This is an attempt to document some equations used to derive equations for torsional friction based on Hertzian contact. It is drawn largely from Popov's Contact Mechanics and Friction (2010).

It is shown by Popov in Section 5.1(a) that a rigid sphere (radius $R$) in contact with an elastic half-space (elastic modulus $E$, Poisson ratio $\nu$) with a circular contact patch of radius $a$ generates a pressure distribution

$p(r) = p_0 \sqrt{1 - (\frac{r}{a})^2}$

where $r$ is the distance from the center of the contact patch and $0 \le r \le a$. The total normal force $F_z$ is also computed as

$F_z = \frac{2}{3} p_0 \pi a^2$

In section 5.2, the parameters $p_0$ and $a$ are solved in terms of other parameters and the maximum penetration depth $d$

$a^2 = Rd$

$E^* = \frac{E}{1 - \nu^2}$

$p_0 = \frac{2}{\pi} E^* \sqrt{\frac{d}{R}}$

so the normal force can be expressed as

$F_z = \frac{4}{3} E^* R^{0.5} d^{1.5}$

In Section 8.2.4, the case of torsional tangential loading is presented for a rigid sphere (radius $R$) in contact with an elastic half-space (elastic modulus $E$, Poisson ratio $\nu$) with a circular contact patch of radius $a$. That example uses a shear stress distribution corresponding to "stiction", a uniform rotation of all components of the contact patch.

This analysis uses a different shear stress distribution that corresponds to the maximum torsional moment during sliding, with a direct relation to the normal pressure distribution.

The torsional loading at any point in the contact patch (specified in polar coordinates by radius $r$ and angle $\phi$) consists of tangential forces directed perpendicular to the polar radius $r$ with stresses

$\sigma_{zx} = -\tau(r) \sin \phi$

$\sigma_{zy} = \tau(r) \cos \phi$

The stresses at a given position are combined into the vector

$\boldsymbol{\sigma}_z = (\sigma_{zx},\sigma_{zy},0)^T$

and with the incremental area at a given point defined as

$dA = r dr d\phi$

the incremental force at a given point is

$d\textbf{F}_z = \boldsymbol{\sigma}_z dA$.

The position vector

$\textbf{r} = (r \cos \phi,r \sin \phi,0)^T$

is then used to compute the incremental torsional moment with respect to the center of the contact patch

$dM_z = \textbf{r} \times d\textbf{F}_z = r^2 \tau(r) dr d\phi$

The full torsional moment is computed by integrating over the entire circular area of the contact patch:

$M_z = \int dM_z = \int_0^{2\pi}\int_0^{a} r^2 \tau(r) dr d\phi$

with the following simplification due to angular symmetry:

$M_z = 2\pi \int_0^{a} r^2 \tau(r) dr$

For sliding torsion, $\tau(r)$ has a similar form to the normal pressure $p(r)$ generated by a rigid sphere contacting an elastic half-space (see the previous section):

$\tau(r) = \tau_0 \sqrt{1 - (\frac{r}{a})^2}$

The torsional moment integral then becomes:

$M_z = 2\pi \tau_0 \int_0^{a} r^2 \sqrt{1 - (\frac{r}{a})^2} dr$

and is simplified with a substitution $s = \frac{r}{a}$ and $dr = a ds$:

$M_z = 2\pi \tau_0 \int_0^{1} (as)^2 \sqrt{1 - s^2} a ds$

or

$M_z = 2 a^3 \pi \tau_0 \int_0^{1} s^2 \sqrt{1 - s^2} ds$

From integrals.wolfram.com:

$\int s^2 \sqrt{1 - s^2} ds = \frac{1}{8} (s \sqrt{1-s^2} (2s^2 -1) + \sin^{-1} s)$

The first term evaluates to $0$ at both $s=0$ and $s=1$, so only the second term needs to be considered. The torsional moment then evaluates to

$M_z = 2 a^3 \pi \tau_0 \frac{1}{8} (\sin^{-1} 1 - \sin^{-1} 0)$

or

$M_z = 2 a^3 \pi \tau_0 \frac{1}{8} (\pi / 2)$

or

$M_z = \frac{\pi}{8} a^3 \pi \tau_0$

The ratio between the torsional moment and normal force $M_z / F_z$ is then computed as

$M_z = \frac{\pi}{8} a^3 \pi \tau_0$

$F_z = \frac{2}{3} p_0 \pi a^2$

$\frac{M_z}{F_z} = \frac{3\pi}{16} a \frac{\tau_0}{p_0}$

and labelling the stress ratio $\frac{\tau_0}{p_0}$ as $\mu$,

$\frac{M_z}{F_z} = \frac{3\pi}{16} a \mu$

This analysis indicates that the sliding torsional moment is equivalent to a tangential friction force acting at a radius of about 59% of the contact patch radius. It also matches a result reported in equation (10) of "Soft Finger Model with Adaptive Contact Geometry for Grasping and Manipulation Tasks (2007)" (DOI 10.1109/WHC.2007.103 and pdf here).