fem: hybrid plane strain and axisymmetric elements

One constant pressure per element (mean dilatation, Nagtegaal, Parks & Rice 1974; the hoop strain in the divergence) for quad4, quad8 and tri6: the linear stiffness, stresses and thermal load, and the finite- and small-strain forms of nonlinear steps with their consistent tangent. The linear triangle and plane stress are refused by name. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
2026-10-14 16:30:44 -04:00
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+# Inflation of a stretched Mooney–Rivlin tube, axisymmetric
+
+The tube of `nl-mooney-rivlin-tube` (A = 10, B = 20, one unit long; E = 3, ν = 0.4995,
+C₀₁ = 0.2) as an axisymmetric model of 12 × 2 hybrid quad4 (one constant pressure each, the mean
+dilatation with the hoop strain): the end y = 0 on a symmetry line, the end y = 1 stretched to
+λ_z = 1.5 while the follower pressure p = 0.540850 inflates the bore. The closed form puts the
+bore at a = 15 and the outer wall at b = 20.615528 (see `nl-mooney-rivlin-tube`); this case checks
+the nearly incompressible 2D hybrid elements at large strain against it.
@@ -0,0 +1,53 @@
+{
+ "title": "Inflation of a stretched Mooney–Rivlin tube (axisymmetric hybrid quad4)",
+ "kind": "theory",
+ "source": "Finite inflation and extension of an incompressible Mooney–Rivlin tube (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals)",
+ "quantities": [
+  {
+   "id": "ua",
+   "quantity": "radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Inflate",
+     "at": [
+      [
+       10.0,
+       0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0"
+   },
+   "value": 5.0,
+   "tolerance": {
+    "relative": 0.01
+   },
+   "reason": "16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube"
+  },
+  {
+   "id": "ub",
+   "quantity": "radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii",
+   "measure": {
+    "command": "solve.probe",
+    "params": {
+     "step": "Inflate",
+     "at": [
+      [
+       20.0,
+       0,
+       0.0
+      ]
+     ]
+    },
+    "pointer": "/nodes/0/displacement/0"
+   },
+   "value": 0.6155281280883038,
+   "tolerance": {
+    "absolute": 0.04
+   },
+   "reason": "0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there"
+  }
+ ]
+}
@@ -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": {
+    "ua": {
+      "band": 5.03287906011305e-9,
+      "value": 5.03287906011305
+    },
+    "ub": {
+      "band": 0.000040000644728979273,
+      "value": 0.6447289792693816
+    }
+  }
+}
@@ -0,0 +1,14 @@
+[
+ {"command": "mesh.block", "params": {"body": "Pipe", "kind": "quad4", "corners": [[10.0, 0, 0], [20.0, 0, 0], [20.0, 1, 0], [10.0, 1, 0]], "divisions": [12, 2], "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": "bottom", "geometry": [{"body": "Pipe", "edge": "bottom"}]}},
+ {"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Pipe", "edge": "top"}]}},
+ {"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.4995}, "hyperelastic": {"type": "mooney_rivlin", "c01": 0.2}}},
+ {"command": "model.section.create", "params": {"name": "pipe", "set": "all", "material": "Rubber", "kind": {"type": "axisymmetric"}, "formulation": {"hybrid": true}}},
+ {"command": "model.boundary_condition.create", "params": {"name": "sb", "type": "symmetry", "target": {"set": "bottom"}, "plane": "y"}},
+ {"command": "model.boundary_condition.create", "params": {"name": "stretch", "type": "displacement", "target": {"set": "top"}, "values": [null, 0.5, null, null, null, null]}},
+ {"command": "model.load.create", "params": {"name": "p", "type": "pressure", "set": "bore", "magnitude": 0.5408503415684837}},
+ {"command": "model.step.create", "params": {"name": "Inflate", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sb", "stretch"], "loads": ["p"]}},
+ {"command": "solve.nonlinear", "params": {"step": "Inflate"}, "id": "solve"}
+]