Hertz Contact Mechanics — Elastic Spheres

Contact radius, pressure distribution, and deformation under normal load

Parameters







Hertz Results

Contact radius a
Indentation δ
Peak pressure p₀
Mean pressure p̄

Hertz Formulae

a = (3FR*/4E*)^(1/3)
δ = a²/R*
p₀ = 3F/(2πa²)
p(r) = p₀√(1−r²/a²)

1/R* = 1/R₁ + 1/R₂

Physics

Heinrich Hertz (1882) solved the elastic contact problem for two spheres. The contact area grows as F^(1/3) — nonlinear due to geometry. The pressure distribution is ellipsoidal (not uniform). Stiffness k = dF/dδ ∝ F^(1/3) is load-dependent. Foundational for tribology, ball bearings, AFM tips, and granular media.