1−V_f) (η_o = orientation efficiency)
- Filler buttons — swap the filler's Young's modulus E_f (single-wall CNT ≈1000 GPa, graphene ≈1050 GPa, exfoliated nanoclay ≈170 GPa, glass fiber ≈72 GPa for scale).
- Aspect ratio L/d — governs η_l directly; below roughly L/d≈20–50 most nanofillers can't reach useful axial stress before their tips run out of matrix to shear against.
- Volume fraction V_f — raises E꜀ roughly linearly but also tightens filler spacing (R/r), which stiffens the shear-lag response.
- Applied strain ε — the far-field matrix strain driving load transfer; scales σ_f and τ_i together.
- Filler alignment — interpolates the orientation efficiency η_o from 1 (perfectly aligned with load) toward ≈0.2 (randomly oriented, Krenchel factor for 3D random), matching how real nanocomposites lose stiffness when fillers aren't aligned by processing (extrusion shear, electric/magnetic field alignment, etc).
The center rod's color shows σ_f(x)/E_f (blue→teal→amber→red = low→high axial stress) while the small halo of background fillers illustrates the surrounding composite at the chosen volume fraction and alignment.