Rubber wear: Experiment and theory Open Access

Wear is the progressive loss of material from a solid body due to its contact and relative movement against a surface.^1–7,9 Rubber wear is of great practical importance, e.g., tires and conveyor belts.^9–11 Tire wear is the largest source of polymer (plastic) particles, and this source may increase with the increasing use of electric vehicles, as they are generally heavier than combustion engine vehicles.¹² A recent UN environment program brief examines the negative effects of tire abrasion particles.⁸ Wear particles produced on road surfaces span a wide range of scales, from nanometers to millimeters, but the largest mass fraction consists of particles with diameters in the range of 1–1000 μm. Particles of this size typically result from the detachment of rubber fragments from surfaces through crack propagation. Smaller particles, particularly nanoparticles, may form due to stress corrosion, where bond-breaking barriers are lowered by interactions with foreign molecules, such as oxygen or ozone.¹³ 

There are several limiting cases of rubber wear, known as fatigue wear, abrasive wear, and smearing wear. When a rubber block slides on a rigid countersurface with “smooth roughness” (to be defined in Sec. VII), the stress concentrations in asperity contact regions are relatively low, and many contacts with substrate asperities are needed to remove rubber particles. This results in fatigue failure rather than tensile failure, and the abrasion of rubber caused by this failure mode is called fatigue wear.

Abrasive wear occurs when a rubber block slides against surfaces with sharp asperities. In this case, stress concentrations generated by the sharp points of contact cut into the rubber, potentially reaching the material’s limiting strength and leading to micro-cutting or scratching on the rubber surface. This process produces longitudinal scratches parallel to the sliding direction, known as score lines.

When rubber compounds are abraded under mild conditions, a sticky, gooey transfer layer often forms on the rubber and countersurface.¹⁴ The abrasion failure in this case is a type of degradation process referred to as smearing. This smearing is likely due to some form of rubber decomposition and may result from stress corrosion, involving reactions with oxygen or ozone at crack tips. This is supported by the observation that no smearing is observed in a vacuum or in a nitrogen atmosphere.¹⁵ It is worth noting that very small (e.g., nanoscale) particles tend to adhere to almost any surface, which can macroscopically appear as a sticky smear film.¹⁶ 

Wear particles from tires on road surfaces often contain not only rubber but also road wear particles and dust (e.g., pollen or sand particles), making it challenging to compare wear studies on road surfaces with theoretical predictions. Wear particles may vary in size depending on how they are generated and collected. For instance, airborne wear particles tend to be smaller than those found on road surfaces. In addition, wear particles produced in water tend to be smaller than those generated in dry conditions. This difference may result from the influence of water on wear processes, which is supported by the observation that the wear rate in water can be very different from that in dry conditions. Furthermore, in water, there is a lower probability of particle agglomeration compared to the dry state, where larger wear particles often consist of agglomerates of smaller particles.

Here, we focus on rubber wear particles produced under well-defined conditions in laboratory environments.¹⁶ An interesting study on this topic was presented in Ref. 10, where detailed results were provided for the wear of three tire tread compounds on three different sandpaper surfaces at three different humidity levels. Most wear particles were in the size range of 10–400 μm, but the wear rate varied significantly depending on the system. Specifically, the wear rate increased with increasing surface roughness, decreased with decreasing humidity, and was lower for two carbon black-filled compounds compared to a silica-filled compound.

Rubber crack propagation is crucial for understanding the origin of rubber wear. The crack or tearing energy γ (usually denoted by T, but here we use γ to avoid confusion with temperature) is defined as the energy per unit area required to separate surfaces at a crack tip.¹⁷ For rubber-like materials, γ can be substantial, typically ranging from $∼102$ to $∼105Jm−2$⁠, depending on the crack tip velocity and temperature. This should be compared to the crack energy for (brittle) crystalline solids, which is on the order of $∼1Jm−2$⁠, even for solids with strong covalent bonds like diamonds. The large γ in rubber-like materials arises partly from the energy required to stretch polymer chains at the crack tip before breaking the (strong) covalent bonds and partly from viscoelastic energy dissipation in the region ahead of the moving crack tip.

In the literature, the crack energy γ has been studied in detail in two cases: for crack tips moving at a constant velocity¹⁸ and for crack propagation in response to an oscillating strain.^19,22,23 Both sets of experiments yield similar results. In the case of an oscillating strain, we present schematic results for a rubber compound in Fig. 1. The small value of Δx indicates that, unless the applied strain (or stress) is large enough to bring γ close to the ultimate tear strength, several stress cycles (resulting from the interaction with road asperities) may be needed to remove a particle from a rubber surface.

Fukahori et al.^24–26 studied crack propagation and wear rate as a razor blade (with a tip radius of curvature of 100 μm) was slid on a smooth rubber surface. They observed the nucleation of some cracks after one sliding pass, but the initial wear rate was extremely small, reaching steady state only after several thousand sliding contacts with the rubber surface. This indicates that many stress cycles are needed for the cracks to grow and become large enough to remove rubber particles.

The crack-tip movement Δx and the wear rate are much smaller for carbon black-filled compounds compared to unfilled compounds. This is expected, as experiments have shown that Δx in oscillatory tearing studies is much smaller for carbon black-filled rubber compounds than for unfilled compounds.

Fukahori et al. also observed the formation of a wear pattern after the run-in period and noted that the steel blade exhibited stick-slip motion. The latter is expected, as the friction force typically decreases with increasing sliding speed above ∼1 cm s⁻¹, and the experiments were performed at a sliding speed of v = 2 cm s⁻¹.

The tearing energy is usually measured in macroscopic rubber samples with a linear size of ∼1 cm, which may not be valid at the small length scales involved in rubber wear, where particles as small as $∼$ 1 μm may be removed. Specifically, the viscoelastic contribution to the tearing or crack energy may be reduced due to finite-size effects.²⁷ In addition, during sliding, the asperity-induced deformation frequencies ω ≈ v/r₀ depend on the sliding speed v and the size r₀ of the contact region, resulting in a broad range of frequency values, while the tearing energy is usually measured at a fixed frequency.

In this paper, we study the wear rate of a tire rubber tread compound (carbon black-filled natural rubber used for bus and truck tires) sliding on concrete surfaces under dry and wet conditions, at different nominal contact pressures (σ₀ = 0.12, 0.29, and 0.43 MPa) and sliding speeds ranging from 1 μm s⁻¹ to 1 cm s⁻¹. We find that the wear rate is proportional to the normal force and independent of the sliding speed. Sliding in water and soapy water results in wear rates that are much lower than in dry conditions. The experimental data are analyzed using a new theoretical approach that predicts wear rates and wear particle sizes consistent with the experimental observations.

We measured the friction coefficient and wear rate using the experimental setup shown in Fig. 2. A rectangular rubber block, 3 cm in length along the sliding direction and 7 cm in the perpendicular direction, is glued into the milling groove of the sample holder, which is attached to the force cell (red box in the figure). The rubber specimen can move vertically with the carriage to adapt to the substrate profile. The normal load can be changed by adding additional steel weights on top of the force cell. The substrate sample is mounted on the machine table, which is moved in a translational manner by a servo drive through a gearbox. Here, we control the relative velocity between the rubber specimen and the substrate, while the force cell records the normal force and friction.

To study the velocity and pressure dependency of the friction coefficient and wear rate, we slide the rubber sample on the surfaces of concrete blocks (concrete pavers) at different velocities and normal forces (loads). Each sliding cycle consists of 20 cm of forward and 20 cm of backward motion. The wear rate is determined from the mass change (the difference in the mass of rubber blocks before and after sliding) using a high precision balance (Mettler Toledo analytical balance, model MS104TS/00) with a sensitivity of 0.1 mg. Except for the run-in, we replace the concrete block after each sliding cycle so that each measurement is conducted on a fresh concrete surface.

Most experiments were performed using concrete surfaces, which are very stable. Before the wear studies, we “prepared” the concrete surfaces by sliding wood plates on them to remove concrete particles weakly attached to the surfaces. After that, the surfaces were cleaned with a brush.

All experiments have been performed using a bus/truck tread compound consisting of natural rubber with carbon black (98 phr natural rubber, less than 1 phr polybutadiene rubber, and 54 phr carbon black). The shore A hardness was not measured but is expected to be between 65 and 70. To reduce the influence of frictional heating, all tests were performed at low sliding speeds. We begin by describing the run-in process of the rubber surfaces, followed by a detailed study of the velocity and load dependency of friction and wear rate. In addition, we present results for sliding in water and soapy water.

Figure 3 shows the ratio F_x/F_z between the tangential and normal force during run-in on a concrete surface. Each run-in sliding cycle consists of 20 cm of forward and 20 cm of backward motion. The sliding speed v = 3 mm s⁻¹, the temperature T = 23 °C, and the nominal contact pressure p = 0.1 MPa.

Figure 4 shows the friction coefficient μ, here defined as the average of |F_x|/F_z during one sliding cycle, as a function of the number of run-in cycles. The friction coefficient stabilized at μ ≈ 1 after $∼6$ sliding cycles.

Figure 5 shows the friction coefficient as a function of total sliding distance for several sliding speeds. Each data point corresponds to the friction averaged over a sliding cycle (20 cm forward and 20 cm backward motion) on a fresh concrete surface (i.e., a new concrete block for each sliding cycle). Note that the friction coefficient remains nearly constant at $∼1$ as the sliding speed varies between 0.1 and 10 mm s⁻¹ but drops to $∼0.75$ at a sliding speed of 5 μm s⁻¹.

Figure 6 shows the cumulative rubber wear in mg as a function of total sliding distance for the same sliding cycles as in Fig. 5. Note that within the noise of the measured data and the tested sliding speed range, the wear rate is independent of sliding speed, and $≈2.5mgm−1$ or ΔV/F_NL = 0.012 mm³ N⁻¹ m⁻¹.

Figure 7 shows the friction coefficient as a function of sliding distance for nominal contact pressures of p = 0.120, 0.293, and 0.432 MPa at a sliding speed of v = 1 mm s⁻¹. Note that within the studied pressure range, the friction coefficient is independent of the nominal contact pressure.

Figure 8 shows the cumulative (rubber wear mass over pressure) (m/p) in mg MPa⁻¹ as a function of sliding distance for the same sliding cycles as in Fig. 7. The red, green, and blue symbols represent contact pressures of p = 0.120, 0.293, and 0.432 MPa, respectively. Note that within the studied pressure range, the wear rate is proportional to the nominal contact pressure.

Figure 9 shows the mass of the sample (rubber plus aluminum plate) as a function of sliding distance for sliding in dry conditions (red), distilled water (green), and soapy water (blue). Each data point represents the result of ten sliding cycles (each consisting of 20 cm forward and 20 cm backward motion), totaling 4 m of sliding on fresh concrete surfaces. After each sliding cycle in water or soapy water, the rubber was dried with a paper towel, and the mass was measured immediately (open squares) and a second time after being left for 10 min in ambient conditions (filled squares). Note that the mass of the block at the start of sliding in water or soapy water is nearly the same as at the start of sliding, indicating little or no mass loss for wet conditions. The dry rubber wear experiment was conducted approximately one month after the wet conditions experiment, suggesting some mass loss occurred during the waiting period.

Figure 9 shows that the wear rate in water and soapy water is much lower than in dry conditions, and much longer sliding distances are needed to determine the wear rate (if it exists) in water and soapy water.

We studied the size of the rubber wear particles using an optical microscope. The particles were picked up from the rubber surface using adhesive tape. A rubber block was first slid on a clean, dry concrete surface, after which adhesive tape was pressed onto the rubber block surface. Figure 10 shows the rubber particles collected by the adhesive tape. The typical size of the rubber particles is about 1–100 μm, which aligns with findings from other studies. However, it is possible that some of the larger wear particles are agglomerates of smaller particles.

We will first assume that whenever U_el > U_cr is obeyed, a wear particle of size r₀ is removed. This is not the case in rubber wear, where in general many contacts between a rubber crack and road asperities are needed to remove a rubber particle, and we will later “correct” this fact.

In what follows we will treat the rubber surface as smooth and assume only roughness on the road surface. In most tire applications, this is a good approximation, as even for a worn rubber surface, the road surface has higher roughness than the rubber surface. We will denote a road asperity where the shear stress is high enough to remove a particle of size r₀ as a wear-asperity and the corresponding contact region as the wear-contact region.

When the crack (or fracture) energy γ is independent of the crack tip speed, one expects a wear particle to form whenever the stored elastic energy U_el is larger than the fracture energy $Ucr≈γπr02$⁠. However, for rubber materials, the crack energy increases strongly with the crack tip speed v_cr. In this case, if the stored elastic energy is close to $γ(vcr=0)πr02$⁠, the crack tip will move only a very short distance Δx ≪ r₀. Therefore, many interactions between the crack and wear asperities are needed to remove a wear particle of size r₀.

Here we will assume that if the elastic energy U_el > U_el0, where $Uel0=γ(vcr=0)πr02$⁠, the crack tip will move with the velocity v_cr, thus $Uel=γ(vcr)πr02$⁠. The crack-tip displacement Δx = v_crΔt during the time Δt in which the crack interacts with an asperity. Since Δt ≈ r₀/v, with v the sliding speed, this argument would indicate that the displacement Δx depends on the sliding speed, but we have found that this is not the case (see Sec. III). One explanation for this could be that the rubber at the interface does not slip uniformly with the applied (or average) driving speed but performs stick-slip motion at the asperity level involving slip speeds independent of the driving speed. In what follows we will assume that Δx does not depend on the sliding speed but only as a function of the tearing energy γ. This assumption cannot be exact and needs further study.

The number of contacts needed to remove a particle N_cont ≈ r₀/Δx depends on the crack energy γ and could be a large number (10³ or more) if the macroscopic relation between the tear-energy γ and Δx would also hold at the length scale of the wear particles (see Sec. VIII). However, for the small r₀ of interest, N_cont may be smaller because of a reduction in the viscoelastic dissipation in front of the crack tip (finite-size effect).

Note that if r₀/Δx is large, a long run-in distance would be needed before the wear reaches a steady state. This is particularly true if the nominal contact pressure is small, where the distance between the wear asperity contact regions may be large. However, since the contact regions within the macroasperity contacts are densely distributed and independent of the nominal contact pressure, there may, in some cases, be enough wear-asperity contact regions within the macroasperity contact regions to reach the N_cont needed for wear particle formation even over a short sliding distance.

Finally, we note that the theory presented earlier is very general, covering both mild (or fatigue) wear involving large N_cont or strong wear, where one single contact (N_cont = 1) can remove a particle.

We first present numerical results for the wear area A_wear and the number of wear particles N, assuming N_cont = 1, as in severe wear. Unless otherwise stated, we assume the friction coefficient μ = 1, Young’s modulus of E = 10 MPa, crack energy of 1330 J m⁻², and nominal contact pressure of σ₀ = 0.12 MPa, as in the experiments reported in Sec. III. The assumption N_cont = 1 corresponds to severe wear involving the ultimate tear strength. In Sec. VI, where we compare the theory with the experiments, we will assume Δx/r₀ ≪ 1, as expected for mild rubber wear.

Figure 15 shows (a) the wear area A_wear in units of the nominal area A₀ [Eq. (7)], and (b) the number of wear particles per unit sliding length N/LA₀ [Eq. (15) with N_cont = 1], as a function of the wear particle radius r₀ (log–log scale). Results are shown for crack energies γ = 50, 100, 1000, and 2000 J m⁻². For the number of wear particles, we assume that for each asperity contact region with radius r₀, one wear particle is generated during sliding over a distance equal to the diameter 2r₀ of the contact region. Therefore, N/LA₀ is given by (15) with N_cont = 1 (or r₀/Δx = 0).

Figure 16 presents the same data as in Fig. 15 for Young’s modulus of E = 10, 20, and 30 MPa (with a Poisson ratio of ν = 0.5) and with γ = 1330 J m⁻². Note the strong dependency of the wear rate on both the crack energy γ (Fig. 15) and the elastic modulus E (Fig. 16). This dependency becomes weaker as A_wear increases, but for small A_wear, it is determined by the large stress tail of the probability distribution P(σ, ζ), which is highly sensitive to σ_c(ζ).

Figure 17 shows the wear area A_wear in units of the nominal area A₀ as a function of the wear particle radius r₀ (in mm) for nominal contact stresses of σ₀ = 0.12, 0.4, and 1 MPa, with the wear area scaled by 1/σ₀. The figure shows that for σ₀ < 0.4 MPa, the wear rate is nearly proportional to σ₀, but it increases slightly faster than linear at higher contact pressures. In tire applications, the nominal pressure in the tire-road footprint is typically below 0.5 MPa, so we expect a wear rate that is roughly linear with the load for low slip velocities (e.g., v < 1 mm s⁻¹), where frictional heating is negligible.

Our experiments (see Sec. III) show that the wear rate is independent of sliding speed, which is remarkable. Although slower speeds mean longer contact times between the rubber and substrate asperities, this does not result in greater crack propagation distances as one might anticipate. This independence may arise if cracks propagate in discrete steps rather than continuously; specifically, if they follow a stick-slip pattern. During sliding, strain energy accumulates at the crack tip until it reaches a critical threshold, after which the crack advances incrementally by a displacement Δx. This process results in a wear rate governed primarily by the cumulative number of asperity contacts per unit sliding distance, rather than by contact time.

Another case where one would expect a wear rate independent of sliding speed is if wear particles form at each contact where the stored elastic energy exceeds that required to form a particle. In this case, the wear rate would depend only on the sliding distance and not on the sliding time. Formation of a wear particle in each contact with a wear-asperity corresponds to severe wear and may occur on surfaces with very sharp roughness, where the elastic energy release rate matches the ultimate tear strength γ_c (see Fig. 1). However, this does not occur on the concrete surface used in this study.

The lowest crack energy γ for tire tread rubber is typically $∼102Jm−2$⁠, and the highest is $∼104−105Jm−2$⁠. Figure 15 shows that rubber wear particles are expected to range in size from μm to mm, which agrees with experimental observations. Tire wear particles have also been observed in the nm-size range, but these particles cannot result from unperturbed crack propagation and must result from a different physical process than considered here, such as slow crack motion due to reactions with foreign molecules (e.g., oxygen or ozone) at the crack tip (stress corrosion).

We will now compare the theoretical predictions to the experimental results for the bus–truck tread compound, which is based on natural rubber. For this compound, we measured a wear rate of $≈2.5mgm−1$ (see Fig. 6). Assuming a rubber mass density of 1250 kg m⁻³, this gives a wear volume per unit sliding distance of V/L ≈ 2 mm³ m⁻¹. In the experiment, the nominal contact pressure was σ₀ ≈ 0.12 MPa, the normal force was 250 N, and the nominal contact area was A₀ = F_N/σ₀ ≈ 20 cm². In the calculations below for the concrete surface, we use a friction coefficient μ = 0.9 unless otherwise stated. This is a typical friction coefficient observed on concrete (see, e.g., Fig. 7).

To compare the measured wear rate to the theory prediction, we need to know the effective modulus E, which depends on the sliding speed (see  Appendix B). The characteristic deformation frequency when a rubber block slides in contact with a road asperity is ω ≈ v/r₀, where ∼ r₀ is the linear size of the contact region. In the present case, the contact radius is $≈0.1mm$⁠, and the sliding speeds range from v = 1–10⁴ μm s⁻¹, giving deformation frequencies between 0.01 and 100 s⁻¹. In this frequency range, the low-strain modulus varies from ∼27 to 36 MPa. For the large strains relevant here, we show in  Appendix B that the effective (secant) modulus is E ≈ 15 ± 5 MPa.

The particle size distribution shown in Fig. 18(b) is consistent with the optical image of the wear particles in Fig. 10. Therefore, the simple theory presented earlier aligns with the experimental observations in terms of both the wear rate (⁠$∼2mm3m−1$⁠) and the observed particle sizes $(∼3−100μm)$⁠.

The small deviation between the calculated and measured wear rate may be due to approximations in the theory; e.g., the factor β in (4) is not exactly 1 as assumed earlier, and the way we separate particle sizes in steps of factors of 2 is non-unique. In addition, the relationship between γ and Δx for micrometer-sized cracks may differ from that of macroscopic cracks; e.g., the frequency of pulsating deformations will differ (see Sec. VII).

In Fig. 19(a), we show the used relationship between the crack tip displacement Δx per oscillation and the tearing (or crack) energy γ (log–log scale). In Fig. 19(b) (red lines), we show the integrand in (21) as a function of γ. The figure consists of numerous curves for different magnifications (or wavenumber cutoffs), corresponding to different particle sizes.

The integration variable in (21) is the pressure, but each pressure corresponds to the tearing energy as given by (13). The red area in Fig. 19(b) is the superposition of many curves for different magnifications or particle radii. In addition, shown (green area) is the result for the sandpaper P100 to be studied in Sec. VII. The sandpaper gives much higher wear rates and involves larger tearing energies or, equivalently, larger Δx corresponding to faster crack propagation. Note that both cases only involve the tearing energy in the region where Δx increases linearly with γ.

For the concrete surface, the center-of-mass of the curves is around γ ≈ 100 J m⁻², relatively close to the fatigue threshold tearing strength (γ₀ ≈ 66 J m⁻²), as expected for mild rubber wear. For surfaces with sharper roughness, such as sandpaper, the elastic energy stored in the asperity contact regions is greater, and particle removal involves higher tearing energies and much higher wear rates.

We will now show how the wear rate depends on the different parameters that enter into the theory. We note that it is in general not possible to design experiments where only one material property is changed; e.g., it is not possible to modify the elastic modulus without changing the relation between the tearing energy γ and Δx, so the theory results presented here may not be easy to test experimentally.

Figure 20 shows the same results as in Fig. 18(a), for the same parameters except for the green and blue curves, which are for E = 20 and 15 MPa. In all the calculations, we used the Δx(γ) relation shown in Fig. 19(a). For E = 15 MPa, the calculated wear rate is very close to the measured value. Figure 21 shows the calculated wear rate as a function of the elastic modulus E (with ν = 0.5) for the rubber block sliding on the concrete surface.

Figures 22 and 23 show the dependency of the wear rate on the friction coefficient and on the magnitude of the surface roughness. In the latter case, we have scaled the height profile with the indicated number κ, which corresponds to scaling the power spectrum of the concrete surface with the factor κ². Note the strong variation of the wear rate with the parameters E, μ, and κ in Figs. 21–23.

Figure 24 shows the calculated wear rate as a function of the nominal contact pressure (log–log scale) for the rubber block sliding on the concrete surface. For contact pressures σ₀ < 0.4 MPa, the wear rate is proportional to the contact pressure and, therefore, to the normal (loading) force. For higher pressures, however, the wear rate increases rapidly as $σ03$⁠. This is primarily due to the formation of very large wear particles, in addition to the 1–100 μm particles formed at lower nominal contact pressures. This additional contribution to the wear rate is referred to as cut-chip-chunk (CCC) wear.^28–31 It occurs because elastic energy scales with the size of the system as $r03$⁠, while the fracture energy is proportional to $r02$⁠; therefore, at large enough length scales, there will always be more elastic energy stored than needed to form wear particles.

To illustrate that the additional contribution is due to the formation of large wear particles, in Fig. 25(a), we show the integrand in the ξ-integral in the wear volume integral (21), and in (b) the number distribution of wear particles, as a function of the logarithm of the radius r₀ of the wear particle. Results are shown for the concrete surface at nominal contact pressures σ₀ = 0.15, 0.30, and 1.0 MPa.

The CCC contribution to the wear volume increases with the wear particle radius up to the cutoff determined by the smallest wavenumber q₀ for which the surface roughness power spectrum was determined, $(r0)max=π/q0$⁠. Since q₀ is defined by the scan length L of the topography measurement, q₀ = π/L, it follows that $(r0)max=L$⁠. In reality, there is always some physical length scale that determines the largest rubber fragments removed. For tires, this may be the size of the tread blocks or the thickness of the rubber layer on the steel cord of slick tires (see below).

Since CCC wear involves removing macroscopic chunks of rubber, the relationship between the crack tip displacement Δx and the tearing energy γ measured for macroscopic rubber samples may be more relevant than the relationship used in this study for the removal of small $(∼10μm)$ particles, where the displacement Δx is assumed to be enhanced by a reduction in viscoelastic screening and strain crystallization (see below).

It is also worth noting that tires are designed such that the nominal pressure when driving on normal road surfaces is not high enough for CCC wear to occur. However, when driving off-road, inhomogeneities such as gravel or roots can generate nominal pressures high enough to induce CCC wear. This is often the case for truck tires in off-road applications (see Fig. 26), and it is also commonly observed in conveyor belts.

Studying the wear integrand as a function of γ, as in Fig. 19(b), shows that the CCC wear rate involves roughly the same range of γ values (γ < 300 J m⁻²) for both σ₀ = 0.30 and 1.0 MPa. This indicates that the CCC wear fragments involve similar crack tip displacements Δx as those required to remove much smaller wear particles. Hence, large wear fragments will be removed very infrequently but may still correspond to the largest wear mass.

In Fig. 25(b), we show the number of wear particles produced, assuming a nominal contact area of A₀ = 20 cm². Note that for all nominal contact pressures, the N/L curves have maxima at the same particle radius, r₀ ≈ 21.8 μm. In fact, the wear particle distribution in the 1–100 μm size range appears not to depend on the contact pressure, except for a scaling with the magnitude of the applied normal force. This observation is in qualitative agreement with optical images of particle distributions at different normal loads.

Figure 25(b) also shows that the sliding distance required to remove a particle in a 2-interval around r₀ ≈ 2.9 mm is about 26 m, while removing a particle in a 2-interval around r₀ ≈ 1 cm requires nearly 1 km of sliding. In the calculations, we used the relationship between γ and Δx shown in Fig. 19. However, the CCC wear region involves macroscopic-sized cracks, where Δx is smaller, so even larger sliding distances than those calculated earlier may be needed to remove large chunks of rubber.

To summarize, there are the following three size regions in rubber wear:

• At very short length scales (nanoscale), the stored elastic energy in asperity contact regions is insufficient to propagate cracks and remove nanoscale wear particles. In this regime, wear particles are produced by stress corrosion. In stress corrosion, molecules from the surrounding atmosphere, adsorbed molecules, or thin contamination films react with rubber chains that are stretched due to the frictional shear stress. This stretching lowers the barriers for chemical reactions, enabling a stress-aided thermally activated process.

• At length scales of around 1–100 μm, the elastic deformation energy in some asperity contact regions during slip exceeds the fracture energy, i.e., the energy per unit area needed to break the bonds at the crack tip. This generates wear particles on the length scale of 1–100 μm.

• Since the stress required to propagate cracks scales as 1/r₀ with the size of the stressed region, see (3) and (4), it follows that at large enough length scales (typically $∼1cm$⁠), σ_c is below the nominal shear stress, allowing the wear of removing large chunks of rubber. That is, at large enough length scales, the elastic deformation energy $∼r03$ is larger than the fracture energy $∼r02$⁠, enabling crack propagation to remove a rubber fragment. The scaling of the stored elastic energy $∼r03$ only holds for systems with linear sizes larger than r₀, and in real applications, some physical constraints will determine (or limit) the size of the removed rubber fragments.

The removal of large chunks of rubber, i.e., cut-chip-and-chunk (CCC) wear, is well-known for tires and conveyor belts and is included in our rubber wear theory. In an infinite system, the removed rubber fragments could theoretically be infinitely large, but in real applications, some physical constraints limit their size. For tires, for example, one would expect the fragments to be smaller than the tread block size or, in the case of slick tires, the thickness of the rubber layer covering the steel cord.

After the study reported earlier was completed, we decided to further test the theory by measuring the wear rate on several different sandpaper surfaces. It is known that rubber wear rates on sandpaper are much higher than on road surfaces, and a crucial test of the theory is to determine if it can account for this difference.

Sandpaper has sharper roughness than the concrete surface we used, which is easily observed by sliding a finger over the concrete and sandpaper surfaces. However, why is this the case despite the fact that the rms-roughness of the concrete surface is similar to that of the sandpaper surfaces (59 μm for the concrete surface and 32, 31, and 47 μm for P180, P100, and P80 sandpaper surfaces, respectively)? Both concrete and sandpaper consist of particles bound together with a binder (epoxy resin for sandpaper and calcium silicate hydrates for concrete), but the concrete surface feels smoother to the touch than sandpaper because it is molded against a flat surface, resulting in a surface where the tops of the stone particles are at (nearly) equal heights, while this is not the case for sandpaper. In sandpaper, particles are deposited (electrostatically) as a monolayer on top of the nominally flat surface covered by the resin binder. The particles have various shapes and sizes (with sandpaper particle diameters typically fluctuating by ∼50% around their average value), resulting in a surface where the particle heights fluctuate by a similar amount as the average particle size.

Newly prepared asphalt road surfaces (which consist of stone particles with a bitumen binder) are similarly smooth to the concrete surface used in this study. This results from the use of heavy rollers, which deform the asphalt surface so that the tops of the stone particles at the top surface are at (nearly) the same height.

We will refer to surfaces, such as the concrete surface, where the tops of the highest asperities are of (nearly) equal height, as having smooth roughness, while surfaces where the tops fluctuate randomly will be described as having sharp roughness. Note that smooth or sharp roughness is unrelated to the rms-roughness; for example, the rms-roughness of the concrete surface is higher than that of the sandpaper surfaces.

In applications involving rubber in contact with very rough surfaces, such as road or concrete surfaces or sandpaper, the rubber only interacts with the roughness above the average surface plane. To account for this, one should use the top power spectra in theoretical calculations, which assume randomly rough surfaces. The top power spectra are obtained by replacing the roughness below the average plane with a roughness that has the same statistical properties as that above the average plane. Similarly, one can define the bottom power spectra. For sandpaper surfaces, all three power spectra (top, bottom, and full) are nearly identical, but for the concrete surface, which is smoother above the average plane than below, this is not the case.

To illustrate this, Fig. 27 shows the measured surface height h(x) as a function of distance x along a straight 25 mm track on the concrete surface (red) and on the sandpaper P80 (blue) and P180 (green) surfaces. Note that the upper part of the height profile on the concrete surface is smoother than the lower part. The horizontal dashed lines indicate the average surface plane ⟨h⟩ = 0, and the data for the P80 and P180 surfaces are shifted downward by 0.4 and 0.2 mm, respectively.

Figure 28 shows the full (red), bottom (blue), and top (green) surface roughness power spectrum as a function of the wavenumber for the concrete surface. In the calculations in Secs. V and VI, we used the top power spectrum; using the full power spectrum would result in a wear rate ∼20 times higher than reported in Sec. V and of similar magnitude to that observed for the sandpaper surfaces (see below). This remarkable result shows the importance of using the top power spectrum for wear and friction calculations.

Figure 29 shows the full surface roughness power spectrum as a function of the wavenumber for sandpaper surfaces P80 (red), P100 (green), and P180 (blue). The top power spectrum, which is used in the calculations below, is nearly identical to the full power spectra, as expected from the topography images in Fig. 27.

In Fig. 30, we show the logarithm of the wear rate (in mg m⁻¹) for rubber blocks sliding on the concrete surface and on the sandpaper surfaces P80, P100, and P180. The blue squares are the experimental results, and the red squares show the theoretical predictions (at a nominal contact pressure of 0.12 MPa). We have used the friction coefficients μ = 0.9 (concrete), 1.19 (P80), 1.06 (P100), and 1.11 (P180). The load on the rubber block is F_N = 250 N. The experiments on sandpaper were performed at a load of 118 N, but we assume that the wear rate is proportional to the normal force and scaled the wear rate by 250/118 for comparison with the theory and the experiments on the concrete surface. The results for the concrete surface, 2.5 and 1.7 mg m⁻¹, are from Figs. 6 and 9, respectively. We also studied the wear for a SBR-based tread compound, under the same conditions as stated earlier, and found a slightly larger wear rate, $≈4mgm−1$⁠.

In a very interesting paper, Tanaka et al.²¹ have studied the friction and wear for a rubber block sliding at low speed (1 mm s⁻¹) on a grinding wheel. The wear rate they observed (about 2 mm³ m⁻¹, when scaled to the same load as we used) is very similar to what we observed on concrete. Grinding wheels are produced in a mold by squeezing together hard particles and a binder (e.g., a resin) under high pressure between two smooth, parallel surfaces. This process results in a surface where the particle tops are nearly at the same height, creating what we refer to as smooth roughness in this paper, despite the large rms-roughness.

We consider the agreement between theory and experiment in Fig. 30 remarkable. We note that the results depend sensitively on the power spectra, which are somewhat noisy as they only involve averaging over three line scans, each 25 mm long.

The theory presented in Sec. V depends on the crack tip displacement Δx induced by the interaction with a single wear-asperity. The displacement Δx determines the number of contacts N_cont ≈ r₀/Δx needed to remove a wear particle. Δx depends on the (oscillatory) driving force or tearing energy γ.

For natural rubber with carbon black filler at room temperature, when γ ≈ 200 J m⁻², the displacement Δx is only $≈1nm$ per oscillation cycle (see Ref. 20). If this result also holds at the length scale of wear particles, for crack energy of γ ≈ 200 J m⁻², one would get that N_cont ≈ (10 μm)/(1 nm) = 10⁴ contacts would be needed to remove a rubber particle of ∼10 μm size, with even more required for larger particles. We found that using a Δx ∼50 times larger (about 50 nm) gives wear rates in agreement with experimental results. Therefore, around N_cont ≈ 100 contacts with wear-asperities are needed to remove a rubber particle of $∼10μm$ size.

The larger Δx required to match the observed wear rate may be due to a reduction in the viscoelastic contribution to the crack propagation energy, as expected for the small-scale systems relevant to rubber wear. In addition, strain crystallization, which reduces Δx, may also play a role. While strain crystallization has been observed in macroscopic experiments for natural rubber,³³ it might behave differently at the microscale.

The region where strain crystallization occurs in macroscopic experiments is very large, extending $∼1.6mm$ from the crack tip in the study reported on in Ref. 32. It has been observed even when the applied tensile strain is below the critical strain required for crack growth initiation. It is also important to note that strain crystallization takes time. In macroscopic experiments, ∼0.1 s is needed for strain crystallization to occur.³⁴ In the wear process, the rubber-road asperity interaction time is of the order Δt ≈ r₀/v, and for the highest sliding speed in our study, v = 1 cm s⁻¹, the interaction time is ∼10⁻³ s (using r₀ = 10 μm). Therefore, at this sliding speed, no strain crystallization is expected. If strain crystallization is absent, natural rubber may exhibit a similar Δx as observed for other types of rubber; e.g., styrene-butadiene (SB) rubber has Δx ≈ 20 nm at γ = 200 J m⁻².

An enhancement of the crack tip displacement Δx was also observed by Southern and Thomas,^35,36 who studied rubber abrasion by scraping the surface with a razor blade. They formulated a theory of abrasion based on the crack growth characteristics of rubber. This theory was successful for non-crystallizing rubbers; however, for natural rubber, the rate of abrasion was higher (by a factor of $∼20$⁠) than anticipated from macroscopic tearing experiments. The authors suggested that this discrepancy may be due to the absence of strain crystallization in the wear experiments.

In a study to be reported on elsewhere,³⁷ we studied the wear for PMMA sliding on three different surfaces (a tile surface, polished steel, and a sandpaper surface). In this study we used the measured relation between Δx and γ and observed good agreement between the theory and the experiment, both for the wear rate and the size of the wear particles, which are predicted to be 10 times smaller than for the rubber wear study.

At high crack tip speeds, viscoelastic energy dissipation occurs further from the crack tip. However, for a very small (circular) crack, the stress field in the vicinity of the crack tip follows the singular form ∼ r^−1/2 only out to distances on the order of the crack size, limiting the volume of the region where viscoelastic dissipation occurs. This results in a lower [1 + f(v, T)] factor than would be expected for an infinitely long crack in an infinitely extended solid.

Figure 31(a) shows an optical image of the surface of a rubber block after it had been sliding on the sandpaper P100 surface at a nominal contact pressure of σ₀ ≈ 0.6 MPa. Figure 31(b) shows rubber particles on an adhesive film that was pressed against the surface in (a). Note the alignment of the wear particles along the lines. This could indicate that the wear particles are not formed randomly on the surface but rather in rows where particularly sharp asperities from the sandpaper cut into the rubber. An alternative explanation is that the wear particles form randomly on the rubber surface after multiple interactions with road asperities but become aligned in lines due to “combing” by the road asperities.

Figure 31(a) shows what appear to be a few linear wear tracks on the same rubber surface. If these tracks resulted from road asperities that cut and removed rubber particles in single asperity contacts (such that N_cont = 1), it would imply local stresses so high that the energy release rate corresponds to a tearing energy γ close to the ultimate tear strength γ_c. Assuming that γ_c for cracks at the micrometer length scale is similar to that at the macrometer scale (where γ_c ≈ 10⁴–10⁵ J m⁻²), this value is much higher than the elastic energy release rate predicted by the theory.

An alternative explanation for the wear tracks could be that cracks on the rubber surface form and grow through interactions with multiple road asperities, with the final removal occurring mainly from a “few” of the highest and sharpest asperities. In this case, one would expect to see wear tracks on the rubber surface. If this explanation is correct, one would expect a run-in period with reduced wear until the distribution of surface cracks reaches its final steady-state configuration.

This behavior has been observed by Fukahori et al.^24–26 in wear studies where a razor blade was slid on a smooth rubber surface. In this case, the initial wear is very small, and ∼1000 sliding contacts are needed to reach a steady state where the wear rate becomes independent of the number of contacts. Similar effects have been observed for sliding on randomly rough surfaces, but in these cases, it is difficult to estimate the number of contacts.

More studies are needed to gain further insight into this problem.

Archard has derived a simple wear law assuming that the contact area and the wear rate are proportional to 1/σ_P, where σ_P is the penetration hardness determined by plastic flow.⁹ This theory neglects the influence of the elastic energy (stored in the asperity contact regions) in the particle removal process. Hence there is no reason for the wear rate to scale with 1/σ_P in general.³⁹ Rubber does not yield plastically, so the concept of “hardness” as used in the Archard theory is not relevant for rubber wear.

We have found that the wear rate increases linearly with the load for a carbon-filled natural rubber (NR) compound sliding on a concrete surface at low sliding speeds (v ≤ 1 cm s⁻¹). Similar results were obtained in an earlier study for an SB rubber compound (unpublished). We conclude that, at low sliding speeds and nominal contact pressures relevant to many practical applications, rubber wear is proportional to the normal force. This is the result one intuitively expected, as the real contact area is proportional to the normal force (for sufficiently low normal forces), and in a large enough system, as the normal force increases, new contacts form in such a way that the probability distribution of contact sizes remains unchanged.

However, this finding disagrees with earlier studies, which show a wear rate that increases faster than linear with the normal force. We attribute this discrepancy to frictional heating, which becomes significant at sliding speeds above $∼1cms−1$⁠. Frictional heating shifts the viscoelastic modulus E(ω) master curve to higher frequencies, reducing the viscoelastic contribution to the tearing energy and thereby increasing the wear rate.

We have also found that for our NR compound on concrete, the rubber wear rate in water (and soapy water) is much lower than in the dry state, despite only a small change in the friction coefficient. This result differs qualitatively from an earlier (unpublished) study for an SB compound under the same conditions, where the wear rate was approximately five times higher in water than in the dry state, despite a similarly small change in the friction coefficient. One explanation for the increased wear in water could be a reduction in the frictional shear stress,⁴⁰ which would allow the rubber to penetrate deeper into roughness cavities (see Fig. 32), resulting in stronger tearing forces acting on the rubber. This explanation is consistent with the observation that it is easier to cut rubber with a sharp knife blade when the contact is lubricated by water or soapy water; this could be due to a reduction in adhesion and frictional shear stress between the blade and the rubber, which concentrates more of the normal force at the sharp knife edge. However, we cannot explain why the same effect does not increase the wear rate in water for the NR compound used in the present study.

We have found that the wear rate is independent of the sliding speed for the range of sliding speeds used in this study (from 1 μm  s⁻¹ to 1 cm s⁻¹). This was also observed in an earlier study for an SB rubber compound. This result is surprising, as one would expect that if a crack is subjected to a constant driving force, where the elastic energy release rate exceeds the value γ₀ required for crack propagation, the crack tip will move at a constant velocity depending on the energy release rate. Since contact time is inversely proportional to sliding speed, one would anticipate increased wear at lower sliding speeds; however, this is not observed.

As previously mentioned, this behavior is only possible if either (a) the crack tip motion is of the stick-slip-stick type, where, after a crack tip displacement Δx, the elastic energy release rate drops enough to prevent further crack motion until another road asperity makes contact with the crack, or (b) a rubber particle is removed with each new asperity contact (with the number of new asperity contacts depending only on sliding distance). We believe the first scenario (a) may be correct, unless the roughness is very sharp, in which case (b) may hold. In either case, however, the macroscopic relationship between γ and Δx may not hold at the asperity contact level, as it might predict a crack tip displacement that is too low.

The theory (and experimental results) presented earlier can be used to estimate the contribution of wear to rubber friction. The energy dissipated during sliding a distance L is μF_NL. If A_w denotes the total surface area of the formed wear particles, then the energy required to form the wear particles is A_wγ, where γ is an effective (or average) fracture energy. Therefore, the fraction of the total dissipated energy needed for the wear process is η = (A_w/L)γ/(μF_N).

For our sliding block, with F_N = 250 N, μ ≈ 1, γ ≈ 500 J m⁻², and L = 1 m, our model calculations give a fracture area of A_w ≈ 4 × 10⁻⁵ m². A_w can also be obtained directly from the experiments: the observed wear volume after sliding L = 1 m is ΔV ≈ 1 mm³. An average wear particle has a size of r₀ ≈ 20 μm. The number of wear particles is $N=ΔV/(2πr03/3)$⁠, and the fracture surface area is $Aw=N×2πr02$⁠, giving A_w ≈ 3ΔV/r₀ ≈ 10⁻⁴ m².

Using A_w ≈ 4 × 10⁻⁵ m² and F_N = 250 N, μ ≈ 1, γ ≈ 500 J m⁻², and L = 1 m, we get η ≈ 10⁻⁴, indicating that the contribution of wear to the friction coefficient is negligible.

A complete theory of rubber wear must accurately predict the sizes of the wear particles produced. This depends on the variation of the crack-tip displacement Δx with the tearing (or crack) energy γ, as discussed in Sec. IV. It is also influenced by crack-tip shielding. If a “long” crack exists (which may ultimately result in the removal of a large particle), it is unlikely that smaller cracks will propagate or extend in its vicinity. This is because a long crack reduces the stress in its surrounding area, thereby decreasing the driving force for smaller cracks.

Molecular dynamics simulations reveal that two nearby contact junctions interact elastically when the distance between them is on the order of the junction diameter.⁴¹ These elastic interactions result in crack shielding, so that during wear particle formation, not all cracks can fully develop as they are unloaded by nearby propagating cracks, leading to the formation of larger wear particles.

Wear particles are removed in asperity contact regions, and the sizes of these contact regions are of central importance. In the wear particle theory presented in Sec. IV, the sizes were determined by the magnification. Here, we present a different approach where the effective size of the contact regions is determined by the stress–stress correlation function. The treatment below is inspired by the studies of Müser et al. on the size of contact regions and its relation to the stress–stress correlation function.^42,43

We begin with a qualitative discussion on the nature of asperity contact regions. At low magnification ζ, relatively large and compact contact regions can be observed, as illustrated in Fig. 33(a). An asperity contact region is considered compact if no non-contact regions can be observed within it. Generally, an asperity contact region that appears compact at magnification ζ becomes non-compact at the highest magnification ζ₁, as shown in (b), often breaking up into several separated regions as in (c).

The magnification ζ* is defined such that (on average) a contact region that appears compact at magnification ζ* reaches the percolation threshold at the highest magnification. We can determine ζ* using A(ζ₁)/A(ζ*) ≈ 0.42. Here, we use the fact that for a randomly rough surface, the non-contact area percolates when A(ζ)/A₀ ≈ 0.42. We refer to the contact regions observed at magnification ζ* as the macroasperity contact regions.

At low magnification (but not so low that the contact area percolates) and at sufficiently low nominal contact pressure, the asperity contact regions are compact and well separated. In this limit, the size of the asperity contact regions is well-defined. However, as illustrated in Fig. 33, at high magnification it is less clear how to define the size of the asperity contact regions. One approach is to use the stress–stress correlation function.

We define the effective asperity contact radius r₀ using the condition g(r₀) = αg(0), where α < 1.

Note that g(r) depends on the range of surface roughness included in the calculation. If q₀ and q₁ are the smallest and largest roughness wavenumbers used in calculating g(r), we include the roughness with wavenumbers q < ζq₀, where 1 < ζ < q₁/q₀. Therefore, g(r) = g(r, ζ) will depend on the magnification ζ.

Figure 33 shows the radius of the asperity contact area as a function of the cutoff wavenumber for α = 0.25, 0.5, and 0.75. Note that for α = 0.25 and 0.5, the radius changes very weakly with increasing wavenumber (or magnification ζ = q/q₀) for q > q*, or ζ > ζ*, where ζ* is defined by A(ζ₁)/A(ζ*) = 0.42, where ζ₁ = q₁/q₀ is the highest magnification. For α = 0.5, the radius r₀ ≈ 0.1 mm, at the magnification ζ = π/q₀r₀ ≈ 2 × 10⁴, where A_wear(r₀) is maximal. Note that, remarkably, the magnification where A_wear(r₀(ζ)) is maximal is almost the same as ζ*.

We have studied the contact between the rubber block and the concrete surface using a pressure-sensitive film (Fujifilm, Super Low-Pressure film, 0.5–2.5 MPa pressure range, λ = 30 μm lateral resolution⁴⁵). Figure 35 shows the optical image of the contact region on two different surface areas of the concrete block after 5 s of contact time. The nominal contact pressure σ₀ = 0.12 MPa is the same as that used in the rubber wear experiments.

Figure 34 shows magnified views of two contact regions from Fig. 35. The effective (or average) diameter of the (macroasperity) contact regions is 2r₀ ∼ 0.2–0.4 mm, which agrees with the predicted size of the macroasperity contact regions (indicated by line B in Fig. 36), which equals 2r₀ ≈ 0.34 mm if α = 0.5.

Estimation of the relative contact area in Fig. 35 gives A/A₀ ≈ 0.015, which is close to the calculated result A/A₀ ≈ 0.017 obtained from Fig. 36(b) for q = π/λ, where λ = 30 μm is the lateral resolution. In the calculation, we assumed a modulus of E = 10 MPa; however, this is not accurately known as it depends on the strain in the asperity contact regions and the contact time due to viscoelastic relaxation.

We have presented an experimental and theoretical study of the wear rate for a rubber block sliding on concrete surfaces. The most important results are as follows:

1. The measured wear rate is independent of the nominal contact pressure (or load) as it varies from σ₀ = 0.12–0.43 MPa. It is also independent of sliding speeds from 1 μm s⁻¹ to 1 cm s⁻¹.

2. The wear rate in water and soapy water is too small to be detected using our approach, which measures the weight of the rubber block before and after a given sliding action.

3. The independence of the wear rate on the sliding speed indicates that the crack tips do not move continuously when exposed to the stress field induced by a road asperity. Instead, during sliding, stress builds up at the crack tip, but the crack does not move until the stored elastic energy reaches a critical value. After this point, crack tip motion occurs at a speed unrelated to the sliding speed and hence unrelated to the rubber-road asperity interaction time.

   This observation was also made in an earlier (unpublished) study involving a SB rubber compound. This result shows that what matters for wear is not the contact time but rather the sliding distance. Although slower speeds result in longer contact times between the rubber and substrate asperities, this does not result in greater crack propagation distances as one might anticipate. This independence may arise if cracks propagate in discrete steps rather than continuously; specifically, they follow a stick-slip pattern. During sliding, strain energy builds up at the crack tip until it reaches a critical threshold, after which the crack advances incrementally by a displacement Δx. This process results in a wear rate governed primarily by the cumulative number of asperity contacts per unit sliding distance, rather than by contact time.

4. A theory for rubber wear is developed based on the probability distribution of stress acting at the interface, as determined by the Persson contact mechanics theory. Rubber wear particles can form at an asperity contact region when the stored elastic energy due to the shear stress exceeds the energy required for bond breaking and the viscoelastic dissipation involved in the formation of the wear particle.

5. A theory and numerical results for the size of the rubber-concrete contact regions as a function of magnification were presented. The theory predicts macroasperity contact regions with a radius of r ≈ 0.1 mm, which is nearly the same as the theoretical predictions for the radius of the wear particles that gives the biggest contribution to the wear volume.

The predicted probability distribution for the size of the wear particles has a maximum at $≈20μm$⁠, which appears consistent with photos of wear particles removed from the rubber surface using adhesive film.

The wear rates we calculate depend rather sensitively on most of the parameters used in the theory. This may explain why, when repeating a wear experiment days or months later, we usually obtain somewhat different wear rates even if the experiments are conducted under nominally identical conditions. For example, we measured the wear rate for the same rubber compound on nominally identical concrete blocks and found different values at different times (separated by a few months): 2.5, 1.7, and 1.0 mg m⁻¹. These variations may reflect small changes in the concrete block surface topography (some blocks were bought at different times and may be from different batches), or fluctuations in humidity or temperature, or aging of the rubber samples (which were kept in a refrigerator at +4 °C when not in use).

To improve the accuracy of wear calculations using the theory presented earlier, it is necessary to study how the relationship between the tearing energy γ and the crack tip displacement Δx is modified at short-length scales. We suggest that this could be performed by examining the propagation of small cracks (length $∼10μm$⁠) in thin rubber sheets (thickness $∼10μm$⁠). It is also important to determine the parameter β in Eq. (3) by comparing theoretical wear predictions with experimental data. Although β is expected to be ∼1, even a small variation in this parameter could significantly impact the wear rate.

We acknowledge A. Almqvist, M. Müser, and R. Stocek for useful comments on the text. We acknowledge M. Müser for suggesting defining the size of contact regions using the stress-stress correlation function. This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB0470200.

The authors have no conflicts to disclose.

All authors have contributed equally.

B. N. J. Persson: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). R. Xu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). N. Miyashita: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Consider the stress at the interface between an elastic half-space and a rigid surface with random roughness. When we study the interface at the magnification ζ, we only observe surface roughness with wavenumber q < ζq₀ (or wavelength λ > λ₀/ζ with q = 2π/λ and q₀ = 2π/λ₀), where q₀ is the wavenumber of the longest wavelength roughness included in the study. Let P(σ, ζ) denote the probability distribution of stress, which depends on the magnification. The expression (5) for P(σ, ζ) can be derived from the Persson contact mechanics theory as is described elsewhere and briefly reviewed here.

The characteristic deformation frequency when a rubber block slides in contact with a road asperity is ω ≈ v/r₀, where ∼ r₀ is the linear size of the contact region. In the present case, the contact radius is $≈0.01mm$⁠, and sliding speeds v = 1–10⁴ μm s⁻¹ give deformation frequencies between 0.1 and 1000 s⁻¹. In this frequency range, the low strain modulus changes from $≈27$ to 36 MPa (see Fig. 37). For the large strain relevant here, we now show that the effective relevant modulus is E ≈ 10–20 MPa.

The stress–strain relation for filled rubbers is strongly non-linear. We define the effective (secant) modulus E as the ratio between the (physical) stress and the strain E = σ/ϵ. Here σ = F/A, where F is the elongation force and A is the rubber block cross section of the rubber strip, which depends on the strain, A = A₀(1 + ϵ), where A₀ is the cross section of the rubber strip before elongation. The strain ϵ is defined in the usual way ϵ = (L − L₀)/L₀, where L and L₀ are the lengths of the rubber strip before and after applying the force F. We need this secant modulus for the typical strain prevailing in the rubber-road asperity contact regions.

The stress in the asperity contact regions can be estimated using (4), which with E* ≈ 15 MPa, γ ≈ 200 J m⁻², and r₀ ≈ 10 μm gives σ ≈ 25 MPa. Figure 38(a) shows the stress–strain relation for the elongation of a strip of the natural rubber used in the present study. The stress σ ≈ 25 MPa corresponds to the strain ϵ ≈ 1.8. The effective (secant) modulus E for this strain is shown in Fig. 38(b) and is about E = 14 MPa.

The most important quantity of a rough surface is the surface roughness power spectrum.^46,47 The two-dimensional (2D) surface roughness power spectrum C(q), which enters in the Persson contact mechanics theory, can be obtained from the height profile z = h(x, y) measured over a square surface unit. However, for surfaces with roughness having isotropic statistical properties, the 2D power spectrum can be calculated from the 1D power spectrum obtained from a line-scan z = h(x).

The wear experiments have been performed on concrete and sandpaper surfaces. The concrete blocks (concrete pavers) were obtained in a large number from a “Do-It-Yourself” shop. In most cases, every new wear experiment was performed on a new concrete block. We have used these concrete blocks in most of our earlier friction studies. They are very stable (no or negligible concrete wear), and concrete blocks obtained from the same batch have all the same nominal surface roughness. For each surface, we measured at least three tracks at different locations, each 25 mm long.

We have measured the surface topography using a Mitutoyo Portable Surface Roughness Measurement Surftest SJ-410 equipped with a diamond tip having a radius of curvature R = 1 μm and with the tip-substrate repulsive force F_N = 0.75 mN. The step length (pixel) is 0.5 μm, the scan length L = 25 mm, and the tip speed v = 50 μm s⁻¹. The top power spectra shown in Fig. 39 (solid line) were obtained by averaging over three measurements. The dotted line is the linearly extrapolated power spectrum. The linear extrapolated region corresponds to the Hurst exponent H ≈ 1. The power spectra of the sandpaper surfaces were shown in Sec. VI and were extrapolated linearly to the same large wavenumber cutoff q₁ as in Fig. 39.