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 <[email protected]>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

docs/oracle.md

@@ -2,7 +2,7 @@
 
 Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](verification.md). Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its `expected.json` cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's `fixture.json`: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.
 
-**149 of 155 quantities pass** (6 known failure(s)) in 44 active cases: 35 theory, 9 published benchmark, 0 experiment; 9 pending cases wait for capabilities still being built.
+**163 of 169 quantities pass** (6 known failure(s)) in 49 active cases: 40 theory, 9 published benchmark, 0 experiment; 9 pending cases wait for capabilities still being built.
 
 ## Theory: closed-form solutions, patch tests and mesh invariants
 
@@ -31,8 +31,10 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | sixth frequency (the last rigid-body mode) | 0 | ≈ 0 | ≈ 0 | ± 8.1264e-5 | pass | matches |
 | [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | first free–free bending frequency (mode 7) | 0.081264 | 0.080618 | -0.795% | 1.5% | pass | matches |
 | [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | total mass | 20 | 20 | < 1e-7% | 1e-12 rel. | pass | matches |
-| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | axial stress back at zero strain after a pull to ε_a = 0.01 | -247.52 | -247.52 | < 1e-7% | 1e-6 rel. | pass | blessed |
-| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | accumulated equivalent plastic strain | 0.016089 | 0.016089 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | axial stress back at zero strain after a pull to ε_a = 0.01 | -247.52 | -247.52 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | accumulated equivalent plastic strain | 0.016089 | 0.016089 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | axial stress, ΔT = 300 (α ΔT = 3.6·10⁻³, 2.9 times the yield strain) | -254.65 | -254.65 | < 1e-7% | 1e-8 rel. | pass | blessed |
+| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | equivalent plastic strain | 0.0023267 | 0.0023267 | < 1e-7% | 1e-7 rel. | pass | blessed |
 | [j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/) | axial Cauchy stress at a logarithmic strain of 0.2 | 643.56 | 643.57 | 0.000174% | 1% | pass | matches |
 | [j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/) | equivalent plastic strain | 0.19678 | 0.19678 | -0.00000113% | 1% | pass | matches |
 | [kirsch-plate](../tests/oracle/kirsch-plate/) | σ_xx at the top of the hole (0, 1, 0) over the remote stress | 3 | 3.0121 | 0.402% | 2% | pass | matches |
@@ -59,8 +61,15 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 5 | 0.38763 | 0.38955 | 0.495% | 2% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 10 | 0.81061 | 0.81421 | 0.444% | 1% | pass | matches |
 | [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 10 | 0.555 | 0.558 | 0.541% | 2% | pass | matches |
-| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | blessed |
-| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | equivalent plastic strain | 0.19678 | 0.19678 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | equivalent plastic strain | 0.19678 | 0.19678 | < 1e-7% | 1e-6 rel. | pass | matches |
+| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | axial Cauchy stress, uniaxial stretch λ = 2 | 2.7366 | 2.7366 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | lateral displacement at (1, 1, 1), uniaxial: l − 1 | -0.28647 | -0.28647 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | in-plane Cauchy stress, equibiaxial stretch λ = 1.5 | 2.8427 | 2.8427 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress along the stretch, pure shear λ = 2 | 3.5261 | 3.5261 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress across, in the held direction, pure shear λ = 2 | 1.5067 | 1.5067 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5 | 5 | 5.0311 | 0.622% | 1% | pass | blessed |
+| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii | 0.61553 | 0.64339 | 2.8e-2 | ± 0.04 | pass | blessed |
 | [nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/) | axial Cauchy stress at the corner (1, 1, 1), stretch λ = 2 | 8.0601 | 8.0601 | < 1e-7% | 1e-6 rel. | pass | matches |
 | [nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/) | lateral displacement u_y at the corner (1, 1, 1), l − 1 | -0.18686 | -0.18686 | < 1e-7% | 1e-6 rel. | pass | matches |
 | [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | u_x at inner node 22 | 0.13583 | 0.13583 | < 1e-7% | 1e-8 rel. | pass | matches |
@@ -76,6 +85,11 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | u_y at inner node 22: the rotation's | 0.078591 | 0.078591 | -5.6e-17 | ± 1.0000e-9 | pass | matches |
 | [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | u_z at inner node 22: the rotation's | -0.067441 | -0.067441 | -1.2e-16 | ± 1.0000e-9 | pass | matches |
 | [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | von Mises stress at inner node 22 | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | matches |
+| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | displacement of the corner (1, 1, 1) along x, free, α ΔT = 0.2 | 0.2 | 0.2 | < 1e-7% | 1e-8 rel. | pass | blessed |
+| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the free body (σ_xx at the corner) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | blessed |
+| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the held block, σ_xx (hydrostatic) | -3.5108 | -3.5108 | < 1e-7% | 1e-8 rel. | pass | blessed |
+| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | axial Cauchy stress, stretch λ = 2 | 3.1743 | 3.1743 | < 1e-7% | 1e-6 rel. | pass | blessed |
+| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | lateral displacement at (1, 1, 1): l − 1 | -0.28545 | -0.28545 | < 1e-7% | 1e-6 rel. | pass | blessed |
 | [patch-hex20](../tests/oracle/patch-hex20/) | u_x at the corner (1, 1, 1) | -2.5000e-4 | -2.5000e-4 | < 1e-7% | 1e-9 rel. | pass | matches |
 | [patch-hex20](../tests/oracle/patch-hex20/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-7% | 1e-9 rel. | pass | matches |
 | [patch-hex20](../tests/oracle/patch-hex20/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
@@ -125,9 +139,9 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 | [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_yy at the inner node (0.46, 0.55, 0.53) | 2 | 2 | < 1e-7% | 1e-9 rel. | pass | matches |
 | [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_zz at the inner node (0.46, 0.55, 0.53) | 3 | 3 | < 1e-7% | 1e-9 rel. | pass | matches |
 | [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
-| [plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/) | tip force at a tip deflection of 15 (4.5 times the first yield's) | -312.5 | -322.61 | -3.23% | 5% | pass | blessed |
-| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250) | 0.024355 | 0.024358 | 0.0089% | 0.2% | pass | blessed |
-| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | 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 | 12572 | 12572 | 0.00000178% | 0.1% | pass | blessed |
+| [plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/) | tip force at a tip deflection of 15 (4.5 times the first yield's) | -312.5 | -322.61 | -3.23% | 5% | pass | matches |
+| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250) | 0.024355 | 0.024358 | 0.0089% | 0.2% | pass | matches |
+| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | 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 | 12572 | 12572 | 0.00000178% | 0.1% | pass | matches |
 | [pure-bending-coupling](../tests/oracle/pure-bending-coupling/) | bending stress σ_xx at mid-span, top fibre (10, 0.5, 1) | 15 | 15 | -0.0000103% | 0.5% | pass | matches |
 | [pure-bending-coupling](../tests/oracle/pure-bending-coupling/) | rotation of the reference point about y | 0.3 | 0.30355 | 1.18% | 1% | known failure: 1.2% high: the distributing coupling weights every node of the end face equally, corner and mid-side nodes of hex20 alike, which is not the consistent linear traction; the end face deforms locally and the fitted rotation includes it. Area-consistent weights would remove it | — |
 | [rotating-cylinder-axisymmetric](../tests/oracle/rotating-cylinder-axisymmetric/) | radial stress σ_r on the axis (r = 0) | 0.38571 | 0.38608 | 0.0958% | 1% | pass | matches |
@@ -181,6 +195,7 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 - **[cdt-stepped-block](../tests/oracle/cdt-stepped-block/)** (theory): Constrained Delaunay mesh of a stepped block: exact volume, valid elements. Source: mesh invariants: a tetrahedralization of a polyhedron fills its volume exactly (Si, TetGen paper ACM TOMS 41(2) 2015 for the definition of a CDT; Diazzi et al., ACM TOG 42(6) 2023); the volume is 60·40·10 + 40·40·10 + 20·40·10 − 10·10·10. Tolerances — volume: a planar boundary is tessellated exactly and the constrained Delaunay mesh fills it exactly: only rounding; invalid: every element of a valid mesh has positive volume; undetermined: a linear tetrahedron's sign is exact
 - **[free-beam-frequencies](../tests/oracle/free-beam-frequencies/)** (theory): Natural frequencies of a free beam: rigid-body modes and the first bending mode. Source: Euler–Bernoulli free–free beam, β₁L = 4.7300; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3. Tolerances — rigid_6: a free body has six zero frequencies; numerical rigid-body frequencies below 10⁻³ of the first elastic one (eigenvalues below 10⁻⁶ of it) are rounding of the shifted solve; f7: Euler–Bernoulli neglects shear and rotary inertia, about 0.8% at L/h = 20 for the free–free mode; mass: the mass matrix integrates the density exactly
 - **[j2-bauschinger](../tests/oracle/j2-bauschinger/)** (theory): Bauschinger effect of linear kinematic hardening: tension, then back to zero strain. 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). Tolerances — residual_stress: 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); plastic_strain: (σ_a − σ_y)/H forward plus E/(E + H) (ε_a − 2σ_y/E) in reverse
+- **[j2-heated-bar](../tests/oracle/j2-heated-bar/)** (theory): A steel bar held at its ends and heated past yield (small strain). Source: Uniaxial J2 plasticity with linear hardening (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) with the additive thermal strain: the bar's total axial strain is held at zero, so its mechanical strain is −α ΔT and σ = −(σ_y + E_t (α ΔT − σ_y/E)) once α ΔT exceeds σ_y/E. Tolerances — sigma: a homogeneous uniaxial state the elements represent exactly; plastic_strain: (\|σ\| − σ_y)/H, as for the stress
 - **[j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (explicit). Source: von Mises plasticity with linear isotropic hardening under uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3); logarithmic strain for a hypoelastic–plastic law under coaxial stretching. Tolerances — sigma: uniaxial J2 with linear hardening: σ = σ_y + E_t (ε − σ_y/E) in logarithmic strain; the stretch takes 10⁴ wave transits, so inertia and the ringing of the start (ρcv ≈ 0.1% of σ) stay well under 1%; plastic_strain: ε_p = ε − σ/E, as for the stress
 - **[kirsch-plate](../tests/oracle/kirsch-plate/)** (theory): Plate with a circular hole in tension (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, Heywood's width correction about +0.1%, and the loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
 - **[kirsch-plate-plane-stress](../tests/oracle/kirsch-plate-plane-stress/)** (theory): Plate with a circular hole in tension, 2D plane stress (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
@@ -198,9 +213,13 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
 - **[nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/)** (theory): Inflation of a thick neo-Hookean cylinder by a follower pressure (plane strain). Source: Finite inflation of an incompressible neo-Hookean thick-walled tube in plane strain: r² = R² + a² − A², σ_θ − σ_r = μ (λ² − λ⁻²) with λ = r/R, integrated through the wall: p = (μ/2)[(ln λ_a² − λ_a⁻²) − (ln λ_b² − λ_b⁻²)] (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, §5.3.2). Tolerances — ua: the closed form is incompressible and our material nearly so (κ/μ = 1000, a volume change of order p/κ ≈ 4·10⁻⁴); hybrid hex20, 6 × 16, follow the wall's strain gradient well inside; ub: as at the bore
 - **[nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/)** (theory): Large-deflection cantilever under a dead tip load against the elastica (solids). 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. Tolerances — v1: 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; u1: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v2: 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; u2: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v5: 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; u5: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v10: 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; u10: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's
 - **[nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain). 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). Tolerances — sigma: a homogeneous deformation the element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding; plastic_strain: ε_p = (τ − σ_y)/H, as for the stress
+- **[nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/)** (theory): Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension. 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). Tolerances — uniaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; uniaxial_lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; equibiaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_xx: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_yy: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
+- **[nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/)** (theory): Inflation of a stretched Mooney–Rivlin tube (follower pressure). 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). Tolerances — ua: 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; ub: 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
 - **[nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/)** (theory): Uniaxial stretch of a neo-Hookean block to twice its length. Source: Closed form of the compressible neo-Hookean model W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² in uniaxial stress (G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000, §6.5, eq. 6.148 with the volumetric term of §6.4): σ_yy = μ J^(−5/3)(l² − Ī/3) + κ(J − 1) = 0 fixes the lateral stretch l, then σ_xx = μ J^(−5/3)(λ² − l²). Tolerances — sigma: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance; lateral: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance
 - **[nl-patch-hex8](../tests/oracle/nl-patch-hex8/)** (theory): Nonlinear patch test, hex8: a homogeneous finite deformation of a distorted mesh. Source: The patch test (B. M. Irons, A. Razzaque, in A. K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, 557–587) at finite strain: a boundary moved by x = F X makes F the solution everywhere (J. Bonet, R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed., Cambridge 2008, §8); the neo-Hookean Cauchy stress σ = μ J^(−5/3) dev b + κ (J − 1) I of that F. Tolerances — ux: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxx: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; szz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance
 - **[nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/)** (theory): A quarter turn strains and stresses nothing (objectivity). Source: Objectivity of the total Lagrangian formulation and of the hyperelastic stress: a rigid motion x = R X + c gives C = I and σ = 0 whatever the rotation (Bonet & Wood 2008, §3.4 and §5.2; T. Belytschko, W. K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, 2nd ed., Wiley 2014, §5.5). Tolerances — ux: the inner nodes follow the rigid rotation exactly; uy: the inner nodes follow the rigid rotation exactly; uz: the inner nodes follow the rigid rotation exactly; von_mises: zero up to rounding of stresses of order E = 1000: a 90° turn leaves no strain
+- **[nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/)** (theory): Free and fully constrained thermal expansion at finite strain. Source: The multiplicative split F = F_m F_θ with F_θ = (1 + α ΔT) I (V. A. Lubarda, Appl. Mech. Rev. 57(2):95–108, 2004): a free body follows F = F_θ stress-free; held on every face, F = I and F_m = ϑ⁻¹ I, so the compressible neo-Hookean material carries the pressure of its volume ratio, σ = κ (ϑ⁻³ − 1). Tolerances — free_u: a homogeneous stretch every element represents exactly; free_stress: rounding only; held_stress: a homogeneous state; E = 10, ν = 0.3 (κ = 8.3333), ϑ = 1.2
+- **[nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/)** (theory): Yeoh block in uniaxial tension. Source: Uniaxial tension of the decoupled compressible Yeoh energy W = C₁₀ (Ī₁ − 3) + C₂₀ (Ī₁ − 3)² + C₃₀ (Ī₁ − 3)³ + κ/2 (J − 1)² (O. H. Yeoh, Rubber Chem. Technol. 66(5):754–771, 1993): σ = (λ/J) ∂W/∂λ with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.49, C₁₀ = μ/2, C₂₀ = −0.02, C₃₀ = 0.004). Tolerances — sigma: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
 - **[patch-hex20](../tests/oracle/patch-hex20/)** (theory): Constant-stress patch test, hex20. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
 - **[patch-hex20-reduced](../tests/oracle/patch-hex20-reduced/)** (theory): Constant-stress patch test, hex20-reduced. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
 - **[patch-hex8](../tests/oracle/patch-hex8/)** (theory): Constant-stress patch test, hex8. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)