Research ArticleAn Image-Based Finite Element Approach forSimulating Viscoelastic Response of Asphalt Mixture

Wenke Huang1 Xiaoning Zhang1 and Yingmei Yin2

1School of Civil Engineering and Transportation South China University of Technology Guangzhou 510641 China2School of Civil and Transportation Engineering Guangdong University of Technology Guangzhou 510006 China

Correspondence should be addressed to Xiaoning Zhang prozxn163com

Received 18 June 2016 Accepted 10 August 2016

Academic Editor Robert Cerny

Copyright copy 2016 Wenke Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents an image-based micromechanical modeling approach to predict the viscoelastic behavior of asphalt mixtureAn improved image analysis technique based on the OTSU thresholding operation was employed to reduce the beam hardeningeffect in X-ray CT images We developed a voxel-based 3D digital reconstruction model of asphalt mixture with the CT imagesafter being processed In this 3D model the aggregate phase and air void were considered as elastic materials while the asphaltmastic phase was considered as linear viscoelastic material The viscoelastic constitutive model of asphalt mastic was implementedin a finite element code using the ABAQUS user material subroutine (UMAT) An experimental procedure for determining theparameters of the viscoelastic constitutive model at a given temperature was proposed To examine the capability of the model andthe accuracy of the parameter comparisons between the numerical predictions and the observed laboratory results of bending andcompression tests were conducted Finally the verified digital sample of asphalt mixture was used to predict the asphalt mixtureviscoelastic behavior under dynamic loading and creep-recovery loading Simulation results showed that the presented image-baseddigital sample may be appropriate for predicting the mechanical behavior of asphalt mixture when all the mechanical propertiesfor different phases became available

1 Introduction

Asphalt mixture is the most widely used road constructionmaterial in the paving industry The mechanical behaviorof the asphalt mixture is complex due to the heterogeneityof the asphalt composite material and the time-dependentviscoelastic binder In the aim of optimizingmaterials designit is necessary to study the behavior of this compositematerialThe traditional trial and error approach in industrialpractice has focused on empirical experiments that developcorrelations between the macro phenomena and the materialcharacteristics However this traditional approach is cost-intensive high-resource-consuming and time-consumingThe developed correlations are not sufficient to gain insightinto the performance of asphalt mixture To overcome thesedifficulties numerical techniques are developed to gain someinsight into the problems

Many numerical attempts have been made to investigatethe internal behavior of asphalt mixture or to estimate

the test indices that can represent the macro phenomenaunder certain conditions [1ndash5] But these researches con-sider the asphalt mixture as being homogenous materialwith equivalent properties When using these homogenousnumerical models the influences of the microstructure suchas angularity of the aggregates percentage of the air voids anddistributions of the asphalt mastic are not taken into accountThemechanical behavior of asphaltmixture strongly dependson the particular components including aggregates asphaltmastic and air voids In order to gain accurate predictionsof the mechanical behavior for asphalt mixture particularattention has to be paid to the study of its microstructure andto the mechanical behaviors of its different phases

Anumber of researchers have paid attention to employingthe discrete element model (DEM) in studying the microme-chanical behaviors of asphalt mixtures Chang and Meegoda[6] used a micromechanical model based on the discrete ele-ment method (DEM) to investigate the viscoelastic behaviorof the internal structure of HMA Collop et al [7 8] applied

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 7428623 11 pageshttpdxdoiorg10115520167428623

2 Advances in Materials Science and Engineering

the discrete element method (DEM) to generate a three-dimensional (3D) idealizedmixture model for simulating thebehavior of a highly idealized asphalt mixture under uniaxialand triaxial compressive creep tests Wu et al [9] usedthe discrete element method (DEM) to simulate constantstrain rate compressive tests for an idealized asphalt mixturecomprising approximately single-sized sand particles Caiet al [10 11] used the discrete element method (DEM)to simulate the uniaxial compression tests of a realisticasphalt mixture under constant strain rate In these studiesthe produced numerical samples are either highly idealizedmodel or randomly generated model The most importantadvantages of these discrete element samples are the abilityto handle complex and changing contact geometries andlarge displacements However these discrete element sampleswere still not adequate enough to describe the complicatedmicrostructure of asphalt mixture with highly irregularshaped components Another disadvantage of these discreteelement samples is the relatively high computational cost

To overcome this shortcoming many researchers devel-oped micromechanical models to investigate the behaviorof asphalt mixture by using the finite element method(FEM) Dai [12 13] developed a two-dimensional (2D)microstructure finite element model of the asphalt mixturegenerated from the scanned image to predict the viscoelasticproperties of the asphalt mixtures where elastic propertieswere assigned to aggregates and viscoelastic properties wereassigned tomastic phase Kim and Lutif [14] proposed a com-putational micromechanics modeling approach to predictdamage-dependent constitutive behavior of asphalt mixturesArshadi and Bahia [15] developed the two-dimensional (2D)images of mastic and mortar scales artificially and used themto characterize the properties of those scales And thenthe 2D scanned images of asphalt mixtures are utilized tostudy the asphalt mixture behavior under uniaxial creepand recovery loading But the majority of the image-basedmicrostructure finite element models is limited to two-dimensional samples Lacking of three-dimensional infor-mation in FE framework makes them less able to representcomplicated morphology of asphalt mixture when used forsimulating

The nondestructive X-ray CT technique is used asan effective method to capture images of the internalmicrostructure of asphalt mixtureThere have been a numberof recent attempts to use X-ray CT images to reconstruct athree-dimensional (3D) finite element specimen to predictthe mechanical properties of asphalt mixture [16ndash19] Theimage-based finite element method requires the image analy-sis techniques in order to segment the CT slices consisting ofdifferent compositions themechanical properties of differentphases and the use of mechanical testing to identify thematerial parameters The majority of these microstructurenumerical samples for asphalt mixture faces challenges withregard to accurate geometry of different phases and precisedescription of asphalt mastic behavior

This paper presents an image-based micromechanicalmodeling approach for simulating the viscoelastic responseof asphaltmixture at a given temperature In order to enhancethe segmentation of different phases in the CT image we

developed an improved image analysis technique based onthe OTSU thresholding operation for use Then the three-dimensional viscoelastic constitutive model is applied tobetter describe themechanical behavior of asphaltmastic andimplemented in a finite element code using the ABAQUSuser material subroutine (UMAT) After that we proposean experimental procedure for identifying and verifying theviscoelastic parameters at a given temperature in the linearviscoelasticity range Finally the image-based numericalsample combinedwith theABAQUSusermaterial subroutinewill be applied to predict the viscoelastic behavior of a realmicrostructure asphalt mixture specimen

2 Microstructural Reconstruction Based onX-Ray CT Images

Asphalt mixture is a particulate composite material con-sisting of aggregate mastic and air voids In this sectiona cylindrical HMA mixture (AC-20) lab specimen wasprepared for capturing the internal microstructure with thenondestructive industrial X-ray CT technique Then thegrayscale thresholds for dividing aggregate matrix and airvoids were chosen based on the improved OTSU methodAdditionally a pixel-based 3D image model of the specimenwas constructed In Sections 21 and 22 the reconstructedprocess of the microstructure model is described in detail

21 Microstructure Acquisitions and Digital Image ProcessingX-ray CT is a commonly used nondestructive technique forcharacterizing the internalmicrostructure of asphaltmixtureThe basic image-forming principle of the industrial X-ray CTis that suppose the X-ray attenuation coefficient of the idealmaterial is120583 If the original X-ray strength is 1198680 the attenuatedX-ray strength is 119868 For homogeneousmaterial the attenuatedX-ray strength follows the Bill Exponential Law describedbelow

119868 = 1198680 exp (minus120583119909) (1)

For heterogeneous materials the attenuated X-raystrength 119868 can be calculated by dividing the object into smallunits So we can get the following equation

ln(1198680119868 ) = 12058311199091 + 12058321199092 + 12058331199093 + sdot sdot sdot (2)

The original X-ray strength 1198680 and the attenuated X-raystrength 119868 can be easily measured and the right side of (2) canbe calculated by numericalmethod In this study the YXLONCompact-225 CT X-ray scanner is used to obtain the detailedmicroscopic structure of the asphalt mixture specimen

Generally asphalt mixture is a three-phase compositematerial including aggregate matrix and air voids ScannedCT images of asphalt mixture have different grayscale inten-sities between 0 and 255 where denser materials have higherintensity The OTSU algorithm is a popular thresholdingmethod in adaptive optimal threshold selection for imagesegmentation

Beam hardening effect is a common phenomenon in X-ray CT images where the edge is brighter than the center

Advances in Materials Science and Engineering 3

(a) (b)

(c)

(d) (e)

Figure 1 Image segmentation of asphalt mixture CT slices (a) the original CT image (b) the segmented image using OTSUmethod (c) thesubimages decomposed by the improved OTSUmethod (d) the segmented image using improved OTSUmethod and (e) phase-segmentedimage

as presented in Figure 1(a) Figure 1(b) shows the phase-segmented HMA mixture (AC-20) image using the OTSUmethod and it can be noticed that the OTSU thresholdingoperation only captures the aggregates at the edges of theimage To reduce the beam hardening effect in the CT imagean improved OTSU method developed by Liu and Li [20]is used for this purpose In this method the CT image is

decomposed into a series of circular subimages with 50overlap between each subimage given in Figure 1(c) Thenthe OTSU thresholding operation is applied for each circularsubimage to segment the three phases Figure 1(d) shows theaggregate phase segmented from the original CT image usingthe improvedOTSUmethod and the phase-segmented imageindicated with different colors can be seen in Figure 1(e)

4 Advances in Materials Science and Engineering

X

Y

Z

Voxel

Node

Figure 2 Sketch map of voxel definition

22 Numerical Model Generation A binary image is repre-sented by an 119872 times 119873 logical matrix where pixel values are 1(true) or 0 (false) The voxel defined in Figure 2 is generatedby expanding the pixel into three-dimensional space (3D)A voxel-based 3D digital reconstruction model of asphaltmixture is constructed when every pixel in the consecutiveCT images is converted into voxel as shown in Figures 3(a)and 3(b)

In order to input the element and node informationinto the finite element software such as ABAQUS the nodeand element numbering rules for generating the voxel-basednumericalmodel are defined as follows the node and elementnumbering sequences start from the lower left corner of thematrix and then go to the right side of each line and thelast number of each line is followed by the next line theposition of corresponding element number in the first imagediffers from that of the adjacent image by 119872 times 119873 while theposition of corresponding node number in the first imagediffers from that of the adjacent image by (119872 + 1) times (119873 + 1)The element and node information is generated with theMATLAB programming and written into an input file ofBAQUS for numerical simulations Figure 3(c) presents thevoxel-based numerical model of asphalt mixture

3 Modeling of Asphalt Mastic

The coarse aggregate phase is basically considered as elasticmaterial in nature The asphalt mastic phase is a typicalviscoelastic material which gives the asphalt mixture itsrheological characteristics The challenge in the modelingof micromechanical finite element model for asphalt con-crete includes the time temperature and rate-dependentbehavior of the asphalt mastic [16] In this section thethree-dimensional viscoelastic constitutive model is used torepresent the behavior of asphalt mastic phase Then theincremental formulations of the constitutive model imple-mented in a finite element code are developed in details

31 Viscoelastic Constitutive Model For linear viscoelasticmaterials including asphalt mastic the stressndashstrain consti-tutive relation is expressed by convolution integrals In thecase of strain response at constant stress the convolution rela-tions is explained as follows for one-dimensional problems

120576 (119905) = 1198630120590 + int1199050Δ119863(120593119905 minus 120593120591) 119889 (120590)119889120591 119889120591 (3)

where the1198630 is the instantaneous elastic compliance120593120591 is thereduced time and Δ119863 is the transient compliance It is givenby

Δ119863 (120593) = 119873sum119899=1

119863119899 (1 minus exp (minus120582119899120593)) (4)

where 119863119899 is the 119899th coefficient of the Prony series and 120582119899 isthe 119899th retardation time

For stress response at constant strain the convolutionrelations can be represented as follows

120590 (119905) = 1198640120576 + int1199050Δ119864 (120593119905 minus 120593120591) 119889 (120576)119889120591 119889120591 (5)

Δ119864 (120593) = 119873sum119899=1

119864119899 exp (minus120588119899120593) (6)

where the 1198640 and Δ119864 are the instantaneous elastic modulusand the transient modulus respectively 119864119899 is the 119899th coeffi-cient of the Prony series and 120588119899 is the 119899th relaxation time

For three-dimensional problems (3) can be decomposedinto deviatoric and volumetric components such that

119890119905119894119895 = 121198690119878119905119894119895 + 12 int119905

0Δ119869 (120593119905 minus 120593120591) 119889 (119878120591119894119895)119889120591 119889120591

120576119905119896119896 = 131198610120590119905119896119896 + 13 int119905

0Δ119861 (120593119905 minus 120593120591) 119889 (120590120591119896119896)119889120591 119889120591

120576119905119894119895 = 119890119905119894119895 + 13120576119905119896119896120575119894119895(7)

where 119890119905119894119895 and 120576119905119896119896 are the deviatoric strain and volumetricstrain respectively 120576119905119894119895 is the total strain and 120575119894119895 is the Kro-necker delta 1198690 and 1198610 are the instantaneous effective elasticshear and bulk compliances respectively Δ119869 and Δ119861 are thetransient shear compliance and bulk compliance respect-ively

Similarly (5) can be decomposed into deviatoric andvolumetric components such that

119878119905119894119895 = 21198660119890119905119894119895 + 2int1199050Δ119866 (120593119905 minus 120593120591) 119889 (119890120591119894119895)119889120591 119889120591

120590119905119896119896 = 31198700120598119905119896119896 + 3int1199050Δ119870(120593119905 minus 120593120591) 119889 (120576120591119896119896)119889120591 119889120591

120590119905119894119895 = 119878119905119894119895 + 13120590119905119896119896120575119894119895(8)

Advances in Materials Science and Engineering 5

(a) (b) (c)

Figure 3 Numerical model generation of asphalt mixture (a) series of consecutive CT images (b) gray unit net of multiple CT images and(c) voxel-based numerical model of asphalt mixture

where 119878119894119895 and 120590119896119896 are the deviatoric stress and volumetricstress respectively 120590119905119894119895 1198660 and 1198700 are the total stressthe instantaneous effective elastic shear modulus and bulkmodulus respectively Δ119866 and Δ119870 are the transient shearmodulus and bulk modulus respectively

32 Numerical Implementation for the Constitutive ModelThe finite element method (FEM) is actually an incrementalapproach for numerical analysis Current stress and strain atintegration points of each element at every time increment areobtained from the stress and strain over the previous loadinghistory So the three-dimensional incremental deviatoric andvolumetric formulations can be derived with some algebraicmanipulations

321 Stress-Based Incremental Formulations For stress-based incremental deviatoric and volumetric formulationsthe results are expressed as

Δ119890119905119894119895 = 119890119905119894119895 minus 119890119905minusΔ119905119894119895= 119869 (119905) Δ119878119894119895 minus 12

119873sum119899=1

119869119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899 119869 (119905)

= 12 1198690 +119873sum119899=1

119869119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)] 119902119905119894119895119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119878119894119895 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119896119896 = 120576119905119896119896 minus 120576119905minusΔ119905119896119896

= 119861 (119905) Δ120590119896119896 minus 13119873sum119899=1

119861119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899 119861 (119905)

= 13 1198610 +119873sum119899=1

119861119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)]

119902119905119896119896119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120590119896119896 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119894119895 = Δ119890119905119894119895 + 13Δ120576119905119896119896120575119894119895

(9)

where Δ119890119905119894119895 and Δ120576119905119896119896 are the incremental shear and bulkstrains respectively the variables 119902119905119894119895119899 and 119902119905119896119896119899 are theshear and volumetric hereditary integrals for every Pronyseries term 119899 at previous time 119905 respectively the hereditaryintegrals are updated at the end of every converged timeincrement for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119878119894119895((1 minus exp(minus120582119899Δ119905))120582119899Δ119905) and Δ120590119896119896((1 minusexp(minus120582119899Δ119905))120582119899Δ119905) respectivelyΔ120576119905119894119895 is the total incrementalstrain

322 Displacement-Based Incremental Formulations Obvi-ously a similar derivation procedure may be carried out forthe case of stress response at constant strainThe incrementalformulations are given by

Δ119878119905119894119895 = 119878119905119894119895 minus 119878119905minusΔ119905119894119895= 119866 (119905) Δ119890119894119895 + 2 119873sum

119899=1

119866119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899

119866 (119905) = 21198660 minus 119873sum119899=1

119866119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119894119895119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119890119894119895 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119896119896 = 120590119905119896119896 minus 120590119905minusΔ119905119896119896

= 119870 (119905) Δ120576119896119896 + 3 119873sum119899=1

119870119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

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Experimental reaction forceNumerical reaction forceDisplacement curve

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Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

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Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Biomaterials

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NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 Advances in Materials Science and Engineering

the discrete element method (DEM) to generate a three-dimensional (3D) idealizedmixture model for simulating thebehavior of a highly idealized asphalt mixture under uniaxialand triaxial compressive creep tests Wu et al [9] usedthe discrete element method (DEM) to simulate constantstrain rate compressive tests for an idealized asphalt mixturecomprising approximately single-sized sand particles Caiet al [10 11] used the discrete element method (DEM)to simulate the uniaxial compression tests of a realisticasphalt mixture under constant strain rate In these studiesthe produced numerical samples are either highly idealizedmodel or randomly generated model The most importantadvantages of these discrete element samples are the abilityto handle complex and changing contact geometries andlarge displacements However these discrete element sampleswere still not adequate enough to describe the complicatedmicrostructure of asphalt mixture with highly irregularshaped components Another disadvantage of these discreteelement samples is the relatively high computational cost

To overcome this shortcoming many researchers devel-oped micromechanical models to investigate the behaviorof asphalt mixture by using the finite element method(FEM) Dai [12 13] developed a two-dimensional (2D)microstructure finite element model of the asphalt mixturegenerated from the scanned image to predict the viscoelasticproperties of the asphalt mixtures where elastic propertieswere assigned to aggregates and viscoelastic properties wereassigned tomastic phase Kim and Lutif [14] proposed a com-putational micromechanics modeling approach to predictdamage-dependent constitutive behavior of asphalt mixturesArshadi and Bahia [15] developed the two-dimensional (2D)images of mastic and mortar scales artificially and used themto characterize the properties of those scales And thenthe 2D scanned images of asphalt mixtures are utilized tostudy the asphalt mixture behavior under uniaxial creepand recovery loading But the majority of the image-basedmicrostructure finite element models is limited to two-dimensional samples Lacking of three-dimensional infor-mation in FE framework makes them less able to representcomplicated morphology of asphalt mixture when used forsimulating

The nondestructive X-ray CT technique is used asan effective method to capture images of the internalmicrostructure of asphalt mixtureThere have been a numberof recent attempts to use X-ray CT images to reconstruct athree-dimensional (3D) finite element specimen to predictthe mechanical properties of asphalt mixture [16ndash19] Theimage-based finite element method requires the image analy-sis techniques in order to segment the CT slices consisting ofdifferent compositions themechanical properties of differentphases and the use of mechanical testing to identify thematerial parameters The majority of these microstructurenumerical samples for asphalt mixture faces challenges withregard to accurate geometry of different phases and precisedescription of asphalt mastic behavior

This paper presents an image-based micromechanicalmodeling approach for simulating the viscoelastic responseof asphaltmixture at a given temperature In order to enhancethe segmentation of different phases in the CT image we

developed an improved image analysis technique based onthe OTSU thresholding operation for use Then the three-dimensional viscoelastic constitutive model is applied tobetter describe themechanical behavior of asphaltmastic andimplemented in a finite element code using the ABAQUSuser material subroutine (UMAT) After that we proposean experimental procedure for identifying and verifying theviscoelastic parameters at a given temperature in the linearviscoelasticity range Finally the image-based numericalsample combinedwith theABAQUSusermaterial subroutinewill be applied to predict the viscoelastic behavior of a realmicrostructure asphalt mixture specimen

2 Microstructural Reconstruction Based onX-Ray CT Images

Asphalt mixture is a particulate composite material con-sisting of aggregate mastic and air voids In this sectiona cylindrical HMA mixture (AC-20) lab specimen wasprepared for capturing the internal microstructure with thenondestructive industrial X-ray CT technique Then thegrayscale thresholds for dividing aggregate matrix and airvoids were chosen based on the improved OTSU methodAdditionally a pixel-based 3D image model of the specimenwas constructed In Sections 21 and 22 the reconstructedprocess of the microstructure model is described in detail

21 Microstructure Acquisitions and Digital Image ProcessingX-ray CT is a commonly used nondestructive technique forcharacterizing the internalmicrostructure of asphaltmixtureThe basic image-forming principle of the industrial X-ray CTis that suppose the X-ray attenuation coefficient of the idealmaterial is120583 If the original X-ray strength is 1198680 the attenuatedX-ray strength is 119868 For homogeneousmaterial the attenuatedX-ray strength follows the Bill Exponential Law describedbelow

119868 = 1198680 exp (minus120583119909) (1)

For heterogeneous materials the attenuated X-raystrength 119868 can be calculated by dividing the object into smallunits So we can get the following equation

ln(1198680119868 ) = 12058311199091 + 12058321199092 + 12058331199093 + sdot sdot sdot (2)

The original X-ray strength 1198680 and the attenuated X-raystrength 119868 can be easily measured and the right side of (2) canbe calculated by numericalmethod In this study the YXLONCompact-225 CT X-ray scanner is used to obtain the detailedmicroscopic structure of the asphalt mixture specimen

Generally asphalt mixture is a three-phase compositematerial including aggregate matrix and air voids ScannedCT images of asphalt mixture have different grayscale inten-sities between 0 and 255 where denser materials have higherintensity The OTSU algorithm is a popular thresholdingmethod in adaptive optimal threshold selection for imagesegmentation

Beam hardening effect is a common phenomenon in X-ray CT images where the edge is brighter than the center

Advances in Materials Science and Engineering 3

(a) (b)

(c)

(d) (e)

Figure 1 Image segmentation of asphalt mixture CT slices (a) the original CT image (b) the segmented image using OTSUmethod (c) thesubimages decomposed by the improved OTSUmethod (d) the segmented image using improved OTSUmethod and (e) phase-segmentedimage

as presented in Figure 1(a) Figure 1(b) shows the phase-segmented HMA mixture (AC-20) image using the OTSUmethod and it can be noticed that the OTSU thresholdingoperation only captures the aggregates at the edges of theimage To reduce the beam hardening effect in the CT imagean improved OTSU method developed by Liu and Li [20]is used for this purpose In this method the CT image is

decomposed into a series of circular subimages with 50overlap between each subimage given in Figure 1(c) Thenthe OTSU thresholding operation is applied for each circularsubimage to segment the three phases Figure 1(d) shows theaggregate phase segmented from the original CT image usingthe improvedOTSUmethod and the phase-segmented imageindicated with different colors can be seen in Figure 1(e)

4 Advances in Materials Science and Engineering

X

Y

Z

Voxel

Node

Figure 2 Sketch map of voxel definition

22 Numerical Model Generation A binary image is repre-sented by an 119872 times 119873 logical matrix where pixel values are 1(true) or 0 (false) The voxel defined in Figure 2 is generatedby expanding the pixel into three-dimensional space (3D)A voxel-based 3D digital reconstruction model of asphaltmixture is constructed when every pixel in the consecutiveCT images is converted into voxel as shown in Figures 3(a)and 3(b)

In order to input the element and node informationinto the finite element software such as ABAQUS the nodeand element numbering rules for generating the voxel-basednumericalmodel are defined as follows the node and elementnumbering sequences start from the lower left corner of thematrix and then go to the right side of each line and thelast number of each line is followed by the next line theposition of corresponding element number in the first imagediffers from that of the adjacent image by 119872 times 119873 while theposition of corresponding node number in the first imagediffers from that of the adjacent image by (119872 + 1) times (119873 + 1)The element and node information is generated with theMATLAB programming and written into an input file ofBAQUS for numerical simulations Figure 3(c) presents thevoxel-based numerical model of asphalt mixture

3 Modeling of Asphalt Mastic

The coarse aggregate phase is basically considered as elasticmaterial in nature The asphalt mastic phase is a typicalviscoelastic material which gives the asphalt mixture itsrheological characteristics The challenge in the modelingof micromechanical finite element model for asphalt con-crete includes the time temperature and rate-dependentbehavior of the asphalt mastic [16] In this section thethree-dimensional viscoelastic constitutive model is used torepresent the behavior of asphalt mastic phase Then theincremental formulations of the constitutive model imple-mented in a finite element code are developed in details

31 Viscoelastic Constitutive Model For linear viscoelasticmaterials including asphalt mastic the stressndashstrain consti-tutive relation is expressed by convolution integrals In thecase of strain response at constant stress the convolution rela-tions is explained as follows for one-dimensional problems

120576 (119905) = 1198630120590 + int1199050Δ119863(120593119905 minus 120593120591) 119889 (120590)119889120591 119889120591 (3)

where the1198630 is the instantaneous elastic compliance120593120591 is thereduced time and Δ119863 is the transient compliance It is givenby

Δ119863 (120593) = 119873sum119899=1

119863119899 (1 minus exp (minus120582119899120593)) (4)

where 119863119899 is the 119899th coefficient of the Prony series and 120582119899 isthe 119899th retardation time

For stress response at constant strain the convolutionrelations can be represented as follows

120590 (119905) = 1198640120576 + int1199050Δ119864 (120593119905 minus 120593120591) 119889 (120576)119889120591 119889120591 (5)

Δ119864 (120593) = 119873sum119899=1

119864119899 exp (minus120588119899120593) (6)

where the 1198640 and Δ119864 are the instantaneous elastic modulusand the transient modulus respectively 119864119899 is the 119899th coeffi-cient of the Prony series and 120588119899 is the 119899th relaxation time

For three-dimensional problems (3) can be decomposedinto deviatoric and volumetric components such that

119890119905119894119895 = 121198690119878119905119894119895 + 12 int119905

0Δ119869 (120593119905 minus 120593120591) 119889 (119878120591119894119895)119889120591 119889120591

120576119905119896119896 = 131198610120590119905119896119896 + 13 int119905

0Δ119861 (120593119905 minus 120593120591) 119889 (120590120591119896119896)119889120591 119889120591

120576119905119894119895 = 119890119905119894119895 + 13120576119905119896119896120575119894119895(7)

where 119890119905119894119895 and 120576119905119896119896 are the deviatoric strain and volumetricstrain respectively 120576119905119894119895 is the total strain and 120575119894119895 is the Kro-necker delta 1198690 and 1198610 are the instantaneous effective elasticshear and bulk compliances respectively Δ119869 and Δ119861 are thetransient shear compliance and bulk compliance respect-ively

Similarly (5) can be decomposed into deviatoric andvolumetric components such that

119878119905119894119895 = 21198660119890119905119894119895 + 2int1199050Δ119866 (120593119905 minus 120593120591) 119889 (119890120591119894119895)119889120591 119889120591

120590119905119896119896 = 31198700120598119905119896119896 + 3int1199050Δ119870(120593119905 minus 120593120591) 119889 (120576120591119896119896)119889120591 119889120591

120590119905119894119895 = 119878119905119894119895 + 13120590119905119896119896120575119894119895(8)

Advances in Materials Science and Engineering 5

(a) (b) (c)

Figure 3 Numerical model generation of asphalt mixture (a) series of consecutive CT images (b) gray unit net of multiple CT images and(c) voxel-based numerical model of asphalt mixture

where 119878119894119895 and 120590119896119896 are the deviatoric stress and volumetricstress respectively 120590119905119894119895 1198660 and 1198700 are the total stressthe instantaneous effective elastic shear modulus and bulkmodulus respectively Δ119866 and Δ119870 are the transient shearmodulus and bulk modulus respectively

32 Numerical Implementation for the Constitutive ModelThe finite element method (FEM) is actually an incrementalapproach for numerical analysis Current stress and strain atintegration points of each element at every time increment areobtained from the stress and strain over the previous loadinghistory So the three-dimensional incremental deviatoric andvolumetric formulations can be derived with some algebraicmanipulations

321 Stress-Based Incremental Formulations For stress-based incremental deviatoric and volumetric formulationsthe results are expressed as

Δ119890119905119894119895 = 119890119905119894119895 minus 119890119905minusΔ119905119894119895= 119869 (119905) Δ119878119894119895 minus 12

119873sum119899=1

119869119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899 119869 (119905)

= 12 1198690 +119873sum119899=1

119869119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)] 119902119905119894119895119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119878119894119895 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119896119896 = 120576119905119896119896 minus 120576119905minusΔ119905119896119896

= 119861 (119905) Δ120590119896119896 minus 13119873sum119899=1

119861119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899 119861 (119905)

= 13 1198610 +119873sum119899=1

119861119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)]

119902119905119896119896119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120590119896119896 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119894119895 = Δ119890119905119894119895 + 13Δ120576119905119896119896120575119894119895

(9)

where Δ119890119905119894119895 and Δ120576119905119896119896 are the incremental shear and bulkstrains respectively the variables 119902119905119894119895119899 and 119902119905119896119896119899 are theshear and volumetric hereditary integrals for every Pronyseries term 119899 at previous time 119905 respectively the hereditaryintegrals are updated at the end of every converged timeincrement for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119878119894119895((1 minus exp(minus120582119899Δ119905))120582119899Δ119905) and Δ120590119896119896((1 minusexp(minus120582119899Δ119905))120582119899Δ119905) respectivelyΔ120576119905119894119895 is the total incrementalstrain

322 Displacement-Based Incremental Formulations Obvi-ously a similar derivation procedure may be carried out forthe case of stress response at constant strainThe incrementalformulations are given by

Δ119878119905119894119895 = 119878119905119894119895 minus 119878119905minusΔ119905119894119895= 119866 (119905) Δ119890119894119895 + 2 119873sum

119899=1

119866119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899

119866 (119905) = 21198660 minus 119873sum119899=1

119866119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119894119895119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119890119894119895 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119896119896 = 120590119905119896119896 minus 120590119905minusΔ119905119896119896

= 119870 (119905) Δ120576119896119896 + 3 119873sum119899=1

119870119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 3

(a) (b)

(c)

(d) (e)

Figure 1 Image segmentation of asphalt mixture CT slices (a) the original CT image (b) the segmented image using OTSUmethod (c) thesubimages decomposed by the improved OTSUmethod (d) the segmented image using improved OTSUmethod and (e) phase-segmentedimage

as presented in Figure 1(a) Figure 1(b) shows the phase-segmented HMA mixture (AC-20) image using the OTSUmethod and it can be noticed that the OTSU thresholdingoperation only captures the aggregates at the edges of theimage To reduce the beam hardening effect in the CT imagean improved OTSU method developed by Liu and Li [20]is used for this purpose In this method the CT image is

decomposed into a series of circular subimages with 50overlap between each subimage given in Figure 1(c) Thenthe OTSU thresholding operation is applied for each circularsubimage to segment the three phases Figure 1(d) shows theaggregate phase segmented from the original CT image usingthe improvedOTSUmethod and the phase-segmented imageindicated with different colors can be seen in Figure 1(e)

4 Advances in Materials Science and Engineering

X

Y

Z

Voxel

Node

Figure 2 Sketch map of voxel definition

22 Numerical Model Generation A binary image is repre-sented by an 119872 times 119873 logical matrix where pixel values are 1(true) or 0 (false) The voxel defined in Figure 2 is generatedby expanding the pixel into three-dimensional space (3D)A voxel-based 3D digital reconstruction model of asphaltmixture is constructed when every pixel in the consecutiveCT images is converted into voxel as shown in Figures 3(a)and 3(b)

In order to input the element and node informationinto the finite element software such as ABAQUS the nodeand element numbering rules for generating the voxel-basednumericalmodel are defined as follows the node and elementnumbering sequences start from the lower left corner of thematrix and then go to the right side of each line and thelast number of each line is followed by the next line theposition of corresponding element number in the first imagediffers from that of the adjacent image by 119872 times 119873 while theposition of corresponding node number in the first imagediffers from that of the adjacent image by (119872 + 1) times (119873 + 1)The element and node information is generated with theMATLAB programming and written into an input file ofBAQUS for numerical simulations Figure 3(c) presents thevoxel-based numerical model of asphalt mixture

3 Modeling of Asphalt Mastic

The coarse aggregate phase is basically considered as elasticmaterial in nature The asphalt mastic phase is a typicalviscoelastic material which gives the asphalt mixture itsrheological characteristics The challenge in the modelingof micromechanical finite element model for asphalt con-crete includes the time temperature and rate-dependentbehavior of the asphalt mastic [16] In this section thethree-dimensional viscoelastic constitutive model is used torepresent the behavior of asphalt mastic phase Then theincremental formulations of the constitutive model imple-mented in a finite element code are developed in details

31 Viscoelastic Constitutive Model For linear viscoelasticmaterials including asphalt mastic the stressndashstrain consti-tutive relation is expressed by convolution integrals In thecase of strain response at constant stress the convolution rela-tions is explained as follows for one-dimensional problems

120576 (119905) = 1198630120590 + int1199050Δ119863(120593119905 minus 120593120591) 119889 (120590)119889120591 119889120591 (3)

where the1198630 is the instantaneous elastic compliance120593120591 is thereduced time and Δ119863 is the transient compliance It is givenby

Δ119863 (120593) = 119873sum119899=1

119863119899 (1 minus exp (minus120582119899120593)) (4)

where 119863119899 is the 119899th coefficient of the Prony series and 120582119899 isthe 119899th retardation time

For stress response at constant strain the convolutionrelations can be represented as follows

120590 (119905) = 1198640120576 + int1199050Δ119864 (120593119905 minus 120593120591) 119889 (120576)119889120591 119889120591 (5)

Δ119864 (120593) = 119873sum119899=1

119864119899 exp (minus120588119899120593) (6)

where the 1198640 and Δ119864 are the instantaneous elastic modulusand the transient modulus respectively 119864119899 is the 119899th coeffi-cient of the Prony series and 120588119899 is the 119899th relaxation time

For three-dimensional problems (3) can be decomposedinto deviatoric and volumetric components such that

119890119905119894119895 = 121198690119878119905119894119895 + 12 int119905

0Δ119869 (120593119905 minus 120593120591) 119889 (119878120591119894119895)119889120591 119889120591

120576119905119896119896 = 131198610120590119905119896119896 + 13 int119905

0Δ119861 (120593119905 minus 120593120591) 119889 (120590120591119896119896)119889120591 119889120591

120576119905119894119895 = 119890119905119894119895 + 13120576119905119896119896120575119894119895(7)

where 119890119905119894119895 and 120576119905119896119896 are the deviatoric strain and volumetricstrain respectively 120576119905119894119895 is the total strain and 120575119894119895 is the Kro-necker delta 1198690 and 1198610 are the instantaneous effective elasticshear and bulk compliances respectively Δ119869 and Δ119861 are thetransient shear compliance and bulk compliance respect-ively

Similarly (5) can be decomposed into deviatoric andvolumetric components such that

119878119905119894119895 = 21198660119890119905119894119895 + 2int1199050Δ119866 (120593119905 minus 120593120591) 119889 (119890120591119894119895)119889120591 119889120591

120590119905119896119896 = 31198700120598119905119896119896 + 3int1199050Δ119870(120593119905 minus 120593120591) 119889 (120576120591119896119896)119889120591 119889120591

120590119905119894119895 = 119878119905119894119895 + 13120590119905119896119896120575119894119895(8)

Advances in Materials Science and Engineering 5

(a) (b) (c)

Figure 3 Numerical model generation of asphalt mixture (a) series of consecutive CT images (b) gray unit net of multiple CT images and(c) voxel-based numerical model of asphalt mixture

where 119878119894119895 and 120590119896119896 are the deviatoric stress and volumetricstress respectively 120590119905119894119895 1198660 and 1198700 are the total stressthe instantaneous effective elastic shear modulus and bulkmodulus respectively Δ119866 and Δ119870 are the transient shearmodulus and bulk modulus respectively

32 Numerical Implementation for the Constitutive ModelThe finite element method (FEM) is actually an incrementalapproach for numerical analysis Current stress and strain atintegration points of each element at every time increment areobtained from the stress and strain over the previous loadinghistory So the three-dimensional incremental deviatoric andvolumetric formulations can be derived with some algebraicmanipulations

321 Stress-Based Incremental Formulations For stress-based incremental deviatoric and volumetric formulationsthe results are expressed as

Δ119890119905119894119895 = 119890119905119894119895 minus 119890119905minusΔ119905119894119895= 119869 (119905) Δ119878119894119895 minus 12

119873sum119899=1

119869119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899 119869 (119905)

= 12 1198690 +119873sum119899=1

119869119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)] 119902119905119894119895119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119878119894119895 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119896119896 = 120576119905119896119896 minus 120576119905minusΔ119905119896119896

= 119861 (119905) Δ120590119896119896 minus 13119873sum119899=1

119861119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899 119861 (119905)

= 13 1198610 +119873sum119899=1

119861119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)]

119902119905119896119896119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120590119896119896 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119894119895 = Δ119890119905119894119895 + 13Δ120576119905119896119896120575119894119895

(9)

where Δ119890119905119894119895 and Δ120576119905119896119896 are the incremental shear and bulkstrains respectively the variables 119902119905119894119895119899 and 119902119905119896119896119899 are theshear and volumetric hereditary integrals for every Pronyseries term 119899 at previous time 119905 respectively the hereditaryintegrals are updated at the end of every converged timeincrement for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119878119894119895((1 minus exp(minus120582119899Δ119905))120582119899Δ119905) and Δ120590119896119896((1 minusexp(minus120582119899Δ119905))120582119899Δ119905) respectivelyΔ120576119905119894119895 is the total incrementalstrain

322 Displacement-Based Incremental Formulations Obvi-ously a similar derivation procedure may be carried out forthe case of stress response at constant strainThe incrementalformulations are given by

Δ119878119905119894119895 = 119878119905119894119895 minus 119878119905minusΔ119905119894119895= 119866 (119905) Δ119890119894119895 + 2 119873sum

119899=1

119866119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899

119866 (119905) = 21198660 minus 119873sum119899=1

119866119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119894119895119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119890119894119895 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119896119896 = 120590119905119896119896 minus 120590119905minusΔ119905119896119896

= 119870 (119905) Δ120576119896119896 + 3 119873sum119899=1

119870119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Nano

materials

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Journal ofNanomaterials

4 Advances in Materials Science and Engineering

X

Y

Z

Voxel

Node

Figure 2 Sketch map of voxel definition

22 Numerical Model Generation A binary image is repre-sented by an 119872 times 119873 logical matrix where pixel values are 1(true) or 0 (false) The voxel defined in Figure 2 is generatedby expanding the pixel into three-dimensional space (3D)A voxel-based 3D digital reconstruction model of asphaltmixture is constructed when every pixel in the consecutiveCT images is converted into voxel as shown in Figures 3(a)and 3(b)

In order to input the element and node informationinto the finite element software such as ABAQUS the nodeand element numbering rules for generating the voxel-basednumericalmodel are defined as follows the node and elementnumbering sequences start from the lower left corner of thematrix and then go to the right side of each line and thelast number of each line is followed by the next line theposition of corresponding element number in the first imagediffers from that of the adjacent image by 119872 times 119873 while theposition of corresponding node number in the first imagediffers from that of the adjacent image by (119872 + 1) times (119873 + 1)The element and node information is generated with theMATLAB programming and written into an input file ofBAQUS for numerical simulations Figure 3(c) presents thevoxel-based numerical model of asphalt mixture

3 Modeling of Asphalt Mastic

The coarse aggregate phase is basically considered as elasticmaterial in nature The asphalt mastic phase is a typicalviscoelastic material which gives the asphalt mixture itsrheological characteristics The challenge in the modelingof micromechanical finite element model for asphalt con-crete includes the time temperature and rate-dependentbehavior of the asphalt mastic [16] In this section thethree-dimensional viscoelastic constitutive model is used torepresent the behavior of asphalt mastic phase Then theincremental formulations of the constitutive model imple-mented in a finite element code are developed in details

31 Viscoelastic Constitutive Model For linear viscoelasticmaterials including asphalt mastic the stressndashstrain consti-tutive relation is expressed by convolution integrals In thecase of strain response at constant stress the convolution rela-tions is explained as follows for one-dimensional problems

120576 (119905) = 1198630120590 + int1199050Δ119863(120593119905 minus 120593120591) 119889 (120590)119889120591 119889120591 (3)

where the1198630 is the instantaneous elastic compliance120593120591 is thereduced time and Δ119863 is the transient compliance It is givenby

Δ119863 (120593) = 119873sum119899=1

119863119899 (1 minus exp (minus120582119899120593)) (4)

where 119863119899 is the 119899th coefficient of the Prony series and 120582119899 isthe 119899th retardation time

For stress response at constant strain the convolutionrelations can be represented as follows

120590 (119905) = 1198640120576 + int1199050Δ119864 (120593119905 minus 120593120591) 119889 (120576)119889120591 119889120591 (5)

Δ119864 (120593) = 119873sum119899=1

119864119899 exp (minus120588119899120593) (6)

where the 1198640 and Δ119864 are the instantaneous elastic modulusand the transient modulus respectively 119864119899 is the 119899th coeffi-cient of the Prony series and 120588119899 is the 119899th relaxation time

For three-dimensional problems (3) can be decomposedinto deviatoric and volumetric components such that

119890119905119894119895 = 121198690119878119905119894119895 + 12 int119905

0Δ119869 (120593119905 minus 120593120591) 119889 (119878120591119894119895)119889120591 119889120591

120576119905119896119896 = 131198610120590119905119896119896 + 13 int119905

0Δ119861 (120593119905 minus 120593120591) 119889 (120590120591119896119896)119889120591 119889120591

120576119905119894119895 = 119890119905119894119895 + 13120576119905119896119896120575119894119895(7)

where 119890119905119894119895 and 120576119905119896119896 are the deviatoric strain and volumetricstrain respectively 120576119905119894119895 is the total strain and 120575119894119895 is the Kro-necker delta 1198690 and 1198610 are the instantaneous effective elasticshear and bulk compliances respectively Δ119869 and Δ119861 are thetransient shear compliance and bulk compliance respect-ively

Similarly (5) can be decomposed into deviatoric andvolumetric components such that

119878119905119894119895 = 21198660119890119905119894119895 + 2int1199050Δ119866 (120593119905 minus 120593120591) 119889 (119890120591119894119895)119889120591 119889120591

120590119905119896119896 = 31198700120598119905119896119896 + 3int1199050Δ119870(120593119905 minus 120593120591) 119889 (120576120591119896119896)119889120591 119889120591

120590119905119894119895 = 119878119905119894119895 + 13120590119905119896119896120575119894119895(8)

Advances in Materials Science and Engineering 5

(a) (b) (c)

Figure 3 Numerical model generation of asphalt mixture (a) series of consecutive CT images (b) gray unit net of multiple CT images and(c) voxel-based numerical model of asphalt mixture

where 119878119894119895 and 120590119896119896 are the deviatoric stress and volumetricstress respectively 120590119905119894119895 1198660 and 1198700 are the total stressthe instantaneous effective elastic shear modulus and bulkmodulus respectively Δ119866 and Δ119870 are the transient shearmodulus and bulk modulus respectively

32 Numerical Implementation for the Constitutive ModelThe finite element method (FEM) is actually an incrementalapproach for numerical analysis Current stress and strain atintegration points of each element at every time increment areobtained from the stress and strain over the previous loadinghistory So the three-dimensional incremental deviatoric andvolumetric formulations can be derived with some algebraicmanipulations

321 Stress-Based Incremental Formulations For stress-based incremental deviatoric and volumetric formulationsthe results are expressed as

Δ119890119905119894119895 = 119890119905119894119895 minus 119890119905minusΔ119905119894119895= 119869 (119905) Δ119878119894119895 minus 12

119873sum119899=1

119869119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899 119869 (119905)

= 12 1198690 +119873sum119899=1

119869119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)] 119902119905119894119895119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119878119894119895 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119896119896 = 120576119905119896119896 minus 120576119905minusΔ119905119896119896

= 119861 (119905) Δ120590119896119896 minus 13119873sum119899=1

119861119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899 119861 (119905)

= 13 1198610 +119873sum119899=1

119861119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)]

119902119905119896119896119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120590119896119896 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119894119895 = Δ119890119905119894119895 + 13Δ120576119905119896119896120575119894119895

(9)

where Δ119890119905119894119895 and Δ120576119905119896119896 are the incremental shear and bulkstrains respectively the variables 119902119905119894119895119899 and 119902119905119896119896119899 are theshear and volumetric hereditary integrals for every Pronyseries term 119899 at previous time 119905 respectively the hereditaryintegrals are updated at the end of every converged timeincrement for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119878119894119895((1 minus exp(minus120582119899Δ119905))120582119899Δ119905) and Δ120590119896119896((1 minusexp(minus120582119899Δ119905))120582119899Δ119905) respectivelyΔ120576119905119894119895 is the total incrementalstrain

322 Displacement-Based Incremental Formulations Obvi-ously a similar derivation procedure may be carried out forthe case of stress response at constant strainThe incrementalformulations are given by

Δ119878119905119894119895 = 119878119905119894119895 minus 119878119905minusΔ119905119894119895= 119866 (119905) Δ119890119894119895 + 2 119873sum

119899=1

119866119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899

119866 (119905) = 21198660 minus 119873sum119899=1

119866119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119894119895119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119890119894119895 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119896119896 = 120590119905119896119896 minus 120590119905minusΔ119905119896119896

= 119870 (119905) Δ120576119896119896 + 3 119873sum119899=1

119870119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

(a) (b) (c)

Figure 3 Numerical model generation of asphalt mixture (a) series of consecutive CT images (b) gray unit net of multiple CT images and(c) voxel-based numerical model of asphalt mixture

where 119878119894119895 and 120590119896119896 are the deviatoric stress and volumetricstress respectively 120590119905119894119895 1198660 and 1198700 are the total stressthe instantaneous effective elastic shear modulus and bulkmodulus respectively Δ119866 and Δ119870 are the transient shearmodulus and bulk modulus respectively

32 Numerical Implementation for the Constitutive ModelThe finite element method (FEM) is actually an incrementalapproach for numerical analysis Current stress and strain atintegration points of each element at every time increment areobtained from the stress and strain over the previous loadinghistory So the three-dimensional incremental deviatoric andvolumetric formulations can be derived with some algebraicmanipulations

321 Stress-Based Incremental Formulations For stress-based incremental deviatoric and volumetric formulationsthe results are expressed as

Δ119890119905119894119895 = 119890119905119894119895 minus 119890119905minusΔ119905119894119895= 119869 (119905) Δ119878119894119895 minus 12

119873sum119899=1

119869119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899 119869 (119905)

= 12 1198690 +119873sum119899=1

119869119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)] 119902119905119894119895119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119878119894119895 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119896119896 = 120576119905119896119896 minus 120576119905minusΔ119905119896119896

= 119861 (119905) Δ120590119896119896 minus 13119873sum119899=1

119861119899 [exp (minus120582119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899 119861 (119905)

= 13 1198610 +119873sum119899=1

119861119899 [1 + 1120582119899Δ119905 (exp (minus120582119899Δ119905) minus 1)]

119902119905119896119896119899 = exp (minus120582119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120590119896119896 1 minus exp (minus120582119899Δ119905)120582119899Δ119905 Δ120576119905119894119895 = Δ119890119905119894119895 + 13Δ120576119905119896119896120575119894119895

(9)

where Δ119890119905119894119895 and Δ120576119905119896119896 are the incremental shear and bulkstrains respectively the variables 119902119905119894119895119899 and 119902119905119896119896119899 are theshear and volumetric hereditary integrals for every Pronyseries term 119899 at previous time 119905 respectively the hereditaryintegrals are updated at the end of every converged timeincrement for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119878119894119895((1 minus exp(minus120582119899Δ119905))120582119899Δ119905) and Δ120590119896119896((1 minusexp(minus120582119899Δ119905))120582119899Δ119905) respectivelyΔ120576119905119894119895 is the total incrementalstrain

322 Displacement-Based Incremental Formulations Obvi-ously a similar derivation procedure may be carried out forthe case of stress response at constant strainThe incrementalformulations are given by

Δ119878119905119894119895 = 119878119905119894119895 minus 119878119905minusΔ119905119894119895= 119866 (119905) Δ119890119894119895 + 2 119873sum

119899=1

119866119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119894119895119899

119866 (119905) = 21198660 minus 119873sum119899=1

119866119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119894119895119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119894119895119899 + Δ119890119894119895 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119896119896 = 120590119905119896119896 minus 120590119905minusΔ119905119896119896

= 119870 (119905) Δ120576119896119896 + 3 119873sum119899=1

119870119899 [exp (minus120588119899Δ119905) minus 1] 119902119905minusΔ119905119896119896119899

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

Table 1 Prony series coefficients of creep compliance and relaxation modulus for asphalt mastic

119894 120582119899s 119869119899MPaminus1 119861119899MPaminus1 119866119899MPa 119870119899MPa1 10minus4 02396 115076 11876 002122 10minus3 00476 21289 82732 021243 10minus2 00423 02913 71921 089384 10minus1 00111 00359 393043 1032265 100 00034 00108 326392 2022226 101 00034 00108 289523 266257 102 00034 00108 738451 2954538 103 00034 00108 3070718 9177361198690 = 221119864 minus 04 1198610 = 727119864 minus 04 1198660 = 28219 1198700 = 00714

119870 (119905) = 31198700 minus 119873sum119899=1

119870119899 1120588119899Δ119905 (exp (minus120588119899Δ119905) minus 1)

119902119905119896119896119899 = exp (minus120588119899Δ119905) 119902119905minusΔ119905119896119896119899 + Δ120576119896119896 1 minus exp (minus120588119899Δ119905)120588119899Δ119905 Δ120590119905119894119895 = Δ119878119905119894119895 + 13Δ120590119905119896119896120575119894119895

(10)

whereΔ119878119905119894119895 andΔ120590119905119896119896 are the incremental shear and bulk stressrespectively for the initial increment the variables 1199021199051198941198951 and1199021199051198961198961 are set to Δ119890119894119895((1 minus exp(minus120588119899Δ119905))120588119899Δ119905) and Δ120576119896119896((1 minusexp(minus120588119899Δ119905))120588119899Δ119905) respectively Δ120590119905119894119895 is the total incrementalstress

The stress-based and displacement-based three-dimensional numerical constitutive models are implementedwithin the FE code using FORTRAN languageTheABAQUSuser material subroutine (UMAT) is applied for this purpose

4 Identification and Verification

41 Identification of the Material Parameters According tothe elasticndashviscoelastic correspondence principle each of thetensile creep compliance 119863 the shear 119869 compliance and thebulk compliance 119861 can be obtained from the other two usingLaplace transform The tensile creep compliance 119863 and theshear compliance 119869 can be easily determined directly fromuniaxial tensile tests and torsion tests respectively Then thebulk compliance 119861 can be calculated from the tensile creepcompliance 119863 and the shear compliance 119869 A similar set ofequations may be formulated for the bulk modulus 119870 whenthe tensile modulus 119864 and the shear 119866 compliance are avail-able For displacement-based linear viscoelastic constitutivemodel an indirect method verified by previous research [21]is applied to determine the fundamental relaxation modulus119864 and 119866 from the known compliance function

Asphalt mastic comprises of fine aggregates and asphaltbinder The asphalt binder content in asphalt mastic is thesame as the full HMAmixture (AC-20) excluding the binderabsorbed by coarse aggregates (larger than 236mm) Thefinished experimental beams cut from a cylindrical specimen

have a dimension of 10mm times 10mm times 50mm in lengthwidth and height respectively

Uniaxial tensile tests and torsion tests were applied todetermine the material parameters in three-dimensionalviscoelastic constitutive model at a temperature of 20∘C Inthe theory of linear viscoelasticity the strain response toany applied stress is independent of the stress magnitudeThis characteristic can be adopted to the static creep test bymonitoring the creep compliance as stress increases If thecreep compliance curves vary a little subjected to a range ofloading levels linear viscoelasticity holds The tensile creepdata and shear creep data obtained from uniaxial tensile testsand torsion tests respectively were employed in determiningProny series coefficients of creep compliances Then theProny-based series coefficients of relaxation modulus can beconverted by themethod proposed in previous study [21]Theresults of Prony series coefficients are shown in Table 1

42 Verifications of the Incremental Constitutive ModelThe capability of the numerical constitutive model can beexamined by comparing the numerical predictions with theobserved laboratory tests In this paper two basic loadingpaths (bending loading and compression loading) were uti-lized to conduct a series of displacement-based tests at atemperature of 20∘C to verify the capability of the incrementalconstitutive model

Beams with a dimension of 30mm times 30mm times 120mm inlength width and height were applied to conduct bendingflexural tests Two primary modes of displacement-basedloading bending tests were performed on the beams For theverification purpose three-dimensional numerical constitu-tivemodelmentioned above was introduced to reproduce thebending tests The comparisons between the predictions andthe corresponding experimental results are given in Figure 4

From Figure 4 it can be seen that the numerical predic-tions exhibit consistent trend with the experimental resultsThe differences of the vertical reaction force between thenumerical predictions and measured results for the twoprimary modes of displacement-based loading are 21and 22 respectively Obviously a rather good agreementbetween the simulations and experimental measurementscan be obtained

As for compression loading cylindrical asphalt masticspecimens having a diameter and height of 100 times 100mm

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

Experimental reaction forceNumerical reaction forceDisplacement curve

0

4

8

12

16

20Ve

rtic

al re

actio

n fo

rce (

N)

0

01

02

03

04

05

06

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 1200Times (s)

(a)

0

5

10

15

20

25

30

Vert

ical

reac

tion

forc

e (N

)

00102030405060708

Vert

ical

disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

Experimental reaction forceNumerical reaction forceDisplacement curve

(b)

Figure 4 Simulated and experimental bending results of displacement-based loading (a) loading and unloading and (b) loading and holdingsteady-state

Numerical reaction forceExperimental reaction forceDisplacement curve

0

40

80

120

160

200

Uni

axia

l rea

ctio

n fo

rce (

N)

0

002

004

006

008

01

012

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 1200Time (s)

(a)

Numerical reaction forceExperimental reaction forceDisplacement curve

0

100

200

300

400

500

600U

niax

ial r

eact

ion

forc

e (N

)

000501015020250303504

Uni

axia

l disp

lace

men

t (m

m)

20 40 60 80 100 120 140 160 1800Time (s)

(b)

Figure 5 Simulated and experimental compression results of displacement-based loading (a) loading and unloading and (b) loading andholding steady-state

were utilized for testing The same shapes of loading wereused for compression responses to arbitrary displacement-based loading step history of mastic asphalt The experimen-tal results and the testing results are given in Figure 5

It can be observed that the predicted results obtainedfrom the numerical constitutivemodel show good agreementwith the corresponding experimental data This suggests thatonce the appropriate viscoelastic constitutive parameters forasphalt mastic at a constant temperature are available thenumerical constitutive model is capable of describing thematerial response under compression loading

5 Numerical Simulations

The purpose of this section is to conduct the micromechan-ical simulations of the digital sample reconstructed fromX-ray CT slices to predict the viscoelastic properties underdynamic loading modes Asphalt mixture specimen with a

dimension of 100mm diameter by 200mm height is usuallyrecommended for the dynamic modulus tests and the creeptests tominimize edge effects (AASHTOTP 62-03 AASHTOTP-70) Usually the thickness of a typical asphalt surface layeris far less than 150mm The indirect tensile test (IDT) usedon field cores is the most effective method to evaluate themechanical properties of the existing pavement In this studyan asphalt mixture specimen with a dimension of 100mmdiameter by 30mm height was utilized for the numericalsimulations The specimen was scanned by the X-ray CTdevice with a resolution of 1024 times 1240 All the CT slices wereconverted to low-resolution images to reduce the number ofelements The numerical model of the asphalt mixture wasgenerated from the image pixels using the MATLAB code asshown in Figure 3(c)

The eight-node 3D solid integration elements (C3D8)with a unit thickness were used in constructing the meshThe aggregates contained a total of 35611 elements thematrix

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

phase had 44181 elements and the remaining elements werepart of the air voids inclusions with a total number of 179elements The aggregates and the steel loading strip areconsidered as a linear elastic material and the modulus ofelasticity and Poissonrsquos ratio for the aggregate and the steelloading strip are assumed to be 25GPa and 025 80GPa and02 respectively The air void will be included in this digitalsample with an elastic modulus of 05MPa and Poissonrsquosratio of 03 The matrix is considered as a linear viscoelasticmaterial Parameters for the viscoelastic constitutive of thematrix are determined in Section 41 and the numericalviscoelastic constitutive model is implemented within theFE code using FORTRAN The ABAQUS user materialsubroutine (UMAT) is applied for this purpose

51 The Dynamic Test In the Mechanistic Empirical Pave-ment Design Guide (MEPDG) the dynamic modulus ofasphalt mixture is used as an important input parameter tocharacterize the temperature and frequency dependentbehavior for pavement design and construction [22] Thestandard dynamic modulus determination procedure con-sists of the uniaxial partial sinusoidal compressive test andthe indirect partial sinusoidal tension test while the stressstate of the indirect tension test specimen subjected to verticalloading is very similar to that of the field It is apparentthat dynamic modulus measured from the indirect tensiontest can better characterize the in-situ behavior of asphaltmixture

The dynamic modulus simulations in indirect tensionmode under six partial sinusoidal cycle loading frequencies(01 05 1 5 10 and 25Hz) were conducted to showthe utility of the developed microstructure digital modelof asphalt mixture containing the three main phases Thedisplacement-based loading was imposed to the top of thesteel loading strip and the steel loading strip distributedthe applied load on the top surface of the numerical modelIn order to ensure that the asphalt mixture behaves as alinear viscoelastic material the vertical strain was confinedto 001 The displacement-based incremental constitutivemodel for asphalt mastic derived in Section 322 would beincorporated in ABAQUS user material subroutine (UMAT)to model the effective asphalt mastic behavior

For frequencies of 01 05 and 1Hz six displacement-based loading cycles were used and ten displacement loadingcycles were applied for the frequencies of 5 10 and 25HzTo balance the computational cost and the smoothness ofthe stressstrain-time response curves 20 time increments(computation points) were applied for each loading cycleIn the dynamic modulus simulation the reacting force andthe vertical displacement of the whole model were recordedStress is defined as the reacting force divided by the crosssection area of the digital specimen Strain is the change in thevertical deformation of the digital specimen divided by theinitial heightThe dynamic modulus is calculated by dividingthe peak stress amplitude with the peak strain amplitude

1003816100381610038161003816119864lowast1003816100381610038161003816 = 12059001205760 (11)

where |119864lowast| is the dynamic modulus 1205900 is the peak-to-peakstress amplitude and 1205760 is the peak-to-peak strain amplitude

In this study the averages of the reacting force and thevertical displacement of the last two loading cycles were usedto calculate the dynamic modulus The stressstrain-timeresponds under different loading frequencies were presentedin Figure 6

It is clearly shown that the overall stress respondsunder different loading frequencies drop as the loading timeincreases and the peak strain lags behind the peak loadingunder the same loading cycle which demonstrates that thedigital sample behaves as a viscoelastic material under cycleloading The dynamic modulus was calculated with (11) asshown in Figure 7 It shows that the dynamic modulusincreases with the loading frequencies The results indicatethat the developed digital microstructure sample is capableto predict the macroscopic viscoelastic behaviors of asphaltmixture with experimental material parameters

52 Repeated Creep-Recovery Test A typical vehicle load inflexible pavement is a cyclic load with loading and unloadingperiods So in the laboratory tests repeated creep-recoverytest is more representative to what the asphalt mixtureexperiences under traffic loading Figure 8 shows the stressinput history for the repeated creep-recovery test simulationwith a 6-second loading duration and a 2-second unloadingduration

Repeated creep-recovery test simulation is conductedusing the same microstructure digital model of asphaltmixture utilized for dynamic simulations The vertical stressapplied to the top of the steel loading strip is 12MPa dueto the high stiffness for AC-20 at a temperature of 20∘C Tobalance the computational cost and the smoothness of thestrain-time response curves 40 time increments (computa-tion points) were applied for each loading cycle The force-based incremental constitutive model for matrix derivedin Section 321 would be incorporated in ABAQUS usermaterial subroutine (UMAT) to model the effective asphaltmixture behavior

Figure 9 shows the vertical displacement and strain con-tour at 60th time step respectively It is found in Figure 9(a)that the vertical displacement decreases from the top to thebottom but not left-right symmetric due to the heterogeneityof the asphalt mixture digital model Figure 9(b) presents thatthe local vertical strain distributions are largely in the state ofcompression The vertical compressive strain mainly occursin the vicinity of the vertical central axis This is because theaggregates around the central axis will be extruded into askeleton to withstand the compressive loading It also can beseen from Figure 9(b) that the matrix phase suffers greaterdeformation due to the large difference in stiffness betweenaggregate and matrix

The corresponding strain-time response under verticalstress for 64 time steps is presented in Figure 10 It can clearlybe noticed that the primary and secondary creep regionscan be observed from this figure However damage wasnot considered in this study and the tertiary creep regiondoes not occur as the numbers of loading cycle increasesThis trend is in close agreement with experimental data

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 9

Stra

in (1

times10minus6)

020406080100120

StressStrain

0015

03045

06075

09St

ress

(MPa

)

5 10 15 20 25 30 35 40 45 50 55 600Time (s)

(a) Frequency = 01 Hz

Stra

in (1

times10minus6)

102 84 6 120 5 93 111 7Time (s)

002040608

1

Stre

ss (M

Pa)

020406080100120

StressStrain

(b) Frequency = 05Hz

Stra

in (1

times10minus6)

020406080100120140

51 42 3 60 25 4515 5505 35Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(c) Frequency = 1Hz

Stra

in (1

times10minus6)

020406080100120140

02 04 06 08 1 12 14 16 18 20Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(d) Frequency = 5Hz

Stra

in (1

times10minus6)

020406080100120140

01 02 03 04 05 06 07 08 09 10Time (s)

002040608

112

Stre

ss (M

Pa)

StressStrain

(e) Frequency = 10Hz

Stra

in (1

times10minus6)

020406080100120140

008 016 024 032 040Time (s)

002040608

11214

Stre

ss (M

Pa)

004 012 02 028 036

StressStrain

(f) Frequency = 25Hz

Figure 6 The stressstrain-time responds of the dynamic simulation under different frequencies

885

99510

10511

11512

Com

plex

mod

ulus

(GPa

)

1 10 10001lg (frequencies) (Hz)

Figure 7 The dynamic modulus simulations under different load-ing frequencies

regarding the linear viscoelastic behavior of asphalt mixtureunder repeated loading in the previous studies [23 24] It isclear that the three-dimensional (3D) image-based numerical

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

002040608

11214

Stre

ss (M

Pa)

Figure 8 Applied stress-time diagram

model and the verified incremental constitutivemodel can beused to predict the mechanical response of asphalt mixtureunder repeated creep loading

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 Advances in Materials Science and Engineering

Y Z

Xminus3478e minus 04+1302e minus 03+2952e minus 03+4602e minus 03+6252e minus 03+7902e minus 03+9552e minus 03+1120e minus 02+1285e minus 02+1450e minus 02+1615e minus 02+1780e minus 02+1945e minus 02+2110e minus 02+2275e minus 02+2440e minus 02+2605e minus 02+2770e minus 02+2935e minus 02+3100e minus 02+3265e minus 02

uu1

(a)

YZ

X minus1876e minus 02

minus1720e minus 02

minus1565e minus 02

minus1410e minus 02

minus1255e minus 02

minus1100e minus 02

minus9450e minus 03

minus7899e minus 03

minus6348e minus 03

minus4797e minus 03

minus3246e minus 03

minus1695e minus 03

minus1435e minus 04

+1408e minus 03

+2959e minus 03

(avg

75

)

12057612057611

(b)

Figure 9 Displacement and local stain distribution of the digital sample (time steps = 62) (a) vertical displacement 1199061 and (b) local normalstrain 12057611

050

100150200250300350400450

Mic

rostr

ain

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 640Time (s)

Figure 10 Strain-time response

6 Conclusions

In this paper the nondestructive industrial X-ray CT tech-nique was employed to capture the internal microstructureof asphalt mixture The CT images were segmented with theimprovedOTSUmethod to reduce the beamhardening effectwhich is a common phenomenon in X-ray CT images Theimage-based three-dimensional (3D) FE model includingaggregates asphalt mastic and air voids was developed fornumerical predictions In this 3D model the aggregate phaseand air void were considered as elastic materials while theasphalt mastic phase was considered as linear viscoelasticmaterial We derived the displacement-based and stress-based incremental formulations of the viscoelastic constitu-tive models for asphalt mastic with some algebraic manipu-lations and then implemented the incremental formulationsin a finite element codeThematerial parameters in the three-dimensional viscoelastic constitutivemodel were determinedby applying the uniaxial tensile tests and torsion testsFurthermore the capability of the incremental constitutivemodel was examined by comparing the numerical predic-tions with the observed laboratory tests at a temperature of20∘C Finally the image-based three-dimensional (3D) FEmode incorporatedwith viscoelastic asphaltmastic phase andelastic aggregates and air voids was used for the dynamic andthe repeated creep-recovery test simulations

The improved OTSU method presented in this studycan better identify the CT images segmentation and thedeveloped image-based three-dimensional (3D) FE model

can provide geometric information for the aggregates airvoids and asphalt mastic Comparison between the numer-ical predictions and the experimental results shows that theincremental constitutive model has considerable promise forpredicting the asphalt mastic mechanical properties undertwo basic loading paths It can be concluded that once theappropriate viscoelastic constitutive parameters for asphaltmastic at a constant temperature are available the numericalconstitutive model is capable of describing the materialresponse Simulation results of the dynamic test and the creeptest presented in this study showed that the 3D finite elementmodels provided reasonable predictions of the complexmodulus and creep characteristics It is feasible to utilizethe proposed approach to conduct numerical simulations formechanical responses of the asphaltmixtureOur futureworkwill focus on extending this approach to conduct triaxial testsof microstructural asphalt mixture under complex loadingpath as well as further generalizing the high-resolutionnumerical model including aggregate-mastic interface

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This publication is supported by Project (51508109) researchon dynamic mechanical behavior of asphalt mixture and keyparameters prediction based on energymethod sponsored byNational Natural Science Foundation of China The authorsthank all those who contributed in the experimental part ofthis study

References

[1] H Kim M Arraigada C Raab and M N Partl ldquoNumericaland experimental analysis for the interlayer behavior of double-layered asphalt pavement specimensrdquo Journal of Materials inCivil Engineering vol 23 no 1 pp 12ndash20 2011

[2] C-H Ho and P Romero ldquoUsing asphalt mixture beams inthe bending beam rheometerrdquo Road Materials and PavementDesign vol 12 no 2 pp 293ndash314 2011

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 11

[3] J Gajewski and T Sadowski ldquoSensitivity analysis of crackpropagation in pavement bituminous layered structures using ahybrid system integrating Artificial Neural Networks and FiniteElement Methodrdquo Computational Materials Science vol 82 pp114ndash117 2014

[4] P Liu D Wang and M Oeser ldquoApplication of semi-analyticalfinite elementmethod coupledwith infinite element for analysisof asphalt pavement structural responserdquo Journal of Traffic andTransportation Engineering (English Edition) vol 2 article 482015

[5] E Masad A Scarpas K R Rajagopal E Kassem S Koneruand C Kasbergen ldquoFinite element modelling of field com-paction of hot mix asphalt Part II applicationsrdquo InternationalJournal of Pavement Engineering vol 17 no 1 pp 24ndash38 2016

[6] G K Chang and J N Meegoda ldquoMicromechanical model fortemperature effects of hot-mix asphalt concreterdquoTransportationResearch Record vol 1687 pp 95ndash103 1999

[7] A C Collop G R McDowell and Y W Lee ldquoModellingdilation in an idealised asphalt mixture using discrete elementmodellingrdquo Granular Matter vol 8 no 3-4 pp 175ndash184 2006

[8] A C Collop G RMcDowell andY Lee ldquoOn the use of discreteelement modelling to simulate the viscoelastic deformationbehaviour of an idealized asphalt mixturerdquo Geomechanics andGeoengineering vol 2 no 2 pp 77ndash86 2007

[9] J Wu A C Collop and G R McDowell ldquoDiscrete elementmodeling of constant strain rate compression tests on idealizedasphalt mixturerdquo Journal of Materials in Civil Engineering vol23 no 1 pp 2ndash11 2011

[10] W Cai G R McDowell and G D Airey ldquoDiscrete elementmodelling of uniaxial constant strain rate tests on asphaltmixturesrdquo Granular Matter vol 15 no 2 pp 163ndash174 2013

[11] W Cai G R McDowell and G D Airey ldquoDiscrete elementvisco-elastic modelling of a realistic graded asphalt mixturerdquoSoils and Foundations vol 54 no 1 pp 12ndash22 2014

[12] Q Dai and Z You ldquoMicromechanical finite element frameworkfor predicting viscoelastic properties of asphalt mixturesrdquoMaterials and Structures vol 41 no 6 pp 1025ndash1037 2008

[13] Q Dai ldquoPrediction of dynamic modulus and phase angleof stone-based composites using a micromechanical finite-element approachrdquo Journal ofMaterials in Civil Engineering vol22 no 6 pp 618ndash627 2010

[14] Y Kim and J Lutif ldquoComputational micromechanics modelingfor damage-induced behavior of asphalt mixtures consideringviscoelasticity and cohesive zone fracturerdquo in Pavements andMaterials pp 17ndash25 American Society of Civil Engineers 2008

[15] A Arshadi and H Bahia ldquoDevelopment of an image-basedmulti-scale finite-element approach to predict mechanicalresponse of asphalt mixturesrdquo Road Materials and PavementDesign vol 16 supplement 2 pp 214ndash229 2015

[16] Q Dai ldquoTwo- and three-dimensional micromechanical vis-coelastic finite element modeling of stone-based materialswith X-ray computed tomography imagesrdquo Construction andBuilding Materials vol 25 no 2 pp 1102ndash1114 2011

[17] H Ying M A Elseifi L N Mohammad and M M HassanldquoHeterogeneous finite-element modeling of the dynamic com-plex modulus test of asphalt mixture using X-ray computedtomographyrdquo Journal of Materials in Civil Engineering vol 26no 9 Article ID 04014052 2014

[18] T You R K Abu Al-Rub E A Masad E Kassem andD N Little ldquoThree-dimensional microstructural modelingframework for dense-graded asphalt concrete using a coupled

viscoelastic viscoplastic and viscodamage modelrdquo Journal ofMaterials in Civil Engineering vol 26 no 4 pp 607ndash621 2014

[19] T Schuler R Janicke and H Steeb ldquoNonlinear modeling andcomputational homogenization of asphalt concrete on the basisof XRCT scansrdquo Construction and Building Materials vol 109pp 96ndash108 2016

[20] J-H Liu and Z Li ldquoImage segmentation and its effect of asphaltmixtures using computed tomography images methodrdquo Journalof Chongqing Jiaotong University (Natural Science) vol 30 no6 pp 1335ndash1338 2011

[21] H Wen-ke Z Li-juan Z Xiao-ning and S Shen-shenldquoResearch on the conversion relationship between relaxationmodulus and creep compliance of asphalt mixture based onProny seriesrdquo Journal of Transport Science and Engineering vol31 p 7 2015 (Chinese)

[22] Applied Research Associates ARA ldquoGuide for mechanistic-empirical design of new and rehabilitated pavement structuresrdquoFinal Report National CooperativeHighwayResearch Program(NCHRP) Project 1ndash37A Albuquerque New Mexico 2004

[23] M Arabani and N Kamboozia ldquoThe linear visco-elastic behav-iour of glasphalt mixture under dynamic loading conditionsrdquoConstruction and Building Materials vol 41 pp 594ndash601 2013

[24] L Zhang X Zhang X Liu and Y Luo ldquoViscoelastic model ofasphalt mixtures under repeated loadrdquo Journal of Materials inCivil Engineering vol 27 no 10 Article ID 04015007 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials