The 3D simulator only tracks elastic instability: it treats the hull as an idealised thin membrane and asks whether hydrostatic pressure exceeds the classical (Zoelly) buckling pressure. It never asks whether the material itself would yield first.
This 2D version instead solves the exact Lamé thick-hollow-sphere elasticity equations for the radial and hoop (tangential) stress at every point through the wall, given the true internal cabin pressure pi (1 atm) and external hydrostatic pressure po(h):
σ_r(r) = C − (a³b³(p_o−p_i)) / (r³(b³−a³))
σ_θ(r) = C + (a³b³(p_o−p_i)) / (2r³(b³−a³))
C = (p_i·a³ − p_o·b³) / (b³−a³)
a = R−t (inner radius), b = R (outer radius)
The equivalent (von Mises) stress at radius r reduces to σ_vm(r) = |σ_r(r) − σ_θ(r)| because σ_θ = σ_φ by spherical symmetry. Stress concentrates at the inner wall (r=a) — the classical result for external pressure alone is σ_θ(a) = −(3/2)·P·b³/(b³−a³), roughly 1.5× the thin-membrane estimate. This 2D solver watches that concentration factor grow as the wall gets thicker relative to the radius, and directly compares two independent failure criteria as depth increases:
- Material yield — max von Mises stress through the wall reaches the material's yield strength (or, for brittle acrylic, its tensile failure strength). This can happen well before the idealised elastic-buckling pressure is ever reached, especially for thicker-walled hulls where stress concentration dominates.
- Elastic buckling — the same Zoelly thin-shell criterion as the 3D simulator, computed independently here for comparison.
- Wall cross-section (top canvas) — a schematic ring from inner cabin wall to outer hull wall, colour-mapped by local von Mises stress (wall thickness is exaggerated for visibility; the underlying radii used in the physics are the real values shown in the readouts).
- Stress-vs-radius graph (bottom canvas) — the actual σ_r(r) and σ_θ(r) curves through the wall thickness, with the material's yield/failure strength drawn as a reference line.
Verified standalone: the formulas reproduce the exact boundary conditions σ_r(a)=−p_i, σ_r(b)=−p_o; match the classical inner-surface stress-concentration formula for p_i=0; converge to thin-wall membrane theory (σ≈−PR/2t) as t/R→0; and for a titanium R=1.00 m, t=60 mm hull, correctly show material yield (~9.9 km) occurring well before Zoelly buckling (~20 km) — a real, distinct failure race the 3D model cannot represent.