MicromechanicalPredictionModelofViscoelasticPropertiesfor...
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Research Article Micromechanical Prediction Model of Viscoelastic Properties for Asphalt Mastic Based on Morphologically Representative Pattern Approach Zhichen Wang, 1 Naisheng Guo , 1 XuYang , 2 and Shuang Wang 3 1 School of Transportation Engineering, Dalian Maritime University, Dalian 116026, Liaoning, China 2 Department of Civil and Environmental Engineering, Michigan Technological University, Houghton, MI 49931-1295, USA 3 School of Computer Science, Lyceum of the Philippines University, Manila, Batangas 4200, Philippines Correspondence should be addressed to Naisheng Guo; [email protected] Received 16 April 2020; Accepted 3 June 2020; Published 28 June 2020 Guest Editor: Meng Guo Copyright © 2020 Zhichen Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is devoted to the introduction of physicochemical, filler size, and distribution effect in micromechanical predictions of the overall viscoelastic properties of asphalt mastic. In order to account for the three effects, the morphologically representative pattern (MRP) approach was employed. e MRP model was improved due to the arduous practical use of equivalent modulus formula solution. en, a homogeneous morphologically representative model (H-MRP) with the explicit solution was established based on the homogenization theory. Asphalt mastic is regarded as a composite material consisting of filler particles coated structural asphalt and free asphalt considering the physicochemical effect. An additional interphase surrounding particles was introduced in the H-MRP model. us, a modified H-MRP model was established. Using the proposed model, a viscoelastic equation was derived to predict the complex modulus and subsequently the dynamic modulus of asphalt mastic based on the elastic-viscoelastic correspondence principle. e dynamic shear rheological tests were conducted to verify the prediction model. e results show that the predicted modulus presents an acceptable precision for asphalt mastic mixed with 10% and 20% fillers volume fraction, as compared to the measured ones. e predicted modulus agrees reasonably well with the measured ones at high frequencies for asphalt mastic mixed with 30% and 40% fillers volume fraction. However, it exhibits underestimated modulus at low frequencies. e reasons for the discrepancy between predicted and measured dynamic shear modulus and the factors affecting the dynamic shear modulus were also explored in the paper. 1.Introduction Asphaltic material is a typical rheological material. Asphalt mastic is regarded as a composite material consisting of asphalt and fillers, which plays an essential role in the binding between aggregates in asphalt mixture. Dynamic shear modulus is the leading indicator to evaluate the vis- coelastic properties of asphalt mastic. A large number of dynamic shear rheological tests have been conducted to explore the influence of different asphalt and fillers on the dynamic shear modulus of asphalt mastic [1–3]. However, plenty of factors are influencing the viscoelastic properties of asphalt mastic, and it is unrealistic and uneconomic to analyze these factors through a large amount of tests. us, it is necessary to propose the prediction methods that can be used to obtain the dynamic shear modulus of asphalt mastic. e empirical prediction models were proposed under the support of NCHRP, including the Witczak 1–37A model, improved Witczak model, and NCHRP 1–40D model [4–6]. In order to provide the parameters of asphalt binder for the prediction model of asphalt mixtures, an empirical pre- diction model for the dynamic shear modulus of asphalt binder was established by Bari and Witczak [7]. However, the model was adopted under specific conditions in the United States, and it may not be applicable in other regions. erefore, it is necessary to get rid of the empirical method and predict the macroscopic properties from the volume and mechanical properties of the components. e Hindawi Advances in Materials Science and Engineering Volume 2020, Article ID 7915140, 12 pages https://doi.org/10.1155/2020/7915140
Transcript of MicromechanicalPredictionModelofViscoelasticPropertiesfor...
Research ArticleMicromechanical Prediction Model of Viscoelastic Properties forAsphalt Mastic Based on Morphologically RepresentativePattern Approach
Zhichen Wang1 Naisheng Guo 1 Xu Yang 2 and Shuang Wang3
1School of Transportation Engineering Dalian Maritime University Dalian 116026 Liaoning China2Department of Civil and Environmental Engineering Michigan Technological University Houghton MI 49931-1295 USA3School of Computer Science Lyceum of the Philippines University Manila Batangas 4200 Philippines
Correspondence should be addressed to Naisheng Guo naishengguodlmueducn
Received 16 April 2020 Accepted 3 June 2020 Published 28 June 2020
Guest Editor Meng Guo
Copyright copy 2020 ZhichenWang et al(is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
(is paper is devoted to the introduction of physicochemical filler size and distribution effect in micromechanical predictions ofthe overall viscoelastic properties of asphalt mastic In order to account for the three effects the morphologically representativepattern (MRP) approach was employed (e MRP model was improved due to the arduous practical use of equivalent modulusformula solution(en a homogeneousmorphologically representativemodel (H-MRP) with the explicit solution was establishedbased on the homogenization theory Asphalt mastic is regarded as a composite material consisting of filler particles coatedstructural asphalt and free asphalt considering the physicochemical effect An additional interphase surrounding particles wasintroduced in the H-MRP model (us a modified H-MRP model was established Using the proposed model a viscoelasticequation was derived to predict the complex modulus and subsequently the dynamic modulus of asphalt mastic based on theelastic-viscoelastic correspondence principle (e dynamic shear rheological tests were conducted to verify the prediction model(e results show that the predicted modulus presents an acceptable precision for asphalt mastic mixed with 10 and 20 fillersvolume fraction as compared to the measured ones(e predictedmodulus agrees reasonably well with the measured ones at highfrequencies for asphalt mastic mixed with 30 and 40 fillers volume fraction However it exhibits underestimated modulus atlow frequencies (e reasons for the discrepancy between predicted and measured dynamic shear modulus and the factorsaffecting the dynamic shear modulus were also explored in the paper
1 Introduction
Asphaltic material is a typical rheological material Asphaltmastic is regarded as a composite material consisting ofasphalt and fillers which plays an essential role in thebinding between aggregates in asphalt mixture Dynamicshear modulus is the leading indicator to evaluate the vis-coelastic properties of asphalt mastic A large number ofdynamic shear rheological tests have been conducted toexplore the influence of different asphalt and fillers on thedynamic shear modulus of asphalt mastic [1ndash3] Howeverplenty of factors are influencing the viscoelastic properties ofasphalt mastic and it is unrealistic and uneconomic toanalyze these factors through a large amount of tests(us it
is necessary to propose the prediction methods that can beused to obtain the dynamic shear modulus of asphalt mastic
(e empirical prediction models were proposed underthe support of NCHRP including theWitczak 1ndash37Amodelimproved Witczak model and NCHRP 1ndash40D model [4ndash6]In order to provide the parameters of asphalt binder for theprediction model of asphalt mixtures an empirical pre-diction model for the dynamic shear modulus of asphaltbinder was established by Bari and Witczak [7] Howeverthe model was adopted under specific conditions in theUnited States and it may not be applicable in other regions(erefore it is necessary to get rid of the empirical methodand predict themacroscopic properties from the volume andmechanical properties of the components (e
HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 7915140 12 pageshttpsdoiorg10115520207915140
micromechanics method of composite materials provides areliable method for this work
(e classical micromechanical methods of compositematerials including composite sphere model [8] self-con-sistent model [9] generalized self-consistent model(Christensen Lo model) [10] and Mori Tanaka method [11]have been widely used in the equivalent properties pre-diction of the particle and fiber-reinforced composite ma-terials Some scholars applied these classicalmicromechanical methods to establish the micromechanicalmodel of asphalt mixtures Pang et al utilized a compositesphere model to predict the elastic modulus of the asphaltmixture [12] Luo et al adopted the self-consistent modeland generalized self-consistent model to predict the dynamicmodulus of the asphalt mixture [13 14] Zhu et al presenteda micromechanical model considering the effect of the in-terface between asphalt and aggregate [15 16]
Asphalt mastic was considered as a known matrix in theabove prediction models (e modulus test of asphalt masticshould be conducted repeatedly to provide the input pa-rameters for the models due to the variety of the fillersconcentration (erefore it is also necessary to establish themicromechanical model of asphalt mastic Asphalt mastic isregarded as a composite material composed of spherical fillerparticles embedded in asphalt binder A lot of differentmethods have been presented to establish the micro-mechanical model of asphalt mastic Yin et al applied fourmicromechanics methods that is dilution model self-consistent model generalized self-consistent model andMori Tanaka method to predict the dynamic shear modulusof asphalt mastic [17] Underwood and Kim presented 12existing micromechanical methods to predict the dynamicshear modulus of asphalt mastic [18] However the resultsshow that the micromechanical model considerably un-derestimates the viscoelastic properties of asphalt mastic
(e percolation effect of fillers and asphalt is consideredby Shashidhar and Shenoy [19] (ere is an increasingprobability of the particles touching one another and thefiller becoming percolated then a modified method wasproposed for the prediction model by percolation theory(ey use the calculation to suggest that percolation phe-nomenon onset occurs at 2ndash10 However the me-chanical data compiled by multiple scholars suggest that afiller concentration of approximately 42 is more appro-priate for asphalt mastics [20ndash22] Li et al utilized Ju-Chenmodel to predict the dynamic shear modulus of asphaltmastic considering the interparticle interaction [23]However the interaction between fillers is difficult tomeasure and it is not easy to judge whether it actuallyoccurred (erefore it is necessary to ascertain the interiorstructural components and interaction mechanism of as-phalt mastic and to establish a targeted micromechanicalmodel
An experiment was conducted by Davis and Castorenausing atomic force microscopy [24] (e microscopic var-iation of asphalt near the filler particle surface was foundedwhich indicated that the physicochemical effect has occurredbetween the fillers and asphalt (e physicochemical processof selective adsorption of the asphalt binder compounds by
the aggregate particles occurring in asphaltic composites isalso widely reported [25ndash27] A structural asphalt layer willbe formed on the surface of the fillers under the physico-chemical effect thus a four-phase micromechanical modelof asphalt mastic was proposed by Underwood and Kim [18](e model is based on the classical n-layer sphere modelderived from point-approaches according to which theRepresentative Volume Element (RVE) is defined throughsome statistical information on points belonging to such orsuch phase It is meaningless to endow them with any in-dividual geometrical or physical specific property of thesephase elements such as their size their mutual distances ortheir superficial area (us the influence of the particle sizeand distribution of fillers were not taken into considerationin the four-phase micromechanical model
In order to consider the distribution and size effects inmicromechanical predictions model a morphologicallyrepresentative pattern (MRP) approach has been proposedfor composite materials [28 29] which could be applied forasphalt mastic (e objective of this paper strives to establisha new micromechanical model considering the physico-chemical filler size and distribution effect based on theMRPapproach (e new proposed model is used to predict thedynamic shear modulus of asphalt mastic and the predictionresults were compared with the experiment results of theasphalt mastic with different filler volume fractions to val-idate its applicability In additions the influence of modelparameters on the prediction results is also analyzed
2 Micromechanical H-MRP Model
Marcadon et al proposed a morphology representativepattern (MRP) model considering distribution and sizeeffects in micromechanical predictions of the overall elasticmoduli of particulate composite materials [28] (e com-position of MRP model comes from the composite spheremodel [8] However the spherical particles with differentparticle sizes are replaced by a series of spherical particleswith the same particle size (e MRP model consists ofmatrix-coated spheres (two-phase pattern) and theremaining matrix (pure matrix pattern) see Figure 1
(e far-field uniform strain of two patterns is equal tothe total strain langεrang E Marcadon et al obtained theequivalent modulus solution formula of the MRP modelcharacterized by nonlinear equations [28] Althoughmathematical software such as Mathematica can be used forauxiliary calculation some negative solutions appear fre-quently which make it difficult to be applied (e ho-mogenization method is employed to improve the MRPmodel (e two-phase pattern in the MRP model is ho-mogenized into an equivalent medium by using the gen-eralized self-consistent method Subsequently the mediumis put into the remaining matrix (e self-consistent methodis applied for homogenization into the equivalent compositematerial (us an improved model is established calledhomogenized MRP model (H-MRP) as shown in Figure 2
According to theMRP approach each pattern consists ofone particle surrounded by a concentric shell of the matrix ofvariable thickness depending on the packing density with an
2 Advances in Materials Science and Engineering
additional pattern of the residual pure matrix (e shellthickness may then be correlated with the mean distancebetween nearest-neighbor particles say λ (e first twopatterns are made of two concentric spheres with a particleat the core and a shell with the thickness Rm minus Ri (λ2)constituted with the pure matrix (erefore the volumefraction of particles in the two-pattern approach ci is cal-culated as
ci fi
cH
R3
i
R3m
R3
i
Ri +(λ2)1113872 11138733
11 + λ 2Ri( 1113857
31113872 11138731113872 1113873
(1)
where fi is the volume fraction of inclusions cH is the volumefraction of two-phase pattern Ri and Rm are the particle andmatrix radius respectively From equation (1) it is apparentalready that the effective properties simultaneously dependon the particle size Ri and on the mean distance λ throughthe ratios (λ(2Ri)) so that (λ(2Ri)) is defined as theparticle distribution coefficient It is an important parameterreflecting particle distribution characteristics
According to the composite sphere (CS) model [8] thebulk modulus of the equivalent medium KH can be obtainedby
KH Km +ci Ki minus Km( 1113857 3Km + 4Gm( 1113857
3Km + 4Gm + 3 1 minus ci( 1113857 Ki minus Km( 1113857 (2)
where Ki and Km are the bulk moduli of inclusions andmatrix respectively Gm is the shear modulus of the matrix
According to the generalized self-consistent (GSC)model [10] the shear modulus of the equivalent medium GHis given by
AGH
Gm
1113888 1113889
2
+ 2BGH
Gm
1113888 1113889 + C 0 (3)
where A B and C are functions related to the modulus andvolume fraction of particles and matrix expressed as
A 8Gi
Gm
minus 11113888 1113889 4 minus 5]m( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
minus 50Gi
Gm
minus 11113888 1113889 7 minus 12]m + 8]2m1113872 1113873η2ci + 4 7 minus 10]m( 1113857η2η3
(4)
B minus 2Gi
Gm
minus 11113888 1113889 1 minus 5]m( 1113857η1c103i
+ 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i minus 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 75Gi
Gm
minus 11113888 1113889 3 minus ]m( 1113857η2]mci +32
15]m minus 7( 1113857η2η3
(5)
Equivalent composite
ParticleMatrix Two-phase pattern Pure matrix pattern
Equivalent composite
λ
Figure 1 MRP model
λ
Equivalent composite
Equivalentmedium
RmRi
λλ2
Figure 2 Homogenized MRP model (H-MRP model)
Advances in Materials Science and Engineering 3
C 4Gi
Gm
minus 11113888 1113889 5]m minus 7( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 25Gi
Gm
minus 11113888 1113889 ]2m minus 71113872 1113873η2ci minus 7 + 5]m( 1113857η2η3
(6)
η1 49 minus 50]i]m( 1113857Gi
Gm
minus 11113888 1113889 + 35Gi
Gm
]i minus 2]m( 1113857 + 2]i minus ]m( 11138571113890 1113891
(7)
η2 5]i
Gi
Gm
minus 81113888 1113889 + 7Gi
Gm
+ 41113888 1113889 (8)
η3 Gi
Gm
8 minus 10]m( 1113857 + 7 minus 5]m( 1113857 (9)
where ]m is Poissonrsquos ratio of the matrix(e equivalent shear modulus can be written in an
explicit expression as
GH
B2 minus AC
radicminus B
AGm (10)
(e equivalent medium is embedded in the remainingmatrix From equation (1) the volume fraction of equivalentmedium cH is given by
cH fi
ci
fi 1 +λ2Ri
1113888 1113889
3
(11)
According to the self-consistent (SC) model [9] the bulkmodulus Kc and shear modulus Gc of the composite can beobtained as follows
Kc Km +cH KH minus Km( 1113857 3Kc + 4Gc( 1113857
3KH + 4Gc
(12)
Gc Gm +5cHGc GH minus Gm( 1113857 3Kc + 4Gc( 1113857
3Kc 3Gc + 2GH( 1113857 + 4Gc 2Gc + 3GH( 1113857 (13)
It can be found that the explicit expressions of the GSCmodel and SC model are used to obtain the equivalentmodulus of the composite which is convenient for engi-neering application
(e spherical particles and the matrix in the composite areregarded as elastic materials Let Youngrsquos modulus ratio
(EiEm) 10 andPoissonrsquos ratio ]i ]m 03 theMRPmodelwas employed to predict the equivalent modulus of compositewith different particle distribution coefficient by Majewski et al[29] (e predicted results of H-MRP model are obtained byusing the same model parameters as the MRP model (ecomparison results are shown in Figures 3(a) and 3(b)
It can be seen from Figure 3 that the prediction results ofH-MRP model proposed in this study are consistent with theMRP model (e predicted modulus of the H-MRP model isbetween the GSC model and SC model Since the predictedresults of the H-MRP model are related to the mean mini-mum distance between nearest-neighbor particles λ there aretwo limit cases (1) When λ 0 the thickness of the particle-coated matrix layer in the composite sphere is 0 (eequivalent material is still spherical particles after homoge-nization which is equivalent to the self-consistent model(erefore the upper limit of the H-MRP model is the SCmodel (2) When the particles in the composite sphere coatedthe overall matrix there is no remaining matrix expressed as
λ2Ri( 1113857
fminus (13)i minus 1 (14)
It is equivalent to the GSC model so the lower limit ofthe H-MRP model is the GSC model
3 Micromechanical Model of Asphalt Masticconsidering the Physicochemical Effect
A structural asphalt layer on the surface of fillers should beproduced by physicochemical interaction between asphaltand fillers and the outside of it is a free asphalt layer withoutphysicochemical effect (erefore it is proposed to trans-form the two-phase pattern of spherical filler particles coatedasphalt in H-MRP model into a three-phase pattern of fillerscoated structural and free asphalt (e modified H-MRPmodel is shown in Figure 4
When studying distribution effects in Section 2 we al-ready noticed (see Figure 2) that a GSCmodel and SC modelare chosen A similar conclusion can be drawn with the 4-phase model [30] when coated particles are dealt with (seeFigure 4) Let R1 R2 and R3 be respectively the radii of thethree different spheres in the composite sphere in Figure 4by beginning the numbering from the center of the com-posite sphere
(e effective moduli Keff and Geff of the equivalentmedium are then determined from equations (46) and (51)of Herve and Zaoui [30]
Keff K3 +3K3 + 4G3( 1113857R3
2 K1 minus K2( 1113857R31 3K3 + 4G2( 1113857 + K2 minus K3( 1113857R3
2 3K1 + 4G2( 11138571113858 1113859
3 K2 minus K1( 1113857R31 R3
2 3K3 + 4G2( 1113857 + 4R33 G3 minus G2( 11138571113858 1113859 + 3K1 + 4G2( 1113857R3
2 3R32 K3 minus K2( 1113857 + R3
3 3K3 + 4G3( 11138571113858 1113859 (15)
Geff
Bprime2
minus AprimeCprime1113969
minus Bprime
AprimeG3
(16)
4 Advances in Materials Science and Engineering
where R1 R2 and R3 are the radii of fillers structural asphaltand free asphalt respectively Aprime Bprime and Cprime are given by
Aprime 4R103 1 minus 2]3( 1113857 7 minus 10]3( 1113857Z12 + 20R
73 7 minus 12]3 + 8]231113872 1113873Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 + 16 4 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(17)
Bprime 3R103 1 minus 2]3( 1113857 15]3 minus 7( 1113857Z12 + 60R
73 ]3 minus 3( 1113857]3Z42
minus 24R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
minus 40R33 1 minus 2]3( 1113857
2Z13 minus 8 1 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(18)
Cprime minus R103 1 minus 2]3( 1113857 7 + 5]3( 1113857Z12 + 10R
73 7 minus ]231113872 1113873]3Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 minus 8 7 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(19)
Zαβ 1113957P(2)
α11113957P
(2)
β2 minus 1113957P(2)
β11113957P
(2)
α2 (20)
P(n) 1113945
3
k1Μ(k)
(21)
Μ(k) Lminus 1
k+1 Rk( 1113857Lk Rk( 1113857 (22)
λ(2Ri) = 005
00 02 04 06 08 100
2
4
6
8
10
GG m
fi
λ(2Ri) = 0005
GSCH-MRP
SC
λ(2Ri) = 0025
(a)
00 02 04 06 08 100
2
4
6
8
10
λ(2Ri) = 005
λ(2Ri) = 0025
SCGSCH-MRP
GG m
fi
λ(2Ri) = 0005
(b)
Figure 3 Predicted results of themicromechanicalmodel under different particle distribution coefficients (a)MRPmodel [29] (b)H-MRPmodel
λ
R2
Structuralasphalt
Filler
R1
Free asphalt
Equivalentmedium
Equivalent asphalt mastic
λ2
R3
λ
Figure 4 Modified H-MRP model
Advances in Materials Science and Engineering 5
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
micromechanics method of composite materials provides areliable method for this work
(e classical micromechanical methods of compositematerials including composite sphere model [8] self-con-sistent model [9] generalized self-consistent model(Christensen Lo model) [10] and Mori Tanaka method [11]have been widely used in the equivalent properties pre-diction of the particle and fiber-reinforced composite ma-terials Some scholars applied these classicalmicromechanical methods to establish the micromechanicalmodel of asphalt mixtures Pang et al utilized a compositesphere model to predict the elastic modulus of the asphaltmixture [12] Luo et al adopted the self-consistent modeland generalized self-consistent model to predict the dynamicmodulus of the asphalt mixture [13 14] Zhu et al presenteda micromechanical model considering the effect of the in-terface between asphalt and aggregate [15 16]
Asphalt mastic was considered as a known matrix in theabove prediction models (e modulus test of asphalt masticshould be conducted repeatedly to provide the input pa-rameters for the models due to the variety of the fillersconcentration (erefore it is also necessary to establish themicromechanical model of asphalt mastic Asphalt mastic isregarded as a composite material composed of spherical fillerparticles embedded in asphalt binder A lot of differentmethods have been presented to establish the micro-mechanical model of asphalt mastic Yin et al applied fourmicromechanics methods that is dilution model self-consistent model generalized self-consistent model andMori Tanaka method to predict the dynamic shear modulusof asphalt mastic [17] Underwood and Kim presented 12existing micromechanical methods to predict the dynamicshear modulus of asphalt mastic [18] However the resultsshow that the micromechanical model considerably un-derestimates the viscoelastic properties of asphalt mastic
(e percolation effect of fillers and asphalt is consideredby Shashidhar and Shenoy [19] (ere is an increasingprobability of the particles touching one another and thefiller becoming percolated then a modified method wasproposed for the prediction model by percolation theory(ey use the calculation to suggest that percolation phe-nomenon onset occurs at 2ndash10 However the me-chanical data compiled by multiple scholars suggest that afiller concentration of approximately 42 is more appro-priate for asphalt mastics [20ndash22] Li et al utilized Ju-Chenmodel to predict the dynamic shear modulus of asphaltmastic considering the interparticle interaction [23]However the interaction between fillers is difficult tomeasure and it is not easy to judge whether it actuallyoccurred (erefore it is necessary to ascertain the interiorstructural components and interaction mechanism of as-phalt mastic and to establish a targeted micromechanicalmodel
An experiment was conducted by Davis and Castorenausing atomic force microscopy [24] (e microscopic var-iation of asphalt near the filler particle surface was foundedwhich indicated that the physicochemical effect has occurredbetween the fillers and asphalt (e physicochemical processof selective adsorption of the asphalt binder compounds by
the aggregate particles occurring in asphaltic composites isalso widely reported [25ndash27] A structural asphalt layer willbe formed on the surface of the fillers under the physico-chemical effect thus a four-phase micromechanical modelof asphalt mastic was proposed by Underwood and Kim [18](e model is based on the classical n-layer sphere modelderived from point-approaches according to which theRepresentative Volume Element (RVE) is defined throughsome statistical information on points belonging to such orsuch phase It is meaningless to endow them with any in-dividual geometrical or physical specific property of thesephase elements such as their size their mutual distances ortheir superficial area (us the influence of the particle sizeand distribution of fillers were not taken into considerationin the four-phase micromechanical model
In order to consider the distribution and size effects inmicromechanical predictions model a morphologicallyrepresentative pattern (MRP) approach has been proposedfor composite materials [28 29] which could be applied forasphalt mastic (e objective of this paper strives to establisha new micromechanical model considering the physico-chemical filler size and distribution effect based on theMRPapproach (e new proposed model is used to predict thedynamic shear modulus of asphalt mastic and the predictionresults were compared with the experiment results of theasphalt mastic with different filler volume fractions to val-idate its applicability In additions the influence of modelparameters on the prediction results is also analyzed
2 Micromechanical H-MRP Model
Marcadon et al proposed a morphology representativepattern (MRP) model considering distribution and sizeeffects in micromechanical predictions of the overall elasticmoduli of particulate composite materials [28] (e com-position of MRP model comes from the composite spheremodel [8] However the spherical particles with differentparticle sizes are replaced by a series of spherical particleswith the same particle size (e MRP model consists ofmatrix-coated spheres (two-phase pattern) and theremaining matrix (pure matrix pattern) see Figure 1
(e far-field uniform strain of two patterns is equal tothe total strain langεrang E Marcadon et al obtained theequivalent modulus solution formula of the MRP modelcharacterized by nonlinear equations [28] Althoughmathematical software such as Mathematica can be used forauxiliary calculation some negative solutions appear fre-quently which make it difficult to be applied (e ho-mogenization method is employed to improve the MRPmodel (e two-phase pattern in the MRP model is ho-mogenized into an equivalent medium by using the gen-eralized self-consistent method Subsequently the mediumis put into the remaining matrix (e self-consistent methodis applied for homogenization into the equivalent compositematerial (us an improved model is established calledhomogenized MRP model (H-MRP) as shown in Figure 2
According to theMRP approach each pattern consists ofone particle surrounded by a concentric shell of the matrix ofvariable thickness depending on the packing density with an
2 Advances in Materials Science and Engineering
additional pattern of the residual pure matrix (e shellthickness may then be correlated with the mean distancebetween nearest-neighbor particles say λ (e first twopatterns are made of two concentric spheres with a particleat the core and a shell with the thickness Rm minus Ri (λ2)constituted with the pure matrix (erefore the volumefraction of particles in the two-pattern approach ci is cal-culated as
ci fi
cH
R3
i
R3m
R3
i
Ri +(λ2)1113872 11138733
11 + λ 2Ri( 1113857
31113872 11138731113872 1113873
(1)
where fi is the volume fraction of inclusions cH is the volumefraction of two-phase pattern Ri and Rm are the particle andmatrix radius respectively From equation (1) it is apparentalready that the effective properties simultaneously dependon the particle size Ri and on the mean distance λ throughthe ratios (λ(2Ri)) so that (λ(2Ri)) is defined as theparticle distribution coefficient It is an important parameterreflecting particle distribution characteristics
According to the composite sphere (CS) model [8] thebulk modulus of the equivalent medium KH can be obtainedby
KH Km +ci Ki minus Km( 1113857 3Km + 4Gm( 1113857
3Km + 4Gm + 3 1 minus ci( 1113857 Ki minus Km( 1113857 (2)
where Ki and Km are the bulk moduli of inclusions andmatrix respectively Gm is the shear modulus of the matrix
According to the generalized self-consistent (GSC)model [10] the shear modulus of the equivalent medium GHis given by
AGH
Gm
1113888 1113889
2
+ 2BGH
Gm
1113888 1113889 + C 0 (3)
where A B and C are functions related to the modulus andvolume fraction of particles and matrix expressed as
A 8Gi
Gm
minus 11113888 1113889 4 minus 5]m( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
minus 50Gi
Gm
minus 11113888 1113889 7 minus 12]m + 8]2m1113872 1113873η2ci + 4 7 minus 10]m( 1113857η2η3
(4)
B minus 2Gi
Gm
minus 11113888 1113889 1 minus 5]m( 1113857η1c103i
+ 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i minus 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 75Gi
Gm
minus 11113888 1113889 3 minus ]m( 1113857η2]mci +32
15]m minus 7( 1113857η2η3
(5)
Equivalent composite
ParticleMatrix Two-phase pattern Pure matrix pattern
Equivalent composite
λ
Figure 1 MRP model
λ
Equivalent composite
Equivalentmedium
RmRi
λλ2
Figure 2 Homogenized MRP model (H-MRP model)
Advances in Materials Science and Engineering 3
C 4Gi
Gm
minus 11113888 1113889 5]m minus 7( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 25Gi
Gm
minus 11113888 1113889 ]2m minus 71113872 1113873η2ci minus 7 + 5]m( 1113857η2η3
(6)
η1 49 minus 50]i]m( 1113857Gi
Gm
minus 11113888 1113889 + 35Gi
Gm
]i minus 2]m( 1113857 + 2]i minus ]m( 11138571113890 1113891
(7)
η2 5]i
Gi
Gm
minus 81113888 1113889 + 7Gi
Gm
+ 41113888 1113889 (8)
η3 Gi
Gm
8 minus 10]m( 1113857 + 7 minus 5]m( 1113857 (9)
where ]m is Poissonrsquos ratio of the matrix(e equivalent shear modulus can be written in an
explicit expression as
GH
B2 minus AC
radicminus B
AGm (10)
(e equivalent medium is embedded in the remainingmatrix From equation (1) the volume fraction of equivalentmedium cH is given by
cH fi
ci
fi 1 +λ2Ri
1113888 1113889
3
(11)
According to the self-consistent (SC) model [9] the bulkmodulus Kc and shear modulus Gc of the composite can beobtained as follows
Kc Km +cH KH minus Km( 1113857 3Kc + 4Gc( 1113857
3KH + 4Gc
(12)
Gc Gm +5cHGc GH minus Gm( 1113857 3Kc + 4Gc( 1113857
3Kc 3Gc + 2GH( 1113857 + 4Gc 2Gc + 3GH( 1113857 (13)
It can be found that the explicit expressions of the GSCmodel and SC model are used to obtain the equivalentmodulus of the composite which is convenient for engi-neering application
(e spherical particles and the matrix in the composite areregarded as elastic materials Let Youngrsquos modulus ratio
(EiEm) 10 andPoissonrsquos ratio ]i ]m 03 theMRPmodelwas employed to predict the equivalent modulus of compositewith different particle distribution coefficient by Majewski et al[29] (e predicted results of H-MRP model are obtained byusing the same model parameters as the MRP model (ecomparison results are shown in Figures 3(a) and 3(b)
It can be seen from Figure 3 that the prediction results ofH-MRP model proposed in this study are consistent with theMRP model (e predicted modulus of the H-MRP model isbetween the GSC model and SC model Since the predictedresults of the H-MRP model are related to the mean mini-mum distance between nearest-neighbor particles λ there aretwo limit cases (1) When λ 0 the thickness of the particle-coated matrix layer in the composite sphere is 0 (eequivalent material is still spherical particles after homoge-nization which is equivalent to the self-consistent model(erefore the upper limit of the H-MRP model is the SCmodel (2) When the particles in the composite sphere coatedthe overall matrix there is no remaining matrix expressed as
λ2Ri( 1113857
fminus (13)i minus 1 (14)
It is equivalent to the GSC model so the lower limit ofthe H-MRP model is the GSC model
3 Micromechanical Model of Asphalt Masticconsidering the Physicochemical Effect
A structural asphalt layer on the surface of fillers should beproduced by physicochemical interaction between asphaltand fillers and the outside of it is a free asphalt layer withoutphysicochemical effect (erefore it is proposed to trans-form the two-phase pattern of spherical filler particles coatedasphalt in H-MRP model into a three-phase pattern of fillerscoated structural and free asphalt (e modified H-MRPmodel is shown in Figure 4
When studying distribution effects in Section 2 we al-ready noticed (see Figure 2) that a GSCmodel and SC modelare chosen A similar conclusion can be drawn with the 4-phase model [30] when coated particles are dealt with (seeFigure 4) Let R1 R2 and R3 be respectively the radii of thethree different spheres in the composite sphere in Figure 4by beginning the numbering from the center of the com-posite sphere
(e effective moduli Keff and Geff of the equivalentmedium are then determined from equations (46) and (51)of Herve and Zaoui [30]
Keff K3 +3K3 + 4G3( 1113857R3
2 K1 minus K2( 1113857R31 3K3 + 4G2( 1113857 + K2 minus K3( 1113857R3
2 3K1 + 4G2( 11138571113858 1113859
3 K2 minus K1( 1113857R31 R3
2 3K3 + 4G2( 1113857 + 4R33 G3 minus G2( 11138571113858 1113859 + 3K1 + 4G2( 1113857R3
2 3R32 K3 minus K2( 1113857 + R3
3 3K3 + 4G3( 11138571113858 1113859 (15)
Geff
Bprime2
minus AprimeCprime1113969
minus Bprime
AprimeG3
(16)
4 Advances in Materials Science and Engineering
where R1 R2 and R3 are the radii of fillers structural asphaltand free asphalt respectively Aprime Bprime and Cprime are given by
Aprime 4R103 1 minus 2]3( 1113857 7 minus 10]3( 1113857Z12 + 20R
73 7 minus 12]3 + 8]231113872 1113873Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 + 16 4 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(17)
Bprime 3R103 1 minus 2]3( 1113857 15]3 minus 7( 1113857Z12 + 60R
73 ]3 minus 3( 1113857]3Z42
minus 24R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
minus 40R33 1 minus 2]3( 1113857
2Z13 minus 8 1 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(18)
Cprime minus R103 1 minus 2]3( 1113857 7 + 5]3( 1113857Z12 + 10R
73 7 minus ]231113872 1113873]3Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 minus 8 7 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(19)
Zαβ 1113957P(2)
α11113957P
(2)
β2 minus 1113957P(2)
β11113957P
(2)
α2 (20)
P(n) 1113945
3
k1Μ(k)
(21)
Μ(k) Lminus 1
k+1 Rk( 1113857Lk Rk( 1113857 (22)
λ(2Ri) = 005
00 02 04 06 08 100
2
4
6
8
10
GG m
fi
λ(2Ri) = 0005
GSCH-MRP
SC
λ(2Ri) = 0025
(a)
00 02 04 06 08 100
2
4
6
8
10
λ(2Ri) = 005
λ(2Ri) = 0025
SCGSCH-MRP
GG m
fi
λ(2Ri) = 0005
(b)
Figure 3 Predicted results of themicromechanicalmodel under different particle distribution coefficients (a)MRPmodel [29] (b)H-MRPmodel
λ
R2
Structuralasphalt
Filler
R1
Free asphalt
Equivalentmedium
Equivalent asphalt mastic
λ2
R3
λ
Figure 4 Modified H-MRP model
Advances in Materials Science and Engineering 5
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
additional pattern of the residual pure matrix (e shellthickness may then be correlated with the mean distancebetween nearest-neighbor particles say λ (e first twopatterns are made of two concentric spheres with a particleat the core and a shell with the thickness Rm minus Ri (λ2)constituted with the pure matrix (erefore the volumefraction of particles in the two-pattern approach ci is cal-culated as
ci fi
cH
R3
i
R3m
R3
i
Ri +(λ2)1113872 11138733
11 + λ 2Ri( 1113857
31113872 11138731113872 1113873
(1)
where fi is the volume fraction of inclusions cH is the volumefraction of two-phase pattern Ri and Rm are the particle andmatrix radius respectively From equation (1) it is apparentalready that the effective properties simultaneously dependon the particle size Ri and on the mean distance λ throughthe ratios (λ(2Ri)) so that (λ(2Ri)) is defined as theparticle distribution coefficient It is an important parameterreflecting particle distribution characteristics
According to the composite sphere (CS) model [8] thebulk modulus of the equivalent medium KH can be obtainedby
KH Km +ci Ki minus Km( 1113857 3Km + 4Gm( 1113857
3Km + 4Gm + 3 1 minus ci( 1113857 Ki minus Km( 1113857 (2)
where Ki and Km are the bulk moduli of inclusions andmatrix respectively Gm is the shear modulus of the matrix
According to the generalized self-consistent (GSC)model [10] the shear modulus of the equivalent medium GHis given by
AGH
Gm
1113888 1113889
2
+ 2BGH
Gm
1113888 1113889 + C 0 (3)
where A B and C are functions related to the modulus andvolume fraction of particles and matrix expressed as
A 8Gi
Gm
minus 11113888 1113889 4 minus 5]m( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
minus 50Gi
Gm
minus 11113888 1113889 7 minus 12]m + 8]2m1113872 1113873η2ci + 4 7 minus 10]m( 1113857η2η3
(4)
B minus 2Gi
Gm
minus 11113888 1113889 1 minus 5]m( 1113857η1c103i
+ 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i minus 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 75Gi
Gm
minus 11113888 1113889 3 minus ]m( 1113857η2]mci +32
15]m minus 7( 1113857η2η3
(5)
Equivalent composite
ParticleMatrix Two-phase pattern Pure matrix pattern
Equivalent composite
λ
Figure 1 MRP model
λ
Equivalent composite
Equivalentmedium
RmRi
λλ2
Figure 2 Homogenized MRP model (H-MRP model)
Advances in Materials Science and Engineering 3
C 4Gi
Gm
minus 11113888 1113889 5]m minus 7( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 25Gi
Gm
minus 11113888 1113889 ]2m minus 71113872 1113873η2ci minus 7 + 5]m( 1113857η2η3
(6)
η1 49 minus 50]i]m( 1113857Gi
Gm
minus 11113888 1113889 + 35Gi
Gm
]i minus 2]m( 1113857 + 2]i minus ]m( 11138571113890 1113891
(7)
η2 5]i
Gi
Gm
minus 81113888 1113889 + 7Gi
Gm
+ 41113888 1113889 (8)
η3 Gi
Gm
8 minus 10]m( 1113857 + 7 minus 5]m( 1113857 (9)
where ]m is Poissonrsquos ratio of the matrix(e equivalent shear modulus can be written in an
explicit expression as
GH
B2 minus AC
radicminus B
AGm (10)
(e equivalent medium is embedded in the remainingmatrix From equation (1) the volume fraction of equivalentmedium cH is given by
cH fi
ci
fi 1 +λ2Ri
1113888 1113889
3
(11)
According to the self-consistent (SC) model [9] the bulkmodulus Kc and shear modulus Gc of the composite can beobtained as follows
Kc Km +cH KH minus Km( 1113857 3Kc + 4Gc( 1113857
3KH + 4Gc
(12)
Gc Gm +5cHGc GH minus Gm( 1113857 3Kc + 4Gc( 1113857
3Kc 3Gc + 2GH( 1113857 + 4Gc 2Gc + 3GH( 1113857 (13)
It can be found that the explicit expressions of the GSCmodel and SC model are used to obtain the equivalentmodulus of the composite which is convenient for engi-neering application
(e spherical particles and the matrix in the composite areregarded as elastic materials Let Youngrsquos modulus ratio
(EiEm) 10 andPoissonrsquos ratio ]i ]m 03 theMRPmodelwas employed to predict the equivalent modulus of compositewith different particle distribution coefficient by Majewski et al[29] (e predicted results of H-MRP model are obtained byusing the same model parameters as the MRP model (ecomparison results are shown in Figures 3(a) and 3(b)
It can be seen from Figure 3 that the prediction results ofH-MRP model proposed in this study are consistent with theMRP model (e predicted modulus of the H-MRP model isbetween the GSC model and SC model Since the predictedresults of the H-MRP model are related to the mean mini-mum distance between nearest-neighbor particles λ there aretwo limit cases (1) When λ 0 the thickness of the particle-coated matrix layer in the composite sphere is 0 (eequivalent material is still spherical particles after homoge-nization which is equivalent to the self-consistent model(erefore the upper limit of the H-MRP model is the SCmodel (2) When the particles in the composite sphere coatedthe overall matrix there is no remaining matrix expressed as
λ2Ri( 1113857
fminus (13)i minus 1 (14)
It is equivalent to the GSC model so the lower limit ofthe H-MRP model is the GSC model
3 Micromechanical Model of Asphalt Masticconsidering the Physicochemical Effect
A structural asphalt layer on the surface of fillers should beproduced by physicochemical interaction between asphaltand fillers and the outside of it is a free asphalt layer withoutphysicochemical effect (erefore it is proposed to trans-form the two-phase pattern of spherical filler particles coatedasphalt in H-MRP model into a three-phase pattern of fillerscoated structural and free asphalt (e modified H-MRPmodel is shown in Figure 4
When studying distribution effects in Section 2 we al-ready noticed (see Figure 2) that a GSCmodel and SC modelare chosen A similar conclusion can be drawn with the 4-phase model [30] when coated particles are dealt with (seeFigure 4) Let R1 R2 and R3 be respectively the radii of thethree different spheres in the composite sphere in Figure 4by beginning the numbering from the center of the com-posite sphere
(e effective moduli Keff and Geff of the equivalentmedium are then determined from equations (46) and (51)of Herve and Zaoui [30]
Keff K3 +3K3 + 4G3( 1113857R3
2 K1 minus K2( 1113857R31 3K3 + 4G2( 1113857 + K2 minus K3( 1113857R3
2 3K1 + 4G2( 11138571113858 1113859
3 K2 minus K1( 1113857R31 R3
2 3K3 + 4G2( 1113857 + 4R33 G3 minus G2( 11138571113858 1113859 + 3K1 + 4G2( 1113857R3
2 3R32 K3 minus K2( 1113857 + R3
3 3K3 + 4G3( 11138571113858 1113859 (15)
Geff
Bprime2
minus AprimeCprime1113969
minus Bprime
AprimeG3
(16)
4 Advances in Materials Science and Engineering
where R1 R2 and R3 are the radii of fillers structural asphaltand free asphalt respectively Aprime Bprime and Cprime are given by
Aprime 4R103 1 minus 2]3( 1113857 7 minus 10]3( 1113857Z12 + 20R
73 7 minus 12]3 + 8]231113872 1113873Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 + 16 4 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(17)
Bprime 3R103 1 minus 2]3( 1113857 15]3 minus 7( 1113857Z12 + 60R
73 ]3 minus 3( 1113857]3Z42
minus 24R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
minus 40R33 1 minus 2]3( 1113857
2Z13 minus 8 1 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(18)
Cprime minus R103 1 minus 2]3( 1113857 7 + 5]3( 1113857Z12 + 10R
73 7 minus ]231113872 1113873]3Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 minus 8 7 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(19)
Zαβ 1113957P(2)
α11113957P
(2)
β2 minus 1113957P(2)
β11113957P
(2)
α2 (20)
P(n) 1113945
3
k1Μ(k)
(21)
Μ(k) Lminus 1
k+1 Rk( 1113857Lk Rk( 1113857 (22)
λ(2Ri) = 005
00 02 04 06 08 100
2
4
6
8
10
GG m
fi
λ(2Ri) = 0005
GSCH-MRP
SC
λ(2Ri) = 0025
(a)
00 02 04 06 08 100
2
4
6
8
10
λ(2Ri) = 005
λ(2Ri) = 0025
SCGSCH-MRP
GG m
fi
λ(2Ri) = 0005
(b)
Figure 3 Predicted results of themicromechanicalmodel under different particle distribution coefficients (a)MRPmodel [29] (b)H-MRPmodel
λ
R2
Structuralasphalt
Filler
R1
Free asphalt
Equivalentmedium
Equivalent asphalt mastic
λ2
R3
λ
Figure 4 Modified H-MRP model
Advances in Materials Science and Engineering 5
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
C 4Gi
Gm
minus 11113888 1113889 5]m minus 7( 1113857η1c103i
minus 2 63Gi
Gm
minus 11113888 1113889η2 + 2η1η31113890 1113891c73i + 252
Gi
Gm
minus 11113888 1113889η2c53i
+ 25Gi
Gm
minus 11113888 1113889 ]2m minus 71113872 1113873η2ci minus 7 + 5]m( 1113857η2η3
(6)
η1 49 minus 50]i]m( 1113857Gi
Gm
minus 11113888 1113889 + 35Gi
Gm
]i minus 2]m( 1113857 + 2]i minus ]m( 11138571113890 1113891
(7)
η2 5]i
Gi
Gm
minus 81113888 1113889 + 7Gi
Gm
+ 41113888 1113889 (8)
η3 Gi
Gm
8 minus 10]m( 1113857 + 7 minus 5]m( 1113857 (9)
where ]m is Poissonrsquos ratio of the matrix(e equivalent shear modulus can be written in an
explicit expression as
GH
B2 minus AC
radicminus B
AGm (10)
(e equivalent medium is embedded in the remainingmatrix From equation (1) the volume fraction of equivalentmedium cH is given by
cH fi
ci
fi 1 +λ2Ri
1113888 1113889
3
(11)
According to the self-consistent (SC) model [9] the bulkmodulus Kc and shear modulus Gc of the composite can beobtained as follows
Kc Km +cH KH minus Km( 1113857 3Kc + 4Gc( 1113857
3KH + 4Gc
(12)
Gc Gm +5cHGc GH minus Gm( 1113857 3Kc + 4Gc( 1113857
3Kc 3Gc + 2GH( 1113857 + 4Gc 2Gc + 3GH( 1113857 (13)
It can be found that the explicit expressions of the GSCmodel and SC model are used to obtain the equivalentmodulus of the composite which is convenient for engi-neering application
(e spherical particles and the matrix in the composite areregarded as elastic materials Let Youngrsquos modulus ratio
(EiEm) 10 andPoissonrsquos ratio ]i ]m 03 theMRPmodelwas employed to predict the equivalent modulus of compositewith different particle distribution coefficient by Majewski et al[29] (e predicted results of H-MRP model are obtained byusing the same model parameters as the MRP model (ecomparison results are shown in Figures 3(a) and 3(b)
It can be seen from Figure 3 that the prediction results ofH-MRP model proposed in this study are consistent with theMRP model (e predicted modulus of the H-MRP model isbetween the GSC model and SC model Since the predictedresults of the H-MRP model are related to the mean mini-mum distance between nearest-neighbor particles λ there aretwo limit cases (1) When λ 0 the thickness of the particle-coated matrix layer in the composite sphere is 0 (eequivalent material is still spherical particles after homoge-nization which is equivalent to the self-consistent model(erefore the upper limit of the H-MRP model is the SCmodel (2) When the particles in the composite sphere coatedthe overall matrix there is no remaining matrix expressed as
λ2Ri( 1113857
fminus (13)i minus 1 (14)
It is equivalent to the GSC model so the lower limit ofthe H-MRP model is the GSC model
3 Micromechanical Model of Asphalt Masticconsidering the Physicochemical Effect
A structural asphalt layer on the surface of fillers should beproduced by physicochemical interaction between asphaltand fillers and the outside of it is a free asphalt layer withoutphysicochemical effect (erefore it is proposed to trans-form the two-phase pattern of spherical filler particles coatedasphalt in H-MRP model into a three-phase pattern of fillerscoated structural and free asphalt (e modified H-MRPmodel is shown in Figure 4
When studying distribution effects in Section 2 we al-ready noticed (see Figure 2) that a GSCmodel and SC modelare chosen A similar conclusion can be drawn with the 4-phase model [30] when coated particles are dealt with (seeFigure 4) Let R1 R2 and R3 be respectively the radii of thethree different spheres in the composite sphere in Figure 4by beginning the numbering from the center of the com-posite sphere
(e effective moduli Keff and Geff of the equivalentmedium are then determined from equations (46) and (51)of Herve and Zaoui [30]
Keff K3 +3K3 + 4G3( 1113857R3
2 K1 minus K2( 1113857R31 3K3 + 4G2( 1113857 + K2 minus K3( 1113857R3
2 3K1 + 4G2( 11138571113858 1113859
3 K2 minus K1( 1113857R31 R3
2 3K3 + 4G2( 1113857 + 4R33 G3 minus G2( 11138571113858 1113859 + 3K1 + 4G2( 1113857R3
2 3R32 K3 minus K2( 1113857 + R3
3 3K3 + 4G3( 11138571113858 1113859 (15)
Geff
Bprime2
minus AprimeCprime1113969
minus Bprime
AprimeG3
(16)
4 Advances in Materials Science and Engineering
where R1 R2 and R3 are the radii of fillers structural asphaltand free asphalt respectively Aprime Bprime and Cprime are given by
Aprime 4R103 1 minus 2]3( 1113857 7 minus 10]3( 1113857Z12 + 20R
73 7 minus 12]3 + 8]231113872 1113873Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 + 16 4 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(17)
Bprime 3R103 1 minus 2]3( 1113857 15]3 minus 7( 1113857Z12 + 60R
73 ]3 minus 3( 1113857]3Z42
minus 24R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
minus 40R33 1 minus 2]3( 1113857
2Z13 minus 8 1 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(18)
Cprime minus R103 1 minus 2]3( 1113857 7 + 5]3( 1113857Z12 + 10R
73 7 minus ]231113872 1113873]3Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 minus 8 7 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(19)
Zαβ 1113957P(2)
α11113957P
(2)
β2 minus 1113957P(2)
β11113957P
(2)
α2 (20)
P(n) 1113945
3
k1Μ(k)
(21)
Μ(k) Lminus 1
k+1 Rk( 1113857Lk Rk( 1113857 (22)
λ(2Ri) = 005
00 02 04 06 08 100
2
4
6
8
10
GG m
fi
λ(2Ri) = 0005
GSCH-MRP
SC
λ(2Ri) = 0025
(a)
00 02 04 06 08 100
2
4
6
8
10
λ(2Ri) = 005
λ(2Ri) = 0025
SCGSCH-MRP
GG m
fi
λ(2Ri) = 0005
(b)
Figure 3 Predicted results of themicromechanicalmodel under different particle distribution coefficients (a)MRPmodel [29] (b)H-MRPmodel
λ
R2
Structuralasphalt
Filler
R1
Free asphalt
Equivalentmedium
Equivalent asphalt mastic
λ2
R3
λ
Figure 4 Modified H-MRP model
Advances in Materials Science and Engineering 5
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
where R1 R2 and R3 are the radii of fillers structural asphaltand free asphalt respectively Aprime Bprime and Cprime are given by
Aprime 4R103 1 minus 2]3( 1113857 7 minus 10]3( 1113857Z12 + 20R
73 7 minus 12]3 + 8]231113872 1113873Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 + 16 4 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(17)
Bprime 3R103 1 minus 2]3( 1113857 15]3 minus 7( 1113857Z12 + 60R
73 ]3 minus 3( 1113857]3Z42
minus 24R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
minus 40R33 1 minus 2]3( 1113857
2Z13 minus 8 1 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(18)
Cprime minus R103 1 minus 2]3( 1113857 7 + 5]3( 1113857Z12 + 10R
73 7 minus ]231113872 1113873]3Z42
+ 12R53 1 minus 2]3( 1113857 Z14 minus 7Z23( 1113857
+ 20R33 1 minus 2]3( 1113857
2Z13 minus 8 7 minus 5]3( 1113857 1 minus 2]3( 1113857Z43
(19)
Zαβ 1113957P(2)
α11113957P
(2)
β2 minus 1113957P(2)
β11113957P
(2)
α2 (20)
P(n) 1113945
3
k1Μ(k)
(21)
Μ(k) Lminus 1
k+1 Rk( 1113857Lk Rk( 1113857 (22)
λ(2Ri) = 005
00 02 04 06 08 100
2
4
6
8
10
GG m
fi
λ(2Ri) = 0005
GSCH-MRP
SC
λ(2Ri) = 0025
(a)
00 02 04 06 08 100
2
4
6
8
10
λ(2Ri) = 005
λ(2Ri) = 0025
SCGSCH-MRP
GG m
fi
λ(2Ri) = 0005
(b)
Figure 3 Predicted results of themicromechanicalmodel under different particle distribution coefficients (a)MRPmodel [29] (b)H-MRPmodel
λ
R2
Structuralasphalt
Filler
R1
Free asphalt
Equivalentmedium
Equivalent asphalt mastic
λ2
R3
λ
Figure 4 Modified H-MRP model
Advances in Materials Science and Engineering 5
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
with
Lk(r)
r minus6]k
1 minus 2]k
r3 3
r45 minus 4]k
1 minus 2]k
1r2
r minus7 minus 4]k
1 minus 2]k
r3
minus2r4
2r2
Gk
3]k
1 minus 2]k
Gkr2
minus12r5
Gk
2 ]i minus 5( 1113857
1 minus 2]k
Gk
r3
Gk minus7 + 2]k
1 minus 2]k
Gkr2 8
r5Gk
2 1 + ]k( 1113857
1 minus 2]k
Gk
r3
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(23)
(e viscoelastic properties of the equivalent medium canbe directly converted from the elastic solutions from themicromechanical models using the elastic-viscoelastic cor-respondence principle (e correspondence principle statesthat the effective complex material properties for a visco-elastic material can be obtained by replacing the elasticmaterial properties by the Laplace transformed materialproperties [31 32] It is reasonable to assume that structuralasphalt and free asphalt are viscoelastic while fillers areelastic Based on the elastic-viscoelastic correspondenceprinciple equations (15) and (16) can be expressed in thefrequency domain as follows
Klowasteff Klowast3 +
3Klowast3 + 4Glowast3( 1113857R32 K1 minus Klowast2( 1113857R3
1 3Klowast3 + 4Glowast2( 1113857 + Klowast2 minus Klowast3( 1113857R32 3K1 + 4Glowast2( 11138571113858 1113859
3 Klowast2 minus K1( 1113857R31 R3
2 3Klowast3 + 4Glowast2( 1113857 + 4R33 Glowast3 minus Glowast2( 11138571113858 1113859 + 3K1 + 4Glowast2( 1113857R3
2 3R32 Klowast3 minus Klowast2( 1113857 + R3
3 3Klowast3 + 4Glowast3( 11138571113858 1113859 (24)
Glowasteff
BPrime2 minus APrimeCPrime
1113968minus BPrime
APrimeGlowast3 (25)
where Klowasteff and Glowasteff are the complex moduli of the equivalentmedium Klowast2 Glowast2 Klowast3 and Glowast3 are the complex moduli of thestructural asphalt and free asphalt respectively APrime BPrime andCPrime are derived from equations (17) to (23) by replacing Gk
by Gklowast and ]k by ]k
lowastFromequations (12) and (13) the complex bulkmodulusKlowastms
and complex shear modulus Glowastms of asphalt mastic are given by
Klowastms K
lowast3 +
ceff Klowasteff minus Klowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowasteff + 4Glowastms (26)
Glowastms G
lowast3 +
5ceffGlowastms Glowasteff minus Glowast3( 1113857 3Klowastms + 4Glowastms( 1113857
3Klowastms 3Glowastms + 2Glowasteff1113872 1113873 + 4Glowastms 2Glowastms + 3Glowasteff1113872 1113873
(27)
where ceff is the volume fraction of equivalent mediumcalculated as
ceff f1
c1 f11 +
λ2R1( 1113857
3⎛⎝ ⎞⎠ (28)
where f1 is the volume fraction of fillers in asphalt mastic c1is the volume fraction of fillers in three-phase pattern
(e storage modulus Gprime and loss modulus GPrime of asphaltmastic are calculated from equations (26) and (27) by usingthe complex modulus relation Glowast Gprime + iGPrime (e dynamicshear modulus of asphalt mastic |Glowast| is calculated as
Glowast11138681113868111386811138681113868111386811138681113868
Gprime( 11138572
+ GPrime( 11138572
1113969
(29)
4 Viscoelastic Properties Prediction andValidation for Asphalt Mastic
(e dynamic shear modulus of asphalt mastic with differentfiller volume fractions was collected from the literaturereported by Underwood and Kim [18] (e volume fractions
of fillers are respectively 10 (MS10) 20 (MS20) 30(MS30) and 40 (MS40) Based on the time-temperatureequivalence principle the master curves of the dynamicshear modulus of asphalt mastic |Glowastms| are shown in Figure 5
(e fillers in asphalt mastic are stone materials and theelastic modulus and Poissonrsquos ratio of the filler could bevalued as E1 56GPa and ]1 025 [32] Di Benedetto ob-tained that Poissonrsquos ratio of the asphalt varies with frequencyfrom 048 to 05 and the phase angle is from minus 018 to minus 129deg[33] Approximate values of 049 and 0deg are used as Poissonrsquo sratio and phase angle of asphalt for computing convenience
Assuming that the fillers are uniformly distributed asbody-centered cubic the fillers distribution coefficient isshown as [29]
λ2R1( 1113857
3
radicπ
8f1( 11138571113890 1113891
13
minus 1 (30)
(e fillers are regarded as spherical the cumulativepassing percentage of particle size di of the fillers is expressedas Pi (i 1 m) (e average particle size of sieve i isrepresented by (di+ diminus 1)4 (e number of filler particles ofsieve i in unit mass Ni is expressed as
Ni Pi minus Piminus 1( 1113857
ρf(43)π di + diminus 1( 11138574( 11138573 (31)
where ρf is the density of fillersFrom equation (31) the specific surface area of filler
particles Sf is given by
Sf 1113944
m
i14π
di + diminus 1
41113888 1113889
2
Ni (32)
(e number of filler particles in asphalt mastic is N therelationship between the average radius R of all sphericalparticles and the specific surface area S is calculated as
6 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
S 4πR
2N
(43)πR3ρfN
3
Rρf (33)
From equation (33) the average radius of filler particlesis given by
R 3
Sρf (34)
(e dynamic shear moduli of structural asphalt inequations (24) and (25) are assumed to satisfy the loga-rithmic mean value relationship between filler and asphalt(e complex shear modulus of structural asphalt can beexpressed as
Glowast2 10 lgG1+lgGlowastba( )2( ) (35)
where G1 and Glowastba are the complex shear moduli of fillers andasphalt respectively
(emixed model proposed by Underwood and Kim [18]is employed to characterize the interaction between struc-tural asphalt and free asphalt the complex shear modulus offree asphalt can be obtained
Glowast3
11138681113868111386811138681113868111386811138681113868
Glowastba1113868111386811138681113868
1113868111386811138681113868 times Glowast21113868111386811138681113868
1113868111386811138681113868 times c3
Glowast21113868111386811138681113868
1113868111386811138681113868 minus Glowastba
11138681113868111386811138681113868111386811138681113868 times c2
(36)
where c2 and c3 are the volume fractions of structural asphaltand free asphalt in the total asphalt
Figure 6 shows the dynamic shear moduli of structuralasphalt and free asphalt calculated from equations (35) and(36) It can be seen that the structural asphalt absorbs thepolar component of the matrix asphalt until it is saturatedwhich leads to a significant increase in the complexmodulusA small number of polar components are lost in the freeasphalt resulting in a slight decrease in the complexmodulus
(e thickness of the structural asphalt layer is related tothe physicochemical reaction between asphalt and fillersUnderwood et al have obtained that the thickness ofstructural asphalt layer is 02ndash063 μm when the volume
fraction of fillers in asphalt mastic is 10ndash60 by micro-scopic analysis (erefore the volume fraction of structuralasphalt was determined as csa 30 and the structuralasphalt thickness ds and volume fraction of each componentare calculated as shown in Table 1
(e H-MRP and modified H-MRPmodels are applied topredict the dynamic shear modulus of asphalt mastic |Glowastms|
with different filler volume fractions Figure 7 presents thepredicted and measured |Glowastms| values for asphalt mastic Itcan be seen that when the volume fraction of fillers is 10and 20 the predicted results of the two models are close tothe measured ones However the predicted modulus of themodified H-MRP model is higher than the measuredmodulus at high frequency It shows that when the contentof fillers is low the physicochemical effect between asphaltand fillers may be weak (erefore the volume fraction ofstructural asphalt assumed as csa 30 overestimates thephysicochemical effects (e volume fraction of structuralasphalt should be lower than 30 however when thevolume fraction of structural asphalt is 0 the modified H-MRP model is equivalent to the H-MRP model It can beseen from Figures 7(a) and 7(b) that the modulus predictedby the H-MRP model is closer to the measured modulusthan the modified H-MRP model (erefore it is suggestedto apply H-MRP model to predict the viscoelasticity ofasphalt mastic while the volume fraction is less than 20
When the volume fractions of fillers are 30 and 40the predicted value of the modified H-MRP model is closerto the measured ones compared with the H-MRP modelwhich could be attributed to the fact that the physico-chemical effect is strengthened when the volume fractions offillers are high so it is reasonable to assume csa 30 (epredicted dynamic modulus is much closer to the measuredones at high frequencies than at low frequencies It indicatedthat the proposed models underestimated the viscoelasticeffect of asphalt at low frequencies asphalt mastic can beconsidered as asphalt with the addition of fillers in it andfillers are basically elastic With the elastic fillers added toasphalt the proportions of the elastic and viscous
1E ndash 05 1E ndash 03 1E ndash 01 1E + 01 1E + 031E ndash 01
1E + 01
1E + 03
1E + 05
Pure asphaltMS10MS20
MS30MS40
|Glowast m
s| kP
a
ω (rads)
Figure 5 Master curves of the dynamic shear modulus of asphaltmastic
1E ndash
04
1E ndash
02
1E +
00
1E +
02
1E +
04
1E + 00
1E + 02
1E + 04
1E + 06
Pure asphaltStructural asphaltFree asphalt
|Glowast| k
Pa
ω (rads)
Figure 6 Dynamic shear modulus of structural asphalt and freeasphalt
Advances in Materials Science and Engineering 7
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
components in asphalt mastic are different from those inpure asphalt which will definitely cause the change in theelastic and viscous components of the complex modulus(erefore asphalt mastic should exhibit higher modulusthan that of pure asphalt due to the addition of elastic fillers(e dynamic modulus is not a linear elasticity but a vis-coelasticity (e equivalent modulus of elasticity is directlytransferred to viscoelasticity by using the elastic-viscoelasticcorrespondence principle which weakens the effect of
viscosity Asphalt mastic behaves more like a viscous ma-terial at low frequencies (erefore the difference betweenpredicted and measured moduli was larger With the in-crease in the loading frequency asphalt mastic shows moreelastically and the predicted modulus was much closer to themeasured value
In order to analyze the effect of the structural asphaltvolume fraction on the prediction modulus the structuralasphalt volume fraction csa was taken from 10 to 60 for
Table 1 (ickness of structural asphalt layer and volume fraction of each component
Mastic types ds (μm)Asphalt mastic Asphalt
Fillers () Structural asphalt () Free asphalt () Structural asphalt () Free asphalt ()MS10 222 10 271 629 301 699MS20 122 20 241 559 301 699MS30 079 30 212 488 302 698MS40 054 40 182 418 303 697
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(a)
ω (rads)1E
ndash 0
5
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(b)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
Measured
Modified H-MRP predictedH-MRP model predicted
(c)
ω (rads)
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Measured
Modified H-MRP predictedH-MRP model predicted
(d)
Figure 7 Comparison of predicted and measured |Glowastms| values (a) MS10 (b) MS20 (c) MS30 (d) MS40
8 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
comparative analysis Due to space limitation the effect of csaon the dynamic modulus of MS30 is shown here only seeFigure 8 (e modulus of structural asphalt is higher thanthat of free asphalt so the predicted modulus increases as thevolume fraction of structural asphalt increases It can be seenthat the increase in the volume fraction of structural asphaltis an effective method to increase the dynamic modulus ofasphalt mastic through the improvement of the physico-chemical effect between fillers and asphalt From the analysisof Figure 7 the underprediction at low frequencies is causedby the elastic-viscoelastic conversion which is not related tothe structural asphalt volume fraction When the structuralasphalt volume fraction is 40ndash50 the predictionmodulusis still lower than the measured value at low frequency It canbe found from Figure 8 that the measured modulus of as-phalt mastic is among the predicted values when the volumefraction of structural asphalt is 20ndash50 the volumefraction of structural asphalt of different types of fillers andasphalt will be measured by nano-micro experiments infuture research
5 Parameters Influencing the Dynamic ShearModulus of Asphalt Mastic
51 Effect of Fillers Distribution (e prediction results inFigure 7 were obtained under the assumption that thedistribution of fillers is uniform but the fillers are non-uniformly distributed in the asphalt mastic actually In orderto analyze the influence of fillers distribution on the pre-diction results of the model the distribution coefficients are0025 005 and 025 for comparative analysis Due to spacelimitation the influence of the filler distribution coefficienton the prediction results of MS30 is given here only seeFigure 9
(e decrease in the distribution coefficient means thatthe particles are dense From the predicted results in Fig-ure 3 it can be observed that the distribution coefficientdecreases and the predicted modulus increases HoweverFigure 9 shows an opposite prediction trend It can be foundthat the distribution coefficient decreases and the prediction|Glowastms| decreases (is is due to the use of different predictionmodels (e prediction results in Figure 3 are obtained byMRP model and the structural asphalt layer by physico-chemical effect is not considered in the model (e distri-bution coefficient decreases which results in the decrease ofthe thickness of the asphalt in the adsorption structure of thefillers considering physicochemical effect Whenλ(2R1) 025 there is a small amount of accumulation offillers and the predicted value slightly decreases Whenλ(2R1) 005 and 0025 a large number of agglomerationsoccur in fillers and the predicted value decreases obviously(erefore in order to improve the viscoelasticity of asphaltmastic the agglomeration of fillers should be avoided in theactual production process of asphalt mastic
52 Effect of Fillers Gradation In order to analyze the in-fluence of the fillers grade on the predicted results of thedynamic shear modulus of asphalt mastic three different
grades of fine medium and coarse are used for comparativeanalysis see Table 2 Here only the predicted dynamic shearmodulus of the MS30 is shown see Figure 10
It can be seen from Figure 10 that the fillers gradation hasa considerable influence on the predicted |Glowastms| of MS30which indicated that the H-MRPmodel proposed in this studycould be well considered for the effect of fillers particle sizeAsphalt mastic mixed with fine-grade fillers has the highestpredicted modulus it is due to the fact that the increase ofthe specific surface area of fillers will enhance the cementationof structural asphalt so that the modulus value increased
53 Effect of Youngrsquos Modulus and Poissonrsquos Ratio of Fillers(e influences of Youngrsquos modulus E1 and Poissonrsquos ratio υ1 offillers on the predicted |Glowastms| of MS30 are shown in Figures 11
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
Measuredcsa = 10csa = 20csa = 30
csa = 40csa = 50csa = 60
ω (rads)
|Glowast m
s| kP
a
Figure 8 Effect of structural asphalt volume fraction on theprediction modulus of MS30
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
ω (rads)
MeasuredUniform distributionλ(2R1) = 025
λ(2R1) = 0025λ(2R1) = 005
Figure 9 Effect of fillers distribution on the predicted |Glowastms|
Advances in Materials Science and Engineering 9
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
and 12 It can be seen that the dynamic modulus of the asphaltmastic increases as the modulus and Poissonrsquos ratio of fillersincreasewithin a certain range themagnitude of the increase in|Glowastms| is comparably small which means that the contributionof aggregate to the dynamic modulus improvement of themixture is limited given the fixed portion of fillers
6 Conclusion
(1) A micromechanical prediction model of asphaltmastic was established based on the morphologically
representative pattern (MRP) approach (e influ-ences of physicochemical effect between fillers andasphalt distribution of fillers and the filler particlesize were considered in the model An explicit so-lution was derived to predict the dynamic modulusof asphalt mastic
(2) (e DSR test was conducted to verify the predictioneffect of the model When the fillers volume fractionswere 10 and 20 the predicted value was relativelyclose to the experimental value When the volumefractions of fillers were 30 and 40 the predictedvalue was lower than the measured ones at lowfrequencies the reasons for the underpredictionwere investigated
(3) (e micromechanical model developed in this paperwas able to reflect the effect of factors during aparameter influencing analysis Based on the sensi-tivity analysis the use of uniformly distributed fillersand fine fillers was an effective way to increase thedynamic modulus of asphalt mastic
(4) In the proposed model the volume fraction of thestructural asphalt and the distribution coefficient of thefillers were assumed the nano-micro test would beconducted to obtain these parameters in the futureresearch to improve the model proposed in this paper
Data Availability
(e data used to support the findings of this study are in-cluded within the article
Measuredυ1 = 020
υ1 = 025υ1 = 030
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 12 Effect of Poissonrsquos ratio of fillers on predicted |Glowastms|
Table 2 Different types of fillers gradation
Mastic typesSieve opening mm ()
0075 0030 0023 0017 00077 00038 00029 00013Coarse 100 50 43 29 14 8 3 0Medium 100 57 50 36 21 15 10 7Fine 100 64 57 43 28 22 17 10
MeasuredFine
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Coarse Medium
Figure 10 Effect of fillers gradation on predicted |Glowastms|
MeasuredE1 = 10GPa
E1 = 56GPaE1 = 100GPa
1E ndash
05
1E ndash
03
1E ndash
01
1E +
01
1E +
03
ω (rads)
1E ndash 01
1E + 01
1E + 03
1E + 05
|Glowast m
s| kP
a
Figure 11 Effect of modulus of fillers on predicted |Glowastms|
10 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
Conflicts of Interest
(e authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
(is study was funded by the Liaoning Highway Admin-istration Bureau under Grant no 201701 and NationalNatural Science Foundation of China under Grant no51308084
References
[1] M C Liao G Airey and J S Chen ldquoMechanical properties offiller-asphalt masticsrdquo International Journal of PavementResearch amp Technology vol 6 no 5 pp 576ndash581 2013
[2] J P Zhang J Z Pei and B G Wang ldquoMicromechanical-rheology model for predicting the complex shear modulus ofasphalt masticrdquo Advanced Materials Research vol 168ndash170pp 523ndash527 2010
[3] G G Al-Khateeb T S Khedaywi and M F Irfaeya ldquoMe-chanical behavior of asphalt mastics produced using wastestone sawdustrdquo Advances in Materials Science and Engi-neering vol 2018 Article ID 5362397 10 pages 2018
[4] M W Witczak and O A Fonseca ldquoRevised predictive modelfor dynamic (complex) modulus of asphalt mixturesrdquoTransportation Research Record Journal of the TransportationResearch Board vol 1540 no 1 pp 15ndash23 1996
[5] J Bari and M W Witczak ldquoDevelopment of a new revisedversion of the Witczak Elowast predictive model for hot mix as-phalt mixturesrdquo Journal of the Association of Asphalt PavingTechnologists vol 75 pp 381ndash423 2006
[6] M Witczak M El-Basyouny and S El-Badawy ldquoIncorpo-ration of the new (2005) Elowast predictive model in the MEPDGNCHRP1-40Drdquo Final report Arizona State UniversityTempe AZ USA 2007
[7] J Bari and M Witczak ldquoNew predictive models for viscosityand complex shear modulus of asphalt binders for use withmechanistic-empirical pavement design guiderdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2001 pp 9ndash19 2007
[8] Z Hashin ldquo(e elastic moduli of heterogeneous materialsrdquoJournal of Applied Mechanics vol 29 no 1 pp 143ndash150 1962
[9] R Hill ldquoA self-consistent mechanics of composite materialsrdquoJournal of the Mechanics and Physics of Solids vol 13 no 4pp 213ndash222 1965
[10] R M Christensen and K H Lo ldquoSolutions for effective shearproperties in three phase sphere and cylinder modelsrdquo Journalof the Mechanics and Physics of Solids vol 27 no 4pp 315ndash330 1979
[11] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica vol 21 no 5 pp 571ndash574 1973
[12] S-S Pang Y Li and J B Metcalf ldquoElastic modulus predictionof asphalt concreterdquo Journal of Materials in Civil Engineeringvol 11 no 3 pp 236ndash241 1999
[13] R Luo and R L Lytton ldquoSelf-consistent micromechanicsmodels of an asphalt mixturerdquo Journal of Materials in CivilEngineering vol 23 no 1 pp 49ndash55 2011
[14] E Aigner R Lackner and C Pichler ldquoMultiscale predictionof viscoelastic properties of asphalt concreterdquo Journal ofMaterials in Civil Engineering vol 21 no 12 pp 771ndash7802009
[15] X-y Zhu X Wang and Y Yu ldquoMicromechanical creepmodels for asphalt-based multi-phase particle-reinforcedcomposites with viscoelastic imperfect interfacerdquo Interna-tional Journal of Engineering Science vol 76 pp 34ndash46 2014
[16] X-Q Fang and J-Y Tian ldquoElastic-adhesive interface effect oneffective elastic moduli of particulate-reinforced asphaltconcrete with large deformationrdquo International Journal ofEngineering Science vol 130 pp 1ndash11 2018
[17] H M Yin W G Buttlar G H Paulino and H D BenedettoldquoAssessment of existing micro-mechanical models for asphaltmastics considering viscoelastic effectsrdquo Road Materials andPavement Design vol 9 no 1 pp 31ndash57 2008
[18] B S Underwood and Y R Kim ldquoA four phase micro-me-chanical model for asphalt mastic modulusrdquo Mechanics ofMaterials vol 75 pp 13ndash33 2014
[19] N Shashidhar and A Shenoy ldquoOn using micromechanicalmodels to describe dynamic mechanical behavior of asphaltmasticsrdquoMechanics of Materials vol 34 no 10 pp 657ndash6692002
[20] W G Buttlar and R Roque ldquoEvaluation of empirical andtheoretical models to determine asphalt mixture stiffnesses atlow temperaturesrdquo Asphalt Paving Technology Association ofAsphalt Paving Technologists Proceedings of the TechnicalSessions vol 65 pp 99ndash141 1996
[21] B S Underwood and Y R Kim ldquoExperimental investigationinto the multiscale behaviour of asphalt concreterdquo Interna-tional Journal of Pavement Engineering vol 12 no 4pp 357ndash370 2011
[22] B Delaporte H D Benedetto P Chaverot et al ldquoLinearviscoelastic properties of bituminous materials from bindersto masticsrdquo Asphalt Paving Technology Association of AsphaltPaving Technologists Proceedings of the Technical Sessionsvol 76 pp 455ndash494 2007
[23] R Li Z Fan and P Wang ldquoMicromechanics prediction ofeffective modulus for asphalt mastic considering inter-particleinteractionrdquo Construction and Building Materials vol 101pp 209ndash216 2015
[24] C Davis and C Castorena ldquoImplications of physico-chemicalinteractions in asphalt mastics on asphalt microstructurerdquoConstruction and Building Materials vol 94 pp 83ndash89 2015
[25] C Clopotel R Velasquez and H Bahia ldquoMeasuring physico-chemical interaction in mastics using glass transitionrdquo RoadMaterials and Pavement Design vol 13 no 1 pp 304ndash3202012
[26] Y Veytskin C Bobko and C Castorena ldquoNanoindentationand atomic force microscopy investigations of asphalt binderand masticrdquo Journal of Materials in Civil Engineering vol 28no 6 Article ID 04016019 2016
[27] M Guo Y Tan J Yu Y Hue and L Wang ldquoA directcharacterization of interfacial interaction between asphaltbinder and mineral fillers by atomic force microscopyrdquoMaterials amp Structures vol 50 no 2 pp 1411ndash14111 2017
[28] V Marcadon E Herve and A Zaoui ldquoMicromechanicalmodeling of packing and size effects in particulate compos-itesrdquo International Journal of Solids and Structures vol 44no 25-26 pp 8213ndash8228 2007
[29] M Majewski M Kursa P Holobut and K Kowalczyk-Gajewska ldquoMicromechanical and numerical analysis ofpacking and size effects in elastic particulate compositesrdquoComposites Part B Engineering vol 124 pp 158ndash174 2017
[30] E Herve and A Zaoui ldquoinclusion-based micromechanicalmodellingrdquo International Journal of Engineering Sciencevol 31 no 1 pp 1ndash10 1993
Advances in Materials Science and Engineering 11
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
[31] S W Park and R A Schapery ldquoMethods of interconversionbetween linear viscoelastic material functions Part I-a nu-merical method based on Prony seriesrdquo International Journalof Solids and Structures vol 36 no 11 pp 1653ndash1675 1999
[32] X Shu and B Huang ldquoDynamic modulus prediction of hmamixtures based on the viscoelastic micromechanical modelrdquoJournal of Materials in Civil Engineering vol 20 no 8pp 530ndash538 2008
[33] H Di Benedetto F Olard C Sauzeat and B DelaporteldquoLinear viscoelastic behaviour of bituminous materials frombinders to mixesrdquo Road Materials and Pavement Designvol 5 no 1 pp 163ndash202 2004
12 Advances in Materials Science and Engineering
