Rigid Cone on Elastic Flat: Online Calculator

TriboNet

May, 27 2021

conical indenter

Conical Indenter

The conical intender is characterized by its inclination angle $\theta$ . After applying a rigid body motion $d$ (resulting in normal load $F_N$ ) on an elastic substrate, the intender deforms the substrate and creates elastic deformation and effective contact radius $a$ as shown in the figure.

Displacement depth $d$ , inclination angle $\theta$ and contact radius $a$ are all related to other geometrically. The contact radius $a$ can be calculated using the relation:

(1) $\begin{align*} d(a) = \frac{\pi}{2} a \tan(\theta) \\ F_N(a) = \frac{\pi a^2}{2} E^* tan\theta \end{align*}$

The profile of the stress $\sigma_{zz}(r)$ and the displacement $\omega(r)$ are determined by the formulas below ([1]):

(2) $\begin{align*} \sigma_{zz}(r;a) &= -p_0 \arccosh\left(\frac{a}{r}\right), \hspace{1cm} r\leq a\\ \omega(r;a) &= a \tan(\theta)\left[ \arcsin \left(\frac{a}{r} + \frac{\sqrt{r^2-a^2} -r}{a} \right)\right], \hspace{1cm} r > a \end{align*}$

where $p_0$ is the average pressure which is given by the equation:

(3) $\begin{align*} p_0 = \frac{1}{2} E^* \tan(\theta) \end{align*}$

while $E^*=\eta E$ is the elasticity of the substrate or the conical intender as well.
These formulas are restricted to relatively small $\theta$ values.

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]

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References:

[1] Valentin L. Popov, Hanbook of Contact Mechnics, Exact Solutions of Axisymmetric Contact Problems, pg. 13

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