Elliptical Hertz Contact

Elliptical Hertz Contact (structures/fem/elliptical-hertz-contact)

Use when the task is computing the general Hertz elliptical contact patch between two elastic bodies pressed together with UNEQUAL principal radii (ball in a conforming groove or raceway, crossed unequal cylinders): the per-plane curvature sums from the signed principal curvatures, the eccentricity of the contact ellipse from the complete elliptic integrals, the major and minor semi-axes a and b, the peak pressure p0 = 3P/(2 pi a b), the approach and the yield-limit margin. This is the general elliptical-patch solution of the classical Hertz theory (Johnson, Contact Mechanics, ch. 4 style, paraphrased, never reproduced). Pairing leaves: hertzian-contact-stress owns the circular-patch (equal radii) and line-contact (strip) degeneracies only and explicitly fences "Unequal crossed radii give the general elliptical patch of the Hertz elliptic integrals, out of scope here"; this leaf claims only the unequal-principal- curvature elliptical patch and evaluates the a = b limit as an identity check.

Domain quick reference

• Equivalent elastic modulus: 1/E* = (1 - nu1^2)/E1 + (1 - nu2^2)/E2. Steel on steel (E = 207 GPa, nu = 0.3): E* = 113.73626373626374 GPa.
• Per-plane curvature sums from the signed principal curvatures of both bodies in their own frames (convex positive, concave negative, flat infinite), combined at the contact-angle phi between the principal planes: A and B, with B/A >= 1 ordering. A + B is the combined curvature, B - A its difference.
• Eccentricity e of the contact ellipse solves the classical relation between the curvature ratio B/A and the complete elliptic integrals K(e), E(e) (bounded deterministic bisection on the standard relation).
• Semi-axes from the elliptic-integral functions of e; the patch is elongated (a > b) when the curvatures differ strongly (a/b = 1/sqrt(1 - e^2) in the aligned-frame convention).
• Peak pressure p0 = 3P/(2 pi a b); approach delta and patch area follow the classical elastic forms.
• First-yield pressure bands (point arm p0_yield = 1.6 * sigma_y for the circular limit, line arm 3.2 GPa convention at 2000 MPa reference) bracket the elliptical first-yield pressure; yield margin = p0_yield/p0 - 1 with the pass/fail verdict. The bands are reported reference-only, never enforced.

Workflow

1. Fix the material pair and geometry: E1, nu1, E2, nu2 and the signed principal radii (r1_a, r1_b) and (r2_a, r2_b) of the two bodies in their own frames, with the contact angle phi between the principal planes. Convex surfaces enter positive, concave negative, flat infinite. Non-physical inputs are rejected with ValueError.
2. Compute the equivalent modulus with equivalent_modulus(e1, nu1, e2, nu2): E* from the two material pairs.
3. Form the per-plane curvature sums with curvature_sums(r1_a, r1_b, r2_a, r2_b, phi): A and B ordered so B/A >= 1.
4. Solve the contact-ellipse eccentricity with eccentricity(a_curv, b_curv): the classical elliptic-integral relation solved by bounded deterministic bisection over e in (0, 1), returning e with K(e) and E(e). Equal curvatures (A = B) return the circular limit e = 0.
5. Compute the patch with elliptical_patch(load, e_star, a_curv, b_curv): semi-axes a and b, peak pressure p0 = 3P/(2 pi a b), approach delta and patch area.
6. Assess yield with yield_limit_pressure(sigma_y, line_arm=False) for the first-yield pressure band, yield_limit_load(p0_yield, a, b) for the yield-limit load, and yield_margin(p0, sigma_y, line_arm=False) for the margin and pass/fail verdict.
7. Confirm the deterministic checks with the contract test scripts/test_elliptical_hertz_contact.py.

Worked example

All values are REAL outputs of the logic module at the spec anchor cases (steel on steel, E = 207 GPa, nu = 0.3, E* = 113736263736.26373 Pa).

Case A, ball in a conforming groove raceway (12.7 mm ball R_b = 6.35e-3 m in a groove of transverse radius r_g = 7.9375e-3 m, concave, entered negative; track radius R_track = 20.0e-3 m convex; P = 4450 N, sigma_y = 2000 MPa):

• curvature_sums(6.35e-3, 6.35e-3, 20.0e-3, -7.9375e-3, 0.0): A = 15.748031496062993 1/m, B = 103.74015748031496 1/m, B/A = 6.5875.
• Eccentricity e = 0.95709921797354292 (K = 2.6606358985778242, E = 1.0913512644153627): a strongly elongated patch.
• Semi-axes a = 0.0012666442469398131 m (1.26664 mm), b = 0.00036702333828792785 m (0.367023 mm), a/b = 3.4511272575972742.
• Peak pressure p0 = 4570387901.1152277 Pa (4.5704 GPa); patch area = 1.460488725338001e-06 m^2; approach delta = 3.9240382445150354e-05 m (39.2404 microns).
• Yield slice: point arm margin 1.4440787396600459 (pass), line arm margin 0.70015938892608276 (fail); yield-limit loads 6426.15 N (point arm) and 3115.71 N (line arm).

Case B, crossed unequal cylinders (r1 = 25.0e-3 m and r2 = 40.0e-3 m at right angles, phi = pi/2, P = 2000 N, sigma_y = 1200 MPa):

• curvature_sums(25.0e-3, inf, 40.0e-3, inf, pi/2): A = 12.5 1/m, B = 20 1/m, B/A = 1.6.
• Eccentricity e = 0.68212392640371822 (K = 1.8257367044193258, E = 1.367914901795005): a mild ellipse.
• Semi-axes a = 0.00087105204631354651 m (0.871052 mm), b = 0.00063694512714920456 m (0.636945 mm), a/b = 1.3675464481722965.
• Peak pressure p0 = 1721175903.0748203 Pa (1.7212 GPa); patch area = 1.7429944229643266e-06 m^2; approach delta = 1.7598127742320018e-05 m (17.5981 microns).
• Yield slice: point arm margin 2.3007526383129111 (pass), line arm margin 1.115516430697169 (pass).

Verification

• Deterministic stdlib math only (complete elliptic integrals by the arithmetic-geometric mean, bounded bisection for the eccentricity); no RNG, no network.
• ValueErrors: non-positive load, radii or modulus; zero curvature sum; eccentricity out of (0, 1); non-physical material inputs.
• Identities: equal-curvature limit reproduces the circular-patch radius of the sibling closed form within the approximation tolerance; K(e) >= E(e) region checks; b/a = sqrt(1 - e^2) in the aligned-frame convention; complete-elliptic-integral values match the published K = 1.6857503548125966 and E = 1.4674622093394273 at modulus 0.5 (parameter m = 0.25) to 1e-15.

Related leaves

• structures/fem/hertzian-contact-stress (circular-patch and line-contact degeneracies; the a = b and line limits of this leaf)
• structures/fem/contact-analysis (FE penalty-method contact, contact stiffness, penetration)
• structures/loads/lug-joint-analysis and the bearing-stress leaves (nominal bearing stress P/(Dt) and lug margins)

Pitfalls

• Do not claim the circular-patch or strip deliverables (p0 = 3P/(2 pi a^2), p0 = 2P/(pi b L), the 0.62 p0/0.557 p0 subsurface factors or the 3.3/1.6 yield relations as owned output): those are the sibling arms of hertzian-contact-stress, and this leaf claims only the unequal- principal-curvature elliptical patch.
• Concave surfaces enter with negative radius; flat surfaces are infinite. Getting the sign of the groove radius wrong inverts the curvature difference and changes the eccentricity solve.
• The conforming-groove case nearly cancels the transverse curvature (B/A = 6.59 for the 12.7 mm ball in the 15.875 mm groove): the patch is strongly elongated and the a != b distinction is the whole point of the leaf.
• Do not use the single-word generic tags hertz, contact, ellipse, pressure, patch, ball, groove, raceway, cylinder, load, stress or yield alone, and never the sibling tokens hertzian-contact-stress or contact-analysis: they steal corpus tasks from the sibling owners.
• The yield bands are reference-only reporting conventions that bracket the elliptical first-yield pressure; they are never enforced as material limits.

Behavior contract (gate 3)

The contract test scripts/test_elliptical_hertz_contact.py (30 methods, stdlib unittest, offline, deterministic) verifies: the complete elliptic integrals against the published K(0.5) and E(0.5) values to 1e-15; the equivalent modulus for steel pairs; the curvature sums and B/A ordering for both worked-example geometries; the eccentricity solve including the equal-curvature circular limit; both worked-example patches (semi-axes, peak pressure, approach, area) within 1e-9 relative; the yield margins and verdicts; the ValueError rejections; and determinism of repeated calls. The test passes under both /usr/bin/python3 and the pyenv 3.13 interpreter; no exact-float equality is asserted on computed sums.

Compliance

STANDARDS-REF, gated false. FAR-25 and CS-25 (reference-only) frame airframe structural substantiation; the Hertz solution itself is paraphrased public contact-mechanics theory (Johnson, Contact Mechanics, ch. 4 style) and is never reproduced verbatim.