This calculator considers the indentation of an elastic half-plane by a rigid body with sinusoidal profile. This is a two-dimensional case. Equations are given below the calculator.

It is assumed here that the sinusoidal profile is given by the following equation:

(1)    \begin{eqnarray*} S = h_0 cos{\frac{2\pi x}{L} }\\ \end{eqnarray*}

where  h_0 is the amplitude of the profile, L is the period of the function.

In this case, the contact half-length is given as follows:

(2)    \begin{eqnarray*} sin^2(\frac{\pi a}{L}) = \frac{p^a}{p^*}  \\ \end{eqnarray*}

where  p^a is applied pressure (load divided by the period),  p^* is pressure at which the full contact would occur.

Pressure  p^* is given as follows:

(3)    \begin{eqnarray*} p^*= \frac{\pi E' h_0}{L}  \\ \end{eqnarray*}

where  1/{E'}= {1 - {\nu_1}^2}/{E_1} + {{\nu_2}^2}/{E_2} is the reduced elastic modulus,  E_1, E_2   and \nu_1, \nu_2 are the Young’s moduli and Poisson’s ratios of the bodies.

This solution is known as Westergaard’s solution [1].

[1]. Contact Mechanics, J.R. Barber, 2018.