Two bodies, one gap, an elliptic pressure

Increase the approach δ until the gap closes. Hertz then gives a finite contact half-width a and a smooth pressure that peaks at the centre and vanishes at the edges. Unlike a sharp crack tip there is no stress singularity — only complementarity: either a positive gap or a positive pressure, never both.

Separated

δ = 0

Contact pressure p(x)

p(x) = p₀ √(1 − (x/a)²) ····· g ≤ 0, p ≥ 0, p g = 0

δ 0.000 mm Config

What the figure shows

Controlled approach δ closes the gap between two elastic bodies.
Gold bars sketch the Hertz pressure profile in the patch.
The side plot is the elliptic law p(x) = p₀ √(1 − (x/a)²).

What to notice

Separated: gap open, a = 0, pressure zero.
In contact: pressure peaks at the centre and is zero at x = ±a.
Line contact: a ∝ √P′; sphere: F ∝ δ^3/2.
See also the double-cantilever ContactFEA mesh.