Hertz Theory: Main Assumptions and Formulas

Hertz contact theory describes the local stresses and elastic deformations that arise when two curved elastic bodies are pressed together over a small contact region.

Before loading, the bodies may touch at a point or along a line. Elastic deformation under the applied normal force creates a finite contact area with a nonuniform pressure distribution.

Main assumptions

• The materials are homogeneous, isotropic, and linearly elastic.
• Deformations are small.
• The dimensions of the contact region are small compared with the characteristic radii of curvature of the bodies.
• The contacting surfaces are smooth in the local contact region.
• Classical Hertz theory neglects tangential traction and treats the load transfer as frictionless normal contact.

Reduced elastic properties

For two bodies with Young's moduli \(E_1,E_2\) and Poisson's ratios \(\nu_1,\nu_2\), the reduced modulus is defined by

\[\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}.\]

For spherical surfaces in the simplest external-contact case, the reduced radius is

\[\frac{1}{R^*}=\frac{1}{R_1}+\frac{1}{R_2}.\]

Circular Hertz contact

For axisymmetric contact between spherical surfaces, or a sphere and an elastic half-space, the radius of the circular contact area is

\[a=\left(\frac{3FR^*}{4E^*}\right)^{1/3}.\]

The maximum contact pressure at the center is

\[p_0=\frac{3F}{2\pi a^2}.\]

The pressure distribution over the contact radius is

\[p(r)=p_0\sqrt{1-\frac{r^2}{a^2}},\qquad 0\le r\le a.\]

The pressure is therefore highest at the center and decreases smoothly to zero at the edge of the contact area.

What controls Hertz contact stresses?

The contact dimensions and pressure level depend on the normal load, local surface curvature, and elastic properties of both materials. Increasing load enlarges the contact region while also changing the maximum pressure.

Applications and limits

Hertz theory is widely used as a first model for rolling-element bearings, gears, wheel-rail contact, rollers, cams, and other components with localized elastic contact.

The classical solution does not account for plastic indentation, strong frictional effects, adhesion, surface roughness, or large deformation. Such cases require more advanced contact models.