Hertzian Stress (Contact Stress) Online Calculator

What is Hertzian Stress?

The area of contact, when two curved surfaces are in contact, is theoretically zero, so this will cause infinite stress and immediate failure of the object. However, on the application of load, a real area of contact is formed due to the deformation of the mating surfaces. Thus, when two bodies having curved surfaces are pressed together, the point or line contact changes to area contact, and three-dimensional stresses are developed in the two bodies. The size of the contact area depends on the load and the materials of objects. These stresses were first studied by the German physicist Heinrich Hertz in 1881. In this honor, the name of this stress is given as Hertzian Stress (Contact Stress).

Hertz’s analysis of contact stress involves the following set of assumptions:
> The surfaces are smooth and frictionless.
> The contact area is small compared to the size of the bodies.
> The bodies are isotropic and elastic.

Static Contact Stresses (Spherical and Cylindrical contact)

When 2 surfaces are in pure rolling contact or primarily in rolling contact with a small percentage of sliding a different surface failure mechanism comes into play known as surface Fatigue. A practical example of such applications is Ball and Roller Bearing, Cam-Follower, Nip Roller, Tooth contact of Spur & Helical Gears, and Wheel rolling on the rails. If load conditions are ideal then all Ball & Roller Bearing, Cam-followers have hardly 1 % sliding motion. But Gear tooth contact has a significant amount of sliding motions at the tooth contact point.  Other types of gear such as Worm, Spiral Bevel, Hypoid, etc. have majorly sliding motion at the contact point.

The 2 most common cases of rolling motion surfaces in contact are Sphere-on-Sphere and Cylinder-on-Cylinder.

Sphere-on-Sphere- In the case of a sphere in contact with another sphere the contact patch is the point contact with zero dimension. But in reality, when the load is applied on the sphere and material deformation takes place (within the elastic limit) then the contact patch becomes an elliptical shape. Example- Ball Bearings

Cylinder-on-Cylinder- In the case of a cylinder in contact with another cylinder the contact patch is the line contact with zero width. In the case of cylinders, when the load is applied, and material deformation takes place (within elastic limit) then the contact patch becomes rectangular in shape. Example- Rolling mill for changing the thickness of the material, Roller Bearings, cam-roller followers, etc

How Hertzian Stress (Contact Stress) is generated?

Consider a ball rolling on a straight line (or a cylinder rolling on a flat surface) against a flat surface and under a contact load. If the load is below the yield point then defection in the contact patch will be elastic and the surface will return to its original curved geometry after passing through the contact. The same spot in the ball or cylinder will come in stress in each revolution. The resulting stress in the contact patch is called contact stress or Hertzian Stress.  The contact stress in this small contact patch (either ellipse or rectangle) is repeated at the frequency of rotation. This leads to a fatigue-loading situation that eventually leads to a surface fatigue failure.

This repeated loading in the contact patch has compressive stress and shear stress in combination. This shear stress is the cause of crack formation after many stress cycles. Crack growth eventually results in failure by Pitting-   the fracture and dislodgement of the material from the surface. Once the surface begins to pit, its surface finish is compromised and it rapidly proceeds to failure by Spalling– the loss of large pieces of surface. 

If the load is above compressive yield strength, then contact patch deflection will lead to permanent deformation or indentation marks on the surface which is known as False Brinelling (the name is derived from Brinell hardness testing as indentation marks have a similar appearance)  

Stress Distribution in Spherical Contact (Pure Rolling)-

The pressure at the contact patch creates a three-dimensional stress state in the material. The applied stress σx, σy, and σz are compressive and are maximal at the sphere’s surface in the center of the patch. These stresses diminish rapidly with depth and with distance from the axis of contact. These applied stresses are also known as principal normal stresses. There is also the principle of shear stress induced by these normal stresses. However, unlike normal stresses, this shear stress is not maximum at the surface but at some small distance z below the surface. This shear is the main cause of the sub-surface initiated fatigue (which means crack initiation below the contact surface)   

Stress Distribution in Parallel Cylindrical Contact (Pure Rolling)-

Stress distribution in the material is similar to those for Spherical contact (except σx & σy are always equal in spherical contact). Also, shear stress does not vary greatly with dept (it almost remains constant)

Online Calculator for Hertzian Stress (Contact Stress)

Below are the 2 calculation programs for Hertzian Stress. Following things should be taken care of while using these programs.
> For a Flat surface, the radius of curvature is infinity (use a very big number such as 1,00,00,00 as the radius). For example, a Sphere on a flat surface
> For Convex surface use radius as a positive number ( Say 100)
> For the Concave surface use radius as a negative number (Say -100). For example, in the case of the sphere in a circular race

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