The following calculation presents an analysis of Hertzian contact stresses when a sphere is pressed against a surface (flat, convex and concave surfaces are considered). Hertzian contact problems require the dimensions of the area of contact are small compared to the radii of curvature of the contacting surfaces near the region of contact. Hertzian bearing pressure distibution is assumed to be a quadratic functions in the region of contact.
Hertzian contact stress refers to the localized stresses that
develop as two curved surfaces come in contact and deform slightly
under the imposed loads. This amount of deformation is dependent on the
elasticity of the the material in contact, i.e., its modulus of elasticity.It
gives the contact stress as a function of the normal contact force,the
radii of curvature of both bodies and the modulus of elasticity of both
bodies.
In gears and bearings in operation, these contact stresses are
cyclic in nature and over time lead to sub-surface fatigue cracks.
Hertzian contact stress forms the foundation for the equations for
load bearing capabilities in bearings, gears, and any other bodies
where two surfaces are in contact.
Calculation Reference
Roark's Formulas for Stress and Strain
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