This calculator allows calculating pressure p, displacement (elastic deformation) w, stiffness k_z, average pressure p_0 and resultant force F_N for the case of indentation of elastic half-space by a rigid cylindrical indenter. In this case, the intender is rigid and of cylindrical shape. The substrate is elastic, the geometric parameters d – rigid body displacement, a – radius of the cylinder and material parameters E^*=E/(1-\nu^2) are the inputs (see the figure and definitions below). The stress distribution in the contact area and the displacements of the half-space outside the contact are:

(1)   \begin{align*} \sigma_{zz}(r;d) &= - \frac{E^*d}{\pi \sqrt{a^2 - r^2}}, \hspace{1cm}r \leq a \\ w(r;d) &= \frac{2d}{\pi} \arcsin(\frac{a}{r}), \hspace{1cm} r > a \end{align*}

where d is the maximum displacement due to the deformation, a is the radius of the punch and r is the radial coordinate.

The contact stiffness can be calculated as follows:

(2)   \begin{align*} k_z &= 2E^*a, \hspace{1cm}\\ \end{align*}

Equations were taken from [1].


Poisson’s ratio \nu dimensionless,
Young’s modulus of elasticity E, [Pa],
Equivalent elastic constant  E^* = \left( \frac{1-\nu^2}{E} \right)^{-1}, [Pa],
Normal load F, [N]


[1] Valentin L. Popov, Hanbook of Contact Mechnics, Exact Solutions of Axisymmetric Contact Problems, pg. 11