verification: nonlinear oracle cases and the nonlinear solver's documentation

Theory cases for the nonlinear solver, each frozen as a fixture: uniaxial
stretch of a neo-Hookean block to twice its length (closed form), the
nonlinear patch test on a distorted hex8 mesh, a quarter turn of the same
mesh (objectivity: zero stress), the elastica of a hex20 cantilever at
P L²/(E I) = 1, 2, 5, 10 against Bisshopp–Drucker (0.2–0.5 %), and the
inflation of a thick neo-Hookean tube by a follower pressure against
Rivlin's closed form (0.08 % at the bore). A finite-strain Cook's membrane
is pending its transcribed reference. docs/fem.md describes the
formulation, the material point interface, the follower-pressure choice and
what nonlinear steps refuse; the literature, parity and roadmap pages
follow.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

tests/oracle/nl-elastica-cantilever/expected.json

@@ -0,0 +1,199 @@
+{
+ "title": "Large-deflection cantilever under a dead tip load against the elastica (solids)",
+ "kind": "theory",
+ "source": "The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits",
+ "quantities": [
+  {
+   "id": "v1",
+   "quantity": "tip deflection δ_v/L at P L²/(E I) = 1",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K1",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/2",
+    "scale": -0.1
+   },
+   "value": 0.301721,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
+  },
+  {
+   "id": "u1",
+   "quantity": "tip shortening δ_h/L at P L²/(E I) = 1",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K1",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0",
+    "scale": -0.1
+   },
+   "value": 0.056433,
+   "tolerance": {
+    "relative": 0.02
+   },
+   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
+  },
+  {
+   "id": "v2",
+   "quantity": "tip deflection δ_v/L at P L²/(E I) = 2",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K2",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/2",
+    "scale": -0.1
+   },
+   "value": 0.493457,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
+  },
+  {
+   "id": "u2",
+   "quantity": "tip shortening δ_h/L at P L²/(E I) = 2",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K2",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0",
+    "scale": -0.1
+   },
+   "value": 0.160642,
+   "tolerance": {
+    "relative": 0.02
+   },
+   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
+  },
+  {
+   "id": "v5",
+   "quantity": "tip deflection δ_v/L at P L²/(E I) = 5",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K5",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/2",
+    "scale": -0.1
+   },
+   "value": 0.713792,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
+  },
+  {
+   "id": "u5",
+   "quantity": "tip shortening δ_h/L at P L²/(E I) = 5",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K5",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0",
+    "scale": -0.1
+   },
+   "value": 0.387628,
+   "tolerance": {
+    "relative": 0.02
+   },
+   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
+  },
+  {
+   "id": "v10",
+   "quantity": "tip deflection δ_v/L at P L²/(E I) = 10",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K10",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/2",
+    "scale": -0.1
+   },
+   "value": 0.810609,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
+  },
+  {
+   "id": "u10",
+   "quantity": "tip shortening δ_h/L at P L²/(E I) = 10",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "K10",
+     "at": [
+      [
+       10.0,
+       0,
+       0.25
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0",
+    "scale": -0.1
+   },
+   "value": 0.554996,
+   "tolerance": {
+    "relative": 0.02
+   },
+   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
+  }
+ ]
+}