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ACOMPUTATIONAL FRAMEWORK FOR EVALUATING
TIRE-ASPHALT HYSTERETIC FRICTION INCLUDING PAVEMENT
ROUGHNESS
Ivana Ban*1,Jacopo Bonari†2,3, and Marco Paggi‡3
1Faculty of Civil Engineering, University of Rijeka, Trg Bra´
ce Maˇ
zurani´
ca 10, 51000 Rijeka, Croatia
2Institute for the Protection of Terrestrial Infrastructures, German Aerospace Center (DLR), Rathausallee 12, 53757
Sankt Augustin, Germany
3IMT School for Advanced Studies Lucca, Piazza San Francesco 19, 56100 Lucca, Italy
April 3, 2025
ABS TRAC T
Pavement surface textures obtained by a photogrammetry-based method for data acquisition and
analysis are employed to investigate if related roughness descriptors are comparable to the frictional
performance evaluated by finite element analysis. Pavement surface profiles are obtained from 3D
digital surface models created with Close-Range Orthogonal Photogrammetry. To characterize the
roughness features of analyzed profiles, selected texture parameters were calculated from the pro-
file’s geometry. The parameters’ values were compared to the frictional performance obtained by
numerical simulations. Contact simulations are performed according to a dedicated finite element
scheme where surface roughness is directly embedded into a special class of interface finite ele-
ments. Simulations were performed for different case scenarios and the obtained results showed a
notable trend between roughness descriptors and friction performance, indicating a promising po-
tential for this numerical method to be consistently employed to predict the frictional properties of
actual pavement surface profiles.
Keywords pavement roughness ·digital surface models ·photogrammetry ·hysteretic friction ·finite elements ·
viscoelasticity
1 Introduction
Pavement surface friction is one of the key functional properties that guarantees safe driving conditions. It affects
the contact interface between vehicle tire and asphalt pavement, providing vehicle stability while steering and grip
during the braking maneuver [1]. It is quantified by the coefficient of friction, a dimensionless number indicating the
ratio between the tangential and the normal force. An important feature in friction phenomenon is the limited portion
of the interface where the contact actually takes place, a quantity highly affected by the roughness properties of the
surface [2]. A close look at most of natural and artificial surfaces reveals a rough topology, that determines a true
contact area smaller than it would be experienced for an idealized smooth contact. This feature directly affects the
value of the resulting friction coefficient [3].
Asphalt pavement surface manifests roughness at many scales of observation, each characterized by several differ-
ent scales of specific amplitude and wavelength, which in turn are connected to functional pavement properties [1].
∗ivana.ban@gradri.uniri.hr
†jacopo.bonari@dlr.de, corresponding author
‡marco.paggi@imtlucca.it
arXiv:2504.01511v1 [cs.CE] 2 Apr 2025
The macro scale of pavement surface texture, or macro-texture, is usually defined within an amplitude ranging from
0.1 mm to 20 mm and a wavelength ranging from 0.05 mm to 50 mm. This texture scale is relevant for tire-asphalt in-
teraction and is considered as the governing parameter for functional properties like friction for speeds above 50 km/h,
aquaplaning, noise, and splash or spray effects. Micro-texture scale is the smallest texture scale for pavements, with
defined amplitude limits of 0.001 mm to 0.05 mm and wavelengths below 0.5 mm. Micro-texture results from the
properties of the aggregate grain material and its morphology is responsible for low speed friction and vehicle tire
wear [1].
An important consideration when dealing with pavement friction is related to the material properties of the bodies in
contact. For the inelastic materials employed in tires production, rubber friction theory recognizes that the frictional
force results from two distinct phenomena, adhesion and hysteresis, that make a different contribution whose com-
bination results in the overall frictional response. Adhesion results from inter-molecular contact between different
materials through Van der Waals dipole forces [4, 5], and is highly dependent on the true contact area size. Hysteresis
is a loss of energy that takes place in rubber materials during deformation and recovery. When such a material is
stretched and then released, it does not immediately return to its original shape; instead, some of the energy is dissi-
pated as heat due to internal friction within its polymer chains. The energy loss determines a lag between the applied
force and the rubber’s response that in turn causes a significant change in the interface contact forces, finally leading
to the arousal of tangential forces opposing the sliding motion [6]. Important contributors to hysteretic friction are
sliding speed and rubber temperature [1].
The non-conforming rough interface between rigid pavement surface and tire rubber makes the true contact area a-
priori unknown, and its value varies during the sliding motion, which renders the problem of rubber sliding over a rough
profile as highly non-linear [7]. Given the problem complexity, no closed form expressions can be derived to predict the
rubber coefficient of friction. To tackle the problem, numerical methods are usually exploited to model rough contact
problems including friction. In the research community, two wide spread and well-established methods are usually
exploited: the finite element method (FEM) [8] and the boundary element method (BEM) [9], each characterized
by their own strengths and weaknesses. FEM requires discretization of the whole domain, including the interior,
while BEM, only requires discretization of the boundaries of the domain, thus reducing the dimensionality of the
problem. BEM is often more efficient for problems with infinite or semi-infinite domains, since it reduces the number
of elements needed. On the other hand, BEM is mostly restricted to simple material laws and geometries, mostly
in a small displacement setting. Quite the opposite, FEM is better suited for complex, non-linear, or multi-material
problems because it can easily handle a variety of diverse material properties and complex geometries, though more
computationally demanding.
Given its characteristics, numerical solutions of pavement-tire interaction problems in the context of friction analysis
are most commonly observed within a FEM framework. The physical problem is simulated by modeling the contact
between vehicle tires or vehicle tires subsets and the pavement structure [10]. Modeling of vehicle tire load requires
the discretization of the full tire or of a representative sample, the definition of the specific viscoelastic material
model, and often includes anisotropic effects due to the different stiffness of tire constituent materials, i.e., tire rubber
and inner embedded steel belt, and specification of loading scenario [11]. Pavement structure modeling requires the
discretization of both surface and bulk, which is more complex if surface roughness is accounted in its actual form and
not simplified. Furthermore, asphalt pavement is characterized as well by a viscoelastic rheology and even though it
is mostly considered rigid in comparison to the tire, an adequate material constitutive law also for the pavement can
be taken into account in pursuit of a more realistic modeling approach [12].
In the course of the years, FEM analysis of tire-pavement interaction in terms of frictional behavior has been inves-
tigated from different research angles. The FEM framework was exploited for the prediction of friction performance
to derive the so-called NUS skid resistance model [13]. The NUS model is a theoretical model for pavement skid
resistance which uses solid mechanics and hydrodynamic theories developed by National University of Singapore
(NUS) research group. The model solved the coupled tire-fluid-pavement interaction problem assuming a Coulomb
friction law to model the frictional behavior of the system and the Navier-Stokes equations equipped with a stan-
dard k−εturbulent model to describe the fluid dynamics. However, the work does not consider the influence of
pavement surface morphology on the resultant forces, but rather only the influence of the water film thickness and
applied sliding speed. Concurrently, [14] analyzed the influence of different pavement surface morphological prop-
erties on the frictional response of the system. They performed several FEM simulations applying different loading
pressures and sliding velocities on three different types of pavements. Excerpts of vehicle tires were simulated as a
rectangular viscoelastic rubber block, while the pavement structure was modeled as both rigid or deformable, in either
case with a fully discretized texture. Contact was enforced using a surface-to-surface contact algorithm and finite
displacements were considered. The sliding response was analyzed through the calculation of a friction coefficient as
the ratio between the resultant applied tangential and normal loads. Research performed by [12] focused on the influ-
ence of different elastic properties (recoverable resilient deformation) of various pavement structures on the braking
2
performance by setting up a 3D tire-pavement interaction model. They investigated the influence of different system
variables such as tread deformation at the contacting interface, actual contact area and the braking force applied within
the dynamic friction contact analysis, concluding that pavement elasticity and deformation influence the real contact
area. However, they did not study the effect of surface texture on the frictional response of the system. Authors [15]
performed a multi-scale FEM analysis to calculate low-speed sliding friction on rough rigid pavement surfaces, em-
phasizing the utmost importance of the two rubber friction components, i.e., adhesion and hysteresis. The simulations
were performed in 2D to reduce the computational complexity caused by including in the model real rough surfaces
representations and a scale separation hypothesis was applied to account for the effects of micro and macro texture
on the frictional response. Hysteretic friction component was accounted by using a viscoelastic material model and a
robust surface-to-surface contact algorithm, while the adhesive contribution was added at the macroscopic level only
tailored for different surface contact conditions based on a tuning coefficient α, whose value ranged from zero to
one depending on the surface conditions (zero for surface covered in soap-water mixture and one for dry surface).
In [16], tire-pavement interaction was investigated within a 3D FEM framework, by emphasizing the effect of high
temperatures on the frictional response of the model. Pavement structure was scanned by x-ray tomography and pro-
cessed to distinguish mixture phases, i.e., aggregates, bitumen, and air voids, and then modeled using micro-structure
FEM meshes where aggregates were described as elastic while binder was supposed to be viscoelastic. The authors
then modeled the contact behavior using, again, a surface-to-surface contact algorithm and calculated the frictional
response from the theory of hysteresis induced energy dissipation. The numerical results were compared to the field
measurements of skid resistance showing promising and comparable results. Research done by [10] focused on FEM
simulations of frictional response using 3D pavement surface model reconstructed from high resolution pavement tex-
ture data in order to explore the influence of texture characteristics on interface friction parameters. They assumed
that the pavement was non-deformable, and the rubber characterized as a hyperelastic and viscoelastic block. They
assumed an exponential decay friction model proposed by [17] taking place at the interface and calculated the friction
coefficient from the resultant tangential force over normal applied load. In the study, they explored the water effect
on frictional response by lowering the exponential decay friction to mimic the presence of a thin lubricant layer at
the interface. The friction model parameters were determined with a binary search back-calculation algorithm, which
adjusted the parameters of the FEM simulation such that the simulated and measured skid resistance values resulted
comparable. In [18] an integrated tire-vehicle model for the prediction of frictional performance of wet pavements
was proposed. They exploited Persson’s friction theory [19] for the calculation of the kinetic friction coefficient, based
only on the pavement surface power spectra and the rubber rheology specifications. Furthermore, they performed a
FEM analysis of the hydroplaning effect to calculate the hydrodynamic forces acting, and determined the effect of
presence of water on different tire movements. The characteristics of tire movements were obtained by integrating the
calculated friction coefficient into the FEM hydroplaning model and further used for the vehicle dynamics simulation
in a dedicated software. This approach considered the influence of pavement texture on the frictional performance, but
also linked the frictional response to different vehicle movements while emphasizing the importance of a high friction
coefficient during braking or steering maneuvers.
In this research, a FEM approach is emploiyed to simulate the frictional response of tire-pavement interaction consid-
ering real pavement texture roughness features in input. To account for the actual geometry of the pavement surface
involved, the novel MPJR (eMbedded Profile for Joint Roughness) approach is utilized. The approach enables to dra-
matically simplify the modeling procedure usually required in the definition of a FEM problem involving contacting
surface, without introducing geometrical approximations or smoothing. The solution was first introduced in [20, 7]
as an approach to address rough contact problems involving a rigid and a deformable body. The method introduces a
novel zero-thickness interface finite element which stores the actual information describing the interface geometry and
encodes roughness information directly at each element quadrature nodes, thus allowing to replace the actual complex
interface geometry with an equivalent smooth interface, while retaining all the problem’s original characteristics.
The method proved to be accurate in the solution of contact problems involving complex interfaces described by ana-
lytical expressions and has been further developed to also account for friction in presence of microscopic sliding [21],
finite frictional sliding of harmonic profiles over viscoelastic bodies [22] and adhesive contact problems, both fric-
tionless and with friction [23]. In an ongoing study the method has also been employed to reproduce the results of
a contact mechanics challenge [24], a demanding test case in which researchers in the field of tribology were invited
to solve the contact problem between a synthetic large-scale and high-resolution rigid rough surface and an elastic
half-space. The MPJR method shows a high degree of accuracy as compared to the benchmark solution provided by
the proposers of the challenge.
In the present study, the MPJR framework is further extended to analyze finite sliding scenarios simulating the sliding
of viscoelastic bodies over complex rough profiles, whose geometry is directly extracted from 3D pavement surface
models obtained with a photogrammetry-based method developed and verified in [25]. The object of the study is to
test and investigate the possibility of employing the MPJR approach for the evaluation of the hysteretic coefficient
3
of friction, and to draw relationships between the parameter obtained through numerical simulations, describing the
frictional performance of the rough profiles, and some selected roughness parameters obtained from the same profiles
employed in the numerical simulations. A positive outcome of this preliminary investigation would entitle the proposed
method as a valid investigation tool to be consistently employed in this field of research, and would provide motivation
for further extended analysis capabilities including, but not limited to, large scale 3D and finite deformation.
2 Rough profiles description
To obtain surface profiles, a novel method for experimental pavement texture data acquisition and analysis is utilized.
The method is based on photogrammetry as a technique for object reconstruction from captured digital images [26]. A
special case of photogrammetry employed in this research is close-range photogrammetry, where objects of interest are
captured from a close distance to investigate the morphology of the surfaces under examination. To obtain the surface
images, the digital camera was mounted 50 cm above the pavement surface with camera lens parallel to the surface
plane. The method was christened Close-Range Orthogonal Photogrammetry or CROP method. The development of
CROP method and its verification procedure are thoroughly described in the doctoral thesis [25], while some important
highlights are summarized here:
1. Images were captured with a single Nikon D500 DSLR 20.1 Mpix digital camera, with AF Nikkor 50 mm
f1.8 D single lens. The camera was mounted on a tripod at a fixed height of 50 cm and the camera lens
positioned parallel to the pavement surface.
2. The surface of interest was 50 cm ×50 cm large, marked by a custom made reference frame for precise
digital surface reconstruction and calibration. The frame was created for the method’s verification procedure
to determine the deviations of dimensions in digital surface model crated by CROP method and true object’s
dimensions.
3. Each surface was photographed twenty-five times, with consecutive images overlap equal to 60% side and
80% forward overlap. The camera was moved in parallel rows with respect to the markings on the reference
frame, where, in each row, five images were captured.
4. Images were captured in .raw format to preserve image information in the original form without compression
and stored in .tiff format after brightness and contrast optimization, with pixel size 4.45 µm×4.45 µm.
5. Acquired surface images were used for reconstructing a digital surface model in the specialized photogram-
metry software Agisoft Metashape v1.5 Pro. Surface reconstruction procedure was done automatically, where
captured images and reconstruction parameters were user-defined inputs. The workflow consisted of image
alignment based on seek-and-match procedure of common points in captured images, resulting in a sparse
point cloud (SPC) entity. The SPC object served as a basis for 3D object reconstruction, subjected to filtering
procedure by selected error reduction features: reprojection error, reconstruction uncertainty and projection
accuracy. The purpose of error reduction features is to remove the outlier points in captured images based
on camera geometry of the images. In this way, the initial SPC entity is filtered and the best-fit tie points are
extracted for further reconstruction procedure. The threshold values of error reduction features are specified
in [25]. The best-fit points filtered from SPC were further used in dense point cloud (DPC) object creation,
consisting of a finite number of points with XY Z coordinates, describing the object’s geometry and sur-
face morphology. The DPC can be the final object in the reconstruction procedure or it can be used for the
reconstruction of 3D mesh object.
An example of data acquisition by CROP method is given in Fig. 1. While the camera arrangement is shown in Fig. 1a,
the created pavement digital surface models (DSMs) can be observed in Fig. 1b and 1c. Once the DSMs were created,
they were subjected to further processing in the open-source software for dense point cloud analysis Cloud Compare,
v2.11.3. The DSMs of investigated surfaces were subjected to initial adjustments: leveling, so the 3D models were
parallel to the horizontal plane; scaling, to correspond to millimeters unit; and filtering, so the DPC points falling
outside the area of interest or relevant texture scale were removed. The final DPC model of each surface was subjected
to profile segmentation to investigate the roughness properties of analyzed surfaces.
To determine the roughness properties of pavement surfaces that could be compared to the friction performance eval-
uated by the proposed numerical method, an analysis of profile roughness features was conducted. The profiles were
segmented from the DSMs of investigated surfaces in Cloud Compare software. To segment the profiles, it has been
necessary to define a segmentation area, a segment orientation, a segment length (i.e., profile length), and a lateral
distance and thickness of the segmented DPC sections. The segmentation area was equal to the pavement surface area
inside the reference frame and the segments were oriented to be parallel to a horizontal x-axis. The length of the seg-
ment in the x-direction was set to be 100 mm, to correspond to the profile’s length for the calculation of the traditional
4
(a) Pavement surface data
acquisition by CROP
method, with orthogonal
positioned digital camera
and target surface marked
with reference frame.
(b) Digital surface model of pave-
ment surface obtained by CROP
method application.
(c) An example of digital pavement surface model pro-
cessed in the Cloud Compare software
Figure 1: Pavement digital surface acquisition model.
pavement texture indicator mean profile depth (MPD) following the EN ISO 13473-1 standard norm. The segments’
lateral distance was set to 10 mm, so that each investigated surface was described by ten profiles, as depicted in Fig. 2.
The depth in the direction orthogonal to the average surface mean plane of the segmented section was set to be
0.01 mm, selected after the comparison of profile’s number of points for different section depth values settings. This
value determines the width of the DPC segment from which the profile is created. If the selected depth values were too
wide, the profile would consist of points having a single xcoordinate and multiple zcoordinates corresponding to the
same xvalue. This would require dataset interpolation to obtain a 2D profile and such approximation could influence
the true geometry of the surface. On the other hand, for a too narrow section depth, the number of points in a profile
would not be sufficient to properly describe all roughness features as such profiles would have gaps, requiring, again,
extrapolation and true roughness features approximation. An example of a profile segmented with three different
section depths is given in Fig. 3, obtained from the Cloud Compare software as result of the DPC data analysis.
For a selected section depth of 0.01 mm, the obtained profiles showed to have an average point-to-point distance
in horizontal direction below 0.01 mm. This value implies that extracted profiles have sub-millimeter resolution in
horizontal direction, which enables the roughness features analysis on both micro- and macro- texture scales relevant
for the analysis of pavement friction phenomenon. The profiles extracted with the previously described settings in the
segmentation procedure had roughly 12.000 points on average. The exact number of points in each profile differs due
to the morphology varieties of inspected surfaces in the positions where the profiles were segmented. For the purpose
of roughness features comparison of extracted profiles, all profile points coordinates were converted from a relative
to an absolute reference coordinate system so that each profile starts with a point having xcoordinate equal to zero,
Figure 2: Profiles segmented from the digital surface model with defined segmentation properties: profile length is
100 mm and profiles’ lateral distance is 10 mm.
5
(a) Profile segment with section thickness equal to 0.005 mm.
(b) Profile segment with section thickness equal to 0.010 mm.
(c) Profile segment with section thickness equal to 0.050 mm.
Figure 3: The same 10 mm length profile segmented with three different section thicknesses.
while height coordinates were adjusted so that the minimum height value was set to zero. After the profiles’ coordinates
were corrected, the adjusted profiles were imported to Mountains Map, v9.0 Lab Premium software, for analysis of
the roughness features. The software enables automatic calculation of profile’s texture roughness features based on
the EN ISO 21920-2 standard. Roughness features can be calculated on profiles with or without scale separation,
a feature that enables the exclusion of specific texture scales, if desired. In this research, no scale separation was
applied and the texture parameters were determined on primary profiles containing both the relevant texture scales.
To remove any remaining profile vertical slope, the profiles were automatically leveled to a horizontal plane. A
Gaussian S-filter was applied to the leveled profiles to remove the elements in lateral direction which were not below a
threshold of 2.5µm. In this way, the profile dataset was de-noised, while the relevant profile points within the defined
horizontal resolution of 0.01 mm were preserved. Profile related roughness parameters were selected from the group of
amplitude- or height-related parameters and feature parameters group, defined in the EN ISO 21920-2 standard. They
are the most common non-standard roughness parameters groups explored in pavement texture analysis in relation
to frictional performance [27, 28, 29, 30]. The amplitude parameters are a sub-group of field parameters, related to
the full length profiles and calculated from the profile heights z(x). Feature parameters are calculated for a given
section of a full length profile, describing amplitude or wavelength features of profiles’ sections within specific section
lengths. Additionally, the traditional pavement texture profile-related parameter MPD was calculated for the analyzed
profiles to compare its values to the values of the obtained non-standard parameters. Table 1 provides an overview of
the roughness parameters calculated for the analyzed profiles, with a description of their physical meaning.
2.1 Selection of profiles for numerical simulation
The aim of the research is to assess if texture roughness features calculated from the processed pavement surface
profiles relate to the friction response obtained for the same profiles, when they are utilized as input sliding support in
the numerical simulations. For this purpose, three profiles were selected arbitrarily from the database with different
values of calculated texture parameters, extracted from three pavement surface models obtained by the CROP method.
The surfaces were previously chosen for their friction performance after a standard pavement friction measurement
campaign performed with the aid of a static skid resistance tester (SRT), test EN ISO 13036-4. This device is used for
the measurement of low-speed friction performance of pavements based on the pendulum principle [1]. The obtained
6
Table 1: Profile roughness parameters determined by following EN ISO 21920-2 standard.
Parameter (mm) Group Description
Paamplitude arithmetic mean height for full
profile length
Pqamplitude root mean square height for full
profile length
Ptamplitude total height difference for full
profile length
Ppt feature (profile peak) maximum peak height for all
profile sections
Pvt feature (profile peak) maximum pit depth for all pro-
file sections
Pzfeature (profile peak) mean value of maximum total
height difference on all profile
sections
Psm feature (profile element) mean profile element spacing in
horizontal direction for all pro-
file elements
Psmx feature (profile element) maximum profile element spac-
ing in horizontal direction for
full profile length
Pcfeature (profile element) mean profile element height for
all profile elements
Pcx feature (profile element) maximum profile element
height for all profile elements
M P D traditional parameter mean profile depth from two
highest profile peaks in the first
and second half of the full pro-
file length
friction values are expressed as SRT, a dimensionless number ranging from 0to 150, where higher values indicate
better friction performance of the inspected surface. Three surfaces selected from the database showed different
measured values of SRT: surface P06 exhibited SRT of 75.2, surface P12 SRT was 85.4and, for surface P20, the
evaluated SRT was 94.4. To be able to compare the roughness properties of the surfaces evaluated by the SRT device,
the CROP method was later applied on the exact same surface area. The size of the inspected surface was 125 mm ×
75 mm, corresponding to the sliding distance of the pendulum’s skid and the skid’s width. The analyzed profiles were
selected from the central part of the surface model, as the CROP method verification procedure showed that the best
precision of a DSM is obtained in the center of the model. All profiles were extracted from the DSMs by following the
same procedure as described in the previous section, with resulting profile properties: profile length 100 mm, profile
resolution 0.01 mm, obtained with a low-pass filter of 2.5µm.
The geometry of selected profiles is provided in Fig. 4 and the resulting roughness parameters values on the inspected
primary profiles, calculated following the relevant standards, are given in Tab. 2. From Tab. 2 it can also be seen
that profile P12 obtained the lowest values of roughness parameters in most of the amplitude and feature parameters
group. The exception are values for Psm and Psmx feature parameters, which are related to the profile’s morphology in
horizontal direction. The traditional texture parameter M P D is the lowest for P12 profile and highest for P20 profile.
Profile P20 obtained the highest values for all calculated parameters, in coincidence with the highest SRT value
determined for the surface from which this profile was segmented. However, the measured friction values are surface-
related and therefore cannot be a reliable representation of profile’s roughness characteristics analyzed in this work.
Amplitude parameters Paand Pqcalculated as mean height values for the full profile length showed less variation in
values among profiles in comparison to feature parameters describing the peak profile features, such as Pzor Pvt. The
highest variation of parameters values was obtained for Pcx profile element parameter, characterizing the maximum
profile element height.
7
0 20 40 60 80 100
x(mm)
−2.50
−1.25
0.00
1.25
2.50
z(mm)
P06 P12 P20
Figure 4: Profiles elevation fields.
Table 2: Measured friction values expressed in SRT (−)and calculated profile roughness parameter values for three
analyzed profiles; all roughness parameters values are expressed in (mm).
Profile P06 P12 P20
SRT 75.2 85.4 94.4
M P D 1.091 0.563 1.184
Pa0.352 0.211 0.663
Pq0.459 0.258 0.809
Pt2.665 1.233 3.285
Ppt 1.180 0.702 1.214
Pvt 1.486 0.531 2.070
Pz1.374 0.724 2.393
Psm 12.319 12.586 11.802
Psmx 27.522 40.029 18.518
Pc1.120 0.565 2.100
Pcx 2.094 0.844 3.285
Considering the obtained values of roughness parameters calculated for the three segmented profiles, it can be con-
cluded that the profiles have significantly different morphology. Consequently, it is expected for them to show different
friction performance in the following numerical simulations.
3 Computational framework
The computational framework chosen to assess correspondences between sample roughness parameters and friction
arising between asphalt and tire leverages on a novel interface finite element introduced in [20] and called MPJR,
eMbedded Profile for Joint Roughness. The method hinges on a special class of finite elements originally derived for
cohesive zone models applied in the field of non-linear fracture mechanics. It allows, in its basic formulation, to easily
analyze contact problems involving rough entities coming into contact. In subsequent extensions, the method has
been further developed to account for infinitesimal sliding with Coulomb friction [21] in two and three dimensions,
for both non adhesive and adhesive contact problems [23], with the extended possibility of considering any arbitrary
geometry for the rough surface, whose height field could be defined according to analytic functions or synthetic or
experimental data. Furthermore, the context of finite sliding has been investigated as well in [22], where a seminal
approach has been tested to simulate finite sliding of a linear viscoelastic block over a rigid surface with a simple
harmonic geometry.
In this study, the last two show-cased features of finite sliding of a linear viscoelastic body on real rough profiles, i.e.,
whose geometry is defined through an experimental campaign of data acquisition, are combined together for the first
time, resulting in a step further in the field of high fidelity simulations of tire-asphalt interaction. The phenomenon
is governed, at the micro-scale, by the interplay of different friction mechanisms. On one hand there is Coulomb
or dry friction, which is proportional to the normal force that brings the surfaces in contact and is characterized by
a coefficient of friction that remains relatively constant regardless of the contact area and the sliding velocity [2].
In contrast, rubber friction involves more complex interactions, due to rubber’s viscoelastic properties. This type of
friction is highly dependent on the texture of the contacting surface, on the applied pressure, on the sliding velocity and
8
on the temperature [19]. Rubber friction itself consists, in turn, of two main components: adhesion, since rubber tends
to adhere to the contacting surface, and hysteresis, an energy loss due to the inelastic deformation of the polymeric
chain composing the microstructure of the rubber that results in heat dissipation. As a result, rubber friction can vary
significantly with changes in these factors, making it less predictable and more complex than Coulomb friction.
Since the proposed computational approach already proved reliable in the analysis of both Coulomb friction and
adhesive problems, the main scope of the current investigation is devoted to the analysis of frictional effects due
to hysteresis alone. A positive outcome of the current investigation will then pave the way for the derivation of a
comprehensive interface model capable of bringing together all the aforementioned features, namely, Coulomb friction
and rubber friction due to both adhesion and hysteresis.
To the scope, a simulation framework has been set up, where the tire has been modeled as a linear viscoelastic block
pressed against a rigid rough profile and moved under the action of an imposed horizontal act of motion, characterized
by specific values of velocity. The numerical solution is obtained in terms of the hysteretic friction coefficient for
different sliding velocities employing, as sliding support, the different profiles obtained in Sec. 2.
3.1 Rheological model for rubber block
Real viscoelastic materials exhibit a transient mechanical response that varies significantly across multiple orders of
magnitude of time and intensity. As a result, a simplistic model comprising a single linear Hookean spring in con-
junction with a Newtonian dashpot fails to accurately represent their behavior. For engineering applications, typically
at least three terms in the Prony series are required to yield meaningful stress analysis predictions. In our case, the
classic expression for the Young’s relaxation modulus has ben employed:
E(t) = E0+
3
X
n=1
Enexp −t
τn.(1)
The rheological properties of the material can be investigated in the frequency domain performing a Fourier transform
of the relaxation modulus defined in Eq. (1). The complex modulus in the frequency domain reads:
ˆ
E(ω) = E0+
n
X
i=1
Ei
τ2
iω2
1 + τ2
iω2+ı
n
X
i=1
Ei
τiω
1 + τ2
iω2.(2)
Starting from the complex modulus, the storage modulus can be defined as the real part ℜˆ
E(ω)of Eq. (1), and it
describes the ability of the material to store elastic energy. Specularly, the loss modulus is defined as the imaginary
part ℑˆ
E(ω)of ˆ
E(ω)and describes the part of energy which is dissipated as heat as the material is subjected to
deformation.
Due to the limitation imposed by the FEM solver in terms of material parameters definition, a model characterized
by a Prony series with three arms has been chosen. To obtain a realistic response, the material model employed
both in [6] and [31], which in turns is based on real data of Styrene Butadiene Rubber (SBR) acquired during an
experimental campaign performed at the German Institute for Rubber Technology, has been fitted using Eq. (1). A
comparison between our model and the model employed in the two aforementioned studies is depicted in Fig. 5. More
specifically, Fig. 5a shows a comparison between two different realization of the relaxation modulus characterized
by three arms, continuous red line, and six arms, red circular markers. On the other hand, Figs. 5b and 5c shows a
comparison between averaged real data, a three arms model, continuous lines, and a six arms model, circular markers.
In all the plots, green color refer to the loss modulus, while red color to the storage modulus. It can be noticed how
the models characterized by a different numbers of the terms composing the Prony series deliver identical results at
higher frequency responses, while differing at low excitation frequencies.
3.2 Friction due to hysteresis
Let us assume a linear viscoelastic body pressed against a rough profile and sliding over it under the action of a
tangential load. If both adhesive and Coulomb frictional effects are neglected, the contact tractions will always be
orthogonal to the deformed surface. In the framework of a linear theory, a global time varying horizontal contact force
Q(t)can then be determined via a geometric relation, as the integral of the projection of the normal contact tractions
pnover the surface gradient ∂u(x, t)/∂x. If P(t)is the overall time varying vertical reaction force, a global averaged
coefficient of friction can then be defined as:
9
10−10 10−810−610−410−2100
t(s)
10−5
10−4
10−3
E(MPa)
three arms
six arms
(a) Response modulus for three, solid red line, and six
terms, circular markers, of the Prony series.
1001021041061081010 1012
ω(Hz)
100
101
102
103
104
<ˆ
E(MPa)
three arms
six arms
experimental data
(b) Storage modulus for the three arms model, solid curve, the
six arms model, circular markers, and for experimental data,
dotted line.
1001021041061081010 1012
ω(Hz)
10−3
10−2
10−1
100
101
102
103
104
=ˆ
E(MPa)
three arms
six arms
experimental data
(c) Loss modulus for the three arms model, solid curve, the
six arms model, circular markers, and for experimental data,
dotted line.
Figure 5: Rheological diagrams for rubber like material.
µ=1
TZT
0
Q(t)
P(t)dt=1
TZT
0RΓpn∂u(x, t)/∂ x dx
RΓpndxdt. (3)
For problems characterized by simple geometrical and mechanical features, an analytical expression can be derived
for Eq. (3). For example, let us consider a layer of viscoelastic material, characterized by a single relaxation time τ,
that extends indefinitely in the horizontal direction and has a finite depth l. If it makes full contact with a sinusoidal
profile of wavelength λand amplitude a, under the action of an imposed vertical far-field displacement u0, and slides
over it with constant velocity v, the average coefficient of friction can be derived as:
µ=πE1a2ωτ
E∞u0λ(1 + ω2τ2),(4)
where ω= 2πv/λ is the excitation frequency and E1and E∞=PiEiare the long term and the instantaneous elastic
modulus of the material, respectively. From this preliminary simple model, an important property of the viscoelastic
coefficient of friction can be deduced, i.e., its marked dependency on the sliding velocity v. For this simple case it
is straightforward to conclude that the maximum value of µis reached when the system is excited with a frequency
ω⋆= 1/τ. In a more general scenario, characterized by additional complexities such as more complex material laws,
partial contact or more complex support sliding surfaces, no closed form expression can be derived and the mean value
of the hysteretic friction coefficient must be evaluated numerically according to Eq. (3).
10
Table 3: Mechanical parameters for the three arms Maxwell Model.
E(MPa) τ(s)
9.77 −
5.41 ×1002 1.85 ×10−06
1.16 ×1003 8.09 ×10−08
1.19 ×1003 4.22 ×10−10
3.3 MPJR interface finite element for normal contact
In its 2D formulation, the MPJR interface finite element is a four nodes quadrilateral element originally defined to
model interfaces in the context of nonlinear fracture mechanics, addressed using a cohesive zone model [32, 33] and
later on extended to contact mechanics [20]. This framework allows to solve contact problems involving deformable
rough surfaces in contact in a straightforward fashion, avoiding to explicitly model the features of geometrically
complex entities, exploiting the concepts of composite topography and composite mechanical parameters [34, §2.2.3].
Within a small deformation setting, the normal contact of two elastic bodies characterized by a complex interface
can be reformulated as a contact problem involving a rigid rough body, whose shape is a combination of the two
original geometries, and a linear elastic body with a flat interface whose mechanical parameters are a combination
of the original ones. If the original bodies are already rough-rigid and flat-deformable, this reformulation step is
unnecessary. Since this condition is coincident with the current case, and no composite mechanical parameters are
involved, any material law can be defined for the deformable bodies, and in our case a linear viscoelastic material is
employed to model the rubber rheology.
Since one of the body is rigid, it has not to be explicitly modeled and the domain definition is limited to the deformable
body and the contact interface, whereby the former can be discretized employing standard finite elements and the
latter a single array of four nodes MPJR interface finite element. In each element, the two lower nodes are tied to the
discretization of the deformable body, while the two upper nodes belong to the rough profile.
The kinematics of the element is shown in Fig. 6. An array of unknown horizontal and vertical nodal displacements
u= [u1, v1, . . . , u4, v4]⊺is introduced for the evaluation of the gap function g= [gt, gn]⊺across the interface, that
reads:
g=QNLu,(5)
where Lis a linear operator that computes the relative nodal displacements across the interface in horizontal and
vertical direction, Nis a matrix that collects standard first order shape functions and Q= [ˆ
t,ˆ
n]⊺is a rotation matrix
that maps the gap function from a local reference system, aligned with the element, to a global Ω(x1, x2)reference
frame, where ˆ
tand ˆ
nare the unit tangential and normal vectors aligned with the element centerline and still evaluated
in Ω(x1, x2).
According to this formulation, the element is suitable for the solution of contact problems involving smooth conformal
interfaces. At this point, a correction in the nodal vector uis performed by hard coding the position of the two upper
nodes v⋆
3and v⋆
4to match the actual position of the rigid rough surface considered. The result of this operation is
a corrected gap function whose normal component g⋆
ndescribes the kinematics of the real geometrically complex
problem, now reading:
g⋆
n=ˆ
n·N1(u4−u1) + N2(u3−u2)
N1(v⋆
4−v1) + N2(v⋆
3−v2)(6)
Figure 6: Kinematics of the 2D quadrilateral MPJR element.
11
Finally, the externally applied contact tractions can be evaluated, e.g., with a standard penalty approach governed by a
penalty parameter εn:
pn=εng⋆
nif g⋆
n<0
0if g⋆
n≥0(7)
A sketch of the whole process is qualitatively shown in Fig. 7 below. It has to be remarked that in the context under
examination the degrees of freedom uirelated to horizontal displacements do not play any active role in the simulation,
since the layer of the interface finite elements is disposed on a horizontal line according to the global reference frame,
so that no related term appears in Eq. (6), and no Coulomb friction is considered acting at the interface level. For a
complete derivation of the method, together with validation and quality assessment of the results, the interested reader
is referred to [21].
3.4 MPJR interface finite element for predicting finite sliding
In the current study, the introduced MPJR approach is extended with an additional feature that allows to consider
a relative finite sliding involving a rough profile expressed as a set of discrete elevations over a viscoelastic body.
Starting from the derivation given in [22], this is possible thanks to the main hypotheses on which the formulation
hinges, which are:
1. rigidity of the rough profile;
2. apriori knowledge of the profile’s act of motion;
3. small displacements analysis.
Given these assumptions, it is always possible to express, for every analysis time step, the relative position of the
viscoelastic body and the sliding profile, and therefore the relative position of the two leading and trailing closest
points of the profile with respect to the correspondent nodes of the boundary of the deformable body, i.e., the two
lower nodes of each interface finite element.
The sliding viscoelastic body, hereinafter labeled as skid, is modeled as a rectangular domain, in the reference frame
Ω(x1, x2)set in correspondence of the skid’s top-left corner and with the x1axis aligned to its top side. The analysis
is developed over a time window T= [t0, tf]and is divided in two main phases, a Phase I involving vertical loading,
in which the rough profile is pressed against the skid, and a Phase II, where a horizontal velocity is imposed, which
determines the sliding motion. Section 4.2 offers a detailed description of the two different phases of the loading
process.
Sliding is supposed to take place between the rough profile and the top side of the skid, with the rough profile moving
in accordance with a predefined rigid translating act of motion yo(t)=[y1(t), y2(t)]⊺characterized by a given
horizontal and vertical velocity. As stated in Sec. 2.1, every profile geometry is stored as a set of discrete points
yp= [yp1,yp2 ]⊺, whose first coordinate is set so that, considering a local reference frame O(y1, y2)solidal with the
rough profile, the first point of the ensemble is characterized by yp1 = 0.0. At t= 0, the rough profile makes contact
with the skid at its minimum point only, without exerting any contact pressure. In the skid’s global reference frame,
the position of each point of the rough profile can be then expressed as xp(t) = yo(t) + yp.
To reproduce the right contact conditions, an array of interface finite elements is defined over the skid’s top side,
where, in accordance with the MPJR paradigm, the lower pair of nodes is tied to the FEM discretization of the skid’s
boundary, while the normal gap gnis corrected to account for the variation of the position of the rough profile, Fig. 8.
(a) Original problem setting. (b) Discretized setting.
Figure 7: Sketch of the MPJR approach.
12
Figure 8: Correct nodes elevation according to their actual position during the sliding process. The deformable bulk is
shaded in gray, while the array of MPJR elements is highlighted on its top by its four nodes. For each element, the two
top nodes, highlighted in green, do not enter the calculations, while the actual top nodes’ position is inferred at each
time step. Cross markers indicate the sampled points on the rough profile, and the nodal elevation used to correct the
normal gap is evaluated from a spline interpolation between the leading and the trailing points next to each node. In
the figure, this operation is highlighted for node (a). The leading and trailing points are highlighted in blue, while the
interpolated portion of the profile is depicted in red. The sliding process is visualized by the superposition of different
profiles snapshots in different shades of gray, each referred to a single time instant, the lightest being related to the
time step furthest in time.
While for a profile defined by an analytical expression the correction of the normal gap can be performed straight-
forwardly via a parametrization in time of the function describing the profile’s elevation, cfr. Eq. (12) in [22], the
same operation can not be extended to the current case, since the elevation is known only at discrete points, thus
allowing to correct the normal gap only if the horizontal position of a profile’s point is coincident with the horizontal
position of the skid’s boundary node. To overcome the problem, a cubic spline interpolation is performed once in a
first offline stage, to sample the intervals between yp1,i and yp1,i+1 and, for every point i, the interpolation coefficients
ci,j for j= [1,...,4] are stored beside the profile’s horizontal position and elevation.
For every interface finite element considered, two elevation values have to be identified, one for each of the base nodes
of the interface finite element. The correct elevation value is then inferred identifying the closest trailing and leading
correspondent points from the profile’s set. Once they have been identified, the correct elevation value is retrieved
evaluating the interpolation in correspondence of the horizontal position of the interface finite element’s node.
To keep the profile’s horizontal velocity as generic as possible, the identification of the neighboring points is performed
at each time step, but since the elevations are stored as an ordered list, a convenient binary search algorithm can be
enforced to solve the problem. For each node, this operation must be carried out only once, since when the trailing
point has been identified, the leading point is the following one, and vice versa. Once the elevation of the rough profile
in correspondence of the interface node under examination has been carried out, its value is plugged into the expression
of the corrected normal gap, cfr. Eq. (6) and Eq. (2) in in [22], and the resulting corrected gap can be employed to
restore the true contact condition of the original problem.
4 Numerical experiments setup
In this section, a first set of numerical experiments is performed to analyze the sliding of a linear viscoelastic block
over a simple rigid sinusoidal shape. The profile is characterized by a wavelength λ= 2π/320 mm and amplitude
g0= 2.0×10−3mm. The results in terms of µ(t)and µare then compared with Eq. (4) and the results presented
in [31]. Each feature of the model setup is then discussed in detail. Concurrently, these preliminary results are useful to
tailor the model for the subsequent use of the rough profiles obtained during the campaign of experimental sampling.
4.1 Rubber block geometry and FEM discretization
This part of the analysis focuses on a rectangular block with length bb=λand hb= 0.75λ. The domain is dis-
cretized employing standard linear quadrilateral elements, while an array of MPJR interface finite elements is placed
in correspondence of the contact interface and stores the shape of the moving sinusoidal profile. To test the effect of
an increasing refinement of the contact zone over the final value of the friction coefficient, the mesh is refined in corre-
spondence of the contact side, where a different and progressively increasing number of elements has been employed.
13
The results of the mesh convergence study are detailed in Sec. 4.3, while a sketch of the problem under examination
is depicted in Fig. 9a, together with the typical mesh arrangement employed shown in an excerpt of a simulation run
shown in Fig. 9b.
The zone interested by mesh refinement, where bulk elements have the smallest size, is characterized by a certain
depth. Below, the size of the mesh elements starts to increase hierarchically. Numerical experiments shows that this
buffer zone must be characterized by a minimum depth value in order to achieve regular results in terms of contact
tractions. By successive refinement, this minimum value has been qualitatively identified to be two times the size of the
coarsest element in the mesh. In an analogous way, the penalty parameter εnhas been increased until the simulation
results were not affected anymore by larger values, yet still avoiding divergence due to a too high εn. The final value
εn= 102E∞/hbhas been employed, being E∞the long term Young’s modulus of the viscoelastic bulk.
4.2 Loading process
The simulation is divided in two main stages. During the first, referred to as Phase I, the block is pressed against the
rigid profile by a constant uniform pressure p0applied to the block on the opposite side with respect to the contact
interface. In the following loading phase, namely Phase II, a uniform horizontal velocity vis imposed on the top side,
while p0is kept constant.
4.2.1 Phase I
The most important parameter governing Phase I is the velocity of application of p0, quantified by the total duration T1
of the vertical loading phase. It has been verified in [31] that if the response of the system is more deformable during
Phase I than during Phase II, then the system will need more time to reach a steady state during the sliding process.
This will negatively affect the friction coefficient averaging process, since a wider analysis time window would be
required. Therefore, a maximum value of T1has to be identified and, to this scope, the same approach used in [31]
has been employed. There, the desired value has been identified by solving the following optimization equation:
n
X
k=1
Ekexp (−T1/τk) =
n
X
k=1
Ek
(ωτk)2
1+(ωτk)2,(8)
where T1is obtained as a function of the applied velocity and ¯ω= (2πv)/λ. The higher v, the lower must be T1,
i.e., the faster must the vertical load be applied. In case of simulations involving more sliding velocities, the more
demanding condition has been addressed, coincident with the highest value of v. The related value T1, i.e., the lower
one resulting from the different velocities analyzed, has been used in every instance. After T1has been identified, a
number of time steps ts1 sufficient to ensure the convergence of the solver has been set. Unless differently specified,
a value of ts1 = 100 is hereinafter used. The trend of the optimal value for T1in function of the block excitation
frequency can be observed in Fig. 10a for different values of ω.
(a) Sketch of the problem under examination. (b) Excerpt from numerical simulation at a selected repre-
sentative time step.
Figure 9: Graphical representation of the addressed problem.
14
4.2.2 Phase II
At the end of Phase I, the onset of sliding takes place and Phase II begins. In every simulation, a constant value of
sliding velocity is considered, while a transition phase with duration equal to T1is defined to guarantee a smooth
acceleration of the sample obtained through a cubic fillet in the time velocity diagram. The duration of Phase II is
determined by the overall sliding length necessary to correctly evaluate µ, a condition which is fulfilled by the full
development of steady state sliding conditions. The adequate number of time steps to be considered depends on a
proper sampling in time of the sliding profile. To ensure that, for each time step, the variation of the profile geometry
is correctly captured, the number of time steps employed in the sliding phase is such that for each time increment, an
advancement of the block on the profile of 0.2bb/mxtakes place, where mxcoincides with the number of interface
finite element used to discretize the interface and bb/mxis the distance between two consecutive points at which the
height is sampled. If the total sliding distance is then expressed as a multiple nλof the profile wavelength, the number
of time steps necessary to cover the required distance results then in ts2 = 5mxnλ
4.3 Results of the benchmark tests
In a first benchmark test, a sinusoidal profile has been employed. Two different conditions have been tested. In the
first test case, a linear viscoelastic block with a single relaxation time has been employed. During Phase I, a uniform
vertical load p0= 10.0 MPa has been applied in an overall time T1= 0.16150688 s, evaluated in accordance with
the criterion expressed by Eq. (8), and then horizontal sliding is enforced by the imposition of a constant horizontal
velocity field vto the profile.
In this very first preliminary test, a different number of interface finite element has been employed to discretize the
interface and therefore the sinusoidal profile. Concurrently, different values of the penalty parameter have been tested,
additionally. The results of the convergence study are presented in Fig. 10b for mx= 16,32,64,128 and 256
interface finite elements employed. The penalty parameter εnhas been increased until the resulting traction field was
not affected anymore by further increases, resulting in a value of 10E∞/h, being hthe domain’s height. In all the
subsequent simulations, a value of mx= 128 interface elements per wavelength has been employed, since further
refinements did not result in any appreciable accuracy gain in terms of µ.
The geometry of the block and the shape of the profile are the same listed at the beginning of Sec. 4.2, and periodic
boundary conditions are enforced in correspondence of the two vertical sides. The linear viscoelastic material em-
ployed for the test is characterized by a single relaxation time, with the following mechanical properties: ν= 0.3,
E0= 4.17 MPa,E1= 1.72 MPa,τ1= 0.01134034 s. Different values of the applied velocity have been tested,
logarithmically centered around the critical value identified by ω⋆, i.e., v⋆=λ/(2πτ1). On the other hand, T1has
been imposed to be correspondent to the most demanding condition, i.e., for the highest sliding velocity considered
of 10λ/(2πτ1). The same value is then used in all the other cases, being them less demanding. Results are plotted in
Fig. 11a in terms of µ(t), for the case v=v⋆. The results of the averaged value µare plotted in Fig. 11b in function
105106107
ω(s−1)
10−7
10−6
10−5
T1(s)
(a) Optimal values for the duration of the vertical loading
phase for different values of excitation pulsation ω.
16 32 64 128 256
mx
0.24
0.26
0.28
0.30
0.32
0.34
0.36
µ
(b) Convergence of the average friction coefficient µfor
different interface discretization levels and εn= 10E∞/h.
Figure 10: Optimization in terms of vertical load velocity application and mesh refinement is performed to achieve
reliable simulation results.
15
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
L/λ
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
µ(t)
(a) Instant value of the friction coefficient at critical sliding
speed.
10−1100101
ωτ
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
µ
FEM model SDOF analytic
(b) Averaged values of the friction coefficient for different
applied velocities.
Figure 11: Benchmark test, friction coefficient for different sliding velocities over a sliding distance L= 4λ.
of every velocity considered, together with the values provided by Eq. (4). Good accordance is found, given the many
different assumptions related to the two different models, above all: the analytical solution refers to a single degree of
freedom model, while the numerical results to the discretization of a distributed setting.
In a second test, the same simulation setting has been employed, with a full viscoelastic model with three terms in the
Prony series used to model the block’s rheological properties. The material parameters are those identified in Sec. 3.1
and listed in Tab. 3. Leaving the applied pressure unchanged, a velocity equal to 100 mm/s, close to the critical
value, is applied to the sliding profile. The related plot for µ(t)is shown in Fig. 12. A direct confrontation with the
results from [31, §6.1] can be performed. The plot is compared with the related curve extracted from Fig. 10 (c) of
the reference, for the case h/λ = 0.75. A perfect accordance can be found. It has to be remarked that such good
accordance is to be expected only in correspondence of the highest viscoelastic response of the deformable material,
since this is the region where the two different material models manifest the same rheological characteristics.
5 Rough profile sliding
In this section, the hysteretic frictional properties of asphalt pavement surfaces are investigated using the extracted
profiles as sliding support. Three different sets of simulations are performed, one set for each of the profiles analyzed,
whose complete height field is depicted in Fig. 4. Each simulation set is characterized by several test sliding velocities.
0 2 4 6 8 10 12
L/λ
0.34
0.35
0.36
0.37
0.38
0.39
µ(t)
Figure 12: Sliding test for the imposed velocity causing a high dissipative effect, for an applied pressure of p0=
10.0 MPa. The correspondent average value from [31] is superposed as a dash-dotted line. The average value µ≈0.36
corresponds to the result obtained in [31, §6.1] for the same values of applied pressure and imposed velocity.
16
5.1 Profiles processing and resolution convergence study
Given the high resolution of the sampling process, a direct correspondence between the acquired points and the in-
terface finite elements storing the related height information rapidly leads to very high computational costs. For this
reason, the effect of a down-sampling is analyzed first. For every profile, sub-profiles have been extracted by collecting
one point every ρnodes, with ρ= 1,2,5, and 10, respectively, where ρ= 1 corresponds to a condition for which
every point of the profile is taken into consideration, thus leading to the highest achievable level of accuracy. The
result of the down-sampling process can be observed in Fig. 13a for a representative magnified portion of a sample
profile and different chosen values of ρ. Since the skid discretization is, in turn, function of the number of points
employed to describe the profile, the down-sampling process results in a mesh characterized by mx= 64,128,256,
and 512 interface finite elements, respectively. Results in terms of µ(t)for the four different discretization levels are
reported in Fig. 13b, for a simulation that considers a short test sliding length of L= 2band for an applied velocity of
v= 7 ×103mm/s, while the other modeling assumptions and parameters are in line with the ones exposed in Sec. 4.
It can be noticed how a chosen value of ρ= 2, implying the use of mx= 256 interface finite elements, results in a
value of µ, dash dotted lines in Fig. 13b, very close to the one obtained with the finest grained resolution, which is
obtained for ρ= 1. This chosen level of discretization will be employed in the final simulations related to the full
scale models, i.e., for a sliding process that takes place over the whole length of the profiles.
5.2 Duration of Phase I for rough profiles sliding
Since the duration of Phase I is an important parameter to guarantee a rapid extinction of the transient period during
the horizontal sliding, special care must be taken for its evaluation. For generic profiles a closed form expression
analogous to Eq. (8) can not be obtained. If a rough profile is considered, it is not possible to follow the same strategy
applied for the identification of an optimal T1, as done for the sinusoidal profile. Nonetheless, a limiting value for the
excitation frequency can still be identified if we consider the hypothetical profile with the shortest wavelength that can
be taken into account given the discretization of the block’s interface. If we consider this harmonic profile as a sampled
time signal, then the shortest wavelength that can be sampled corresponds to its Nyquist frequency, i.e., two times the
sampling interval, which in turn is equal to λmin = 2∆x, where ∆xis the distance between two sampled points.
If we now plug these values, related to different discretization levels, into Eq. (8) for a specific sliding velocity, the
correspondent maximum durations of the normal loading phase to be taken into account, that minimizes the transient
during the sliding process can be obtained. For the chosen discretization level of mx= 256, a correspondent value of
approximately T1= 2.0×10−6sis obtained and employed in the simulations that follow. It has to be remarked that
different sliding velocities lead to different T1. For the sake of simplicity the more demanding values, obtained for the
highest velocity employed will be considered in all the simulations. Higher values of T1, related to different possible
discretization levels and sliding velocities, will not be taken into account.
0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625
x(mm)
0.0990
0.0995
0.1000
0.1005
0.1010
0.1015
0.1020
0.1025
0.1030
z(mm)
ρ= 10
ρ= 5
ρ= 2
ρ= 1
sampled points
(a) Circular markers represent the original dataset along-
side different level of down-sampling.
0 2 4 6 8 10 12
L(mm)
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
µ(t)
mx= 64
mx= 128
mx= 256
mx= 512
(b) Different realizations of µ(t)function for different sam-
pling resolutions.
Figure 13: The down-sampling of the points acquired on the profiles reflects on the instantaneous value µ(t).
17
5.3 Additional model parameters
A range of different sliding velocities is applied during Phase II to all the selected profiles. For each of them,
twenty different values have been chosen, logarithmically centered around v0= 104mm/s, and bounded, by
vmin = 103mm/sand vmax = 105mm/s. The intensity of v0has been chosen to approximately resemble the
standard sliding velocity usually employed in the experimental British Pendulum friction tests (BPT) [35], while the
bounding values of the range have been designed to be far enough from the critical speed in order to clearly assess the
maximum value of µand its sensitivity to the changes in skid’s velocity. A sliding length coincident with the complete
development of the available profiles has been used in every case analyzed, leading to an overall value sliding length
of L≈90 mm, resulting, in turn, in a ratio with the skid length of L/b ≈9. It has to be remarked that if a non-
periodic profile is involved, the sliding length plays an important role in the determination of the average coefficient
of friction since it has to be long enough for all the relevant roughness features to be taken into consideration. Finally,
a value p0= 2.0 MPa for the vertical applied pressure has been employed, equal for all the three different profiles
analyzed. Even though theoretical and experimental evidences [19, 36] for its concurrent influence on the resulting
viscoelastic coefficient of friction exist, its analysis goes beyond this preliminary study and it is therefore left for
further investigations.
5.4 Results and discussion
Simulation results are exposed in Fig. 14. Figure 14a shows the instant values of viscoelastic coefficient of friction for
the three different profiles in correspondence of the velocity value which determines the highest µ. It can be noticed
how the average values are in line with mean values obtained in literature, cfr., e.g., [10]. Figure 14b shows the average
values of the viscoelastic coefficient of friction µfor every sliding velocity analyzed and for each of the three profiles.
In every case, a critical value for the velocity range can be clearly identified, which starts around 10 m/sfor P12 and
then slightly increases for P20 and P06.
Profile roughness parameters analysis exposed in Sec. 2 showed that the profile labeled as P20 has the highest values
for all the texture parameters considered. The surface from which this very same profile was extracted also shows the
highest frictional performance measured by the SRT device. Even though this feature is not reproduced in the numer-
ical simulation in terms of µfor most of the velocities under consideration, cfr. Fig. 14b, the profile is remarkably
characterized by the highest instantaneous values of µ(t), as clearly visible from Fig. 14a, a response localized around
u≈30 mm.
Profile P12 has the lowest roughness parameters values, however the surface from which the profile was sectioned
did not share this characteristic, i.e., was not characterized by the lowest friction performance determined by SRT
measurements. On the other hand, for what concerns profile’s parameters, the trend is well captured by the numerical
10 20 30 40 50 60 70 80
u(mm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
µ(t)
P06
P12
P20
(a) Instant value of the viscoelastic coefficient of friction
for three different profiles in correspondence of the critical
value of the sliding velocity.
103104105
v(mm/s)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
µ
P06
P12
P20
(b) Average value of the viscoelastic coefficient of friction, µ
for every sliding velocity analyzed. For each profile a critical
value close to the sliding velocity employed in the BTP test
can be observed.
Figure 14: Simulation results in terms of viscoelastic coefficient of friction.
18
framework, which delivers the lowest values of µfor all the velocities under consideration, and also the lowest values
of maximum µ(t)for the sliding velocities considered.
For P06, good accordance is found between the SRT values related to the surface and the profile roughness parameters,
since they are both characterized by the lowest indicators. Unfortunately, this trend is not captured at the simulation
stage, which delivers unexpectedly high µresults. In this case the simulation was not able to capture the expected low
average values for the hysteretic coefficient of friction. A possible explanation is related to missing features of the
numerical model employed which is characterized by small displacements, linear material assumptions, an interface
model which is not comprehensive of adhesion and Coulomb friction and a 2D setting.
Furthermore, a comparison between profile roughness parameters and measured SRT surface values is not straight-
forward, as the SRT values are related to the measured surface area and not to a single profile. As the choice of the
profiles extracted from the digital surface models was arbitrary, it was not guaranteed that the measured frictional
performance would follow the same trend as the roughness parameters values for the single profile. Pavement surfaces
were evaluated for the friction performance to gain a general representation of surface roughness characteristics and to
distinguish between the surfaces having significantly different friction performance. This was the key for the selection
of surfaces for rough profiles extraction, which were later used in numerical simulations. Concerning this discrepancy,
an explanation can be provided invoking missing additional mechanical effects related to the viscoelasticity of the bulk
that, in conjunction with surface roughness, gives rise to a trend that cannot be explained according to the geometrical
features of profile roughness alone.
6 Conclusion
This research has extended the MPJR approach to address the complexities of tire-pavement interaction, by simulat-
ing the hysteretic frictional response of real pavement texture roughness features. By embedding detailed pavement
profile models into the finite element method, the study avoids the need of geometrical approximations, maintaining
high fidelity in representing profile interactions. The MPJR framework, previously validated for various contact prob-
lems, proves to be robust and reliable in capturing the characteristics of finite sliding scenarios, as highlighted by the
employment of both simple sinusoidal and complex real rough profiles.
The findings of the study suggest that the MPJR approach is accurate and practical for evaluating the hysteretic co-
efficient of friction, delivering values which are in line with the literature in terms of magnitude. In spite a perfect
correlation with specific roughness parameters derived from the analysis of the profiles could not be identified, this
capability opens new possibilities for using the MPJR method as a consistent investigative tool in the field of tribology
for applications requiring adequate modeling of rough contact interactions. The additional features provided by the
current model are a necessary building block required to derive a comprehensive framework capable of delivering a
better correlation with surface roughness parameters.
Profile-related roughness parameters calculated from the extracted pavement digital surface models showed to have
a certain discrepancy with the experimentally measured friction performance, determined by the SRT device. The
surface roughness evaluated by SRT device was approximated with a single 2D profile and its roughness features,
generalizing the overall roughness of inspected surface. A better coincidence between calculated roughness parameters
and experimentally validated skid resistance might be possible in case of 3D surface roughness parameters calculation
from pavement digital surface model, which is planned in further research of pavement friction investigations.
After this preliminary study, future work will build on these results by incorporating large-scale 3D analyses, intu-
itively based on surface interpolation; more complex deformation scenarios and interface friction laws; and thermal
effects. Moreover, the setting is prone to be embedded in extended analysis tools like time-series synthetic generation
to artificially recreate wider surfaces for considering wider sliding ranges, thus leveraging on a surrogate modeling
approach that would relieve expensive experimental data acquisition campaigns. Furthermore, model order reduction
(MOR) techniques could also be put in place to reduce the high computational costs required by 3D simulations, also
leveraging on HPC resources. All these new features will further consolidate the MPJR approach use for advanced
frictional performance studies.
References
[1] N. A. of Sciences Engineering, Medicine, Guide for Pavement Friction (2009).
URL https://doi.org/10.17226/23038
[2] F. P. Bowden, D. Tabor, The area of contact between stationary and moving surfaces, Proceedings of the Royal
Society of London. Series A. Mathematical and Physical Sciences 169 (938) (1939) 391–413. doi:10.1098/
19
rspa.1939.0005.
URL https://doi.org/10.1098/rspa.1939.0005
[3] L. V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Heidelberg, 2010.
URL https://link.springer.com/book/10.1007/978-3-662-53081-8
[4] N. Yu, A. A. Polycarpou, Adhesive contact based on the Lennard–Jones potential: a correction to the value of the
equilibrium distance as used in the potential, Journal of Colloid and Interface Science 278 (2) (2004) 428–435.
doi:10.1016/j.jcis.2004.06.029.
URL https://linkinghub.elsevier.com/retrieve/pii/S0021979704005454
[5] R. A. Sauer, P. Wriggers, Formulation and analysis of a three-dimensional finite element implementation for
adhesive contact at the nanoscale, Computer Methods in Applied Mechanics and Engineering 198 (49-52) (2009)
3871–3883. doi:10.1016/j.cma.2009.08.019.
URL https://linkinghub.elsevier.com/retrieve/pii/S0045782509002631
[6] P. Wriggers, J. Reinelt, Multi-scale approach for frictional contact of elastomers on rough rigid surfaces, Com-
puter Methods in Applied Mechanics and Engineering 198 (21-26) (2009) 1996–2008. doi:10.1016/j.cma.
2008.12.021.
URL https://linkinghub.elsevier.com/retrieve/pii/S0045782509000267
[7] M. Paggi, D. Hills, Modeling and Simulation of Tribological Problems in Technology, Springer, 2020.
URL https://link.springer.com/book/10.1007/978-3-030-20377-1
[8] O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method - Volume 1: The basis, 5th Edition, Butterworth-
Heinemann, Oxford, 2002.
[9] C. A. Brebbia, The birth of the boundary element method from conception to application, Engineering Analysis
with Boundary Elements 77 (2017) iii–x. doi:10.1016/j.enganabound.2016.12.001.
URL https://linkinghub.elsevier.com/retrieve/pii/S0955799716304453
[10] P. Yi, J. Q. Li, Y. Zhan, K. C. P. Wang, G. Yang, Finite element method-based skid resistance simulation using
in-situ 3d pavement surface texture and friction data, Materials 12 (23) (2019) 3821.
URL https://doi.org/10.3390/ma12233821
[11] M. Yu, Z. You, G. Wu, L. Kong, C. Liu, J. Gao, Measurement and modeling of skid resistance of asphalt
pavement: A review, Construction and Building Materials 260 (2020) 119878.
URL https://doi.org/10.1016/j.conbuildmat.2020.119878
[12] M. Yu, G. Wu, L. Kong, Y. Tang, Tire-pavement friction characteristics with elastic properties of asphalt pave-
ments, Applied Sciences 7 (11) (2017) 1123.
URL https://doi.org/10.3390/app7111123
[13] T. Fwa, Skid resistance determination for pavement management and wet-weather road safety, International
Journal of Transportation Science and Technology 6, 3 (2017) 217–227.
URL https://doi.org/10.1016/j.ijtst.2017.08.001.
[14] S. K. Srirangam, K. Anupam, C. Kasbergen, A. T. Scarpas, Analysis of asphalt mix surface-tread rubber interac-
tion by using finite element method, Journal of Traffic and Transportation Engineering (English Edition) Volume
4, Issue 4 (2017) 395–402.
URL https://doi.org/10.1016/j.jtte.2017.07.004
[15] P. Wagner, P. Wriggers, L. Veltmaat, H. Clasen, C. Prange, B. Wies, Numerical multiscale modelling and exper-
imental validation of low speed rubber friction on rough road surfaces including hysteretic and adhesive effects,
Tribology International 111 (2017) 243–253.
URL https://doi.org/10.1016/j.triboint.2017.03.015
[16] T. Tang, K. Anupam, C. Kasbergen, R. Kogbara, A. S. E. Masad, Finite element studies of skid resistance under
hot weather condition, Transportation Research Record 2672(40) (2018) 382–394.
URL https://doi.org/10.1177/0361198118796728
[17] J. Oden, J. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in
Applied Mechanics and Engineering 52 (1-3) (1985) 527–634.
URL https://doi.org/10.1016/0045-7825(85)90009-X
[18] X. Liu, Q. Cao, H. Wang, J. Chen, X. Huang, Valuation of vehicle braking performance on wet pavement surface
using an integrated tire-vehicle modeling approach, Transportation Research Record 2673(3) (2019) 295–307.
URL https://doi.org/10.1177/0361198119832886
[19] B. N. J. Persson, Hysteresis friction of sliding rubbers on rough and fractal surfaces, Rubber Chemistry and
20
Technology 70 (1) (1997) 1–55.
URL https://meridian.allenpress.com/rct/article/70/1/1/92419/
Hysteresis-Friction-of-Sliding-Rubbers-on-Rough?utm_source=chatgpt.com
[20] M. Paggi, J. Reinoso, A variational approach with embedded roughness for adhesive contact problems, Me-
chanics of Advanced Materials and Structures 27 (20) (2020) 1731–1747. doi:10.1080/15376494.2018.
1525454.
[21] J. Bonari, M. Paggi, J. Reinoso, A framework for the analysis of fully coupled normal and tangential contact
problems with complex interfaces, Finite Elements in Analysis and Design 196 (2021) 103605. doi:10.1016/
j.finel.2021.103605.
URL https://linkinghub.elsevier.com/retrieve/pii/S0168874X21000895
[22] J. Bonari, M. Paggi, Viscoelastic Effects during Tangential Contact Analyzed by a Novel Finite Element Ap-
proach with Embedded Interface Profiles, Lubricants 8 (12) (2020) 107. doi:10.3390/lubricants8120107.
URL https://www.mdpi.com/2075-4442/8/12/107
[23] J. Bonari, M. Paggi, D. Dini, A new finite element paradigm to solve contact problems with roughness, Interna-
tional Journal of Solids and Structures 253 (2022) 111643. doi:10.1016/j.ijsolstr.2022.111643.
URL https://linkinghub.elsevier.com/retrieve/pii/S0020768322001640
[24] M. H. M¨
user, W. B. Dapp, R. Bugnicourt, P. Sainsot, N. Lesaffre, T. A. Lubrecht, B. N. J. Persson, K. Harris,
A. Bennett, K. Schulze, S. Rohde, P. Ifju, W. G. Sawyer, T. Angelini, H. Ashtari Esfahani, M. Kadkhodaei,
S. Akbarzadeh, J.-J. Wu, G. Vorlaufer, A. Vernes, S. Solhjoo, A. I. Vakis, R. L. Jackson, Y. Xu, J. Streator,
A. Rostami, D. Dini, S. Medina, G. Carbone, F. Bottiglione, L. Afferrante, J. Monti, L. Pastewka, M. O. Robbins,
J. A. Greenwood, Meeting the Contact-Mechanics Challenge, Tribology Letters 65 (4) (2017) 118. doi:10.
1007/s11249-017-0900-2.
URL http://link.springer.com/10.1007/s11249-017-0900-2
[25] I. Ban, A model for skid resistance prediction based on non-standard pavement surface texture parameters, Ph.D.
thesis, University of Rijeka Faculty of Civil Engineering (2023).
[26] T.Luhmann, S.Robson, S.Kyle, I.Harley, Close Range Photogrammetry:Principles, techniques and applications
(2006).
[27] G. Bitelli, A. Simone, F. Girardi, C. Lantieri, Laser scanning on road pavements: A new approach for character-
izing surface texture, Sensors 12 (2012) 9110–9128.
URL https://doi.org/10.3390/s120709110
[28] S. C. Callai, M. D. Rose, P. Tataranni, C. Makoundou, C. Sangiorgi, R. Vaiana, Microsurfacing pavement so-
lutions with alternative aggregates and binders: A full surface texture characterization, Coatings 12(12) (2022)
1905.
URL https://doi.org/10.3390/coatings12121905
[29] N. Zuniga-Garcia, J. A. Prozzi, High-definition field texture measurements for predicting pavement friction,
Transportation Research Record 2673(1) (2019) 246–260.
URL https://doi.org/10.1177/0361198118821598
[30] R. B. Kogbara, E. A. Masad, D. Woodward, P. Millar, Relating surface texture parameters from close range
photogrammetry to grip-tester pavement friction measurements, Construction and Building Materials 166 (2018)
227–240.
URL https://doi.org/10.1016/j.conbuildmat.2018.01.102.
[31] L. De Lorenzis, P. Wriggers, Computational homogenization of rubber friction on rough rigid surfaces, Compu-
tational Materials Science 77 (2013) 264–280. doi:10.1016/j.commatsci.2013.04.049.
URL https://linkinghub.elsevier.com/retrieve/pii/S0927025613002218
[32] M. Paggi, P. Wriggers, A nonlocal cohesive zone model for finite thickness interfaces – Part I: Mathematical
formulation and validation with molecular dynamics, Computational Materials Science 50 (5) (2011) 1625–
1633. doi:10.1016/j.commatsci.2010.12.024.
URL https://linkinghub.elsevier.com/retrieve/pii/S0927025610006956
[33] M. Paggi, P. Wriggers, A nonlocal cohesive zone model for finite thickness interfaces – Part II: FE implemen-
tation and application to polycrystalline materials, Computational Materials Science 50 (5) (2011) 1634–1643.
doi:10.1016/j.commatsci.2010.12.021.
URL https://linkinghub.elsevier.com/retrieve/pii/S0927025610006920
[34] J. Barber, Contact Mechanics, Vol. 250 of Solid Mechanics and Its Applications, Springer International Publish-
ing, Cham, 2018. doi:10.1007/978- 3-319-70939-0.
URL http://link.springer.com/10.1007/978-3-319-70939-0
21
[35] Y. Liu, T. F. Fwa, Y. S. Choo, Finite-element modeling of skid resistance test, Journal of Transportation
Engineering-asce 129 (2003) 316–321.
URL https://api.semanticscholar.org/CorpusID:108936458
[36] G. Fortunato, R. D. Lorenzetti, A. Bellini, E. Ciulli, On the dependency of rubber friction on the normal force or
load: Theory and experiment, arXiv preprint (2015). arXiv:1512.01359.
URL https://arxiv.org/abs/1512.01359?utm_source=chatgpt.com
22
