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Home/Engineering/Hertzian Contact Stress

Hertzian Contact Stress

Hertz contact theory estimates the elastic pressure field when two smooth curved bodies touch over a small patch. This simulator covers a sphere-on-flat point contact and a cylinder-on-flat line contact. The combined material stiffness is represented by the effective modulus 1/E* = (1−ν1²)/E1 + (1−ν2²)/E2. For a sphere the contact radius is a = (3FR/4E*)^(1/3) and p0 = 3F/(2πa²); for a cylinder the half-width is b = sqrt(4F′R/(πE*)) and p0 = 2F′/(πb). The page also shows elastic approach and a rule-of-thumb location for maximum subsurface shear. It is a teaching model only: real contacts involve roughness, friction, lubrication, tangential traction, plastic yield, residual stress, heat, wear, and finite geometry.

Who it's for: Machine design, bearings, gears, tribology, contact mechanics, materials engineering, and mechanical design courses.

Key terms

  • Hertzian contact
  • Contact pressure
  • Effective modulus
  • Contact patch
  • Subsurface shear

Hertz theory assumes smooth elastic bodies, small contact patches, frictionless normal loading, and no plasticity. Real contacts add roughness, tangential traction, lubrication, residual stress, wear, and yield limits.

Live graphs

Contact geometry

2.5 kN
25 mm
40 mm

Materials

210 GPa
0.3
210 GPa
0.3

Measured values

Effective modulus E*115.4GPa
Contact radius a0.741mm
Peak pressure p02176MPa
Contact area1.723mm²
Elastic approach21.941µm
Max shear estimate674.6MPa

How it works

Hertz contact calculator for a sphere or cylinder on a flat: contact patch size, peak pressure p0, elastic approach, and approximate subsurface shear maximum.

Key equations

1/E* = (1−ν1²)/E1 + (1−ν2²)/E2
Sphere: a=(3FR/4E*)^(1/3), p0=3F/(2πa²); Cylinder: b=sqrt(4F′R/(πE*)), p0=2F′/(πb)

Frequently asked questions

Why is the contact pressure not uniform?
In elastic Hertz contact the surfaces deform together smoothly, so pressure is highest at the center and falls to zero at the edge of the contact patch.
Does high Hertz pressure mean immediate failure?
Not by itself. Hertz pressure must be compared with material yield, fatigue, lubrication, roughness, and load cycles. The simulator shows elastic contact stresses, not a complete bearing or gear design check.