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In the Pre-Analysis step, we will review the following:

• Assumptions:  Assumptions for classical Hertz contact mechanics are discussed.
• Mathematical model: The fundamental governing  Governing equations and boundary conditions, as well as additional relations will be discussed.
• FEM approach: We  We will discuss solution strategy used in solving a nonlinear problem in FEM.

Assumptions

This problem is a classic example of Hertz Contact Mechanics[1|file:///J:/ANSYS%20Research/Spherical%20Textbook%20Problem/Scripts%20stuff%20for%20Hertz/Problem%20Specification.docx#_ftn1], and hence, makes the following assumptions:

Mathematical Model

Assumptions

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1. Surfaces are continuous and non-conforming, which means that initial contact is a point or a line.  In our example of sphere-plate, the initial contact interface is in a form of a point.
2. Strains are small.
3. Solids are elastic.  This means that the material response of stress and strain behaves linearly.
4. Surfaces are frictionless and cannot penetrate into each other.

For analytical solution, the following additional assumption is made.

5. Both objects (in our case, sphere and plate) are semi-infinitely large bodies (R1, R2 >> a, where a is contact radius).

_Reference_: \[1\] S. Timoshenko and J.N. Goodier: "Theory of Elasticity" \--  Chap. 13: Sect. 125, "Pressure between Two Spherical Bodies in Contact

Mathematical Model

As in any static analysis, the fundamental governing equations that we must keep in mind are the stress equilibrium equations (i.e. governing equation).

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