Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness
The fatigue damage characteristics of an asphalt mixture were analyzed, a new fatigue damage model was constructed based on the Schapery pseudovariable theory and continuum damage mechanics. According to the Schapery theory and the generalized elastic-viscoelastic correspondence principle, the exper...
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irk-123456789-1739952020-12-29T01:26:38Z Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness Wang, X.R. Yin, B.Y. Luo W.B. Научно-технический раздел The fatigue damage characteristics of an asphalt mixture were analyzed, a new fatigue damage model was constructed based on the Schapery pseudovariable theory and continuum damage mechanics. According to the Schapery theory and the generalized elastic-viscoelastic correspondence principle, the experimental stress–strain curves of the asphalt mixture are transformed into the stress vs. pseudostrain curves. The relation between stress and pseudostrain is shown to be linear, which eliminates the viscoelastic hysteresis effect of the asphalt mixture and its time dependence. Выполнен анализ характеристик усталостного повреждения битумной смеси. Построена новая модель усталостного повреждения на основании псевдопараметрической теории Шапери и механики повреждения сплошных сред. В соответствии с теорией Шапери и обобщенным принципом соответствия между упругими и вязкоупругими деформациями экспериментальные кривые напряжение–деформация для битумной смеси преобразуются в кривые напряжение–псевдодеформация. Показано, что последняя линейна, что устраняет вязкоупругий эффект гистерезиса битумной смеси и его временную зависимость. 2018 Article Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness / X.R. Wang, B.Y. Yin, W.B. Luo // Проблемы прочности. — 2018. — № 5. — С. 89-97. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173995 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Wang, X.R. Yin, B.Y. Luo W.B. Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness Проблемы прочности |
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The fatigue damage characteristics of an asphalt mixture were analyzed, a new fatigue damage model was constructed based on the Schapery pseudovariable theory and continuum damage mechanics. According to the Schapery theory and the generalized elastic-viscoelastic correspondence principle, the experimental stress–strain curves of the asphalt mixture are transformed into the stress vs. pseudostrain curves. The relation between stress and pseudostrain is shown to be linear, which eliminates the viscoelastic hysteresis effect of the asphalt mixture and its time dependence. |
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Article |
| author |
Wang, X.R. Yin, B.Y. Luo W.B. |
| author_facet |
Wang, X.R. Yin, B.Y. Luo W.B. |
| author_sort |
Wang, X.R. |
| title |
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness |
| title_short |
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness |
| title_full |
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness |
| title_fullStr |
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness |
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Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness |
| title_sort |
fatigue damage analysis of an asphalt mixture based on pseudostiffness |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2018 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/173995 |
| citation_txt |
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness / X.R. Wang, B.Y. Yin, W.B. Luo // Проблемы прочности. — 2018. — № 5. — С. 89-97. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT wangxr fatiguedamageanalysisofanasphaltmixturebasedonpseudostiffness AT yinby fatiguedamageanalysisofanasphaltmixturebasedonpseudostiffness AT luowb fatiguedamageanalysisofanasphaltmixturebasedonpseudostiffness |
| first_indexed |
2025-07-15T10:52:08Z |
| last_indexed |
2025-07-15T10:52:08Z |
| _version_ |
1837709895890108416 |
| fulltext |
UDC 539.4
Fatigue Damage Analysis of an Asphalt Mixture Based on Pseudostiffness
X. R. Wang,
a
B. Y. Yin,
a
and W. B. Luo
a,b,1
a College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, China
b Hunan Key Laboratory of Geomechanics and Engineering Safety, Xiangtan University, Xiangtan,
China
1 luowenbo@xtu.edu.cn
The fatigue damage characteristics of an asphalt mixture were analyzed, a new fatigue damage
model was constructed based on the Schapery pseudovariable theory and continuum damage
mechanics. According to the Schapery theory and the generalized elastic-viscoelastic correspondence
principle, the experimental stress–strain curves of the asphalt mixture are transformed into the stress
vs. pseudostrain curves. The relation between stress and pseudostrain is shown to be linear, which
eliminates the viscoelastic hysteresis effect of the asphalt mixture and its time dependence.
Pseudostiffness decreases gradually with the number of fatigue cycles. The damage variable is
defined by the change in pseudostiffness. The parameters of the fatigue damage model are verified by
fatigue tests up to 1000 cycles, and the fatigue damage for 2000 cycles is predicted and compared
with the test results, which are in good agrement between themselves.
Keywords: fatigue damage, pseudostrain, fatigue lifetime, viscoelasticity, pseudostiffness.
Introduction. Asphalt pavement has excellent performance of driving comfort and
noise reduction and has been used widely in road engineering. Under the cyclic action of
vehicle load, the asphalt pavement will have varying degrees of damage such as cracking,
raveling, potholing, and rutting [1]. The fatigue damage of asphalt mixture has been an
important issue for scholars. Lee proposed a fatigue performance prediction model of
asphalt concrete [2], which has been developed from a uniaxial constitutive model based on
the elastic-viscoelastic correspondence principle and continuous damage mechanics by use
of mathematical simplifications. Kim et al. [3] analyzed the fatigue cracking and failure of
inelastic heterogeneous asphalt concrete mixtures with the finite element method and
proposed a calculation model to predict the fatigue damage behavior of asphalt mixture
under cyclic loading. Li et al. [4] used the fracture energy density of asphalt mixture from
indirect tension test to propose a simple fatigue prediction model. Guan et al. [5] introduced
the viscoelastic fatigue damage model of asphalt mixture under cyclic loading on the bases
of the viscoelastic constitutive model and strain equivalent assumption. After studying the
fatigue damage under cyclic loading, Zhu et al. [6] have put forward the damage model of
interaction between fatigue and creep for asphalt mixture by using fatigue-creep coupling
damage theory. In the above-mentioned fatigue damage models, some of these models are
based on the energy dissipation principle. The others are phenomenological models relating
the damage to the fatigue cycles. However, the damage variable is lack of mechanical
meaning.
Developing a realistic mathematical model of the mechanical behavior of asphalt
mixture with accumulation damage is complicated work. The complexity is attributed to the
viscoelastic hereditary effects of the binder, the complex nature of describing the damage,
the plastic and viscous flow of the binder, the friction of aggregate particles and the
coupling between these mechanisms [7], However, how can these complex problems be
simplified and which should be the researchers’ thinking. Based on the damage mechanics,
the essence of fatigue is the accumulation of material damage with the increase of cyclic
loading. Thus, in this paper, the cyclic loading times in the fatigue test were contacted from
© X. R. WANG, B. Y. YIN, W. B. LUO, 2018
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 89
the macro view, and according to the strain response of asphalt mixture under cyclic
loading, the stress vs. pseudostrain curves was obtained. Pseudostiffness has been used to
characterize the damage degree of the asphalt mixture. By use of the pseudostiffness, the
mechanical properties of the asphalt mixture are related to a cyclic loading time to construct
a new kind of intuitive and straightforward damage model.
1. Pseudovariable Theory. The mechanical properties of viscoelastic materials reveal
time or rate dependence. The response of asphalt mixture depends not only on the current
state of input but also on all history of input [8]. If the material has linear viscoelastic, its
constitutive equation can be constructed by the Boltzmann superposition principle as in [9]:
� �
��
��
�ij ijkl
kl
t
E t d� �� ( ) ,
0
(1)
where � ij and �kl are stress and strain, respectively, Eijkl is the stress relaxation
function of the material, and t is physical time.
By using elastic-viscoelastic correspondence principle, the viscoelastic problem can
be transformed into a linear elastic problem. Schapery proposed the extended elastic-
viscoelastic correspondence principle which can apply to both linear and nonlinear
viscoelastic materials [10]. It was suggested that the constitutive equation of viscoelastic
media such as asphalt mixture was identical to that in the elastic case. Stresses and strains
are not necessarily physical quantities in the viscoelastic body. Instead, they are the
pseudovariables. The pseudostrain �kl
R is defined by Eq. (2):
� �
��
��
�kl
R
R
ijkl
t
kl
E
E t d� ��
1
0
( ) , (2)
where ER is a reference modulus. According to Eq. (2), Eq. (1) can be rewritten as
� �ij R kl
RE� , (3)
which revealed the linear relationship between the stress and pseudostrain.
2. Calculation of Pseudostrain Based on Cyclic Loading Test.
2.1. Relaxation Modulus Determined from the Dynamic Mechanical Test. From the
above, we can see that the relationship between stress, strain and time can be obtained.
However, to find the pseudostrain, the constitutive model is needed to be obtained, which
can characterize the nature of the asphalt mixture. The relaxation modulus of the material
can be deduced from the Wiechert model and described in the form of Prony series [11] as
shown below:
E t E E ee n
t
n
N
n( ) ,
/
�
�
�
�
1
(4)
where Ee is the long-time equilibrium modulus, and it is the steady state value of modulus
E t( ) when t approaches infinity, � �n n nE� is the relaxation time of the nth Maxwell
unit, and N is the total number of the Maxwell units.
Based on the Wiechert model, the relaxation modulus coefficient can be determined
from the storage modulus. First of all, using the relationship between dynamic modulus and
phase angle in dynamic modulus experiment [12], we can get the storage modulus. Then
the Fourier transform of the constitutive relation of the Wiechert model is used to transform
X. R. Wang, B. Y. Yin, and W. B. Luo
90 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
the time domain into the frequency domain, and then the storage modulus can be obtained
by elastic-viscoelastic correspondence principle [13], as shown in Eq. (5):
�
�
E E
E
e
n n
nn
N
( ) .�
� �
� �
2 2
2 2
1 1
(5)
The storage modulus at each temperature and its master curve are shown in Fig. 1.
Because the direct solution of relaxation time can produce nonlinear problems [14], it is
hugely inconvenient. It is necessary to solve the corresponding parameters for a given
relaxation time in advance. The relaxation time in this paper is determined according to
Eq. (6) and then the remaining parameters in the storage modulus are determined by
Levenberg–Marquardt + global optimization. The fitting decision coefficient (DC) is 0.998.
The fitting results are shown in Fig. 2, and the corresponding parameters are shown in
Table 1. Furthermore, the respective coefficients of the relaxation modulus expressed by the
Prony series are also known as
�n
n ck� �
�10( ) ( , , , ... , ),n N� 1 2 3 (6)
where k is a constant of 2, and according to the experimental time range and the
requirements of fitting analysis, N � 8 and c� 5.
2.2. Stress and Strain Characterization under Cyclic Loading Experiment. In this
work, the semi-positive vector cyclic loading test under stress control was conducted on the
asphalt mixture specimen with the material testing machine UTM. The stress input �( )t is
shown in Fig. 3. The corresponding strain response curve is given in Fig. 4.
Fatigue Damage Analysis of an Asphalt Mixture ...
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T a b l e 1
The Fitted Parameters in Eq. (5)
n 1 2 3 4 5 6 7 8 Ee DC
�n 10 4� 10 3� 10 2� 10 1� 100 101 102 103
109.3 0.998
En 4328.5 4276.7 4184.5 3610.4 1996.7 878.3 183.8 73.7
Fig. 1 Fig. 2
Fig. 1. Frequency sweep of storage modulus at various temperatures and its master curve.
Fig. 2. Storage modulus master curve and its model fit.
AC-13 type dense-graded asphalt mixture sample is used in this test [15]. The asphalt
used is AH70# ordinary road asphalt. The ratio of oil to stone is 4.6%. The sample is
compacted using a rotary compactor CONTROLS 76-PV2522. The gradation of AC-13
dense-graded asphalt mixture is shown in Table 2.
According to the “highway engineering asphalt and asphalt test procedures” norms
[16], the tests were carried out with UTM. A pre-loading of 0.3 kN was conducted for 1 min
to avoid the impact of mechanical clearance. To weaken the end friction, PTFE films were
put on the contact surfaces. The experiment was divided into three groups, and all tests
were conducted with a loading frequency of 0.1 Hz and a stress amplitude of 40 kPa, the
test temperature was controlled at 25�C. The cyclic loading curve is shown in Fig. 3. The
strain response curve is shown in Fig. 4, only revealing the last five cycles.
The loading stress history applied in the test is given below:
� � � � �( ) ( ) sin( . ) ,t H t ta� �0 0 2 2 (7)
where H t( ) is the Heaviside step function, and � 0 55� kPa, � a � 40 kPa.
According to the Boltzmann superposition principle, the total strain response to the
applied stress history is decomposed into two parts: the static strain response �1 and the
sinusoidal strain response �2:
�
� � � � �
�
1
2 0 2 2
( ) ,
( ) sin( . ),
t Ct
t ta
�
� �
�
�
�
(8)
where C and � are model constants, �a is the strain amplitude corresponding to the
cyclic stress amplitude � a , and � is the phase angle due to viscoelasticity. Thus, the total
strain is given by
� � � � �
�( ) sin( . ).t Ct ta� � 0 2 2 (9)
92 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
X. R. Wang, B. Y. Yin, and W. B. Luo
T a b l e 2
Grading of AC-13 Dense-Graded Asphalt Mixture
Aggregate gradation
sieve sizes (mm)
Percentage passing (%)
16 13.2 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075
Sieve grading 100 96.3 77.3 54.1 37.9 27.6 19.1 14.7 10.4 6.2
Fig. 3. Stress input in cyclic loading. Fig. 4. Strain response and fitting curve.
By combining the Levenberg–Marquardt (LM) method with the global optimization
method, the strain response curves were fitted by Eq. (9). The model parameters are
obtained and shown in Table 3. The model fitting curve is also plotted in Fig. 4, which is in
perfect consistency with the test data in the last five cycles.
2.3. Calculation of Pseudostrain under Cyclic Loading. According to the relationship
between stress and time, the relationship between strain and time, the relationship between
stress and strain can be obtained as shown in Fig. 5. From the stress–strain curve, it can be
seen that the asphalt mixture has apparent hysteretic behavior under cyclic loading, and the
curve gradually moves towards the right side with the increases in cycles.
When the pseudostrain is calculated according to the definition of pseudostrain in
Eq. (2), the stress vs. strain curve is converted to the stress vs. pseudostrain curve, as
shown in Fig. 6. It can be seen from the stress vs. pseudostrain curve that the material’s
hysteretic behavior disappears, moreover, the stress vs. pseudostrain curve presents a
simple linear relationship, which also satisfies the expression of Eq. (3).
The pseudostiffness, S R , is defined by
S
d
d
R
R
�
�
�
. (10)
3. Pseudovariables and Fatigue Damage.
3.1. Pseudostiffness under Cyclic Loading. The above pseudostrain is based on the
theory of viscoelasticity and various ideal conditions and assumptions [17]. The effect of
heavy cyclic loading on the damage of asphalt mixture is not considered, and the damage of
asphalt mixture is the process of microcracks and/or microvoids initiation and expanding
during the cyclic loading. Due to the randomness of these microcracks and microvoids,
their propagation mechanism is very complicated. Therefore, it is difficult to deduce the
fatigue damage law directly through theoretical derivation.
As can be seen from the stress vs. pseudostrain curve in Fig. 6, the pseudostiffness S R
is approximately 1 in magnitude, in such case the damage is negligible during the cyclic
loading because the maximum stress is less than 100 kPa.
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Fatigue Damage Analysis of an Asphalt Mixture ...
T a b l e 3
Strain Response Fitting Parameters
�0 � , rad C � Decision coefficient
63 92 10 6. �
�
�0.25 74 37 10 6. �
� 0.31 0.9979
Fig. 5. Stress–strain curve under cyclic loading. Fig. 6. Stress–pseudostrain curve.
To study the damage evolution in cyclic deformation of asphalt mixtures, the
stress-controlled cyclic tests with a stress amplitude of 400 kPa at a frequency of 0.1 Hz
were conducted up to 2000 cycles. The stress vs. strain curves at 1st, 10th, 100th, 1000th
and 2000th cycles were recorded and converted into stress vs. pseudostrain curves
according to the above method, as shown in Fig. 7. Linear fitting to Eq. (3) with the stress
vs. pseudostrain curves, as shown in Fig. 7, provides the pseudostiffness for different load
cycles, which is listed in Table 4.
3.2. Fatigue Damage Model Based on Pseudorigidity. The damage in the material
should be divided into two parts [18] corresponding to static damage and fatigue damage
respectively. According to phenomenological law, the static damage variable Ds is
established as below:
D D es s
bN
� �
�
( ),1 (11)
where Ds� and b are material parameters depending on the loading stress amplitude,
frequency, and temperature.
The fatigue damage D f can be established according to Miner damage model, as
shown in Eq. (12):
D
N
N
f
f
� , (12)
where N f is the ideal fatigue life and N is the load cycles.
The ideal fatigue life is the cycles up to failure under the cyclic load of the perfect
state. However, the process from fatigue damage to fatigue failure is the process of energy
release. The process is concise and is affected by the experimental conditions, operational
errors and the anisotropy between samples in each group. As a result, the actual fatigue life
of each specimen does not reach the ideal fatigue life. Thus the establishment of the
relationship between the real fatigue life and the perfect fatigue life is shown in Eq. (13):
94 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
X. R. Wang, B. Y. Yin, and W. B. Luo
T a b l e 4
The Pseudostiffness at Various Load Cycles
Load cycle 0 cycle 1st cycle 10th cycle 100th cycle 1000th cycle 2000th cycle
S R 1 0.9817 0.9323 0.8791 0.7927 0.7086
Fig. 7. Stress vs. pseudostrain curves: test and model fit.
N mNf rf� , (13)
where m� 1.1 is the compensation factor and N rf is the actual fatigue life. Thus the total
damage D can be obtained by summing the fatigue damage and the static damage as
below:
D N D N D N D e
N
N
s f s
bN
rf
( ) ( ) ( ) ( )
.
,� � �
�
�1
11
0� �N N rf . (14)
It is evident from Table 4 that the pseudostiffness decreases with the increase in load
cycles. We define the damage variable D by the change of the pseudostiffness as below:
D
S S
S
R R
R
�
�0
0
, (15)
where S R is the pseudostiffness at any load cycle and S R
0 is the initial pseudostiffness. It
is seen from Table 4 that S R
0 � 1. Using Eq. (15) we can calculate the damage degree for
different load cycles, and the calculation results are shown in Fig. 8.
3.3. Model Verification and Fatigue Damage Analysis. By use of the Levenberg–
Marquardt (LM) method and generalized global optimization method, the damage for 1, 10,
100, and 1000 cycles in Fig. 8 were fitted with Eq. (14), and the parameters in Eq. (14) are
determined to be that Ds� � 0.1070, b� 0.0907, and N rf � 9441. Therefore, the damage
evolution equation can be described by Eq. (16):
D N e
NN( ) . ( ) ..
� �
�0 10701 1
10385
0 0907
(16)
It is clear that Eq. (16) satisfies the initial condition, in which case D( )0 0� .
Moreover, the sample will be damaged to fracture when it is loaded by N rf cycles.
According to Eq. (16), D N Drf( ) ( )� �9441 1.01, which meet the failure criterion.
The damage at the 2000th cycle is not used in the model fitting of Eq. (16). To check
the validity of the model, the damage in the 2000th cycle is predicted using Eq. (16),
D( )2000 � 0.2996, while the corresponding observed damage in Fig. 8 is 0.2914. The
prediction is in good agreement with the test, as shown in Fig. 9.
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Fatigue Damage Analysis of an Asphalt Mixture ...
Fig. 8. Observed damage variation with load cycles.
C o n c l u s i o n s
1. Based on the Schapery pseudovariable theory and Wiechert model, the experimental
stress vs. strain curves of asphalt mixture under semi-positive vector cyclic loading are
transformed into stress vs. pseudostrain curves. The stress vs. pseudostrain curve eliminates
the hysteresis effect and load history dependence and presents a simple linear relationship.
2. The slope of the stress vs. pseudostrain curve gradually decreases with the increase
of fatigue cycles. The damage variable is defined by the change of the pseudostiffness.
Meanwhile, a damage evolution equation is constructed by considering the superposition of
the static damage and the fatigue damage. The prediction by the proposed damage model is
in good agreement with the test data.
Acknowledgments. This work was supported by the National Natural Science
Foundation of China (11572275), the Key Laboratory of Road Structure and Material of
Ministry of Transportation (Beijing) and the Hunan Provincial Innovation Foundation for
Postgraduate, China (CX2018B074).
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Received 05. 03. 2018
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 97
Fatigue Damage Analysis of an Asphalt Mixture ...
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