Hertzian contact stress

Under the Hertzian pressure ( after Heinrich Hertz ) is the largest voltage that prevails in the middle of the area of ​​contact of two elastic bodies.

When two elastic bodies pressed with a curved surface (cylinder or sphere) against each other, then they touch only linear or punctiform idealized case. By the elasticity, but in real case of the point of contact results in a flattened and a contact surface. On the contact surface is formed in both bodies, a characteristic distribution of stress ( surface pressure), the voltage is always the highest in the middle.

Yourself by touching two spheres, a sphere and a plane, or two crossed cylinders, the result is a Berührellipse. In case of contact of two parallel cylinders or a cylinder with a plane produces a rectangular, elongated contact surface; it is also called by roller compaction.

According to the theories of the German physicist Heinrich Hertz size and shape of the contact surfaces and the amount and distribution of the mechanical stresses can be calculated using the contact surfaces. After Hertz is the highest voltage that prevails in the center of the contact area, also called Hertzian pressure.

The height of the Hertzian pressure depends on the force with which the two bodies are pressed against each other by their radii of curvature and their modulus of elasticity.

Calculation

Conditions for the calculation of the surface pressure according to the Hertz equations are

  • Linear-elastic, homogeneous and isotropic materials
  • Contact surface is flat and small ( compared to the dimensions of the body )
  • No friction, no shear stresses in the contact area
  • The body can be considered as an elastic half- space

Generally

The Hertzian pressure on contact of curved surfaces calculated by the following equation:

Wherein:

- Force between the bodies

- E- module

The following applies:

- Poisson's ratio (also: Poisson's ratio ) body 1, body 2

- Modulus of elasticity of the material body 1, the body 2

- Coefficients after Hertz for contact of curved surfaces

- curvature

Point contact ball - ball

For the simple case of contact ball - ball ( or level ) is considered:

And

As well as

With

- Ball radii ball 1, ball 2; Special case level: and thus

Line contact cylinder - cylinder

For the simple touch case cylinder - cylinder ( or level ) is considered:

And

As well as

With

- Contact length of the cylinder

- Normal force acting in N

- Cylinder radii cylinder 1, cylinder 2; Special case level: and thus

Plane (geometry) Half-space (geometry) Contact (mechanics)
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