A. M. Bahadori,1 A. Mansourkhaki,2 and M. Ameri3

A Phenomenological Fatigue PerformanceModel of Asphalt Mixtures Based on FractureEnergy Density

Reference

Bahadori, A. M., Mansourkhaki, A., and Ameri, M., “A Phenomenological Fatigue Performance Model of

Asphalt Mixtures Based on Fracture Energy Density,” Journal of Testing and Evaluation, Vol. 43, No. 1,

2015, pp. 133–139, doi:10.1520/JTE20130057. ISSN 0090-3973

ABSTRACT

In this study the fatigue performance of asphalt mixtures was interpreted based on fracture

energy density. Recent studies indicate that the fracture energy from indirect tension tests

correlates with the field performance of asphalt concrete. The fracture energy density was

obtained from indirect tensile strength tests, and the fatigue performance of asphalt

mixtures was evaluated with four-point bending beam fatigue tests implemented at three

different strain levels. Two different asphalt mixtures with varying binder contents were

tested during this study. Test results showed that the fracture energy density could be an

appropriate material property in phenomenological fatigue models. Thus a

phenomenological fatigue model based on fracture energy density is presented, and this

approach could be advantageous because a simple fatigue model based on fracture energy

density does not require time-consuming fatigue tests. For comparison purposes, the

various types of fatigue models were evaluated, and fatigue models based on fracture

energy density and dissipated energy showed rather high prediction accuracy. In general,

fatigue models based on the energy concept have the least dependence on material

properties and can predict the fatigue lives of asphalt mixtures without changing

coefficients.

Keywords

asphalt mixtures, fracture energy density, indirect tensile strength test, fatigue models

Manuscript received March 8, 2013;

accepted for publication January 10,

2014; published online June 18, 2014.

1 M.Sc. Student, Department of Civil

Engineering, Iran University of Science

and Technology, Tehran 16846-13114,

Iran, (Corresponding author) e-mail:

amir.mohamad.bahadori@gmail.com

2 Associate Professor, Department of Civil

Engineering, Iran University of Science

and Technology, Tehran 16846-13114,

Iran, e-mail: mansourkhaki@iust.ac.ir

3 Professor, Department of Civil

Engineering, Iran University of Science

and Technology, Tehran 16846-13114,

Iran, e-mail: ameri@iust.ac.ir

Copyright VC 2014 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. 133

Journal of Testing and Evaluation

doi:10.1520/JTE20130057 / Vol. 43 / No. 1 / January 2015 / available online at www.astm.org

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Introduction

Flexible pavements usually experience three major distresses: fa-

tigue cracking, low-temperature cracking, and rutting. Fatigue

cracking is caused by repeated loading and can result in signifi-

cant reductions in the serviceability of flexible pavements [1].

The prediction of asphalt pavement performance is an im-

portant topic in pavement design and rehabilitation. There are

various approaches that can be used to predict the fatigue per-

formance of asphalt mixes. Most of these approaches investigate

the crack initiation threshold in asphalt mixtures [2]. Tradi-

tional fatigue models are based on the correlation between fa-

tigue life and strain level at the bottom of the asphalt layer. In

these types of models coefficients are strongly related to the ma-

terial property, test type, and loading conditions. The

mechanistic-empirical pavement design guide (MEPDG) fatigue

model is based on traditional fatigue models. The following

equation shows the MEPDG fatigue model [3]:

Nf ¼ Ck11et

� �k2 1E

� �k3

(1)

k1 ¼1

0:000398þ 0:0036021þ e11:02�3:49hac

(2)

C ¼ 10M(3)

M ¼ 4:84Vb

Va þ Vb� 0:69

� �(4)

where:

Nf ¼ number of repetitions to fatigue failure,

et ¼ tensile strain at critical location,

E¼modulus of material,

k1; k2; k3¼ regression coefficients,

C¼ lab-to-field shift adjustment factor,

hac¼ thickness of asphalt layer, and

Va;Vb¼ in situ air void and effective binder content.

This model is based on an Asphalt Institute fatigue model.

Recent studies showed that a dissipated energy approach

could be a reliable alternative to traditional fatigue models. The

latest dissipated energy fatigue model was presented by Ghuzlan

and Carpenter [4]. They indicated that the ratio of dissipated

energy change (RDEC) between cycles in a fatigue test provides

a better indication of damage being done to the asphalt concrete

from one cycle to another. Furthermore, dissipated energy

change is a function of damage growth during sequential cycles.

The graph of RDEC versus the number of cycles to fatigue con-

sists of three stages; in the first stage RDEC decreases rapidly,

whereas in the second stage of the RDEC–Nf graph RDEC is

constant for a long period of time. The plateau value that is

defined as a constant value of RDEC in the second stage has a

unique relation with the number of cycles to fatigue failure. The

plateau value is a comprehensive material property, and

investigations show that it could be a reliable parameter for

studying the fatigue performance of asphalt mixes [5].

Zhang introduced the threshold concept. In this concept

for cyclic loading, the dissipated creep strain energy is assumed

to be the threshold between macro- and micro-cracks. Further

investigations have shown that the dissipated creep strain

energy is one of the key parameters controlling top-down crack-

ing in asphalt pavements [6].

The fracture energy is equal to the area under the load-

deflection curve up to the maximum failure load. Kim and Wen

[7] indicated that fracture energy from indirect tension tests

correlates with the field performance of asphalt concrete. The

fracture energy of an asphalt mix is the sum of the strain energy

(elastic energy) and dissipated energy due to structural change,

and the resistance of asphalt concrete to fatigue cracking is con-

trolled by the resistance to deformation and damage. Therefore,

fracture energy could be a reasonable material property for ana-

lyzing the fatigue performance of asphalt mixes [7].

As mentioned above, traditional fatigue models strongly

relate to the material type and the test condition. Furthermore,

models with a dynamic modulus in their structure need creep

compliance tests that require complex laboratory facilities. In

order for the fatigue response of an asphalt concrete to be uti-

lized in a design procedure, a simple fatigue model must be pre-

sented. Recent studies indicate that the fracture energy of

asphalt mixes could be a good material property for use in the

prediction of asphalt fatigue performance.

During this study, two types of asphalt mixes with two dif-

ferent gradations and three asphalt contents (low, optimum,

and high) were tested. Fracture energies of the asphalt mixes

were obtained from indirect tension (IDT) tests, and the fatigue

responses of the mixes were obtained through four-point bend-

ing beam fatigue tests.

Experimental

The aggregates selected for this study were crushed limestone.

Table 1 presents the engineering properties of the aggregates

used. Two different gradations with nominal maximum aggre-

gate sizes of 9.5 and 12.5mm were used in this study. Figures 1

and 2 show the gradations of designated aggregates.

One type of binder (PG 64-22) chosen in this study was

from Tehran Pasargade Oil Company. Its properties are pre-

sented in Table 2.

TABLE 1 Engineering properties of aggregate sources.

Aggregate type Result Test Method

Bulk specific gravity 2.493 ASTM C127 [15]

Absorption coarse aggregate, % 2.2 ASTM C127 [15]

Absorption fine aggregate, % 4.2 ASTM C128 [16]

Los Angeles abrasion loss, % 22.3 AASHTO T96 [17]

Two fractured faces, % 94 ASTM D5821 [18]

Journal of Testing and Evaluation134

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Sample Preparation

Samples were prepared with three different asphalt contents

(optimum, low, and high). The optimum asphalt content was

determined using the Marshall mix design method [8]. The

asphalt content was then varied by �0.7 % (low) and þ0.07 %

(high) from the optimum asphalt content for each of the grada-

tions developed for testing. Table 3 shows the asphalt contents

of the mixes.

To produce the fatigue test specimens, the asphalt mixtures

were prepared and compacted in the laboratory using a rolling

wheel compactor. Two different gradations were combined with

three different asphalt contents (low, optimum, and high). The

combinations and conditioning details were based on AASHTO

PP3 and SHRP A-404 [9,10]. The compaction was done by a

steel roller, and samples were made with 56 0.5 % air voids. To

achieve the desired air void content, the amount of mix that

was placed in the mold was varied for each combination. In

other words, different gradations and asphalt contents cause dif-

ferent compaction energies per weight. After the compaction

process, asphalt concrete slabs were cut into dimensions of

63mm by 50mm by 380mm according to AASHTO T321 [11].

Three replicates were tested for each type of mixture.

Samples for indirect tensile strength tests were compacted

using a gyratory compactor with the same specifications

(gradation and asphalt content) as mentioned above. Specimens

101.6mm in diameter and 50.8mm in height were used in this

test, and three replicates were fabricated for each combination.

Testing Program

INDIRECT TENSILE STRENGTH TEST

The tensile property of asphalt concrete is evaluated through

tensile strength tests. IDT tests following ASTM D6931-12 were

performed at a constant rate of 50.8mm/min and a temperature

of 20�C [12].

FRACTURE ENERGY DENSITY

To determine the fracture energy density from indirect tensile

strength tests, the fracture energy, defined as the area under the

load-deflection curve up to maximum failure load, is divided by

the volume of mixture [13]. Figure 3 represents the fracture

energy curve, and the fracture energy can be calculated accord-

ing to the following equation:

FE ¼Ð dmax

0 P dð ÞddV

(5)

where:

FE¼ fracture energy density, MPa,

P¼ applied load, N,

V¼ volume of asphalt mixture, mm3, and

d¼ deformation.

FOUR-POINT BEAM FATIGUE TEST

A beam fatigue test according to AASHTO T321-07 [11] was

utilized to evaluate the fatigue life of asphalt mixtures. The test

was performed at three strain levels, 600 le, 800le, and

FIG. 1 Aggregate gradation for type A.

FIG. 2 Aggregate gradation for type B.

TABLE 2 Asphalt binder properties.

Property Bitumen 60/70 Test Method

Specific gravity at 25�C 1.03 ASTM D70 [19]

Penetration at 25�C 64 ASTM D5 [20]

Softening point, �C 52 ASTM D36 [21]

Ductility at 25�C 102 ASTM D113 [22]

Flash point, �C 305 ASTM D92 [23]

Fire point, �C 317 ASTM D70 [19]

TABLE 3 Asphalt content of samples.

Gradation Asphalt Content

A Low 4

Optimum 4.7

High 5.4

B Low 4.5

Optimum 5.2

High 5.9

BAHADORI ET AL. ON FATIGUE PERFORMANCE 135

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1000 le, at a temperature of 20�C. A repeated sinusoidal load

was applied at a frequency of 10 Hz without rest periods. Three

specimens were tested for each type. The test setup is shown in

Fig. 4.

Results and Discussion

The average values of fracture energy density for three speci-

mens are shown in Fig. 5. It can be seen that the fracture energy

of samples with nominal maximum aggregate size (NMAS) of

9.5mm was more than the fracture energy density of samples

with 12.5mm NMAS. Fracture energy is the sum of elastic

energy and dissipated energy; an increase in asphalt content

enhances the elastic energy, and aggregates with small size in

combination with binder increase the amount of dissipated

energy during the fracture process. Table 4 shows the results of

two-way analysis of variance (ANOVA) for fracture energy den-

sities. The results of two-way ANOVA indicated that different

asphalt contents, different gradations, and their interaction have

a significant effect on the fracture energy density.

FATIGUE TEST RESULTS

In this study, in order to determine the fatigue behavior of the

asphalt samples, a four-point beam fatigue test from IPC Global

was employed. Figures 6, 7, and 8 indicate the fatigue lives of

asphalt mixes under strain levels of 600le, 800 le, and 1000 le.

Mixes with low asphalt contents had the lowest fatigue life, and

an increase in asphalt content caused an increase in fatigue life.

This indicates that the strain-controlled fatigue test is not an

appropriate test for achieving the optimum asphalt content at

which the fatigue life value is maximized. Furthermore, samples

with aggregate size B had higher fatigue life values than samples

prepared with aggregate size A. Table 5 presents phenomenolog-

ical fatigue models for six different mix types. It can be seen

that a phenomenological fatigue model’s coefficients are very

sensitive to the material type, and it is impossible to select a

unique model for different types of mixes.

DISSIPATED ENERGY APPROACH

A ratio of dissipated energy change was developed to describe

an asphalt mixture’s fatigue damage [4]. This ratio can be repre-

sented as

FIG. 4 Four-point beam fatigue apparatus.

FIG. 5 Fracture energy density.

TABLE 4 Results of two-way ANOVA for fracture energy density.

Source of Variation Sum of Squares df Mean Square F Sig

Binder 2.200E-5 2 1.100E-5 43.964 0.000

Gradation 8.352E-6 1 8.352E-6 33.375 0.000

Interaction 2.377E-6 2 1.189E-6 4.750 0.03

Error 3.003E-6 12 2.502E-7 — —

Total 0.017 18 — — —

FIG. 3 Fracture energy from IDT strength test.

TABLE 5 Phenomenological fatigue models.

GradationBinderContent

PhenomenologicalFatigue Model R2

A Low Nf ¼ 1:218� 1014 1=eð Þ3:435 0.947

A Optimum Nf ¼ 5:304� 1016 1=eð Þ4:27 0.914

A High Nf ¼ 1:835� 1017 1=eð Þ4:348 0.91

B Low Nf ¼ 1:526� 1015 1=eð Þ3:773 0.886

B Optimum Nf ¼ 5:633� 1017 1=eð Þ4:505 0.875

B High Nf ¼ 9:608� 1017 1=eð Þ4:526 0.913

Journal of Testing and Evaluation136

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RDEC ¼ DEnþ1 �DEn

DEn(6)

where:

RDEC¼ ratio of dissipated energy change,

DEn¼ dissipated energy produced in load cycle n, and

DEnþ1¼ dissipated energy produced in load cycle nþ 1.

In this approach the percentage of dissipated energy that

causes damage in the asphalt mixture is calculated for consecu-

tive cycles. Figure 9 shows the values of RDEC versus load cycle

for one of the samples. This graph has three different stages.

RDEC decreases rapidly in the first stage. Then, during the sec-

ond stage, RDEC is constant for a long period of time, and

finally in the beginning of stage three RDEC increases rapidly,

which indicates the initiation of macro-cracks in the mixture

and shows failure of the specimen. In this study a reduction of

initial stiffness to 50 % was assumed as a failure criterion.

Therefore, samples did not enter stage 3. A plateau value based

on the average RDEC in stage 2 was calculated, and Eq 7 shows

the fatigue model based on the plateau value for all samples.

This equation was obtained based on nonlinear regression.

PV ¼ 0:913ðNf Þ�0:883(7)

where:

PV¼ plateau value, and

Nf¼ fatigue life of asphalt mixture.

The coefficient of determination for this model is 0.913.

RELATION BETWEEN FATIGUE TEST RESULTS AND

FRACTURE ENERGY DENSITY

Fracture energy is a property of asphalt mixes that correlates

with damage characteristics of asphalt concrete. Fracture energy

is the sum of elastic energy and dissipated energy due to dam-

age. Elastic energy that results in the deformation of the mate-

rial and dissipated energy due to damage constitute the portion

of fracture energy that indicates the progress of damage in

asphalt concrete [7]. Figures 10, 11, and 12 demonstrate the rela-

tion between fracture energy and fatigue life in asphalt mixes at

three different strain levels.

It can be seen that greater fracture energy densities corre-

spond to a greater number of cycles to fatigue failure. The linear

relation between fracture energy density and fatigue life is

authentic for different strain amplitudes. The correlation

between fatigue life and fracture energy density signifies that

fracture energy density is a proper performance parameter for

mix designs.

As discussed earlier, the plateau value (PV) has a unique

relation with asphalt mixture fatigue life. Figure 13 demonstrates

the PV–fracture energy curve for asphalt mixes at different

strain levels. PV is the ratio of the variation in dissipated energy

due to damage to the viscoelastic dissipated energy [14]. A small

PV is related to a high fracture energy value, and this means

low damage propagation during cycles.

PHENOMENOLOGICAL FATIGUE MODEL BASED ON

FRACTURE ENERGY DENSITY

Equation 8 shows the basic form of the phenomenological fa-

tigue model based on fracture energy density. This equation is

very similar to the MEPDG model in which the dynamic modu-

lus of asphalt concrete is replaced by the fracture energy density

of an asphalt mix. The dynamic modulus is the linear visco-

elastic property of an asphalt mix, and during the fatigue pro-

cess of asphalt concrete pavements, we need the property of

FIG. 7 Fatigue life of samples at 800 strain level.

FIG. 8 Fatigue life of samples at 1000 strain level.FIG. 6 Fatigue life of samples at 600 strain level.

BAHADORI ET AL. ON FATIGUE PERFORMANCE 137

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asphalt concrete that correlates with crack initiation in asphalt

concrete; therefore, the fracture energy that is related to the

cracking resistance of a material could be an appropriate mate-

rial property in the fatigue model.

Nf ¼ a11e

� �a2

ðFEÞa3(8)

where:

Nf¼ fatigue life of asphalt mixture,

e¼ strain amplitude, and

FE¼ fracture energy density of asphalt mixture.

Based on the results of the fatigue test and the fracture

energy density of the asphalt mixes, a phenomenological fatigue

model was obtained by means of nonlinear regression as shown

by Eq 9. The coefficient of determination (R2) is 0.933 for this

model.

Nf ¼ 1037:891e

� �4:13

ðFEÞ14:126(9)

Another phenomenological fatigue model (strain based) based

on the results of all the fatigue tests is presented in Eq 10 where

the coefficient of determination for this model is 0.59. This indi-

cates that the coefficient of a phenomenological fatigue model

based on strain is strongly dependent on the material type, and

we cannot present a unique equation for different asphalt mixes.

Unlike the phenomenological model based on strain, the dissi-

pated energy model and phenomenological fatigue model based

on fracture energy density show acceptable coefficients of

determination.

Nf ¼ 2:935� 10161e

� �4:13

(10)

FIG. 11 Relation between fracture energy and fatigue life at 800 strain level.

FIG. 10 Relation between fracture energy and fatigue life at 600 strain level.

FIG. 9 Relation between RDEC and load cycles.

FIG. 12 Relation between fracture energy and fatigue life at 1000 strain

level.

FIG. 13 Relation between PV and fracture energy density.

Journal of Testing and Evaluation138

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Conclusion

1. There is a linear relation between fatigue life at differentstrain levels and the fracture energy densities of asphaltmixes. Mixes with greater fracture energy values have reli-able resistance to fatigue cracking.

2. Mixes with high fracture energy have lower values of dis-sipated energy change due to damage, and a decrease indissipated energy change due to damage propagationresults in high fatigue life.

3. It was observed from the verification study that the pro-posed phenomenological fatigue model based on fractureenergy reliably predicts the fatigue life of asphalt mixes,and we could use one unique fatigue model for all mixeswith different gradations and asphalt contents.

4. Fracture energy density is not only a reliable criterion forselecting a fatigue-resistant mix but also an appropriatematerial property in phenomenological fatigue models,and it could be an acceptable alternative to the dynamicmodulus in fatigue life prediction models.

5. A dissipated energy fatigue model requires cyclic fatiguetesting in order to acquire a PV, and such testing is verytime consuming and requires complex facilities. In contrast,for a phenomenological fatigue model based on fractureenergy, only a simple indirect strength test must be done.

ACKNOWLEDGMENTS

The writers sincerely thank Dr. Amir Izadi at the Shomal Uni-

versity and Hamed Reyhani.

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