verification: a nearly incompressible hybrid block under prescribed stretch

nl-hybrid-incompressible: 27 hybrid hex8 at ν = 0.4999 pulled to λ = 2 by a prescribed
displacement, against the compressible neo-Hookean closed form (1.4·10⁻⁹); its first
iterations need the LDLᵀ factorization. ROADMAP and parity updated, docs/oracle.md regenerated.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed 2026-10-14 12:42:12 -04:00
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{
"title": "Nearly incompressible hybrid block under prescribed stretch",
"kind": "theory",
"source": "Uniaxial stress of the decoupled compressible neo-Hookean energy W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² (P. J. Flory, Trans. Faraday Soc. 57:829–838, 1961): σ = μ J^(−5/3) (λ² − I₁/3) + κ (J − 1) with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.4999: μ = 1.0000667, κ = 5000)",
"quantities": [
{
"id": "sigma",
"quantity": "axial Cauchy stress, stretch λ = 2, ν = 0.4999",
"measure": {
"command": "solve.probe",
"params": {
"step": "Uniaxial",
"at": [
[
1,
1,
1
]
]
},
"pointer": "/nodes/0/stress/0"
},
"value": 3.498756466061613,
"tolerance": {
"relative": 1e-06
},
"reason": "a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding"
},
{
"id": "lateral",
"quantity": "lateral displacement at (1, 1, 1): l − 1",
"measure": {
"command": "solve.probe",
"params": {
"step": "Uniaxial",
"at": [
[
1,
1,
1
]
]
},
"pointer": "/nodes/0/displacement/1"
},
"value": -0.29281075714096483,
"tolerance": {
"relative": 1e-06
},
"reason": "a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding"
}
]
}