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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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# Mooney–Rivlin block in Treloar's three modes
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One hybrid hex20 unit cube on three symmetry planes, E = 3, ν = 0.49 (μ = 1.006711,
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κ = 50), Mooney–Rivlin with C₀₁ = 0.2 (C₁₀ = μ/2 − C₀₁ = 0.303356), in three nonlinear
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static steps:
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- **Uniaxial**: the face x = 1 pulled to λ = 2, the others free.
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- **Equibiaxial**: x = 1 and y = 1 pulled to λ = 1.5, z free.
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- **Pure shear**: x = 1 pulled to λ = 2, y = 1 held (λ_y = 1), z free.
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The references are the compressible decoupled energy's principal Cauchy stresses
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σᵢ = (λᵢ/J) ∂W/∂λᵢ, by differences of W alone, with the free stretch found by bisection so the
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free faces carry no stress. The deformation is homogeneous, so the element represents it
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exactly.
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{
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"title": "Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension",
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"kind": "theory",
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"source": "Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁)",
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"quantities": [
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{
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"id": "uniaxial",
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"quantity": "axial Cauchy stress, uniaxial stretch λ = 2",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Uniaxial",
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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": 2.736633402084229,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
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},
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{
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"id": "uniaxial_lateral",
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"quantity": "lateral displacement at (1, 1, 1), uniaxial: l − 1",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Uniaxial",
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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/1"
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},
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"value": -0.28647206688129256,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
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},
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{
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"id": "equibiaxial",
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"quantity": "in-plane Cauchy stress, equibiaxial stretch λ = 1.5",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Equibiaxial",
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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": 2.842734818108988,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
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},
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{
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"id": "shear_xx",
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"quantity": "Cauchy stress along the stretch, pure shear λ = 2",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Shear",
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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": 3.526124960232999,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
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},
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{
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"id": "shear_yy",
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"quantity": "Cauchy stress across, in the held direction, pure shear λ = 2",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Shear",
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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/1"
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},
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"value": 1.506681177907714,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
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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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"equibiaxial": {
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"band": 2.8427348183579755e-9,
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"value": 2.8427348183579753
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},
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"shear_xx": {
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"band": 3.5261249603786212e-9,
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"value": 3.526124960378621
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},
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"shear_yy": {
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"band": 1.5066811779077142e-9,
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"value": 1.5066811778673217
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},
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"uniaxial": {
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"band": 2.736633402109491e-9,
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"value": 2.736633402109491
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},
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"uniaxial_lateral": {
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"band": 2.8647206688145424e-10,
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"value": -0.2864720668814542
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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": [1, 1, 1], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1", "5": "y1"}}},
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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.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.49}, "hyperelastic": {"type": "mooney_rivlin", "c01": 0.2}}},
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{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Rubber", "kind": {"type": "solid"}, "formulation": {"hybrid": true}}},
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{"command": "model.boundary_condition.create", "params": {"name": "sx", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sy", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
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{"command": "model.boundary_condition.create", "params": {"name": "sz", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
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{"command": "model.boundary_condition.create", "params": {"name": "pull_u", "type": "displacement", "target": {"set": "x1"}, "values": [1.0, null, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "pull_e", "type": "displacement", "target": {"set": "x1"}, "values": [0.5, null, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "pull_ey", "type": "displacement", "target": {"set": "y1"}, "values": [null, 0.5, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "pull_s", "type": "displacement", "target": {"set": "x1"}, "values": [1.0, null, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "hold_y", "type": "displacement", "target": {"set": "y1"}, "values": [null, 0.0, null, null, null, null]}},
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{"command": "model.step.create", "params": {"name": "Uniaxial", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_u"], "loads": []}},
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{"command": "model.step.create", "params": {"name": "Equibiaxial", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_e", "pull_ey"], "loads": []}},
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{"command": "model.step.create", "params": {"name": "Shear", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_s", "hold_y"], "loads": []}},
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{"command": "solve.nonlinear", "params": {"step": "Uniaxial"}, "id": "uniaxial"},
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{"command": "solve.nonlinear", "params": {"step": "Equibiaxial"}, "id": "equibiaxial"},
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{"command": "solve.nonlinear", "params": {"step": "Shear"}, "id": "shear"}
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]
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