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 
, the inclination angle 
, the effective contact radius 
and the deformation depth 
(rigid body indentation). This solution is correct without restrictions only for rigid indenters (see [1]).
The contact radius can be deduced numerically using the relation that connects the geometrical parameters to each other:
(1) 
such that the angle 
is introduced as:
(2) 
Other parameters, such as load force 
, displacement 
and the stress profile 
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*}](https://quicklatex.com/cache3/c7/ql_dc20bbc92f283ab44bec81c9ac3648c7_l3.png "Rendered by QuickLaTeX.com")
(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*}](https://quicklatex.com/cache3/a4/ql_53d39626113cd679a1a0ce5f7a931ea4_l3.png "Rendered by QuickLaTeX.com")
(5) 
Definitions:
Poisson’s ratio of the substrate 
dimensionless,
Young’s modulus of elasticity 
of the substrate, [Pa],
Equivalent elastic constant 
, [Pa],
Normal load , [N]
References:
[1] Valentin L. Popov, Hanbook of Contact Mechnics, Exact Solutions of Axisymmetric Contact Problems, pg. 13, pg.29
