Finite Element Analysis of Tyre Contact Interaction Considering Simplified Pavement with Different Aggregate Sizes

Abstract

This study considered the effect of pavement aggregate grain size on tyre–pavement contact interaction during the late stages of pavement skid resistance. First, hemispherical shells 7, 9, and 13 mm in diameter were used to simulate coarse pavement aggregates. Subsequently, a three-dimensional finite element tyre–pavement contact model developed using ABAQUS was employed to analyse the contact interaction between each simplified pavement type and the tyre under steady–state rolling and braking conditions. Finally, the concept of occlusal depth was proposed and applied to characterise pavement skid resistance. The results showed that under steady–state rolling conditions, the peak contact stress of the simplified pavement increased with the pavement mean texture depth, whereas the contact area decreased. Under steady–state braking conditions, the effect of the contact interaction between the tyre and simplified pavement aggregates was ranked in order of superiority as aggregate grain sizes of 9, 7, and 13 mm, indicating that aggregate grain size did not exhibit any correlation with tyre–pavement contact interaction. Additionally, the squares of linear correlation coefficients between the pavement cumulative occlusal depth and horizontal braking force reached 0.921, 0.941, 0.889, and 0.894 for vehicle speeds of 30, 60, 90, and 120 km/h, respectively, indicating that they could be used to assess pavement skid resistance.

Full text

Citation: Li, Z.; Chen, W.; Li, Y.; Wu,
W. Finite Element Analysis of Tyre
Contact Interaction Considering
Simplified Pavement with Different
Aggregate Sizes. Appl. Sci. 2023,13,
12011. https://doi.org/10.3390/
app132112011
Academic Editors: Edgar Sokolovskij
and Vidas Žuraulis
Received: 9 October 2023
Revised: 28 October 2023
Accepted: 31 October 2023
Published: 3 November 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
applied
sciences
Article
Finite Element Analysis of Tyre Contact Interaction Considering
Simplified Pavement with Different Aggregate Sizes
Zhi Li, Weiyong Chen, Yinghui Li and Wenliang Wu *
School of Civil Engineering and Transportation, South China University of Technology,
Guangzhou 510640, China; lizhi@scut.edu.cn (Z.L.); 202121010428@mail.scut.edu.cn (W.C.);
202221010106@mail.scut.edu.cn (Y.L.)
*Correspondence: ctwlwu@scut.edu.cn
Abstract:
This study considered the effect of pavement aggregate grain size on tyre–pavement
contact interaction during the late stages of pavement skid resistance. First, hemispherical shells
7, 9, and 13 mm in diameter were used to simulate coarse pavement aggregates. Subsequently,
a three-dimensional finite element tyre–pavement contact model developed using ABAQUS was
employed to analyse the contact interaction between each simplified pavement type and the tyre
under steady–state rolling and braking conditions. Finally, the concept of occlusal depth was proposed
and applied to characterise pavement skid resistance. The results showed that under steady–state
rolling conditions, the peak contact stress of the simplified pavement increased with the pavement
mean texture depth, whereas the contact area decreased. Under steady–state braking conditions, the
effect of the contact interaction between the tyre and simplified pavement aggregates was ranked
in order of superiority as aggregate grain sizes of 9, 7, and 13 mm, indicating that aggregate grain
size did not exhibit any correlation with tyre–pavement contact interaction. Additionally, the squares
of linear correlation coefficients between the pavement cumulative occlusal depth and horizontal
braking force reached 0.921, 0.941, 0.889, and 0.894 for vehicle speeds of 30, 60, 90, and 120 km/h,
respectively, indicating that they could be used to assess pavement skid resistance.
Keywords:
finite element model; aggregate grain size; tyre–pavement contact interaction; occlusal depth
1. Introduction
The skid resistance of an asphalt pavement is a critical factor affecting road traffic
safety [
1
–
3
]. Whilst driving, vehicle tyres are in direct contact with the pavement; thus,
tyre–pavement contact interaction is a crucial aspect determining asphalt pavement skid
resistance. The primary factors affecting tyre–pavement interaction are the tyre character-
istics (tyre tread, tyre pressure, driving speed, etc.), pavement characteristics (aggregate
properties, aggregate grain size, pavement macrotexture, etc.), and external environment
(temperature, water, snow, etc.) [4–6].
Many researchers have studied tyre–pavement contact interactions. Zhang et al. [
7
,
8
]
used a pressure–sensitive, electrically conductive rubber sensor to measure the contact
pressure between the tyre and pavement and successively analysed the corresponding
tyre–pavement contact interaction under static and dynamic conditions; however, they
ignored the effects of the tyre texture and pavement macrotexture. Chen et al. [
9
] measured
the contact pressure between a tyre and pavements of different macrotextures under static
conditions using a pressure film and showed that the pavement macrotexture and tyre load
had a greater effect on tyre–pavement contact than the tyre inflation pressure; however, the
study lacked a dynamic analysis of tyre–pavement contact.
Indoor testing is a vital research method; however, it can be easily affected by external
factors and consumes considerable manpower and material resources. At the same time,
tyre–pavement contact analysis involves complex nonlinear mechanical problems. With the
rapid development of computers, finite element analysis allows for intuitive and accurate
Appl. Sci. 2023,13, 12011. https://doi.org/10.3390/app132112011 https://www.mdpi.com/journal/applsci

Appl. Sci. 2023,13, 12011 2 of 15
solutions to complex non-linear mechanical problems. Consequently, finite element nu-
merical simulation has become a more effective means of studying tyre–pavement contact
interactions. Liu and Al-Qadi [
10
,
11
] used a deep learning approach to predict 3D tyre–
pavement contact stresses based on finite element generated tyre contact stress datasets
and developed ContactNet and ContactGAN models for the fast prediction of 3D tyre
pavement stresses. Xie and Yang [
12
], He et al. [
13
], and Tang et al. [
14
] used the ABAQUS
finite element-based software package to develop a three-dimensional (3D) tyre–pavement
contact model and analyse the impact of driving speed, tyre load, and tyre pressure on tyre–
pavement contact interactions under different driving conditions; however, the pavement
model applied during their study ignored the pavement macrotexture.
As road researchers, we would prefer to focus on the impacts of factors such as the
pavement macrotexture and aggregate properties on tyre–pavement contact interaction. Lin
and Wang [
15
] analysed the effect of fine-aggregate angularity (FAA) on asphalt pavement
skid resistance using the British Pendulum Number test. The results showed that the
pavement FAA had a considerable effect on skid resistance at the macrotexture level.
Xiao et al. [16]
analysed the contributions of different texture wavelengths on the aggregate
surface to tyre–pavement interaction. The results showed that the correlation between
textures with small-scale wavelengths and tyre–pavement interaction decreased with
increasing sliding speed; the correlation changed in the opposite direction for textures with
large-scale wavelengths. Riahi et al. [
17
] considered the interaction between the pavement
macrotexture and tyre rolling process to propose a model based on the dissipation of energy
by the rubber block during the cyclic deformation of the pavement in concave and convex
cycles. Yu et al. [
18
] selected seven parameters to characterise pavement texture features
and analysed the impact of those features on pavement skid resistance. Sharma et al. [
19
]
developed a multi-physics tyre–pavement contact model based on the temperature and
pavement texture, focusing on the impact of pavement macrotexture on tyre–pavement
interaction. Finite element simulation technology can be applied to further analyse tyre–
pavement contact interactions; however, in analysing the factors affecting tyre–pavement
contact interactions, previous studies considered the tyre factors but ignored the effect of
the pavement macrotexture and considered the effect of the aggregate surface properties
but paid little attention to the effect of the aggregate grain size after the aggregate texture
was abraded.
Consequently, this study focused on the impact of aggregate grain size on tyre–
pavement contact interaction. Three groups of simplified pavements comprising single-
diameter hemispherical shells were designed. A 3D tyre–pavement contact model devel-
oped using ABAQUS 2021 was subsequently employed to analyse the contact interaction
between each simplified pavement and a tyre under steady–state rolling and braking con-
ditions. Finally, we summarised the change law of the contact interaction between tyres
and aggregates of different grain sizes in simplified pavements and proposed reasonable
evaluation indices to guide the design of asphalt pavement skid resistance.
2. Contact Theory
During the normal vehicle driving process, tyre–pavement contact can be divided into
three general regions, as shown in Figure 1. Region I denotes before contact (where the
tyre and pavement are about to make contact); Region II denotes stable contact (where
the tyre and pavement are in full contact); and Region III denotes after contact (where the
tyre and pavement experience contact detachment). When the vehicle brakes, the impact
force generated in contact Region I and the adhesion friction generated in contact Region
III contribute less to the tyre’s overall friction, and the vehicle’s braking force comes from
contact Region II. Consequently, this study focused on the contact behaviour in Region II
(full contact state).

Appl. Sci. 2023,13, 12011 3 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 16
contact Region II. Consequently, this study focused on the contact behaviour in Region II
(full contact state).
Figure 1. Schematic of the tyre–pavement contact region.
The contact behaviour in Region II primarily comprises tyre–pavement aggregate
contact interaction. Because the tyre’s internal inflatable structure and tread material stiff-
ness are considerably lower than those of the pavement aggregate, the tyre produces a
localised wrapping of the raised aggregate on the pavement during the rolling process.
When the vehicle brakes, the speed of the outer edge of the tyre line is less than the vehi-
cle’s driving speed; consequently, the tyre and raised aggregate produce an “occlusal
braking” effect. This effect can be explained by the hysteresis effect (F
h
) of the aggregate
as well as the adhesion (F
a
) between the tread and pavement materials [20,21], as shown
in Figure 2. The hysteresis effect is primarily a result of the resistance provided by the
occlusion of the tread with the raised pavement aggregates and the surface friction gener-
ated by the relative slip between them. Because the adhesion interaction between materi-
als has a small effect on the overall friction behaviour, its magnitude is negligible [22,23].
Consequently, this study ignored the effect of adhesion and focused on the contact inter-
action between the tyre and pavement aggregates.
Figure 2. Analysis of tyre–pavement contact interaction.
Additionally, research and practice have shown that during the early stages of road
use, different pavement types can provide excellent skid resistance, although their macro-
textures vary considerably. With repeated traffic loads, pavement aggregate angles and
surface microtextures are repeatedly abraded to present a smoother surface; therefore, the
skid resistances of different pavements will exhibit different degrees of aenuation. How-
ever, pavement aggregate grain size is a critical factor affecting pavement macrotexture
Figure 1. Schematic of the tyre–pavement contact region.
The contact behaviour in Region II primarily comprises tyre–pavement aggregate
contact interaction. Because the tyre’s internal inflatable structure and tread material
stiffness are considerably lower than those of the pavement aggregate, the tyre produces
a localised wrapping of the raised aggregate on the pavement during the rolling process.
When the vehicle brakes, the speed of the outer edge of the tyre line is less than the vehicle’s
driving speed; consequently, the tyre and raised aggregate produce an “occlusal braking”
effect. This effect can be explained by the hysteresis effect (F
h
) of the aggregate as well as
the adhesion (F
a
) between the tread and pavement materials [
20
,
21
], as shown in Figure 2.
The hysteresis effect is primarily a result of the resistance provided by the occlusion of
the tread with the raised pavement aggregates and the surface friction generated by the
relative slip between them. Because the adhesion interaction between materials has a small
effect on the overall friction behaviour, its magnitude is negligible [
22
,
23
]. Consequently,
this study ignored the effect of adhesion and focused on the contact interaction between
the tyre and pavement aggregates.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 16
contact Region II. Consequently, this study focused on the contact behaviour in Region II
(full contact state).
Figure 1. Schematic of the tyre–pavement contact region.
The contact behaviour in Region II primarily comprises tyre–pavement aggregate
contact interaction. Because the tyre’s internal inflatable structure and tread material stiff-
ness are considerably lower than those of the pavement aggregate, the tyre produces a
localised wrapping of the raised aggregate on the pavement during the rolling process.
When the vehicle brakes, the speed of the outer edge of the tyre line is less than the vehi-
cle’s driving speed; consequently, the tyre and raised aggregate produce an “occlusal
braking” effect. This effect can be explained by the hysteresis effect (F
h
) of the aggregate
as well as the adhesion (F
a
) between the tread and pavement materials [20,21], as shown
in Figure 2. The hysteresis effect is primarily a result of the resistance provided by the
occlusion of the tread with the raised pavement aggregates and the surface friction gener-
ated by the relative slip between them. Because the adhesion interaction between materi-
als has a small effect on the overall friction behaviour, its magnitude is negligible [22,23].
Consequently, this study ignored the effect of adhesion and focused on the contact inter-
action between the tyre and pavement aggregates.
Figure 2. Analysis of tyre–pavement contact interaction.
Additionally, research and practice have shown that during the early stages of road
use, different pavement types can provide excellent skid resistance, although their macro-
textures vary considerably. With repeated traffic loads, pavement aggregate angles and
surface microtextures are repeatedly abraded to present a smoother surface; therefore, the
skid resistances of different pavements will exhibit different degrees of aenuation. How-
ever, pavement aggregate grain size is a critical factor affecting pavement macrotexture
Figure 2. Analysis of tyre–pavement contact interaction.
Additionally, research and practice have shown that during the early stages of road
use, different pavement types can provide excellent skid resistance, although their macro-
textures vary considerably. With repeated traffic loads, pavement aggregate angles and
surface microtextures are repeatedly abraded to present a smoother surface; therefore,
the skid resistances of different pavements will exhibit different degrees of attenuation.
However, pavement aggregate grain size is a critical factor affecting pavement macrotex-
ture [
24
–
26
]. Consequently, attention must be paid to the effect of aggregate grain size on
pavement skid resistance. To analyse the contact interaction between a tyre and aggregates
of different grain sizes in this study, hemispherical shells were used to simulate abraded
coarse aggregates, as shown in Figure 3.

Appl. Sci. 2023,13, 12011 4 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 16
[24–26]. Consequently, aention must be paid to the effect of aggregate grain size on pave-
ment skid resistance. To analyse the contact interaction between a tyre and aggregates of
different grain sizes in this study, hemispherical shells were used to simulate abraded
coarse aggregates, as shown in Figure 3.
Figure 3. Schematic of occlusal depth using a hemispherical shell.
To model the local package behaviour produced by the process of rolling tyre defor-
mation on pavement raised aggregates, this study proposed the concept of “occlusal
depth”, which refers to the package depth of a tyre on pavement raised aggregate. Within
the occlusal range of the tyre and aggregate, the occlusal depth (H) of a single aggregate
is expressed using the difference between the vertex elevation value of the aggregate (
111
(, )
Z
xy) and the critical point elevation of the aggregate ( 222
(, )
Z
xy), as shown in Figure 3.
It can be calculated as follows:
11 1 2 2 2
(, ) (, )
H
Zxy Zx y=− (1)
where
H
denotes the occlusal depth of a single aggregate, 111
(, )
Z
xy denotes the vertex
elevation in the occlusal range, and 222
(, )
Z
xy denotes the critical point elevation in the
occlusal range. The mean occlusal depth of the simplified pavement can be calculated as
follows:
1
n
i
i
H
Hn
=
=

(2)
where
H
denotes the mean occlusal depth of the simplified pavement, 1
n
i
i
H
=
 denotes the
cumulative occlusal depth of the simplified pavement, and n denotes the number of ag-
gregates in contact with the tyres.
We initially assumed that a greater degree of tyre tread deformation (larger occlusal
depth) would result in beer contact interaction between the tyre and pavement raised
aggregate; this assumption was investigated using the 3D tyre–pavement contact finite
element model presented in Section 4.
3. Methods: Finite Element Modelling
3.1. Tyre Model
3.1.1. Tyre Type and Material Parameters
The tyre type selected for this study was a 175/80 R14 radial tyre (where 175 denotes
a tyre section width of 175 mm, 80 denotes an aspect ratio of 80%, R denotes a radial tyre
type, and 14 denotes a nominal rim diameter of 14 in). A radial tyre is a complex composite
comprising rubber and cord skeleton materials. The rubber material includes the tread
rubber, crown rubber, and carcass rubber, and the cord skeleton material includes two
belt cord layers and one carcass cord layer, as shown in Figure 4.
Figure 3. Schematic of occlusal depth using a hemispherical shell.
To model the local package behaviour produced by the process of rolling tyre deforma-
tion on pavement raised aggregates, this study proposed the concept of “occlusal depth”,
which refers to the package depth of a tyre on pavement raised aggregate. Within the
occlusal range of the tyre and aggregate, the occlusal depth (H) of a single aggregate is ex-
pressed using the difference between the vertex elevation value of the aggregate (
Z1(x1
,
y1)
)
and the critical point elevation of the aggregate (
Z2(x2
,
y2)
), as shown in Figure 3. It can be
calculated as follows:
H=Z1(x1,y1)−Z2(x2,y2)(1)
where
H
denotes the occlusal depth of a single aggregate,
Z1(x1
,
y1)
denotes the vertex
elevation in the occlusal range, and
Z2(x2
,
y2)
denotes the critical point elevation in the
occlusal range. The mean occlusal depth of the simplified pavement can be calculated
as follows:
H=
n
∑
i=1Hi
n(2)
where
H
denotes the mean occlusal depth of the simplified pavement,
n
∑
i=1Hi
denotes
the cumulative occlusal depth of the simplified pavement, and ndenotes the number of
aggregates in contact with the tyres.
We initially assumed that a greater degree of tyre tread deformation (larger occlusal
depth) would result in better contact interaction between the tyre and pavement raised
aggregate; this assumption was investigated using the 3D tyre–pavement contact finite
element model presented in Section 4.
3. Methods: Finite Element Modelling
3.1. Tyre Model
3.1.1. Tyre Type and Material Parameters
The tyre type selected for this study was a 175/80 R14 radial tyre (where 175 denotes
a tyre section width of 175 mm, 80 denotes an aspect ratio of 80%, R denotes a radial tyre
type, and 14 denotes a nominal rim diameter of 14 in). A radial tyre is a complex composite
comprising rubber and cord skeleton materials. The rubber material includes the tread
rubber, crown rubber, and carcass rubber, and the cord skeleton material includes two belt
cord layers and one carcass cord layer, as shown in Figure 4.

Appl. Sci. 2023,13, 12011 5 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 16
Figure 4. Material composition of the tyre.
The neo-Hookean and linear elasticity intrinsic models were used to numerically sim-
ulate the tyre rubber and cord skeleton materials, respectively. The strain energy function
is expressed as follows:
2
10 1
1
1
(3) ( 1)
e
UCI J
D
=−+ −
where
U
denotes strain potential energy, C
10
denotes the shear characteristics of the ma-
terial,
e
J
denotes the elastic volume ratio,
1
I
denotes the deformation of the material,
and D
1
denotes the compression characteristics of the material.
The tyre industry does not make the exact material properties and structural design
of tyres available to the general public. At the same time, the workload and cost of testing
for tyre materials are relatively large. Therefore, the mechanical parameters of these ma-
terials in Yu et al. [25] are used, as shown in Tables 1 and 2.
Table 1. Rubber material parameters for the 175/80 R14 radial tyre.
Material Neo-Hookean Model
C
10
(MPa) D
1
(MPa
−1
) Density (g/cm
3
)
Tread rubber 0.835 0.024 1.12
Crown rubber 0.869 0.04 1.12
Carcass rubber 1.0 0.02 1.15
Table 2. Cord skeleton material parameters for the 175/80 R14 radial tyre.
Material Cross-Sectional
Area (mm
2
)
Spacing
(mm) Angle (°) You n g ’ s M o d u -
lus (10
5
MPa)
Poisson’s
Ratio
Density
(g/cm
3
)
Belt 1 0.212 1.16 70 1.722 0.3 5.9
Belt 2 0.212 1.16 110 1.722 0.3 5.9
Carcass cord 0.421 1.0 0 9.87 0.3 1.5
3.1.2. 2D and 3D Tyre Modelling
This study focused on the contact interaction between the rubber tread material and
the different grain size aggregates when the tyre is running in the longitudinal direction;
ignoring the transverse tyre tread paern had lile effect on the analysis results. Mean-
while, to save computational time, the tyre modelling ignored the transverse tyre tread
paern and retained only the two longitudinal grooves. As the work of tyre modelling is
two-dimensional (2D), the 3D tyre finite element model was obtained by rotating the 2D
Figure 4. Material composition of the tyre.
The neo-Hookean and linear elasticity intrinsic models were used to numerically
simulate the tyre rubber and cord skeleton materials, respectively. The strain energy
function is expressed as follows:
U=C10(I1−3) + 1
D1(Je−1)2
where
U
denotes strain potential energy, C
10
denotes the shear characteristics of the material,
Je
denotes the elastic volume ratio,
I1
denotes the deformation of the material, and D
1
denotes the compression characteristics of the material.
The tyre industry does not make the exact material properties and structural design of
tyres available to the general public. At the same time, the workload and cost of testing for
tyre materials are relatively large. Therefore, the mechanical parameters of these materials
in Yu et al. [25] are used, as shown in Tables 1and 2.
Table 1. Rubber material parameters for the 175/80 R14 radial tyre.
Material Neo-Hookean Model
C10 (MPa) D1(MPa−1)Density (g/cm3)
Tread rubber 0.835 0.024 1.12
Crown rubber 0.869 0.04 1.12
Carcass rubber 1.0 0.02 1.15
Table 2. Cord skeleton material parameters for the 175/80 R14 radial tyre.
Material Cross-Sectional
Area (mm2)
Spacing
(mm) Angle (◦)Young’s Modulus
(105MPa) Poisson’s Ratio Density
(g/cm3)
Belt 1 0.212 1.16 70 1.722 0.3 5.9
Belt 2 0.212 1.16 110 1.722 0.3 5.9
Carcass cord 0.421 1.0 0 9.87 0.3 1.5
3.1.2. 2D and 3D Tyre Modelling
This study focused on the contact interaction between the rubber tread material and
the different grain size aggregates when the tyre is running in the longitudinal direction;
ignoring the transverse tyre tread pattern had little effect on the analysis results. Meanwhile,
to save computational time, the tyre modelling ignored the transverse tyre tread pattern
and retained only the two longitudinal grooves. As the work of tyre modelling is two-
dimensional (2D), the 3D tyre finite element model was obtained by rotating the 2D tyre

Appl. Sci. 2023,13, 12011 6 of 15
model. To reduce the modelling workload, only half of the tyre profile was selected for
modelling. The specific modelling process employed was as follows:
First, we drew a 2D profile of the tyre based on its size and material composition
using AutoCAD and exported the *.iges file format to the HyperMesh 2021 pre-processing
software for meshing. Considering that mesh quality has a major influence on the accuracy
and convergence of the model results, the rubber material was divided into quadrilateral
CGAX4 cells with triangular CGAX3 cells used only at the sharp corners of the material
boundaries, and the cord skeleton material was represented by the SFMGAX1 cell type.
In ABAQUS 2021, the material mechanical parameters were subsequently set based on
the intrinsic tyre model, and a uniform pressure was applied to the inside of the tyre to
simulate its inflation pressure. The resulting 2D tyre finite element model is shown in
Figure 5.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 16
tyre model. To reduce the modelling workload, only half of the tyre profile was selected
for modelling. The specific modelling process employed was as follows:
First, we drew a 2D profile of the tyre based on its size and material composition
using AutoCAD and exported the *.iges file format to the HyperMesh 2021 pre-processing
software for meshing. Considering that mesh quality has a major influence on the accu-
racy and convergence of the model results, the rubber material was divided into quadri-
lateral CGAX4 cells with triangular CGAX3 cells used only at the sharp corners of the
material boundaries, and the cord skeleton material was represented by the SFMGAX1
cell type. In ABAQUS 2021, the material mechanical parameters were subsequently set
based on the intrinsic tyre model, and a uniform pressure was applied to the inside of the
tyre to simulate its inflation pressure. The resulting 2D tyre finite element model is shown
in Figure 5.
Second, the 2D finite element model of the tyre was rotated 360° around its rotation
axis in ABAQUS 2021 to obtain half of the 3D finite element model, as shown in Figure 6.
As the use of a dense tyre mesh can increase model calculation overhead but the use of a
sparse tyre mesh can affect the accuracy of the results, mesh encryption was performed
only in the contact interval between the tyre and pavement, and the mesh sizes of the tyre
and pavement were matched to the greatest extent possible.
Third, half of the 3D finite element model was mirrored to obtain the completed 3D
finite element model of the tyre, as shown in Figure 7. An analytical rigid body was used
at the inner edge of the tyre bead to form the rim.
Figure 5. Two-dimensional finite element tyre model.
Figure 6. Half of the 3D finite element tyre model.
Figure 5. Two-dimensional finite element tyre model.
Second, the 2D finite element model of the tyre was rotated 360
◦
around its rotation
axis in ABAQUS 2021 to obtain half of the 3D finite element model, as shown in Figure 6.
As the use of a dense tyre mesh can increase model calculation overhead but the use of a
sparse tyre mesh can affect the accuracy of the results, mesh encryption was performed
only in the contact interval between the tyre and pavement, and the mesh sizes of the tyre
and pavement were matched to the greatest extent possible.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 16
tyre model. To reduce the modelling workload, only half of the tyre profile was selected
for modelling. The specific modelling process employed was as follows:
First, we drew a 2D profile of the tyre based on its size and material composition
using AutoCAD and exported the *.iges file format to the HyperMesh 2021 pre-processing
software for meshing. Considering that mesh quality has a major influence on the accu-
racy and convergence of the model results, the rubber material was divided into quadri-
lateral CGAX4 cells with triangular CGAX3 cells used only at the sharp corners of the
material boundaries, and the cord skeleton material was represented by the SFMGAX1
cell type. In ABAQUS 2021, the material mechanical parameters were subsequently set
based on the intrinsic tyre model, and a uniform pressure was applied to the inside of the
tyre to simulate its inflation pressure. The resulting 2D tyre finite element model is shown
in Figure 5.
Second, the 2D finite element model of the tyre was rotated 360° around its rotation
axis in ABAQUS 2021 to obtain half of the 3D finite element model, as shown in Figure 6.
As the use of a dense tyre mesh can increase model calculation overhead but the use of a
sparse tyre mesh can affect the accuracy of the results, mesh encryption was performed
only in the contact interval between the tyre and pavement, and the mesh sizes of the tyre
and pavement were matched to the greatest extent possible.
Third, half of the 3D finite element model was mirrored to obtain the completed 3D
finite element model of the tyre, as shown in Figure 7. An analytical rigid body was used
at the inner edge of the tyre bead to form the rim.
Figure 5. Two-dimensional finite element tyre model.
Figure 6. Half of the 3D finite element tyre model.
Figure 6. Half of the 3D finite element tyre model.

Appl. Sci. 2023,13, 12011 7 of 15
Third, half of the 3D finite element model was mirrored to obtain the completed 3D
finite element model of the tyre, as shown in Figure 7. An analytical rigid body was used at
the inner edge of the tyre bead to form the rim.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 16
Figure 7. Complete 3D finite element tyre model.
3.2. Simplified Pavement Model
3.2.1. Simplified Pavement Consisting of a Single Grain Size Aggregate
Coarse aggregates larger than 4.75 mm can have a major impact on pavement skid
resistance [27,28]; consequently, hemispherical shells with diameters of 7, 9, and 13 mm
were used to represent different coarse aggregate grain sizes in three simplified pave-
ments, each comprising a single grain size aggregate. The planar size of this simplified
pavement was 200 × 200 mm, and the 7, 9, and 13 mm hemispherical shells were arranged
in a matrix such that they numbered 28 × 28, 22 × 22, and 15 × 15, respectively. To ensure
the convergence of the model and retain a sufficient pavement height, the heights of the
simplified pavements with aggregate grain sizes of 7, 9, and 13 mm were set to 3, 4, and 5
mm, respectively. The 3D models of the simplified pavements were drawn using the
CATIA V5R20 software, as shown in Figure 8.
(a) (b) (c)
Figure 8. Three-dimensional model of simplified pavement. (a) Aggregate grain size 7 mm. (b) Ag-
gregate grain size 9 mm. (c) Aggregate grain size 13 mm.
The pavement mean texture depth (MTD) was used to characterise the differences
between the macrotextures of the simplified pavements and is expressed as follows:
12
VV
MTD
A
−
= (3)
where 1
Vdenotes the cuboid volume within the height of the pavement bulge, 2
Vde-
notes the volume of the pavement bulge, and
A
denotes the planar area of the pavement.
Figure 7. Complete 3D finite element tyre model.
3.2. Simplified Pavement Model
3.2.1. Simplified Pavement Consisting of a Single Grain Size Aggregate
Coarse aggregates larger than 4.75 mm can have a major impact on pavement skid
resistance [
27
,
28
]; consequently, hemispherical shells with diameters of 7, 9, and 13 mm
were used to represent different coarse aggregate grain sizes in three simplified pavements,
each comprising a single grain size aggregate. The planar size of this simplified pavement
was 200
×
200 mm, and the 7, 9, and 13 mm hemispherical shells were arranged in a
matrix such that they numbered 28
×
28, 22
×
22, and 15
×
15, respectively. To ensure
the convergence of the model and retain a sufficient pavement height, the heights of the
simplified pavements with aggregate grain sizes of 7, 9, and 13 mm were set to 3, 4, and
5 mm, respectively. The 3D models of the simplified pavements were drawn using the
CATIA V5R20 software, as shown in Figure 8.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 16
Figure 7. Complete 3D finite element tyre model.
3.2. Simplified Pavement Model
3.2.1. Simplified Pavement Consisting of a Single Grain Size Aggregate
Coarse aggregates larger than 4.75 mm can have a major impact on pavement skid
resistance [27,28]; consequently, hemispherical shells with diameters of 7, 9, and 13 mm
were used to represent different coarse aggregate grain sizes in three simplified pave-
ments, each comprising a single grain size aggregate. The planar size of this simplified
pavement was 200 × 200 mm, and the 7, 9, and 13 mm hemispherical shells were arranged
in a matrix such that they numbered 28 × 28, 22 × 22, and 15 × 15, respectively. To ensure
the convergence of the model and retain a sufficient pavement height, the heights of the
simplified pavements with aggregate grain sizes of 7, 9, and 13 mm were set to 3, 4, and 5
mm, respectively. The 3D models of the simplified pavements were drawn using the
CATIA V5R20 software, as shown in Figure 8.
(a) (b) (c)
Figure 8. Three-dimensional model of simplified pavement. (a) Aggregate grain size 7 mm. (b) Ag-
gregate grain size 9 mm. (c) Aggregate grain size 13 mm.
The pavement mean texture depth (MTD) was used to characterise the differences
between the macrotextures of the simplified pavements and is expressed as follows:
12
VV
MTD
A
−
= (3)
where 1
Vdenotes the cuboid volume within the height of the pavement bulge, 2
Vde-
notes the volume of the pavement bulge, and
A
denotes the planar area of the pavement.
Figure 8.
Three-dimensional model of simplified pavement. (
a
) Aggregate grain size 7 mm. (
b
) Ag-
gregate grain size 9 mm. (c) Aggregate grain size 13 mm.
The pavement mean texture depth (MTD) was used to characterise the differences
between the macrotextures of the simplified pavements and is expressed as follows:
MTD =V1−V2
A(3)

Appl. Sci. 2023,13, 12011 8 of 15
where
V1
denotes the cuboid volume within the height of the pavement bulge,
V2
denotes
the volume of the pavement bulge, and Adenotes the planar area of the pavement.
The pavement MTD values corresponding to aggregate grain sizes of 7, 9, and 13 mm
were calculated using Equation (3); the results are shown in Table 3.
Table 3. Pavement mean texture depth (MTD) with aggregate grain sizes of 7, 9, and 13 mm.
Aggregate Grain Size V1(mm3)V2(mm3)A(mm2)MTD (mm)
7 mm 120,000 55,417 40,000 1.61
9 mm 160,000 77,040 40,000 2.07
13 mm 200,000 85,412 40,000 2.86
3.2.2. Simplified Pavement Modelling
The primary objective of simplified pavement modelling is to divide the mesh of the
pavement model. The simplified pavement model drawn using CATIA V5R20 was output
in a saturated file format, then imported into HyperMesh 2021 for meshing. The pavement
model was divided into quadrilateral cells to ensure mesh quality, as shown in Figure 9.
Finally, the simplified pavement with sufficient meshing was output in the *.inp file format
to import into ABAQUS 2021.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 16
The pavement MTD values corresponding to aggregate grain sizes of 7, 9, and 13 mm
were calculated using Equation (3); the results are shown in Table 3.
Table 3. Pavement mean texture depth (MTD) with aggregate grain sizes of 7, 9, and 13 mm.
Aggregate Grain Size 1
V (mm3) 2
V (mm3) A (mm2) MTD (mm)
7 mm 120,000 55,417 40,000 1.61
9 mm 160,000 77,040 40,000 2.07
13 mm 200
,
000 85
,
412 40
,
000 2.86
3.2.2. Simplified Pavement Modelling
The primary objective of simplified pavement modelling is to divide the mesh of the
pavement model. The simplified pavement model drawn using CATIA V5R20 was output
in a saturated file format, then imported into HyperMesh 2021 for meshing. The pavement
model was divided into quadrilateral cells to ensure mesh quality, as shown in Figure 9.
Finally, the simplified pavement with sufficient meshing was output in the *.inp file for-
mat to import into ABAQUS 2021.
(a) (b) (c)
Figure 9. Three-dimensional mesh model of simplified pavement. (a) Aggregate grain size 7 mm.
(b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
3.3. Tyre–Pavement Contact Model
3.3.1. Contact and Boundary Seings
The stiffness of the simplified pavement model used in this study was considerably
larger than that of the radial tyre model; consequently, the simplified pavement model
could be regarded as a rigid body. The contact constraint of the tyre–pavement system
was accordingly set to reflect the finite slip contact between rigid and deformed bodies in
ABAQUS 2021. The tyre–pavement contact type was set to face-to-face contact, with the
pavement set as the “master surface” and the tyre tread set as the “slave surface”. For the
contact property seing, the normal property was set as “hard contact”, and the tangential
property was set as the Coulomb friction model.
In ABAQUS 2021, the pavement was set to be stationary by constraints, and the travel
speed of a tyre was simulated by seing the linear and angular velocities at the reference
points of the rim. The tyre load was set in the negative direction of the Z-axis at the rim
reference point. The 3D contact finite element model of the tyre and simplified pavement
is shown in Figure 10.
Figure 9.
Three-dimensional mesh model of simplified pavement. (
a
) Aggregate grain size 7 mm.
(b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
3.3. Tyre–Pavement Contact Model
3.3.1. Contact and Boundary Settings
The stiffness of the simplified pavement model used in this study was considerably
larger than that of the radial tyre model; consequently, the simplified pavement model
could be regarded as a rigid body. The contact constraint of the tyre–pavement system
was accordingly set to reflect the finite slip contact between rigid and deformed bodies in
ABAQUS 2021. The tyre–pavement contact type was set to face-to-face contact, with the
pavement set as the “master surface” and the tyre tread set as the “slave surface”. For the
contact property setting, the normal property was set as “hard contact”, and the tangential
property was set as the Coulomb friction model.
In ABAQUS 2021, the pavement was set to be stationary by constraints, and the travel
speed of a tyre was simulated by setting the linear and angular velocities at the reference
points of the rim. The tyre load was set in the negative direction of the Z-axis at the rim
reference point. The 3D contact finite element model of the tyre and simplified pavement is
shown in Figure 10.

Appl. Sci. 2023,13, 12011 9 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 16
Figure 10. Three-dimensional contact model of the tyre and simplified pavement.
3.3.2. Contact Analysis
This study focused on the steady–state analysis of vehicle drive wheel and simplified
pavement. Steady–state transport was used in ABAQUS 2021 to simulate the steady–state
dynamic interaction between the tyre and rigid pavement surface. For this, the Eulerian
description of rigid-body rotation and Lagrangian description of deformation were used
to convert the steady–state moving contact problem into a purely spatially dependent
simulation that described the rolling of the tyre as a material flow motion through the
mesh.
To conduct a steady–state analysis, the motion state of the tyre can be set to steady–
state rolling, braking, or driving. In ABAQUS 2021, the speed at which the tyre travels can
be simulated in terms of linear and angular velocities, and the driving state of the tyre can
be changed by transforming their combined values [29]; the relationship between the lin-
ear and angular velocities is given by
v
r
ω
=
(4)
where
ω
denotes the angular velocity,
v
denotes the linear velocity, and
r
denotes the
tyre radius of rotation. When the torque of the tyre around the rotation axis is zero, the
tyre is in a stable rolling state; when the torque is non-zero, the tyre is accelerating or
decelerating.
In the steady–state rolling analysis model, the tyre pressure was 260 kPa, the load
was 2600 N, and the driving speed was 60 km/h, corresponding to a linear speed of 16.67
m/s. The angular velocity was initially calculated to be 57 rad/s based on the radius of the
tyre. When simulated in ABAQUS 2021, the angular velocity was appropriately adjusted
such that the tyre torque was close to zero. Eventually, the linear velocity was determined
to be 16.67 m/s and the angular velocity 56.2 rad/s. Thus, when the linear velocity re-
mained constant, tyres with angular velocities less than 56.2 rad/s were decelerating,
whereas tyres with angular velocities greater than 56.2 rad/s were accelerating.
In the steady–state braking analysis model, the tyre pressure was 260 kPa, the load
was 2600 N, the friction coefficient was 0.3, and the driving initial speeds were 30, 60, 90,
and 120 km/h. Using the methods described in the previous paragraph, the linear and
angular speeds of the tyres were determined for the different speeds during steady–state
braking, and simulation analyses were performed using these values.
Figure 10. Three-dimensional contact model of the tyre and simplified pavement.
3.3.2. Contact Analysis
This study focused on the steady–state analysis of vehicle drive wheel and simplified
pavement. Steady–state transport was used in ABAQUS 2021 to simulate the steady–state
dynamic interaction between the tyre and rigid pavement surface. For this, the Eulerian
description of rigid-body rotation and Lagrangian description of deformation were used
to convert the steady–state moving contact problem into a purely spatially dependent
simulation that described the rolling of the tyre as a material flow motion through the mesh.
To conduct a steady–state analysis, the motion state of the tyre can be set to steady–
state rolling, braking, or driving. In ABAQUS 2021, the speed at which the tyre travels can
be simulated in terms of linear and angular velocities, and the driving state of the tyre can
be changed by transforming their combined values [
29
]; the relationship between the linear
and angular velocities is given by
ω=v
r(4)
where
ω
denotes the angular velocity,
v
denotes the linear velocity, and
r
denotes the tyre
radius of rotation. When the torque of the tyre around the rotation axis is zero, the tyre is in
a stable rolling state; when the torque is non-zero, the tyre is accelerating or decelerating.
In the steady–state rolling analysis model, the tyre pressure was 260 kPa, the load was
2600 N, and the driving speed was 60 km/h, corresponding to a linear speed of 16.67 m/s.
The angular velocity was initially calculated to be 57 rad/s based on the radius of the tyre.
When simulated in ABAQUS 2021, the angular velocity was appropriately adjusted such
that the tyre torque was close to zero. Eventually, the linear velocity was determined to be
16.67 m/s and the angular velocity 56.2 rad/s. Thus, when the linear velocity remained
constant, tyres with angular velocities less than 56.2 rad/s were decelerating, whereas tyres
with angular velocities greater than 56.2 rad/s were accelerating.
In the steady–state braking analysis model, the tyre pressure was 260 kPa, the load
was 2600 N, the friction coefficient was 0.3, and the driving initial speeds were 30, 60, 90,
and 120 km/h. Using the methods described in the previous paragraph, the linear and
angular speeds of the tyres were determined for the different speeds during steady–state
braking, and simulation analyses were performed using these values.

Appl. Sci. 2023,13, 12011 10 of 15
4. Results and Discussion
4.1. Steady–State Rolling Analysis
4.1.1. Contact Stress and Contact Area
To analyse the contact stress and contact area of the simplified pavement according to
aggregate grain size, the 3D tyre–pavement contact model was used to simulate the steady–
state rolling of a vehicle at a driving speed of 60 km/h, as shown in Figures 11 and 12.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16
4. Results and Discussion
4.1. Steady–State Rolling Analysis
4.1.1. Contact Stress and Contact Area
To analyse the contact stress and contact area of the simplified pavement according
to aggregate grain size, the 3D tyre–pavement contact model was used to simulate the
steady–state rolling of a vehicle at a driving speed of 60 km/h, as shown in Figures 11 and
12.
(a) (b) (c)
Figure 11. Contact stresses of simplified pavement during steady–state rolling. (a) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
(a) (b) (c)
Figure 12. Contact areas of simplified pavements during steady–state rolling. (a) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
As shown in Figure 11a–c, the general extent of contact stresses between the tire and
each simplified pavement was similar; however, the different aggregate sizes resulted in
different magnitudes of contact stress on each pavement. The peak stresses on the simpli-
fied pavements with aggregate grain sizes of 7, 9, and 13 mm were 6.77, 8.28, and 9.67
MPa, respectively. As shown in Table 3, the MTD values of the simplified pavements with
aggregate grain sizes of 7, 9, and 13 mm were 1.61, 2.07, and 2.86 mm, respectively. These
results show that both the MTD values and peak contact stresses on the simplified pave-
ments were positively correlated with the aggregate grain size.
As shown in Figure 12a–c, the overall shape of the contact area between the tyre and
each simplified pavement was similar; however, the size of the contact area decreased
with the change in the macrotexture of the simplified pavement corresponding to the in-
crease in aggregate grain size. To quantitatively simplify the contact area characterisation
Figure 11.
Contact stresses of simplified pavement during steady–state rolling. (
a
) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 16
4. Results and Discussion
4.1. Steady–State Rolling Analysis
4.1.1. Contact Stress and Contact Area
To analyse the contact stress and contact area of the simplified pavement according
to aggregate grain size, the 3D tyre–pavement contact model was used to simulate the
steady–state rolling of a vehicle at a driving speed of 60 km/h, as shown in Figures 11 and
12.
(a) (b) (c)
Figure 11. Contact stresses of simplified pavement during steady–state rolling. (a) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
(a) (b) (c)
Figure 12. Contact areas of simplified pavements during steady–state rolling. (a) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
As shown in Figure 11a–c, the general extent of contact stresses between the tire and
each simplified pavement was similar; however, the different aggregate sizes resulted in
different magnitudes of contact stress on each pavement. The peak stresses on the simpli-
fied pavements with aggregate grain sizes of 7, 9, and 13 mm were 6.77, 8.28, and 9.67
MPa, respectively. As shown in Table 3, the MTD values of the simplified pavements with
aggregate grain sizes of 7, 9, and 13 mm were 1.61, 2.07, and 2.86 mm, respectively. These
results show that both the MTD values and peak contact stresses on the simplified pave-
ments were positively correlated with the aggregate grain size.
As shown in Figure 12a–c, the overall shape of the contact area between the tyre and
each simplified pavement was similar; however, the size of the contact area decreased
with the change in the macrotexture of the simplified pavement corresponding to the in-
crease in aggregate grain size. To quantitatively simplify the contact area characterisation
Figure 12.
Contact areas of simplified pavements during steady–state rolling. (
a
) Aggregate grain
size 7 mm. (b) Aggregate grain size 9 mm. (c) Aggregate grain size 13 mm.
As shown in Figure 11a–c, the general extent of contact stresses between the tire and
each simplified pavement was similar; however, the different aggregate sizes resulted
in different magnitudes of contact stress on each pavement. The peak stresses on the
simplified pavements with aggregate grain sizes of 7, 9, and 13 mm were 6.77, 8.28, and
9.67 MPa, respectively. As shown in Table 3, the MTD values of the simplified pavements
with aggregate grain sizes of 7, 9, and 13 mm were 1.61, 2.07, and 2.86 mm, respectively.
These results show that both the MTD values and peak contact stresses on the simplified
pavements were positively correlated with the aggregate grain size.
As shown in Figure 12a–c, the overall shape of the contact area between the tyre and
each simplified pavement was similar; however, the size of the contact area decreased with
the change in the macrotexture of the simplified pavement corresponding to the increase
in aggregate grain size. To quantitatively simplify the contact area characterisation of
the pavement according to aggregate grain size, we outputted the contact area of each
simplified pavement in the software post-processing, as shown in Figure 13.

Appl. Sci. 2023,13, 12011 11 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 16
of the pavement according to aggregate grain size, we outpued the contact area of each
simplified pavement in the software post-processing, as shown in Figure 13.
Figure 13. Contact area of each simplified pavement during steady–state rolling.
As shown in Figure 13, the maximum contact areas of simplified pavements with
aggregate grain sizes of 7, 9, and 13 mm were 1292.5, 1137.2, and 744.3 mm2, respectively.
These results demonstrate that the contact area of the simplified pavement decreased as
its aggregate grain size increased, indicating that pavements with large MTD values will
have small contact areas.
4.1.2. Occlusal Depth
Based on the modelled contact area between the tyre and pavement, the nodal coor-
dinates of the vertices and contact critical points of the aggregate were output using the
post-processing module in ABAQUS 2021. The mean and cumulative occlusal depths of
each simplified pavement model were subsequently calculated using Equations (1) and
(2) with the results shown in Figures 14 and 15.
Figure 14. Mean occlusal depth of simplified pavement according to aggregate grain size.
Figure 13. Contact area of each simplified pavement during steady–state rolling.
As shown in Figure 13, the maximum contact areas of simplified pavements with
aggregate grain sizes of 7, 9, and 13 mm were 1292.5, 1137.2, and 744.3 mm
2
, respectively.
These results demonstrate that the contact area of the simplified pavement decreased as its
aggregate grain size increased, indicating that pavements with large MTD values will have
small contact areas.
4.1.2. Occlusal Depth
Based on the modelled contact area between the tyre and pavement, the nodal coor-
dinates of the vertices and contact critical points of the aggregate were output using the
post-processing module in ABAQUS 2021. The mean and cumulative occlusal depths of
each simplified pavement model were subsequently calculated using Equations (1) and (2)
with the results shown in Figures 14 and 15.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 16
of the pavement according to aggregate grain size, we outpued the contact area of each
simplified pavement in the software post-processing, as shown in Figure 13.
Figure 13. Contact area of each simplified pavement during steady–state rolling.
As shown in Figure 13, the maximum contact areas of simplified pavements with
aggregate grain sizes of 7, 9, and 13 mm were 1292.5, 1137.2, and 744.3 mm2, respectively.
These results demonstrate that the contact area of the simplified pavement decreased as
its aggregate grain size increased, indicating that pavements with large MTD values will
have small contact areas.
4.1.2. Occlusal Depth
Based on the modelled contact area between the tyre and pavement, the nodal coor-
dinates of the vertices and contact critical points of the aggregate were output using the
post-processing module in ABAQUS 2021. The mean and cumulative occlusal depths of
each simplified pavement model were subsequently calculated using Equations (1) and
(2) with the results shown in Figures 14 and 15.
Figure 14. Mean occlusal depth of simplified pavement according to aggregate grain size.
Figure 14. Mean occlusal depth of simplified pavement according to aggregate grain size.
As shown in Figures 14 and 15, the mean occlusal depths of the simplified pavements
increased with increasing aggregate grain size, although the magnitude of the cumulative
occlusal depth did not exhibit any correlation with aggregate grain size. As shown in
Table 3, the MTD values of the simplified pavements with aggregate grain sizes of 7, 9, and
13 mm were 1.61, 2.07, and 2.86 mm, respectively. The results indicate that during the late
stages of pavement skid resistance, a larger aggregate grain size will increase the MTD
and mean occlusal depth of the pavement. However, the cumulative occlusal depth of a

Appl. Sci. 2023,13, 12011 12 of 15
pavement with a larger aggregate grain size will be smaller because of the fewer points of
contact between the pavement and tyre. As the cumulative occlusal depth of the pavement
directly reflects the overall deformation of the tyre tread, under tyre rolling conditions,
pavement with a smaller aggregate grain size will increase the overall deformation of the
tyre tread, thereby producing better contact interaction.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 16
Figure 15. Cumulative occlusal depth of simplified pavement according to aggregate grain size.
As shown in Figures 14 and 15, the mean occlusal depths of the simplified pavements
increased with increasing aggregate grain size, although the magnitude of the cumulative
occlusal depth did not exhibit any correlation with aggregate grain size. As shown in Table
3, the MTD values of the simplified pavements with aggregate grain sizes of 7, 9, and 13
mm were 1.61, 2.07, and 2.86 mm, respectively. The results indicate that during the late
stages of pavement skid resistance, a larger aggregate grain size will increase the MTD
and mean occlusal depth of the pavement. However, the cumulative occlusal depth of a
pavement with a larger aggregate grain size will be smaller because of the fewer points of
contact between the pavement and tyre. As the cumulative occlusal depth of the pavement
directly reflects the overall deformation of the tyre tread, under tyre rolling conditions,
pavement with a smaller aggregate grain size will increase the overall deformation of the
tyre tread, thereby producing beer contact interaction.
4.2. Steady–State Braking Analysis
The effect of the contact interaction between the tyre and simplified pavements with
different aggregate grain sizes can be further analysed in terms of the horizontal braking
force. In this study, the steady–state braking of a vehicle travelling at initial speeds of 30,
60, 90, and 120 km/h was simulated using the established 3D tyre–pavement contact finite
element model. After simulation, the horizontal combined force of the contact force and
friction force is output in the post-processing of the software, which is the horizontal brak-
ing force, as shown in Figure 16. The cumulative occlusal depth of each simplified pave-
ment for vehicles traveling at different speeds was calculated using Equations (1) and (2),
as shown in Figure 17. To verify that the cumulative occlusal depth accurately reflected
the contact interaction between the tyre and pavement aggregate, an analysis of the cor-
relation between the cumulative occlusal depth and horizontal braking force of the sim-
plified pavement was conducted, with the results shown in Figure 18.
Figure 15. Cumulative occlusal depth of simplified pavement according to aggregate grain size.
4.2. Steady–State Braking Analysis
The effect of the contact interaction between the tyre and simplified pavements with
different aggregate grain sizes can be further analysed in terms of the horizontal braking
force. In this study, the steady–state braking of a vehicle travelling at initial speeds of 30,
60, 90, and 120 km/h was simulated using the established 3D tyre–pavement contact finite
element model. After simulation, the horizontal combined force of the contact force and
friction force is output in the post-processing of the software, which is the horizontal braking
force, as shown in Figure 16. The cumulative occlusal depth of each simplified pavement
for vehicles traveling at different speeds was calculated using
Equations (1) and (2)
, as
shown in Figure 17. To verify that the cumulative occlusal depth accurately reflected the
contact interaction between the tyre and pavement aggregate, an analysis of the correlation
between the cumulative occlusal depth and horizontal braking force of the simplified
pavement was conducted, with the results shown in Figure 18.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16
Figure 16. Horizontal braking force of each simplified pavement at different vehicle initial speeds.
Figure 17. Cumulative occlusal depth of each simplified pavement at different vehicle initial speeds.
Figure 18. Correlation analysis between horizontal braking force and cumulative occlusal depth.
Figure 16. Horizontal braking force of each simplified pavement at different vehicle initial speeds.

Appl. Sci. 2023,13, 12011 13 of 15
Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16
Figure 16. Horizontal braking force of each simplified pavement at different vehicle initial speeds.
Figure 17. Cumulative occlusal depth of each simplified pavement at different vehicle initial speeds.
Figure 18. Correlation analysis between horizontal braking force and cumulative occlusal depth.
Figure 17.
Cumulative occlusal depth of each simplified pavement at different vehicle initial speeds.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 16
Figure 16. Horizontal braking force of each simplified pavement at different vehicle initial speeds.
Figure 17. Cumulative occlusal depth of each simplified pavement at different vehicle initial speeds.
Figure 18. Correlation analysis between horizontal braking force and cumulative occlusal depth.
Figure 18. Correlation analysis between horizontal braking force and cumulative occlusal depth.
As shown in Figures 16 and 17, the horizontal braking force and cumulative occlusal
depth for each simplified pavement decreased as the vehicle speed increased, with both
the horizontal braking force and cumulative occlusal depth for the simplified pavements
ranked from largest to smallest as aggregate grain sizes 9, 7, and 13 mm at all four speeds.
As shown in Table 3, the MTD values of the simplified pavements with aggregate grain sizes
of 7, 9, and 13 mm were 1.61, 2.07, and 2.86 mm, respectively. Thus, the effect of the contact
interaction between the tyre and simplified pavement aggregates was ranked in order of
superiority as aggregate grain sizes of 9, 7, and 13 mm, indicating that the aggregate grain
size does not exhibit any correlation with tyre–pavement contact interaction in the later
stages of pavement skid resistance. Though the increase in aggregate grain size resulted in
a larger MTD of the simplified pavement, this does not indicate a better contact interaction
effect between the tyre and pavement.
As shown in Figure 18, the squares of linear correlation coefficients between the
horizontal braking force and pavement cumulative occlusal depth were 0.921, 0.941, 0.889,
and 0.894 for vehicle initial speeds of 30, 60, 90, and 120 km/h, respectively, indicating
that using the pavement cumulative occlusal depth to characterise the effect of contact
interaction between the tyre and pavement aggregate is accurate and can be used to evaluate
asphalt pavement skid resistance.

Appl. Sci. 2023,13, 12011 14 of 15
5. Conclusions
This study considered three simplified pavements comprising aggregates of different
grain sizes as the research object to develop a 3D finite element tyre–pavement contact
model in ABAQUS 2021. First, the contact stress and contact area distribution characteristics
of each simplified pavement and the occlusal depth of the tyre–pavement system were
analysed under steady–state rolling conditions. Subsequently, the horizontal braking force
and cumulative occlusal depth of each simplified pavement were analysed at different
vehicle speeds under steady–state braking conditions, and a correlation analysis between
these parameters was conducted. The results of this study can be summarised as follows:
1.
Under steady–state rolling conditions, the overall shape of the contact area between
the tyre and each simplified pavement was similar; however, the peak contact stresses
on the simplified pavement increased with its MTD, whereas pavements with larger
MTD values exhibited smaller contact areas.
2.
Under steady–state rolling conditions, the mean occlusal depth of the simplified
pavement was positively correlated with the aggregate grain size. However, the
cumulative occlusal depth did not exhibit any correlation with the aggregate grain
size and ranked from largest to smallest as aggregate grain sizes 9, 7, and 13 mm.
These results indicate that a simplified pavement with a smaller aggregate grain size
exhibited a larger overall deformation of the tyre tread, implying that the grain size of
the pavement aggregate bulge affects pavement skid resistance during the later stages.
3.
Under steady–state braking conditions, the effect of the contact interaction between
the tyre and simplified pavement aggregates was ranked in order of superiority as
aggregate grain sizes of 9, 7, and 13 mm, indicating that during the late stages of
pavement skid resistance, the two smaller aggregates provided superior performance.
Consequently, to ensure the durability of pavement skid resistance, more consider-
ation should be given to aggregates with grain sizes in the 7–9 mm range during
pavement design.
4.
The squares of linear correlation coefficients between the horizontal braking force
and pavement cumulative occlusal depth were 0.921, 0.941, 0.889, and 0.894 for
vehicle speeds of 30, 60, 90, and 120 km/h, respectively, indicating that the cumulative
occlusal depth can be confidently used for the assessment of pavement skid resistance.
Note that this study was limited to the use of specific tyre and simplified pavement
models. In future research, experimental studies will be conducted on pavements with
actual aggregate compositions using different types of tyres. These limitations notwith-
standing, the conclusions of this study indicate a clear direction for application and can
contribute to the development of skid resistance design methods for asphalt pavements.
Author Contributions:
Conceptualization, Z.L. and W.C.; methodology, Z.L.; software, W.C.; valida-
tion, W.W., Z.L. and Y.L.; formal analysis, W.C.; investigation, Y.L.; resources, W.W.; data curation,
Y.L.; writing—original draft preparation, W.C.; writing—review and editing, Z.L.; visualization, Y.L.;
supervision, W.W.; project administration, W.W.; funding acquisition, Z.L. All authors have read and
agreed to the published version of the manuscript.
Funding: This study received no external funding.
Data Availability Statement:
The data presented in this study are available on request from the
corresponding author. The data are not publicly available due to privacy.
Conflicts of Interest: The authors declare no conflict of interest.
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