Archard Wear Law Simulator — Tribology of Sliding Contact
Compute Archard's adhesive-wear law dV/dx = K F_N / H in real time. The tool returns the volumetric wear rate dV/dt = K F_N v / H, the mass wear rate (steel rho = 7800 kg/m^3), the thickness reduction over a 10 cm^2 apparent contact area, and the time to a V_max = 1 mm^3 wear volume from the wear coefficient K, normal load F_N, sliding speed v and hardness H. A contact-zone schematic and a K-rate log-log plot make the wear scaling intuitive.
Parameters
Wear coefficient K x10^-6
Normal load F_N
N
Sliding speed v
m/s
Hardness H
MPa
Defaults: K = 10 x 10^-6, F_N = 100 N, v = 1.00 m/s, H = 500 MPa. Steel density rho = 7800 kg/m^3, apparent contact area A_app = 10 cm^2 = 1 x 10^-3 m^2 and target wear volume V_max = 1 mm^3 = 1 x 10^-9 m^3 are fixed. Multiplying K by 10 multiplies the wear rate by 10 and divides the life by 10.
Results
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Volumetric wear rate
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Mass wear rate (steel)
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Thinning rate (A=10 cm^2)
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Life to V_max = 1 mm^3
Contact-zone wear schematic
Top = harder body (blue, pressed by F_N); bottom = softer body (red, the one that wears). Yellow arrows = normal load F_N and sliding speed v. The yellow particles are wear debris whose density scales with the current wear rate.
K-rate plot (log-log)
Horizontal axis = wear coefficient K (log10, 10^-7 to 10^-3); vertical axis = thickness wear rate (micrometres/h, log10). Slope +1 reflects dV/dt ∝ K. The yellow marker is the current K. Coloured bands show typical lubricated, dry and seizure regimes.
Theory & Key Formulas
Adhesive wear in sliding contact follows Archard's wear law:
$$\frac{dV}{dx} = K\,\frac{F_N}{H}$$
$V$ is the worn volume, $x$ the sliding distance, $K$ the dimensionless wear coefficient, $F_N$ the normal load and $H$ the hardness of the softer body. Multiplying both sides by the sliding speed $v = dx/dt$ gives the volumetric wear rate:
$$\frac{dV}{dt} = K\,\frac{F_N\,v}{H}$$
For an apparent contact area $A_{\text{app}}$ the thickness reduction rate is $dh/dt = (dV/dt)/A_{\text{app}}$, and the life to a target wear volume $V_{\max}$ is:
$$t_{\text{life}} = \frac{V_{\max}}{dV/dt}$$
Microscopically the real contact area is $A_r = F_N/H$ from asperity plastic flow, and $K$ is the probability that a contact junction generates a wear particle.
What is the Archard Wear Law Simulator
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Why do sliding parts in machines wear out at all? Surfaces look flat, but rub them and they always lose material. What sets the rate?
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Good question. Microscopically the surfaces are a forest of asperities and only the tips actually touch. Under the load F_N the asperity tips flow plastically until the real contact area equals A_r = F_N/H, where H is the hardness of the softer body. As they slide, junctions weld and tear off as wear particles. Archard captured this with dV/dx = K F_N / H. With the defaults (K = 10 x 10^-6, F_N = 100 N, v = 1 m/s, H = 500 MPa) the tool reports a volumetric wear rate of 7.20 mm^3/h, a thickness rate of 7.20 μm/h and a life of about 8 minutes to lose 1 mm^3.
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10 x 10^-6 sounds tiny — what condition does that represent and how does the answer move if I change K?
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For unlubricated steel-steel contact K is roughly 10^-5 to 10^-4, lubricated drops to 10^-7, and seizure can shoot above 10^-2. The slider here covers 1 to 1000 (in 10^-6 units) to span that range. Move it: K = 1 gives a 1.39 h life, K = 1000 burns through 1 mm^3 in 5 seconds. Because the chart is log-log with slope +1, you can see directly that "10x larger K means 10x faster wear, 1/10 the life".
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So increasing the hardness H should extend the life — how does that play out in real machines?
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Right. Wear rate is proportional to 1/H, so case-hardening a gear surface to H about 8 GPa gives roughly 16x the life of the base material at H = 500 MPa. In the tool, push H from 500 to 5000 MPa and life jumps 10x. But over-hardened materials embrittle and are prone to fatigue spalling and chipping; in practice you balance hardness, modulus, fracture toughness and lubricant film thickness. Archard's law is only a first-order picture and breaks down once the PV (pressure x speed) limit is exceeded.
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How does the sliding speed v come in? Faster or slower changes the wear, right?
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Per unit time, dV/dt is proportional to v (dV/dt = K F_N v / H), so doubling speed halves the life. But the wear volume per unit sliding distance dV/dx does not depend on v (it stays at K F_N / H). Whether to design "per distance" or "per time" matters — the answer changes. At higher speeds frictional heating raises the contact temperature, H softens and K grows: this thermal feedback is exactly the PV-limit behaviour. Car clutches and brakes live in that regime and add cooling to manage it. This tool stays in the linear range up to v = 10 m/s; real systems need an extra thermal model beyond it.
FAQ
Archard's wear law is the standard expression for adhesive wear in sliding contact: dV/dx = K F_N / H. Multiplying by the sliding speed gives dV/dt = K F_N v / H. K is the dimensionless wear coefficient (typically 10^-3 to 10^-7), F_N is the normal load and H is the hardness of the softer body. With the defaults (K = 10 x 10^-6, F_N = 100 N, v = 1.0 m/s, H = 500 MPa) the tool reports a volumetric wear rate of 7.20 mm^3/h, a mass rate of 56.2 mg/h (steel), a thinning rate of 7.20 μm/h and a life of about 0.139 h (8.33 min) to a 1 mm^3 wear volume.
K varies by orders of magnitude with the lubrication regime, material pair and contact state. Typical values are about 10^-7 for lubricated steel-steel, 10^-5 to 10^-4 for unlubricated steel-steel and above 10^-2 for severe seizure. The slider covers 1 to 1000 (in 10^-6 units), and the default 10 x 10^-6 represents mild dry adhesive wear. K is measured in pin-on-disk or block-on-ring tests in practice. Doubling K doubles the wear rate.
Wear rate scales as 1/H, so doubling H halves the wear rate and doubles the life. This comes from the assumption that plastic flow at asperities makes the real contact area F_N/H. Use the hardness of the softer (worn) body. Convert from Vickers HV to MPa via H about 9.8 HV. The default 500 MPa (HV about 51) corresponds to mild carbon steel; quenched high-carbon steel reaches 2-4 GPa and case-hardened gear surfaces above 8 GPa.
(1) Abrasive wear (sand or oxide particles cutting the surface) depends on particle size, shape and concentration in ways K alone cannot capture. (2) Above the PV limit wear becomes nonlinear with an exponential rise in rate up to seizure. (3) Fatigue wear (pitting, spalling) and corrosion wear depend on stress cycling and chemistry and need their own models. (4) Full hydrodynamic lubrication eliminates metal contact and the wear is set by the film thickness rather than H. This tool is restricted to boundary-lubricated to dry adhesive wear.
Real-world applications
Gear and bearing life design: In automotive gearboxes and industrial robot reducers tooth-flank wear largely sets the service life. Designers often work with the specific wear rate k = K/H (mm^3/(N m)) instead of K. The defaults give k = K/H = 10 x 10^-6 / 5 x 10^8 = 2 x 10^-14 m^2/N, i.e. 2 x 10^-8 mm^3/(N m). Allowable tooth-wear depths around 0.1 mm are estimated by measuring K under representative PV, lubrication and temperature conditions and extrapolating linearly with a tool like this one.
Brake pads and clutch discs: Brake pads are deliberately wear-prone composites of resin, fibres, inorganic particles and metal powders, with K about 10^-4 so they absorb braking energy by sacrificing material. Setting K = 100 x 10^-6 here gives a life of only a few minutes, but a real car only brakes for seconds at a time, so cumulative wear lasts tens of thousands of kilometres. Clutches work the same way — the rotor hardness H and pad K together set the feel and durability.
Flank wear of cutting tools: On lathes and mills the flank wear VB on a carbide insert is the main life criterion. For steel turning at v = 200 m/min, feed 0.2 mm/rev and 1 mm depth a typical K is about 10^-5 with VB = 0.3 mm allowable. Plugging F_N = 300 N, v = 3.3 m/s (200 m/min), H = 5000 MPa (carbide) and K = 10 x 10^-6 into this tool gives the right order of magnitude. In practice the Taylor tool-life equation V T^n = C is used together with this kind of estimate.
Hip implants and medical devices: A modern hip implant (UHMWPE cup on CoCr head) targets cumulative wear of about 0.1 mm per year, which forces the design toward K about 10^-7. Setting K = 1 x 10^-6, F_N = 2500 N (about three times body weight), v = 0.05 m/s (gait) and H = 80 MPa (PE) brings the tool's life prediction close to the 15-year in-vivo target. Cross-linked polyethylene and ceramic heads are exactly the technology that pushed K down by another decade.
Common misconceptions and pitfalls
The most common pitfall is the belief that "K is a material constant". In reality K can swing by one or two orders of magnitude with load, speed, temperature, lubrication and atmosphere. The same steel-steel pair gives K about 5 x 10^-5 at 80 percent humidity but K about 2 x 10^-4 in dry nitrogen (no protective oxide film). Treat the K slider here as a sensitivity-analysis knob and always feed real designs with K measured under representative conditions. Textbook values are first-pass estimates only.
A second pitfall is over-extending the law: "Archard works for every kind of wear". It does not. (a) Abrasive wear (sand or oxide cutting the surface) needs particle-based models, (b) fatigue wear (pitting, spalling) is a function of stress cycle count, (c) corrosion wear (tribochemical) is governed by chemical reaction rates. Treat this tool as a quick sensitivity look at the adhesive-wear regime, not a general wear calculator.
The last pitfall is "the linear law extrapolates above the PV limit". Archard's law assumes a roughly constant contact temperature. Once PV = p v (contact pressure times sliding speed) exceeds a material limit, frictional heat raises the contact temperature, H drops, K explodes and you transition to severe wear or seizure. Polymer plain bearings have PV limits around 0.1 MPa m/s, bronze bearings around 1.5 MPa m/s, PTFE-impregnated materials about 0.3 MPa m/s. With the v slider at maximum (10 m/s) and the default load you get p = F_N/A_app = 100 / 10^-3 = 0.1 MPa, so PV = 1 MPa m/s — already in the danger zone for polymer bushings. Always check the PV value alongside the wear-rate prediction.