Hertz Contact Mechanics — Why Nothing Touches Perfectly

1. The Contact Problem

Classical rigid-body mechanics treats colliding spheres as dimensionless points, exchanging momentum instantaneously. Real materials are elastic: when two curved bodies are pressed together, a finite contact patch forms and grows as the load increases. Hertz assumed both bodies are elastic half-spaces, the contact area is small compared to the radii of curvature, and friction inside the patch is negligible — assumptions that hold remarkably well for hard, smooth, dry contacts such as steel balls, ball bearings, and most sports equipment.

2. Contact Radius and Approach

For two spheres of radii R₁ and R₂ pressed together with force F, Hertz theory gives the radius a of the circular contact patch and the mutual approach δ (how much the centres move closer than the sum of the radii):

E* is the "effective" or reduced modulus — the softer material dominates. For a steel ball on a rigid rubber pad, E* is essentially the rubber's modulus even though steel is thousands of times stiffer.

3. Pressure Distribution

Contrary to intuition, pressure is not uniform across the contact patch — it is highest at the centre and falls to zero at the edge, following an ellipsoidal (hemispherical) profile:

4. Nonlinear Contact Stiffness

Because δ ∝ F^(2/3), the force-deflection relationship is nonlinear: F = k_H · δ^(3/2), where the Hertzian stiffness constant is:

This hardening behaviour is why a soft rubber ball feels progressively firmer the harder you squeeze it, and why real impacts are not symmetric sine-wave pulses.

5. Impact Duration and Restitution

Treating the Hertzian contact as a nonlinear spring lets us derive the impact duration and peak force for a sphere striking a massive target at velocity v:

6. Engineering Applications

7. JavaScript Hertzian Impact Simulator

// Hertzian nonlinear-spring impact model (normal impact, no damping)
function effectiveModulus(E1, nu1, E2, nu2) {
  const inv = (1 - nu1**2)/E1 + (1 - nu2**2)/E2;
  return 1 / inv;
}

function hertzImpact({ m1, m2, R1, R2, E1, nu1, E2, nu2, v0, dt = 1e-7 }) {
  const mStar = 1 / (1/m1 + 1/m2);
  const RStar = 1 / (1/R1 + 1/R2);
  const EStar = effectiveModulus(E1, nu1, E2, nu2);
  const k = (4/3) * EStar * Math.sqrt(RStar); // F = k * delta^1.5

  let delta = 0, vRel = v0;
  const trace = [];
  while (vRel > 0 || delta > 0) {
    const F = delta > 0 ? k * delta ** 1.5 : 0;
    const a = F / mStar; // relative deceleration
    vRel -= a * dt;
    delta += vRel * dt;
    trace.push({ delta, F, vRel });
    if (trace.length > 2e6) break; // safety cap
  }
  const Fmax = Math.max(...trace.map(t => t.F));
  return { steps: trace.length, durationS: trace.length * dt, Fmax };
}

// 10 mm steel ball bearing dropped at 1 m/s onto a massive steel plate
const result = hertzImpact({
  m1: 0.0325, m2: 1e6,       // plate treated as effectively infinite mass
  R1: 0.005, R2: 1e6,        // flat plate: R2 -> infinity
  E1: 210e9, nu1: 0.3,
  E2: 210e9, nu2: 0.3,
  v0: 1.0
});
console.log(`Contact duration ${(result.durationS*1e6).toFixed(1)} µs, Fmax ${result.Fmax.toFixed(0)} N`);

8. Limits of the Theory

Hertz's assumptions break down under several common conditions, each requiring an extended model:

• Plasticity: once p₀ exceeds roughly 1.1× the material's yield strength, permanent (plastic) deformation occurs — the domain of elastic-plastic contact models.
• Adhesion: for soft, sticky, or very small contacts, surface energy matters and the JKR (Johnson-Kendall- Roberts) or DMT models replace pure Hertz theory.
• Friction and tangential loading: Hertz is purely normal; combined normal-tangential contact needs the Cattaneo-Mindlin extension for partial slip.
• Large contact patches: when a is no longer small relative to R, the half-space assumption fails and finite-element contact analysis is required instead.