**Paper: ****Mathematical
analysis of the entrapment of solid spherical particles in non-conformal
contacts.**

**Author:
George K.
Nikas **

**Published
in: **Transactions of the American Society of Mechanical
Engineers (ASME), Journal of Tribology, 2001, 123(1), 83-93

**Abstract**

The process of entrapment of small spherical particles in both lubricated and dry, line and point contacts, is mathematically analysed and a criterion is postulated in order to evaluate the possibility of entrapment. The analysis is generalized for non-conformal contacts, operating under mixed rolling-sliding conditions with emphasis given to elastohydrodynamic contacts. This study aims to provide the methods, suitable for easy computer programming, to evaluate the risks of surface damage from small debris particles in machine element applications and to make the selection of proper filtration more confined.

**Some
figures from this work**

A couple of examples from
this paper are shown below. Figure 1 shows the x-components of the
various forces (mechanical __and__ fluid forces) acting on a
spherical particle at the entrance of a typical elastohydrodynamic line
contact (central film thickness = 0.7 microns) as the particle barely
touches the counterfaces of the contact. The figure shows how these
forces vary with the particle diameter and reveals the maximum particle
diameter for entrapment: if a particle has a diameter greater than that
critical value, it will be rejected from that particular contact. The
critical diameter is calculated by checking the sign of the x-resultant
force on the particle: at the point where this resultant force becomes
negative (see Fig. 1), the particle has reached its maximum possible
size for entrapment.

**Fig. 1.** *x*-components of forces on a particle (line contact).

Figure 2 below is the adaptation of Fig. 1 for point
contacts and shows
a contour map of the film thickness in a typical point contact. On the
same map, the “line of entrapment” is drawn. This line gives the
maximum particle diameter for entrapment. It is found that this line
is almost straight, inclined at about 45°
to axes *x* and *y* and that this behaviour is typical for
similar cases. In this example, it gives a maximum particle
diameter for entrapment purposes between 80 and 170 microns, depending
on the position where a particle comes in contact with the counterfaces.

**Fig. 2.** Contour map of film thickness in a point
contact and