Hertz theory fails when contacting surfaces are expressed by a sum of even polynomials of higher powers than two. An alternative analytical solution implies the knowledge of contact area. In the case of elliptical domains, there are some published proposals for the correlation between pressure distribution and surface normal displacement. This paper identifies the class of high order surfaces which lead to elliptical contact domains and solves a contact between fourth order surfaces.
Elliptical Contact Area Between Elastic Bodies Bounded by High Order Surfaces
Diaconescu, EN. "Elliptical Contact Area Between Elastic Bodies Bounded by High Order Surfaces." Proceedings of the World Tribology Congress III. World Tribology Congress III, Volume 1. Washington, D.C., USA. September 12–16, 2005. pp. 333-334. ASME. https://doi.org/10.1115/WTC2005-63646
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