This is the same nanocomposite spar as the 3D version, viewed a different way: an independent 2D cross-section and a strength model the 3D scene never computed. Stiffness still comes from real Halpin-Tsai micromechanics:
η = (Ef/Em − 1) / (Ef/Em + ξ), ξ = 2·(L/d)
E_L = Em·(1 + ξ·η·Vf) / (1 − η·Vf), E_T uses ξ=2
E(θ) ≈ E_T + (E_L − E_T)·cos⁴θ
The top panel is a stereological cross-section: cutting a plane through the spar perpendicular to its span meets each cylindrical nanofiber at an angle, so a fiber tilted θ off the cut normal traces an ellipse whose major axis stretches by 1/cosθ — a real projection effect, not a flattened 3D render. Circle/ellipse count is solved so the covered area actually matches the target Vf, and the achieved area fraction is displayed as a live self-check.
The middle panel is the cantilever's bending-moment diagram M(x) = w·(L−x)²/2, and the bottom panel maps the through-thickness bending stress σ(x,y) = M(x)·y/I as a red/blue heatmap alongside the (exaggerated) deflection curve — a field the 3D scene never rendered.
Strength uses the Kelly-Tyson short-fiber model: a nanofiber only carries its full tensile load once it is longer than a critical length lc = σf·d/(2τi) (shear-lag load transfer through the matrix). Below that length efficiency drops linearly:
η_l = 1 − l_c/(2l) if l ≥ l_c
η_l = l / (2·l_c) if l < l_c
σ_c ≈ η_l·cos⁴θ·Vf·σ_f + (1−Vf)·σ_m
Safety factor = σ_c / σ_max, σ_max = M(0)·(h/2)/I
- Aspect ratio — with σf=4 GPa and interfacial shear strength τi=20 MPa here, the critical aspect ratio is exactly 100: below it a nanofiber slips in the matrix before it snaps, above it the fiber's own tensile strength becomes the limit.
- Alignment angle θ — off-axis fibers both stiffen and strengthen less (cos⁴θ), and their cross-section ellipse visibly elongates.
- Vf and wind speed — drive the same stiffness/load trade-off as the 3D version; the safety factor readout here is the piece that actually tells you whether a lay-up survives the load.
Together the two views make the point real spar designers rely on: Halpin-Tsai modulus alone doesn't guarantee a safe blade — short, poorly aligned nanofibers can raise stiffness while barely raising strength, which is exactly what the safety-factor panel exposes.