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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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- 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.
- Strains are small.
- Solids are elastic. This means that the material response of stress and strain behaves linearly.
- 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).
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_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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