Hertzian Contact Stress

Hertzian contact stress, abbreviated as \( p_{max} \), is the localized stresses that develop when two curved elastic bodies, such as spheres or cylinders, are pressed together under a normal load, causing slight deformation at the contact point.  This theory describes how an initially theoretical point or line contact expands into a small elliptical or rectangular contact area due to elastic deformation, resulting in high compressive stresses that are maximum at the center of the contact zone and decrease rapidly away from it. 

These stresses arise in many engineering applications, including rolling-element bearings, gear teeth, cam-follower mechanisms, and wheel-rail interactions, where they play a critical role in determining load-bearing capacity, fatigue life, and potential surface failures like pitting or spalling.  The theory assumes frictionless, smooth surfaces, small contact areas relative to body sizes, and materials behaving elastically within limits, providing foundational equations for calculating maximum contact pressure based on applied force, radii of curvature, and material properties like elastic modulus.