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 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
Bauschinger effect of linear kinematic hardening
One hex8 cube (E = 200000, ν = 0.3, σ_y = 250, H = 2000, Prager's linear kinematic hardening) held by symmetry on three faces, its fourth face pulled to a strain of 0.01 and brought back to zero within one nonlinear static step without nonlinear geometry, by the load path 0, 0], [0.5, 1], [1, 0 in increments of 0.05.
At ε_a the stress is σ_a = σ_y + E_t (ε_a − σ_y/E) = 267.327. The yield surface keeps its size and moves with the back stress, so unloading is elastic only until σ_a − 2σ_y and then hardens with E_t: back at zero strain σ = σ_a − 2σ_y − E_t (ε_a − 2σ_y/E) = −247.525. Isotropic hardening would yield again only at −σ_a and end at −281.835. The equivalent plastic strain accumulates (σ_a − σ_y)/H forward and E/(E + H)(ε_a − 2σ_y/E) in reverse, 0.0160891.