Analysis of the apparent friction of polymeric surfaces

Abstract

The apparent friction coefficient is the ratio between the tangential force and the normal load applied to moving body in contact with the surface of a material. This coefficient includes a so-called “true local friction” at the interface and a “geometrical friction” which is the ploughing effect. The material underneath a moving tip may display various types of behaviour: elastic, elastic–plastic where elastic and plastic strain are present in the contact area, or fully plastic. As is usual in polymers, the material behaviour is time and temperature dependent and may exhibit strain hardening. A surface flow line model of a scratching tip which links the apparent friction to the local friction and contact geometry was recently proposed. An inverse analysis is used in the present work to estimate the local friction from the measured apparent friction and a knowledge of the contact area and tip shape. The polymer true friction coefficient displays temperature and sliding speed dependency, which may be attributed to the surface thermodynamics. It is shown that the local friction depends on the level of strain in the polymer at the contact interface.

Appendix

Calculation of the ploughing friction for a hyperbolic rear contact edge

The ploughing friction for a zero true local friction is the ratio of the tangential S _t to the normal contact area S _n· S _t is delimited

• (a) by the intersection of the conical shape $x^2+y^2-(z\tan\theta )^2=0$ and the rear plane $x=r\sin\omega$, where S _t is part of the section included in the hyperbola (H)

  $$ -\frac{y^2}{p^2}+\frac{z^2}{q^2}-1=0 $$

  with $p=r\sin\omega$ and $q=r\sin\omega/\tan\theta$, and

• (b) by the plane $z=-r/\tan\theta$.

S _t is calculated from

$$S_{\rm t}=2p\int\limits_{z_1}^{z_2}{\sqrt {\frac{z^2}{q^2}-1}\hbox{d}z}$$

with the integration limits $z_1 =-r/\tan\theta$ and $z_2=-r\sin\omega/\tan\theta$.

If $z=q/\sin\varphi$, then $S_{\rm t}=-2pq\int{\frac{\hbox{d}\varphi}{\sin^3\varphi}+2pq\int{\frac{\hbox{d}\varphi }{\sin\varphi}}}$ and if $t=\tan\varphi/2$ where $\sin\varphi =\frac{2t}{1+t^2}$, then

$$ S_{\rm t}=\frac{2r^2\sin^2\omega}{\tan\theta}\left[{\frac{1}{8}\left( {\tan^2\varphi/2-\frac{1}{\tan^2\varphi/2}} \right)-\frac{1}{2}\hbox{ln}\left|{\tan\varphi/2}\right|}\right]_{z_1 }^{z_2} $$

where

$$\tan\varphi/2=\frac{1-\sqrt{1-\frac{r^2\sin^2\omega }{\tan^2\theta z^2}}}{\frac{r\sin\omega}{\tan\theta z}}$$

The normal contact area is $S_{\rm n}=\left(\pi+2\omega+\sin2\omega \right)r^2/2$ and the general form of Tabor’s ploughing friction coefficient is

$$ \mu_{\rm plough}=4\cot\theta\frac{\sin^2\omega}{\left(\pi+2\omega +\sin2\omega\right)}f(\varphi) $$

where

$$f(\varphi)=\left[{\frac{1}{8}\left({\tan^2\varphi /2-\frac{1}{\tan^2\varphi/2}}\right)-\frac{1}{2}\hbox{ln}\left| {\tan\varphi/2}\right|}\right]_{-r/\tan\theta}^{-r\sin\omega /\tan\theta}$$

like $f(\varphi)$ depends on $\tan\theta$, $f(\varphi)=g(\theta)$. Hence there is no analytical expression for the general case of the exact solution of the ploughing friction coefficient in the form $\mu_{\rm plough}=2/\pi\cot\theta f(\omega)$.