P. Pallav, PhD

Contact stresses arise in the materials of the cusps of the upper and lower teeth, when these come together. Because the surfaces are rather hard, the contacts between the cusps are generally very small. Depending on the exact shape, such a contact area may have a shape anywhere from a very small circle to a very thin line.

A common way to simulate the stresses, is to simplify the setup to a sphere or cylinder of one material touching or pressed against a plane surface of another material.

Assuming that all deformations are fully elastic, i.e. that nowhere in either body the elastic limit is exceeded, then Hertz' formula's (1881) can be applied. In these circumstances the pressure on a contact increases with the third root (sphere) or the square root (cylinder) of the contact force.

The highest stresses in the bodies (both!) occur at the center at a small distance (r) beneath the surface, marked M. This is why surface fatigue initiates beneath the surface. The magnitude of the stress at M is given as

P_max ≈ 1.6 x S_elastic, Equation 1,

Example

A gold alloy cusp (E ≈ 75 GPa) with a (cylindrical) radius of curvature of 5 mm at the 1 mm long area of contact, pressed against a plane surface of composite (E ≈ 20 GPa) with a (static) force of just 1 N (about 0.1 kg or 0.22 lbs), results in a contact pressure of about 95 MPa(!) on a 1 mm x 40 µm contact line.

Plastic Deformation

As the contact force increases beyond Equation 1, the area around M in the figure will undergo plastic deformation and Hertz' formula's cannot be applied anymore. 

It will depend on the material what happens as the force continues to increase. With most metals the plastically deformed area at M will grow until (nearly) all deformations are plastic.
With brittle materials subsurface cracks will initiate and grow depending on the fracture toughness. In these conditions the contact area will become proportional to the contact force. The contact pressure will not increase any further and become equal to the surface hardness (BHN, etc.). 

Hertz' Contact Mechanics

Hertz' equations assume that the elastic limit is not exceeded anywhere in either body. With this assumption, the following equations can be produced for the mathematically simple configurations of a (theoretical) line contact, i.e. a cylinder against a plane surface,

Or of a point contact or a sphere against a plane,

The average pressure (P_av) can be calculated from P_max as follows:

E is a kind of mean elastic modulus of the two materials:

When both surfaces are curved, the appropriate value for R in Equations 2 and 3 can be found with

Hertz H (1881): Ueber die Berührung Fester Elastischer Körper. J Reine Angew Math 92: 155-171.

Poritsky H and Schenectady NY (1950): Stresses and Deflections of Cylindrical Bodies in Contact with Application to Contacts of Gears and Locomotive Wheels. J Appl Mech 17(june): 191-201.