Classical smooth elastic non-conforming contact
Documented Hertz Circular & Parallel-Line Contact Calculator
Calculate effective modulus, signed relative curvature, contact size and maximum pressure for either circular sphere contact or the central region of parallel-cylinder line contact.
Documented Hertz result
Shared compliance and signed curvature
κ = 1/R1 + s/R2 ; R* = 1/κ
s=+1 external convex, s=0 flat, s=−1 internal concave
For the internal case, R2 is a positive magnitude but its curvature is subtracted; R2 must exceed R1 so κ remains positive. Treating a concave groove as another positive convex radius is a different geometry.
Circular point contact
p0 = 3F/(2πa²)
Ac = πa²
δ = a²/R*
The displayed δ is the total normal elastic approach for this circular Hertz model. It is not used for the parallel-cylinder result.
Parallel-cylinder line contact
b = [4wR*/(πE*)]^(1/2)
p0 = 2w/(πb)
projected footprint = 2bL
The line model represents the central region of straight parallel cylinders. The footprint is a nominal rectangle; end pressure and edge effects are not calculated.
Primary published references
The equations are classical engineering Hertz relations, not “ISO formulas.” The NIST Engineering Metrology Toolbox documentation states the smooth, elastic, homogeneous and negligible-friction assumptions and links the published Puttock–Thwaite technical paper. NIST separately implements two external spheres, а sphere in an internal spherical surfaceи parallel cylinders. Its published 4.45 N, two 25.4 mm diameter steel-sphere case gives 0.519 µm compression; the displayed equations reproduce 0.5193248 µm before NIST rounding. For a 25.4 mm ball diameter inside a 50.8 mm spherical diameter with the same properties/load, NIST gives 0.327 µm; subtracting curvature reproduces that result.
Why the former page was changed
The former “ball in groove” preset added both positive radii, although an internal groove requires subtracting curvature. It also advertised approach but did not calculate it, accepted broad gear/bearing/wheel-rail applications that commonly require general elliptical contact, embedded unsourced material/geometry presets, and used partial-number parsing.
Not a strength, fatigue or bearing-life decision: independently verify elastic limits, subsurface stress state, residual stress, hardness/yield relation, material anisotropy, coatings, roughness, lubrication/traction, thermal effects, misalignment, edge loading, plasticity, repeated loading and the applicable component standard or validated numerical model.
