Finnie Erosion of a Ductile Metal

Iain Finnie's 1960 paper "Erosion of surfaces by solid particles" in Wear volume 3, page 87, did something nobody had done properly before: it treated an impacting particle as a machining operation. A hard grain striking a ductile metal at an angle does not smash it — it ploughs in, cuts a chip out of the surface the way a lathe tool does, and exits. Write down the equations of motion of a rigid indenter cutting a plastically flowing solid, integrate over the contact, and the volume removed falls out.

What falls out is $W = (c\,m\,V^2/8p)\,f(\alpha)$. The velocity squared is a kinetic energy argument — the particle's energy divided by the stress it takes to move metal gives a volume. The division by flow stress $p$ says harder targets erode less. The mass $m$ makes it linear in how much erodent arrives, and $c$ is the fraction of particles that actually cut in the idealised way rather than rebounding, embedding, or merely deforming the surface. Finnie found $c$ had to be well below 1 to match his data.

The angle function is the interesting part, and it is the whole personality of the model. For shallow impacts, $f(\alpha) = \sin 2\alpha - 3\sin^2\alpha$; once $\tan\alpha$ exceeds 1/3 — that is, above 18.4° — the particle stops cutting before it leaves the groove and the second branch $\cos^2\alpha/3$ takes over. The function starts at zero for a grazing impact, peaks around 0.303 near 17°, and falls to zero at 90°. A particle striking dead-on has no sideways motion, so it cannot cut, so it removes nothing. Ductile erosion is worst at a shallow angle. Measurements on ductile metals put the real peak nearer 20 to 30° rather than 17, but the shape is right, and it explains everything you see in the field: the outer radius of an elbow, the shoe of a tee, the downstream face of a choke. The flow turns, the particles do not, and they arrive at a shallow angle to the wall.

BRITTLE MATERIALS DO EXACTLY THE OPPOSITE, and this is the mistake worth naming loudly. A ceramic, a hard overlay, a glass-flake coating, a hardened white iron: these erode by cracking and chipping, not by cutting. Damage scales with the normal component of the impact, so erosion RISES toward 90° and is smallest at grazing incidence. Apply Finnie's ductile model to a brittle liner and you get the angle dependence exactly backwards — it will tell you that aiming the flow straight at the surface is safe, which for a brittle material is the worst possible geometry. It also means that the standard fix for a ductile problem, hardening the surface, can make things worse rather than better: a very hard, very brittle surface at a steep impingement angle can wear faster than the soft steel it replaced.

The other honesty items. The velocity exponent is 2 in this model because it is a cutting argument; measured erosion of steels usually runs as $V^{2.3}$ to $V^{2.6}$, so extrapolating a long way from one calibration point underpredicts. Hardness alone is a weak predictor across material classes — a steel three times the hardness of mild steel does not last three times as long. The particle impact velocity is generally well below the bulk flow velocity, because fine particles follow the streamlines around a bend and never touch the wall at all; how much below is a Stokes number question. And most important of all: erosion and corrosion multiply rather than add. The impacts strip the passive film, the bare metal beneath corrodes at its unprotected rate, the film re-forms and is stripped again. The combined loss routinely exceeds the sum of an erosion test and a corrosion test run separately, sometimes by a large factor, and a life estimate built by adding two independently measured rates is optimistic in the direction that hurts.