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Rutting deformations analysis for asphalt pavement base on the
visco-elastoplastic theory
Ping Yang a,b
Department of Civil Engineering, Hunan Technical College of Engineering, Changsha, Hunan, 410151, China
Keywords: Visco-elastpolastic, Asphalt pavement, Rutting deformations; Dynamic load,ANSYS
Abstract. Base on visco-elastoplastic theory, using Laplace transformation the asphalt pavement
parameters of viscoelasticity model and visco-elastoplastic model are converted into parameters of
prony series, the rutting deformations of asphalt pavement are analyzed by using ANSYS, the
influence of load, material temperature and loading cycles are considered. The research shows that
with the increase of loading cycles, the rutting deformations of visco-elastoplastic model gradually
large than the results of viscoelastic model, both the load and temperature have a influence on the
rutting deformations of asphalt pavement.
Introduction
With the rapid development of China's economy, the road traffic increase and overload phenomenon
is common, the rutting deformations phenomenon of asphalt pavement is increasingly prominent, and
become the main disease in asphalt pavement. At present, rutting deformations of asphalt pavement
have not taken into account in the design specification of our country, and still have not established a
mature method to analyze and forecast the rutting deformations of asphalt pavement. At this stage the
viscoelastic model is wildly used for analyzing the rutting deformations of asphalt pavement [1-3].
Most of the studies show that the rutting deformations of asphalt mixture obviously contain
visco-elastpolastic properties [4,5]. Base on the creep experiments, the asphalt mixture are
categorized into three groups as elasticity, viscoelasticity and viscoplasticity [6], and the
visco-elastpolastic model is established, which can describe the mechanical properties of asphalt
mixture. Asphalt mixtures are described as visco-elastoplastic materials for which the responses of
the mixtures to the loads [7], and they are categorized into three groups as elasticity, viscoelasticity
and viscoplasticity, depending on whether they are time dependent and recoverable during the
unloading period. The deformations of asphalt pavement are analyzed by using Perzyna’s theory and
finite element method. Base on the visco-elastoplastic theory, Liao[8] use the finite element method
to analyze the deformations of asphalt pavement with semi-rigid base course. In this paper, base on
the visco-elastoplastic constitutive relationship of asphalt mixture, the rutting deformations of asphalt
pavement structure under cyclic loading are analyzed by using ANSYS, and the research results will
provide theoretical basis for the design of asphalt pavement.
Mechanical models of asphalt mixture
Research shows that: asphalt mixture can be categorized into three groups as elasticity, viscoelasticity
and viscoplasticity in the condition of high temperature, and it is easier to produce irreversible
deformation. At present, a lot of models can be used to describe the mechanics characteristic of
viscoelastic asphalt mixture, such as Maxwell model, Kelvin model and Burgers model et al..
Burgers model is widely used to analyze rutting deformations of asphalt pavement, and the Burgers
model shows in Fig. 1.
Advanced Materials Research Vols. 838-841 (2014) pp 1227-1233Online available since 2013/Nov/08 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.838-841.1227
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 141.117.125.1, Ryerson University Lib, Toronto-02/12/14,21:02:32)
The constitutive equation of Burgers model
2 1 2 1 2 1 2
2
2 1 1 2 1
+E E E E E
η η η η η η ησ σ σ η ε ε + ′ ′′ ′ ′+ + = +
(1)
The creep equation 1
1
0
2 2 1
1 1 1( ) (1 )
Et
t t eE E
ηε ση
− = + + −
(2)
Fig.1. Bursers model
Fig.2. Visco-elastoplastic model
Study shows that the rutting deformations of asphalt mixture obviously contain visco-elastpolastic
properties. In order to accurately analyze the rutting deformations of asphalt mixture, a new
visco-elastpolastic model is given base on Burgers model as shown Fig. 2.
The mechanics property of visco-elastoplastic model can be expressed as
K M B=ε ε ε ε+ +� � �
(3)
And the viscoplastic strain is defined by[9]
B p
B B
n nB B
A
σ σεη ε
= =�
(4)
where B
ε is viscoplastic strain; B
ε� denotes derivative of viscoplastic strain with respect to time t; σ
denotes the stress; A，B，p and n are materials constants. Derivative Eq. (3) and ignore the constant
parts, the creep equation can be written as 1
11 1
1 1 1
0
1 2 1
1 1( ) ( ) [ ( 1)]
Et
n
p p pt e Bt p t
E E A
η
ε σ ση
−
+ + +−= + + + +
(5)
The parameters transformation
(1) The transformation method of viscoelastic parameters to prony series parameters
The Burgers model parameters can not used by ANSYS directly, it is necessary to convert them into
prony series and the method is given by
The shear modulus can be written as
1
12(1 )
EG
µ=
+ , 2
22(1 )
EG
µ=
+
(6)
Then the distorted constitutive equation of Burgers model can be defined by
ij 1 ij 2 ij 1 ij 2 ijs p s p s q e q e+ + = +� �� � ��
(7)
where 1 1 2
1
1 2
pG G
η η η+= + ；
1 2
2
1 2
pG G
η η= ； 1 12q η= ；
1 2
2
2
2q
G
η η=
1228 Civil, Structural and Environmental Engineering
Using Laplace transformation to Eq. (7), we have 2 2
1 2 1 2(1 ) ( ) ( ) ( )p s p s s q s q s sσ ε+ + = +
(8)
Postulating the 0 ( )H tε ε= ,and substitute 0( ) /s sε ε= into Eq. (8) ,we obtain the relaxation
modulus 2
1 2 0
2
1 2
( )
(1 )
q s q s
p s p s s
εσ +=
+ + (9)
where 0( )Y sσ ε=
Then we have 2
1 2
2
1 2
( )(1 )
q s q sY s
p s p s s
+=
+ + (10)
Using inverse Laplace transform to Eq. (10), we obtain the relaxation modulus
1 2 2
2 2
2( ) [( ) ( ) ]t tG G G
Y t e eβ αβ α
α β η η− −= − − −
− (11)
where
2
1 1 2
2
4=
2
p p p
pα β
± −、
The shear modulus can be written as[10]
1 21 2 2
0 1 2
2 2
( ) 0.5 ( ) [( ) ( ) ] ( )
t t
t tG G GG t Y t e e G G g e g e
τ τβ αβ αα β η η
− −− −
∞= = − − − = + +−
(12)
where 0G∞ = ， 0 1G G= ，2
1
2
1( )G
g βα β η
= −−
，2
2
2
1( )G
g αα β η
= − −−
， 1
1τβ
= ， 2
1τα
=
Then we can get the prony series parameters of Burgers model which are inputted ANSYS
( 1 2 1 2, , ,g g τ τ ).
(2) The transformation method of visco-elastoplastic parameters to prony series parameters
According to Eq. (5), E1, E2, η1, η2, A, B, p and n are the parameters of visco-elastoplastic model.
The parameters can not be used by ANSYS directly, them have to be converted into the prony series
parameters. The transfornation method is similar to the upper part.
The shear modulus can be written as
1
12(1 )
EG
µ=
+ 2
22(1 )
EG
µ=
+ (13)
Then the distorted constitutive equation of visco-elastoplastic model can be defined by
0 ij 1 ij 2 ij 1 ij 2 ijijp s s p s p s q e q e′′ + + + = +� �� � �� (14)
where 1
0
B
Bp
ηη
= ；1 1 2
1
1 2
pG G
η η η+= + ；
1 2
2
1 2
pG G
η η= ； 1 12q η= ；
1 2
2
2
2q
G
η η=
Replaced 0 ijp s′′ by ( 1)
0
n
ijp sσ − and Eq. (14) gives
0 ij 1 ij 2 ij 1 ij 2 ijijp s s p s p s q e q e′′ + + + = +� �� � �� (15)
Using Laplace transformation to Eq. (15), we have 2 2
1 2 1 2(1 ) ( ) ( ) ( )p s p s s q s q s sσ ε+ + = + (16)
Postulating the 0 ( )H tε ε= ,and Substitute 0( ) /s sε ε= into Eq. (16) ,we obtain the relaxation
modulus
Advanced Materials Research Vols. 838-841 1229
2
1 2 0
2
1 2
( )
(1 )
q s q s
p s p s s
εσ +=
+ + (17)
where 0( )Y sσ ε=
Then we have 2
1 2
2
1 2
( )(1 )
q s q sY s
p s p s s
+=
+ + (18)
Using inverse Laplace transform to Eq. (18),we obtain the relaxation modulus
1 2 2
2 2
2( ) [( ) ( ) ]t tG G G
Y t e eβ αβ α
α β η η− −= − − −
− (19)
where
2
1 1 2
2
4=
2
p p p
pα β
± −、
The shear modulus can be written as[10]
1 21 2 2
0 1 2
2 2
( ) 0.5 ( ) [( ) ( ) ] ( )
t t
t tG G GG t Y t e e G G g e g e
τ τβ αβ αα β η η
− −− −
∞= = − − − = + +−
(20)
where 0G∞ = ， 0 1G G= ，
2
1
2
1( )G
g βα β η
= −−
，
2
2
2
1( )G
g αα β η
= − −−
， 1
1τβ
= ， 2
1τα
=
Then we can get the prony series parameters of visco-elastoplastic model which are inputted
ANSYS ( 1 2 1 2, , ,g g τ τ ).
Choice of parameters for computations
The structure and materials of asphalt pavement shows in Fig. 3. In this paper the model is analyzed
by using two-dimensional finite element model, its size is 6m×5.73m. The AC-13C, AC-20C and
AC-25C are used element PLANE182, and other are used element PLANE42. The bottom of finite
element model is fixed all constraints, the horizontal displacement of left and right edges are fixed.
The dynamic vehicle load is 0.7 MPa, the loading time is 0.1 s and break time is 0.9s. The finite
element model is shown in Fig. 4, and the properties of asphalt pavement are shown in Table 1.
Fig.3. Structure of asphalt pavement
1
X
Y
Z
JUN 15 2013
08:23:53
ELEMENTS
Fig.4. Finite element model of asphalt pavement
Table 1 Asphalt pavement properties
Materials
Elastic modulus (Mpa) Poisson's ratio Friction angle
(°)
Cohesion force
(kPa)
Density
(kg/m3) Elasto-
plastic
Visco-
elastoplastic
Elasto-
plastic
Visco-
elastoplastic
AC-13C 1900 1900 0.25 0.45 ─ ─ 2600
AC-20C 2400 2400 0.25 0.45 ─ ─ 2500
1230 Civil, Structural and Environmental Engineering
AC-25C 3000 3000 0.25 0.45 ─ ─ 2500
CSM 1500 1500 0.25 0.45 ─ ─ 2400
LFSG 1400 1400 0.25 0.45 ─ ─ 2000
LFS 550 550 0.35 0.45 22 55 1930
Soil 50 48 0.4 0.45 16 30 1900
In this paper, the rutting deformations analysis base on the asphalt pavement creep experimental
data of Hebei road [11]. Table 2 shows the parameters of asphalt mixture at different temperature
which the oil ratio is 3.60%.
Table 2 Parameters of asphalt pavement for computation
Materials
AC-13C AC-20C AC-25C
Temperature (oC)
35 50 60 35 50 60 35 50 60
E1 3000 1200 1000 2200 1450 800 2050 1200 600
E2 583 542 402 461 707 511 1257 790 507
η1 1830425 1240781 1384922 1806501 1333541 1412325 1998943 1275954 1599713
η2 108865 47810 29532 99462 29345 9761 7497 14561 8345
Rutting deformations analysis for two models
Substituting the material parameters of Tables 1 and 2 into Eq. (6)-(20), we obtain the prony series
parameters of Burgers model and visco-elastoplastic model, the results shows in Table 3.
Table 3 Prony series parameters at various temperature
Materials
Temperature
(oC)
AC-13C AC-20C AC-25C
elastoplastic visco-elastoplastic elastoplastic visco-elastoplastic elastoplastic visco-elastoplastic
g1
35 0.149807 0.149807 0.160770 0.160770 0.379008 0.379008
50 0.299931 0.299931 0.321301 0.321301 0.393697 0.393697
60 0.280573 0.280573 0.387775 0.387775 0.456591 0.456591
g2
35 0.850193 0.850193 0.839230 0.839230 0.620992 0.620992
50 0.700069 0.700069 0.678699 0.678699 0.606303 0.606303
60 0.719427 0.719427 0.612225 0.612225 0.543409 0.543409
τ1
35 9768.5 11331.4 12298.8 14266.7 6422.6 7450.2
50 8461.3 9815.1 7084.8 8218.4 6723.9 7799.8
60 12206.6 14159.6 11352.3 13168.7 14575.9 16908.1
τ2
35 72.8962 84.5595 90.0301 104.435 5.6594 6.5649
50 67.3717 78.1512 33.6749 39.0629 18.2169 21.1316
60 52.0929 60.4278 18.5658 21.5363 18.8171 21.8278
Analysis results of two models
(1) Fig. 5 shows the rutting deformations curve for the two models at temperature 60oC and the load
is 0.7MPa.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
6.0E63.0E61.0E61.0E51.0E41.0E31.0E21.0E1
loading cycle N
defo
rmation (
mm
)
Burgers model
visco-elastoplastic model
Fig.5. Comparison of rutting deformations curve for two models
Advanced Materials Research Vols. 838-841 1231
The results show that the change rule of rutting deformations of two models are similar when they
are applying the same load and loading cycles, and due to considering the plastic deformation, the
results base on visco-elastoplastic model gradually large than Burgers model, while the loading cycles
>105. The reasons is that the viscoplastic element start to work as system is in plastic phase, and lead
to the rutting deformations become large as the increase of loading cycles.
(2) The largest results of rutting defomations of two models at load 0.7MPa and various
temperatures is shown in Table 4. The results show that the change rule of two models are the same:
the rutting deformations are gradually increasing with the increase of temperature. The results of
visco-elastoplastic model are larger than the values of Burgers model.
Table 4 Comparison of rutting deformations of two models at various temperature Temperature (oC) Burgers model (mm) visco-elastoplastic model (mm)
35 1.20368 2.2847301
50 2.41578 3.2413781
60 2.82185 3.6741589
(3) Table 5 shows the largest results of rutting deformations of two models at temperature 60℃
and at various loads. The results show that the change rule of two models are the same: the rutting
deformations are gradually increasing with the increase of load. The results of visco-elastoplastic
model are larger than the values of Burgers model.
Table 5 Comparison of rutting deformations of two models at various load Load (MPa) 0.7 0.8 0.9 1.0 1.1
Burgers model (mm) 2.82185 3.81943 4.88372 5.98824 6.85846
visco-elastoplastic model (mm) 3.67416 4.81835 5.96401 7.10217 8.25087
Conclusions
In the paper, two different models of asphalt pavement are analyzed by ANSYS. Through a FEM
analysis and comparison with the results of two different models, the following conclusions can be
made.
1) The results show that the change rule of two models are the same when they are applying the
same load, and the results of visco-elastoplastic model are larger than the values of Burgers
model.The reasons is that viscoplastic element start to work as system is in plastic phase, it leads to
the rutting deformations become large as the increase of loading cycles.
2) The change rules of two models are slimily, when analyze the influence of temperature and load.
With the increase of temperature and load, the rutting deformations are gradually increasing, and
results of visco-elastoplastic model are larger than the values of Burgers model.
According to the study above, when analyze the rutting deformations of asphalt pavement under
vehicle load, the effect of temperature and load should not been ignored, and with the increase of
loading cycles the visco-elastoplastic gradually reflected. Therefore, the visco-elastoplastic property
of asphalt pavement should been taken into account when analyze the rutting deformations of asphalt
pavement.
Acknowledgements
This work was financially supported by the Colleges and Universities in Hunan Province Science and
Technology Research Projects (10C0106).
1232 Civil, Structural and Environmental Engineering
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Advanced Materials Research Vols. 838-841 1233
Civil, Structural and Environmental Engineering 10.4028/www.scientific.net/AMR.838-841 Rutting Deformations Analysis for Asphalt Pavement Base on the Visco-Elastoplastic Theory 10.4028/www.scientific.net/AMR.838-841.1227
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