In Situ Investigation of Load-Dependent Nonlinearities in Tangential Stiffness and Damping of Spherical Contacts

Seemingly stationary (pre-sliding) interfaces between different materials, parts, and components of an engineered structure are major sources of compliance and damping. Interfacial stiffness and damping exhibit nonlinearities due to changes in geometry and physical response of the contact under varying loads [1]. Spherical contacts first visited by Hertz [2] are commonly used to investigate those tribological changes thanks to their resemblance to asperity-scale contacts between rough surfaces (e.g., elastic fully adhered contact [3,4], elastic-plastic frictionless contact [5], elastic-plastic fully adhered contact [6]; all in normal loading). The efforts for resolving the nonlinearities of tangential contact stiffness and energy dissipation of spherical contacts date back to two seminal studies by Cattaneo [7] and Mindlin [8]. They developed a partial-slip-based elastic contact model for the pre-sliding response of spheres under unidirectional tangential loading. In partial-slip contact model, a slip zone at the edge of contact area nucleates upon application of tangential loading and propagates inward as tangential load increases. During loading process, the tractions over the slip zone are bounded by friction, and hence the tangential contact stiffness decreases, i.e., softening contact. Later, Mindlin extended partial-slip model to cyclic tangential loading of pre-sliding spherical contacts [9]. Irreversible frictional losses during this process lead to hysteretic tangential force–displacement loops. Area inside those loops yields energy dissipation per cycle (frictional losses). Mindlin's model predicts a power-law scaling between frictional energy dissipation ΔW and maximum tangential force applied on the contact Q_m, i.e., $ΔW∼Qmn$ with exponents n = 3 for small and n > 3 for large oscillations. In this scaling, n = 2 would offer a linear load-independent damping ratio [10]. Therefore, Mindlin's model yields nonlinear load-dependent damping ratio for all tangential loads.

Experiments starting with Johnson [11] verify those predictions only partially. Johnson's fretting experiments (steel ball-on-flat) deviate from Mindlin's predictions especially under low tangential forces in both nonlinear softening and damping. In those experiments, especially under large normal preloads, linear and sometimes hardening contact stiffness were more prominent as tangential force increases. Besides, Johnson's experiments delivered linear damping for small tangential oscillations and nonlinear damping for large oscillations in contrast to Mindlin's always nonlinear frictional damping prediction. Similar deviations from partial-slip theory were observed in numerous experiments with various sample geometries, materials, and instruments [12–17]. Johnson listed roughness, plasticity, and elastic hysteresis as possible explanations for mismatch as none were accounted for in Mindlin's analysis [11]. The influence of roughness and plasticity on tangential response of contacts was first recognized by Bowden's group [18,19]. Accordingly, in the case that asperity-scale contacts fully adhere, their response to subsequent shear can be determined by the interfacial or material strength, and thus plastic flow. This idea has been further explored by various experimental and numerical works in the last two decades [20–30]. Of particular importance among these works is Etsion's group [21,24,31,32] revisiting the simplifying assumptions of Mindlin's approach. Particularly, they tested copper and steel spheres against sapphire flat under combined normal and tangential loading while monitoring the contact area in situ by an optical microscope. Under heavy preloads, their observations confirmed predominant plastic flow over the contact. Besides, they carefully placed micro-indentation marks at the edge of the contact and traced the distance between them during tangential loading as a measure of relative slip. Finding nearly uniform motion of those marks indicated no measurable slip. Although those tests studied almost fully plastic contacts, which were not intended by Mindlin, Etsion's group demonstrated the utility of in situ imaging for direct observation of changes in contact.

In this work, we investigate load-dependent nonlinearities in tangential contact stiffness and dissipation, through combined normal and tangential loading tests conducted inside a scanning electron microscope (SEM). Mechanical testing in electron microscopy reveals material-scale changes and thus enables discovery of novel constitutive behavior [33–36]. To the best of our knowledge, in situ SEM pre-sliding contact testing is missing in the literature. Although in situ SEM tests cannot reveal contact areas explicitly due to lateral point-of-view, they provide real-time evolution of roughness and plastic deformations at the edge of the contact. When synchronized to hysteretic tangential force–displacement curves (loops), nonlinear behavior in tangential stiffness and energy dissipation can be directly linked to those physical mechanisms. Another advantage of in situ SEM investigation over optical microscopy is its ability to resolve contacts with several micron radii. This range of contact sizes approximates asperity-scale contacts better than larger contacts as featured in in situ optical microscopy studies [21,24,31]. Note that transmission electron microscopy is more suited for nano-scale asperity studies (see Refs. [37,38] for example).

Nominally flat high-density polyethylene (HDPE by Polymershapes) and polyurethane rubber (PUR by Smooth-on, Inc.) samples are prepared by wire cutting (∼5 × 5 mm² surface areas and 1 mm thickness) and attached to the SEM stubs by silver paste. Simpact 85A fast setting elastomer (Smooth-on, Inc.) is used to produce PUR samples by mixing 85 weight units of agent A with 100 weight units of agent B. They are thoroughly mixed and poured into molds. The mixture is cured at room temperature for 24 h. The samples are coated with 5 nm thick silver (Ag) to enable imaging. The root-mean-square (rms) roughness of HDPE and PUR samples averaged over five scans from different locations are 88 nm and 325 nm, respectively (Zygo New View 600 white light interferometer).

Picoindenter inside SEM (PI 88 SEM PicoIndenter, Bruker Inc.) is used to monitor contact mechanics between the nominally flat samples and a sphero-conical diamond probe with a tip-radius of curvature R = 17 μm (Fig. 1(a)). For in situ SEM imaging, the sample holder stage is tilted by 10 deg and an accelerating voltage of 5 kV was used. The SEM images are recorded with 15 and 30 frames per second for HDPE and PUR samples, respectively. Those rates enable sufficient temporal resolution when compared with the loading period (tangential loading period is 6 s). The spatial resolution in those SEM images is 0.02 μm/pixel and 0.01 μm/pixel for HDPE and PUR samples, respectively. Under the preloads applied in the tests, the contact radii range from 6 to 10 μm for HDPE cases and 10 to 15 μm for the PUR cases. Therefore, the spatial resolution of the images facilitates for detailed analyses of changes in the vicinity of the contacts. During the tests, the contact load normal to the surface (preload) is kept constant while tangential displacements are varied periodically. Three different preloads for each sample: 5, 10, and 15 mN for HDPE, and 2, 3.5, and 5 mN for PUR are chosen to capture elastic and elastic-plastic contact response. Note that this range of contact response has not been tested previously by in situ techniques. For example, all of in situ optical microscopy tests cited in Sec. 1 are conducted under heavy preloads leading to fully plastic contacts. Considering shakedown and flattening of asperities under long durations of fretting, elastic-plastic contact conditions are more realistic and suited to test classical contact models based on elasticity, e.g., Cattaneo–Mindlin.

Four trials at four different locations on the sample are performed at each preload. Within the first 5 s of each experiment, normal load is increased linearly to the final preload values (P_max). Then, tangential displacements δ_raw are applied as periodic triangular signals with at least four (mostly five) different maximum amplitudes (Fig. 1(b)) in the descending order. Time period of tangential displacements is set to 6 s for quasistatic loading conditions. Maximum displacement amplitudes are chosen after a few unidirectional tangential loading trials to gage only pre-sliding response of the contact during cyclic loading. Besides, slip-index for each preload and maximum tangential load case is estimated as outlined by Varenberg et al. [39,40]. The slip indices for 23 out of 24 loops studied are found to range from 0.5 to 0.8 (mean: 0.58). According to the criterion established by Varenberg et al. [39,40], this range corresponds to pre-sliding response (either partial slip or transition regimes). One case (fourth trial of 5 mN preload case on HDPE) yields a large slip-index (1.55), which is characterized as gross slip case by the same criterion. For uniformity of number of cases, this loop is also analyzed.

Forces and displacements are sampled at 200 Hz. Loading (normal force P and tangential displacement δ_raw) applied on the PUR sample and representative responses measured (normal displacement w and tangential force Q) are shown in Fig. 1(b) at P_max = 2 mN and five different tangential displacement amplitudes (δ_raw).

Thus, for very small vibrations, the Mindlin model predicts $ΔW∼Qm3$⁠, (the power law with exponent n = 3). The exponent and hence the nonlinearity in damping increases with increasing tangential load. However, regardless of the amplitude of loading, material damping always prevails as was shown by Johnson [11] and others, and even exceeds frictional damping effect for small tangential loads. Material damping can be seen as a cumulative consequence of losses occurring in loading–unloading cycles in the bulk material. Several mechanisms were proposed to explain the origins of material damping: diffusion of defects, dislocations and grains in metals [43], realignment and internal friction of fibers, and chains in polymers and composites [44,45], and fluid–solid interactions in multiphasic materials [46]. Loss tangent ratio of losses to maximum stored elastic energy in a loading cycle is commonly used to quantify the material damping for polymers [47], which is independent of loading amplitude and hence linear (n = 2).

Figure 2 shows representative hysteresis loops; i.e., tangential force as a function of tangential displacement, for HDPE under 5 and 10 mN normal preloads, and PUR under 2 and 3.5 mN normal preloads (also see Appendix  A for Supplementary Videos for three representative cases synchronized to SEM images). Five loops in each plot correspond to five different maximum tangential displacements and possess a few common and peculiar characteristics. Increasing normal forces on contacts suppress dissipation, resulting in narrower loops. Similar decrease in dissipation and slight increase in stiffness is observed for decreasing maximum tangential displacements at a given normal force. Those load-dependent trends in stiffness and damping will be studied more in the subsequent subsections. Since the Picoindenter's transducer assembly is open loop, we can only control the global tangential displacement of the transducer δ_raw and net relative displacement of the probe over the contact depends on the instantaneous tangential force and transducer stiffness (Eq. (2)). Therefore, symmetric global tangential displacements result in smaller and asymmetric net displacements when the loading reverses and stick state dominates the contact initially. Similar asymmetric hysteresis loops are observed by earlier works especially at the initial stages of fretting response while the contact is still experiencing elastic-plastic shakedown [16,48,49].

Two other reasons behind this asymmetry are uneven distribution of plastic deformations over the contact and hysteretic material response. The former causes abrupt stiffness changes during a loading–unloading cycle thanks to plastic deformation (pileups) accumulated more in one direction. This source of asymmetry is applicable to HDPE cases only and will be studied in depth in Sec. 3.2 with synchronized SEM images. The latter is applicable to both samples as polymers are known to exhibit hysteretic mechanical response due to viscoelasticity, irreversible bond breakage, and interactions between soft–hard phases as in polyurethane elastomers [47,50,51]. When combined with finite deformations, the hysteretic material response could deliver asymmetric loading–unloading curves as in Fig. 2. Note that Mindlin's model, cannot explain the asymmetric response as frictional tractions are assumed to remain axisymmetric throughout the loading cycle. Only a finite element model with more realistic constitutive and failure models for HDPE can explain those peculiarities.

Figure 3 shows averaged tangential stiffness (Eq. (3)) as a function of maximum tangential force estimated from the hysteresis loops at the fifth loading cycle. Average stiffness of the loops at a given maximum tangential force increases with increasing normal preloads. This is primarily due to increase in contact radii, which is the length scale determining the stiffness of the contact. The dependence of average tangential stiffness on tangential force is not as uniform. For low preloads, increasing tangential forces lead to reduction in stiffness (softening) as predicted by the classical partial slip model. For high preloads, opposite seems to occur (hardening) with increasing tangential loading. Note that at constant preloads, viscoelastic creep occurs in both materials. Combined with prominent plasticity observed for the HDPE cases (Fig. 10 in Appendix  B), further increase in normal penetrations and thus contact radii can be responsible after those increases with tangential loading.

Next, we compare those stiffness values quantitatively with the prediction of the CM model. In Eq. (5), we had obtained the linearized contact stiffness that holds for fully adhered spherical contacts as $8G*a$⁠, and remaining softening terms. In Fig. 4, we normalize the average tangential stiffness with $8G*a$ and replot them against maximum tangential forces normalized to normal preloads Q_m/P_max. To account for viscoelastic effects, we treat both shear modulus and contact radii as time-dependent variables to be estimated from the normal penetrations as $8G*a=1−νs/2−νs3Pmax/wmean$⁠, where w_mean is the mean penetration depth measured during a corresponding hysteresis loop (fifth cycle) from which the tangential stiffness is extracted, and ν_s is the Poisson's ratio of samples taken as 0.45 [52] and 0.5 [47], respectively, for HDPE and PUR. Note that this estimation follows from the Young's modulus derivation upon normal preloading (indentation) and then linking Young's modulus to the equivalent shear modulus as $G*=E2(1+ν)(2−ν)$ for both HDPE and PUR contact cases [2,41]. The Young's modulus of diamond probe E_dia = 1141 GPa is significantly higher than that of both HDPE and PUR, and thus is dropped from the equivalent contact modulus estimations.

As shown in Fig. 4, the normalized stiffness values cluster around 0.5 for HDPE and 1 for PUR. Softer response observed for HDPE cases is primarily due to plasticity as evident in Fig. 10(a). Finite element modeling of elastic-plastic spherical contacts also delivers similar softening for increasing degrees of plasticity [29,30]. It can be seen for instance in Fig. 4 of [30] that averaged tangential stiffness reduces below 60% of Mindlin's fully adhered contact predictions when normal preloads cause penetrations that are more than ten times the critical penetrations at the onset of yielding. Substantial degree of plasticity can be directly seen in Fig. 10(a) and inferred from mean contact pressures used in our tests. Even for the lightest preload on HDPE, mean contact pressures are 50% higher than yield strength of the material. Therefore, softening in tangential response is largely due to preload-induced plastic deformations in HDPE.

In contrast, stiffness values measured for the PUR samples follow CM predictions closely. This is expected since reversible deformations illustrated in Fig. 10(b) and minute roughness effect in PUR samples match well with the assumptions of the CM model. Small scatter around predicted linearized stiffness can be attributed to nonlinear deformations in PUR contacts as the indentation strains reach 15%, i.e., $0.2a/R∼0.15$ [53]. Viscoelasticity-corrected normalized stiffness of HDPE samples exhibits more dependence on tangential loading than that of PUR samples. Increase in tangential forcing leads to softening under low normal preload (5 mN) and hardening under high normal preloads (10 and 15 mN) for HDPE. Note that the partial slip contact model predicts only softening response irrespective of normal preloads (see Eq. (5)). Hardening observed at high normal forces is due to pileups at the edge of the contacts for HDPE samples. Pileups occur mainly after plastic shearing and wear of asperities within the contact and subsequent accumulation of them at the contact edge [54]. SEM images of the contacts taken at the end of tangential loading under highest normal forces (Fig. 5) clearly show plastic deformation and pileup in HDPE case while none for the PUR case. Plastic flow is more prominent at higher normal preloads where mean contact pressures (∼50 MPa) exceeds HDPE's strength limits [52].

Plasticity also increases contact area via junction growth during shearing [6,19,30,31]. To quantify possible junction growth, we utilize the results of numerical simulations of fully adhered elastic-plastic spherical contacts. Brizmer et al.'s finite element simulations [22] proposed a correction to Tabor's empirical relation for junction growth as A/A₀ = 1 + 0.6(Q/P_max)⁴ for moderate plastic flows within the contact. When we substitute the maximum tangential forces measured for hardening HDPE cases into this expression $(Qm/Pmax∼0.7)$⁠, increase in contact area is limited to 14% and thus, increase in contact radii can be assumed bounded within 7%. Therefore, even if HDPE's relaxation is neglected, the junction growth fails to explain 20–30% hardening observed in HDPE cases at 10 and 15 mN normal preloads (Fig. 4(a)).

Next, we study the SEM images of representative HDPE cases along with the hysteresis loops to clarify the role of plastic deformations and uneven pileups. Figures 6(a) and 6(b) show SEM images taken at the leftmost position within a loading cycle of the maximum and minimum tangential displacement cases, respectively. Asymmetric pileup on the left side is clear in Fig. 6(a). While the probe is plowing through that pileup at the leftmost position, the hysteresis loops exhibit stiffening (see the inset in Fig. 6(c)). It can be observed that the hysteresis loop in this region converges into a single line, i.e., stiffening with very little energy dissipation is expected here. The instantaneous stiffness in the circled region is found as $2150N/m$⁠, and this estimation is a combination of the contact stiffness and the bearing stiffness of the pileup zone.

For the minimum tangential displacement case, the probe does not reach the pileup region and thus pileup stiffness has no contribution in the average stiffness (see Figs. 6(b)–6(d)). Therefore, the normalized contact stiffness (Fig. 4(a)) at higher normal preloads increases with increasing tangential forces. In summary, increasing tangential forces lead to softening in contacts under normal preloads below materials strength limits for both materials as expected from CM model. In contrast, tangential stiffness increases at higher preloads due to significant plastic deformations and pileups over the contact as seen in HDPE cases.

Lastly, we study the energy dissipation and damping ratio per cycle of tangential loading. Figure 7 shows energy dissipation per loading cycle values as a function of maximum tangential forces. Except for a few HDPE cases, increasing maximum tangential forces at a fixed normal preload results in more energy dissipation. As summarized in Sec. 2.2, CM model predicts a power-law scaling between energy dissipation and maximum tangential force, i.e., $ΔW∼Qmn$ with exponents n = 3. Similar power-law scaling is found for all the PUR cases (Fig. 7(b)) with n ranging from ∼2.2 at the highest normal preload to ∼2.8 at the lowest. This preload-dependence in power-law exponents is expected since frictional slip is suppressed more at higher preloads and material damping still prevails. Hence, the power-law exponents approach the linear damping limit of n = 2. At lower normal preloads, frictional slip is more prominent and so the exponents approach to partial-slip model prediction of n = 3. Starting with Johnson's [11] and Goodman and Brown's [12] experiments, various experiments with different contact geometries, materials, and testing equipment reported power-law exponents within similar ranges as in PUR cases [13–17].

Dissipation trends of HDPE cases are more irregular than PUR cases (Fig. 7(a)) primarily thanks to substantial plasticity as shown in Figs. 5 and 6. As noted by Refs. [26,49], plastic deformations in fully adhered spherical contacts can undergo shakedown, i.e., contact approaches to an elastic state after a few cycles of loading and no further dissipation occurs due to plasticity. The maps provided by Patil and Eriten [26] predict the shakedown to occur at normalized tangential forces lower than 0.2, i.e., Q_m/P_max ≤ 0.2. As seen in Fig. 7(a), the dissipation values at small tangential forces fall in this shakedown regime, where material damping such as viscoelastic losses and frictional slip prevail as the only sources of energy dissipation. Moreover, since frictional losses vanish at small tangential forcing, material damping dominates and hence the power-law exponents approach 2 as in PUR cases.

Plasticity and frictional slip contribute more to the energy losses observed at higher normal preloads and tangential forces in HDPE contacts. Trends in energy dissipation become more irregular under those loading conditions thanks to plastic deformations and uneven pileups. As discussed in Fig. 6, the hysteresis loops for HDPE cases possess a stiffness-dominated region thanks to combination of contact and bearing stiffnesses over pileup zones. In that region, there is negligible energy dissipation, and thus increasing tangential forces to plow through the pileup zone does not result in increasing energy dissipation. Once this stiffness-dominated zone is plowed through, energy dissipation increases sharply with further increase in tangential forces (see for instance 5 mN preload case). In this sharp increase zone, plastic shearing and friction contribute significantly to the dissipation, and thus increase damping capacity of the contact nonlinearly with large power-law exponents, n ≫ 3 as illustrated in Patil and Eriten [26]. Note that this sharp increase zone is not reached for 15 mN preload case as maximum tangential force needed is beyond the transducer's limits.

Note that the specific damping capacity is a measure of mechanical dissipation in dynamic loading of both linear and nonlinear materials and can be related to loss coefficients of linear viscoelastic materials via η = ψ/2π [55]. Figure 8 shows the specific damping capacities with respect to the tangential forces normalized to the normal preloads. Specific damping capacities for PUR steadily increase with increasing tangential forces, while for HDPE the trend is more erratic, reflecting almost perfectly the trends in measured energy dissipations. At normalized tangential forces lower than 0.2, i.e., Q_m/P_max ≤ 0.2, further plastic shearing does not occur and thus specific damping capacities reflect mostly material damping for HDPE. Within this mild loading regime, ψ ranges from 0.04 to 0.3 for HDPE, which correspond to loss coefficients (ψ/2π) ranging from 0.007 to 0.05. For most of the glassy polymers at 30 °C as HDPE, the loss coefficients are reported to range from 0.007 to 0.07 [55], and thus confirms the dominance of material damping at small tangential forces. At large tangential forces, the specific damping capacity increases tenfold for HDPE. As elaborated above, this highly nonlinear and load-dependent damping behavior is due to plastic shearing and frictional losses. For elastomers like PUR samples, reported loss coefficients range from 0.1 to 1 [55]. The maximum loss coefficients obtained from hysteresis loops of PUR samples is 0.16 (Fig. 8(b)). Even this value is comparable with the material losses (loss coefficients) reported. Therefore, pre-sliding tests conducted on PUR samples can be considered as shearing of almost fully adhered spherical probe, with negligible frictional losses at small tangential forces and dominant material damping for all loading cases. Note that this explains the power-law exponents lower than CM model predictions.

To the best of our knowledge, in situ SEM study of pre-sliding response of contacts has never been reported elsewhere. Therefore, future research can benefit from a short discussion featuring the challenges we encountered in those tests. In particular, we study pre-sliding response of a rigid spherical probe on a thermoplastic polymer (HDPE) and an elastomer. Similar low-moduli materials (polymethyl-methacrylate (PMMA)) were utilized by others to study onset of frictional sliding [56–58]. Practical challenges about pre-sliding tests and picoindenter's limitations lead to our choice of materials. Metals, for instance, would be an obvious candidate for those tests as nonlinear compliance and damping is commonly observed in assembled metallic components. [1,59–61]. Structural metallic alloys possess high moduli and thus form contacts that are too stiff for the picoindenter transducer. To exemplify, we assume a representative shear modulus of 100 GPa for a metallic alloy to be tested and a sufficiently large contact radius, say 1 µm to resolve under SEM. Those values yield tangential contact stiffnesses of about $Kt=8G*a∼106N/m$ in the linear elastic response regime. This stiffness is orders of magnitude higher than transducer's tangential stiffness (k_tr = 2252 N/m in Eq. (2)). Therefore, nearly all of the raw tangential displacements applied by the transducer will perturb transducer only rather than the contact until the metallic contact softens considerably. Given the contrast in stiffnesses, such a softening would coincide with the onset of sliding in the contact. Confirming this quantitative argument, our efforts in testing pre-sliding response of structural steel and aluminum alloys in the picoindenter system always resulted in gross sliding response of the contact. Even with the stiffer of the materials we were able to test, the pileup at the edge of the contacts lead to instantaneous tangential stiffness values of $2150N/m$ (see Fig. 6(c)). This stiffness being comparable with transducer's already limited range of pre-sliding response, we could explore for HDPE under heavy loads.

The materials we chose to test pose additional challenges for an in situ SEM study. Both HDPE and PUR samples are insulators. Therefore, a thin metallic coating (5 nm silver) and relatively low beam energies (5 kV) were needed for reliable imaging while limiting charge buildup in the samples. Particularly, limiting charging of samples was essential in preserving mechanical integrity of our samples. Similar combinations of metallic coating and beam energies are commonly used to reduce those SEM artifacts and deliver reliable imaging of polymers without structural changes [62]. Another concern on performance of polymers and elastomers under mechanical loading is their time–temperature dependence. Especially under contact and vibration applications, where mechanical energy is localized space and time, those materials exhibit frequency-dependent mechanical response [42]. In our tests, loading frequencies are rather low (<1 Hz) for substantial frequency-dependence to be observed (and loading frequencies cannot be increased due to SEM imaging limitations). However, long durations in our pre-sliding tests (∼200 s in Fig. 1(b)) lead to creep and relaxations in the samples. We attempted to account for this effect in stiffness estimations via normalization against mean penetration within a steady-state loading cycle (w_mean in Sec. 3.2). Note that this normalization assumes linear viscoelastic response and successfully accounted for time-dependence in PUR samples under light loading. Under heavy loading, we observe nonlinear inelastic response, especially for HDPE samples (see the hysteresis loops in Sec. 3.1). The authors are currently working on a finite element model with finite deformations and time-dependent constitutive models to explain the nonlinear response.

In this work, we conducted pre-sliding contact tests with a rigid spherical probe and two materials (HDPE and polyurethane elastomer) and measured the tangential stiffness and damping capacities inside a SEM. In particular, the pre-sliding tests were conducted under a fixed normal preload followed by cyclic tangential loading at different amplitudes. Hysteresis loops obtained in those tests delivered the trends in tangential stiffness, magnitude of energy dissipation, and damping capacities as a function of maximum tangential force. Utilizing in situ SEM images synchronized to the hysteresis loops enabled quantification of the degree of plasticity and geometric alterations in the vicinity of contact, and their influence on the tangential stiffness and damping capacities. The trends were then compared with the predictions of the classical partial-slip-based (Cattaneo–Mindlin) model. Major findings are:

• Contact radii extracted from the SEM images deviate from elastic contact predictions primarily due to plasticity in HDPE and contact-edge-detection error in PUR samples;

• PUR samples behave elastically whereas HDPE exhibit significant plastic deformations under the combined normal and tangential loading used in pre-sliding tests;

• Increasing tangential forces are found to cause softening in contacts under normal preloads below materials strength limits for both materials, and this is in agreement with the classical Cattaneo–Mindlin solution;

• In contrast, tangential stiffness increases with tangential forcing at higher normal preloads thanks to plastic deformations and pileups over the contact in HDPE cases;

• Linear material damping is predominant under small tangential forces for both materials, yet, at large tangential forces, the specific damping capacity increases tenfold for HDPE; and

• This highly nonlinear and load-dependent damping behavior is due to plastic shearing and frictional losses. In contrast, nearly fully adhered elastic contact in polyurethane samples results in dominant material damping.

Classical theory of Cattaneo–Mindlin neglects nonlinear constitutive behavior, geometric alterations (wear, pileup), plasticity, and material damping, and thus cannot predict the observed load-dependent nonlinearities in tangential stiffness and damping. Similar deviations from the theory were previously observed in many different contact pairs (mostly metallic) in the literature. We propose that a combination of those neglected factors is the major reason for those deviations reported in the literature. To explain those deviations, and stiffness and damping nonlinearities, more sophisticated models featuring finite deformations and temporal evolution of contact and material properties are needed. We anticipate that such models will outperform the classical models at the expense of added complexity and number of parameters to be determined. Note that uncertainties associated to each additional parameter can accumulate to shadow the predictive capacity. Therefore, choice and parameter estimation of sophisticated models should be guided by a given application, material set in contact and pre-sliding test conditions. Nevertheless, predictive modeling of pre-sliding contacts still remains as a challenging and fertile research task.

This work is partially supported by the National Science Foundation CMMI-CAREER Grant No. 1554146. The authors would like to thank Chris Getz, Douglas Stauffer, Joe Lefebvre, Sanjit Bhowmick, and Jacob Noble of Bruker, Inc. for their help with the settings of PI 88 system used in data collection.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party listed in Acknowledgment. No data, models, or code were generated or used for this paper.

For Supplementary SEM videos, see².

Three representative SEM videos are provided for the HDPE under the lightest (5 mN) and heaviest normal preloads (15 mN), and for the PUR under the heaviest normal preload (5 mN). The videos are with 2× actual speed and compressed to lower resolutions for better accessibility (higher resolution with actual speeds are available upon request). The SEM videos with self-descriptive file names are synchronized to the measured hysteresis loops for all five tangential loading levels and are shown side-by-side with the corresponding hysteresis loops. HDPE videos demonstrate significant plastic deformation and pileup especially at the heaviest normal preload, whereas very little plastic deformation results for the PUR case even under the heaviest normal preload.

Figure 9 shows comparison of the contact radii estimated from SEM images and calculated by the Hertzian theory (Eq. (1)) under different normal preloads. The contact radii estimated from SEM pictures are 32–67% higher for HDPE and 14–33% higher for PUR when compared with Hertzian theory predictions. Manual detection of contact edges from tilted and partial view of the contact in SEM images could lead to this overestimation of contact radii. Other possible reasons for the mismatch are the influence of plasticity and roughness in the experiments as Hertzian theory holds only for smooth elastic contacts. Permanent plastic deformation and failure are clearly seen in the vicinity of the contact for HDPE (see Fig. 10(a) for 10 mN preload case). Failure within the contact features circumferential (ring) and radial cracks. Ring cracks (aka Hertzian cone crack) form after maximum tensile radial stress at the edge of the contact reaches a critical failure threshold [41]. Radial cracks are usually observed after inelastic deformations induced by sharp tip indentation. Residual tensile stresses result as the load over an elastic-plastically deformed contact is decreased. Those tensile stresses over the surface drive the flaws beneath the sharp indenter further in radial direction and form the radial cracks [63]. Therefore, existence of radial cracks is a secondary proof of inelastic deformations in HDPE samples. In contrast, even at the highest normal preloads (5 mN), PUR samples exhibit intact elastic contact (Fig. 10(b) and Appendix  A: Supplementary Videos). Quantitatively, this contrasting response between tested materials is expected. Mean contact pressures even at the lightest preload (≈35 MPa) exceeds yield strength of HDPE reported elsewhere (∼23 MPa in Ref. [64]). In contrast, PUR exhibits elastic response for which polyurethane elastomers are well known up to large strains. Tensile strengths of those elastomers are around 20 MPa ([65]), and contact pressures are below 12 MPa for all PUR cases. Thus, elastic response with negligible damage is expected for PUR samples as confirmed by Fig. 10(b). To quantify the influence of plasticity on contact radii, we also added the truncated contact radii $at=R2−(R−wmax)2$ in Fig. 9. Truncated contact radii is a commonly used approximation for fully plastic spherical contact configurations [66]. Note that truncated contact radii correlate well with the SEM estimations for HDPE, further confirming the predominant plasticity over the contact. Contact radii for PUR cases fall between the fully elastic and plastic limits.

Next, we inspect the validity of the smooth profile assumption of Hertz theory. As evident from the SEM images in Fig. 10, the tested surfaces are rough with average RMS roughness of 88 and 325 nm, respectively, for HDPE and PUR samples (see Sec. 2 for details). In Greenwood et al. [67], the ratio of RMS roughness to normal displacement (penetration) on a nominally spherical contact; $α=σ/w0$ is shown to quantify the validity of smooth contact assumption (similar ratio holds for fully plastic contact [68]). For HDPE, α values are estimated to be less than 0.05 for all the loading cases, and thus smooth contact assumption holds to a large degree. For PUR, maximum α value is around 0.1 (2 mN case), and so up to 20% larger contact radii could be expected when compared with the smooth Hertzian theory.

It is also noteworthy that parabolic profile assumption of Hertzian theory is violated in the experiments since $a∼R$⁠. Nevertheless, contact parameters (pressure, radius, deformation) are shown to remain very close to Hertzian predictions for sphere-to-contact radii ratios R/a larger than 1.5 as is in all the tests we presented here [69].

In summary, contact-edge-detection error and plasticity are the major factors leading to deviations between SEM-detected contact radii and Hertzian theory predictions in HDPE samples. For PUR samples, only edge-detection and roughness are influential. For PUR sample at the maximum normal force (P_max = 5 mN), α = 0.05, and so the roughness is expected to increase the contact radius about 5% compared with Hertzian theory. Thus, one can approximate that the contact-edge detection in SEM leads to less than 10% overestimation. Nearly perfect correlation of SEM estimations and fully plastic truncated contact radii predictions for HDPE cases also suggests minute contact-edge detection error.