Indentation of an Elastic plane by Truncated Cone: Online Calculator

TriboNet

June, 6 2021

normal indentation by a conical indenter

Truncated Conical Indenter

Indentation of an elastic flat by a truncated cone

The truncated conical intender is characterized by the radius of the blunt end $b$ , the inclination angle $\theta$ , the effective contact radius $a$ and the deformation depth $d$ (rigid body indentation). This solution is correct without restrictions only for rigid indenters (see [1]).
The contact radius $a$ can be deduced numerically using the relation that connects the geometrical parameters to each other:

(1) $\begin{equation*} d(a) = a \tan(\theta) \arccos\left(\frac{b}{a}\right) = \phi_0 a \tan(\theta) \end{equation*}$

such that the angle $\phi_0$ is introduced as:

(2) $\begin{equation*} \phi_0 := \arccos\left(\frac{b}{a}\right) \end{equation*}$

Other parameters, such as load force $F_N$ , displacement $\omega(r;a)$ and the stress profile $\sigma_{zz}(r;a)$ are given by the following equations:

(3) $\begin{align*} F_N(a) &= E^* \tan(\theta) a^2 \left[ \arccos\left(\frac{b}{a}\right) + \frac{b}{a} \sqrt{1 - \frac{b^2}{a^2} } \right]\\ &= E^* \tan(\theta) a^2 \left( \phi_0 + \cos \phi_0 \sin \phi_0 \right) \end{align*}$

(4) $\begin{align*} \sigma_{zz}(r;a) &= \nonumber \\ &= - \frac{E^* \tan(\theta)}{\pi} \begin{cases} \int_{b}^{a} \left[ \frac{b}{\sqrt{x^2 - b^2}} + \arccos\left(\frac{b}{x}\right)\right] \frac{dx}{\sqrt{x^2 - r^2}} & r \leq b, \\ \int_{r}^{a} \left[ \frac{b}{\sqrt{x^2 - b^2}} + \arccos\left(\frac{b}{x}\right)\right] \frac{dx}{\sqrt{x^2 - r^2}} & b < r \leq a. \end{cases} \end{align*}$

(5) $\begin{align*} \omega(r;a) &= \frac{2\tan(\theta)}{\pi} \left\{ \phi_0 a \arcsin\left(\frac{a}{r}\right) - \int_b^a x \arccos\left( \frac{b}{x}\right) \frac{dx}{\sqrt{r^2 - x^2}} \right\, r>a} \end{align*}$

Definitions:

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