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

Here,  r _ r represents the radial position away from the axis of symmetry and_ a_ 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.

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