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

tests/oracle/plastic-thick-cylinder/README.md

@@ -1,3 +1,14 @@
-# Elastic–perfectly plastic thick cylinder: collapse pressure
+# Elastic–perfectly plastic thick cylinder: plastic zone and collapse pressure
 
-The Lamé cylinder of `lame-cylinder`, elastic–perfectly plastic, loaded to collapse.
+The Lamé cylinder of `lame-cylinder-axisymmetric` (a = 10, b = 20, an axisymmetric slice held
+axially on both faces, so plane strain), now elastic–perfectly plastic: E = 200000,
+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:
+
+- **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.

tests/oracle/plastic-thick-cylinder/expected.json

@@ -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"
   }
  ]
-}
+}

tests/oracle/plastic-thick-cylinder/fixture.json

@@ -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
+    }
+  }
+}

tests/oracle/plastic-thick-cylinder/recipe.json

@@ -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"}
+]