Contact stresses

The shape of the contact area between two elastic bodies is usually circular, elliptical or rectangular. This is valid for ideal elastic bodies.

The size of the contact area depends on the modulus of elasticity of the balls, their radii and the compressive force F. It is proportional to F^1/3. Absolute value of the contact normal stress s₃ is at a maximum at the contact surface. The two other components are 80% of s₃. Maximum shear stress acts inside the ball at 0.4a from the surface.

For compression of two balls the contact stress is at a maximum in the center of the contact spot. The stress decreases to zero at the edges of the contact region. For the two other loading situations, the stress is not the highest in the center.

The theory of elasticity predicts the maximum contact stress for ball bearings. The smaller contact stress, the better. The smaller the modulus of elasticity E, the smaller the contact stress. The hardness, not the contact stress, is more important in choosing the material for ball bearings.

Shear stress in a rail is approximately equal to zero at the surface (z=0) and is at a maximum at a distance 0.45a from the contact zone, where a is size of contact region.

The distance between the centers of the cylinders decreases with an increase in force. There is no linear dependence for the elastic displacement D.

Time of contact for impact between two steel balls in a ball mill is proportional to the radius of the balls R. Increase in the modulus of elasticity or in velocity decreases the time of contact.
The bigger the size of the balls the longer the time of contact.