Verification and Validation

 This section contains a few formulae, which made the listed assumptions, found in the Pre-Analysis & Start-Up page.

The analytical formula for computing the radius of contact zone (a) is given as follows:

Using this value of contact radius, we can also compute the normal pressured induced at the contact zone.  Theoretically, the maximum pressure (p_max) is induced along the y-axis, as expected, and is given by the following formula:

Furthermore, we can derive the following formula for the normal stresses σ_z and σ_r = σ_θ along the z-axis.

Here we note that the principal normal stresses σ₁ = σ₂ = σ_r = σ_θ since the out-of-plane shear stresses, τ_rz = τ_θz = 0 and σ₃ = σ_z.  And we can deduce that τ_max = |τ₁|=|τ₂|=|(σ₁-σ₂) / 2|.  The effective stress (using the Von-Mises criterion) along the y-axis can be computed as the following:

Lastly, we also confirm that the applied load at the top vertex of the sphere matches our numerical contact pressure, integrated along the interface.

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