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Reference* 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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Each of the eight nodes shown above can be described by displacement vectors (translational and rotational components, depending on the element type) and by force vectors.  Finite element method first solves for the nodal displacement field with the specified boundary conditions.  The underlying system of equations that ANSYS solves for is shown below.

Here, \[*K*\] is also referred to as the global stiffness matrix, and contains _n_ by _n_ components, where _n_ is equal to the total degrees of freedom of the system.  On the other hand, {*u*} and {*f*} are column matrices with _n_ components, which represent nodal displacement fields and nodal force fields, respectively.  After specifying the appropriate boundary condition in ANSYS, it then solves for these displacement and force fields simultaneously.

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