The 3D twin renders true roll-bite geometry and estimates the total roll force from a single closed-form "friction hill" formula. This 2D version instead numerically solves the actual slab-method pressure equation along the contact arc — a genuinely different computation, not the same scene flattened.
Slab equilibrium (Bland-Ford / Orowan form):
d(p·h)/dx = sign·2·τ, τ = min(μ·p, k)
k = Y_f/√3 (shear yield — Coulomb friction
saturates to "sticking" near the peak)
Boundary condition: p = Y_f at both the entry (x=-L)
and exit (x=0) edges, where the strip is not yet
compressed above its flow stress.
Integrating forward from the entry edge (sign=+1, roll surface outruns the strip — backward slip) and separately backward from the exit edge (sign=−1, strip outruns the roll — forward slip) produces two rising pressure branches. Where they cross is the neutral (no-slip) point — found here directly from the physics, not from the 3D twin's small-angle approximation φₙ ≈ α/2 − Δh/(4μR).
Checked against the 3D twin's formula: at typical drafts the two neutral-point estimates and total forces agree to within roughly 10–50% — exactly the known gap between a full slab-method integration and the classic closed-form Ekelund-style estimate in rolling-mechanics textbooks. Both readings are shown side by side so you can see that gap directly rather than assume the two models must match.
Mass-conservation velocity is unchanged from the 3D
sim: v(x)·h(x) = const, evaluated at the NUMERIC
neutral point's own h, not the closed-form one.
- Draft Δh — bigger reduction steepens the bite and raises both the peak pressure and the force.
- Roll radius R — a bigger roll gives a longer, gentler contact arc — more force, but spread over a longer L.
- Friction μ — higher friction raises the hill (more sticking) and moves the neutral point toward the entry side.
- Roll speed — sets the strip's absolute entry/exit speed via mass conservation; it does not change the pressure distribution shape (this quasi-static slab model has no rate dependence).