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Along the frictionless contact interface, we specify the following boundary conditions.

Here,  r _represents the radial position away from the axis of symmetry and _a denotes the contact radius.  We note that, due to the nonlinear nature of our problem, the contact radius a will change throughout the loading process.  Even though the contact interface between the sphere and the surface is initially a single point, the contact interface will grow to become a surface as the sphere deforms.

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Along the free surface of the sphere, the boundary condition may be specified as follows.

With symmetry condition, the following boundary condition is prescribed along the axis of symmetry.

By identifying the governing equations and defining the boundary conditions, we have set up the mathematical model.  We will now establish several additional relationships, which are used in the postprocessing step for computing stress and strain fields using these nodal displacements.  These relationships are commonly referred to as the constitutive equations.  One of these equations is the strain-displacement relationship.

Second relationship is called Hooke's law.  For our model, we assume isotropic material under plane stress, and so further simplifying the Hooke's law results in the following equations.

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