Direct Evaluation of Pressure Distribution of Frictionless Axisymmetric Indentation Problems

The pressure distribution between a rigid frictionless axisymmetric punch and an elastic half-space can be evaluated if the punch shape can be expressed by a polynomial function. However, there is a lack of investigation on how to calculate the pressure distribution when the punch shape cannot be expressed by a polynomial formula. This paper shows that with the help of a mathematical software, the pressure distribution can be evaluated directly from its corresponding analytical solutions. Using this technique, we evaluate the pressure distributions of a cosine punch and a hyperbolic cosine punch, and compare the results with Hertz’s solution.

An Extension of Hertz’s Theory in Contact Mechanics

Hertz’s theory, developed in 1881, remains the foundation for the analysis of most contact problems. In this paper, we consider the axisymmetric normal contact of two elastic bodies, and the body profiles are described by polynomial functions of integer and noninteger positive powers. It is an extension of Hertz’s solution, which concerns the contact of two elastic spheres. A general procedure on how to solve this kind of problem is presented. As an example, we consider the contact between a cone and a sphere. The relations among the radius of the contact area, the depth of the indentation, the total load, and the contact pressure distribution are derived.