Area of contact between stationary surfaces – Bowden and Tabor

Introduction

Hertz’s 1881 work established theoretical equations describing elastic deformation and the contact area between curved surfaces under load. Subsequent researchers, like Bidwell and Meyer, investigated electrical conductance in contacting materials, with Meyer linking contact resistance to pressure, though later studies suggested surface contamination affected his results. Later Binder (1912) demonstrated that conductance was smaller than expected, proposing that actual contact occurred over a small fraction of the surface. Experimental verification followed, with varying interpretations, such as Pedersen’s idea of a high-resistance transitional layer. Finally, Holm’s extensive research from 1922 onwards showed that contact resistance for clean metals adheres to Ohm’s Law and arises from “spreading resistance” due to current constriction in small contact areas. He concluded that flat surfaces consist of numerous small contact points.

The modified derivation

When a metal is very soft and subjected to high pressure, the contact area can approach the size of the entire metal surface. In such cases, the spreading resistance decreases, and according to the conditions where the radius of the contact area (a) equals the radius of the metal surface (r), the spreading resistance becomes zero. Hence the modified equation for electrical conductance (Λ) can be written as.

Λ = 2λ (a*r/(r-a))

Experimental Setup by Bowden and Tabor

Two crossed cylinders were arranged with their axes at right angles. Electrical conductance (A) was measured using a current-potential method. A known current (i) was passed through the contact, and the potential difference (v) was measured using a high-resistance potentiometer or galvanometer. The conductance was calculated as A=i/v. The load on the contact surfaces was varied by: Using water or mercury to adjust the weight (load range: 20 g to 6000 g). Applying larger loads (up to 1000 kg) with a lever machine and a spring balance. The contact surfaces were prepared through fine grinding, polishing, chemical etching, or scraping. Although surface preparation had minimal impact on results, scraping improved reproducibility.

Figure-1 Figure of the cylinder arrangement [1]

Important Results and discussion

1. The measured conductance values between metal surfaces are similar to the specific conductivities of the metals, indicating that the resistance observed is mainly metallic (or carbon/carbon) and not due to oxides or surface contaminant films. It would be highly unlikely for the electrical conductivities of oxides or other films to match the metals’ conductivity order. If contaminating films are present, they must be very thin, contributing minimally to the electrical resistance.
2. The results clearly show that conductance varies in an orderly fashion with the applied load. If elastic deformation occurs at the contact, the conductance should be proportional to the cube root of the load. However, if plastic flow occurs, the conductance would be proportional to the square root of the load. Due to uncertainties in the conductance measurements, distinguishing between these two behaviors can be difficult.
3. For surfaces that make contact in a single region (e.g., crossed cylinders or sphere-on-flat), the measured conductance values suggest that intimate contact occurs across the entire deformed area. Despite the radius of curvature of the spherical surface varying by a factor of at least ten, the conductance values and the slope of the curves are almost identical. This suggests that, as long as the contact is localized in a single continuous area, the conductance—and therefore the real contact area—remains largely unaffected by the shape and radius of curvature of the surfaces.
4. It is observed that, despite the potential contact area being vastly larger for the flat surface compared to the curved surface, the measured conductance is of the same order of magnitude. However, experiments show that the actual conductance is only about twice as large. This indicates that only a small fraction of the flat surface is in intimate contact, with contact occurring at several “legs” or “bridges.” It is also noted that the conductance for both sets of flat surfaces is nearly the same, even though their apparent areas differ by a factor of 30. This leads to the conclusion that the conductance is largely independent of the apparent surface area and is primarily influenced by the applied load.
5. In the case of flat contacts, it is challenging to precisely estimate the real contact area based solely on conductance measurements. The conductance depends on both the size and the number of metallic bridges. Since the spreading resistance of each bridge is inversely proportional to its diameter, and the contact area is proportional to the square of the diameter, it follows that for a given conductance, the contact area is inversely proportional to the number of bridges. While the exact number of bridges is uncertain, it is clear that for flat surfaces, the number of contact points cannot be fewer than three.

Conclusion

It is clear that for flat surfaces, the real contact area, even under significant loads, is only a small fraction of the apparent area. The flat surfaces are kept apart by small surface irregularities that create metallic bridges. Under applied pressure, these bridges either flow or their number increases until their total cross-section is large enough to support the applied load. In general, the load-conductance curve for flat contacts appears to be steeper than for curved surfaces. However, it is challenging to propose a satisfactory quantitative hypothesis to fully explain this behavior.

Reference

[1] Bowden, F.P. and Tabor, D., 1939. The area of contact between stationary and moving surfaces. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 169(938), pp.391-413.