International Journal of Solids and Structures 50 (2013) 2901–2913

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International Journal of Solids and Structures

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Constitutive modeling of fatigue damage response of asphalt concretematerials with consideration of micro-damage healing

0020-7683/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijsolstr.2013.05.007

⇑ Corresponding author. Tel.: +971 2810 9162; fax: +971 2810 9901.E-mail address: rabualrub@masdar.ac.ae (R.K. Abu Al-Rub).

Masoud K. Darabi a, Rashid K. Abu Al-Rub b,⇑, Eyad A. Masad a,c, Dallas N. Little a

a Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USAb Mechanical Engineering, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emiratesc Mechanical Engineering Program, Texas A&M University at Qatar, Doha, Qatar

a r t i c l e i n f o

Article history:Received 10 May 2012Received in revised form 18 March 2013Available online 16 May 2013

Keywords:Fatigue damageMicro-damage healingViscoelasticityViscoplasticityAsphalt concrete

a b s t r a c t

A continuum mechanics-based viscodamage (VD) constitutive relationship is proposed to model fatiguedamage of asphalt concrete. The form for the evolution of the viscodamage function is postulated basedon the damage density which was determined from uniaxial constant strain rate tests that were per-formed at different strain rates. The proposed viscodamage model is coupled with Schapery’s nonlinearviscoelastic (VE), Perzyna’s viscoplastic (VP), and micro-damage healing (H) models to simulate the non-linear mechanical response of asphalt concrete during fatigue. Numerical algorithms are implemented inthe finite element code Abaqus via the user material subroutine UMAT. The proposed model is validatedagainst extensive experimental data including constant strain rate, cyclic displacement controlled, andcyclic stress controlled tests over a range of temperatures, strain rates, loading frequencies, and stress/strain levels/amplitudes. The model predictions show that the VE–VP–VD–H model is capable of predict-ing the fatigue damage response of asphalt concrete subjected to different loading conditions. The resultsdemonstrate that micro-damage healing occurs not only during the rest period, but also during the cyclicstrain controlled tests even in the absence of the resting time.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fatigue damage and permanent deformation are the distressesthat cause the most concern of asphalt concrete as they affect bothsafety and serviceability over the performance life of asphaltpavements. The repetitive actions of traffic loading cause theprogressive evolution of micro-damage (i.e. micro-cracks andmicro-voids) and permanent strain in asphalt concrete pavementsleading to fatigue damage and rutting (i.e. permanent deforma-tion), respectively. Rutting is more dominant at higher tempera-tures while fatigue damage prevails at lower and intermediatetemperatures (e.g. Morrison et al., 1994; Uzan, 1996; Tashmanet al., 2005). This paper focuses on modeling the fatigue damage re-sponse of asphalt concrete at low and intermediate temperatures.Modeling the permanent deformation response of asphalt concretepavements at high temperatures has been discussed thoroughly inrecent papers by the authors (e.g. Masad et al., 2007; Huang et al.,2011a, 2011b; Abu Al-Rub et al., 2012; Darabi et al., in press,2012d).

Schapery (1969) viscoelastic model and Perzyna (1971) visco-plastic model are among the classical theories that have been usedextensively by several researchers to predict the viscoelastic and

viscoplastic responses of asphalt concrete materials, respectively(e.g. Seibi et al., 2001; Masad et al., 2005; Huang et al., 2007,2011b; Saadeh et al., 2007; Abu Al-Rub et al., 2009; Darabi et al.,2011). However, these models should be coupled with micro-dam-age evolution in order to capture specific phenomena such as fati-gue damage, tertiary creep, and post-peak behavior of stress–strainresponse.

Several researchers (e.g. Kim and Little, 1990; Park et al.,1996; Lee et al., 2000) have coupled Schapery (1975a, b) damagemodel with VE–VP models using the extended correspondenceprinciple (Schapery, 1984). However, models based on this ap-proach have only been used to predict damage behavior undertensile loading conditions. Therefore, they cannot be used inmodeling the damage evolution for general cases when the mate-rial exhibits inherent nonlinear viscoelastic–viscoplastic behavior(Abbas et al., 2004; Huang et al., 2007; Delgadillo et al., 2012)and also when the material is subjected to multi-axial state ofstresses occurring in the asphalt pavement structures subjectedto traffic loadings (e.g. Gibson et al., 2009; Wang and Al-Qadi,2010).

Models based on continuum damage mechanics (CDM) can beconsidered as alternatives to model the effect of micro-damageevolution on the mechanical response of materials subjected tomulti-axial stress states (e.g. Kachanov, 1958; Rabotnov, 1969;Lemaître, 1992). The simplicity associated with the coupling of

2902 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

the damage models to the complex constitutive models as well asthe straightforward procedure associated with their numericalimplementation techniques are the primary reasons for the popu-larity of continuum damage mechanics-based models. This ap-proach has recently been used by the authors to formulate acoupled VE–VP–VD constitutive model that can be used for pre-dicting the damage response of asphalt concrete under differentloading conditions (e.g. Abu Al-Rub et al., 2010; Darabi et al., 2011).

Micro-damage healing is another important factor that shouldbe included in constitutive models to predict the fatigue responseof asphalt concrete materials more accurately. During the rest per-iod between the loading cycles, some induced micro-damage healsleading to the recovery of the material strength. The models thatdo not consider micro-damage healing and the strength recoveryduring the rest period underestimate the fatigue life of asphaltpavements subjected to realistic cyclic loading (Abu Al-Rub et al.,2010).

2. Objectives and methodologies

The above discussion shows that fatigue damage in asphalt con-crete is a result of several interacting mechanisms. On one hand,damage evolution and growth degrades the mechanical propertiesof the material and shortens the fatigue life. On the other hand,healing of micro-damage surpasses the rate of the damage evolu-tion and contribute to extending the fatigue life. Finally, stressrelaxation and the viscoelastic nature of asphalt concrete materialsadd to the time-dependency of micro-damage evolution in asphaltconcrete. These mechanisms are coupled and occur concurrentlyduring general loading conditions. Therefore, the objective of thisstudy is to develop a constitutive model that can be used to realis-tically predict the fatigue damage response of asphalt concretesubjected to general loading conditions by considering these inter-acting mechanisms while maintaining a straightforward system-atic ability to characterize these interacting mechanisms.

The following steps are taken to achieve this objective:

- Extend the continuum-damage mechanics theory to the contin-uum damage-healing mechanics theory.

- Discuss the mechanism of micro-damage healing during cyclicloadings and demonstrate that healing doesn’t only occur dur-ing rest periods but also residual (internal) compressive stressescan aid in the micro-damage healing of asphalt materials.

- Propose a general continuum-based viscodamage model thatcan capture the response of asphalt concrete subjected to gen-eral loading conditions

- Present a simple procedure for identification of the damagemodel parameters.

- Validate the model against different cyclic and monotonic load-ing conditions at different temperatures.

3. Continuum damage-healing mechanics

Kachanov (1958), Odqvist and Hult (1961), and Rabotnov(1969) pioneered continuum damage models by introducing theeffective (undamaged) stress space concept. They introduced thedamage variable / as the reduction of the area due to micro-dam-age, such that:

/ ¼ 1� n ¼ ðA� ~AÞ=A ¼ AD=A ð1Þ

where AD is the area of micro-damage (i.e. micro-cracks and voids)such that AD ¼ A� ~A; ~A and A being the undamaged area and totalcross-section of the material, respectively The damage densitystarts from / = 0 (even for initially damaged materials) and endswith / = 1 indicating the complete failure. The stresses in the

damaged and effective configurations as a function of the damagedensity can be expressed as:

r ¼ ð1� /Þ~r ð2Þ

where r and ~r are the stress tensors in the damaged and effectiveconfigurations, respectively; with superscript ‘‘�’’ designating theeffective configuration. The inherent assumption in Eq. (2) is thatthe damage density is irreversible, such that an increasing functionwith time is commonly postulated to describe the evolution of thedamage density / (e.g. Kachanov, 1958; Lemaître, 1992; Krajcinov-ic, 1996). However, this assumption is not accurate for materialsthat have the potential to heal and recover parts of their strengthand stiffness under specific conditions such as resting periods dur-ing the fatigue loading and under hydrostatic pressure.

Recently, Abu Al-Rub et al. (2010) and Darabi et al. (2012a) ex-tended the well-known framework of the continuum damagemechanics to the continuum damage-healing mechanics in orderto enhance the ability of these theories to model the micro-damagehealing phenomenon in the materials that tend to heal. They intro-duced the healing natural configuration for self-healing materialsas the extension of the effective configuration by incorporatingthe contribution of the healed micro-damage areas in carryingand transferring loads and the stresses. They expressed the stresstensors in the damaged (nominal) and healing configurations as afunction of the damage density / and a physically defined healingvariable h, such that:

r ¼ ½1� /ð1� hÞ��r; / ¼ AD=A; h ¼ Ah

=AD ð3Þ

where the superimposed ‘‘–’’ designates the healing configuration;and Ah is the area of the healed micro-damage. The term h repre-sents the healed fraction of the total damaged area ranging from0 to 1 with h = 0 indicating no healing and h = 1 indicating that alldamage is healed. Comparing Eqs. (2) and (3) makes it possible todefine the effective damage density �/ in the continuum damage-healing mechanics as the counterpart of the classical damage den-sity in classical CDM, /, such that:

�/ ¼ /ð1� hÞ ð4Þ

Therefore, �/ can be substituted for / in Eq. (2) to consider theeffects of both damage and micro-damage healing on mechanicalresponse of materials.

A common argument when using effective stress space in CDMis that once the material becomes damaged, further loading onlyaffects the undamaged material skeleton. This argument impliesthat the effective stress is only determined as a function of theundamaged area and is independent of the crack shape, geometry,and the configuration around the cracks. However, it has theadvantage of simplifying the implementation of such theoriesand it avoids the complexities associated with direct coupling ofdamage models to other mechanisms. This paper presents a similarargument stating that once the material becomes damaged, furtherloading only affects the intact and healed portion of the material(i.e. the healing configuration). Therefore, the constitutive equa-tions are expressed in the healing configuration (i.e. in terms of�r). However, a transformation hypothesis is required in order torelate the strain tensors in the damaged and healing configurations(refer to Darabi et al., 2012a for more details on transformationhypothesis in CDHM). This paper assumes the strain equivalencehypothesis to simplify the implementation of the constitutivemodels, such that:

e ¼ �e ð5Þ

where e and �e are strain tensors in the damaged and healing config-urations, respectively.

M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913 2903

4. Micro-damage healing during cyclic strain/displacementcontrolled loading

This section presents the significance of including the micro-damage healing effect during the cyclic displacement controlled(CDC) loading for viscoelastic materials. During the CDC test, a cyc-lic displacement is applied at the end of a specimen. The healingmechanism during the CDC test is illustrated with the aid ofFig. 1, which shows part of strain and stress responses during theCDC test in tension at 19 �C (Kim et al., 2008) with a loading fre-quency of 4 cycles/s. In the CDC test, the stress remains tensile dur-ing a large portion of the loading history. During the unloading (i.e.when the strain rate is negative), however, compressive stressesare induced in the specimen. These compressive stresses causethe micro-crack free surfaces to wet (i.e. get close to each other)and undergo healing and crack closure. This effect can be explainedbased on the fading memory properties of viscoelastic materialsand the lag between stress and strain responses, the term d asshown in Fig. 1. As explained by Coleman (1964b), a fading mem-ory is referred to the assertion that the deformations experienced

-400

-200

0

200

400

600

800

40.25 4

Stre

ss (k

Pa)

Tim

250

500

750

1000

40.25 4

Stra

in (m

icri

stra

in)

Tim

A

The region during which damage oc

The region during which micro-dam

0a

Damageth

δ

ε

Fig. 1. Schematic representation of the strain and stress responses for tensile CDC testclosure/healing occur; d being the lag between stress and strain responses.

in the distant past should have less effect on the present valuesof the stress than deformations that occurred in the recent past(e.g. Coleman and Noll, 1961; Coleman, 1964a, b).

In order to illustrate the effect of the fading memory, it is hypo-thetically assumed that the viscoelastic medium contains a singlemicro-crack of length a0 at point ‘‘A’’ and the damage growth oc-curs when the tensile strain exceeds a threshold value (i.e.eDamage

th ). Moving from point ‘‘A’’ to point ‘‘B’’ in Fig. 1, the strain re-mains larger than eDamage

th and the induced stresses in the viscoelas-tic media are also tensile. Therefore, the micro-crack lengthincreases by DaL, such that the crack length at point ‘‘B’’ reachesa0 + DaL. However, the fading memory effect causes the stresses in-duced in the material to relax and fade away with time. The in-crease in the strain level during the preceding loading stage hasless effect on the stress level than the decrease in the strain levelduring the current unloading stage. Obviously, the negative incre-ment in the stress due to the decrease in the strain level during theunloading stage is more than the positive increment in the stressdue to the increase in the strain level during the preceding loadingstage. Therefore, the viscoelastic medium feels compressive stres-

0.5 40.75

e (sec)

0.5 40.75e (sec)

B

curs

age healing occurs

0La a

0L ULa a a

B C

+ Δ+ Δ − Δ

and the stages during which micro-damage nucleation/growth and micro-damage

2904 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

ses at some point during the unloading stage although the totalstrain is still tensile (see the stress response in Fig. 1 when movingfrom point ‘‘B’’ to point ‘‘C’’). Therefore, the crack length decreaseswhen moving from point ‘‘B’’ to point ‘‘C’’ by the increment of DaUL.The induced stresses are compressive during a small portion of theloading and deformation history for the tensile CDC test, such thatDaL is always greater than DaUL (i.e. DaL > DaUL). Therefore, micro-cracks still propagate and increase in length when moving frompoint ‘‘A’’ to point ‘‘C’’. However, accurate estimation of the damagedensity requires the incorporation of both DaL and DaUL

components.As shown in the following sections, the accurate prediction of

the fatigue response of asphalt concrete under CDC test requiresthe inclusion of the micro-damage healing.

5. Viscoelastic and viscoplastic constitutive equations

This section recalls the key equations for the nonlinear Scha-pery (1969) viscoelastic and Perzyna-type (1971) viscoplasticmodels that have been used previously to model the mechanicalresponse of asphalt concrete (e.g. Masad et al., 2005; Huanget al., 2007; Abu Al-Rub and Darabi, 2012). However, detailed for-mulation and numerical implementation of the coupled VE-VPconstitutive model will not be presented. The readers are referredto previous papers for detailed presentation of the formulation andnumerical implementation techniques (e.g. Heeres et al., 2002;Haj-Ali and Muliana, 2004; Abu Al-Rub et al., 2010; Huang et al.,2011b; Darabi et al., 2012b). It should be noted that this paper pre-sents VE–VP equations in the healing configurations to facilitatethe coupling of the damage and healing models to the rest of theconstitutive model.

Small deformation theory is postulated, such that total straintensor e can be additively decomposed into the viscoelastic straintensor eve and the viscoplastic strain tensor evp, such that:

e ¼ eve þ evp ð6Þ

The Schapery’s one-dimensional single-integral nonlinear vis-coelastic model is written as follows:

eve;t ¼ g0�D0 �rt þ g1

Z t

0D�Dðw

t�wsÞ dðg2 �rsÞds

ds; wt ¼Z t

0

d1aT

ð7Þ

where superscripts t and s designate the response at specific times;g0, g1, g2 are nonlinear viscoelastic parameters; wt is the reducedtime; aT is the time–temperature shift factor; �D0 and D�D are theinstantaneous and transient creep compliances in the healing natu-ral configuration which are equivalent to their values for the intactstate of the material. For numerical purposes, the transient creepcompliance can be expressed using the Prony series as:

D�Dwt ¼XN

n¼1

Dn 1� expð�knwtÞ

� �ð8Þ

where N is the number of terms; and Dn and kn are the nth coeffi-cients of the Prony series. Decomposing the viscoelastic strain ten-sor eve into its deviatoric tensor eve and a volumetric component eve

kk

facilitates the presentation of the Schapery’s viscoelastic model forthe general three-dimensional problems (Lai and Bakker, 1996),such that:

eveij ¼ eve

ij þ13eve

kkdij ¼�J2

�Sij þ�B3

�rkkdij ð9Þ

where �J and �B are the shear and bulk compliances of the intactmaterial; �Sij ¼ �rij � �rkkdij=3 are the components of the deviatoricstress tensor in the healing configuration; and dij is the Kroneckerdelta. The deviatoric viscoelastic strain tensor and the volumetricviscoelastic strain can be expressed in terms of �J and �B, such that:

eveij ¼

12

g0�J0

�Stij þ

12

g1

Z t

0D�Jðw

t�wsÞdðg2�Ss

ijÞ;

evekk ¼

13

g0�B0 �rt

kk þ13

g1

Z t

0D�Bðw

t�wsÞdðg2 �rskkÞ ð10Þ

where �J0 and �B0 are the instantaneous shear and bulk compliancesin the healing configuration, respectively; and D�J and D�B are thetransient shear and bulk compliances in the healing configuration,respectively. The compliances �J and �B are related to the creep com-pliance �D through the following relations:

�J0 ¼ 2ð1þ mÞ�D0; D�J ¼ 2ð1þ mÞ�D; �B0 ¼ 3ð1� 2mÞ�D0;

D�B ¼ 3ð1� 2mÞD�D ð11Þ

where m is the Poisson’s ratio, which is assumed to be time-inde-pendent. In this study, it is assumed that the Poisson’s ratio is con-stant in order to simplify the constitutive model and to reduce thenumber of testing required for characterization of the viscoelasticresponse of asphalt concrete materials. The Poisson’s ratio of theviscoelastic materials could be a function of time, temperature,and stress/strain-history (refer to Hilton and Yi, 1998; Hilton,2011; Kassem et al., in press for more details).

The rate of the viscoplastic strain tensor _evp is defined throughthe following non-associative viscoplastic flow rule (Perzyna,1971), such that:

_evp ¼ _cvp @F@�r

ð12Þ

where F is the viscoplastic potential function and _cvp is the visco-plastic multiplier which can be expressed in terms of an overstressfunction U and the viscoplasticity fluidity parameter Cvp, such that:

_cvp ¼ CvphUðf ÞiM ; Uðf Þ ¼ fr0

yð13Þ

where M is the viscoplasticity rate-sensitivity parameter; f is theviscoplastic yield function and r0

y is a yield stress quantity usedto normalized the overstress function and can be assumed unity;and h i is the Macaulay brackets defined by hUi = (U + |U|)/2.

Due to the non-associative nature of viscoplastic response of as-phalt concrete, modified Drucker-Prager-type functions are usedfor f and F (e.g. Seibi et al., 2001; Dessouky, 2005; Masad et al.,2007), such that:

f ¼ �svp � a�I1 � jðpÞ; F ¼ �svp � b�I1; �svp

¼ffiffiffiffiffiffiffi3�J2

p2

1þ 1dvp þ 1� 1

dvp

� �3�J3ffiffiffiffiffiffiffi

3�J32

q264

375 ð14Þ

where a and b are the pressure-sensitivity material parameters;�I1 ¼ �rkk is the first stress invariant; �svp is the effective shear stress;�J2 ¼ 1

2�Sij

�Sij and �J3 ¼ 12

�Sij�Sjk

�Ski are the second and the third deviatoricstress invariants in the healing configuration, respectively; j(p) isthe isotropic hardening function which is a function of the effective(equivalent) viscoplastic strain p (Lemaître and Chaboche, 1990),such that:

jðpÞ ¼ j0 þ j1½1� expð�j2pÞ� ð15Þ

where j0, j1, and j2 are material parameters.This VE–VP constitutive model was calibrated against the Fed-

eral Highway Administration’s, FHWA’s, Accelerated Load Facility(ALF) data (e.g. Kim et al., 2008). The dynamic modulus test at dif-ferent temperatures in tension is used to identify the viscoelasticmodel parameters as well as the time–temperature shift factorsaT. The repeated creep-recovery test at various stress levels in com-pression was used to identify the viscoplastic model parametersonce the viscoelastic model parameters were identified. Readers

Table 1Summary of the tests used for identification of the model parameters.

Test Temperature(�C)

Strain rate(s�1)

Strain amplitude(le)

Identified model parameters

Complex Modulus test �10,10,35,55 – – aT ;Dn; kn Eqs. (7), (8)Repeated creep-recovery test at various stress level (RCRT–VS) in

compression55 200 – Cvp,M,j0,j1,j2 Eqs. (13) and

(15)Constant strain rate test in tension 5 7 � 10�6,

2.1 � 10�5,3 � 10�5,5.5 � 10�5

– Cvd, q, k Eq. (26)

Cyclic displacement controlled test (CDC) in tension 19 - 1200 Ch, b1, b2 Eq. (29)

M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913 2905

are referred to Huang et al. (in press) and Darabi et al. (2012c) formore details on the identification of the VE–VP model parameters.

Table 1 list the tests from which the model parameters are iden-tified as well as the identified model parameter. The tests whichhave been used to validate the coupled VE–VP–VPD–H model arelisted in Table 2. The identified VE–VP model parameters are listedin Table 3.

6. Viscodamage evolution model

In previous studies, the authors proposed a viscodamage modelthat can be used to predict the mechanical response of asphalt con-crete subjected to different loading conditions (Darabi et al., 2011,2012b). The damage evolution function was defined as a functionof the damage driving force �Y , the total strain eeff, and a historyterm (1 � /)2, such that (Darabi et al., 2011):

_/ ¼ Cvd�Y

Y0

� �q

expðkeeff Þð1� /Þ2; �Y ¼ �svd � a�I1 ð16Þ

where Cvd is the viscodamage fluidity parameter; Y0 is the referencedamage force which can be assumed to be unity; k is a materialparameter; and eeff is the effective total strain defined aseeff ¼

ffiffiffiffiffiffiffiffiffieijeijp

. The damage force �Y is assumed to have aDrucker–Prager type form, such that �svd is defined as:

Table 2Summary of the tests used for validation of the viscodamage model.

Test Temperature (�C) Strain rate (s�1)

Uniaxial constant strain rate 12 2.7 � 10�4, 4.6 � 1025 5 � 10�4, 1.5 � 10�

40 3 � 10�4, 10�3, 3 �Cyclic stress controlled 5 –

19 –Cyclic displacement controlled (CDC) 5 –

19 –

Table 3Identified viscoelastic and viscoplastic model parameters at the reference temperature T0

Viscoelastic model parameters

n 1 2 3 4

kn (s�1) 104 102 10 1Dn (kPa�1) 1 � 10�8 3 � 10�8 2 � 10�8 1 � 10�7

D0 (kPa�1) 3 � 10�8

Time–temperature shift factors (T0 = 10 �C)T (�C) �10 10 35 55

1.26 � 103 1 6.3 � 10�4 10�5

Viscoplastic model parametersa b r0

y (kPa) Cvp (s�1) M

0.25 0.2 100 2.4 � 10�8 1

�svd ¼ffiffiffiffiffiffiffi3�J2

p2

1þ 1

dvdþ 1� 1

dvd

� �3�J3ffiffiffiffiffiffiffi3�J3

2

q264

375 ð17Þ

Eqs. (14) and (16) state that damage and viscoplasticity havesimilar driving forces with different evolution functions. Moreover,Eq. (17) possesses a model parameter (i.e. dvd) distinguishing thedamage responses in extension and contraction loading conditions.Finally, the damage driving force of Eq. (16) is sensitive to thehydrostatic pressure. This damage model has been calibrated andvalidated against extensive experimental data (e.g. Darabi et al.,2011, 2012b). The calibration was based on using the tertiary partof the creep response of asphalt concrete.

The subsequent sections present a novel approach to identifythe proper viscodamage constitutive model for asphalt concrete.The constant strain rate tests in tension at 5 �C were analyzed toidentify the damage evolution response. A viscodamage model isproposed based on the determined damage evolution. At 5 �C,one can reasonably assume negligible viscoplastic deformation.To examine this assumption, the identified VE–VP model parame-ters listed in Table 3 are used to simulate the response of asphaltconcrete at 5 �C over a rage of strain rates. The predicted viscoplas-tic strain over a range of strain rates (i.e. 1 � 10�3 � 1 � 10�5/s) isless than 1% of the total applied strain which reasonably verifiesthe assumption of neglecting the viscoplastic strain at 5 �C.

Stress amplitude (kPa) Strain amplitude (le)

�4 – –3, 4.5 � 10�3, 1.35 � 10�2 – –10�3 – –

1525 –250, 750 –– 1750– 1500

= 10 �C and the time–temperature shift factors.

5 6 7 8 9

10�1 10�2 10�3 10�4 10�5

1.5 � 10�7 9 � 10�7 1 � 10�6 5 � 10�6 6 � 10�6

j0 (kPa) j1 (kPa) j2

50 1800 135

-7.5

-6.5

-5.5

-4.5

1 1.5 2 2.5 3

Ln

[Dam

age

rate

]

Ln [Effective damage force/Y0]

Strain level=0.10%Strain level=0.15%Strain level=0.20%Strain level=0.25%Strain level=0.30%

(a)

-9

-8

-7

-6

-5

-4

-3

-8 -7 -6 -5 -4

Ln

[Dam

age

rate

]

Ln [Strain]

Effective damage force/Y0=3

Effective damage force/Y0=5

Effective damage force/Y0=7

Effective damage force/Y0=9

(b)

67 10 / sec

52.1 10 / sec

53 10 / sec

55.5 10 / sec

k1

67 10 / sec

52.1 10 / sec

53 10 / sec

55.5 10 / sec

q1

−×

−×

−×

−×

−×

−×

−×

−×

Fig. 2. (a) Plot of the damage density rate versus the normalized effective damage force at constant strain levels which is used to identify q and Cvd parameters; (b) plot of thedamage density rate versus the effective strain at constant effective damage forces which will is used to identify k and Cvd parameters.

Table 4Identified viscodamage model parameters at the reference temperature T0 = 10 �C.

Viscodamage model parameters

Cvd (s�1) Y0 (kPa) q k

2.14 � 10�11 1000 2.41 �2.53

2906 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

For the constant strain rate test, the strain input can be ex-pressed as follows:

eðtÞ ¼ Ct ð18Þ

where C is a constant representing the strain rate in the uniaxialconstant strain rate test. The uniaxial stress output in the healing

configuration can be obtained using the superposition principlefor linear viscoelastic materials, such that:

�rðtÞ ¼ EðtÞeð0Þ þZ t

0EðsÞ _eðt � sÞds;

�rðtÞ ¼ Eð0ÞeðtÞ þZ t

0Eðt � sÞ _eðsÞds ð19Þ

where E(t) is the relaxation modulus that is already known from thedynamic modulus test results. Substituting the strain input for theuniaxial constant strain rate test (Eq. (18)) into Eq. (19) yields thestress response during the constant strain rate test, such that:

�rðtÞ ¼ CZ t

0EðsÞds ð20Þ

M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913 2907

It should be noted that healing does not occur during the uniax-ial constant strain rate test (h = 0) since the strain increases mono-tonically and there is no rest period. Using the concept of thecontinuum damage-healing mechanics, the damage density vari-able / can be determined as:

/ðtÞ ¼ 1� rðtÞ�rðtÞ ð21Þ

where r(t) is the stress in the damaged configuration which is mea-sured experimentally. This procedure is followed for several uniax-ial constant strain rate tests in tension at 5 �C to calculate thedamage density variable.

Guided by the evolution function in Eq. (16), a general form isassumed for the damage evolution function as a function of thedamage force �Y and effective strain eeff, such that:

_/ ¼ Cvdf1

�YY0

� �f2ðeeff Þ ð22Þ

where �Y , eeff, Y0, and Cvd are defined in Eqs. (16), (17). The mainobjective is to identify the forms of functions f1 and f2 based onthe experimentally calculated damage densities.

Taking the natural logarithm of both sides of Eq. (22) yields:

Lnð _/Þ ¼ LnðCvdÞ þ Ln f1

�YY0

� �� �þ Ln f2ðeeff Þ

� �ð23Þ

0

0.03

0.06

0.09

0.12

0.15

0.18

1 10 100 1000 10000

Stra

in a

mpl

itud

e (%

)

Time (sec)

-1000

-500

0

500

1000

0.04 0.05 0.06 0.07 0.08 0.09 0.1

Stre

ss (k

Pa)

Strain (%)

Experimental measurements

Model prediction without micro-damage healing

Model prediction using micro-damage healing

(a)

(c)

Fig. 3. Responses during CDC test at 19 �C when the amplitude of the averaged strain apLVDTs; (b) stress strain response for the experimental measurements as compared to mothe experimental measurements as compared to model predictions for cycles 2200–225significantly enhances the prediction of the stress response, stress amplitude, and the d

The rate of the damage density _/ versus the effective total straineeff can be plotted once the damage density is calculated accordingto Eq. (21). The first and the third terms in the right-hand-side ofEq. (23) are constants at a fixed effective strain level eeff. To identifythe function f1, the values of the normalized damage force in thehealing configuration (i.e. �Y=Y0) at four strain levels (i.e. 0.1, 0.15,0.2, 0.25, and 0.3%) are plotted versus the damage rate inFig. 2(a) for several strain rate tests. Fig. 2(b) clearly suggests apower law function for f1, which agrees with Eq. (16), such that:

f1

�YY0

� �¼

�YY0

� �q

ð24Þ

where q is the slope of the lines as shown in Fig. 2(a). The intercep-tion of each strain level line with the vertical axis in Fig. 2(a) is thesummation of the first and the third terms in the right side of Eq.(23). This interception at different strain levels is used to identifythe viscodamage fluidity parameter Cvd. The values of Cvd identifiedat different strain levels are similar suggesting that Cvd is constantand does not depend on the strain and/or stress levels. The outlinedprocedure allows the identification of the parameter q independentof other viscodamage parameters.

Similarly, function f2 can be identified by plotting _/ versus thenormalized damage force (i.e. �Y=Y0). The first and second termson the right hand side of Eq. (23) remain constant at a fixednormalized damage force. To characterize f2 independently, _/ is

-1500

-1000

-500

0

500

1000

1500

2000

0.00 0.02 0.04 0.06 0.08

Stre

ss (k

Pa)

Strain (%)

Experimental measurements

Model prediction using VE-VP-VD model

0

500

1000

1500

2000

1 10 100 1000 10000

Stre

ss a

mpl

itud

e (k

Pa)

Time (sec)

Experimental measurements

VE-VP-VD model prediction

VE-VP model prediction

VE-VP-VD-H model prediction

(b)

(d)

plied to the specimen is 1200 le. (a) Averaged amplitude of the strain measured atdel predictions using VE–VP–VD model for cycles 1–50; (c) stress strain response for0; (d) measured and predicted stress amplitudes. The micro-damage healing modelissipated energy.

0

1000

2000

3000

4000

0.00 0.10 0.20 0.30 0.40 0.50

Stre

ss (k

Pa)

Strain (%)

44.6 10 / sec

42.7 10 / sec

Experimental measurementsVE-VP-VD model prediction

−×

−×

Fig. 4. Model predictions and experimental measurements for the constant strainrate tests in tension at 12 �C when strain rates are 2.7 � 10�4/s; (b) 4.6 � 10�4/s.

2908 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

plotted versus eeff at fixed normalized damage forces, as shown inFig. 2(b). As shown in Fig. 2(b), the fitted lines at different normal-ized damage force are parallel, such that f2 can be expressed as:

f2ðeeff Þ ¼ ðeeff Þk ð25Þ

0

1000

2000

3000

0.00 0.20 0.40 0.60 0.8 0

Stre

ss (

kPa)

Strain (%)

(a)

Experimental measurements VE-VP-VD model prediction

34.5 10 / sec−×

45 10 / sec−×

31.5 10 / sec−×

0

100

200

300

400

500

600

0.0 0.2 0.4 0.6 0.8

Stre

ss (

kPa)

Strain (%)

(c)

33 10 / sec

310 / sec

43 10 / sec

Fig. 5. (a) Model predictions and experimental measurements for the constant strain rapredicted damage density at 25 �C; (c) stress–strain response at 40 �C; (b) predicted dexperimental measurements at different strain rates. Also, at the same strain level, the daconcrete materials are more prone to damage as the rate of loading increases.

where k is the slope of ln _/� ln eeff curves at different normalizeddamage forces. The interception of the lines in Fig. 2(b) correspondto the summation of the first two terms in the right side of Eq. (23)from which Cvd can be identified at different normalizeddamage force. The viscodamage fluidity parameter Cvd identifiedat different strain and normalized damage force levels (Figs. 2(a)and (b)) range between 10�12 and 6 � 10�12/s with the average of4 � 10�12/s.

It should be noted that the viscodamage model presented in thispaper (i.e. Eqs. (22), (24), and (25)) is slightly different from theviscodamage model of Darabi et al. (2011) as presented in Eq.(16). The main difference between these two models is that theexponential term of the effective strain in Eq. (16) is substitutedwith a power law function for the effective strain in Eq. (25).

Eqs. (22), (24), and (25) yield:

_/ ¼ Cvd�Y

Y0

� �q

ðeeff Þk ð26Þ

The dynamic viscodamage loading/unloading surface can be ob-tained by rearranging Eq. (26), such that:

vvd ¼ �Y � Y0ðeeff Þ�k=q_/

Cvd

!1q

¼ 0 ð27Þ

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.20 0.40 0.60 0.80

Dam

age

dens

ity

Strain (%)

(b)

Strain rate increases from 4 35 10 4.5 10 / sec− −× − ×

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

Dam

age

dens

ity

Strain (%)

(d)

Strain rate increases from 4 33 10 3 10 / sec

te tests in tension at different strain rates; (a) stress–strain response at 25 �C; (b)amage density at 40 �C. The viscodamage model predictions agree well with themage density has higher values as the strain rate increases showing that the asphalt

- 4

- 3

- 2

- 1

1

0 10 20 30 40 50

Log

[T

ime -

tem

pera

ture

shi

ft f

acto

r]

T (oC)

VE-VP time-temperature shift factor

VD time-temperature shift factor

Fig. 6. Comparison of the viscodamage time–temperature shift factors and theviscoelastic–viscoplastic time–temperature shift factors (identified from dynamicmodulus tests) when the reference temperature is 10 �C. The viscodamage time–temperature shift factors are almost the same as those of the viscoelastic–viscoplastic model. Therefore, the asphalt concrete materials studied in this workcan be considered as the thermorheologically simple materials for coupledviscoelastic, viscoplastic, and viscodamage mechanisms.

M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913 2909

where vvd is the viscodamage dynamic growth surface and can alsobe regarded as the damage initiation and evolution condition, suchthat the micro-damage evolution occurs only if vvd P 0.

The viscodamage model parameters are identified at 5 �C,whereas, the viscoelastic and viscoplastic model parameters areidentified at 10 �C. The temperature-dependency is incorporatedin the viscodamage model using the time–temperature shifting.Therefore, the time increment in Eq. (22) is substituted by the re-duced time increment (i.e. reduced by the time–temperature shiftfactor aT), such that Eq. (22) can be expressed as:

D/ ¼ Cvd�Y

Y0

� �q

ðeeff ÞkDtaT

� �¼ Cvd

aT

!�Y

Y0

� �q

ðeeff ÞkDt ð28Þ

Eq. (28) suggests that using the time temperature shift factor isequivalent to use the viscodamage fluidity parameter normalizedby aT. For the consistency purposes, the viscodamage modelparameters are shifted to the reference temperature T0 = 10 �Csimply by scaling Cvd, such that Cvd(10 �C) = Cvd(5 �C)� aT(5 �C,T0 = 10 �C). It is assumed that the time–temperature shift factoris the same for all the components (i.e. VE, VP, VD, and H) of thepresented constitutive model. The validity of this assumption isinvestigated in the subsequent sections. Table 4 lists the identifiedviscodamage model parameters.

(kPa)σ

0 0.5 1.

maxσ

Fig. 7. Schematic representation of loading history

7. Micro-damage healing model parameters

This section recalls the micro-damage healing model recentlyproposed by Abu Al-Rub et al. (2010), such that:

_h ¼ Chð1� /Þb1 ð1� hÞb2 ð29Þ

where Ch is the micro-damage healing fluidity parameter whichcontrols the rate of the micro-damage healing and b1 and b2 aredamage and micro-damage healing history parameters, respec-tively. The model parameters associated with the micro-damagehealing model are usually determined using the repeated creep-recovery tests in tension with the rest period between the loadingcycles (e.g. Abu Al-Rub et al., 2010). However, such tests are notavailable in the current experimental database. Therefore, to reducethe number of model parameters, the simplest evolution functionfor the micro-damage healing model is considered, such that:

_h ¼ Ch ð30Þ

As emphasized in Section 3 of this paper, micro-damage healingcan occur during the cyclic displacement controlled test. This testis, therefore, used to identify the micro-damage healing fluiditymodel parameter Ch .The displacement input during the cyclic dis-placement controlled test (CDC) can be written as:

D ¼ Dmax=2½�1þ cosð2pftÞ� ð31Þ

where f is the frequency. The term Dmax is the displacement ampli-tude applied at the end plates, such that Dmax/‘ can be considered asthe amplitude of the averaged strain applied to the specimen; ‘being the specimen height (i.e. 6 inches). However, the averagedLVDTs’ strains should be considered as the input strain for the val-idation and simulation purposes since it measures the strain in themiddle of the specimen where stress and strain can be assumed tobe uniformly distributed. Fig. 3(a) shows the LVDT strain input forthe CDC test at 19 �C when the amplitude of the averaged speci-men’s strain is 1200 le (Kim et al., 2008). Fig. 3(b) shows that theVE–VP–VD model predictions are in agreement with the experi-mental measurements during the initial loading cycles. However,the coupled VE–VP–VD significantly underestimates the stress out-put as the number of loading cycles increases, Fig. 3(c).

As shown in Fig. 3(d), the model predictions using VE–VP com-ponents significantly overestimate the stress amplitude while thepredictions using VE–VP–VD model significantly underestimatesthe measured stresses.

Model predictions using the viscodamage model predicts pre-mature failure for the material while the experimental measure-ments show that the material is capable of sustaining loads formuch longer time. As discussed in Section 3, this discrepancy is re-lated to the inability of the model to account for crack closure and

Time (sec)0 tf-0.5 tf

for the cyclic stress controlled test in tension.

0

200

400

600

800

0.00 0.01 0.02 0.03 0.04 0.05

Stre

ss (k

Pa)

Strain (%)

Experimental data

Model prediction using VE-VP model

0

200

400

600

800

0.10 0.15 0.20 0.25 0.30 0.35

Stre

ss (k

Pa)

Strain (%)

Experimental data

Model prediction using VE-VP model

0

200

400

600

800

0.28 0.29 0.30 0.31 0.32

Stre

ss (k

Pa)

Strain (%)

Experimental data

Model prediction using VE-VP-VD model

0.0

0.4

0.8

1.2

1.6

10 100 1000 10000

Stra

in (%

)

Time (sec)

(a) (b)

(c) (d)

Stress amplitude: 750kPa Stress amplitude:

250kPa

Experimental measurements VE-VP-VD model prediction

Fig. 8. Model predictions as compared to experimental data for cyclic stress controlled test at 19 �C. (a) VE–VP model predictions for cycles 1–30 when the stress amplitude is750 kPa; (b) VE–VP model predictions for cycles 970–980 when the stress amplitude is 750 kPa; (c) VE–VP–VD model prediction when the stress amplitude is 750 kPa; (d)VE–VP–VD model predictions for the strain responses when the stress amplitudes are 250 and 750 kPa.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 100 10000

Stra

in (%

)

Time (sec)

Experimental measurementsVE-VP-VD model prediction

Stress amplitude:1525kPa

Fig. 9. VE–VP–VD model predictions for the strain response at 5 �C when the stressamplitude is 1525 kPa.

0

1000

2000

3000

1 10 100 1000 10000

Stre

ss a

mpl

itud

e (k

Pa)

Time (sec)

19oC; strain amplitude:1500 ;Frequency:4cycle/sec

5oC; strain amplitude:1750 ;Frequency:10 cycle/sec

Experimental measurementsVE-VP-VD-H model prediction

VE-VP-VD model prediction

Fig. 10. Experimental measurements and model predictions for the stress outputduring the CDC test: (a) at 19 �C when the amplitude of the averaged specimen’sstrain is 1500 le; (b) at 5 �C when the amplitude of the averaged specimen’s strainis 1750 le.

2910 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

the micro-damage healing during the cyclic strain controlled tests.The fluidity model parameter Ch, which characterizes the micro-damage healing rate, is identified using the deviation betweenthe prediction of VE–VP–VD model and the experimental measure-ments for the CDC test at 19 �C when the amplitude of the aver-aged specimen’s strain is 1200 le. The micro-damage healingfluidity parameter is identified to be Ch = 6.5 � 10�3/s.

Fig. 3(c) shows that the micro-damage healing model signifi-cantly enhances the prediction of the stress response as well asthe dissipated energy during the intermediate cycles. Fig. 3(d)shows that the model without micro-damage healing underesti-mates the stress level and predicts premature failure almost by a

M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913 2911

factor of four, while the model prediction with the micro-damagehealing agrees well with experimental measurements.

8. Model validation

The identified model parameters are used in this section to val-idate the proposed viscodamage model against uniaxial constantstrain rate, cyclic stress controlled, and cyclic strain controlledtests at different temperatures, strain rates, and stress/strain lev-els/amplitudes. Table 2 lists the tests that have been used for thevalidation of the proposed viscodamage model. It should be notedthat these tests have not been used in the calibration process.

8.1. Uniaxial constant strain rate tests

The uniaxial constant strain rate tests at different strain ratesand temperatures are used to further validate the viscodamagemodel. Fig. 4 shows the VE–VP–VD model predictions as comparedto the experimental data at 12 �C for the constant strain rate at dif-ferent strain rates. Fig. 4 clearly shows that the model is capable ofpredicting the rate-dependent damage response of asphaltconcrete.

Fig. 5(a) presents the viscodamage model predictions as com-pared to experiments at four strain rates at 25 �C. Fig. 5(a) clearlyshows that the viscodamage model predictions agree well withthe experimental measurements. Fig. 5(b) shows the increase inthe damage susceptibility of the asphalt concrete as the strain rateincreases; the damage density has higher values at the same strainlevel as the strain rate increases. Fig. 5(b) suggests that the modelis capable of capturing the commonly observed increase in thedamage susceptibility as the strain rate and rate of loading in-creases. The model is further validated against the uniaxial strainrate tests in tension at 40 �C as shown in Fig. 5(c) and (d). Fig. 4and 5 present the model capabilities in capturing rate- and temper-ature-dependent response of asphalt concrete during the mono-tonic loading.

In this paper, the damage model parameters are determined at5 �C. The responses at other temperatures are obtained throughusing the time–temperature shift factor. The time–temperatureshift factors for the viscodamage model are assumed to be slightlydifferent from the time–temperature shift factor identified fromthe dynamic modulus test. However, this difference is negligiblesuch that the asphalt concrete used in this study can be assumedto be a thermorheologically simple material for the coupled VE–VP–VD mechanisms, Fig. 6. This finding is in agreement with thepreviously reported experimental observations (e.g. Schwartzet al., 2002).

8.2. Cyclic stress controlled tests

In this section, the proposed viscodamage model is validatedagainst the cyclic stress controlled tests in tension. The cyclic stresscontrolled tests are conducted at 5 �C and 19 �C and for multiplestress amplitudes. This test applies a cyclic stress input with thefrequency of 4 cycles/s, such that the stress input can be writtenas follows:

r ¼ rmax=2� ½�1þ cosð8ptÞ� ð32Þ

where rmax is the stress amplitude. Fig. 7 schematically presentsthe stress history for the cyclic stress controlled test where tf canbe considered as the failure time.

Unlike the cyclic displacement controlled test, the micro-dam-age healing is not important during the cyclic stress controlled testin tension as both strain and stress responses remain tensile.

Fig. 8(a) and (b) present the VE–VP model predictions andexperimental measurements for initial and intermediate loadingcycles at 19 �C when the stress amplitude is 750 kPa. As shownin Fig. 8(a), VE–VP model predictions agree well with experimentaldata for the initial loading cycles. However, as the number of load-ing cycles increases, VE–VP model significantly underestimates thestrain level as shown in Fig. 8(b). This deviation manifests the ini-tiation and evolution of the micro-damage as the number of load-ing cycles increases. Once the viscodamage and micro-damagehealing components are also included in the model, the model pre-dictions agree well with the experiments, Fig. 8(c). Fig. 8(c) alsoshows that the viscodamage model properly predicts the dissi-pated energy during the cyclic loading since the predicted hyster-esis loop is very similar to the experimental measurements. Thestrain responses for the cyclic stress controlled tests at 19 �C whenthe stress amplitude is 250 and 750 kPa are presented in Fig. 8(d).The damage model is further validated at 5 �C when the stressamplitude is 1525 kPa, see Fig. 9. Figs. 8 and 9 show that the modelpredictions without damage significantly deviate from the experi-mental data as number of loading cycles increases. However, mod-el predictions agree well with the experimental measurements upto thousands of loading cycles once the viscodamage component ofthe model is activated.

8.3. Cyclic strain controlled tests

In this subsection, the model is further validated against cycliccontrolled displacement test (CDC). To further validate the model,the CDC test is repeated at 19 �C and 5 �C at different strain ampli-tudes. Fig. 10 show the stress output for the CDC test at 19 �C and5 �C when the strain amplitudes are, respectively, 1500 and1750 le. Fig. 10 shows that when the micro-damage healing isnot included, the model predicts premature failure during theCDC test. However, model predictions using the calibrated VE–VP–VD–H model agrees reasonably well with the experimentalmeasurements. The predictions presented in this section clearlyshow that the model can reasonably predict the time-, tempera-ture-, and rate-dependent response of asphalt concrete under bothmonotonic and cyclic loading conditions.

9. Conclusions

In this paper, a viscodamage model that can reasonably predictthe fatigue damage response of asphalt concrete materials sub-jected to different loading conditions is presented. The specificform of the viscodamage evolution function for the asphalt con-crete is motivated by the experimentally measured damage den-sity values using different loading conditions.

It is shown that asphalt concrete materials, and in general theviscoelastic materials with healing potential, have a micro-damagehealing potential during the cyclic strain controlled loading condi-tions. This phenomenon is related to the fading memory character-istics of the viscoelastic materials. Therefore, this paper concludesthat the existence of a rest period between the loading cycles is notthe only condition for the micro-damage healing of the asphaltconcrete materials. Residual (internal) compressive stresses canalso aid in the micro-damage healing of asphalt materials.

The proposed concept of the continuum damage-healingmechanics significantly simplifies the coupling between the visco-elastic–viscoplastic mechanisms with the viscodamage and micro-damage healing mechanisms. Therefore, expressing the viscoelas-tic and viscoplastic models in the healing stress space is physicallysound and at the same time simplifies the numerical implementa-tion techniques.

2912 M.K. Darabi et al. / International Journal of Solids and Structures 50 (2013) 2901–2913

The viscodamage model parameters can be identified using asystematic procedure based on the application of uniaxial constantstrain rate tests in tension at several strain rates. The viscoelastic,viscoplastic, viscodamage, and micro-damage healing model pre-dictions show that the proposed model can accurately capturethe mechanical response of asphalt concrete materials subjectedto different loading conditions at a range of temperatures. Modelpredictions agree well with experimental measurements includinguniaxial constant strain rate, cyclic stress controlled, and cyclicstrain/displacement controlled tests at different temperatures,strain rates, and stress/strain levels/amplitudes. These compari-sons clearly show the capabilities of the model in capturing the fa-tigue response of asphalt concrete materials subjected to differentloading conditions.

Acknowledgements

The authors acknowledge the financial support provided by theAsphalt Research Consortium through the US Federal HighwayAdministration. R.K. Abu Al-Rub and E. Masad also would like toacknowledge the financial support provided by Qatar National Re-search Fund (QNRF) through the National Priority Research Pro-gram project 08-310-2-110. The QNRF funding supported thedevelopment of the viscodamage model for predicting the fatiguedamage in asphalt pavements (Section 4 of the paper). Finally,the authors acknowledge Dr. Richard Kim from North CarolinaState University for providing the ALF experimental data used inthis study.

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