A new comprehensive analysis framework for fatigue characterization of asphalt binder using the Linear Amplitude Sweep test

Construction and Building Materials 171 (2018) 1–12

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

A new comprehensive analysis framework for fatigue characterization of asphalt binder using the Linear Amplitude Sweep test Wei Cao a,⇑, Chao Wang b a b

Department of Civil and Environmental Engineering, Louisiana State University, 4101 Gourrier Avenue, Baton Rouge, LA 70820, USA Department of Road and Railway Engineering, Beijing University of Technology, Beijing 100124, PR China

h i g h l i g h t s
The existing analysis framework for LAS test was critically reviewed.
The revised formulation was rigorously derived based on the VECD theory.
The proposed failure definition provided results consistent with experiments.
The proposed failure criterion unified LAS and TS tests under various test conditions.

a r t i c l e

i n f o

Article history: Received 13 November 2017 Received in revised form 26 February 2018 Accepted 16 March 2018

Keywords: Fatigue resistance Viscoelastic continuum damage Failure definition Failure criterion Linear Amplitude Sweep Time Sweep

a b s t r a c t The Linear Amplitude Sweep (LAS) test has become an efficient tool to characterize fatigue resistance of asphalt binders. The existing analysis scheme for this test based on the linear viscoelastic continuum damage (VECD) theory was critically reviewed. Based on the rigorous VECD formulation and experimental evidences, a new comprehensive analysis framework was developed which includes damage characteristic relationship, failure definition, and failure criterion. Compared to the existing approach, the proposed framework provided a more reasonable agreement with experimental observations on fatigue life and a more reliable mathematical description of fatigue failure, and was also capable of unifying both LAS and Time Sweep fatigue tests. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue failure has been one of the primary distress types plaguing asphalt pavement. It generally occurs at intermediate temperatures under repeated traffic loading. In addition to traffic and environmental factors, performance of asphalt pavement critically depends on fatigue resistance of asphalt mixture. As a composite of aggregate, asphalt cement, and air void, asphalt mixture’s fatigue characteristics is dominated by the fatigue properties of asphalt binder. As such, a standard test method and analysis framework are necessitated for evaluating fatigue resistance of asphalt binders. In the 1990s, the Strategic Highway Research Program (SHRP) developed the Superpave fatigue parameter, denoted as |G⁄|sind, which can be obtained using a dynamic shear rheometer (DSR).

⇑ Corresponding author. E-mail addresses: [email protected] (W. Cao), [email protected] (C. Wang). https://doi.org/10.1016/j.conbuildmat.2018.03.125 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

However, this parameter has been found to be an inadequate indicator of fatigue resistance of asphalt binders, which is partly explained by the fact that the measurement is taken in the linear viscoelastic range without interference of damage [1]. In order to simulate the process of damage accumulation in asphalt under repeated loading, the Time Sweep (TS) test was proposed in which the strain input is oscillated with a prescribed constant amplitude [1,2]. This test has proved a reasonable fatigue evaluation tool for asphalt binders with various compositions [3] using the concept of dissipated energy ratio in analysis [4]. However, the TS test has been considered time-consuming and thus is not practically favored as a routine specification test [5]. The Linear Amplitude Sweep (LAS) test was then developed as a substitute for the TS test [5,6]. In this test, the damage process is accelerated via a systematically increasing strain amplitude in a stepwise manner, which reduces the testing time from several hours down to minutes. Subsequently, a machine compliance issue was identified that certain DSR equipment was not capable of making abrupt changes in strain amplitude between loading steps.

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

Given this, a variation of the LAS loading profile was proposed which implemented the linear ramping [7]. This modification avoided the compliance issue and also facilitated analytical modeling [8]. Analysis on the LAS data has been performed using the linear viscoelastic continuum damage (VECD) theory. This theory has seen a great success in fatigue characterization of asphalt mixtures and performance prediction of asphalt pavements [9,10]. Promising correlations were observed between fatigue prediction results of binders and cracking performance of field pavements [5,8]. On the other hand, the VECD theory is certainly also applicable to TS test for which the loading profile resembles that of fatigue test on asphalt mixtures performed in the uniaxial tension mode [11]. The GR failure criterion [12] originally developed for asphalt mixtures has also found its success in asphalt binders, and moreover this failure criterion has demonstrated its capability in unifying both LAS and TS tests [8,13]. Based on the VECD theory, alternative analysis methods have also been proposed to characterize fatigue failure in TS and LAS tests using the concepts of dissipated strain energy and stored pseudo-strain energy, respectively [14]. It has been recognized that other mechanisms may be present and interacting with fatigue damage during continuous fatigue loading of binders, such as thixotropy, nonlinear viscoelasticity, and steric hardening [15–18]. Such mechanisms have not been fully understood and are highly challenging to be experimen tally/mechanistically characterized. On the other hand, the application of the linear viscoelastic continuum damage theory to fatigue characterization and performance prediction of asphalt binders and mixtures has proved its effectiveness and versatility in the literature. This paper presents a critical review of the existing fatigue characterization scheme for asphalt binders using the LAS test and the VECD theory, followed by the development and validation of a new analysis framework that is theoretically more rigorous and practically more reliable. It should be noted that the work undertaken is confined to the linear viscoelastic continuum damage theory while the extra complicating effects such as thixotropy, nonlinearity, and steric hardening are excluded from consideration.

2. Objectives and scope The objectives of this research were threefold:
To reveal and demonstrate flaws in the existing analysis scheme for the LAS test;
To propose a rigorous formulation for the LAS test within the framework of linear viscoelastic continuum damage theory;
To propose a reasonable failure definition and a robust failure criterion which unify both LAS and TS tests. Note that even though the focus of this research is on LAS, it appears unavoidable to involve TS test given their common nature in the VECD analysis and also given the unifying nature of the proposed failure definition and failure criterion, as will be demonstrated. To achieve the above objectives, a critical review of the existing VECD formulation, failure definition, and failure criterion for LAS was first conducted. The revised formulation was obtained following rigorous derivation, and was verified using LAS data from various test conditions. The irrationality of the existing failure definition was illustrated using the LAS data on asphalt binders with three different aging conditions. A new failure definition was proposed which was aimed to provide more reasonable conformance with general engineering experience and experimental observations. A series of LAS and TS tests with various test

conditions were designed and performed on additional asphalt binders to verify the validity and unifying nature of the proposed failure definition and failure criterion. 3. Theoretical background Application of the VECD theory to fatigue characterization of asphalt materials yields the so-called damage characteristic relationship, which is a function relating material integrity (represented by C) and damage intensity (S). This function prescribes the path following which material loses its structural integrity as a result of damage accumulation under repeated loading. The damage characteristic relation C(S) obtained for a given material is unique in that it is independent of test conditions (e.g., temperature, load level, frequency, and control mode). The VECD theory and thus the C(S) relation are applicable to damaged material states prior to the occurrence of damage localization or macrocracking. Hence, the analysis framework should be completed experimentally with a proper failure definition and mathematically with a failure criterion function. This section presents a brief review of the existing formulation of the damage characteristic relationship, failure definition, and failure criterion as the background for further exploration. To facilitate subsequent discussion, it is considered important to first clarify the two above-mentioned concepts necessary in fatigue analysis: failure definition and failure criterion. Failure definition prescribes when material failure occurs, and therefore determines fatigue life in testing. In general, selection of failure definition should be carefully determined with comprehensive and balanced considerations on experimental, theoretical, and analytical factors, as will be discussed subsequently. Failure criterion is typically represented by a function correlating two variables: one is associated with the material responses while the other is related to the load input. Identification of failure criterion needs a proper failure definition as the prerequisite. 3.1. Existing formulation As previously mentioned, the LAS test was proposed as a surrogate of the TS test as an accelerated fatigue characterization method [6]. The analysis approach adopted then was already developed for evaluating fatigue behaviors of asphalt matrix under oscillatory distortion [19]. However, a close investigation of the derivation in [19] and comparison with the original VECD formulation [9,20] for asphalt mixtures expose a fundamental flaw, which lies in the use of dissipated strain energy in the damage evolution law:


a dS @W ¼  dn @S

ð1Þ

where S represents damage intensity, n is reduced time, a is damage evolution rate, and W denotes the dissipated strain energy which is given by

W ¼ pc2 jG j sin d

ð2Þ

⁄

where c, |G |, and d denote shear strain amplitude, dynamic shear modulus (damaged), and phase angle of a cycle in interest. It then followed that the damage increment DS was given by

DSi ¼



1þa a

pc2i ðjG ji1  sin di1  jG ji  sin di Þ

1

 ðni  ni1 Þ1þa

ð3Þ

where i denotes cycle number. Additionally, it is worth mentioning that the use of dissipated strain energy led to the material integrity being represented by |G⁄|sind, i.e., the loss modulus (damaged), or alternatively by its normalized form.

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It should be realized that the VECD theory is founded on the pseudo-strain based elastic-viscoelastic correspondence principle [21]. Consequently, pseudo-quantities such as pseudo-strain and (dissipated) pseudo-strain energy should be used in place of corresponding physical variables. Furthermore, use of Eq. (1) violates the fact that only a portion of the dissipated strain energy serves as the driving force for damage growth; the dissipated strain energy is also responsible for viscous dissipation. The damage evolution law originally proposed as part of the theory should read [22]:

dS ¼ dn

@W R  @S

!a ð4Þ

where WR is pseudo-strain energy. Recently, this fundamental blemish was acknowledged and the formulation has been updated by closely following the same logic as in deriving the VECD equations used for asphalt mixtures [8,23]. In this updated formulation, material integrity is represented by the normalized dynamic shear modulus:

C¼

jG j jG jLVE  DMR

ð5Þ

where |G⁄|LVE represents the undamaged modulus measured in the linear viscoelastic range, and DMR stands for dynamic modulus ratio (typically in the range of 0.9–1.1) which is used to quantify specimen-to-specimen variability. In addition, Eq. (4) is adopted as the damage evolution raw in which the pseudo-strain energy is expressed as

WR ¼

1 2 DMR  CðSÞðcR Þ 2

ð6Þ

where cR denotes pseudo-strain amplitude. With the steady state assumption, the pseudo-strain time history can be expressed as

cR ðnÞ ¼ c  jG jLVE  sinðxr nÞ

ð7Þ

where xr is reduced angular frequency. Based on Eqs. (5)–(7), the latest expression for damage increment DS reads [8,23]:

D Si ¼

 1þa a  2 1 1 DMR  cRi ðC i1  C i Þ  ðni  ni1 Þ1þa 2

ð8Þ

Eqs. (5) and (8) constitute the basis of the VECD modeling approach currently used for LAS test (and in fact also for TS test). 3.2. Existing failure definition Defining material failure under fatigue loading is always challenging. Ideally, fatigue failure should be determined based on physical characteristics of intensity and distribution of damage within the material body. However, it is unrealistic, if not impossible at all, to capture and monitor material’s internal physical state during the course of loading. The alternative definition from phenomenological perspective has been commonly adopted by researchers and practitioners. In general, developing a phenomenological failure definition calls for a comprehensive and balanced consideration of factors such as experimental observations, theoretical soundness, and analytical convenience. The traditional definition of fatigue failure simply employs a certain percentage of reduction in observed stiffness or modulus. For example, the four-point bending beam test conventionally adopted in fatigue characterization of asphalt mixtures has been using 50% drop in flexural stiffness as the threshold for material failure [24]. In the current AASHTO standard for the beam fatigue test, failure is defined as the point at which the product of flexural stiffness and cycle number reaches the maximum [25]. This alter-

native definition yields a more convenient identification of fatigue life and appears to be indicative of localization of cracking [26]. Given the similar nature of loading in bending beam test and TS test (both under sinusoidal oscillation with constant amplitude), similar failure definitions have been applied in TS [8,27], i.e., 50% drop in modulus, and peak of C  N where N denotes cycle number and C is the normalized modulus as in Eq. (5). When the LAS test was first proposed, Johnson [6] determined 35% reduction in material integrity (as represented by |G⁄|sind) as the threshold of fatigue failure. This level of reduction was selected because good agreement was observed between LAS and TS test results, and also because a satisfactory correlation was found between an LAS-based parameter and fatigue performance of field pavements [6]. Despite the fact that a limited number and types of binders were investigated, this failure definition as well as the problematic formulation summarized by Eqs. (1) through (3) have been standardized in an AASHTO specification and are still retained in the current version [28]. Alternatively, fatigue failure in LAS has been determined by identifying the maximum shear stress amplitude s, and by the peak of C  N as already employed for the TS test. With the linear ramping LAS version that is currently adopted and favored by researchers, these two definitions have been shown to be equivalent [8]. Hence, the peak of C  N can be considered as the first unified definition of fatigue failure for LAS and TS tests. More recently, Wang et al. [8] proposed another unified failure definition, the peak of s  N. It was demonstrated that for the TS test this definition reduces to the peak of C  N given the constant strain amplitude. Meanwhile, for LAS this definition is equivalent to peak of WR(N), which suggests that failure occurs when material starts to lose its capability in storing more pseudo-strain energy. 3.3. Existing failure criterion As is well known, the results of beam fatigue test on asphalt mixtures have been traditionally represented by a power-law relation between the control parameter (e.g., strain or stress level) and the observed number of cycles to failure. This relation can be deemed as an early version of failure criterion, and has been used to discriminate mixtures based on fatigue resistance and to make predictions on fatigue performance of asphalt pavements. However, a major drawback of this methodology lies in that the obtained relationship is dependent on the mode of control (e.g., strain control versus load control) and test temperature. Recently, Sabouri and Kim [12] developed the so-called GR failure criterion which unified all possible test control modes (load, displacement, and on-specimen strain controls) in uniaxial fatigue test on asphalt mixture. It has been demonstrated in a number of studies that this unified failure criterion also appeared to be independent of test temperature and load level [12,29,30]. The GR failure criterion states that for a given asphalt mixture, there exists a unique power-law relationship between GR and the number of cycles to failure:

GR ¼ kðNf Þf

ð9Þ

where k and 1 are regression constants, Nf denotes the number of cycles to failure, and GR is defined as

GR 

W Rr;sum N2f

with W Rr;sum ¼

Nf X

W Rr

ð10Þ

1

in which,

W Rr ¼

1 2 DMR  ð1  C i ÞðcRi Þ 2

ð11Þ

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

where W Rr denotes released pseudo-strain energy in a cycle, and W Rr;sum is the total released pseudo-strain energy accumulated up to failure. The GR failure criterion then found its application in fatigue characterization of asphalt binders using LAS and TS tests. Moreover, it has been demonstrated that this failure criterion is able to unify LAS and TS tests in that for a given binder the GR-Nf relations from the two tests fall on the same curve [8,13]. Despite the success of the GR failure criterion as documented in literature so far, the validity and its unifying nature has been in some sense disguised by the way how GR is defined as in Eq. (10). More detailed discussion will be provided subsequently, prior to the proposition of the new failure criterion.

4. Materials and experimental Program In order to achieve the objectives stated previously, a range of asphalt binders were selected from different sources including both neat and modified binders. In general, asphalt binder can be tested at original (unaged) and aged conditions. Aging using a rolling thin-film oven (RTFO) according to AASHTO T 240 [31] is expected to simulate short-term aging occurring during field production and construction of asphalt mixtures. The RTFO aged binder can then be subjected to further aging in a pressurized aging vessel (PAV) according to AASHTO R 28 [32] in order to simulate long-term aging that occurs over the service period of field pavements. These different aging conditions were used in this study to serve the research objectives. For each asphalt binder at each aging condition used in fatigue characterization, frequency sweep test from 0.1 rad/s to 100 rad/s was first conducted at multiple temperatures of 35°, 20°, and 5 °C. The strain level was controlled at 0.1% such that the material

response was within the linear viscoelastic range. The obtained dynamic shear modulus data were used to construct the master curve in order to determine the damage evolution rate a, as will be described in the next section. In the current AASHTO TP 101 standard for the LAS test, a total of 3100 load cycles are performed at a frequency of 10 Hz with the strain amplitude ramping from 0.1% to 30% in a stepwise manner. Given the linear ramping, the loading profile is characterized by increment of strain amplitude per cycle, herein referred to as cyclic strain rate (CSR). In this study, LAS test was conducted with various CSR values ranging from 0.0033% to 0.03% per cycle, yielding a total of 1000–9000 load cycles over the linear ramping strain from 0.1% to 30%. Further, in order to validate the unifying nature of the proposed failure definition and failure criterion, TS test was also performed which included both strain- and stress-controlled modes. Both LAS and TS tests were conducted at intermediate temperatures ranging from 15° to 25 °C given the independence of the VECD theory on temperature. Typically, these tests have been carried out at temperatures between 18° and 20 °C in the literature. All the frequency sweep, LAS, and TS tests were performed using parallel plates with 8-mm diameter and 2-mm gap geometry in a DSR equipment. The fatigue test matrix is given in Table 1. As shown in the table, the asphalt binders used covered a range of performance grade (PG), and included both neat (unmodified) and Styrene-Butadiene-Styrene (SBS) modified binders. 5. Revised formulation This section presents the revised formulation for the damage characteristic relationship following a rigorous VECD derivation. In addition, given that multiple options currently exist in the literature for the definition of damage evolution rate a, an optimum selection will be identified based on experimental evidences.

Table 1 Fatigue Test Matrix. Set #

Binder Designation

Material Information

Aging Condition

Test Description

Note

1

A

Neat, PG 64-22

Original

LAS:
CSR = 0.030%/cycle,
CSR = 0.015%/cycle,
CSR = 0.010%/cycle,
CSR = 0.030%/cycle,
CSR = 0.015%/cycle,
CSR = 0.010%/cycle,

To validate the revised formulation of C(S) and definition of a

18 °C 18 °C 18 °C 15 °C 20 °C 25 °C

2

A B

Neat, PG 64-22 Neat, PG 58-28

Original, RTFO, PAV

LAS:
CSR = 0.010%/cycle, 18 °C

To validate the proposed new failure definition

3

C

SBS modified, PG 76-22

Original

To validate the proposed new failure definition and failure criterion

4

D

SBS modified, PG 70-28

RTFO

5

E

Neat, PG 70-22

RTFO

6

A

Neat, PG 64-22

Original

LAS:
CSR = 0.010%/cycle, 20 °C
CSR = 0.005%/cycle, 20 °C
CSR = 0.0033%/cycle, 20 °C LAS:
CSR = 0.010%/cycle, 19 °C TS:
Controlled-strain at 5%, 19 °C
Controlled-strain at 7%, 19 °C LAS:
CSR = 0.010%/cycle, 19 °C
CSR = 0.0089%/cycle, 19 °C TS:
Controlled-strain at 3%, 19 °C
Controlled-strain at 5%, 19 °C
Controlled-strain at 7%, 19 °C
Controlled-stress at 758 kPa, 19 °C LAS:
As described in Set 1 TS:
Controlled-strain at 3%, 18 °C
Controlled-strain at 5%, 18 °C
Controlled-strain at 7%, 18 °C

W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

5.1. Revised formulation of damage increment The VECD fatigue modeling theory was originally developed for asphalt mixtures subjected to uniaxial cyclic loading. It should be noted that for mixture analysis the computation of C and S is divided into two parts: transient and cyclic analyses [9,20]. Transient analysis refers to the algorithm used for the first half cycle with ascending load which resembles a monotonic tension test. Cyclic analysis is used for subsequent cycles and as such a pair of representative C and S values are obtained for each cycle. Also note that the transient analysis is necessary to be included because significant damage can be induced in the first cycle due to the considerable strain level imposed. The expression for damage increment DS in transient analysis reads [9]

D Si ¼

 1þa a  2 1 1 DMR  eRi ðC i1  C i Þ  ðni  ni1 Þ1þa 2

ð12Þ

where eRi denotes the pseudo-strain amplitude in the i-th cycle of the uniaxial fatigue test of asphalt mixtures. A comparison between the two expressions for DS in Eqs. (8) and (12) reveals that the current LAS analysis uses an algorithm that was developed for each time step in a monotonic loading scenario. Yet, the LAS test is cyclic by nature, which dictates that the cyclic analysis as used in mixture modeling should be followed. The transient analysis is in fact not needed for LAS because for the first load cycle the 0.1% strain level is well within the linear viscoelastic range and thus no damage is expected to be induced. In the revised formulation, the same definition for the normalized modulus as in Eq. (5) is retained as the indicator of material integrity. The rigorous derivation of DS begins with substitution of pseudo-strain energy, Eq. (6), into the damage evolution law, Eq. (4), yielding


  2 dC a dS 1 ¼  DMR  cR ðnÞ dn 2 dS

ð13Þ

Following a similar assumption as in mixture modeling with dC/dS being constant in a cycle, and separating the damage terms, we have

Z

DS ¼


 Z
  2 a DC a 1 DMR  cR ðnÞ dS ¼   dn 2 DS

ð14Þ

which with Eq. (7) simplifies into


1þa a Z 1þ1 a  2 1 2a DS ¼  DMR  cR DC  ðsinðxr nÞÞ dn 2

ð15Þ

Adding back the subscript for cycle number, the above expression can be rewritten as


1þa a  2 1 DMR  cRi ðC i1  C i Þ  Q with 2 Z 1þ1 a 2a Q ðsinðxr nÞÞ dn

D Si ¼

ð16Þ

Note that the integrations in Eqs. (14)–(16) are performed over a loading cycle. Also note that Eq. (16) is applicable to both LAS and TS tests. The obtained C and S data can then be cross-plotted to produce the damage characteristic curve, which can be fitted using the following forms:

C ¼ 1  yS z

ð17Þ

C ¼ expðaSb Þ

ð18Þ

where y, z, a and b are regression coefficients. Use of the power form in Eq. (17) is typically preferred considering its ease provided in analytical manipulation.

5

Comparing the Q term in Eq. (16) and the term ðni  ni1 Þ1=ð1þaÞ in Eq. (8), it can be seen that the rigorous derivation realistically captures the sinusoidal loading feature. Additionally, it is worth mentioning that Eq. (16) follows a similar but simpler form of DS than that for asphalt mixtures [9] due to the absence of the form factor. The form factor is required in modeling uniaxial fatigue testing of asphalt mixtures in order to account for the time period during which damage evolves in a cycle. In LAS and TS tests where asphalt binder is subjected to oscillatory shearing, damage is assumed to grow continuously throughout the loading period of a cycle. Further investigation on Q and ðni  ni1 Þ1=ð1þaÞ yields that for varying reduced angular frequency xr, these two terms are related through a proportionality constant depending solely on a. Hence, use of the two different formulations for DS results in different damage characteristic curves but the horizontal axes are related via a scaling constant for a given asphalt binder. This scaling constant is a function of the material constant a alone and is independent of loading conditions (frequency and temperature). As a result, prior observations and findings in the literature should still hold when the existing formulation is substituted with the revised one. Subsequent analyses are based on the revised formulation. However, it is important to note that issues to be revealed with the existing failure definition and failure criterion using the revised C(S) formulation would also apply to the case using the existing formulation, for the reasons stated above. 5.2. Definition of a In the fatigue modeling of asphalt mixtures, the damage evolution rate a is defined via a combination of theoretical deduction [33] and experimental exploration [20]: a = 1/m + 1 for displacement or strain controlled mode and a = 1/m for load controlled mode, where m is the absolute value of the maximum log-log slope of the linear viscoelastic relaxation modulus. One experimental criterion used in selecting the a definition is to collapse the resulting C(S) curves from different test conditions. In modeling fatigue of asphalt binders in LAS and TS tests, both definitions have been employed with various identifications of the m-value. The definition a = 1/m + 1 has been widely used with m varying from a selection of the absolute value of the maximum (steady-state) log-log slope of the relaxation modulus G(t) [6], the storage modulus G0 (x) [5], or the dynamic modulus |G⁄| [13]. Use of a = 1/m is also available in the literature with m determined as the absolute value of the maximum log-log slope of the dynamic modulus [8,23,34]. It should be pointed out that in the current AASHTO standard for the LAS test [28], a = 1/m + 1 is adopted but the definition of m is questionable. In this standard m is defined as the exponent of a power-law fit of the storage modulus G0 (x) obtained through a frequency sweep prior to the amplitude sweep at the same temperature. Apparently, the resulting m-value and thus a depend on the specific test temperature, which violates the premise of the VECD theory that a is a material constant and should be independent of test conditions. Hintz et al. [5] demonstrated that the m-values obtained using the relaxation modulus G(t) versus the storage modulus G0 (x) exhibited no statistically significant difference, and thus proposed the use of storage modulus to simplify the analysis. For similar reasons, use of relaxation modulus has been further substituted by dynamic modulus |G⁄| which avoids conversion and thus eliminates the need of phase angle. An additional benefit is that the obtained m-value is less affected by experimental errors if only the |G⁄| data are used, as in general |G⁄| can be more accurately measured than phase angle.

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

5.3. Experimental Validation Given the above considerations, in the following analysis m is determined using dynamic modulus |G⁄|, while the two definitions a = 1/m + 1 and a = 1/m will be investigated using the LAS data on the neat binder A with PG 64-22 at the original (unaged) condition, i.e., Set #1 in Table 1. The LAS test covered three CSRs (0.010%, 0.015%, and 0.030% per cycle) and four test temperatures (15°, 18°, 20°, and 25 °C). For each condition, one replicate was employed. The m-value is identified by fitting the dynamic modulus master curve using the CAM model [35]:


g m=g xc jG j ¼ Gg 1 þ

ð19Þ

xr

where g is a regression constant related to the shape of the master curve; Gg is glassy modulus typically taken as 1 GPa; xc is crossover angular frequency; and xr is reduced angular frequency, xr = xaT, where x denotes the physical angular frequency and aT the time-temperature shift factor. The time-temperature shift factor function is represented by

log aT ¼ a1 ðT  T R Þ2 þ a2 ðT  T R Þ

ð20Þ

where TR is reference temperature, and a1 and a2 are regression coefficients. The frequency sweep test data at multiple temperatures on binder A were used to construct the |G⁄| master curve according to Eqs. (19) and (20). For the interest of space, the test results are not presented herein. The identified m-value and the shift factor function were used in the VECD formulation to determine the damage characteristic curve.

CSR=0.030%/cycle, 18°C

CSR=0.015%/cycle, 18°C

CSR=0.010%/cycle, 18°C

CSR=0.030%/cycle, 15°C

CSR=0.015%/cycle, 20°C

CSR=0.010%/cycle, 25°C

1 0.8

C

0.6 0.4

α = 1/m 0.2 0

(a)

0

500

1000

1500

S 1

0.8

C

0.6 0.4

α = 1/m + 1

0.2 0

(b)

0

1000

2000

3000

4000

S

Fig. 1. Damage characteristic curves for the two definitions of a: (a) a = 1/m, and (b) a = 1/m + 1.

The resulting damage characteristic curves from the LAS tests with various test conditions are shown in Fig. 1 for the two definitions of a. Note that the computation of C and S were performed in the reduced-time domain by applying the time-temperature superposition principle with a reference temperature selected at 20 °C. Note that the applicability of the linear viscoelastic timetemperature shift factors to asphalt materials (binders and mixtures) with fatigue damage that is modeled using the VECD theory has been explored and well established in the literature [13,34,36,37]. Compared to existing experimental investigation [13] on the definition of a, this study employed a wider range of test conditions with various CSRs and test temperatures as shown in the figure. The two extreme conditions were 0.03% per cycle CSR at 15 °C and 0.01% per cycle CSR at 25 °C. Fig. 1(b) shows that the use of the definition a = 1/m + 1 was able to collapse all C(S) curves from a range of test conditions. Given this observation, it can be concluded that a = 1/m + 1 should be adopted in analyzing the LAS fatigue data of asphalt binders. In summary, Eqs. (5) and (16) constitute the basis of the revised C(S) formulation, in which the damage evolution rate a is defined as a = 1/m + 1 with m being the absolute value of the maximum log–log slope of the dynamic shear modulus |G⁄|.

6. Failure definition As previously discussed, currently there exist two unified definitions of fatigue failure in both LAS and TS tests: peak of C  N, and peak of s  N. Despite their success in certain applications, use of these definitions may produce results inconsistent with general experimental observations and engineering experience. Such inadequacy will be illustrated using experimental data in this section, followed with the proposed new failure definition. In order to assess the validity of the current failure definitions, the experimental data from Set #2 were employed. In this set, the LAS test with CSR = 0.01% per cycle was conducted on binder A (PG 64-22) and B (PG 58-28) at the temperature of 18 °C. Each binder was evaluated at three aging conditions: original (unaged), RTFO aged, and PAV aged. Two replicates was used for each binder at each aging condition. The obtained data were processed to determine the fatigue life, i.e., the number of cycles to failure Nf. Considering the same test condition, aging is expected to reduce the fatigue resistance or Nf for a given asphalt binder. In addition, at the same aging level, soft binder (i.e., binder B, PG 58-28) would exhibit a longer fatigue life in the LAS test (which is strain controlled). Fig. 2(a) and (b) present the fatigue life results based on the existing two unified definitions of fatigue failure. Error bars representing standard deviation are included to illustrate the test variability. The maximum coefficients of variation were 2.3% and 3.3% for failure definitions using peak of C  N and s  N, respectively. Fig. 3 provides an example of the identified failure points located on the stress amplitude curves. It can be seen from Fig. 2(a) that the failure definition by peak of C  N could not always yield lower Nf values for increased aging levels, and as previously mentioned this definition identified failure at maximum stress points (Fig. 3). On the other hand, the failure definition by peak of s  N resulted in longer fatigue lives for higher aging levels and slightly lower Nf for the soft binder B with PG 58-28, as shown in Fig. 2(b). This observation is contrary to general engineering experience and experimental evidences. Additionally, as illustrated in Fig. 3, this definition identified failure at the point where the stress amplitude curve started to drop abruptly. Given the inadequacies of the existing failure definitions as demonstrated in the above, after an extensive exploration a new failure definition is proposed which identifies fatigue failure at

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

Number of Cycles to Failure

1500

A: PG 64-22

Peak of C×N

B: PG 58-28

1300

1100

900

700

(a)

Original

RTFO

PAV

Number of Cycles to Failure

1500

Peak of τ×N

the peak of C2  N  (1-C). Results based on this new definition are included in Figs. 2(c) and 3 for comparison. It can be seen from Fig. 2(c) that the proposed definition was able to clearly discriminate the two binders. Moreover, it yielded a reasonable trend that increase in the aging level reduced fatigue life. The observed maximum coefficient of variation was 1.6%, which is smaller than those for the two existing failure definitions. As shown in Fig. 3, for the original binder the failure point identified by the proposed failure definition is in the neighborhood of those determined by the two existing definitions. For the RTFO and PAV aged binder, the proposed definition yielded similar but improved failure identification compared to the definition by peak of C  N. Based on the above observation, the proposed failure definition can be considered as a fine-tuned version of the existing definition, peak of C  N. This perspective can be illustrated by rewriting C2  N  (1-C) as C  N  (C-C2), which compared to C  N includes the additional term (C-C2) that varies within a narrow range of 0–0.25.

1300

7. Failure criterion As previously mentioned, the GR failure criterion was originally proposed as a unified criterion in modeling fatigue failure of asphalt mixtures. The unifying nature lies in that the obtained GR-Nf relationship was independent of control mode. The same concept has been adopted to approach fatigue of asphalt binder under oscillatory shearing, and its unifying nature was seen to encompass both LAS and TS tests with failure defined at the peak of s  N. In this section, however, it will be revealed and demonstrated that in certain cases the GR failure criterion may fail to retain as a reliable relationship describing fatigue failure. A new unified failure criterion will be proposed for asphalt binder and then validated using both LAS and TS data covering various test conditions.

1100

(b)

900

700

Original

RTFO

PAV

Number of Cycles to Failure

1500

(c)

Peak of C2×N×(1-C) 1300

7.1. Issue with the existing failure criterion

1100

As can be seen from the defining equation for GR, Eq. (10), GR only serves as an intermediate variable. If Eq. (10) is substituted into (9), the following power-law relation is obtained:

900

W Rr;sum ¼ kðNf Þfþ2

700

Original

RTFO

PAV

Fig. 2. Number of cycles to failure based on the existing failure definitions: (a) peak of C  N, (b) peak of s  N, and the proposed new definition: (c) peak of C2  N  (1-C).

Original

8.E+05

Stress (Pa)

RTFO PAV

6.E+05

Peak of C×N Peak of tau×N Peak of C^2N(1-C)

4.E+05 2.E+05 0.E+00

ð21Þ W Rr;sum

In the above equation, is to be first determined using experimental data for calculating GR. It is critical to point out that, a strong relationship between GR and Nf does not always guarantee an adequate correlation between W Rr;sum and Nf, which will be demonstrated using data (on both asphalt mixtures and binders) collected from literature in what follows. Fig. 4 presents the GR-Nf relations collected from several studies and the corresponding W Rr;sum -Nf relationships. Fig. 4(a) depicts the obtained failure criterion of the mixture labelled as NH6400-opt in Fig. 3(b) of [29]. Fig. 4(c) presents the relationship determined using both LAS and TS test data on a PG 70–22 binder in Fig. 10(d) of [13]. Fig. 4(e) provides the GR-Nf relation using both LAS and TS tests on an aged PG 64-22 binder in Fig. 7(a) of [8]. Apparently, in all these three selected cases GR is strongly correlated to Nf, as evidenced by the high R2 values. However, if GR is multiplied twice by Nf to obtain W Rr;sum , poor correlations are

0

1000

2000

3000

Cycle Number Fig. 3. Failure points identified on the stress amplitude curves (B: PG 58-28) using the existing and proposed failure definitions.

revealed between W Rr;sum and Nf, as shown in Fig. 4(b), (d), and (f). Particularly, the worst scenario is demonstrated in Fig. 4(f) in which W Rr;sum and Nf is essentially not related despite the nearperfect correlation between GR and Nf in Fig. 4(e). A closer investigation suggests that the strong relationship typically observed between GR and Nf can be misleading; it may be dis-

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

1.E+04

1.E+11 R² = 0.996

R² = 0.606

WRr,sum

GR

1.E+03 1.E+02 1.E+01

(a)

1.E+00 1.E+03

1.E+04

1.E+05

Nf

1.E+10 1.E+03

R² = 0.978

R² = 0.715

GR

WRr,sum

1.E+09

1.E+07

(c)

1.E+04

1.E+06

Nf

1.E+15

(d)

1.E-01

1.E+14 1.E+02

1.E+06

Nf

R² = 0.999

R² = 0.071

WRr,sum

GR 1.E-05

(e)

1.E+04

1.E+04

1.E-03

1.E-07 1.E+02

1.E+05

Nf 1.E+16

1.E+11

1.E+05 1.E+02

1.E+04

(b)

1.E+03

1.E+04

1.E+05

Nf

(f)

1.E+03

1.E+02 1.E+02

1.E+03

1.E+04

1.E+05

Nf

Fig. 4. Illustration of inadequacy of the GR failure criterion using the resulting W Rr;sum -Nf relations: (a) and (b) asphalt mixture [29], (c) and (d) asphalt binder with both LAS and TS data [13], and (e) and (f) asphalt binder with both LAS and TS data [8].

guised by the way how GR is defined. It is noted that Nf usually ranges over several orders of magnitude whereas W Rr;sum spans over a much fewer order of magnitude, as can be seen in Fig. 4. Hence, when dividing W Rr;sum twice by Nf to obtain GR, GR would turn out to be strongly affected by and thus correlated with Nf as a result of this data manipulation technique. Validity of this failure criterion should be examined by the W Rr;sum -Nf relation, not indirectly and doubtfully by the relationship between the intermediate variable GR and Nf.

ramps linearly in LAS but remains constant in TS. Therefore, when both LAS and TS data are employed in establishing the failure criterion, they should be treated individually and discreetly according to their different natures in the loading profile. Additionally, it is worth pointing out that the new failure criterion should be developed based on the new failure definition proposed previously, i.e., peak of C2  N  (1-C). Given the above considerations, the new failure criterion is proposed as a power law relating the total pseudo-strain energy WRsum with a variable named Straining Effort (SE) as a function of the loading condition:

7.2. Proposed failure criterion

W Rsum ¼ j  SEl

Given the unreliability of the existing GR failure criterion as illustrated in the above, a new failure criterion is proposed for fatigue modeling of asphalt binder below, and its unifying nature will be demonstrated. (A new failure criterion is still in need for asphalt mixtures but is beyond the scope of this study). In the current application of the GR criterion to LAS and TS tests, fatigue life as designated by Nf is used in correlating GR for both tests without discrimination. It is important to realize that, however, for the same Nf value materials in the two tests undergo totally different loading histories, because the strain amplitude

where j and l are regression constants, and SE is defined by

SE 

Nf X

ð22Þ

!

ci  ðjG jLVE Þ2

ð23Þ

i¼1

where ci denotes the strain amplitude in the i-th cycle, and |G⁄|LVE is the undamaged shear modulus. SE is used to represent the amount of mechanical effort required to deform and damage the material until failure. Note that SE is not the physical work input to the material during the course of fatigue loading. Incorporation of

W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

|G⁄|LVE in SE is for the purpose of accommodating the effects of temperature and loading frequency given the viscoelastic nature of asphalt binders, as more effort is required to deform and damage the material when tested under lower temperatures and/or higher frequencies. Further, the rationale behind the use of exponent 2 on |G⁄|LVE lies in the consistent consideration of the role of |G⁄|LVE in both WRsum and SE; the purpose is to eliminate the temperature and frequency dependence in the resulting WRsum-SE relation. 8. Validation of the proposed failure definition and failure criterion This section is devoted to validating the proposed failure definition and failure criterion. Their applicability and unifying nature will be demonstrated using a series of experimental sets including both LAS and TS tests with various test conditions as listed in Table 1. In what follows, validation is performed progressively in the order with increasing range of test conditions and thus with elevated challenges. For comparison, the existing GR failure criterion is also provided, but in the form of W Rr;sum -Nf relations for the reasons stated previously. Also note that the existing GR failure criterion should be obtained using the existing failure definition, peak of s  N. Likewise, the proposed failure criterion should be established based on the proposed definition which uses peak of C2  N  (1-C).

9

0.010%, 0.005%, and 0.0033% per cycle. At each CSR two replicates were tested. The obtained failure criterion relationships are provided in Fig. 5. Note that one replicate with 0.010% per cycle CSR was identified as an outlier and thus not included in the analysis. As shown in Fig. 5(a), using the existing failure definition, W Rr;sum was poorly correlated with Nf, thereby yielding a weak failure criterion. On the contrary, the new failure criterion relating WRsum and SE proved a strong correlation based on the proposed failure definition, as seen in Fig. 5(b). 8.2. LAS and TS data at a single temperature The experimental data of Set #4 in Table 1 were used to verify the capability of the failure definitions and failure criteria in unifying both LAS and TS tests. This experimental set employed an RTFO aged, SBS modified binder D with PG 70-28. The LAS test was conducted at a single CSR of 0.010% per cycle with two replicates. The TS test was performed at the strain-controlled mode with 5% and 7% strain levels, each with one replicate. All LAS and TS testing was carried out at the temperature of 19 °C. Fig. 6 presents the obtained failure criteria. It is noted that the existing and proposed analysis schemes resulted in failure criteria with similar adequacies. 8.3. LAS and TS data at a single temperature with mixed control modes

8.1. LAS data at a single temperature The experimental data of Set #3 in Table 1 were first used to examine the validity of the existing and proposed failure definitions and failure criteria. In this set, LAS test was performed at 20 °C on the SBS modified binder C with PG 76-22 at the original condition. The LAS loading profile covered three different CSRs:

The experimental data of Set #5 in Table 1 were used to further validate the unifying nature of the proposed analysis framework. This experimental set employed an RTFO aged, neat binder E with PG 70-22. The LAS test was conducted at two CSRs: 0.010% and 0.0089% per cycle, each with one replicate. The TS test was performed at both strain-controlled (3%, 5%, 7%) and stresscontrolled (758 kPa) modes with one replicate for each condition. All testing was carried out at a temperature of 19 °C.

WRr,sum

1.E+08

R² = 0.335

(a)

1.E+07 1.E+03

Nf

1.E+04

WRsum

1.E+08

1.E+07

R² = 0.974

(b)

1.E+06 1.E+08

1.E+09

1.E+10

SE

Fig. 5. Validation using Set #3 with LAS data at a single temperature: (a) existing failure criterion, and (b) proposed failure criterion.

Fig. 6. Validation using Set #4 with both LAS and TS data at a single temperature: (a) existing failure criterion, and (b) proposed failure criterion.

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W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

Fig. 7 presents the obtained failure criteria using the existing and proposed analysis schemes. It is obviously seen from Fig. 7(a) that the existing failure definition and failure criterion presented a fairly weak relationship between W Rr;sum and Nf. Use of the proposed analysis framework, however, was able to accommodate both LAS and TS tests and both stress and strain controlled modes via a single WRsum-SE relationship, as show in Fig. 7(b). 8.4. LAS and TS data at multiple temperatures

Fig. 7. Validation using Set #5 with both LAS and TS data including mixed control modes at a single temperature: (a) existing failure criterion, and (b) proposed failure criterion.

The experimental data of Set #6 in Table 1 were employed as the final step in validating the proposed failure definition and failure criterion. This data set consists of Set #1 and additional TS test results on the same binder, neat binder A with PG 64-22 at the original condition. The additional TS tests were conducted at 18 °C with strain-controlled mode at 3%, 5%, and 7% strain levels, each with one replicate. Hence, Set #6 covers both LAS and TS data from various loading conditions with multiple temperatures. The identified failure criteria are given in Fig. 8. Note that the analysis was performed at a reference temperature of 20 °C, for this purpose the time-temperature shift factor obtained from constructing the dynamic shear modulus master curve has been applied. It is clearly demonstrated that use of this existing failure definition and failure criterion was not able to accommodate LAS data across different temperatures; see Fig. 8(a). Further, it is observed that even at the same temperature, 18 °C, the LAS and TS data points followed totally different trends. Use of the proposed failure definition and failure criterion, however, proved their excellent capability and versatility in unifying LAS and TS tests at multiple temperatures, as illustrated in Fig. 8(b).

Fig. 8. Validation using Set #6 with both LAS and TS data at multiple temperatures: (a) existing failure criterion, and (b) proposed failure criterion.

W. Cao, C. Wang / Construction and Building Materials 171 (2018) 1–12

11

9. Summary and conclusions

References

In this study, the existing analysis framework for fatigue characterization of asphalt binders using the LAS test was critically reviewed. Issues in the existing formulation for the damage characteristic relationship were pointed out, and the inadequacy of the currently adopted failure definition and failure criterion was revealed using LAS and TS data generated in this study and those collected from literature. The objectives were to derive a rigorous formulation following the linear viscoelastic continuum damage theory, and to propose a reasonable failure definition and a robust failure criterion which unify both LAS and TS tests. The developed analysis framework was experimentally validated. The following conclusions can be drawn based on the findings in this study:

[1] H.U. Bahia, H. Zhai, M. Zeng, Y. Hu, P. Turner, Development of binder specification parameters based on characterization of damage behavior, J. Assoc. Asphalt Paving Technol. 70 (2001) 442–470. [2] H.U. Bahia, D.I. Hanson, M. Zeng, H. Zhai, M.A. Khatri, R.M. Anderson, Characterization of Modified Asphalt Binders in Superpave Mix Design. NCHRP Report 459 (Project 9-10), Transportation Research Board, Washington, D.C., 2001. [3] K.S. Bonnetti, K. Nam, H.U. Bahia, Measuring and defining fatigue behavior of asphalt binders, Transp. Res. Rec. 2002 (1810) 33–43. [4] A.C. Pronk, P.C. Hopman, Energy Dissipation: The Leading Factor of Fatigue, Proceedings of the Conference of the United States Strategic Highway Research Program, London, 1990. [5] C. Hintz, R. Velasquez, C. Johnson, H. Bahia, Modification and validation of Linear Amplitude Sweep test for binder fatigue specification, Transp. Res. Rec. 2207 (2011) 99–106. [6] C.M. Johnson, Estimating Asphalt Binder Fatigue Resistance Using an Accelerated Test Method Ph.D. Dissertation, University of WisconsinMadison, Madison, WI, 2010. [7] C. Hintz, H.U. Bahia, Simplification of Linear Amplitude Sweep test and specification parameter, Transp. Res. Rec. 2370 (2013) 10–16. [8] C. Wang, C. Castorena, J. Zhang, Y.R. Kim, Unified failure criterion for asphalt binder under cyclic fatigue loading, J. Assoc. Asphalt Paving Technol. 84 (2015) 269–299. [9] W. Cao, L.N. Mohammad, M. Elseifi, Assessing the effects of RAP, RAS, and warm-mix technologies on fatigue performance of asphalt mixtures and pavements using viscoelastic continuum damage approach, Road Mater. Pavement Des. 18 (s4) (2017) 353–371. [10] H.J. Park, M. Eslaminia, Y.R. Kim, Mechanistic evaluation of cracking in inservice asphalt pavements, Mater. Struct. 47 (2014) 1339–1358. [11] AASHTO. Standard method of test for determining the damage characteristic curve of asphalt mixtures from direct tension cyclic fatigue tests. AASHTO TP 107, 2016, Washington D.C. [12] M. Sabouri, Y.R. Kim, Development of a failure criterion for asphalt mixtures under different modes of fatigue loading, Transp. Res. Rec. 2447 (2014) 117– 125. [13] F. Safaei, C. Castorena, Y.R. Kim, Linking asphalt binder fatigue to asphalt mixture fatigue performance using viscoelastic continuum damage modeling, Mech. Time-Depend. Mater. 20 (2016) 299–323. [14] R. Micaelo, A. Pereira, L. Quaresma, M.T. Cidade, Fatigue resistance of asphalt binders: assessment of the analysis methods in strain-controlled tests, Constr. Build. Mater. 98 (2015) 703–712. [15] F.E. Pérez-Jiménez, R. Botella, R. Miró, Differentiating between damage and thixotropy in asphalt binder’s fatigue tests, Constr. Build. Mater. 31 (2012) 212–219. [16] E. Santagata, O. Baglieri, L. Tsantilis, D. Dalmazzo, Evaluation of self-healing properties of bituminous binders taking into account steric hardening effects, Constr. Build. Mater. 41 (2013) 60–67. [17] F. Canestrari, A. Virgili, A. Graziani, A. Stimilli, Modeling and assessment of self-healing and thixotropy properties for modified binders, Int. J. Fatigue 70 (2015) 351–360. [18] B.S. Underwood, A continuum damage model for asphalt cement and asphalt mastic fatigue, Int. J. Fatigue 82 (3) (2016) 387–401. [19] Y. Kim, H.J. Lee, D.N. Little, Y.R. Kim, A simple testing method to evaluate fatigue fracture and damage performance of asphalt mixtures, J. Assoc. Asphalt Paving Technol. 75 (2006) 755–787. [20] B.S. Underwood, Y.R. Kim, M.N. Guddati, Improved calculation method of damage parameter in viscoelastic continuum damage model, Int. J. Pavement Eng. 11 (2010) 459–476. [21] R.A. Schapery, Correspondence principles and a generalized J-integral for large deformation and fracture analysis of viscoelastic media, Int. J. Fract. 25 (1984) 195–223. [22] S.W. Park, Y.R. Kim, R.A. Schapery, A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete, Mech. Mater. 24 (1996) 241–255. [23] F. Safaei, J.-S. Lee, L.A.H. do Nascimento, C. Hintz, Y.R. Kim, Implications of warm-mix asphalt on long-term oxidative ageing and fatigue performance of asphalt binders and mixtures, Road Mater. Pavement Des. 15 (2014) 45–61. [24] S. Shen, S. Carpenter, Development of an asphalt fatigue model based on energy principles, J. Assoc. Asphalt Paving Technol. 76 (2007) 525–573. [25] AASHTO. Standard method of test for determining the fatigue life of compacted asphalt mixtures subjected to repeated flexural bending. AASHTO T 321, 2014, Washington D.C. [26] G.M. Rowe, P. Blankenship, M.J. Sharrock, T. Bennert. The fatigue performance of asphalt mixtures in the four point bending beam fatigue test in accordance with AASHTO and ASTM analysis methods. 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The existing formulation for damage intensity (S) only applies to a monotonic loading profile, not for repeated fatigue loading as in the LAS and TS tests. The revised formulation was rigorously derived from the VECD theory and captures the cyclic nature of fatigue loading.
Experimental evidences suggested that the damage evolution rate, a, should be defined as a = 1/m + 1, where m denotes the absolute value of the maximum log-log slope of the dynamic shear modulus.
Fatigue lives identified according to the existing failure definitions, peak of C  N and s  N, were not able to reasonably differentiate and rank asphalt binders with different performance grades and with different aging conditions.
The proposed failure definition, peak of C2  N  (1-C), provided fatigue life results consistent with general engineering experience and experimental observations. Based on this new definition, soft asphalt is more fatigue resistant than stiff binder (in strain controlled loading conditions) and increase in the aging level would deteriorate fatigue resistance.
The existing GR failure criterion proved to be unreliable in certain cases due to the use of the intermediate variable GR, in spite of the high R2 values typically observed. The validity of this failure criterion should be examined by the relationship between the total released pseudo-strain energy W Rr;sum and fatigue life Nf.
The Straining Effort (SE) was established as a new variable to represent the mechanical effort required to deform and damage materials up to failure under various test conditions.
The new failure criterion was proposed as a power law relating the total pseudo-strain energy WRsum and SE. Using the experimental data covering a range of materials, the proposed failure criterion proved capable and versatile to accommodate and unify both LAS and TS tests with various test conditions (test temperature, loading rate, load level, and control mode). Future work is expected to include further development of the proposed analysis framework for performance evaluation and prediction using laboratory and field fatigue data of asphalt materials and pavements. Disclosure statement No potential conflict of interest was reported by the authors. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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[29] M. Sabouri, T. Bennert, J.S. Daniel, Y.R. Kim, Fatigue and rutting evaluation of laboratory-produced asphalt mixtures containing reclaimed asphalt pavement, Transp. Res. Rec. 2506 (2015) 32–44. [30] W. Cao, A. Norouzi, Y.R. Kim, Application of viscoelastic continuum damage approach to predict fatigue performance of Binzhou perpetual pavements, J. Traffic Transp. Eng. 3 (2016) 104–115. [31] AASHTO. Standard method of test for effect of heat and air on a moving film of asphalt (rolling thin-film oven test). AASHTO T 240, 2017, Washington D.C. [32] AASHTO. Standard practice for accelerated aging of asphalt binder using a pressurized aging vessel (PAV). AASHTO R 28, 2016, Washington D.C. [33] R.A. Schapery, A theory of crack initiation and growth in viscoelastic media II. Approximate method of analysis, Int. J. Fracture 11 (1975) 369–388.

[34] C. Wang, C. Castorena, J. Zhang, Y.R. Kim, Application of time-temperature superposition principle on fatigue failure analysis of asphalt binder, J. Mater. Civil Eng. 29 (2017), https://doi.org/10.1061/(ASCE)MT.1943-5533.0001730. [35] M.O. Marasteanu, D.A. Anderson, Improved model for bitumen rheological characterization, Eurobitume Workshop - Performance Related Properties of Bituminous Binders, Luxembourg, May 1999. [36] G.R. Chehab, Y.R. Kim, R.A. Schapery, M.W. Witczak, R. Bonaquist, Timetemperature superposition principle for asphalt concrete with growing damage in tension state, J. Assoc. Asphalt Paving Technol. 71 (2002) 559–593. [37] B.S. Underwood, Y.R. Kim, M.N. Guddati, Characterization and performance prediction of ALF mixtures using a viscoelastoplastic continuum damage model, J. Assoc. Asphalt Paving Technol. 75 (2006) 577–636.