Hertz elliptical contact

Contact semi-axes, mean and maximum contact pressure for two bodies in elliptical (general curved) contact, with independent radii of curvature in two perpendicular planes (Hertz theory).

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Hertz elliptical contact — original diagram from /calculators/e2_2.htm

Young’s modulus body 1, E₁GPaPoisson body 1, ν₁Young’s modulus body 2, E₂GPaPoisson body 2, ν₂Radius body 1, plane x, R₁ₓ (0 = flat)mmRadius body 1, plane y, R₁ᵧ (0 = flat)mmRadius body 2, plane x, R₂ₓ (0 = flat)mmRadius body 2, plane y, R₂ᵧ (0 = flat)mmNormal force, FN

Semi contact width, a0.3 mm

Semi contact width, b0.39 mm

Mean pressure, pₘ1.38 GPa

Max pressure, p₀2.07 GPa

Reduced modulus, E*232.6 GPa

Reduced radius, R*6 mm

Method & references

Hertz contact theory for elastic bodies in elliptical (general curved) contact.
Brewe & Hamrock approximation for the elliptic-integral ratio K used to relate the semi-axes a, b to the reduced radius.
A radius of 0 in a given plane represents a flat surface in that plane.

The computation is a literal port of the original client-side JavaScript from /calculators/e2_2.htm. Results are verified equal to the original by an automated golden-master test. No formula has been re-derived or “improved”.

Hertz elliptical contact — Engineering ABC