verification: implicit plasticity oracle cases and documentation

Four cases, each against its closed form: uniaxial tension of a J2 cube to a
logarithmic strain of 0.2 (Kirchhoff stress exact; 0.13 % from the explicit
solver's hypoelastic Cauchy stress, the elastic volume change between the two
laws), the Bauschinger effect of linear kinematic hardening on a load path
back to zero strain (exact), Hill's elastic-perfectly plastic thick cylinder
(bore displacement at a plastic radius of 1.5a within 0.009 %, collapse
pressure within 2e-8; the pending case is now active) and a cantilever's
plastic collapse, converged 3.1 % above beam theory's hinge, the bound a
continuum exceeds. docs/fem.md, the literature, ROADMAP (M15), parity and the
oracle page follow.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed 2026-10-14 06:03:36 -04:00
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# Plastic collapse of a cantilever
A steel bar 200 long with a 10 × 10 section (E = 200000, ν = 0.3, elastic–perfectly plastic at
σ_y = 250), reduced hex20 20 × 2 × 4, in a nonlinear static step without nonlinear geometry. The
root x = 0 is a plane of symmetry (no rotation, the section free to contract laterally), held in
Y, Z and against twist on average through a distributing coupling; a coupling at the tip pushes
the end down by 15, 4.5 times the deflection at first yield.
Beam theory's collapse is a plastic hinge at the root: M_p = σ_y b h²/4 = 62500 and
P_c = M_p/L = 312.5. The solid's collapse lies above it, since a continuum cannot rotate about a
hinge (that needs a jump of the normal velocity across the section): refined to 8 elements
through the depth or 160 along the length, the force settles at 322.3, 3.1 % above. The case
checks that the implicit plasticity reaches this plateau and holds it.