Numerical multiscale modelling and experimental validation of low speed rubber friction on rough road surfaces including hysteretic and adhesive effects

Author’s Accepted Manuscript Numerical multiscale modelling and experimental validation of low speed rubber friction on rough road surfaces including hysteretic and adhesive effects Paul Wagner, Peter Wriggers, Lennart Veltmaat, Heiko Clasen, Corinna Prange, Burkhard Wies www.elsevier.com/locate/jtri

PII: DOI: Reference:

S0301-679X(17)30132-9 http://dx.doi.org/10.1016/j.triboint.2017.03.015 JTRI4643

To appear in: Tribiology International Received date: 3 November 2016 Revised date: 16 February 2017 Accepted date: 11 March 2017 Cite this article as: Paul Wagner, Peter Wriggers, Lennart Veltmaat, Heiko Clasen, Corinna Prange and Burkhard Wies, Numerical multiscale modelling and experimental validation of low speed rubber friction on rough road surfaces including hysteretic and adhesive effects, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2017.03.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Numerical multiscale modelling and experimental validation of low speed rubber friction on rough road surfaces including hysteretic and adhesive effects Paul Wagnera,∗, Peter Wriggersa , Lennart Veltmaata , Heiko Clasenb , Corinna Prangeb , Burkhard Wiesb a

Institute of Continuum Mechanics, Gottfried Wilhelm Leibniz Universit¨ at Hannover, Appelstraße 11, 30167 Hannover, Germany b Continental Reifen Deutschland GmbH, J¨ adekamp 30, 30419 Hannover, Germany

Abstract A multiscale finite element framework for the calculation of sliding rubber samples on rough surfaces is proposed. The two essential physical contributions hysteresis and adhesion are modelled. Hysteresis originating from the viscoelastic nature of rubber materials is included directly by incorporating the rough surface in the calculation with microscopic details and without transformations or a reconstruction of the surface. The adhesive interaction is introduced by a macroscopic shear stress law coupled to the hysteresis simulation by the evaluation of the relative contact area. The assumed mechanisms and the framework are explained in detail and are used to validate the method for experimental results of different materials for low sliding speeds. Keywords: Rubber friction, Hysteretic friction, Adhesion, Multiscale contact homogenization

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1. Introduction

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1.1. Motivation and goals The contact of car tires with road tracks is a research field of high practical importance since grip properties during tire-road interaction have a direct impact on safety issues. In order to improve tire properties, a deep

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∗

Corresponding author Email address: [email protected] (Paul Wagner)

Preprint submitted to Tribology International

March 11, 2017

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understanding of the tire-road contact mechanics and the underlying rubber friction physics are essential. The major goal of our work is to predict performances of different rubber materials on rough road tracks. Besides, our numerical studies provide access to data that is not reachable in experiments like local pressures, dissipated energies, or contact area. As a result, our method can give additional insight into the physics of rubber friction. 1.2. Background of rubber friction The frictional response of sliding rubber is determined by the rubber material properties, the rough counter surface including its condition (dry or wet for example), and the global test parameters (normal load, velocity, temperature). These statements are based on a lot of experimental studies starting with early works revealing pressure-dependence (cf. [1]). In [2], nonlinear velocity- and temperature-dependence of rubber friction is reported. Further experiments performed with modern test rigs for high velocities and temperature measurements can be found in [3, 4, 5, 6]. Experimental and analytical studies also link the measured responses to physical effects. They provide theories and statements about the shares of each effect on the overall frictional response. In [7, 8], the velocitydependence of the frictional response is linked and correlated to the viscoelastic material properties of elastomers providing a first theory for hysteretic friction. In contrast to hysteretic friction as a volumetric effect, adhesive friction is a surface effect induced by intermolecular effects. For an overview of adhesive interactions see [9]. Details of both effects are explained in section 2. These two effects (hysteresis and adhesion) can be considered as the main effects contributing to the overall friction force. Additionally lubrication, cohesion and interlocking are often observed in experiments. Viscous forces (lubrication), wear- (cohesion) and temperature-effects play a major role for large sliding velocities, which are out of scope of the present work. Furthermore the validation is carried out for low macroscopic velocities where these effects are assumed to be negligible and a large contribution of adhesion is expected (cf. [10, 11]). 1.3. Modelling of rubber friction In [12, 13, 14], analytical approaches dealing with hysteretic friction are proposed. These works are based on an analysis of the dissipated energy induced by the rough surface revealing the possibility to study elastomer 2

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friction for a wide parameter range. Due to the fractal nature of the rough surface (cf. [15]), a power spectral density (PSD) or a height difference correlation (HDC) can be used to represent the surface. Furthermore, different analytical contact theories are introduced in [13] and [14], see also [16] for an overview. The approach introduced in [14] is enhanced by an adhesive contribution in [17] adding an adhesive coefficient of friction to the hysteretic response based on the contact area and an interfacial shear stress representing a free parameter. In [18] a theory is provided that states a bell-shaped character of the shear stress over increasing velocity. Following the introduced assumptions and theories for adhesion in [10, 11, 19] different velocity-dependent laws for the shear stress are introduced, fitted to experiments and compared for different global conditions. Especially, in [11] the proposed law for the shear stress (see section 4) is linked to the assumption of rubber chains undergoing bonding-stretching-debonding cycles with the rough surface (details are provided in section 2). Furthermore, for example in [17] assumptions for the share of hysteresis and adhesion for different surface conditions are provided proposing that adhesion is activated on dry surfaces and suppressed on rough surfaces covered with a water-detergent film, see section 2 for further details. Different rubber materials in the small velocity range are studied in [11] providing good agreement of the theory with respect to friction measurements enhanced by an expansion of the law for different temperatures by the use of an Arrhenius-like shift factor. The proposed adhesion law is used in this work without the expansion to different temperatures in order to validate small velocity measurements at room temperature T = 20◦ C, see section 6. Due to the presence of many different length scales in real rough surfaces, multiscale approaches are especially suited to tackle these kinds of contact problems. Numerical multiscale approaches for rubber friction using finite element models are developed in [20, 21, 22]. The common feature of all cited multiscale frameworks is a decomposition of a complex contact problem into separate length scales and the use of homogenization techniques. A coupling between two scales is established through passing contact values (pressure, velocity) of the larger scale (macroscopic scale) to the smaller scale (microscopic scale). A microscopic coefficient of friction originating from the microscopic details of the surface is gained by homogenization and passed back to the macroscopic scale. Contact homogenization approaches can be used if the scales are separated meaning that the characteristic macroscopic length scale is a few times bigger than the microscopic length scale. In [20] 3

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the complex dynamic contact interaction of a rubber block with microscopic moving particles is studied accomplished by homogenization studies varying rubber block dimensions and further parameters. The contact homogenization technique is transferred for the first time in [21] to the contact of a rubber block with stationary rough road surfaces. Nevertheless, the rough surface is decomposed by a certain number of sinusoidal functions approximating the HDC function of the real rough surface. The treatment of the rough surface reveals problematic properties with respect to converging global results, because a small change in the used sinusoidal functions leads to large changes in the gained macroscopic coefficient of friction. Therefore, in [22] a different treatment of the rough surface is proposed reconstructing the surface with a PSD-transformation including more frequencies and amplitudes in the analysis. As a next step in this work, the rough surface is directly included in the numerical setups without idealizations like in [21]. A lot of microscopic aspects are investigated in several works studying parameters that influence the outcome of the homogenization procedure. For example in [23] the difference between displacement and traction driven boundaries is investigated. Among other aspects for homogenization on sinusoidal profiles, in [24] the compression time is identified as an influencing parameter that needs to be adopted to avoid large oscillations of the resulting coefficient of friction. In [25] complex geometries of micro particles are introduced followed by the introduction of rough surface profiles in [26, 27] providing the basis for the later proposed multiscale method with direct inclusion of rough surface profiles. In addition, in [28] a comparison between an analytical and a numerical calculation on sinusoidal functions with different parameters is executed revealing the benefit of numerical approaches by introducing geometrical and material non-linearity. A complex thermo-mechanical numerical multiscale setup for rough surface contact problems based on homogenization is introduced in [29]. Furthermore, this framework is extended to isogeometric analysis in [30] and to dissipative interactions in [31]. Numerical investigations dealing with the determination of the thermal contact conductivity are explored in [32, 33]. Additionally, in [34, 35, 36] the previously mentioned analytical approaches are extended to thermal interactions between rubber and rough surfaces. The cited studies dealing with thermal interactions would provide a basis for an extension of the proposed multiscale framework to larger velocities with non-negligible temperature effects. The proposed numerical multiscale framework in this work is based on 4

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the publications [20, 21, 22]. A separation of rough surface length scales and time homogenization in a two-dimensional plane strain setup is applied. Some changed and adapted features are described in detail in section 3. A macroscopic adhesion law adapted from [10, 11] is implemented in the finite element multiscale setup and compared to experiments. Direct numerical modelling of adhesion with complex interface kinematics is for example investigated in [37, 38], an overview of numerical approaches for adhesion can be found in [39].

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1.4. Outline First, hysteretic friction including the possibilities for an implementation within a finite element framework, the adhesion mechanism, and the shares of each effect on the final friction response are explained in section 2. The multiscale methodology for hysteretic friction is outlined in section 3. The used assumptions for adhesion and the inclusion of an adhesion law into the multiscale setup are described in section 4. A mechanical study of hysteretic friction on the microscopic scale is evaluated in section 5, providing important information for the following validation. The validation of the numerical approach with experiments is performed in section 6 using three different rubber materials for low sliding speeds. Finally, concluding remarks are given in section 7.

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2. Rubber friction mechanisms and modelling aspects

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In this section a background for the modelled contributions, hysteretic and adhesive friction, is provided in detail. Assumptions and important aspects for the solution of the problem with finite elements are given.

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2.1. Modelling hysteretic friction Viscoelastic materials show the effect of energy dissipation inside the material during cyclic loading and unloading. Furthermore, the amount of dissipated energy depends on the applied frequency of the cyclic loading and the used rubber material. In the case of a sliding rubber block on a rough surface, asperities induce hysteresis inside the rubber material. The material response leads to a horizontal resistance force, called hysteretic friction, see Fig. 1 (a). A global velocity change induces the excitation of different frequencies. Therefore, a velocity-dependence of the global frictional response is observed in 5

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experiments, compare subsection 1.2. Additionally, [12, 13] point out that microscopic surface asperities are responsible for high local strains in the rubber and contribute significantly to the global response. Thus, the microscopic structure of the surface has to be included in any analysis of the sliding process. There exists a significant difference in pattern size of the rubber block (very small) and the rough surface. Due to this fact, the rubber block surface is idealized as a surface without any pattern and modelled in this way. In order to model hysteretic friction directly in a finite element framework, two aspects are essential: an adequate material description and a robust contact algorithm, see Fig. 1 (b). Fundamentals of non-linear finite element calculations can be found in [40], detailed information on the used software ABAQUS and its implemented features can be found in [41, 42]. A viscoelastic material model that is able to capture large deformations is applied in this work. A common approach is to use a finite linear theory based on the displayed rheological model (Fig. 1 (b)). The model consists of a hyper-elastic equilibrium part represented by a spring and a parallel series of Maxwell elements given by a spring-damper-combination called nonequilibrium part. The hyper-elastic part is modelled with a Mooney-Rivlin material model based on a strain energy function including the parameters C10 , C01 , D1 , see [41, 42, 43, 44]. This part is responsible for the static longtime behaviour whereas the series of Maxwell-elements is used for the timedependent material response. The stress strain curves are measured in a quasi static test in order to determine the material parameters. After preconditioning, the rubber sample is loaded with tension and compression. Based on the hysteretic response the results for the Mooney-Rivlin parameters for each individual rubber compound are obtained. The concept of over-stresses is applied to calculate the complete stress of the viscoelastic material, see [45, 46, 47] for details. The Maxwell-elements i are characterized by an elastic modulus Ei and a relaxation time τi . Due to the incompressible behaviour of rubber a hybrid formulation is used in our work with bilinear shape functions for the displacement and a constant pressure in order to avoid locking, see [40] for details. Measurements of the material properties and the rough surface are performed in order to link the physical data to the finite element models used in this study, see Fig. 1 (c). The viscoelastic material response is determined in a dynamic mechanical analysis (DMA) evaluating the frequency-dependent storage- and loss-modulus, i.e. E  (ω) and E  (ω). In detail, a cylindrical 6

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prestrained rubber sample is exposed to a harmonically oscillating load with a small strain amplitude. The measurements are repeated for different temperatures (−40◦ C up to +70◦ C) in the frequency range (0.01Hz up to 50Hz). Afterwards, the temperature is eliminated by using the time-temperature superposition principle (see [48] for further details) in order to achieve a master curve for E  (ω) and E  (ω) for the master temperature of T = 20◦ C over a wide frequency range. A fitting procedure is used to adapt the moduli Ei and relaxation times τi of the Maxwell-elements to the measured curves (cf. [22, 49]) by using the following equations: E  (ω) = E∞ +

n  Ei τi2 ω 2 i=1

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E  (ω) = 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221

n 

1 + τi2 ω 2

Ei τi ω , 2 2 i=1 1 + τi ω

,

(1)

(2)

with n Maxwell elements and the long time modulus E∞ . Analyzing the used surface structure in combination with the applied velocity range (see section 6), an excited frequency domain of 10− 2Hz to 104 Hz for the rubber material was obtained. The fit of the later used material data for three compounds was done for this domain using 19 Maxwell elements. An explicit representation of the rough surface is necessary to generate the hysteretic response inside the rubber material. A robust contact algorithm is necessary because large deformations are expected at the contact interface. Overviews of numerical contact mechanics can be found in [50, 51]. In [22], a mortar contact formulation is proposed for a calculation on a rough surface. However, in this work the software ABAQUS is utilized focusing on the construction of the multiscale framework with Python-scripts. Thus, from the available options in ABAQUS the surface to surface contact algorithm is chosen for rough surface contact calculations, see [41] for details. The road surface is modelled as a discrete rigid body fixed in space reducing computational effort. The rough surface is measured with two optical profilometers using 3D scanning, one device captures the macroscopic details with a nominal lateral resolution of 0.019mm and the second device measures microscopic details with a nominal lateral resolution of 0.003mm. The microscopic scale measurement is performed at the top surface asperities that come into contact with the rubber block. 7

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2.2. Adhesion mechanism Adhesion is an important physical effect in the context of rubber friction. The term adhesion is generally used for frictional effects at the interface of two bodies originating from interaction of molecules or atoms, Fig. 2 (a). In the case of rubber friction it is possible to consider adhesion as the interactions of rubber molecules with atoms of the rough counter surface at nanometer length scale, see [11]. A rubber molecule represented as an elastic chain (Fig. 2 (b)) attaches to the rough counter surface in the case of normal contact without sliding. Applying a global velocity results in a stretching of the rubber molecule leading to a shear force fA (v): fA (v) = c · dx(v).

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(3)

The shear force increases with increasing velocity since each chain is stretched more (elongation dx(v) increases) under a higher velocity with c representing the spring stiffness. Meanwhile, the probability of detachment increases (Fig. 2 (c)) and the probability for reattachment of single chains at the interface decreases rapidly. This circumstance is expressed globally by the number of bonded chains Nb (v) that decreases over velocity. The global shear force is achieved by the multiplication of the single shear force and the number of bonded chains leading to a bell-shaped global shear force FA (v). Dividing this shear force by the current contact area Ac (v) results in the global shear stress τA (v): FA (v) τA (v) = . (4) Ac (v) 2.3. Shares of adhesion and hysteresis to the frictional response The influence of adhesion (and hysteresis) is highly depending on surface conditions. A common assumption is that adhesive bonds are suppressed under wet conditions, see Fig. 3. For rough surfaces covered with a soap-watermixture a pure hysteretic response without an adhesive part is assumed, see Fig. 3 (a) and section 1. This assumption is based on the low surface tension of the mixture covering the whole surface and leading to a separation of the rough surface and the rubber bulk preventing the rubber chains from surface bonding, see Fig. 3 (b). Experiments on the same rough surface with water display a higher friction level (Fig. 3 (a)), since the hysteretic response is enhanced with adhesive friction on dry parts of the surface where water is wiped away, see Fig. 3 (c). A complete dry surface leads to the largest contribution of adhesion since no separation medium is present at the interface 8

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while the hysteretic part remains at the same level as for wet conditions, Fig. 3 (a,d). Following for example [10, 19] the total coefficient of friction μ(v) for different surface conditions is built up with a basic hysteretic part, and an adhesive part additionally adapted by the coefficient α for different surface conditions: μ(v) = μH (v) + αμA (v), α ∈ [0, 1] . (5) For all surface conditions the same amount of the hysteretic contribution is assumed since the rubber material is able to follow the surface asperities as well under wet conditions. Consequently, no suppression of upper surface asperities (the parts that get into contact with the rubber block) by water is assumed in this work. The introduced coefficient α provides the possibility to adopt the adhesive contribution for the different surface conditions by setting α to zero for a surface covered with a soap-water mixture in a first step. This parameter is later (see sections 4 and 6) adopted whenever an adhesive contribution has to be added to the hysteretic response introducing a value between zero and one for a wet surface. For a dry surface, the full adhesive contribution to the overall friction force is assumed and thus α is set to one. The hysteretic and adhesive parts of the coefficient of friction depend on the considered rubber compound and thus different materials are investigated in section 6 in order to study this effect. 3. Multiscale finite element approach for hysteretic friction of rubber In this section, essential features of the multiscale finite element framework for hysteretic rubber friction are outlined. Especially changes to previous works are highlighted and explained. 3.1. Framework The multiscale finite element approach is strongly based on the approaches in [20, 21, 22]. The surface is divided into macroscopic and microscopic parts with the possibility to include further scales in the framework, as shown in [21]. Fig. 4 displays two scales and the six most important steps A-F. Although the multiscale approach reduces the computational effort drastically (cf. [22]), three-dimensional calculations would lead to large computational times. Thus, the contact interaction is reduced to a two-dimensional plane strain setup, as in [20, 22]. 9

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First of all, the rough surface is divided in separate scales by band-passfilters using two edge frequencies (wavelengths), (step A). The used upper and lower wavelengths for the filters are directly determined by the modelled problem. The size of the macroscopic problem defines the upper limit λmax whereas the measurement resolution of the microscopic scale or a certain cutoff can be used as a lower limit λmin . Depending on the number of chosen scales n intermediate cutting wavelengths λi are introduced that divide the whole surface range: λmax − λ1 − (..λi ..)i=2−nc − λmin .

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(6)

The band-pass-filters split the surface at nc cutting wavelengths λi , ensuring a continuous spectrum of the rough surface, comparable to the splitting wavelength in [22]. In contrast to [21, 22], the rough surface is not transformed by a HDC or PSD circumventing problems in reconstruction of a rough surface or a loss of surface characteristics. In principle, the number of scales is a free parameter, although at least two scales with one intermediate cutting wavelength λ1 have to be introduced for the method. Depending on the problem size, especially the size of the macroscopic rubber dimensions and the lowest cutoff wavelength, for example some micrometers, the number of intermediate scales is defined for the approach. Since the approach is aimed towards the modelling of real rough surfaces, two scales trying to cover the whole surface spectrum would lead to immense calculation times containing too many details on each scale. Hence, a compromise between the number of scales that increases the effort of the method and the individual calculation time of the finite element setup for each scale has to be found. In general, three or four scales are introduced for such a multiscale setup, compare also to the approach in [21] and section 6. Secondly, in step B all macroscopic contact pressures p1,c are evaluated. These pressures originate from a frictionless (μ1 = 0) macroscopic simulation with the global parameters p1 , v1 . A compression phase is used to apply a pressure value on top of the rubber block and afterwards a dragging phase is realised with a set velocity on the same edge keeping the pressure constant during this step, see [20, 21, 22] for details. Due to the reduction to a twodimensional setup and in order to include some statistical features of the surface, k surface samples are calculated. In contrast to the cited approaches in [20, 21, 22] an averaged macroscopic pressure p1,avg. is identified instead

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of using the local contact pressures: w 

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i=1

p1,c (i) w

.

(7)

The averaging procedure considers all nodal pressure values w from space (the finite element nodes are distributed equidistant at the rubber block bottom) and time to determine the input parameter for the microscopic scale. The use of contact pressures as an input parameter is linked to the assumption of scale separation, see [20]. The scales of a rough surface are not fully separated by applying filters that decompose the rough surface in a continuous spectrum without skipping intermediate surface wavelengths. Therefore, a space and time averaged value is passed on the lower scale representing an intermediate value simultaneously reducing the computational effort. In step C, the pressure values p1,avg. (k) for different surface samples k are averaged in order to gain a definite input parameter pavg. for the microscopic scale depending on the global conditions, the rubber compound, and the surface structure. This applied pressure value includes the surface characteristics in an averaged sense. The microscopic and macroscopic velocities are set to the same value assuming a macroscopic full slip condition. This means that just experimental steady state responses are considered neglecting an acceleration phase at the beginning with a wide range of slip velocities. Frictionless behaviour (μ2 = 0) is assumed on the microscopic scale (lowest scale). Furthermore, periodic boundary conditions are set for the rubber block requiring a periodic microscopic surface sample. Details regarding the construction of such samples can be found in subsection 3.2. The use of a viscoelastic material model induces a hysteretic friction response appearing as a time-dependent coefficient of friction, see step D in Fig. 4. Like in [20, 21, 22], time homogenization μ2,avg. is applied for the j surface sample responses. Further information on the single simulation steps (compression and dragging phase) can be found in [20, 22]. The number of surface samples on each scale has not necessarily to be the same on each scale. The exact number of used samples and effects on the solution are discussed in section 5. In step E, the calculated responses j are averaged and the microscopic coefficient of friction μavg. is directly inserted in the macroscopic tangential contact formulation in step F. In contrast to [20, 21, 22], all local dependencies are removed leading to an approach that is able to use a lot of surface 11

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samples, scales and macroscopic parameters (see section 6). Especially, the difficult choice of one representative surface sample for each rough surface scale (cf. [22]) is avoided. The last evaluation step is the determination of the macroscopic coefficient of friction including also the hysteretic response of the macroscopic scale. This evaluation is gained by time homogenization as it was also done on the microscopic scale in step D.

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3.2. Surface treatment The surface treatment is a crucial step for our approach and therefore we describe some details in this subsection. Fig. 5 displays the steps that are necessary to construct periodic surface samples which are mandatory except for the macroscopic scale. First of all, the surface profiles are divided by the application of band-pass filters as described in the previous subsection (Fig. 5 (b)). The macroscopic surface samples remain in the filtered shape and can directly be used in a finite element simulation. Contact of the rubber block with the rough surface is expected on the top parts of the surface asperities in sliding direction. Thus microscopic surface samples are extracted out of these expected contact spots (Fig. 5 (c)). In Fig. 5 (d), the spline construction that is used to ensure a periodic microscopic sample is shown (dotted line). A few data points before the left and the right end of the chosen surface sample are neglected and a spline is used to construct a smooth transition between these ends. Thus, the profile of the microscopic surface part is slightly changed but periodicity of the surface sample is guaranteed. This procedure is necessary to create the desired boundary conditions at the rubber block and to ensure a response that approaches a steady state solution such that time homogenization can be applied (see step D in Fig. 4).

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4. Inclusion of adhesive friction in the multiscale setup

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Fig. 6 displays the coupling of the described hysteresis multiscale framework with an approach for adhesive friction. The idea is to include a phenomenological approach for adhesion in the multiscale setup motivated by the described molecular chain dynamics (see section 2). The relative contact area calculated by the hysteresis setup and an adhesion law proposed by [11] (see subsections 1.3 and 2.2) are combined with experimental results entering the formulation before the final calculation of the macroscopic scale. The

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friction mechanisms are coupled through the contact area calculated by the non-linear finite element framework. Four scales are applied in the validation (section 6) and already indicated here in Fig. 6. Averaged pressure values pi are passed downwards from the macroscopic scale (i = 1) to the lowest scale (i = 4) in the pure hysteretic part, step A. In addition to the coefficient of friction μi (v), also the current contact area Ai,c (v) is evaluated during the upward steps of the framework by time homogenization, step B. Together with the nominal contact area of each scale Ai,0 , the full velocity-dependent relative contact area reads: 



A2,c (v) A3,c (v) A4,c (v) A1,c (v) Ar (v) = · · · . A1,0 A2,0 A3,0 A4,0 400 401 402 403 404 405 406 407

The current contact area of the macroscopic scale A1,c (v) is taken from the frictionless calculation in the downscaling part. The multiscale setup is interrupted previous to the last macroscopic simulation and therefore the macroscopic contact area based on pure hysteretic friction is not available to be included in the adhesion law. However almost no deviation between the frictionless and frictional macroscopic contact area are detected in comparison after all steps. The adhesion law for the bell-shaped adhesive shear stress ([11]) reads 





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(8)

 2 

,

(9)

and is based on three parameters that determine the width c, offset v0 and peak value τ0 of the function, see step C in Fig. 6. These parameters are linked to the rubber material and the rough surface, dependencies are addressed in section 6. No separate microscopic experiments are available that could provide values for the parameters τ0 , c, v0 . However, in order to adapt the parameters, a macroscopic experimental input is necessary using this methodology, step D. The difference between the dry and soap-water macroscopic friction curve is assumed to represent the full adhesive response in a first step, compare section 2. The adhesive coefficient of friction τA (v) μA (v) = · Ar (v), (10) p1 is fitted by adapting the parameters, see μA (v) in Fig. 6. 13

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In the last step E of the framework, a combined coefficient of friction is inserted in the macroscopic contact formulation: μ∗ (v) = μ2 (v) + α · μA (v),

α ∈ [0, 1] .

(11)

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Here, μ2 (v) includes the hysteretic effects of all scales except the macroscopic one. The adhesive part μA (v) is introduced at the macroscopic level and can be regulated for different surface conditions with α. Furthermore, the adhesive part is suppressed for a wet surface with soap-water by setting the parameter α to zero, whereas a dry surface condition is modelled by the full adhesive interaction gained by α = 1. The overall response μ1 (v) including hysteretic effects induced by the macroscopic surface asperities is gained by homogenization after the simulation with μ∗ (v). All scales in Fig. 6 include calculations on a certain number of surface samples k. An adequate choice of this number is discussed in section 5.

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5. Micro mechanical hysteretic friction study

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In this section an important free parameter of the proposed multiscale setup is discussed: the number of calculated surface samples on each scale. In order to gain some knowledge about statistical quantities and how the choice influences the resulting global coefficient of friction, a large set of microscopic surface samples is calculated. The lowest scale with the largest fluctuations of the height profiles is used for this investigation.

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5.1. Setup and used surface samples The lowest scale and the applied parameters are extracted from the validation study with four scales, used in section 6. In Fig. 7 (a), an exemplary finite element setup with some representations of the lower scale of the rough surface is shown. The edge wavelengths of the applied band-pass filter are set to λstart = 0.04mm and λend = 0.003mm, the latter being the measurement resolution. Features of the finite element model like the use of mixed finite elements and the rheological model are already given in sections 2 and 3. All calculations are performed on periodic microscopic surface samples in a frictionless state and with a pressure of p = 3MPa and an applied velocity of v = 0.01m/s. Fig. 7 (b) shows a few of the chosen surface samples. In total, 287 microscopic surface samples are used and finite element calculations are performed on these samples. The Mooney-Rivlin material parameters and the viscoelastic properties of the used rubber material A can be found in Tab. 1 and Fig. 10 (c). 14

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5.2. Statistical results With the described setup, 287 calculations are performed. Afterwards a time homogenization for the resulting coefficients of friction is applied, compare section 3. Fig. 8 illustrates the evolution of the time homogenized coefficient of friction μ with an increasing number of calculated surface samples n. Eight histograms are shown starting with the first ten evaluated samples n = 1 − 10 up to the full amount of surface samples n = 1 − 287. A trend towards an asymmetric distribution is observed and indicated with a scaled curve in the last histogram. The proposed multiscale setup uses the averaged coefficient of friction for the coupling of the scales, see section 3. Therefore, in addition to the qualitative observation of an approaching statistical distribution, a quantitative evaluation for the mean value is carried out and shown in Fig. 9. The averaged coefficient of friction gained through consideration of all 287 calculations is used as a reference μref. avg. . Furthermore, all results can be displayed and evaluated in a random order using for example the first ten or the last ten calculated samples for μkavg. (n = 10). Fig. 9 (a) shows three different configurations k with a random choice for n averaged surface samples. The results indicate a large standard deviation for a coefficient of friction calculated with one sample. Hence, the choice of one sample for each scale of the multiscale setup would lead to an insufficient representation of surface characteristics. The choice of a representative surface sample for each scale is rather difficult. As a consequence, we propose to include more than one surface sample in order to enhance the multiscale calculation with statistical properties of the rough surface. Fig. 9 (b) shows the standard deviation S(n) for 1000 representations k of μkavg. (n). The standard deviation decreases rapidly until an averaged value over ten surface samples. After a transition phase, a slow decrease of S(n) is observed in Fig. 9 (b). In order to find a compromise between calculation time and accuracy, a number of ten surface samples is chosen for each scale for the following validation setup. 6. Validation of hysteretic and adhesive friction for low sliding speeds The proposed multiscale finite element approach for hysteretic and adhesive friction is validated with experimental results in this section. After a short description of the used setup, a summary of all results is provided.

15

488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516

517 518 519 520 521 522 523 524

6.1. Multiscale setup and used parameters Three experimental rubber compounds (A, B, C) are tested with a macroscopic pressure of p = 2bar, a global temperature T = 20◦ C and low velocities of v = 10−5 − 10−2 m/s on a measured rough surface varying the surface conditions (soap-water, dry). Fig. 10 (a) shows the two three-dimensional measurements of the rough surface applied as an input for the multiscale setup. The first measurement includes the macroscopic details whereas the second one has a focus on smaller length scales (see section 5). The MooneyRivlin material parameters and viscoelastic properties of the used rubber materials can be found in Tab. 1 and Fig. 10 (c). The dimensions of the rubber blocks on each of the four scales are given in Fig. 10 (b) representing simultaneously the used band-pass filter edgewavelengths. Four scales are introduced to cover the whole range of surface wavelengths starting from the macroscopic size of the rubber block (20mm x 8mm) that is used in experiments down to the smallest wavelengths given by the measurement resolution of 0.003mm in this study. Following the descriptions in section 3 four scales are introduced representing a good compromise between communication effort between the scales and the particular calculation time of each finite element setup. A division in three scales would be possible, but since all intermediate surface details are covered with the multiscale approach, the resulting finite element setups would lead to very large and impractical computation times even for a two-dimensional setup. The macroscopic rubber block with a smoothed front edge reflects the geometry of the experimental studies gained after a run-in, wearing down the front edge. Each scale is calculated with ten surface samples in the up-passing branch (compare the conclusion of the last section) indicated by a box with all samples of the scale ”Micro2” in Fig. 10 (b). During the down-passing procedure, a smaller deviation of the averaged pressure value is detected and as a result five samples are used gaining a speed-up for the whole framework. 6.2. Multiscale results The multiscale setup starts with a pure hysteresis calculation and is stopped previous to the final macroscopic simulations. The final relative contact area for all three compounds is evaluated according to Equation (8). The results are shown in Fig. 11. These results are used as an input for Equation (10) while the missing parameters τ0 , c and v0 in Equation (9) are fitted to approach μdif f (v). Like in [11], parameters c and v0 are fixed and τ0 is identified as a rubber material dependent parameter, compare Fig. 11. 16

525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542

543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560

The combination of the contact area gained through a hysteresis simulation and the law for the adhesive shear stress results in a μA (v) that is able to represent material characteristics. The fitted adhesive part is added in the final macroscopic simulations following Equation (11) and the described procedure in section 4. The results are shown in Fig. 12 with α = 0 for the soap-water response calculating the pure hysteretic part, whereas α = 1 is set for the dry part. In accordance with [10, 11] our calculation predicts a small contribution of hysteretic effects and a large contribution of adhesive effects for the used small velocities. An increase of μ with increasing velocity is observed in wet and dry experiments and reproduced with the simulations. With the exception of the highest velocity the order of the materials is also reproduced correctly. Nevertheless, a gap between the absolute values of simulations and experiments is detected. The gap is attributed to a presumable adhesive interaction on a surface covered with soap-water mixture since the indicated ideal separation in Fig. 2 may fail locally. We propose for this reason a modified simulation setup in the next subsection introducing an adhesion fit for the soap-water and the dry surface. 6.3. Modified approach Under the changed assumption of adhesion being already present for surfaces covered with soap-water, the fitting procedure introduced in the last subsection is modified. Instead of adapting the adhesion law to the difference between the wet and dry measurement, the pure hysteresis calculation is used as a reference for the fitting procedure. The difference μdif f (v) is now extracted between the dry measurement and the hysteresis multiscale simulation adjusting this difference with the adhesion law, see Fig. 13 (a-b). Fig. 13 (c) provides the idea how the experimental soap-water measurement is reproduced by setting the coefficient α to a defined value between zero and one. Equation (11) is thus already used for the soap-water measurement and α is again set to one for the dry measurement. Since the hysteretic part of the calculation remains unchanged, the calculated relative areas are equal to the presented values of the last subsection, see Fig. 11. By adapting the parameters for c, v0 and τ0 , again a qualitatively good approximation is achieved, see Fig. 14. Like for the previous approach the values of c and v0 are fixed for all materials whereas τ0 is again adapted separately for all materials leading to the results reported in Fig. 14.

17

569

Finally, the adhesive contribution is added to the hysteretic part on the macroscopic scale (Equation (11)). To represent the dry surface conditions, α = 1 is set. For the simulation of the soap-water α = 0.3 is identified for the current rough surface description and the three rubber materials. In other words, 30 percent of the full adhesive contribution is assumed for a surface covered with soap-water. In Fig. 15, the experimental and the simulation results are shown again. With this modified approach, the ranking of the materials is reproduced correctly by the simulation for every point in the velocity range. The shapes of the curves over increasing velocity fit as well.

570

7. Conclusions

571

We propose a multiscale finite element setup including surface details that are necessary for a direct modelling of the hysteretic response of a sliding rubber block. A special treatment with band-pass filters is suggested for the decomposition of the rough surface using the original surface profiles without any transformations or reconstruction of the surface, like in previous approaches [21, 22]. A processing step is included to ensure the periodicity of the surface samples except for the macroscopic scale. The local pressure distribution is analysed and a representative mean pressure value is passed from the macroscopic scale down to microscopic scales. The calculated multiscale response obtained by passing the coefficients of friction upwards is enhanced by the use of a defined number of surface samples for each scale. Statistical properties of the microscopic scale are investigated and a practically useful number of samples is detected. An efficient multiscale finite element framework is constructed providing a good compromise between calculation time (for different global parameters) and accuracy. We add a macroscopic phenomenological adhesion law proposed by [11] to our numerical setup. The necessary input of the relative contact area is determined on all applied scales depending on the viscoelastic rubber behaviour as well. The adhesive shear stress is motivated by the underlying sub-micron molecular chain dynamics and three parameters are adjusted to model the gap between a dry and wet (soap-water) measurement assuming it as the adhesive part of friction. The shapes of the dry and wet measurement are reproduced with this approach identifying the parameter τ0 as material-dependent.

561 562 563 564 565 566 567 568

572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595

18

616

Nevertheless, a gap between simulation and measurement is detected that is attributed to present adhesive interactions for a wet surface covered with soap-water. Therefore, the proposed assumptions are revised and a modified approach is suggested by introducing an adhesive part already for wet surfaces. The parameters c, v0 and τ0 of the adhesive shear stress are fitted to μdif f (v) being the difference between hysteresis simulation and dry measurement. Again the parameter τ0 is identified as material-dependent. Additionally, the parameter α is adapted equally for all rubber compounds reproducing the experimental soap-water result as well as the dry result. In the future, the prediction of different parameter sets (velocity, pressure, surfaces) could be a desirable goal. Higher velocities remain a challenging topic since viscous effects and thermo-mechanical coupling should be added in the current approach. The proposed approach could also be used to calculate small velocity answers for different global temperatures with a shift of the viscoelastic material properties. This point remains open for future studies. Another aim is to link the parameters of the adhesion law to repeatable experiments providing a direct link to the physical properties of the rubber material and the used rough surface. With microscopic data and experiments, also a direct modelling of adhesion on the lowest scale could be a desirable goal since adhesion is an effect underneath the modelled scales (from a physical point of view).

617

Acknowledgements

618

622

Financial support of Continental Reifen Deutschland GmbH is gratefully acknowledged. We thank Manfred Kl¨ uppel and Andrej Lang from the DIK for carrying out the friction measurements. We thank Dr. Pavel Ignatyev and Dr. Stefan Torbr¨ ugge for many fruitful discussions that helped to improve the model.

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[41] SIMULIA, Abaqus Analysis User’s Guide (6.14), Dassault Syst`emes SIMULIA Corp., Providence Rhode Island USA, 2014. [42] SIMULIA, Abaqus Theory Guide (6.14), Dassault Syst`emes SIMULIA Corp., Providence Rhode Island USA, 2014. [43] M. Mooney, A theory of large elastic deformation, Journal of Applied Physics 11 (9) (1940) 582–592. doi:10.1063/1.1712836. [44] R. S. Rivlin, Large elastic deformations of isotropic materials. iv. further developments of the general theory, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 241 (835) (1948) 379–397. doi:10.1098/rsta.1948.0024. [45] J. C. Simo, On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects, Computer Methods in Applied Mechanics and Engineering 60 (2) (1987) 153–173. doi:10. 1016/0045-7825(87)90107-1. [46] G. A. Holzapfel, J. C. Simo, A new viscoelastic constitutive model for continuous media at finite thermomechanical changes, International Journal of Solids and Structures 33 (20-22) (1996) 3019–3034. doi: 10.1016/0020-7683(95)00263-4. [47] G. A. Holzapfel, J. C. Simo, Entropy elasticity of isotropic rubberlike solids at finite strains, Computer Methods in Applied Mechanics and Engineering 132 (1-2) (1996) 17–44. doi:10.1016/0045-7825(96) 01001-8. [48] P. A. Ignatyev, S. Ripka, N. Mueller, S. Torbruegge, B. Wies, Tire absbraking prediction with lab tests and friction simulations, Tire Science and Technology 43 (4) (2015) 260–275. doi:10.2346/tire.15.430401. [49] J. Reinelt, Frictional contact of elastomer materials on rough rigid surfaces, Phd thesis, Leibniz Universit¨at Hannover (2008). [50] P. Wriggers, Computational contact mechanics, 2nd Edition, Springer, Berlin and New York, 2006. [51] T. A. Laursen, Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer Berlin Heidelberg, Berlin, Heidelberg, 2003. 24

Table 1: Mooney-Rivlin parameters for rubber compounds A, B and C in MPa

Parameter C10 C01 D1

A 0.382 0.300 0.029

B 0.113 0.581 0.029

C 0.374 0.266 0.031

Figure 1: (a) Hysteresis mechanism with internal energy dissipation. (b) Rheological material model and numerical contact modelling. (c) Experimental input for the finite element setup.

25

Figure 2: (a) Adhesion at the rubber-road interface. (b) Adhesion mechanism: chain dynamics at nano-meter length scale. (c) Adhesive shear stress over velocity.

Figure 3: (a) Coefficient of friction over velocity for dry and lubricated (water and soap-water) rough surfaces. Assumed chain dynamics and adhesive interaction at the contact interface: for a surface covered with soap-water (b), for a surface covered with water (c), for a dry surface (d).

26

Figure 4: Multiscale setup with most important steps A-F. (A) Surface decomposition. (B) Contact pressure evaluation. (C) Averaging of macroscopic contact pressure. (D) Homogenization of microscopic friction response. (E) Averaging of microscopic friction response. (F) Incorporation of microscopic response in macroscopic setup.

Figure 5: (a) Three-dimensional measure of surface with extracted profile (dark line). (b) Application of band pass filters on original surface profile generating macroscopic profile (macro) and mesoscopic profile (meso). (c) Choice of mesoscopic surface sample, example. (d) Artificial construction of periodic transition with a spline. (e) Constructed periodic profile.

27

Figure 6: Multiscale setup including a macroscopic adhesion law. (A) Hysteresis calculation. (B) Calculation of current contact area. (C) Adhesive shear stress with fitting parameters. (D) Macroscopic experimental input μdif f (v). (E) Combination of hysteretic friction and adhesive coefficient of friction

Figure 7: (a) Microscopic setup. (b) Four different microscopic surface samples.

28

Figure 8: Evolution of μ-histograms m(μ) with increasing number of calculated samples n.

Figure 9: (a) Averaged coefficient of friction μkavg. (n) over number of surface samples n for three random evaluation orders k. (b) Standard deviation over number of surface samples n for 1000 random evaluations.

29

Figure 10: (a) Rough surface measurement. (b) Multiscale validation setup including four scales. (c) Storage (E  (ω)) and loss modulus (E  (ω)) for rubber compounds A, B and C.

Figure 11: Relative contact areas Ar calculated with the multiscale framework for all materials (top). Fitted adhesive coefficients of friction with the experimental input μdif f (v) (bottom). Parameters c = 0.18 and v0 = 0.02m/s for all materials. a) τ0 = 0.29MPa for Material A . (b) τ0 = 0.25MPa for Material B . (c) τ0 = 0.20MPa for Material C.

30

Figure 12: Experimental results for dry and wet (soap-water) surface conditions (dashed lines). Numerical results calculated with the multiscale finite element framework for three materials A, B and C (thick lines).

Figure 13: (a) Changed definition of μdif f (v). (b) Adhesion fitting. (c) Exemplary simulation results by adapting α for the wet surface (soap-water).

31

Figure 14: Fitted adhesive coefficients of friction with the experimental input μdif f (v) between dry measurement and hysteresis calculation. Parameters c = 0.12 and v0 = 0.11m/s for all materials. (a) τ0 = 0.5MPa for Material A . (b) τ0 = 0.4MPa for Material B . (c) τ0 = 0.3MPa for Material C.

Figure 15: Experimental results for dry and soap-water surface conditions (dashed lines). Numerical results calculated with the multiscale finite element framework for three materials A, B and C (thick lines) with adapted coefficient α for different surface conditions.

32