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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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# Free and constrained thermal expansion at finite strain
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A unit cube of 2 × 2 × 2 hex20, compressible neo-Hookean (E = 10, ν = 0.3) with an expansion
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coefficient α = 10⁻³ from 20 degrees, heated to 220 by a temperature load in two nonlinear static
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steps: **Free** (three symmetry planes) and **Held** (symmetry planes on all six faces). The
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thermal stretch is ϑ = 1 + α ΔT = 1.2. Free, the cube grows to 1.2 along every edge without
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stress; held, F = I, the material's mechanical part is F_m = ϑ⁻¹ I and its stress the neo-Hookean
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pressure κ (J_m − 1) with J_m = ϑ⁻³: −3.510802.
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{
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"title": "Free and fully constrained thermal expansion at finite strain",
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"kind": "theory",
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"source": "The multiplicative split F = F_m F_θ with F_θ = (1 + α ΔT) I (V. A. Lubarda, Appl. Mech. Rev. 57(2):95–108, 2004): a free body follows F = F_θ stress-free; held on every face, F = I and F_m = ϑ⁻¹ I, so the compressible neo-Hookean material carries the pressure of its volume ratio, σ = κ (ϑ⁻³ − 1)",
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"quantities": [
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{
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"id": "free_u",
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"quantity": "displacement of the corner (1, 1, 1) along x, free, α ΔT = 0.2",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Free",
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"at": [
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[
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1,
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1,
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1
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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": 0.2,
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"tolerance": {
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"relative": 1e-08
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},
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"reason": "a homogeneous stretch every element represents exactly"
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},
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{
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"id": "free_stress",
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"quantity": "stress of the free body (σ_xx at the corner)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Free",
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"at": [
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[
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1,
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1,
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1
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]
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]
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},
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"pointer": "/nodes/0/stress/0"
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},
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"value": 0.0,
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"tolerance": {
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"absolute": 1e-08
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},
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"reason": "rounding only"
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},
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{
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"id": "held_stress",
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"quantity": "stress of the held block, σ_xx (hydrostatic)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Held",
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"at": [
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[
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0.5,
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0.5,
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0.5
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]
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]
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},
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"pointer": "/nodes/0/stress/0"
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},
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"value": -3.5108024691358017,
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"tolerance": {
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"relative": 1e-08
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},
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"reason": "a homogeneous state; E = 10, ν = 0.3 (κ = 8.3333), ϑ = 1.2"
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}
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]
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}
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{
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"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
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"values": {
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"free_stress": {
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"band": 1e-8,
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"value": -1.0882345665298319e-14
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},
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"free_u": {
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"band": 2.0000000000000037e-10,
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"value": 0.20000000000000034
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},
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"held_stress": {
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"band": 3.510802469135803e-9,
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"value": -3.5108024691358026
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}
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}
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}
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[
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{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex20", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]], "divisions": [2, 2, 2], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1", "5": "y1", "2": "z1"}}},
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Cube"}]}},
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{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Cube", "face": "x0"}]}},
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{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Cube", "face": "y0"}]}},
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{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Cube", "face": "z0"}]}},
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{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Cube", "face": "x1"}]}},
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{"command": "model.set.create", "params": {"name": "y1", "geometry": [{"body": "Cube", "face": "y1"}]}},
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{"command": "model.set.create", "params": {"name": "z1", "geometry": [{"body": "Cube", "face": "z1"}]}},
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{"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 10.0, "poisson": 0.3}, "hyperelastic": {"type": "neo_hookean"}, "thermal": {"expansion": 0.001, "reference_temperature": 20.0}}},
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{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Rubber", "kind": {"type": "solid"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "sx0", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sy0", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sz0", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sx1", "type": "symmetry", "target": {"set": "x1"}, "plane": "x"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sy1", "type": "symmetry", "target": {"set": "y1"}, "plane": "y"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sz1", "type": "symmetry", "target": {"set": "z1"}, "plane": "z"}},
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{"command": "model.load.create", "params": {"name": "heat", "type": "temperature", "target": {"set": "all"}, "value": 220.0}},
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{"command": "model.step.create", "params": {"name": "Free", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx0", "sy0", "sz0"], "loads": ["heat"]}},
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{"command": "model.step.create", "params": {"name": "Held", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx0", "sy0", "sz0", "sx1", "sy1", "sz1"], "loads": ["heat"]}},
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{"command": "solve.nonlinear", "params": {"step": "Free"}, "id": "free"},
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{"command": "solve.nonlinear", "params": {"step": "Held"}, "id": "held"}
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]
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