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.


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]


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