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Characterization of Fatigue Damage in Asphalt MixturesUsing Pseudostrain Energy
Xue Luo, Ph.D., A.M.ASCE1; Rong Luo, Ph.D., P.E., M.ASCE2; and Robert L. Lytton, Ph.D., P.E., F.ASCE3
Abstract: Fatigue cracking and permanent deformation develop simultaneously in asphalt mixtures under repeated destructive loadingregardless of temperature. Because these two distinctively different damage mechanisms always occur together, it is not possible to obtainpure cracking damage or pure permanent deformation damage in any asphalt mixture by varying the temperature or loading level. This paperdevelops an energy-based method to separate fatigue cracking from permanent deformation in terms of energy consumption. Specifically, thedissipated pseudostrain energy (DPSE) expended on cracking is separated from that expended on permanent deformation. Controlled-strainrepeated direct tension (RDT) tests are performed on asphalt mixture specimens to determine their nondestructive properties, critical non-linear viscoelastic properties, and destructive properties. The total DPSE, which is the amount of energy dissipated for the fatigue damage,including both fatigue cracking and permanent deformation, is calculated based on the critical nonlinear viscoelastic properties in order toseparate both linear and nonlinear viscoelastic effects from the strain energy that is available to cause damage. The energy distribution in anasphalt mixture under destructive loading is investigated at the microscopic level. Balance equations are established between the true stress/strain/energy of the intact material and the apparent stress/strain/energy of the bulk specimen. These balance equations are utilized to com-pletely formulate the DPSE for cracking and the DPSE for permanent deformation, respectively. As a result, the DPSE for cracking isseparated from the DPSE for permanent deformation. The formulations of the two types of DPSE and the RDT data will be used to calculatethe amount of each type of DPSE. The DPSE for cracking will be used to predict the crack growth, average crack size, and number of cracks inthe asphalt mixture under repeated destructive loading. The DPSE for permanent deformation will also be used to predict the growth ofpermanent deformation. This will be introduced in following papers. This paper is one of a series of companion papers that characterizenondestructive and destructive behaviors of an asphalt mixture under tensile loading. DOI: 10.1061/(ASCE)MT.1943-5533.0000633.© 2013 American Society of Civil Engineers.
CE Database subject headings: Fatigue; Cracking; Deformation; Viscoelasticity; Strain; Asphalts; Mixtures; Damage.
Author keywords: Fatigue cracking; Permanent deformation; Nonlinear viscoelastic properties; Pseudostrain energy; Energy distribution;Energy balance.
Introduction
Fatigue cracking and permanent deformation are prevalent dis-tresses of asphalt pavements under repeated traffic loading. Fatiguecracking is induced by crack growth in a small increment in eachloading cycle, while permanent deformation refers to an accumu-lation of plastic strain in the asphalt mixture with the increase ofloading cycles. Both types of distress have been evaluated usinglaboratory experiments. Fatigue tests have been developed to evalu-ate the fatigue resistance of asphalt mixtures. Common fatigue testsinclude the repeated flexural bending test (ASTM 2010), indirecttension test (Roque et al. 1997; Kim et al. 2002), and uniaxial cyclic
test (Si et al. 2002; Walubita et al. 2006). These tests are usuallyconducted at room temperature in accordance with the understand-ing that fatigue cracking is predominant at moderate temperatures.Separate laboratory tests have also been developed to measure per-manent deformation of asphalt mixtures, such as the creep test andrepeated loading test in a temperature range from 40°C to 70°C(Sousa et al. 1994) because yielding of the asphalt mixture andplastic deformation become vital at elevated temperatures. The re-sults of the fatigue tests and permanent deformation tests are inter-preted using models (most of which are empirical) to simulate theevolution of fatigue cracking and permanent deformation, respec-tively. For example, fatigue cracking is modeled using Paris’ law interms of the crack growth rate per cycle as a function of materialproperties obtained from the fatigue tests (Kim 2008). Models forpermanent deformation are usually regression functions illustratingthe relationship between the permanent strain and the number ofloading cycles [Barksdale 1972; Monismith et al. 1975; Kenis1977; National Cooperative Highway Research Program (NCHRP)2002].
As a matter of fact, it is not appropriate to characterize thefatigue cracking and permanent deformation of asphalt mixturesseparately because fatigue cracking and permanent deformationdevelop simultaneously as two companion components of thedamage, regardless of the temperature. In other words, both crack-ing and permanent deformation develop simultaneously under de-structive repeated loading at any temperature. Meanwhile, fatigue
1Graduate Research Assistant, Zachry Dept. of Civil Engineering, TexasA&M Univ., 3136 TAMU, CE/TTI Building 501J, College Station, TX77843 (corresponding author). E-mail: [email protected]
2Associate Research Engineer, Texas Transportation Institute, TexasA&M Univ. System, 3135 TAMU, CE/TTI Building 503C, CollegeStation, TX 77843. E-mail: [email protected]
3Professor, Fred J. Benson Chair, Zachry Dept. of Civil Engineering,Texas A&M Univ., 3136 TAMU, CE/TTI Building 503A, College Station,TX 77843. E-mail: [email protected]
Note. This manuscript was submitted on December 7, 2011; approvedon June 28, 2012; published online on August 27, 2012. Discussion periodopen until July 1, 2013; separate discussions must be submitted for indi-vidual papers. This paper is part of the Journal of Materials in Civil En-gineering, Vol. 25, No. 2, February 1, 2013. © ASCE, ISSN 0899-1561/2013/2-208-218/$25.00.
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cracking and permanent deformation are closely related to eachother. In a viscoelastoplastic material like an asphalt mixture, crack-ing is accompanied by a large amount of plastic deformation in thelocalized regions around the cracks. The localized plastic deforma-tion not only contributes to one part of the permanent deformationbut also initiates the growth of fatigue cracking from air voids. Theother part of the permanent deformation results from the yielding ofthe bulk intact material; when the yielding of the bulk intactmaterial reaches a critical level, more cracks develop in the bulkintact material. Elber (1971) used experimental and analytical tech-niques to prove that the existence of permanent deformation led to aconsiderable reduction of allowable load cycles to fatigue crackingfailure. Consequently, it is more realistic to treat the development offatigue cracking and permanent deformation as one damaging pro-cess rather than treating them separately as two damaging processesusing two separate tests. It is fairly easy to generate both fatiguecracking and permanent deformation in an asphalt mixture speci-men using the destructive repeated loading test. However, it has notbeen practical to simultaneously measure both fatigue cracking andpermanent deformation generated in the same specimen.
An energy-based approach has been used to separate fatiguecracking from permanent deformation in terms of energy consump-tion. Specifically, separation of the two types of damage in an as-phalt mixture specimen can be conducted by distinguishing theenergy consumed for each damage type. The energy expendedon fatigue cracking is then used in crack growth models to predictthe development of fatigue cracking, and the remaining dissipatedenergy is used in plasticity models to predict the development ofpermanent deformation. The sum of these two parts of energy is thetotal energy dissipated for the total damage generated in thematerial, which is defined as the dissipated pseudostrain energy(DPSE). Although the DPSE has been used as an index to charac-terize the damage in asphalt mixtures for more than a decade(Little et al. 1998; Lytton 2000; Kim et al. 2002; Si et al. 2002;Arambula et al. 2007), it is still a challenge to properly separatethe total DPSE into two parts: one for cracking and the otherfor permanent deformation. Masad et al. (2007) proposed partition-ing the DPSE into three components, including (1) a componentassociated with the increase of the phase angle, (2) a componentaccounting for the change in the stiffness, and (3) a component re-lated to permanent deformation. The first and second componentsare considered to be related to fatigue cracking because the increaseof the phase angle and change in the stiffness are due to the crackgrowth in the material. The third component, which is calculated bysubtracting the first and second components from the total DPSE, isthus associated with permanent deformation. However, this parti-tion method is phenomenological rather than mechanistic in natureand does not accurately describe the energy expended for each typeof damage.
In order to address the previously mentioned research gaps, thispaper aims at characterizing the fatigue cracking and permanentdeformation of asphalt mixtures by separating the DPSE for thefatigue cracking from the DPSE for the permanent deformationfrom a mechanistic perspective. This paper is organized as follows:First, the material properties of a typical asphalt mixture underdifferent stress levels are investigated to differentiate the linear vis-coelastic, nonlinear viscoelastic, and damaged regions on thestress-strain curve of the asphalt mixture. This investigation pro-vides the theoretical foundation for the test protocol design. Sec-ond, a series of laboratory tests are conducted to determine thestress-strain curves of lab-mixed-lab-compacted (LMLC) speci-mens as well as the properties of the undamaged and damagedspecimens. Third, the pseudostrain and pseudostrain energy ofthe asphalt mixture specimen are calculated based on the test
results. Based on the calculated pseudostrain and pseudostrain en-ergy, a set of balance equations are developed to separate the mea-sured apparent properties of the tested asphalt mixture specimensfrom their true properties, which are the properties of the intact ma-terials excluding air voids and possible cracks within the asphaltmixtures. Then the DPSE for cracking and DPSE for permanentdeformation are separated from each other, both of which are com-pletely formulated based on the balance equations, respectively.The final section summarizes the major findings of this studyand presents the ongoing research on this subject.
Due to the limits of the paper length, this paper presents only theformulations of the DPSE for cracking and the DPSE for permanentdeformation. The method to calculate the amount of each type ofthe DPSE is presented in Luo et al. (2012).
Material Properties of Asphalt Mixtures underDifferent Stress Levels
Asphalt mixtures exhibit different properties at different levels ofdeformation. The stress-strain curve of a typical paving asphaltmixture consists of the undamaged region and the damaged region,as shown in Fig. 1. These two regions are separated at the criticalnonlinear viscoelastic point (point E in Fig. 1). The undamagedregion includes the linear viscoelastic region and the nonlinear vis-coelastic region, separated by point B in Fig. 1. The nonlinearviscoelastic behavior of an undamaged asphalt mixture is due tothe rotation and translation of aggregates and localized high strainsin the asphalt mastic (Kose et al. 2000). The different properties ofthe asphalt mixture in the three regions can be characterized usingthe controlled-strain repeated direct tension (RDT) test, as detailedin this section.
Controlled-Strain RDT Test
In a typical controlled-strain RDT test, the axial strain of the testedasphalt mixture specimen is controlled to be a standard haversineshape so the axial strain is always positive. When controlling theaxial strain in such a profile, the applied stress has a shifted haver-sine shape consisting of a tensile stress portion and a compressivestress portion in a loading cycle, as shown in Fig. 2. Within a load-ing cycle of the controlled-strain RDT test, the asphalt mixturespecimen is considered to be in tension when it has a tensile strainand is subjected to a tensile stress; if the specimen has a tensilestrain and is under a compressive stress, it is regarded to be inquasi-compression.
σ
ε
Linearviscoelastic
region
Nonlinear viscoelasticregion
Damaged regionUndamaged region
Critical nonlinearviscoelastic point
A
C
D
O
B
E
Fig. 1. Stress-strain curve of typical asphalt mixtures
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It has been found that asphalt mixtures exhibit different proper-ties in tension from those in quasi-compression (Luo et al. 2013). Inorder to distinguish the tensile properties from the quasi-compres-sive properties, the stress and strain data are simulated using thefollowing mathematical models shown in Eqs. (1)–(4).
When the stress is tensile,
σt ¼ σ0t½1 − cosðωtÞ� − σst ð1Þ
ε ¼ ε0½1 − cosðωt − φtÞ� ð2Þ
When the stress is compressive,
σc ¼ σ0c½1 − cosðωtÞ� − σcm ð3Þ
ε ¼ ε0½1 − cosðωt − φqcÞ� ð4Þ
where σ0t = tensile stress amplitude; ω = loading frequency inradians/second; t = loading time; σst = absolute value of the down-ward shift of a standard haversine wave; ε0 = strain amplitude;ϕt = phase angle of the tensile complex modulus; σ0c = compressivestress amplitude; σcm = magnitude of the minimum stress; andφqc = phase angle of the quasi-compressive complexmodulus. Fig. 3illustrates the simulation of the tensile stress portion and the com-pressive stress portion, respectively. The tensile complex modulusE�t has a magnitude of jE�
t j, which is determined using Eq. (5),
jE�t j ¼
σ0t
ε0ð5Þ
and a phase angle φt, which is the time lag between the tensile stresspeak and the strain peak. The quasi-compressive complex modulusE�qc has a magnitude of jE�
qcj that is determined using Eq. (6),
jE�qcj ¼
σ0c
ε0ð6Þ
and a phase angle φqc, which is the time lag between the compres-sive stress peak and the strain valley. It has been proven that bothmagnitude and phase angle of E�
t are different from those of E�qc in
both nondestructive and destructive RDT tests (Luo et al. 2013). Asa result, the calculation of strain energy using Eq. (7) must beconducted for the tensile stress portion and the compressive stressportion separately.
W ¼Z
t2
t1
σðtÞ dεðtÞdt
dt ð7Þ
whereW = strain energy density (energy per unit volume) in a load-ing period [t1, t2]; and σðtÞ and εðtÞ are modeled using Eqs. (1) and(2), respectively, for the tensile stress portion, and they are modeledusing Eqs. (3) and (4), respectively, for the compressive stress por-tion. There are two types of strain energy density in a loading cycleof the RDT test: dissipated strain energy (DSE) and recoverablestrain energy (RSE). The details of the previously mentioned sim-ulation of the stress and strain using Eqs. (1)–(4), statistical evidenceof the difference between E�
t and E�qc, and calculation of the DSE
and RSE using Eq. (7) are presented in Luo et al. (2013), whichserves as the theoretical foundation of this study.
When calculating the DPSE, the energy for both linear and non-linear viscoelastic deformations must be subtracted from the DSE.Unfortunately, in previous studies, the DPSE was not properly cal-culated: only the energy for the linear viscoelastic deformation wassubtracted from the DSE (Kim et al. 2002) or the energy for thelinear viscoelastic deformation and part of the nonlinear viscoelas-tic deformation was removed from the DSE (Si et al. 2002). Amethod of calculating the DPSE will be proposed in the followingsections by determining the threshold strain level that correspondsto the critical nonlinear viscoelastic point on the stress-strain curvein Fig. 1 (point E). The threshold strain level is determined by vary-ing the strain level of the RDT test and by examining the change ofasphalt mixture properties under different strain levels.
Behaviors of Asphalt Mixtures in RDT Tests withDifferent Stress Amplitudes
When changing the stress amplitude in the controlled-strain RDTtest, the deformation of the asphalt mixture may be in the linearviscoelastic region, nonlinear viscoelastic region, or the damagedregion. As a result, the measured E�
t and E�qc may vary because the
asphalt mixture behaves differently in the three regions.1. If the axial strain of the specimen is within the linear viscoe-
lastic region (from point O to point B in Fig. 1),
Fig. 2. Measured stress and strain of a loading cycle in the destructiveRDT test
0 O T/2 N
σ0t
σ0t
σst
σtm
σ
tTt
L
P
O'
0 O
σcm
σ
t
σ0c
T/2σ0c
Tc
L
N
P
O'
(a)
(b)
Fig. 3. Simulation of axial stress curve: (a) tensile stress curve; (b) com-pressive stress curve
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• both E�t and E�
qc remain the same with the increase of thenumber of loading cycles in the controlled-strain RDTtest; and
• neither E�t nor E�
qc change when the strain level varies (aslong as the strain level is low enough to produce only linearviscoelastic strain) in the controlled-strain RDT test.
For example, if the loading curve is from point O to A,jE�
t j and jE�qcj stay constant because curve OB is a straight
line and the slope of OA stays the same. In addition, theunloading curve AO is a straight line because it is withinthe linear viscoelastic region. Therefore, φt and φqc remainthe same. If point A moves to any other location on lineOB, the previous statements remain correct because theslope of line OB remains the same and the unloading curveBO overlies the loading curve OB.
2. If the axial strain of the specimen is in the nonlinear viscoe-lastic region (from point B to point E in Fig. 1),• both E�
t and E�qc remain the same with the increase of the
number of loading cycles in the controlled-strain RDT test,provided that the strain level stays constant; and
• as long as the strain level changes to a higher level that pro-duces a higher strain within the nonlinear viscoelastic re-gion, both jE�
t j and jE�qcj decrease, and both φt and φqc
increase.For example, if the loading curve is OC (solid curve) in
Fig. 1 and the unloading curve is CO (dashed curve), bothmagnitude and phase angle of the complex moduli remainthe same with the increase of the number of the loadingcycles because the tangent to the stress-strain curve at pointC does not vary and the area within the loading and unload-ing loop does not change. When the strain level moves frompoint C to point D in Fig. 1, the tangent to the stress-straincurve at point D is smaller than that at point C and the areawithin the loading and unloading loop OD is larger than thearea within the loading and unloading loop OC. Therefore,jE�
t j and jE�qcj at pointD are smaller than those at pointC, and
both φt and φqc at point D are larger than those at point C.3. If the axial strain of the specimen is in the damaged region
(beyond point E in Fig. 1),• both jE�
t j and jE�qcj decrease and both φt and φqc increase
with the increase of the number of loading cycles in thecontrolled-strain RDT test; and
• when the strain level changes to a higher level, both jE�t j
and jE�qcj decrease and both φt and φqc increase.
Based on the previous investigation on the material properties ofthe asphalt mixture in different deformation regions, a comprehen-sive experimental plan is designed to measure the material proper-ties of asphalt mixtures under different stress levels. An importantobjective of the experimental testing is to determine the strain levelcorresponding to point E in Fig. 1 in order to separate the undam-aged region from the damaged region on the stress-strain curve ofthe asphalt mixture. The undamaged region must be separated fromthe damaged region in calculating the DPSE. The following sectiondetails the laboratory tests conducted on selected asphalt mixturespecimens.
Configuration and Procedure of Laboratory Tests onAsphalt Mixtures
Test Configuration
The asphalt mixture specimens tested in this study are made of anunmodified asphalt binder, which is labeled AAD in the Strategic
Highway Research Program Materials Reference Library (Jones1993), and a common Texas limestone shipped from a quarry inSan Marcos, Texas. These asphalt mixture specimens are densegraded mixtures with a Texas Department of Transportation(TxDOT) Type C gradation (TxDOT 2004). The optimum asphaltbinder content is determined to be 4.5% by weight of the mixtureaccording to the TxDOT specification Tex-204-F (TxDOT 2005).The Superpave gyratory compactor (SGC) is used to compact mix-tures into cylindrical specimens 152 mm in diameter and 178 mmin height. The compacted specimen is then cored and cut to a di-mension of 102 mm in diameter by 152 mm in height as recom-mended by the Superpave Models Team (1999). The total air voidcontent of the 102 by 152 mm specimen is 4.0� 0.5%.
The controlled-strain RDT test is then conducted on the spec-imens using the Material Test System (MTS) at 20°C. A pair of endcaps are glued to the two ends of the test specimen, which is thenset up in the environmental chamber of the MTS. It is critical thatthe ends of the test specimen are smooth and parallel so that theload applied through the loading frame of the MTS is concentricwith the axis of the specimen and with the loading frame to avoidpossible bending moment imposed on the specimen. Three axiallinear variable differential transformers (LVDTs) are mounted120° apart around the middle part of the test specimen. The averagevalue of the measurements from the three LVDTs is used as theaxial deformation. Two replicate specimens are subjected to thesame test procedure, which is detailed as follows.
Test Procedure
A series of consecutive controlled-strain RDT tests with differentstrain levels are performed on the same test specimen, as shown inFig. 4. All tests are conducted at a frequency of 1 Hz. There is a15 min rest period between two consecutive RDT tests in order tocompletely recover any possible deformation.
In the first controlled-strain RDT test, the maximum axial strainis controlled at 40 με to assure that the deformation of the asphaltmixture is within the linear viscoelastic region so that the deforma-tion will fully recover after unloading (Carpenter et al. 2003;Gibson et al. 2003). The data analysis procedure is as follows:1. Calculate jE�
t j, φt, jE�qcj, and φqc of the specimen for every
loading cycle using Eqs. (1)–(6); and2. Perform the ANOVA test on the calculated jE�
t j, φt, jE�qcj, and
φqc to determine whether they vary with the increase of thenumber of loading cycles.
Fig. 4. Sequence of controlled-strain RDT tests
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The null hypothesis H0 and the alternative hypothesis Ha areconstructed for jE�
t j, φt, jE�qcj, and φqc as hypotheses I, II, III,
and IV, respectively, as listed in Table 1. The sample for eachmaterial property population contains five groups (i ¼ 5 in Table 1)and 10 loading cycles in each group, which means that 50 loadingcycles are selected from the 200-cycle nondestructive RDT test.Specifically, the 50 loading cycles are the 1st–10th, 50th–59th,100th–109th, 150th–159th, and 190th–199th. The values of theF-statistic for these 50 loading cycles and the critical F-statisticat a 95% confidence level are automatically calculated using theAnalysis ToolPak in Microsoft Excel and are presented in Table 2.For the first RDT test with a 40 με maximum axial strain, theF-statistic of each material property is smaller than the criticalF-statistic. As a result, the ANOVA test fails to reject the nullhypotheses, and it is concluded that jE�
t j, φt, jE�qcj, and φqc are
approximately the same in each loading cycle. Therefore, thedeformation of the test specimen is within the undamaged regionduring the first RDT test with a 40 με maximum axial strain.Subsequently, the average values of jE�
t j, φt, jE�qcj, and φqc of
the selected 50 loading cycles are computed and listed in Table 3.Because the deformation of the test specimen is proved to be in
the undamaged region in the first RDT test, the maximum axialstrain level is then increased from 40 to 50 με by an incrementof 10 με in order to search for the critical nonlinear viscoelasticpoint (point E in Fig. 1). As a result, the same test specimen isrested for 15 min and is then subjected to the second RDT test with
the maximum axial strain of 50 με. Then the same statistical analy-sis procedure for the first RDT test as described previously appliesto the test data of the second RDT test. The results for hypotheses I,II, III, and IV at the strain level of 50 με are shown in Table 2,which suggests that the deformation of the test specimen is withinthe undamaged region in the second RDT test. The mean values ofjE�
t j, φt, jE�qcj, and φqc at representative loading cycles are then
computed to be the values of these properties, as listed in Table 3.Because the deformation of the test specimen is still in the
undamaged region in the second RDT test, the maximum axialstrain is once again increased by an increment of 10 με to the levelof 60 με in the third RDT test to continue searching for point E inFig. 1. After performing the third RDT test on the same test speci-men, the same data analysis procedure applies to the test data of thethird RDT test. The statistical analysis results shown in Table 2prove that jE�
t j, φt, jE�qcj, and φqc are approximately the same
in each loading cycle of the third RDT test. Therefore, the defor-mation of the test specimen is still in the undamaged region in thethird RDT test. Then the average values of jE�
t j, φt, jE�qcj, and φqc
are calculated based on the corresponding values of representativeloading cycles, and are listed in Table 3.
Although statistical analysis proves that the first three RDT testsare all nondestructive and that the material properties of the testspecimen do not change with the increase of the number of loadingcycles in each test, Table 3 shows that the test specimen exhibitsdifferent properties when the maximum axial strain changes fromone level to the next higher level. Specifically, as the maximumaxial strain increases, the values of jE�
t j and jE�qcj decrease and
the values of φt and φqc increase. These observations are verifiedusing the independent t-test. The null hypothesis H0 and the alter-native hypothesis Ha are constructed and presented for hypothesesV, VI, VII, and VIII in Table 1. The values of the t-statistic andcritical t-statistic are automatically calculated using the AnalysisToolPak in Excel and are listed in Table 4. The absolute valuesof the t-statistics of all four properties are larger than their corre-sponding critical t-statistics at a 95% confidence level. The resultsof the t-tests prove that, when the maximum axial strain increases,both jE�
t j and jE�qcj decrease and both φt and φqc increase. Based on
Table 2. Values of F-Statistic and Critical F-Statistic of ANOVA Test
Materialproperty
F-statistic
40 με 50 με 60 με 70 με 200 με
jE�t j 1.362 1.280 1.373 5.543 157.5
φt 1.740 1.181 0.208 4.080 16.15jE�
qcj 1.056 0.310 1.482 2.302 63.85φqc 0.349 0.531 1.507 2.339 14.88Critical F-statistic(95% confidence level)
2.579 2.246
Table 3. Measured Material Properties in RDT Tests
Material property
Strain level
40 με 50 με 60 με
jE�t j (MPa) 5,027 4,971 4,953
φt (°) 26.28 26.87 27.59jE�
qcj (MPa) 4,412 4,169 4,076φqc (°) 28.65 29.22 30.08
Table 4. Values of t-Statistic and Critical t-Statistic of t-Test
Materialproperty
t-statistic(40 verus50 με)
t-statistic(50 versus60 με)
Critical t-statistic(95% confidence
level)
jE�t j 5.339 4.944 2.132
φt −3.313 −5.638jE�
qcj 16.302 4.540φqc −4.896 −3.283
Table 1. Statistical Analysis on Measured Material Properties
Hypothesis Null hypothesis (H0) Alternative hypothesis (Ha) Statistical test Purpose of test
Ia jE�t j1 ¼ jE�
t j2 ¼ jE�t j3 ¼ · · ·¼ jE�
t ji At least one jE�t j differs from the rest ANOVA Compares means of three or more
groups at one time using variancesIIa φt1 ¼ φt2 ¼ φt3 ¼ · · ·¼ φti At least one φt differs from the rest
IIIa jE�qcj1 ¼ jE�
qcj2 ¼ jE�qcj3 ¼ · · ·¼ jE�
qcji At least one jE�qcj differs from the rest
IVa φqc1 ¼ φqc2 ¼ φqc3 ¼ · · ·¼ φqci At least one φqc differs from the rest
Vb jE�t jk − jE�
t jkþ1 ≤ 0 jE�t jk − jE�
t jkþ1 > 0 t-test Compares means between two groups
VIb ðφtÞk − ðφtÞkþ1 ≥ 0 ðφtÞk − ðφtÞkþ1 < 0
VIIb jE�qcjk − jE�
qcjkþ1≤ 0 jE�
qcjk − jE�qcjkþ1
> 0
VIIIb ðφqcÞk − ðφqcÞkþ1≥ 0 ðφqcÞk − ðφqcÞkþ1
< 0
ajE�t ji, φti, jE�
qcji, φqci are group means of a group of corresponding material property, and i is the number of groups.bk = number of strain level (= 1, 2 : : : ).
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Fig. 1 and the analysis in the previous section, it can be inferred thatthe deformation of the test specimen must be within the nonlinearviscoelastic region in the second and third RDT tests.
After the first three RDT tests, the test specimen is once againrested for 15 min and is subjected to the fourth RDT test with themaximum axial strain of 70 με. The ANOVA test is performed onthe test data; the hypotheses and statistics are listed in Tables 1 and2. The F-statistics of jE�
t j and φt are larger than the criticalF-statistic at the 95% confidence level, while the F-statistics ofjE�
qcj and φqc are slightly smaller than the critical F-statistic atthe 95% confidence level. This fact proves that at least jE�
t j andφt vary with the increase of the number of loading cycles, whichsuggests the occurrence of damage in the specimen. Consequently,the fourth RDT test is a destructive test that introduces damage tothe test specimen, and the deformation of the specimen is within thedamaged region in the fourth RDT test.
Based on the analysis of the test data of the four RDT tests, itcan be concluded that the critical nonlinear viscoelastic point (pointE in Fig. 1) corresponds to a strain level between 60 and 70 με. Forconvenience, the strain level of 60 με is regarded as the criticalnonlinear viscoelastic point (point E in Fig. 1), which is the thresh-old strain level between the nondestructive loading and the destruc-tive loading. Therefore, the material properties at the strain level of60 με are defined as critical nonlinear viscoelastic properties andwill be used as reference properties in order to quantify the damagein the specimen. These critical nonlinear viscoelastic properties in-clude the magnitudes and phase angles of the tensile and quasi-compressive complex moduli, which are denoted as jE�
tNLVEj,φtNLVE, jE�
qcNLVEj, and φqcNLVE, respectively.Once the strain level corresponding to the critical nonlinear vis-
coelastic point is determined, a destructive RDT test with 200 με isconducted on the same test specimen to introduce significant dam-age to the specimen. The test data are used to calculate the materialproperties, including jE�
t j, φt, jE�qcj, and φqc, of the test specimen at
every loading cycle.After the properties of the test specimen at different stress levels
are determined, the test data of the nondestructive and destructiveRDT tests stated previously are used to calculate the pseudostrainand pseudostrain energy of the specimen at different load levels.The following section details the calculation of the pseudostrainand pseudostrain energy.
Calculation of Pseudostrain and PseudostrainEnergy
Calculation of Pseudostrain
Pseudostrain is an essential parameter for separating the viscoelas-tic effect from the damage that is generated in an asphalt mixture. Itis defined based on Schapery’s extended elastic-viscoelastic corre-spondence principle (Schapery 1984) as follows:
εR ¼ 1
ER
Zt
0
Eðt − τÞ dεðτÞdτ
dτ ¼ σVEðtÞER
ð8Þ
where εR = pseudostrain; t = loading time; τ = any arbitrary timebetween 0 and t; EðtÞ = relaxation modulus of undamaged asphaltmixture; ER = reference modulus that is used to achieve a dimen-sionless variable for εR; and σVEðtÞ = viscoelastic stress corre-sponding to the strain history εðtÞ.
In order to calculate εR in the controlled-strain RDT test, σVEðtÞshould be determined first. As detailed in previous sections, a load-ing cycle in any controlled-strain RDT test has two stress portions,a tensile stress portion and a compressive stress portion, in order to
maintain the strain curve with the haversine shape. When the stressis tensile in a loading cycle of the RDT test, the strain, as formu-lated in Eq. (2), can be reformulated as
ε ¼ ε0½1 − cosðωt − φtÞ� ¼ ε0 − ε0 cosðωt − φtÞ ¼ ε1 − ε2 ð9Þwhere ε1 ¼ ε0, which is a constant strain history corresponding tothe viscoelastic stress σVE1; and ε2 ¼ ε0 cos ðωt − φtÞ, which is asinusoidal strain history in response to the viscoelastic stress σVE2.The formulations of σVE1 and σVE2 are given in Eqs. (10) and (11),respectively (Wineman and Rajagopal 2000).
σVE1 ¼ ε0EðtÞ ð10Þ
σVE2 ¼ ε0jE�tuj cosðωt − φt þ φtuÞ ð11Þ
where jE�tuj and φtu = magnitude and phase angle of the tensile
complex modulus of the undamaged asphalt mixture, respectively,or the reference material properties of the undamaged asphalt mix-ture. Based on the identification of the threshold strain level be-tween the nondestructive loading and the destructive loading inthe previous section, the critical nonlinear viscoelastic propertiesat the critical nonlinear viscoelastic point (point E in Fig. 1) areused as the reference material properties in Eq. (11), which thenbecomes
σVE2 ¼ ε0jE�tNLVEj cosðωt − φt þ φtNLVEÞ ð12Þ
The viscoelastic stresses corresponding to ε1 and ε2 are super-imposed to obtain the viscoelastic stress corresponding to ε, asshown in Eq. (13).
σVEt ¼ ε0EðtÞ − ε0jE�tNLVEj cosðωt − φt þ φtNLVEÞ ð13Þ
where σVEt = viscoelastic stress in the tensile stress portion of aloading cycle of the RDT test. When the axial deformation ofthe asphalt mixture is at the critical nonlinear viscoelastic state,which is 60 με in the specific case stated in the previous section,σVEt is calculated by substituting ε0 ¼ ε0;60 and φt ¼ φNLVE intoEq. (13), which results in
σVEt;60 ¼ ε0;60EðtÞ − ε0;60jE�tNLVEj cosðωt − φtNLVE þ φtNLVEÞ
ð14Þwhere ε0;60 = strain amplitude in the RDT test with the maximumaxial strain level of 60 με. The viscoelastic stress at this strain levelis the same as the measured stress (σt;60) because there is no dam-age introduced to the specimen at this strain level and the measuredstress corresponds to the viscoelastic strain. According to Eq. (1),the measured stress can be formulated as
σt;60 ¼ σ0t;60½1 − cosðωtÞ� − σst;60 ð15ÞBased on Eq. (5), σ0t;60 ¼ ε0;60jE�
tNLVEj, which is substitutedinto Eq. (15) to obtain
σt;60 ¼ ε0;60jE�tNLVEj½1 − cosðωtÞ� − σst;60 ð16Þ
Because σt;60 ¼ σVEt;60, Eqs. (14) and (16) are utilized to solvefor EðtÞ as
EðtÞ ¼ jE�tNLVEj − σst;60
ε0;60ð17Þ
As a result, instead of measuring the relaxation modulus [EðtÞ],EðtÞ can be calculated using Eq. (17) based on jE�
tNLVEj, σst;60, andε0;60, all of which are obtained from the nondestructive RDT testwith the maximum axial strain level of 60 με. Consequently, EðtÞ is
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substituted into Eq. (13) to obtain the final expression of σVEt for acontrolled-strain RDT test at any strain level as follows:
σVEt ¼ ε0jE�tNLVEj½1− cosðωt−φtþφtNLVEÞ�− ε0
ε0;60σst;60 ð18Þ
When the strain level is 60 με, σVEt;60 is given in Eq. (16); whenthe strain level is 200 με, σVEt;200 is obtained using Eq. (18).
σVEt;200 ¼ ε0;200jE�tNLVEj½1 − cosðωt − φt;200 þ φtNLVEÞ�
− ε0;200ε0;60
σst;60 ð19Þ
where σVEt;200 = viscoelastic stress in the tensile stress portionof a loading cycle of the RDT test at the strain level of 200 με;ε0;200 = strain amplitude in the RDT test at 200 με; andφt;200 = phase angle of the tensile complex modulus in the RDTtest at 200 με.
Following the same procedure for the tensile stress portionof a loading cycle, the viscoelastic stress in the compressivestress portion of a loading cycle is also formulated as shown inEq. (20).
σVEqc ¼ ε0jE�qcNLVEj½1 − cosðωt − φqc þ φqcNLVEÞ� − ε0
ε0;60σcm;60
ð20Þwhere σVEqc = viscoelastic stress in the compressive stress portionof a loading cycle of the RDT test at any strain level.
Based on the determination of σVEðtÞ, Eq. (8) is used to calcu-late the pseudostrain εR. The reference modulus ER in Eq. (8) isequal to jE�
tNLVEj when the stress is tensile, and is equal tojE�
qcNLVEj when the stress is compressive in a loading cycle. Con-sequently, the pseudostrain is calculated as follows:• When the stress is tensile,
εR ¼ σVEðtÞER
¼ σVEt
jE�tNLVEj
¼ ε0½1 − cosðωt − φt þ φtNLVEÞ� − ε0ε0;60
·σst;60
jE�tNLVEj
ð21Þ
• When the stress is compressive,
εR ¼ σVEðtÞER
¼ σVEqc
jE�qcNLVEj
¼ ε0½1 − cosðωt − φqc þ φqcNLVEÞ� − ε0ε0;60
·σcm;60
jE�qcNLVEj
ð22ÞEqs. (21) and (22) are validated using the stress-strain and
stress-pseudostrain diagrams, or hysteresis loops. The stress-strainhysteresis loops are produced based on Eqs. (1)–(4); the stress-pseudostrain hysteresis loops are generated based on Eqs. (1)and (21) for the tensile stress portion and based on Eqs. (3) and(22) for the compressive stress portion. The generated hysteresisloops are presented in Figs. 5 and 6. Discontinuity is observed be-tween the tensile stress portion and the compressive stress portionon all hysteresis loops. This fact is due to the difference between thetensile properties and quasi-compressive prosperties of the asphaltmixture. The area of the stress-strain hysteresis loop represents theamount of the DSE in the corresponding loading cycle. When thetest specimen is at the critical nonlinear viscoelastic state, the entireamount of the DSE is expended on the viscoelastic effect of theasphalt mixture, which includes both linear and nonlinear visco-elastic effects.
In contrast, the stress-pseudostrain hysteresis loops becomestraight lines because the pseudostrain is in phase with the stressand all energy used for the viscoelastic effect of the material is re-moved from the stress-pseudostrain hysteresis loop. For example,for the first loading cycle of the nondestructive RDT test at 60 με,the stress-pseudostrain hysteresis loop becomes line OM for thetensile stress portion and line ON for the compressive stress por-tion, as shown in Fig. 5. The slope of line OM is different from the
Fig. 5. Hysteresis loop of an asphalt mixture specimen at the criticalnonlinear viscoelastic state in the nondestructive RDT test
Fig. 6. Hysteresis loop of an asphalt mixture specimen at the damagedstate in the destructive RDT test
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slope of line ON because the tensile properties are different fromthe quasi-compressive properties of the asphalt mixture.
When the destructive RDT test is performed on the same speci-men, the DSE is expended not only on the viscoelastic effect butalso to drive the damage. As a result, the stress-pseudostrain hys-teresis loops are no longer straight lines. Because the part of theDSE expended on the viscoelastic effect has been removed fromthe stress-pseudostrain hysteresis loop, the enclosed area of thestress-pseudostrain hysteresis loop represents the DPSE, whichis the part of the DSE used to drive cracking and permanent defor-mation in the asphalt mixture, as shown in Fig. 6.
Calculation of Pseudostrain Energy
Based on the pseudostrain formulated in Eqs. (21) and (22), thepseudostrain energy density (energy per unit volume) is calculatedby integrating the stress and pseudostrain as follows:
WR ¼Z
t2
t1
σðtÞ dεRðtÞdt
dt ð23Þ
where WR = pseudostrain energy density in a loading period½t1; t2�, which is equal to the strain energy density W minusthe energy spent on the linear and nonlinear viscoelastic effects.There are two types of pseudostrain energy density, one of whichis the energy dissipated to develop damage in the specimen, definedas the DPSE, the other of which is the recoverable pseudo-strain energy (RPSE) that is stored and recovered correspondingto the purely elastic effect of the material. The DPSE and RPSEare both calculated using the same method and procedure as thosefor the DSE and RSE, as detailed in Luo et al. (2013). The finalformulation of the DPSE and RPSE are presented in Eqs. (24)and (25).
DPSE ¼ σ0tε0
8>><>>:
sinðφt − φtNLVEÞ½2π − sinðTcωÞ þ sinðTtωÞ − Tcωþ Ttω�4
− cosðφt − φtNLVEÞ½cosðTcωÞ − cosðTtωÞ�4
9>>=>>;
þ ðσ0t − σstÞε0
8>><>>:
sinðφt − φtNLVEÞhsin
�Ttω2
�þ sin
�Tcω2
�i
þ cosðφt − φtNLVEÞhcos
�Ttω2
�þ cos
�Tcω2
�i9>>=>>;
þ σ0cε0
8>><>>:
sinðφqc − φqcNLVEÞ½2π þ sinðTcωÞ − sinðTtωÞ þ Tcω − Ttω�4
þ cosðφqc − φqcNLVEÞ½cosðTcωÞ − cosðTtωÞ�4
9>>=>>;
þðσ0c − σcmÞε0
8>><>>:
− sinðφqc − φqcNLVEÞhsin
�Ttω2
�þ sin
�Tcω2
�i
− cosðφqc − φqcNLVEÞhcos
�Ttω2
�þ cos
�Tcω2
�i9>>=>>;
ð24Þ
RPSE ¼ σ0tε0
8>><>>:
− sinðφt − φtNLVEÞ½2ðφt − φtNLVEÞ þ sinð2φt − 2φtNLVEÞ − sinðTtωÞ − Ttω�4
−cosðφt − φtNLVEÞ
hcos2ðφt − φtNLVEÞ − cos2
�Ttω2
�i2
9>>=>>;
þ ðσ0t − σstÞε0
8>><>>:
sinðφt − φtNLVEÞhsin
�Ttω2
�− sinðφt − φtNLVEÞ
iþ cosðφt − φtNLVEÞ
hcos
�Ttω2
�− cosðφt − φtNLVEÞ
i9>>=>>;
þ σ0cε0
8>><>>:
− sinðφqc − φqcNLVEÞ½2ðφqc − φqcNLVEÞ þ sinð2φqc − 2φqcNLVEÞ − sinðTcwÞ − Tcw�4
−cosðφqc − φqcNLVEÞ
hcos2ðφqc − φqcNLVEÞ − cos2
�Tcw2
�i2
9>>=>>;
þ ðσ0c − σcmÞε0
8>><>>:
− sinðφqc − φqcNLVEÞhsin
�Tcω2
�− sinðφqc − φqcNLVEÞ
i− cosðφqc − φqcNLVEÞ
hcos
�Tcω2
�− cosðφqc − φqcNLVEÞ
i9>>=>>;
ð25Þ
Based on Eqs. (24) and (25), the DPSE and RPSE are calculated for every loading cycle of the destructive RDT test. Fig. 7 shows theevolution of the DPSE and RPSE of a number of representative loading cycles. The DPSE increases with the increase of the number ofloading cycles, indicating the damage accumulation in the asphalt mixture specimen. The rate of damage accumulation can be measured bythe rate of the change of the DPSE with the increase of the number of loading cycles. Specifically, an exponential function is used to fit theDPSE curve using the software MATLAB.
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DPSE ¼ 6.36N0.24 þ 7.19; R2 ¼ 0.998 ð26Þ
in whichN = number of loading cycles. The high R2 value indicatesthe goodness of the model fit. The rate of the change of the DPSEis thus calculated as
dDPSEdN
¼ 1:53N−0.76 ð27Þ
Fig. 7 also shows that the RPSE decreases with the increase ofthe number of loading cycles, implying that the specimen is losingthe ability to store and recover energy or the material is losing itsintegrity due to the damage accumulation.
The calculated DPSE will be further decomposed into two com-ponents: the DPSE for cracking and the DPSE for permanent de-formation. The formulation of the two components of the DPSEwill be detailed in the following section.
Balance of Pseudostrain Energy
In order to separate the two components of the DPSE, this sectionwill first investigate the energy distribution around the cracks in theprocess of damage development at the microscopic level. At themicroscopic level, the asphalt mixture specimen consists of intactmaterial (asphalt binder and aggregates) and air voids, which act asthe initial cracks in the specimen.
Fig. 8(a) presents a schematic sketch of the microlevel view ofan asphalt mixture subjected to a tensile stress σt, which is normallyregarded to be uniformly distributed on the entire cross section ofthe specimen. This is an apparent description of the stress withoutconsidering any air voids or cracks in the material. As a matter offact, due to the existence of air voids (initial cracks), the stress dis-tribution is not uniform, which is represented using force lines(dashed curves) in Fig. 8(a). Denser force lines at the crack tipsindicate local stress concentration, and the curved force lines inthe vicinity of the crack tips illustrate the nonlinear distributionof the stress. This localization of the stress (stress concentrationand nonlinear distribution) results in a localization of strain inthe corresponding regions surrounding the cracks. It has been re-ported that the average value of the localized strain can be from 4 to100 times of the bulk strain in the asphalt mixture (Bahia et al.1999; Kose et al. 2000; Masad and Somadevan 2002). It can be
concluded that the presence of cracks significantly changes thestress and strain in the intact material at the microscopic level.
Because of the variation of the stress and strain distributionaround the cracks, the energy is also redistributed around thecracks. Due to the viscoelastoplastic nature of the asphalt mixture,a large amount of energy is consumed as the plastic works to drivethe advance of the fracture process zone, where material separationoccurs, which creates two new surfaces from the initial crack(Schapery 1984). Fig. 8(b) illustrates the fracture process zoneand the resulting enlargement of the initial crack, i.e., the propa-gation of the initial crack. Consequently, a plastic region developssurrounding the fracture process zone with large local plastic de-formation. This plastic region is labeled 1 and is located approx-imately in the long-dashed and short-dashed circle in Fig. 8(a). Inaddition to the localized plastic deformation, plastic deformationalso develops in the area without any cracks due to yielding ofthe intact material, such as the region labeled 2 in Fig. 8(a). Thesetwo parts of plastic deformation add up to the total plastic defor-mation, or the permanent deformation of the asphalt mixture speci-men. The energy expended on the total plastic deformation is theDPSE for permanent deformation.
The propagation of the initial crack also results in the redistrib-ution of the RPSE because of the relaxation of the intact materialabove and below the crack in the region enclosed in long-dashedand double short-dashed lines labeled 3 in Fig. 8(a). The RPSE isreleased as the intact material in this region is unloaded due to crackpropagation. Part of the released RPSE provides the surface energythat is required to separate the intact material to create new cracksurfaces labeled 4 in Fig. 8(a). This surface energy is absorbed bythe newly created crack surfaces.
σ
σ
Initial crack
Fracture process zone
1
3
42
Force line
1 24
44
cI
cN
3
3 3
z
y
x
Cross section of thespecimen
Initial crack
Fracture process zone
Intact material
New crack
(a)
(b)
Fig. 8. Stress and energy distribution around a crack in a damagedasphalt mixture specimen: (a) side view; (b) plan view
Fig. 7. DPSE and RPSE at various numbers of loading cycles in thedestructive RDT test
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Because of the presence of cracks in the asphalt mixture, thestress, strain, and energy distribution of the intact material are cer-tainly different from those of the bulk material. For example, thestress of the intact material should be equal to the applied load di-vided by the cross-sectional area excluding the crack opening areas(instead of the entire cross section). This stress is defined as the truestress in this paper. Similarly, the strain and energy in the intactmaterial are defined as true strain and true energy, respectively.In contrast, the stress in the bulk material is defined as the apparentstress, which is equal to the applied load divided by the entire cross-sectional area (including crack opening areas). The strain and en-ergy of the bulk material are defined as apparent strain and apparentenergy. The RDT test provides measurements of the apparentstress, strain, and energy, which are related to the true stress, strain,and energy based on the principle of energy balance.
The principle of energy balance states that any kind of true en-ergy within the intact material equals its counterpoint from the ap-parent measurement. This is because only the intact material canstore, release, and dissipate energy, whereas cracks cannot. Conse-quently, the true energy within the intact material must equal theapparent energy within the entire specimen, including the intactmaterial and cracks. According to this principle, energy balanceequations, including strain energy balance equations and pseudo-strain energy balance equations, are formulated between the trueenergy and the apparent energy.
First the strain energy balance equations are formulated for theDSE and RSE. The DSE balance equation states that the apparentDSE equals the true DSE.
DSEA ¼ DSET ð28Þin which the superscript A stands for apparent and T stands for true.Similarly, the RSE balance equation states that the apparent RSEequals the true RSE.
RSEA ¼ RSET ð29ÞSecond, the pseudostrain energy balance equations are con-
structed for damaged asphalt mixtures under destructive loading.The DPSE balance equation is formulated as follows:
DPSEA ¼ DPSET ð30Þin which DPSEA = DPSE calculated using Eq. (24). The RPSE bal-ance equation shown in Eq. (31) is constructed on the basis that theapparent RPSE equals the true RPSE associated with the energyredistribution during crack propagation.
RPSEAVm ¼ RPSETVm − ðRPSETVr − γSÞ ð31ÞRPSEA is the apparent RPSE calculated using Eq. (25). Vm is
the volume of the asphalt mastic in one layer of the asphalt mixturespecimen, whose thickness equals the mean film thickness. Theformulation of Vm is
Vm ¼ Amt ð32Þwhere Am = area of the asphalt mastic on a cross section of thespecimen; and t = mean film thickness, determined based on theaggregate surface area and the effective volume of the asphaltbinder, as formulated in Eq. (33) (Roberts et al. 1996),
t ¼ Vasp
SA ×Wð33Þ
where Vasp = effective volume of the asphalt binder; SA = surfacearea of the aggregate; and W = weight of the aggregate. RPSET
represents the true RPSE of the asphalt mastic before cracks propa-gate from the initial air voids. Vr is the volume of the asphalt masticin region 3 in Fig. 8(a), which is subjected to a relaxation processand releases the RPSE during the crack propagation. The crack isassumed to be penny-shaped (Lytton 2004), as shown in Fig. 8(b).Therefore, Vr approximately equals the difference between the vol-ume of the larger cone and the volume of the smaller cone aboveand the below the crack. If the radius of the initial crack is cI andthe radius of the new crack after crack propagation is cN , the heightof the smaller cone is πcI and the height of the larger cone is πcN(Roylance 2001). Thus Vr is calculated as follows:
Vr ¼ 2 ·
�1
3πc2N · πcN · MN − 1
3πc2I · πcI · MI
�ð34Þ
in which MN = number of new cracks within the volume Vm; andMI = number of initial cracks within the volume Vm. γ is the surfaceenergy density (energy per unit area). S is the total area of the newlycreated crack surfaces corresponding to the region labeled 4 inFig. 8(a) and also the shadowed area in Fig. 8(b). S is calculatedas follows:
S ¼ 2 · ðπc2N · MN − πc2I · MIÞ ð35ÞγS is the amount of the energy absorbed from the released
RPSETVr and restored on the newly created crack surfaces.Because RPSETVr is the energy released around the crack dur-
ing the crack propagation and γS is the surface energy absorbedfrom RPSETVr, the difference between RPSETVr and γS is theenergy dissipated due to crack growth, which is the DPSE forcracking as formulated in Eq. (36).
DPSEcV ¼ RPSETVr − γS ð36Þwhere DPSEc = DPSE for cracking; and V = volume of one layerof the asphalt mixture specimen, calculated by the entire cross-sectional area multiplied by the mean film thickness. Then theDPSE for permanent deformation (DPSEp) is calculated by sub-tracting DPSEc from the total DPSE as follows:
DPSEp ¼ DPSET − DPSEc ð37Þ
As a result, the DPSE expended on cracking and the DPSEexpended on permanent deformation are completely formulated,respectively.
The next step is to calculate the amount of DPSEc and DPSEpbased on the test data using the previous formulation. During thecalculation, the average crack size and the number of cracks arefirst determined using the RPSE balance equation [Eq. (31)].The details about how to solve the RPSE balance equation are pre-sented in Luo et al. (2012).
Conclusions and Future Work
This paper presents a mechanistic method of characterizing thefatigue damage of asphalt mixtures under repeated loading. Thismethod separates the DPSE for cracking from the DPSE forpermanent deformation so that the two types of damage can becharacterized separately.
The material properties of a typical asphalt mixture under differ-ent stress levels are first investigated to differentiate the linear vis-coelastic, nonlinear viscoelastic, and damaged regions on thestress-strain curve of the asphalt mixture. Then the controlled-strainRDT tests are performed on asphalt mixture specimens to deter-mine their nondestructive properties, critical nonlinear viscoelastic
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properties, and destructive properties. The total DPSE, which is theamount of energy dissipated for the fatigue damage, is calculatedbased on the critical nonlinear viscoelastic properties in order toeliminate both linear and nonlinear viscoelastic effects. The criticalnonlinear viscoelastic state is the threshold between the undamagedstate and the damaged state of the asphalt mixtures.
The energy distribution in the asphalt mixture under destructiveloading is investigated at the microscopic level. At this level, theasphalt mixture specimen is further divided into the intact materialand cracks. The stress, strain, and energy in the intact material arewhat actually occur inside the specimen, whereas the stress, strain,and energy measured from the test are apparent representations bythe bulk material. Balance between the true circumstance and theapparent measurement is established on the basis of energy equiv-alence between the intact material and the bulk specimen. Four en-ergy balance equations are established, including the DSE balanceequation, RSE balance equation, DPSE balance equation, andRPSE balance equation. These balance equations are utilized tocompletely formulate the DPSE for cracking and DPSE for perma-nent deformation, respectively. Therefore, the DPSE for cracking isseparated from the DPSE for permanent deformation.
The formulations for the two types of DPSE and the RDT testdata will be utilized in Luo et al. (2012) to calculate the amount ofeach type of DPSE. The DPSE for cracking will be used to predictthe crack growth, average crack size, and number of cracks in theasphalt mixture under repeated destructive loading. The DPSE forpermanent deformation will be used to predict the growth the per-manent deformation.
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