Truncated Paraboloid on Elastic Flat: Online Calculator

14.09.2021

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).

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]