forked from 1b-agent/fieldcraft-upstream
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
1006 B
1006 B
Elastic–perfectly plastic thick cylinder: plastic zone and collapse pressure
The Lamé cylinder of lame-cylinder-axisymmetric (a = 10, b = 20, an axisymmetric slice held
axially on both faces, so plane strain), now elastic–perfectly plastic: E = 200000,
ν = 0.4999 (Hill's solution takes the elasticity incompressible), σ_y = 250, one row of 20
reduced quad8 elements through the wall, in two nonlinear static steps without nonlinear
geometry:
- Pressure: the bore pressure p = k (2 ln(c/a) + 1 − c²/b²) = 180.195 (k = σ_y/√3) that takes the plastic zone to c = 15. The bore then moves u_a = k c²/(2 G a) = 0.024355: the plastic zone moves incompressibly (u r constant) and the elastic ring outside yields at its inner edge.
- Collapse: the bore pushed out by 0.3 (24 times its first-yield displacement), past the pressure at which the wall is plastic through; the radial force on the bore stays at the collapse pressure 2k ln(b/a) = 200.094 times 2πa per unit height.