jyvikoandClaude Opus 5.5
7a69fe5e2f
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
928 B
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.