verification: hyperelastic and thermal oracle cases and documentation

Five cases against closed forms: a Mooney-Rivlin block in Treloar's
uniaxial, equibiaxial and pure-shear modes and a Yeoh block in uniaxial
tension (the compressible energy's stresses, exact); a stretched
Mooney-Rivlin tube inflated by a follower pressure (Rivlin's incompressible
solution, 0.6 % at nu = 0.4995); free and fully held thermal expansion at a
thermal stretch of 1.2 (exact); a bar held at its ends heated past yield
(exact). docs/fem.md, the literature, ROADMAP (M15, M17), 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 07:23:06 -04:00
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{
"title": "A steel bar held at its ends and heated past yield (small strain)",
"kind": "theory",
"source": "Uniaxial J2 plasticity with linear hardening (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) with the additive thermal strain: the bar's total axial strain is held at zero, so its mechanical strain is −α ΔT and σ = −(σ_y + E_t (α ΔT − σ_y/E)) once α ΔT exceeds σ_y/E",
"quantities": [
{
"id": "sigma",
"quantity": "axial stress, ΔT = 300 (α ΔT = 3.6·10⁻³, 2.9 times the yield strain)",
"measure": {
"command": "solve.probe",
"params": {
"step": "Heat",
"elements": [
2
]
},
"pointer": "/elements/0/stress/0/0"
},
"value": -254.65346534653466,
"tolerance": {
"relative": 1e-08
},
"reason": "a homogeneous uniaxial state the elements represent exactly"
},
{
"id": "plastic_strain",
"quantity": "equivalent plastic strain",
"measure": {
"command": "solve.probe",
"params": {
"step": "Heat",
"elements": [
2
]
},
"pointer": "/elements/0/plastic_strain/0"
},
"value": 0.0023267326732673297,
"tolerance": {
"relative": 1e-07
},
"reason": "(|σ| − σ_y)/H, as for the stress"
}
]
}