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]

References:

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