forked from 1b-agent/fieldcraft-upstream
jyvikoandClaude Opus 5.5
d586d84859
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 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
53 lines
2.0 KiB
JSON
53 lines
2.0 KiB
JSON
{
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"title": "Inflation of a stretched Mooney–Rivlin tube (follower pressure)",
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"kind": "theory",
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"source": "Finite inflation and extension of an incompressible Mooney–Rivlin tube (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals)",
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"quantities": [
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{
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"id": "ua",
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"quantity": "radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Inflate",
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"at": [
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[
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10.0,
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0,
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0
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]
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]
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},
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"pointer": "/nodes/0/displacement/0"
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},
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"value": 5.0,
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"tolerance": {
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"relative": 0.01
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},
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"reason": "16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube"
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},
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{
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"id": "ub",
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"quantity": "radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Inflate",
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"at": [
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[
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20.0,
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0,
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0
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]
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]
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},
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"pointer": "/nodes/0/displacement/0"
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},
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"value": 0.6155281280883038,
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"tolerance": {
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"absolute": 0.04
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},
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"reason": "0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there"
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}
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]
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} |