Contact problem for elastic spheres: Applicability of the Hertz theory to non-small contact areas

The problem of normal contact of two identical elastic spheres is considered. The corresponding mixed boundary-value problem is formulated at the surface of the sphere, thereby relaxing the assumption of the Hertz theory about small size of the contact area compared to the sizes of the contacting bodies. A general solution in the form of Legendre series expansions is used to reduce the problem to dual series equations, and, subsequently, to a Fredholm integral equation of the second kind. Analysis of the contact stress and surface displacements of the sphere is carried out. Comparison of the results with the Hertz theory shows that the latter predicts contact stresses with high accuracy even for relatively large contact areas.