This calculator allows calculating pressure p, displacement (elastic deformation) w, average pressure p_0 and resultant force F_N for the case of indentation of elastic half-space by a rigid truncated paraboloid. The substrate is elastic, the geometric parameters, R – radius of curvature, b – radius of the blunt end, indentation depth d and material parameters E^*=E/(1-\nu^2) are the inputs (see the figure and definitions below).

contact of a truncated paraboloid

A truncated paraboloid is characterized by the blunt radius b that covers the base, the contact radius a > b and the curvature R. The intender is described by the profile:

(1)   \begin{align*} f(r) &= \begin{dcases} 0, \hspace{1cm} r\leq b,\\ \frac{r^2 - b^2}{2R}, \hspace{1cm} r > b. \end{dcases} \end{align*}

The contact radius a can be deduced numerically using the relation that connects the geometrical parameters to each other:

(2)   \begin{align*} d(a) &= \frac{a}{R} \sqrt{a^2 - b^2} \end{align*}

Other parameters: Load force F_N, displacement \omega(r;a) and the stress profile \sigma_{zz}(r;a) are given by the following formulas:

(3)   \begin{align*} F_N(a) &= \frac{2E^*}{3R}\left( 2a^2+b^2 \right) \sqrt{a^2 - b^2}. \end{align*}

(4)   \begin{align*} \sigma_{zz}(r;a) &= -\frac{E^*}{\pi R} \nonumber \\ \cdot& \begin{dcases} \int_b^a \ddfrac{(2x^2-b^2)dx}{\sqrt{x^2-b^2}\sqrt{x^2-r^2}}, r\leq b \int_r^a \ddfrac{(2x^2-b^2)dx}{\sqrt{x^2-b^2}\sqrt{x^2-r^2}}, b< r\leq a \end{dcases} \end{align*}

(5)   \begin{align*} \omega(r;a) &= \frac{2a}{\pi R}\sqrt{a^2-b^2}\arcsin\left(\frac{a}{r}\right) \nonumber\\ &-\frac{1}{\pi R}\left[(r^2-b^2)\arcsin\left(\frac{\sqrt{a^2-b^2}}{\sqrt{r^2-b^2}}\right) - \sqrt{a^2-b^2}\sqrt{r^2-a^2}\right], r>a. \end{align*}

Equations were taken from [1].

Definitions:

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]

References:

[1] Valentin L. Popov, Hanbook of Contact Mechnics, Exact Solutions of Axisymmetric Contact Problems, pg. 31

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