Summary
A variational principle governing the frictionless contact between two elastic bodies is established, which is valid both for linear and for non-linear elasticity. In the case of linear elasticity it appears to lead to an infinite dimensional convex quadratic programming problem. It is applied to the half-space geometry in linear elasticity and it is established that non-Hertzian normal half-space contact problems are physically meaningful.
A Hertzian and a non-Hertzian normal contact problem are investigated numerically, to which end the principle is discretised on a triangular network. In the case of the Hertz problem it is found that the exact relationships between penetration, maximum pressure, and total normal force are well satisfied. The form of the contact area is given only crudely, unless the discretisation network is considerably refined. It appeared that such a refinement is only necessary close to the edge, in which case passable results will be obtained.
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Kalker, J.J., Van Randen, Y. A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems. J Eng Math 6, 193–206 (1972). https://doi.org/10.1007/BF01535102
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DOI: https://doi.org/10.1007/BF01535102