forked from 1b-agent/fieldcraft-upstream
verification: implicit plasticity oracle cases and documentation
Four cases, each against its closed form: uniaxial tension of a J2 cube to a logarithmic strain of 0.2 (Kirchhoff stress exact; 0.13 % from the explicit solver's hypoelastic Cauchy stress, the elastic volume change between the two laws), the Bauschinger effect of linear kinematic hardening on a load path back to zero strain (exact), Hill's elastic-perfectly plastic thick cylinder (bore displacement at a plastic radius of 1.5a within 0.009 %, collapse pressure within 2e-8; the pending case is now active) and a cantilever's plastic collapse, converged 3.1 % above beam theory's hinge, the bound a continuum exceeds. docs/fem.md, the literature, ROADMAP (M15), 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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# Bauschinger effect of linear kinematic hardening
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One hex8 cube (E = 200000, ν = 0.3, σ_y = 250, H = 2000, Prager's linear kinematic hardening)
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held by symmetry on three faces, its fourth face pulled to a strain of 0.01 and brought back to
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zero within one nonlinear static step without nonlinear geometry, by the load path
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[[0, 0], [0.5, 1], [1, 0]] in increments of 0.05.
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At ε_a the stress is σ_a = σ_y + E_t (ε_a − σ_y/E) = 267.327. The yield surface keeps its size
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and moves with the back stress, so unloading is elastic only until σ_a − 2σ_y and then hardens
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with E_t: back at zero strain σ = σ_a − 2σ_y − E_t (ε_a − 2σ_y/E) = −247.525. Isotropic hardening
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would yield again only at −σ_a and end at −281.835. The equivalent plastic strain accumulates
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(σ_a − σ_y)/H forward and E/(E + H)(ε_a − 2σ_y/E) in reverse, 0.0160891.
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@@ -0,0 +1,45 @@
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{
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"title": "Bauschinger effect of linear kinematic hardening: tension, then back to zero strain",
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"kind": "theory",
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"source": "J2 plasticity with Prager's linear kinematic hardening in uniaxial stress (W. Prager, Ing.-Arch. 1955; Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3): from σ_a = σ_y + E_t (ε_a − σ_y/E) the material unloads elastically by 2σ_y and then hardens with E_t, so at zero strain σ = σ_a − 2σ_y − E_t (ε_a − 2σ_y/E)",
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"quantities": [
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{
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"id": "residual_stress",
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"quantity": "axial stress back at zero strain after a pull to ε_a = 0.01",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/stress/0/0"
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},
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"value": -247.52475247524754,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "a homogeneous small-strain state the element represents exactly, with increments that end at the load path's turn; isotropic hardening would give −281.835 (it yields again only at −σ_a)"
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},
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{
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"id": "plastic_strain",
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"quantity": "accumulated equivalent plastic strain",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/plastic_strain/0"
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},
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"value": 0.016089108910891083,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "(σ_a − σ_y)/H forward plus E/(E + H) (ε_a − 2σ_y/E) in reverse"
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}
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]
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}
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@@ -0,0 +1,13 @@
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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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"plastic_strain": {
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"band": 1.6089108910891082e-11,
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"value": 0.01608910891089074
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},
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"residual_stress": {
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"band": 2.4752475247524754e-7,
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"value": -247.52475247484173
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}
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}
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}
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@@ -0,0 +1,16 @@
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[
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{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex8", "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"}}},
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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.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 200000.0, "poisson": 0.3}, "plastic": {"type": "bilinear", "yield_stress": 250.0, "hardening_modulus": 2000.0, "hardening": "kinematic"}}},
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{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Steel", "kind": {"type": "solid"}}},
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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", "type": "displacement", "target": {"set": "x1"}, "values": [0.01, null, null, null, null, null]}},
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{"command": "model.step.create", "params": {"name": "Pull", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false, "initial_increment": 0.05, "max_increment": 0.05, "load_path": [[0, 0], [0.5, 1], [1, 0]]}, "boundary_conditions": ["sx", "sy", "sz", "pull"], "loads": []}},
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{"command": "solve.nonlinear", "params": {"step": "Pull"}, "id": "solve"}
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]
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# Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit)
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The cube of `j2-uniaxial-explicit` solved by the implicit nonlinear solver: one fully integrated
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hex8, E = 200000, ν = 0.3, σ_y = 250, H = 2000 (isotropic), held by symmetry on three faces and
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stretched on the fourth to λ = e^0.2 in a nonlinear static step, lateral faces free.
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Simo's exponential return map integrates this proportional path exactly in the logarithmic
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strain, so the Kirchhoff stress is the small-strain curve τ = σ_y + E_t (ε − σ_y/E) = 643.564 at
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ε = 0.2 and the equivalent plastic strain (τ − σ_y)/H = 0.196782. The Cauchy stress is
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σ = τ/J with Hencky's volume law ln J = τ/(3K): 642.737.
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Cross-check with the explicit solver: `j2-uniaxial-explicit` reaches 643.57 for the Cauchy stress
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with the same plastic strain. Its hypoelastic update takes the logarithmic strain rate against
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the Cauchy stress, while the implicit model is hyperelastic in the Kirchhoff stress: the two
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laws differ by the factor J = 1.0013 (the elastic volume change), which is the 0.13 % between
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them; both meet the closed form of their own law.
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@@ -0,0 +1,45 @@
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{
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"title": "Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain)",
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"kind": "theory",
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"source": "J2 plasticity with linear isotropic hardening in uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) in the logarithmic strain, which Simo's exponential return map (CMAME 99:61–112, 1992) integrates exactly on a proportional path: Kirchhoff stress τ = σ_y + E_t (ε − σ_y/E), Cauchy stress σ = τ/J with ln J = τ/(3K) (Hencky's volume law)",
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"quantities": [
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{
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"id": "sigma",
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"quantity": "axial Cauchy stress at a logarithmic strain of 0.2",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/stress/0/0"
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},
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"value": 642.7365391418857,
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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 element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding"
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},
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{
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"id": "plastic_strain",
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"quantity": "equivalent plastic strain",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/plastic_strain/0"
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},
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"value": 0.19678217821782182,
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"tolerance": {
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"relative": 1e-06
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},
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"reason": "ε_p = (τ − σ_y)/H, as for the stress"
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}
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]
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}
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@@ -0,0 +1,13 @@
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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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"plastic_strain": {
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"band": 1.9678217821795454e-10,
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"value": 0.19678217821795452
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},
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"sigma": {
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"band": 6.427365391418857e-7,
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"value": 642.7365390754569
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}
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}
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}
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@@ -0,0 +1,16 @@
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[
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{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex8", "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"}}},
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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.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 200000.0, "poisson": 0.3}, "plastic": {"type": "bilinear", "yield_stress": 250.0, "hardening_modulus": 2000.0, "hardening": "isotropic"}}},
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{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Steel", "kind": {"type": "solid"}}},
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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", "type": "displacement", "target": {"set": "x1"}, "values": [0.22140275816016985, null, null, null, null, null]}},
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{"command": "model.step.create", "params": {"name": "Pull", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull"], "loads": []}},
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{"command": "solve.nonlinear", "params": {"step": "Pull"}, "id": "solve"}
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]
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# Plastic collapse of a cantilever
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A steel bar 200 long with a 10 × 10 section (E = 200000, ν = 0.3, elastic–perfectly plastic at
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σ_y = 250), reduced hex20 20 × 2 × 4, in a nonlinear static step without nonlinear geometry. The
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root x = 0 is a plane of symmetry (no rotation, the section free to contract laterally), held in
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Y, Z and against twist on average through a distributing coupling; a coupling at the tip pushes
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the end down by 15, 4.5 times the deflection at first yield.
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Beam theory's collapse is a plastic hinge at the root: M_p = σ_y b h²/4 = 62500 and
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P_c = M_p/L = 312.5. The solid's collapse lies above it, since a continuum cannot rotate about a
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hinge (that needs a jump of the normal velocity across the section): refined to 8 elements
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through the depth or 160 along the length, the force settles at 322.3, 3.1 % above. The case
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checks that the implicit plasticity reaches this plateau and holds it.
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@@ -0,0 +1,20 @@
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{
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"title": "Plastic collapse of a cantilever: the fully plastic moment at the root",
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"kind": "theory",
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"source": "Limit analysis (B. G. Neal, The Plastic Methods of Structural Analysis, 3rd ed., Chapman & Hall 1977, ch. 1–2): beam theory's plastic hinge at the root, the fully plastic moment M_p = σ_y b h²/4 of a rectangular section, collapses a cantilever of length L at P_c = M_p/L = 312.5 (b = h = 10, L = 200, σ_y = 250); a bound for the solid, whose collapse lies above it (W. Prager, P. G. Hodge, Theory of Perfectly Plastic Solids, Wiley 1951: a continuum mechanism allows no normal velocity jump, which the beam's hinge rotation is)",
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"quantities": [
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{
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"id": "collapse_force",
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"quantity": "tip force at a tip deflection of 15 (4.5 times the first yield's)",
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"measure": {
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"from": "solve",
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"pointer": "/reference_points/1/reaction/2"
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},
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"value": -312.5,
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"tolerance": {
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"relative": 0.05
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},
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"reason": "the solid's collapse lies above beam theory's hinge: refined through the depth (4 to 8 elements) and along the length (40 to 160) the force settles 3.1 % above M_p/L (322.3), so the band is that bound's gap; the root is a symmetry plane (no rotation, free lateral contraction) held on average through a coupling, the tip pushed through another; reduced hex20, 20 × 2 × 4"
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}
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]
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}
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@@ -0,0 +1,9 @@
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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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"collapse_force": {
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"band": 3.2260685847425373e-7,
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"value": -322.6068584742537
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}
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}
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}
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@@ -0,0 +1,17 @@
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[
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{"command": "mesh.block", "params": {"body": "Beam", "kind": "hex20", "corners": [[0, 0, -5.0], [200.0, 0, -5.0], [200.0, 10.0, -5.0], [0, 10.0, -5.0], [0, 0, 5.0], [200.0, 0, 5.0], [200.0, 10.0, 5.0], [0, 10.0, 5.0]], "divisions": [20, 2, 4], "faces": {"6": "root", "4": "tip"}}},
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{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Beam"}]}},
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{"command": "model.set.create", "params": {"name": "root", "geometry": [{"body": "Beam", "face": "root"}]}},
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{"command": "model.set.create", "params": {"name": "tip", "geometry": [{"body": "Beam", "face": "tip"}]}},
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{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 200000.0, "poisson": 0.3}, "plastic": {"type": "bilinear", "yield_stress": 250.0}}},
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{"command": "model.section.create", "params": {"name": "beam", "set": "all", "material": "Steel", "kind": {"type": "solid"}, "formulation": {"integration": "reduced"}}},
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{"command": "model.boundary_condition.create", "params": {"name": "plane", "type": "symmetry", "target": {"set": "root"}, "plane": "x"}},
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{"command": "model.reference_point.create", "params": {"name": "Root", "xyz": [0, 5.0, 0]}},
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{"command": "model.coupling.create", "params": {"name": "support", "reference_point": "Root", "set": "root", "kind": "distributing"}},
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{"command": "model.boundary_condition.create", "params": {"name": "hold", "type": "displacement", "target": {"reference_point": "Root"}, "values": [null, 0, 0, 0, null, null]}},
|
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{"command": "model.reference_point.create", "params": {"name": "Tip", "xyz": [200.0, 5.0, 0]}},
|
||||
{"command": "model.coupling.create", "params": {"name": "end", "reference_point": "Tip", "set": "tip", "kind": "distributing"}},
|
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{"command": "model.boundary_condition.create", "params": {"name": "push", "type": "displacement", "target": {"reference_point": "Tip"}, "values": [null, null, -15.0, null, null, null]}},
|
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{"command": "model.step.create", "params": {"name": "Collapse", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false}, "boundary_conditions": ["plane", "hold", "push"], "loads": []}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Collapse"}, "id": "solve"}
|
||||
]
|
||||
@@ -1,3 +1,14 @@
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# Elastic–perfectly plastic thick cylinder: collapse pressure
|
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# Elastic–perfectly plastic thick cylinder: plastic zone and collapse pressure
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The Lamé cylinder of `lame-cylinder`, elastic–perfectly plastic, loaded to collapse.
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The Lamé cylinder of `lame-cylinder-axisymmetric` (a = 10, b = 20, an axisymmetric slice held
|
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axially on both faces, so plane strain), now elastic–perfectly plastic: E = 200000,
|
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ν = 0.4999 (Hill's solution takes the elasticity incompressible), σ_y = 250, one row of 20
|
||||
reduced quad8 elements through the wall, in two nonlinear static steps without nonlinear
|
||||
geometry:
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|
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- **Pressure**: the bore pressure p = k (2 ln(c/a) + 1 − c²/b²) = 180.195 (k = σ_y/√3) that
|
||||
takes the plastic zone to c = 15. The bore then moves u_a = k c²/(2 G a) = 0.024355: the plastic
|
||||
zone moves incompressibly (u r constant) and the elastic ring outside yields at its inner edge.
|
||||
- **Collapse**: the bore pushed out by 0.3 (24 times its first-yield displacement), past the
|
||||
pressure at which the wall is plastic through; the radial force on the bore stays at the
|
||||
collapse pressure 2k ln(b/a) = 200.094 times 2πa per unit height.
|
||||
@@ -1,14 +1,43 @@
|
||||
{
|
||||
"title": "Elastic–perfectly plastic thick cylinder: collapse pressure",
|
||||
"title": "Elastic–perfectly plastic thick cylinder: plastic zone and collapse pressure",
|
||||
"kind": "theory",
|
||||
"source": "Hill, The Mathematical Theory of Plasticity, Oxford 1950, ch. V §2: plane strain von Mises collapse pressure p = (2/√3)σ_y ln(b/a)",
|
||||
"status": "pending",
|
||||
"requires": "implicit (static) J2 plasticity: explicit J2 is in, a collapse load needs the static solver",
|
||||
"source": "Hill, The Mathematical Theory of Plasticity, Oxford 1950, ch. V §2: a von Mises cylinder in plane strain, elastically incompressible, with the plastic zone a ≤ r ≤ c under the pressure p = k (2 ln(c/a) + 1 − c²/b²), k = σ_y/√3, and the bore displacement u_a = k c²/(2 G a); collapse when c = b at p = 2k ln(b/a)",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "p_collapse",
|
||||
"quantity": "collapse pressure for a = 10, b = 20, σ_y = 250",
|
||||
"value": 200.0944
|
||||
"id": "u_bore",
|
||||
"quantity": "bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250)",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Pressure",
|
||||
"at": [
|
||||
[
|
||||
10.0,
|
||||
0,
|
||||
0
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/0"
|
||||
},
|
||||
"value": 0.02435534068380524,
|
||||
"tolerance": {
|
||||
"relative": 0.002
|
||||
},
|
||||
"reason": "20 reduced quad8 elements through the wall resolve the logarithmic plastic field and ν = 0.4999 stands in for Hill's incompressible elasticity (0.009 % apart); u_a ∝ c², so 0.2 % on u_a is 0.1 % on the plastic radius"
|
||||
},
|
||||
{
|
||||
"id": "collapse_force",
|
||||
"quantity": "radial force on the bore (per radian times 2π, height 1) at a bore displacement of 0.3, 24 times the first yield's: the collapse pressure 2k ln 2 = 200.094 times 2πa",
|
||||
"measure": {
|
||||
"from": "collapse",
|
||||
"pointer": "/reaction_total/0"
|
||||
},
|
||||
"value": 12572.299154203694,
|
||||
"tolerance": {
|
||||
"relative": 0.001
|
||||
},
|
||||
"reason": "once the wall is plastic through, perfect plasticity holds the force at the collapse value (σ_θ − σ_r = 2k everywhere, σ_z the mean of the others as plastic flow dominates); the band admits the quadrature of the logarithmic field"
|
||||
}
|
||||
]
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,13 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"collapse_force": {
|
||||
"band": 0.000012572299377396248,
|
||||
"value": 12572.299377396248
|
||||
},
|
||||
"u_bore": {
|
||||
"band": 2.4357507851220506e-11,
|
||||
"value": 0.024357507851220504
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,15 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Pipe", "kind": "quad8", "corners": [[10.0, 0, 0], [20.0, 0, 0], [20.0, 1, 0], [10.0, 1, 0]], "divisions": [20, 1], "bias": [1, 1], "edges": [{"corners": [1, 4], "name": "bore"}, {"corners": [1, 2], "name": "bottom"}, {"corners": [3, 4], "name": "top"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Pipe", "face": "area"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "bore", "geometry": [{"body": "Pipe", "edge": "bore"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "ends", "geometry": [{"body": "Pipe", "edge": "bottom"}, {"body": "Pipe", "edge": "top"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 200000.0, "poisson": 0.4999}, "plastic": {"type": "bilinear", "yield_stress": 250.0}}},
|
||||
{"command": "model.section.create", "params": {"name": "pipe", "set": "all", "material": "Steel", "kind": {"type": "axisymmetric"}, "formulation": {"integration": "reduced"}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "ends", "type": "symmetry", "target": {"set": "ends"}, "plane": "y"}},
|
||||
{"command": "model.load.create", "params": {"name": "p", "type": "pressure", "set": "bore", "magnitude": 180.19538034924005}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "expand", "type": "displacement", "target": {"set": "bore"}, "values": [0.3, null, null, null, null, null]}},
|
||||
{"command": "model.step.create", "params": {"name": "Pressure", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false}, "boundary_conditions": ["ends"], "loads": ["p"]}},
|
||||
{"command": "model.step.create", "params": {"name": "Collapse", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false}, "boundary_conditions": ["ends", "expand"], "loads": []}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Pressure"}, "id": "pressure"},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Collapse"}, "id": "collapse"}
|
||||
]
|
||||
Reference in new issue
Block a user