Archard Wear Equation (Volume Lost)

In 1953 J. F. Archard published four pages in the Journal of Applied Physics under the title "Contact and Rubbing of Flat Surfaces", and they have run the engineering of wear ever since. The argument is beautifully short. Two surfaces touch only at the tips of their asperities, so the real contact area is a tiny fraction of the apparent one. If those junctions deform plastically, each carries a pressure equal to the material's indentation hardness, and the total real area is therefore $A_r = W/H$. Slide the surfaces, and each junction is sheared away and remade. Suppose that only a small fraction of junctions produce a wear particle, and that the particle is roughly hemispherical with the junction's own diameter. Add it up over a sliding distance $s$ and the volume removed is $V = K W s / H$.

Everything worth knowing about the equation is in what it does and does not contain. It is linear in load and linear in distance, and it contains no area and no speed. Doubling the load doubles the wear; doubling the distance doubles the wear; spreading the same load over twice the apparent area changes nothing at all, for exactly the same reason that Coulomb's friction law does not care about apparent area. Both results come from the same fact: the real contact area is what matters, and the real contact area is set by the load.

Now the honesty, and it is the whole of this page. The wear coefficient $K$ runs from about $10^{-8}$ for a well-lubricated, well-behaved contact to about $10^{-2}$ for severe galling with metal transferring across the interface. That is EIGHT ORDERS OF MAGNITUDE. It is not measurement scatter around some true value — it is a genuine range, and where in it you land is decided mostly by the lubrication regime and hardly at all by which metals are involved. A wear calculation is therefore an order-of-magnitude estimate. Presenting the output as a service-life prediction to three significant figures is not optimism, it is dishonesty, and anybody who has quoted a bushing life from an Archard calculation and watched it come back at a tenth of the number knows why.

The second reason for caution is subtler and worse. Archard assumes the wear mechanism does not change. Real contacts have a transition: raise the load or the speed past some threshold and mild oxidative wear gives way abruptly to severe adhesive wear, with $K$ jumping by two or three orders of magnitude across a very small change in the input. The equation is perfectly linear on both sides of that cliff and gives no hint that the cliff exists. This is why wear testing is done at the conditions of interest rather than extrapolated to them, and why a $K$ borrowed from a test at a tenth of your load may describe a different physical process entirely.

Three input mistakes account for most wrong answers. Hardness is a PRESSURE. $H$ is the indentation hardness — the mean pressure under a hardness indenter at full load — so it belongs in megapascals, not in Brinell numbers. For steels the working bridge is $H \approx 9.8 \times$ HB in MPa, so a 200 HB steel is about 1960 MPa; a Rockwell C number has to go through a conversion table before it means anything numerically. Entering "200" for 200 HB gives an answer wrong by a factor near ten million, and the arithmetic will not complain. The hardness wanted is the SOFTER member's, and for a case-hardened, nitrided or coated part it is the hardness of the skin — which stops being true the moment the skin wears through. And $K$ is not a material property: it belongs to a pair, a lubricant, a temperature, a speed and a geometry, and a coefficient lifted from a different regime is the single most common way to get a wear number that is wrong by a thousand.

Used properly, though, it earns its keep. Fit $K$ from your own test or from a machine you already run, keep every condition the same, and the equation becomes an excellent tool for COMPARISON — this load against that one, this hardness against that one, this duty cycle against that one. What it cannot do is tell you, from first principles and a handbook, how long a part you have never run will last.