Controlling the Friction Coefficient and Adhesive Properties of a Contact by Varying the Indenter Geometry

Abstract

In the present paper, we describe a series of laboratory experiments on the friction between rigid indenters with different geometrical forms and an elastic sheet of elastomer as a function of the normal load. We show that the law of friction can be controlled by the shape of the surface profile. Since the formulation of the adhesive theory of friction by Bowden and Tabor, it is widely accepted and confirmed by experimental evidence that the friction force is roughly proportional to the real contact area. This means that producing surfaces with a desired dependence of the real contact area on the normal force will allow to “design the law of friction”. However, the real contact area in question is that during sliding and differs from that at the pure normal contact. Our experimental studies show that for indenters having a power law profile f(r) = c_nrⁿ with an index n < 1, the system exhibits a constant friction coefficient, which, however, is different for different values of n. This opens possibilities for creating surfaces with a predefined coefficient of friction.

1. Introduction

Friction plays an important role in many industrial processes. Often one seeks to reduce friction in order to lower the energy consumption [1,2]. In other technological applications, the friction coefficient must be high to provide optimal performance, such as in nanostructuring burnishing [3], car braking systems [4], contact of wheels with road surfaces [5], movement transmission [6], etc. Knowledge of the friction behavior in many engineering fields, such as the automotive industry [7], wind turbines [8] or medical devices [9], is of crucial interest.

Adhesion is another important phenomenon that is interesting both as one of the contributing factors to friction and in itself. Adhesion plays an important role in nature and engineering both at the micro- [10,11] and macroscopic scale [12,13]. The pollination process of plants is a biological example [14]. Certain amphibians (tree frogs) and reptiles (geckos) use adhesive forces to climb up or even hang upside down on vertical surfaces. Understanding the underlining mechanisms lead to mimicking their function for engineering purposes [15,16]. An artificial surface with gecko-like adhesive properties was developed by Gorb et al. [17]. In engineering applications, adhesion plays an important role in paints and coatings [18], orthodontics brackets [19] and aeronautical applications [20]. Adhesive contacts have therefore been the subject of intensive research [21,22,23].

Despite decades of research, it is still impossible to deliberately design surfaces with desired frictional or adhesive properties [24]. The most common way to influence the frictional behavior in dry friction contacts is modifying the surface of the contacting bodies [25,26] either by modifying the surface topography or by applying homogeneous or heterogeneous thin coatings [27,28,29,30,31]. This approach, however, is still based on trial and error [24].

In macroscopic contacts, the friction force is often roughly proportional to the normal force and, thus, Amontons’s law F_x = μF_N is valid [32]. On the microscale, on the contrary, many studies show the proportionality of the frictional force to the contact area A, i.e., F_x = τA [33,34,35,36,37,38,39], where τ is the (approximately constant) tangential stress in the contact area necessary to shear the interface. The widely accepted solution of this controversy is that the contact area in macroscopic contacts is roughly proportional to the normal load [40].

Even if Amontons’s law is not valid, one can formally define the coefficient of friction by dividing the tangential force by the normal force, i.e., μ = τA/F_N. The ratio A/F_N strongly depends on the contact geometry, which opens up possibilities for creating friction surfaces with predefined friction laws, even with complex specific dependencies μ(F_N). In normal contacts, the ratio A/F_N can be determined through numerical modeling, for example, by using the boundary element method (BEM) [41] or finite element method (FEM) [42]. However, tangential displacement causes the adhesive contact to lose symmetry. In this case, the contact area usually decreases significantly [21,36,43,44], while the normal force F_N changes only slowly [44,45,46]. Therefore, the ratio A/F_N also takes on different values. Modeling tangential adhesive contact is a complex task, and to date, the most studied case is that of a spherical indenter, for which there are analytical estimates that allow for the contact area under shear to be evaluated [36,45]. For contact geometries differing from the spherical, the situation becomes much more complicated, and therefore, studying arbitrary contact geometries requires conducting real experiments, which we propose in this work. The main goal of this work is to determine the indenter geometry in which, regardless of the magnitude of the normal load, a constant coefficient of friction μ = τA/F_N is realized. We found such a geometry, which opens up the possibility of manufacturing surfaces with a predetermined coefficient of friction.

A systematic design strategy for producing interfaces with preset frictional properties was proposed by Aymard et al. [24]. In a contact between a rough and a smooth interface, the controllable parameter is the topography of the rough surface. In [24], it was suggested to produce a meta interface consisting of spherical asperities. To target a specific friction law, the number of asperities and their shapes, sizes, height distributions and positions have been varied. The authors illustrate three different friction law types that can be obtained by optimizing the asperity heights.

Enhancing adhesive properties can be achieved by designing the shape of fibrillary adhesive microstructures [17]. A similar approach for the design of a surface meeting preset frictional properties by changing the geometric profile of the asperities is described in the present study.

In our previous work [38], the effect of the indentation depth of spherical indenters on the friction coefficient was studied experimentally. The present work extends this study to the frictional properties of indenters with power law profiles f(r) = c_nrⁿ. The experiments show that for 0 < n < 1, the friction coefficient µ remains constant over a wide range of normal loads. However, the value of the coefficient of friction depends on the parameters n and c_n. Contrary to [24], where only spherical asperities were used, our design strategy allows us to control the friction coefficient µ by changing the shape of the indenter. This works even for the case of one single indenter. This approach opens opportunities to create new types of surfaces with predefined coefficients of friction. The experimental setup as well as the experimental techniques used in the present study are similar to those described in [38].

2. Materials and Methods

With the knowledge of the tangential force F_x and contact area A, the averaged shear stress <τ> in the contact is given by:

$$〈\tau 〉=\frac{F_x}{A}.$$

(1)

Often, it is assumed that the shear stress <τ> is equal to some constant value τ₀, independently of the normal load or contact area [33,34,35,36,37]. If this assumption is valid, the friction force only depends on the real contact area A and is calculated as:

$${F_x=\tau }_{0}A.$$

(2)

In particular, the friction law (2) is observed in contacts with strong adhesion, e.g., when a hard indenter is pressed into a soft elastomer. This also applies to friction in contacts that are plastically deformed, where τ₀ represents the yield stress [47].

The friction coefficient μ is defined as:

$$\mu =\frac{F_x}{F_N},$$

(3)

where F_N is the normal force. For most non-adhesive contacts, the friction coefficient μ is approximately constant over some range of normal forces F_N, so that the friction force law takes the classical Amontons form:

$$F_x=\mu F_N.$$

(4)

Equations (2) and (4) can be considered as two limiting cases that are valid for strongly adhesive and non-adhesive contacts, respectively. In the classical works [48,49], it is stated, that there exists a transition from the “adhesive” friction mode (Equation (2)) to the “normal” friction mode (Equation (4)) with increasing normal load F_N. But, in our previous experimental work [38], we did not find such a transition in a wide range of external loads. On the contrary, it was shown that the experimental results of [48] can be well described with the “adhesive” friction law (Equation (2)) in the whole range of normal forces used in [48].

The friction coefficient μ can be formally calculated using the standard definition μ = F_x/F_N (3) independently of whether it is really constant or not. If Equation (2) is valid (adhesive contact case), for a circular contact with radius a, such a formally calculated coefficient of friction is equal to

$$\mu =\frac{{\tau }_{0}A}{F_N}=\frac{{\tau }_0\pi a^2}{F_N}.$$

(5)

According to this equation, the coefficient of friction depends on the relationship between the contact area A and the force F_N. Consider an axially symmetric indenter with the following power law shape:

$$f\left(r\right)=c_n{r}^n,$$

(6)

where r is the radial coordinate. In our previous paper [38], the dependency of the friction coefficient on the normal force F_N was derived for axially symmetric contacts between a hard indenter and an elastic half-space, provided that the friction force is proportional to the contact area, as shown in Equation (2):

$$\mu \left(F_N\right)=\pi {\tau }_0{\left(\frac{(n+1)(1-{\nu }^2)}{2nE{c}_n{\kappa }_n}\right)}^{\frac{2}{n+1}}\times \frac{1}{{\left(F_N\right)}^{\frac{n-1}{n+1}}},$$

(7)

where κ_n are constants that can be found in [38], E is the elastic modulus of the half-space and ν is its Poisson’s ratio. Based on this equation, we can conclude the following:

“1”

For 0 < n < 1, the friction coefficient μ increases with an increase in the applied normal force F_N because the normal force F_N increases slower than the contact area A.

“2”

For n = 1 (conical form), the friction coefficient μ does not depend on the normal force F_N:

$${\mu }_{cone}=\frac{{2\tau }_0(1-{\nu }^2)}{E{c}_n},$$

(8)

despite the fact that the friction force is still given by the equation F_x = τ₀A (2).

“3”

For n > 1 (for instance, for a parabolical indenter with n = 2), the friction coefficient μ decreases with an increase in the normal force F_N.

“4”

For n >> 1, the indenter turns into a flat stamp, where the contact area A does not change and is independent of the normal force F_N. In the limiting case n → ∞, Equation (7) shows an asymptotic behavior:

$$\mu \propto \frac{1}{F_N}.$$

(9)

Note that if the chosen parameter $c_n=a_0^{1-n}$ is inserted into Equation (6), while n → ∞, the profile f(r) describes a cylindrical stamp with a flat base with radius a₀.

In our previous work [38], hypothesis “3” was experimentally proven; it was also shown that there was no transition between the “adhesive” friction and “normal” friction modes in the wide range of normal forces. The present work is dedicated to the further experimental verification of all the above formulated hypotheses for adhesive contacts.

3. Results

All the experiments described below were performed in the same way as in [38] but for indenters with various geometrical shapes. Figure 1 shows the scheme of the experiment (left panel) and a real photo of the contact region of the experimental setup (right panel).

In the experiments, the indenters ((1) in Figure 1) were immersed in a CRG N3005 transparent elastomer sheet with a thickness of h = 5 mm ((2) in Figure 1). This material is a soft thermoplastic polystyrene-type gel produced by TANAC Co. Ltd., Gifu, Japan [50]. The elastomer sheet was placed on a glass substrate, which allowed for the direct observation of the contact area. To analyze the contact area, it needed to be homogeneously illuminated from all sides, which was provided by a surrounding LED light system ((3) in Figure 1). The contact forces (normal, tangential and lateral, i.e., perpendicular to tangential) were measured with a three-axis force sensor ((4) in Figure 1).

In all the experiments, the indenter was moved simultaneously in the normal and tangential directions with velocities of v_n = 0.2 μm/s and v_t = 5 μm/s, respectively. This means that the indenter was immersed in an elastomer sheet under a small angle α = arctan (0.2/5) ≈ 2.29°. In such conditions, the contact could be characterized as a tangential contact (friction) but with a slowly increasing indentation depth and, consequently, normal force. In each experiment, the indenter was immersed in the elastomer up to the maximum distance d_max and was then lifted until the moment of complete detachment. Supplementary videos show the complete experiments (indentation and pull-off phases), but in this article, only indentation phase is shown. Note that there are also experimental works of other authors with similar elastomers to those we used in this article, for example, [51,52].

3.1. Spherical Profiles

We started with a partial repetition of the results for the spherical indenters with different radii of R = 50 mm and R = 100 mm from our previous work [38]. The spherical indenters corresponded to a value of n = 2 in the profile function f (r) (6). In both experiments, the spheres were indented in the elastomeric sheet to the maximal depth d_max = 0.6 mm. This means that both the indenters were shifted in the tangential direction up to a distance of x_max = 15 mm. The results of these experiments can be seen in Figure 2.

The experimental results for both indenters confirmed hypothesis “3” of Section 2, since the dependences μ(t) in Figure 2f showed a decrease in the friction coefficient with time. This means that μ decreased with increasing the normal force. Moreover, for a parabolic indenter n = 2, Equation (7) leads to the following formula [38,53,54]:

$$\mu \left(F_N\right)=\pi {\tau }_0{\left(\frac{3R}{4E^{\mathrm{*}}}\right)}^{\frac{2}{3}}\frac{1}{{\left(F_N\right)}^{1/3}}.$$

(10)

Figure 3b shows the dependences of μ(F_N) in a double-logarithmic scale for all the cases considered in this paper. For convenience, in this figure, several dashed lines are shown, where the upper one shows the dependence μ ~ 1/(F_N)^1/3, and it was close to the trend of the experimental dependences for the spherical indenters (red and blue curves).

As the normal force increased, the friction coefficient measured in the experiment began to decrease faster. The reason may be that Equation (10) was derived for the idealized case of the half-space, while the experiment was carried out with an elastomeric sheet with a final thickness of h = 5 mm. It was previously shown in [38] that for thinner elastomers, the coefficient of friction μ decreases faster with increasing the normal load. In general, however, experiments with special indenters have shown that Equation (10) (or, in general, Equation (7)) is satisfactorily applicable over a wide range of normal loads.

3.2. Cylindrical Profiles

In order to test the following hypothesis “4“ of Section 2, experiments were carried out on indentation of cylindrical stamps with a flat base with diameters of D = 10 mm and 7 mm, as shown in Figure 2. A cylinder with a diameter of 10 mm was indented to a depth of d_max = 0.6 mm, and at the same time, it was shifted tangentially by x_max = 15 mm. A stamp with a smaller diameter of D = 7 mm was plunged to a depth of d_max = 1.0 mm and shifted by x_max = 25 mm. Here and in the following experiments, different indentation depths d_max were selected for the indenters with different geometries to demonstrate the experimental dependencies over a wider range of normal forces, if possible. In the case of a cylinder, for example, the smaller the base diameter, the greater the required indentation depth to achieve a fixed level of normal force.

Equation (2) predicts that in the case of an indenter with a flat base, the contact area A, and hence, the friction force F_x, should not change during tangential motion. However, Figure 2c shows that for both cylindrical indenters, the area A increased to some maximum value at the beginning of the indentation. This was due to the fact that in the experiment, the surfaces of the cylindrical stamp base and the elastomer were not perfectly parallel. Therefore, at the beginning of the indentation, the contact was incomplete, and the contact area built up to the maximal possible value A_max = πD²/4. In this regard, during the initial shear phase, the increase in the friction force F_x (see Figure 2b) can be explained by the increase in the contact area, aligning with Equation (2). However, as evident from the experimental relationships, even after the area reached the maximum value of A_max, the friction force F_x continued to increase, although the rate of its increase decreased significantly. This was partly because the value of the tangential stresses <τ> increased with increasing the normal force. However, it should be noted that we calculated the value of <τ> indirectly, as per the ratio <τ> = F_x/A, while the directly variable data were just the friction force (force sensor data) and the contact area (observed by the video camera). It should be taken into account that not only can the increase in the tangential stresses in the contact zone lead to an increase in the friction force but also the features observed at the edges of the contact. In the case of a spherical indenter, these effects were not significant because the contact zone grew continuously during the indentation. But, in the case of an indenter with a flat base, the effects at the contact edges could make a significant contribution to the friction force, since the elastomer sheet at the contact edges was deformed in such a way that the indenter was kind of situated in a “pit”, from which it must always “climb out” during tangential shear. This can serve as an additional channel for energy dissipation. In Supplementary Videos S1 and S2, it is possible to trace the change in the contact shape over time for both the experiments with cylindrical indenters. The videos show that during the indentation, the contact area increased to a maximum value, after which a light ring was visible at the edges of the contact, which increased in brightness and size throughout the indentation of the indenter into the elastomer. This suggests that the elastomer was highly deformed at the contact edges, which could resist the tangential movement of the indenter and result in an increase in the frictional force. As a result, the formally calculated tangential stresses <τ> = F_x/A increased, as shown in Figure 2e, where in the case of the cylindrical indenters, the stresses <τ> increased faster than in all the other cases.

It is more reasonable to represent the tangential stresses not as a function of the experiment time (which sets the indentation depth d = v_zt) but as a function of the average contact pressure <p> = F_N/A. Eliminating time t from the dependencies <τ>(t) in Figure 2e and <p>(t) in Figure 2d, the tangential stress–pressure relationship shown in Figure 4 was calculated for all the experiments.

It follows from Figure 4 that the dependence of τ(p) in many cases can be described by a power function of the following form [55,56]:

$$\tau ={\tau }_0+\alpha p^{\gamma }$$

(11)

with an exponent of γ ≈ 0.2. However, in the case of a cylindrical indenter with a smaller diameter D = 7 mm, as the normal pressure increased, a point was reached where the tangential stresses began to increase more rapidly. This aligns with the power law (11) characterized with a larger exponent of γ ≈ 1.0.

Note that using τ (11) instead of τ₀ in the friction law (5) results in a two-term friction law of the following form:

$$\mu ={\tau }_0\left(\frac{A}{F_N}\right)+\alpha {\left(\frac{A}{F_N}\right)}^{1-\gamma },$$

(12)

where the expression for the average contact pressure <p> = F_N/A is used. In our experiments (see Figure 4), the exponent was γ ≤ 1. Note that two-term friction laws have been used for a long time. As examples, we can cite the classical Amontons–Coulomb friction law [32,57], Derjaguin’s law [23], the law considering friction at the boundary of adhesive contacts [46], the law describing the friction force in the boundary regime [58,59], and so on.

Returning to the experimental results, we can conclude that during the indentation of the cylinder with a flat base, three phases can be distinguished: (1) an increase in the friction force F_x = <τ>A, mainly due to the increase in the contact area; (2) a further increase in F_x at the maximum contact area A = A_max = πD²/4 due to the growth of tangential stresses according to Equation (11); and (3) an increase in the friction force growth rate due to the even faster growth of the stresses at the subsequent indentation into the elastomer. Note that for the cylinder with diameter D = 7 mm, all three phases were present, and the transitions between them can be seen in the relationships shown in Figure 2b and Figure 4. However, for the indenter with a larger diameter D = 10 mm, only the first two phases were realized since this experiment was performed in a smaller range of contact pressures <p>.

Thus, in a certain range of parameters, hypothesis “4“ of Section 2 for a cylindrical indenter with a flat base is satisfactorily fulfilled. This follows from Figure 3b, according to which both cylindrical indenters showed a range of forces (approximately at F_N > 2 N) in which μ ~ 1/F_N. According to Figure 2, these were the forces at which the contact area was maximized and then remained constant. But, according to the same figure (Figure 2), the friction force F_x in this range of forces consistently increased, which contradicts assumption (2) that was used to derive Equation (9). Therefore, the coefficient of friction shown in Figure 3b decreased slightly slower than μ ~ 1/F_N. However, in the case of the cylindrical indenter, the friction coefficient μ decreased much faster than in the above-described case of a spherical indenter, for which μ ~ 1/(F_N)^1/3.

Note that the dependence μ ~ 1/F_N was violated for the cylindrical indenter with diameter D = 7 mm immediately after the change in trend (in the above-mentioned phase (3)), which can be seen in Figure 2b at t ≈ 50 min. In this phase, the growth rates of the tangential and normal forces were the same, which led to validity of μ = const with a further increase in the normal force, as shown in Figure 3a,b for a cylinder with a diameter of D = 7 mm at a normal force of F_N > 5.5 N. This change in behavior in the μ(F_N) dependence was not caused by a higher value of the normal force but by a higher contact pressure, since for the second indenter with a diameter of D = 10 mm, the μ = const region was not observed despite the similar range of normal forces. In Section 2, hypothesis “2“ was formulated, stating that the constancy of the friction coefficient, μ = const, is expected for conical indenters, as the contact area should increase monotonically with normal indentation, but not for a cylindrical indenter with a constant contact area. Therefore, the situation where μ = const for a cylindrical indenter is unexpected and may indicate a transition between the “adhesive” mode of friction, which is given by Equation (2), and “normal” friction (see Equation (4)) with μ = const. Some classical works, e.g., [48,49], speak about the existence of such a transition, although we did not find such a transition in our recent work [38]. Moreover, it was shown in [38] that the results of [48] can be interpreted by using the single concept of “adhesion” friction over the entire range of experimental parameters. We now find that, for some reason, the cylindrical indenter transitioned to the μ = const regime at high contact pressures. This feature was demonstrated by only one of the indenters, as the second indenter was operated in a smaller pressure range. Therefore, unambiguous conclusions about the presence of the transition between friction modes cannot be made, and the study of this issue requires additional experiments. However, this is beyond the scope of the proposed work.

3.3. Conical Profiles

The next step was to test hypothesis ”2” of Section 2, stating that for conical indenters, the coefficient of friction μ should be constant at all normal loads. The shape of the conical indenter is given by Equation (6) when n = 1, i.e., f (r) = c_nr. Therefore, the angle φ at the base of the cone, expressed in radians, is defined by the following formula:

φ = π − 2arctan(c_n).

(13)

The case of conical indenters was investigated using two indenters made of steel with angles of φ ≈ 160° and φ ≈ 170°. The indenter profiles were measured with a 3D laser scanning confocal microscope Keyence VK-X150 (KEYENCE DEUTSCHLAND GmbH, Neu-Isenburg, Germany) using a 10× magnification objective. Figure 5a,b show the profiles of both indenters.

Figure 5c shows the two-dimensional profiles of the indenters corresponding to the three-dimensional images. The red and blue curves in Figure 5c show the cuts of the profiles in two perpendicular vertical planes. Their overlap** confirms the axial symmetry of the profiles. The tips of both indenters were slightly rounded; the tips of the regular cones shown in Figure 5c are completed by the straight lines shown in black. The angles between these straight lines represent the exact values of the angles φ (Equation (13)), which were 160.16° and 170.89°. In our case, the rounded corners at the base of the indenters did not interfere with the test of hypothesis “2” of Section 2, since the rounding only affected the results obtained at the beginning of the indentation process. Moreover, the use of rounded corners at the top of the conical punches avoided the occurrence of a region with a singular stress concentration [60,61], which can lead to the destruction of the elastomer surface [62], especially at a tangential indenter shift.

Figure 2 shows the experimental data for both conical indenters. Here, the maximum indentation depth for the indenter with angle φ ≈ 160° was d_max = 1.2 mm at its tangential displacement x_max = 30 mm. The indenter with an angle φ ≈ 170° was immersed into the elastomer to a smaller depth of d_max = 0.6 mm, so it was also displaced by a smaller distance x_max = 15 mm. As evident from Figure 3b, in both cases, the coefficient of friction decreased with increasing the external load according to the trend μ ~ 1/(F_N)^0.15 (black and magenta curves). Thus, the rate of the decrease in the friction coefficient here was lower than in the cases of the cylindrical and spherical indenters, but the coefficient of friction still decreased with the normal force and was not constant, as predicted by Equation (8). This was primarily because the formula is derived on the assumption that the contact is axially symmetric, as in the case of purely normal indentation, but due to tangential shear, the contact quickly lost axial symmetry. At the same time, the main contact zone relative to the indenter center was concentrated at the leading edge of motion, as can be seen in Videos S3 and S4. In the case of the conical indenters, the tangential stresses <τ> increased with the pressure <p> according to the same trend as in the cases of the spherical and cylindrical indenters considered before, which follows from Figure 4, namely, <τ> ~ <p>^0.2. However, this build-up came after a decrease in the <τ> value at the beginning of the indentation, which can be seen better in Figure 2e.

Thus, the assumption that the friction coefficient μ = F_x/F_N does not depend on the applied external load in the case of a conical indenter (hypothesis “2” of Section 2) was not confirmed.

3.4. Power Profiles with Exponent n < 1

The last cases considered in this paper were axially symmetric indenters, whose profiles are described by the function f(r) = c_nrⁿ (6) with an exponent n < 1. Hypothesis “1“ of Section 2 suggests that in this case, the coefficient of friction μ should increase with increasing the external load rather than decrease as it does for values of the exponent n > 1. Since we did not have equipment that would have allowed us to mill indenters from steel with an arbitrary value of the exponent n, we decided to make such indenters from plastic. We printed two indenters with values of n = 0.7 and n = 0.5 using a “QIDI TECH I Fast FDM 3D Printer” (DI JIA TECHNOLOGY LIMITED, MONGKOK, Kowloon Hong Kong SAR) from Basicfil PLA material in an orange color. In both cases, the coefficient c_n in Equation (6) was equal to 1.

Profiles of the 3D-printed indenters scanned by a 3D microscope are shown in Figure 5d,e. Figure 5f shows the 2D profiles of these indenters, where lines of different colors (red and blue) show profiles corresponding to cross-sections of the 3D surfaces in two mutually perpendicular directions. Here, the black lines show the power functions that approximate these profiles. In the case of n = 0.7, the real printed indenter was well described by a power function with a similar exponent. However, in the case of n = 0.5, it turned out that in reality, the printed indenter was close to a power function with an exponent of n = 0.54, which seemed to be caused by printing inaccuracies. In Figure 5f, the black curves show both profiles completed to the correct power functions of f(r) = rⁿ, compared to which it can be seen that the indenter peaks were significantly rounded in the center. This was due to the fact that the printer used is not generally capable of printing such narrow areas with a high quality. This feature was similar to that described above with conical indenters. However, as above, the discussed rounding only affected the relationship between the contact forces at the beginning of the indentation, which did not interfere with the objectives of this paper. Figure 5 shows some other features of 3D printing. For example, in the case of n = 0.7 in Figure 5d,f we can clearly see the “step**” of the profile caused by the fact that the printer prints in layers. Moreover, the indenter insignificantly deviated from the axially symmetric shape due to the printing quality.

In the experiment, both indenters were immersed to the same depth of d_max = 2 mm, corresponding to their tangential displacement of x_max = 50 mm. The indenter with n = 0.5 exhibited much higher normal force values, which can be seen in Figure 2a. At the same time, however, both indenters showed similar pressure values over the entire experimental range, as can be seen in Figure 2d. This was due to the different rate of increase in the contact area, which is shown in Figure 2c.

Hypothesis “1“ of Section 2 suggests that for an indenter with a power profile of f(r) = c_nrⁿ with an exponent of n < 1, the friction coefficient μ should increase with increasing the normal force. It was assumed that this should occur since the contact area, and therefore the tangential force, will increase faster than the normal force. This, according to Equation (5), should lead to an increase in the coefficient of friction with increasing the normal force. However, Figure 2f shows that, after an initial decrease, the friction coefficient remained constant. Moreover, the plateau μ = const was rather long, which was due to the fact that the indenters in the experiments were displaced by the largest distance of x_max = 50 mm used in our experiments.

Thus, for indenters with n < 1, the behavior assumed for the conical indenter and formulated in hypothesis “2” in Section 2 was realized. As in the cases described above, the deviation from the assumed behavior was caused by a strong violation of the contact symmetry at the tangential shear of the indenter. The dependencies shown in Figure 2e, on average, indicate an increase in the tangential stresses <τ> with increasing the indentation depth (or pressure). However, they were non-monotonic due to the complex processes of contact rearrangement during sliding, which can be clearly seen in Videos S5 and S6. Note that the value of the stresses <τ> lay in the same range as in all the other cases, despite the fact that both indenters for the case n < 1 were made on a 3D printer from plastic (PLA). In all the other experiments, steel indenters were used.

From Figure 3, which shows the dependencies of the friction coefficients on the normal force, it can be seen that in the discussed case n < 1, after a rapid decrease, the friction coefficient remained constant. This was especially clearly seen for the case n = 0.5, since this experiment was carried out over a larger range of normal forces. Thus, in spite of the fact that, here, the adhesive friction regime was realized, in which the tangential stresses <τ> were close to constant regardless of the load, for this indenter shape, the formally calculated friction coefficient μ = F_x/F_N showed a constant value of μ = const in a wide range of normal forces F_N. Thus the effect of the indenter shape led to the fact that despite the realization of the friction law in the form of ${F_x=\tau }_{0}A$ (2), the classical Amontons law $F_x=\mu F_N$ (4) with a constant friction coefficient μ was also fulfilled.

In this specific case, the coefficient of friction in Equation (4) depended on the geometrical shape of the indenter. For an indenter with an exponent n = 0.5, the coefficient of friction is greater than for an indenter with n = 0.7.

Note that although we assumed constant stresses τ₀ here, the tangential mean stresses increased slightly with pressure, as in all the previous cases. And, as indicated by Figure 4, for both indenters with exponent n < 1, at high pressures <p>, the stresses <τ> increased according to the power law with an exponent of γ ≈ 1.0, as in the previously considered case of a cylindrical indenter with a smaller diameter of D = 7 mm. For a cylinder with a diameter of D = 7 mm, the range μ = const was also unexpectedly observed just when γ ≈ 1.0, i.e., for higher pressures. Therefore, it remains possible that in the cases of indenters with exponent n < 1, as well as in the case of a cylindrical indenter with a diameter of D = 7 mm, we are dealing with a transition between “adhesive” and “normal” friction, the presence of which is indicated by [48,49] but was not detected by us in [38]. The presence of such a transition may be due to the fact that in these three cases, the highest values of the contact pressure <p> were realized, as can be seen in Figure 2d. To verify the presence of such a transition between the friction modes, it is necessary to conduct additional experiments in the range of high contact pressures, which is beyond the scope of this paper.

4. Discussion

The effect described in Section 3.4 is that for indenters with power law profiles with an exponent of n < 1, the regime of a constant friction coefficient μ = const was realized. However, this effect may have also been due to the fact that these indenters were made of plastic (PLA). After all, the properties of the adhesive contact (e.g., the specific work of adhesion) strongly depend on the properties of both contacting surfaces, and all the cases described in this paper for indenters with n ≥ 1 were carried out with indenters made of steel. Therefore, the cases with indenters n ≥ 1 (steel) and n < 1 (PLA) cannot be compared unambiguously, and in order to understand the differences, it is necessary to establish some kind of “bridge” between them, which would help to make sure that it was the shape’s effect and not the material’s effect. An additional experiment with a spherical indenter with a radius of R = 100 mm acted as such a “bridge”. The experiment was completely similar to the one described in Section 3.1 but with the difference that an indenter printed on a 3D printer from the same PLA as the indenters with an exponent of n < 1 (see Section 3.4) was used. The results of this additional experiment are shown in the figures above with dashed lines. The detailed course of the experiment can be seen in Supplementary Video S7. Note that the data corresponding to indenters with radii R = 100 mm made of steel and PLA were different. This was due to the fact that the used 3D printer prints in layers, so the indenter had a stepped shape, as with the indenters shown in Figure 5d,f. However as can be clearly seen in Figure 3, the μ(F_N) dependence of the PLA indenter showed exactly the same trend as that of the steel indenter with the same radius R = 100 mm. Therefore, the discovered effect, i.e., that for indenters with an exponent of n < 1, the friction coefficient does not depend on the external load, was not related to the indenter material but was caused by its specific geometrical shape, in which the contact area increased as fast as the normal force. In this case, according to the expression μ = τ₀A/F_N (5), we have a situation with μ = const.

In our previous work [38], where adhesive contact was investigated, it was hypothesized that a contact-geometry-dependent and, at the same time, normal-force-independent coefficient of friction should be realized in the case of a conical indenter. In this case, μ = const is unique because in it the mutually exclusive laws of friction ${F_x=\tau }_{0}A$ (2) and $F_x=\mu F_N$ (4), where both the tangential stress τ₀ and the coefficient of friction μ are constants, must be simultaneously valid. However, the hypothesis (put forward in [38]) would only be valid if the contact remained axially symmetric during tangential motion. But, in the adhesive contact, the breaking of the axial symmetry at tangential shear violates the hypotheses formulated in [38], which are briefly described in Section 2 of the present paper. However, the general tendency still remains, namely, that with decreasing the exponent n in the profile function f(r) = c_nrⁿ, the friction coefficient μ decreases more and more slowly with increasing the external normal force F_N. Moreover, the form of an indenter for which the laws of friction (2) and (4) are simultaneously valid was found, and it was with an indenter whose profile is given by the power function f(r) = c_nrⁿ with exponent n < 1.

The cases of a constant coefficient of friction, μ = const, for indenters with n < 1, which were studied experimentally in this paper, are of high application value [63]. Here, we studied indenters with two different values of n = 0.5 and n = 0.7. According to Figure 3, in the case of n = 0.7, the coefficient of friction was lower. Thus, there is an indenter shape for which the coefficient of friction on the adhesive contact is constant and does not depend on the applied load. Moreover, by varying the shape of this indenter, the coefficient of friction can be set. One potential application of the found effect is the creation of surfaces that exhibit constant coefficients of friction in adhesive contacts. On these surfaces, at a certain distance from each other, protrusions should be placed whose profiles would geometrically represent the functions f(r) = c_nrⁿ with an exponent of n < 1. Moreover, these can be, among others, microscopic surfaces with nanoscale protrusions. The protrusions under discussion can be located on the surfaces periodically or randomly, which requires additional study.

Note that in this paper, we often talk about contact symmetry breaking in tangential motion without giving pictures of the contact region. However, the detailed evolution of the contact areas for all the experiments performed can be observed in the supplementary videos to this paper, which are an important part of the paper. Moreover, these videos not only show the phase of the indentation of the indenter at an angle into the elastomer but also the pull-off phase the indenter in the normal direction, which was performed after the indenter reached the maximum indentation depth d_max and tangential displacement x_max.

5. Conclusions

This paper experimentally investigated the influence of the geometric shape of an axially symmetric indenter, which is given by the power law f(r) = c_nrⁿ, on the coefficient of friction in an adhesive contact. Spherical (n = 2), conical (n = 1) and indenters with an index of n < 1 (with n = 0.5 and n = 0.7) as well as flat-ended stamps were studied. It was shown that the friction coefficient μ = F_x/F_N decreased with increasing the external force F_N applied to the friction surfaces. This confirms the analytically determined tendency showing that for indenters with smaller indices of n, the friction coefficient decreases slower with increasing the force F_N. It was experimentally shown that in the case when the exponent n < 1, the friction coefficient took a constant value independent of the external load. In this case, two laws of friction, which are normally thought of as mutually exclusive, were simultaneously fulfilled. The first of them says that the friction force is proportional to the contact area, since constant-contact tangential stresses independent of the external load are realized. The second law of friction is the classical Amontons law, according to which the coefficient of friction does not depend on the external load. However, this does not mean that the true Amontons law is realized. In the case n < 1, the adhesive tangential contact, as before, was characterized by a constant value of the tangential stress τ₀, which slightly increased with increasing the pressure in the contact interface. In this case, the friction coefficient μ = F_x/F_N, although constant, depends on the exponent n, so it loses its original meaning as a material constant. The found mode can be used for creating surfaces with a given coefficient of friction. These surfaces should have a large number of protrusions with profiles f(r) = c_nrⁿ with the power n < 1. Thus, we proposed a method of creating surfaces with a given friction coefficient, which can be varied by the geometric shape of the indenters (protrusions or roughness on the surfaces). This work is also of fundamental interest because it advances the understanding of the dependence of the coefficient of friction between rough surfaces on the geometric characteristics of their topography.

The question remains open as to what shape an indenter should have so that the coefficient of friction increases with increasing load. To realize such a situation, the contact area must increase faster than the normal force. In normal contact, this condition is realized for n < 1. But, in the presence of tangential displacement, as this experiment has shown, due to the symmetry breaking of the contact area, the relationship between the contact area and the normal force is broken, so for indenters with n < 1, a constant friction coefficient μ = const is observed.