A Simple Quantitative Method for Identification of Failure Due to Fatigue Damage

A Simple Quantitative Methodfor Identification of Failure due to

Fatigue Damage

GHAZI AL-KHATEEB*,**Department of Civil Engineering, Jordan University of Science

and Technology, Irbid, 22110, Jordan

AROON SHENOY**S.A.I.C.O. President, 1111 Arlington Blvd, Arlington, VA 22209, USA

ABSTRACT: A simple quantitative method is presented which is applicable to anytype of fatigue testing that uses sinusoidal strain/stress input and which will work forexperimentally identifying points of failure due to fatigue damage of any kind ofmaterial being tested. The present work utilizes strain-controlled bending beam fatiguetest on asphalt mixtures to demonstrate the efficacy of this method. Distortions in thehysteresis loop or waveform are tracked to pinpoint the appearance of initial micro-cracks and final point of complete failure due to fatigue damage. Relationship betweenoutput signals for consecutive cycles with reference to initial stable cycle is used forcomputing ‘R2’. The ‘R2’ drops sharply from initial stable value of 1 to less than 0.5and eventually to almost 0 with increasing loading cycles. The number of cycles deter-mined from the fitted equation at ‘R2’ =1 marks the point of first fatigue failure Nfff

and ‘R2’ =0 marks the point of complete fatigue failure Ncff.

KEY WORDS: fatigue damage, repeated loading, hysteresis loop, sinusoidalwaveform, asphalt mixtures.

INTRODUCTION

FATIGUE CRACKING IS one of the major distresses that occur in asphaltpavements and considered in mechanistic-empirical design methods.

*Author to whom correspondence should be addressed. E-mail: [email protected]**The opinions, findings, and conclusions expressed in this paper are those of the authors only and notnecessarily of any organizations that the authors are affiliated with or consultants to in their professionalcapacity.

Figures 4�11 appear in color online: http://ijd.sagepub.com

International Journal of DAMAGE MECHANICS, Vol. 20—January 2011 3

1056-7895/11/01 0003�19 $10.00/0 DOI: 10.1177/1056789509343084� The Author(s), 2011. Reprints and permissions:http://www.sagepub.co.uk/journalsPermissions.nav

Fatigue cracking in asphalt pavements occurs as a result of repeated traffic

loading at intermediate temperature range of � 10�30�C. Substantial fati-gue-related research has been done to develop theoretical models

(Ramsamooj, 1991, 1999; Baburamani, 1999; Lee et al., 2000; Rodrigues,

2000; Zhang and Raad, 1999, 2001; Lundstrom and Isacsson, 2003) exper-

imental laboratory testing techniques (Irwin and Gallaway, 1974; Porter and

Kennedy, 1975) and data analysis methods to predict the fatigue perfor-

mance of asphalt paving mixtures (Pell and Cooper, 1975; Van Dijk,

1975; Hopman et al., 1989; Pronk and Hopman, 1990; Tayebali et al.,

1992, 1993; Rowe, 1993; Kim et al., 1997; Ghuzlan and Carpenter, 2000;

Rowe and Bouldin, 2000; Al-Khateeb and Shenoy, 2004).Fatigue failure or the point of failure due to fatigue damage has been

defined in various ways in the asphalt pavement literature. In some cases

(e.g., Pell and Cooper, 1975; Van Dijk, 1975; Tayebali et al., 1992; Rowe,

1993), fatigue failure was defined arbitrarily; in other cases (e.g., Van Dijk

and Vesser, 1977; Pronk and Hopman, 1990; Tayebali et al., 1992, 1993), it

was based on the mode of loading; and sometimes energy methods (e.g.,

Hopman et al., 1989; Rowe, 1993; Kim et al., 1997; Rowe and Bouldin,

2000; Ghuzlan and Carpenter, 2000) were used to define fatigue failure.

Thus, many researchers considered the reduction in initial stiffness of an

asphalt mixture during fatigue testing as a measure of fatigue failure. Others

observed crack propagation to track fatigue failure. Some others introduced

dissipated energy concepts to define fatigue failure.Fatigue testing can be conducted using the constant stress mode of testing

and the constant strain mode of testing. In the constant stress mode of

testing, some researchers, (e.g., Pell and Cooper, 1975; Tayebali et al.,

1992), defined fatigue failure as the complete fracture at the end of the

fatigue test when the specimen fails due to tensile strains. Other researchers

such as Rowe (1993) defined fatigue failure as occurring when the initial

complex modulus has been reduced by 90%. Van Dijk (1975) defined fatigue

failure as occurring when the initial strain doubled.In the constant strain mode of testing, since the strain stays constant and

the stress decreases during the fatigue test, defining fatigue failure is more

difficult. Several different fatigue failure definitions have been adopted in

this mode. The most common and widely used definition for fatigue failure

in the constant strain mode is the 50% reduction in the initial stiffness

as defined by Pronk and Hopman (1990) and Tayebali et al. (1992, 1993).

A 50% reduction in the initial modulus was also defined as fatigue failure by

Van Dijk and Vesser (1977). Subsequently, the 50% reduction in stiffness

was adopted to define the fatigue failure point by the AASHTO as a

provisional standard TP8-94 (2002).

4 G. AL-KHATEEB AND A. SHENOY

Besides the fatigue failure definition of 50% stiffness reduction, some

researchers have proposed energy-based failure concepts. In the SHRP-

A-404 Report (1994), dissipated energy was used for fatigue analysis. The

dissipated energy per cycle is computed as the area within the stress�strainhysteresis loop. This energy decreases with an increasing number of load

cycles in the strain-controlled fatigue test as the stress decreases; on the other

hand, the dissipated energy per cycle increases as the number of load cycles

increases for the stress-controlled fatigue test.Hopman et al. (1989) proposed the use of an ‘Energy Ratio’ concept to

define fatigue failure in the controlled strain fatigue tests. They plotted the

energy ratio against the number of load cycles. A significant change in the

slope of the curve occurred at a critical number of load cycles, which was

considered the failure point on the curve. The critical number of load cycles

corresponded to a 40% reduction in the complex modulus. Hopman et al.

(1989) found evidence of crack initiation at the point where the significant

change in slope occurred, i.e., deviated from a straight line.Kim et al. (1997) introduced the 50% reduction in pseudo stiffness as a

failure point in fatigue testing, which was believed to be independent of

mode of loading and stress/strain amplitude.Rowe (1993) and Rowe and Bouldin (2000) used the concept of ‘Energy

Ratio’, which was proposed by Hopman et al. [14], to define fatigue failure.

The fatigue failure for the controlled stress and the controlled strain modes

was identified using an ‘energy ratio’ that was defined as the ratio of the

dissipated energy in the first cycle times the number of cycles (N) to the

dissipated energy in the N-th cycle.Ghuzlan and Carpenter (2000) proposed a ‘Dissipated Energy’ concept to

define fatigue failure in asphalt mixtures. They used the ratio of the change

in dissipated energy between two consecutive cycles (N, N+1) to the total

dissipated energy in the load cycle N. This new method was believed to

define fatigue failure independent of the mode of loading in fatigue testing.

Failure was selected as the point where this ratio increased rapidly after a

consistent stable trend for this ratio with load cycles.Al-Khateeb and Shenoy (2004) presented a new distinctive fatigue failure

criterion for asphalt paving mixtures that was based on observing the

load�deformation (or stress�strain) hysteresis loop or tracking the stress

and strain waveforms during fatigue testing. They defined the point of

first fatigue failure as that point at which the hysteresis loop or the response

waveform started to show the first signs of distortion. The point of complete

fatigue failure, on the other hand, was defined as the point at which

the stress response was no longer dependent on the strain input due to the

formation of interconnected fatigue cracks, and that could be identified by

Quantitative Fatigue Failure Criterion 5

the extensive or almost complete distortion of the load�deformation hyster-esis loop or the response waveform.

OBJECTIVE

The main objective of this study is to present a new simple quantitativemethod for identification of fatigue failure based on the changes to theload�deformation (stress�strain) hysteresis loop or the response waveform.The quantitative method presented in this article defines the fatigue failureas the point at which the coefficient of determination (R2) value, for therelationship of the response (stress) signals for progressive loading cycleswith reference to the response signal of initial stable cycle, decreases sharply.

MATERIALS

Aggregate

The aggregate consisted of 92% crushed diabase and 8% quartz andquartzite natural sand as shown in Table 1. The aggregate gradation metthe 1991 Virginia Department of Transportation (VDOT) specifications forsurface mixtures (SM-3).

Binders

Eleven binders were used in this study. They included a PG64-22 and aPG70-22 (unmodified binders), a PG70-28 (air-blown), and eight polymer-modified binders of PG70-28, which consisted of the following polymers:Terpolymer (Elvaloy), Styrene�Butadiene�Styrene Linear Grafted,Styrene�Butadiene�Styrene Linear, Styrene�Butadiene�Styrene RadialGrafted, Ethylene�Vinyl Acetate, Ethylene�Vinyl Acetate Grafted,Ethylene�Styrene Interpolymer, and Chemically Modified Crumb Rubber.The PG numbers shown are based on the Superpave system description forPerformance Grading. All the binders were from the same source, aVenezuelan crude (blend of Boscan and Bachaquero). The air-blownPG70-28 was obtained by noncatalytic air blowing of a PG52-28. Thepolymer-modified grades were obtained by adding various amounts of dif-ferent polymers to the PG64-22, the PG52-28, or a mixture of the PG64-22and the PG52-28 in appropriate proportions; the goal was to achieve aPG70-28 performance grade.


Sample Preparation and Fatigue Testing

The binders were heated to 163�C and mixed with heated aggregates inproportion to achieve a binder content of 4.85% by the total mass of mixture.

All asphalt mixtures were short-term oven aged for 2 h at 135�C accordingto the AASHTO provisional practice PP2-00 (2000) and compacted, using aSlab-PakTM linear kneading compactor, into 180� 500� 50mm slabs. Twobeams, each 63mm wide, 50mm high, and 380mm long with smooth faces,were then cut from each slab. The target air-void level was 7.0±0.5%. Theair-void level in the sawed beam specimens was determined, and those beamshaving air voids outside the specified range were discarded. All asphaltmixtures were tested � 48 h after compaction.

Bending beam fatigue tests were performed according to the AASHTOprotocol (2002). The tests were conducted in the strain-controlled mode at a

Table 1. Aggregate properties for the diabase.

Percent Passing (%)

Sieve size (mm) 92% Diabase 8% Natural sand Blend

25.0 100.0 100.019.0 98.6 98.712.5 73.9 76.09.5 58.7 100.0 62.04.75 39.5 95.8 44.02.36 27.2 88.2 32.11.18 19.4 74.8 23.80.600 14.4 46.0 16.90.300 11.1 14.1 11.30.150 8.2 4.8 7.90.075 5.7 2.9 5.5

Specific gravity and percent absorptionBulk Dry SGa 2.933 2.565 2.892Bulk SSDb 2.956 2.601 2.916Apparent SGa 3.002 2.659 2.961% Absorption 0.8 1.4 0.8

Flat and elongated particles at 3-to-1 length-to-thickness ratio, percent by mass21 NAc

Los Angeles abrasion, percent loss by mass14 NAc

Fine aggregate angularity49 45

aSpecific gravity.bSaturated surface dry.cNot applicable.


strain level of 1000 microstrains and at a test temperature of 19�C. A vertical

repeated sinusoidal displacement was applied at a frequency of 10Hz with

no rest periods. Three to five replicates were used for each asphalt mixture at

the 1000-microstrain level.Two concentrated and symmetrical loads are applied on the fatigue beam

specimen using a standard 4-point bending test, as shown in Figures 1 and 2.The beam specimen is forced back to its original position at the end of

each load pulse after it is subjected to 4-point bending. Free rotation and

horizontal translation are allowed at all load and reaction points, as shown

in Figure 3.

P/2

R R

a = l /3

Crack

l

h

a a

P/2

Figure 2. Loading and geometry.

Specimenclamp

Reaction Reaction

Load Load

Deflection

Specimen

Figure 1. Bending beam fatigue test schematic diagram using a standard 4-point bendingtest.


The data acquisition software (TestStarTM) of the device recorded the

load and the deformation of the specimen. Tensile strains and stresses

were calculated using the following equations:

"t ¼12d� h

3l2 � 4a2ð1Þ

whereet=maximum tensile reference strain; d=maximum vertical deforma-

tion at the center of the beam; h=average specimen height; l=length of

beam between outside clamps; and a=space between inside clamps= l/3.

�t ¼lP

bh2ð2Þ

wherert=maximum tensile reference stress; P=load applied by actuator; and

b=average specimen width.Flexural stiffness (S) was calculated using the equation given below:

S ¼�t"t

ð3Þ

Applied strains, response stresses, and flexural stiffnesses were determined

from the load and deformation amplitude, the geometry of the tested beam,

and the distance between the beam supports.The fatigue data obtained on the various asphalt mixtures were then

analyzed using three procedures:

1. The AASHTO TP8-94 50% reduction in stiffness (2002),2. The Al-Khateeb and Shenoy criterion (2004),3. The quantitative R2-based method presented in this study.

The results for all tested asphalt mixtures are summarized in Table 2.In the AASHTO TP8-94 (2002) method, the number of load cycles to

fatigue failure was determined using the 50% stiffness reduction point.

Figure 3. Freedom conditions of bending beam fatigue test.


The actual number of load cycles to fatigue failure can be established graph-ically from interpolation of fatigue data or calculated after fitting a curve tothe data; the natural logarithm function (ln) of the load cycles in this casewas used to fit the stiffness versus load cycles data (Figure 4).

Table 2. Number of load cycles to fatigue failure.

Number of load cycles to fatigue failure

AASHTO TP 8-94(2002) method

Quantitative failurecriterion [present work]

Qualitative failure criterionAl-Khateeb and Shenoy (2004)

Asphaltmixture

50% Stiffnessreduction

Initial point of sharpchange in R2-Value

Visual distortion ofwaveform or

hysteresis loop

PG 64-28 5,323 7,292 10,000PG 70-22 3,144 3,623 4,000Air-Blown 7,614 7,434 8,400Elvaloy 97,389 197,531 217,000SBS LG 9,911 15,801 20,000SBS L 8,774 12,012 30,000SBS RG 12,372 13,144 10,000EVA 5,905 12,654 36,700EVA G 7,183 11,840 16,700ESI 10,301 34,937 36,700CMCRA 4,158 3,631 8,400

Stiffness = –107.1 Ln(N) + 1673.7

R2 = 0.99

0

245

490

Fle

xura

l stif

fnes

s (M

Pa) 735

980

0

Number of load cycles (N)

50,000 100,000 150,000 200,000 250,000 300,000

Figure 4. Flexural stiffness vs load cycles for Elvaloy.


Al-Khateeb and Shenoy (2004) defined a distinctive fatigue failure crite-rion through visual observations of the load�deformation (stress�strain)hysteresis loop or waveform during fatigue testing. A slight distortion inthe hysteresis loop or the response sinusoidal waveform indicated the firstfatigue failure and an extensive (or almost complete) distortion in the hys-teresis loop or the response waveform pointed to the complete fatigue fail-ure. This failure criterion presented in Al-Khateeb and Shenoy (2004) is,however, a qualitative criterion that is based on visual observation of thehysteresis loop or the response waveform during fatigue testing.

In a fatigue test, when a sinusoidal strain is applied to the sample, asinusoidal stress response is expected even for a heterogeneous materiallike the asphalt concrete. This is valid as long as the test sample is integraland undamaged. The stress paths are all well defined at that time, and theload�deformation (or stress�strain) hysteresis loop is also well defined andsmooth, as shown in Figure 5.

With repeated fatigue loading (strain in this case), the material starts tofatigue and microcracks are induced in the system. These microcracks intro-duce discontinuities in the stress paths, and the stress response starts to dis-tort. This can be clearly seen by observing the load�deformation (orstress�strain) hysteresis loop. Initially the distortion of the hysteresis loopis slight and indicates the onset of the induced defect due to the repeatedstrain. By observing the hysteresis loop for each loading cycle after that point,the progression of the fatigue damage is actually monitored. After the initial

–500

–400

–300

–200

–100

0

100

200

300

400

500

–0.0012

Strain (m/m)

Str

ess

(kP

a)

–0.001 –0.0008 –0.0006 –0.0004 –0.0002 0

Figure 5. Stress�strain hysteresis loop.


slight distortion of the stress�strain loop, the shape buckles slightly, followedby a radical change in the shape of the loop with the progress of the fatiguefailure. The progression of the fatigue failure and the progressive distortionsin the stress�strain hysteresis loops are shown in Figure 6.

Basically, the stress�strain curves are scaled with the load�displacementcurves. Hence, the same information can be obtained from the observation

Complete failure

–300

–200

–100

0

100

200

300

0.0180.0160.014

Strain (m/m)

Str

ess

(kP

a)

First failure

–300

–100

100

300

0.008

Strain (m/m)

Str

ess

(kP

a)

Before failure

–500

–300

–100

100

300

500

–0.002

Strain (m/m)

Str

ess

(kP

a)90,000 110,000 130,000 150,000 170,000

190,000 210,000 230,000

250,000 270,000

(a)

(b)

(c)

0 0.002 0.004 0.006 0.008 0.01

0.01 0.012 0.014 0.016

Figure 6. Stress�strain hysteresis loop through the progress of fatigue failure for Elvaloy.


of the sinusoidal waveform. The input (strain or deformation) signal issmooth and well defined and stays so from the start to the end of the fatiguetest. However, the response (stress or load) sinusoidal waveform startssmooth and well defined, but soon distorts and finally becomes almostflat; the latter indicates that the stress response is no longer dependent onthe strain because of the discontinuities that have emerged from the cracking(Figure 7).

Before failure

–0.8

–0.4

0

0.4

0.8

517.1

Time (s)

Load

/ de

form

atio

n(k

N /

mm

)

Load = response

Deformation = input

First failure

–0.8

–0.4

0

0.4

0.8

1017.6

Time (s)

Load

/ de

form

atio

n(k

N /

mm

)

Load = response

Deformation = input

Complete failure

–0.8

–0.4

0

0.4

0.8

2018

Time (s)

Load

/ de

form

atio

n(k

N /

mm

)

Load = response

Deformation = input

517.2 517.3 517.4 517.5 517.6 517.7

(a)

(b)

(c)

1017.7 1017.8 1017.9 1018 1018.1

2018.1 2018.2 2018.3 2018.4 2018.5 2018.6

Figure 7. Load�deformation relationship through the progress of fatigue failure for EVA grafted.


The transitional point when the sinusoidal stress waveform goes from thesmooth profile to the first level of distortion indicates the initiation of thefirst level of cracking (first failure). In the next few cycles, it is this initiallevel of cracking along with the applied strain that controls the responsestress and maintains the distorted shape of the sinusoidal stress waveform.However, when the material fatigues further and the crack intensity rises,the applied strain can no longer drive the stress response and a flattenedstress or load waveform is obtained, as shown in Figure 7 (complete orultimate fatigue failure).

QUANTITATIVE METHOD FOR FATIGUE FAILURE

IDENTIFICATION

R2 Statistic

The quantitative method for determination of fatigue failure criterion isbased on the R2 statistic for the relationship between the output responsestress signals for consecutive loading cycles with reference to initial stablecycle in a strain-controlled test or for the relationship between the outputresponse strain signals for consecutive cycles with reference to initial stablecycle in a stress-controlled test.

In the present strain-controlled test, R2 value for the relationship betweenthe stress signals for consecutive cycles with reference to the initial stablecycle was computed. The change in the R2 value with the progressive cycleswas monitored as shown in Figure 8.

The R2 value at the beginning of the fatigue test as expected was found tobe approximately 1.0 or close to 1.0. With the progress of the fatigue test asthe sample starts to fatigue and experiences microcracks, the R2 value startsto decrease sharply as shown in Figure 8. A linear relationship was fitted tothe data and two resulting linear lines were obtained: the first line representsthe beginning of the fatigue test before fatigue failure when the R2 value isapproximately equal to 1.0, and the second line represents the stage whenfatigue failure starts to take place.

In Figure 8, at the intersection point between the two lines, the firstfatigue failure point is identified. This point is determined by solving thefitted equation shown in Figure 8 at R2 value of 1.0. The complete fatiguefailure point is determined by solving the same equation at R2 value of 0.0 orin other words at the intersection point with the x-axis. At an R2 valueof 1.0, the relationship between the response (output) signal (waveform)of progressive cycles with respect to the response signal (waveform) ofthe first cycle is ideal. No microcracking or failure is taking place at


this stage. When the R2 value starts to show a sharp decrease as shown inFigure 8, this is an indication of fatigue failure occurrence.

The same result can be obtained if the data are plotted on a semi-log scaleas shown in Figure 9. The logarithmic relationship of the second set of data(after the R2 value starts to decrease sharply) looks linear on the semi-logscale, however, the intersection of this linear-looking line with the first hor-izontal line (Figure 9) identifies the first fatigue failure point. At the pointwhen the second line intersects with the x-axis, is when the complete fatiguefailure takes place. An algorithm for sensing the onset of fatigue failurecan be performed and only requires the computation of the R2 valuedescribed above.

The general form of the fitted equation of the second line in Figure 9 is:

R2Value ¼ A logNþ B ð4Þ

Equating both sides of the equation to 1.0 results in Equation (5) below:

A logNþ B ¼ 1:0 ð5Þ

and therefore,

Nfff ¼ 101�BAð Þ ð6Þ

The Nfff represents the number of load cycles at first fatigue failure andthat is happening when R2 value=1.0.

The complete fatigue failure point is determined by equating both sides ofEquation (4) to 0.0 as shown in the equation below:

A logNþ B ¼ 0:0 ð7Þ

0.00

0.20

0.40

0.60

0.80

1.00

1000

N

R2 -

valu

e

10,000 100,000 1,00,0000

Figure 8. R2 vs N on semi-logarithmic scale for EVA asphalt mixture.


In this case,

Ncff ¼ 10�BAð Þ ð8Þ

The Ncff represents the number of load cycles at complete fatigue failure

and that is occurring when R2 value=0.0.The R2 values were calculated in the same way and plotted against the

number of load cycles for all replicates of the eleven asphalt mixtures used in

the study. The Nfff and Ncff values were determined as well and averaged

over the number of replicates. These values were compared with the values

obtained using the AASHTO TP8-94 (2002) method (the 50% stiffness

reduction) and the values obtained using the qualitative fatigue failure

criterion for Al-Khateeb and Shenoy (2004) as shown in Table 2.The associated stress response values at the three stages for all replicates

of all asphalt mixtures were also computed. In other words, the stress

response was calculated when R2=1.0, R2 started to decrease sharply

(first fatigue failure point=Nfff), and when R2=0.0 (complete fatigue fail-

ure point=Ncff), respectively (Figure 10).In addition, the stress reduction relative to the stress of the initial stable

loading cycle was calculated as the R2 decreased (Figure 11). In this case,

since the fatigue test was conducted in the strain-controlled mode (i.e., the

strain input was constant throughout the test), the stress reduction provided

values for the stiffness reduction at first failure and complete failure as well.Figure 10 shows that the stress decreased sharply at the initial stable

loading cycle until the first fatigue failure point and after that the stress

reduction rate was lower till the point of complete fatigue failure. The initial

0.00

0.20

0.40

0.60

0.80

1.00

1.20

N

R2 -

valu

e

1000 10,000 100,000

Figure 9. R2 vs N for air-blown asphalt mixture.


stress response or stiffness values were naturally different from mixture tomixture, and so the stress response or stiffness values at the point of firstfatigue failure were also different from material to material. At the end,when the material failed completely (i.e., at the point of complete fatiguefailure), the stress response or stiffness values were minimal.

Figure 11 also provided similar information, but in this case about thereduction in stress response or stiffness values relative to the initial stableloading cycle. At the beginning, the stress response or stiffness was stable(0.0% reduction). However, with repeated fatigue loading cycles, the

0

20

40

60

80

100

Str

ess

resp

onse

red

uctio

n (%

)

PG 64-28PG 70-22Air-blownElvaloySBS LGSBS LSBS RGEVAEVA GESICMCRA

Complete failure, R2 = 0

R2 = 1 First failure, R2 decreases

sharply

Figure 11. R2 and associated stress or stiffness reduction values for all asphalt mixtures.

0

300

600

900

1200

1500

Complete failure,R2 = 0

Ass

ocia

ted

stre

ss r

espo

nse

(kP

a)PG 64-28PG 70-22Air-blownElvaloySBS LGSBS LSBS RGEVAEVA GESICMCRA

R2 = 1 First failure, R2 decreases

sharply

Figure 10. R2 and associated stress response values for all asphalt mixtures.


Tab

le3

.S

tre

ssan

dst

rain

valu

es

wit

hass

oc

iate

dc

han

ge

inR

2fo

rE

VA

asp

halt

mix

ture

-re

plic

ate

no

.3

.

Str

ess

valu

es

for

asta

ble

cycle

(kP

a)

Str

ess

valu

es

for

asta

ble

cycle

be

fore

failu

re(k

Pa

)

Actu

al

str

ain

valu

es

for

asta

ble

cycle

be

fore

failu

re(k

e)

Str

ess

valu

es

for

asta

ble

cycle

(kP

a)

Str

ess

valu

es

for

the

firs

tcycle

at

firs

tfa

ilu

re(k

Pa

)

Actu

al

str

ain

valu

es

for

the

firs

tcycle

at

firs

tfa

ilu

re(k

e)

Str

ess

valu

es

for

asta

ble

cycle

(kP

a)

Str

ess

valu

es

for

the

cycle

at

co

mp

lete

failu

re(k

Pa

)

Actu

al

str

ain

valu

es

for

the

cycle

at

co

mp

lete

failu

re(k

e)

�2

73

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0.6

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9.4

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3.8

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reduction in stress response or stiffness became more marked and increasedrapidly till the point of first fatigue failure. The reduction then continued ata lower rate, until the point of complete fatigue failure. The reduction instress response or stiffness at the point of first fatigue failure ranged from48.1% to 82.9%. This result actually pointed to a significant finding that thestiffness reduction should not be an identification or definition of fatiguefailure point due to the fact that each material responds differently to fatigueand the initial stiffness is by default different from material to material.Therefore, the procedures followed in this article and the criterion describedin Al-Khateeb and Shenoy (2004) are indeed the proper way to identify thefatigue failure points for asphalt mixtures and other materials.

Table 3 shows the actual applied strains and the stress responses whenR2=1.00, when first failure started to take place (R2=0.98), and at com-plete failure when R2=0.00 (close to zero). The table provides an ideaabout the change in the stress�strain relationship as R2 value changeswith the progress of fatigue test until complete fatigue failure.

FINDINGS AND CONCLUSIONS

This article shows that there is no need to go through any level of calcula-tions based on formulae for the dissipated energy or energy ratio, or rely ona somewhat arbitrary 50% reduction in stiffness in order to identify thefatigue failure points. The fatigue failure is revealed directly from rawdata by observing the distortion of the load�deformation hysteresis loopor the response waveform at the onset of the first crack appearance. Beforeoccurrence of fatigue failure, the stress and strain signals are strongly cor-related (R2=1) whereas after failure, they are no longer correlated (R2=0).This marks a very significant and clear definition of the points of fatiguefailure. The first fatigue failure point is identified as occurring when theshape of the stress�strain (load�deformation) hysteresis loop starts toshow the first signs of distortion from the original smooth oval-or-ellipticalshape. The initially distorted shape typically lasts for a long period before avery irregular shape of the distorted waveform shows up marking the pointof the complete fatigue failure and the ultimate fatigue failure of the sampleis reached.

In this article, the R2 value for the relationship of each of the response(stress) signals for progressive cycles with reference to the response signal ofa stable cycle was computed. The change in the R2 value with the progressivecycles was monitored as shown in Figure 8. The R2 value at the beginning ofthe fatigue test is expected to be approximately 1.0. With the progress of thefatigue test as the sample starts to fatigue and experiences microcracks, the


R2 value starts to decrease sharply as shown in Figure 8. A linear relation-ship was fitted to the data and two resulting linear lines were obtained: thefirst line represents the beginning of the fatigue test before fatigue failurewhen the R2 value is approximately equal to 1.0, and the second line repre-sents the stage when fatigue failure starts to take place and eventually goesto complete fatigue failure.

The first fatigue failure is obtained at the intersection of the two fittedlinear lines when R2 value is equal to 1.0. On the other hand, the completefatigue failure is obtained at the intersection of the second linear line withthe x-axis when R2 value equals 0.0. This new fatigue failure identificationmethod is based on quantitative measures of the R2 value of the relationshipbetween the response signal or waveform (sine wave) of progressive loadingcycles and the response signal or waveform of initial stable cycle. This newquantitative method accurately describes the true point of fatigue failurebased on fundamental load�deformation (stress�strain) relationship of thematerial and hence only raw data of fatigue testing are needed to determinethe points of first and complete fatigue failure. Though the method wasdeveloped and demonstrated for fatigue testing of asphalt mixture, it isubiquitously applicable to any type of fatigue testing that utilizes a sinusoi-dal strain or stress input, and will work for identification of the points offatigue failure of any kind of material that is being tested.

In conclusion, the method bypasses the need to perform complicatedcalculations for the viscoelastic response of the material and provides asimple quantitative method to determine fatigue damage.

REFERENCES

AASHTO PP2-00 (2000). Standard Practice for Mixture Conditioning of Hot-Mix Asphalt(HMA). AASHTO Provisional Standard.

AASHTO TP8-94 (2002). Method for Determining the Fatigue Life of Compacted Hot-MixAsphalt (HMA) Subjected to Repeated Flexural Bending. AASHTO ProvisionalStandards.

Al-Khateeb, G. and Shenoy, A. (2004). A Distinctive Fatigue Failure Criterion, Journal of theAssociation of Asphalt Paving Technologists, 73: 585�622.

Baburamani, P. (1999). Asphalt Fatigue Life Prediction Models � A Literature Review.Research Report ARR 344, Australia.

Ghuzlan, K.A. and Carpenter, S.H. (2000). Energy-derived, Damage-Based Failure Criterionfor Fatigue Testing, Transportation Research Record: Journal of the TransportationResearch Board, 1723: 141�149.

Hopman, P.C., Kunst, P.A.J.C. and Pronk, A.C. (1989). A Renewed Interpretation Method forFatigue Measurements, Verification of Miner’s Rule. In: 4th Eurobitume Symposium inMadrid, Vol. 1, pp. 557�561.

Irwin, L.H. and Gallaway, B.M. (1974). Influence of Laboratory Test Method on FatigueResults for Asphaltic Concrete. Fatigue and Dynamic Testing of Bituminous Mixtures,ASTM STP, 561: 12�46.


Kim, Y.R., Lee, H.J. and Little, D.N. (1997). Fatigue Characterization of Asphalt ConcreteUsing Viscoelasticity and Continuum Damage Theory, Journal of the Association ofAsphalt Paving Technologists, 66: 520�569.

Lee, H.J., Daniel, J.S. and Kim, Y.R. (2000). Continuum Damage Mechanics�Based FatigueModel of Asphalt Concrete, American Society of Civil Engineering (ASCE) Journal ofMaterials in Civil Engineering, 12: 105�112.

Lundstrom, R. and Isacsson, U. (2003). Asphalt Fatigue Modelling using ViscoelasticContinuum Damage Theory, Road Materials and Pavement Design, 4: 51�75.

Pell, P.S. and Cooper, K.E. (1975). The Effect of Testing and Mix Variables on the FatiguePerformance of Bituminous Materials, Journal of the Association of Asphalt PavingTechnologists, 44: 1�37.

Porter, B.W. and Kennedy, T.W. (1975). Comparison of Fatigue Test Methods for AsphaltMaterials. Research Report 183-4, Project 3-9-72-183, Center for Highway Research,University of Texas at Austin.

Pronk, A.C. and Hopman, P.C. (1990). Energy Dissipation: The Leading Factor of Fatigue.Highway Research: Sharing the Benefits. In: Proceedings of the Conference, the UnitedStates Strategic Highway Research Program, London.

Ramsamooj, D.V. (1991). Fatigue Cracking of Asphalt Concrete Pavements, Journal of Testingand Evaluation, 19: 231�239.

Ramsamooj, D.V. (1999). Prediction of Fatigue Performance of Asphalt Concrete Mixes,Journal of Testing and Evaluation, 27: 343�348.

Rodrigues, R.M. (2000). A Model for Fatigue Cracking Prediction of Asphalt PavementsBased on Mixture Bonding Energy, Internation Journal of Pavement Engineering, 1:133�149.

Rowe, G.M. (1993). Performance of Asphalt Mixtures in the Trapezoidal Fatigue Test, Journalof the Association of Asphalt Paving Technologists, 62: 344�384.

Rowe, G.M. and Bouldin, M.G. (2000). Improved Techniques to Evaluate the FatigueResistance of Asphaltic Mixtures. In: Proceedings of 2nd Eurasphalt & EurobitumeCongress, Barcelona, Spain.

SHRP-A-003A Report (1992). Fatigue Response of Asphalt-Aggregate Mixes, Part 1 � TestMethod Selection by Tayebali, A.A., Deacon, J.A., Coplantz, J.S., Harvey, J.T. andMonismith, C.L. Strategic Highway Research Program, National Research Council,Washington, D.C.

SHRP-A-404 Report (1994). Fatigue Response of Asphalt-Aggregate Mixes.Strategic HighwayResearch Program, National Research Council, Washington, D.C.

Tayebali, A.A., Rowe, G.M. and Sousa, J.B. (1992). Fatigue Response of Asphalt-Aggregate Mixtures, Journal of the Association of Asphalt Paving Technologists, 61:333�360.

Tayebali, A.A., Deacon, J.A., Coplantz, J.S. and Monismith, C.L. (1993). Modeling FatigueResponse of Asphalt-Aggregate Mixes, Journal of the Association of Asphalt PavingTechnologists, 62: 385�421.

Van Dijk, W. (1975). Practical Fatigue Characterization of Bituminous Mixes, Journal of theAssociation of Asphalt Paving Technologists, 44: 38�72.

Van Dijk, W. and Visser, W. (1977). The Energy Approach to Fatigue for Pavement Design,Journal of the Association of Asphalt Paving Technologists, 46: 1�37.

Zhang, T. and Raad, L. (1999). Numerical Methodology in Fatigue Analysis: BasicFormulation, Journal of Transportation Engineering, 125: 552�559.

Zhang, T. and Raad, L. (2001). Numerical Methodology in Fatigue Analysis: Applications,Journal of Transportation Engineering, 127: 59�66.


A Simple Quantitative Method for Identification of Failure Due to Fatigue Damage

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