Construction and Building Materials 18(2004) 399–408

0950-0618/04/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.conbuildmat.2004.03.003

Rutting response of hot-mix asphalt to generalized dynamic shear moduliof asphalt binder

W.G. Wong *, Haifeng Han , Guiping He , Kelvin C.P. Wang , Weimin Lua, a a b c

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, PR Chinaa

Department of Civil Engineering, University of Arkansas, Arkansas 72701, USAb

Department of Road and Traffic Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, PR Chinac

Received 22 January 2004; received in revised form 8 March 2004; accepted 9 March 2004

Abstract

In this paper, a stiffness indicator of Generalized Dynamic Shear Modulus(GDSM) of asphalt binders is introduced to studythe rutting resistance of unmodified and polymer-modified asphalt binders. Seven binders of two unmodified and five SBS-modified binders are tested. At 608C, the data of peak shear stress and corresponding peak shear strain are obtained for eachbinder by the Dynamic Shear Rheometer(DSR). In the linear viscoelastic region of binders, the GDSM is calculated on the basisof the constitutive shear stress–strain equation that is obtained by fitting the Box–Lucas model to experimental data. Two kindsof bituminous mixtures classified by the type of aggregate gradation are used to evaluate the rutting potential. For each gradation,the aggregate is blended with the seven binders. For each mixture, single binder content is employed so that differences in ruttingdepth can be investigated, depending on various GDMS values of the binders. The rutting depth of mixes is tested with theAsphalt Pavement Analyzer(APA) at 60 8C and the correlation between the average rutting depth and the GDSM is established.Results indicate that for two kinds of mixtures, there exists a good correlation between the GDSM and the average rutting depth.It is confirmed that GDSM is useful as a stiffness indicator for the evaluation of rutting resistance of binders. The constitutiveequation of shear stress–strain can be obtained by fitting the Box–Lucas model up to the peak shear stress, which represents anexplicit mechanistic implication.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Generalized dynamic shear modulus; Rutting potential; Asphalt pavement analyzer; Dynamic shear rheometer; Polymer modifiedasphalt; Hot-Mix asphalt; Box–Lucas model; Constitutive shear stress–strain equation

1. Introduction

It has been demonstrated that polymer-modified bind-ers have the ability to improve the viscoelastic propertiesof asphalt mixtures at high temperatures and signifi-cantly reduce the rutting under repeated loadsw1x. TheSuperpave parameter G*ysind was specified as thestiffness indicator for evaluating the rutting resistanceof both unmodified and polymer-modified bindersw2x.However, in some studiesw3,4x, it is reported that thisparameter did not give a good correlation with ruttingperformance of mixtures, especially for the polymer-modified binders. It is therefore necessary to find a new

*Corresponding author. Tel.:q852-2766-6067; fax:q852-2334-6389.

E-mail address: cewgwong@polyu.edu.hk(W.G. Wong).

stiffness indicator for more suitably evaluating andranking the rutting performance of binders.The stiffness indicator should be directly correlated

with the rutting failure of binders in mixtures, and italso should be easy to use and mechanistically explicit.Under wheel loading, shear flow deformation is themain deformation mode of asphalt binders, especiallywhen temperature of the mixture is high. When theshear deformation is only partially recovered afterremoval of loading, the permanent deformation in thebinders then has occurred. It is commonly known thatshear modulus is the most direct stiffness indicator,which reflects the ability of shearing-resistance of bind-ers. The main research in this study is to classify therutting resistance of both unmodified and polymer mod-ified binders, based on the shear stiffness modulus. Toachieve this, the Generalized dynamic shear modulus

400 W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Table 1Physical properties measured and corresponding labels of binders

Type of binder Binder label SBS content Penetration Softeningin binder at 258C point(wt.%) (0.1 mm) (R&B,8C)

AH-70 base bitumen 70�0%SBS 0 67 46AH-70 plus 1% SBS 70�1%SBS 1 60 50AH-70 plus 3% SBS 70�3%SBS 3 49 52AH-90 base bitumen 90�0%SBS 0 88 44AH-90 plus 2% SBS 90�2%SBS 2 77 46AH-90 plus 5% SBS 90�5%SBS 5 62 51AH-90 plus 8% SBS 90�8%SBS 8 48 66

Table 2Aggregate gradations of mixtures

Sieve size(mm) Percent of passing through(%)

Mixture A Mixture B

19.0 100 10016.0 100 97.513.2 97 82.59.5 76 684.75 50 52.52.36 32 411.18 22 29.50.6 15 220.3 11 160.15 7 110.075 5 6

Table 3Physical properties of aggregate

Aggregate class Aggregate type Bulk specific gravity

Coarse Aggregate Gritstone 2.69Fine Aggregate Limestone 2.74

(GDSM) is proposed as the modulus indicator, whichrepresents a shearing stiffness of materials, derived fromthe generalized stress–strain system. The constitutiveshear stress–strain equation is developed by using thedynamic stress–strain testw5x. GDSM reflects the per-manent deformation resistance of viscoelastic materials,including the asphalt binder.

2. Objectives

The objectives of this study are:(1) to propose theGeneralized dynamic shear modulus(GDSM) and tointroduce the procedure for determining this indicator,and (2) to evaluate an applicability of the GDSM torank rutting resistance of unmodified and polymer-modified binders, from the correlation between therutting depth of mixtures and GDSM of binders.

3. Materials selection and specimen preparation

3.1. Binders

Bituminous binders used in the tests were eitherconventional road bitumen or polymer-bitumen. Sevenbinders of two unmodified and five modified binderswere prepared in the laboratory. Two unmodified binderswere AH-70 and AH-90 Taizhou base bitumen from

ZhongHai Petroleum Company in China. Five modifiedbinders were produced by adding styrene-butadiene-styrene(SBS) block copolymer to the two base bitumensat a determined percent by weight of bitumen. Thesewere AH-70 plus 1% SBS, AH-70 plus 3% SBS, AH-90 plus 2% SBS, AH-90 plus 5% SBS and AH-90 plus8% SBS. The primary reason to add polymer to basebitumen at different percentage was to obtain a widerange of moduli of binders. Labels of these binders andphysical properties measured are shown in Table 1.The Dynamic Mechanical Analysis(DMA) of bitu-

minous binders is an important method to study therelationship between binder properties and pavementperformancew6–9x. As a tool of DMA, the dynamicshear rheometer(DSR) was recommended to determinedynamic viscoelastic properties of bituminous bindersin the Strategic Highway Research Program(SHRP). Inthis study, DSR was applied to obtain the dynamic shearstress–strain data. The specimens of binders for DSRwere prepared according to the recommendation bySHRP A-369w10x. The specimen of 25-mm diameterand 1-mm thickness was selected. The binders wereheated in the temperature range 150–1758C for prepar-ing the specimens. The higher temperature was requiredfor binders with higher SBS percentage.

3.2. Mixtures

Bituminous mixtures were prepared for the ruttingtest. Two kinds of mixtures, which were named MixtureA and Mixtures B, respectively, were used, on the basisof the difference in the aggregate gradation. Both kindsof mixtures were dense-graded, which are commonlyused as wearing course in China. The composition ofgradations is given in Table 2. Crushed gritstone andlimestone were used in mixtures as the coarse and fine

401W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Table 4Table 4 Labels of mixtures

Type of mixture Binder label Binder content(wt.%) Label of mixture

A 70�0%SBS 5.0 70�0%SBS-AA 70�1%SBS 5.0 70�1%SBS-AA 70�3%SBS 5.0 70�3%SBS-AA 90�0%SBS 5.0 90�0%SBS-AA 90�2%SBS 5.0 90�2%SBS-AA 90�5%SBS 5.0 90�5%SBS-AA 90�8%SBS 5.0 90�8%SBS-AB 70�0%SBS 5.8 70�0%SBS-BB 70�1%SBS 5.8 70�1%SBS-BB 70�3%SBS 5.8 70�3%SBS-BB 90�0%SBS 5.8 90�0%SBS-BB 90�2%SBS 5.8 90�2%SBS-BB 90�5%SBS 5.8 90�5%SBS-BB 90�8%SBS 5.8 90�8%SBS-B

aggregate respectively. Specific gravity of aggregate isshown in Table 3. The mineral filler used in the mixtureswas limestone powder. Its specific gravity is 2.72. Theaggregate of a determined gradation was blended withthe seven binders. For each kind of mixture, regardlessof the type of binders, single binder content wasemployed so that the differences in rutting depth couldbe attributed to the differences in the GDSM of thebinders. For mixture A, the binder content was 5.0 wt.%,while for Mixture B, the binder content was 5.8 wt.%.Table 4 summarizes the information of all 14 mixtures.After short, term aging at 1358C for 4 h w11x,

cylindrical specimens(approximately 150 mm=75 mm)were compacted by Superpave Gyratory Compactor(SGC). The controlled-height mode was applied duringcompaction. For all specimens, the voids in total mix(VTM) was kept at 7.0%("0.5%).

4. Experimental equipment and testing method

4.1. Dynamic shear rheometer

DSR was used for dynamic measurements of binders.Different from the recommended testing procedures bySHRP A-370w12x, in which the strain sweep and thefrequency sweep are carried out in turn, measurementsof DSR were focused on the dynamic stress sweep. Theterm ‘dynamic’ refers to the shear stresses applied tothe test specimen, which vary with the time frompositive to negative in a sinusoidal manner. All thespecimens were loaded at an assigned frequency of 10rads. In the process of dynamic stress sweep, the stressamplitude was increased gradually from initial 1 Pa toending 15 000 Pa. One hundred Pascals was set as theincrement of stress amplitude from the one loading cycleto the next. The loading cycles were continued untilfailure or the preset ending value of stress amplitude.The tests of binders were conducted at 608C forsimulating the higher pavement temperature condition.

The control of temperature during testing was achievedby means of a water bath surrounding the samples. Testdata were obtained automatically by data acquisitionsystems of DSR. The acquired data included the appliedmaximum peak stress and the resulting maximum peakstrain. Three samples were tested for each label ofbinders at the same conditions.

4.2. Asphalt pavement analyzer

The Asphalt pavement analyzer(APA) was employedto test the rutting potential of bituminous mixtures. APArutting depth was applied to estimate the degree ofpermanent deformation of specimens. The hose pressureof APA was set at 0.69 MPa(100 psi) and the load ofwheels on the rubber hoses was set to 0.44 kN(100 lb)in the process of cyclingw13x. In view of APA testingvariability w14x, for each rut test, six SGC specimenswere prepared for every mixture of the same label. Therutting depth data were collected at 500, 1000, 2000,4000 and 8000 cycles. Each two samples were laidunder one of the three load wheels, two rutting depthdata were collected from each sample, and thus fourrutting depth data were collected from these two sam-ples. An average rutting depth value was calculatedbased on four rutting depths, and so three averagedrutting depths were collected simultaneously in each testof APA. The overall averaged rutting depths werecalculated from these three averaged rutting depths at8000 cycles.Rainfalls can be unusually frequent in many areas

under the warmweather in China. Water exists in voidsof paving materials and the surrounding environment.Therefore, the effect of water on rutting potential ofbituminous mixtures is worth investigation in the lab-oratory research. Cross and Vothw15x studied the influ-ences of sample preconditioning on rutting depths anddrew the conclusion that APA wet rutting depths of 408C soak conditioning was significantly greater than the

402 W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Fig. 1. Data point distribution of peak shear stresses vs. peak shearstrains.

Fig. 2. Box-lucas model constitutive curve.

Fig. 3. Typical result of fitting box-locus model to the peak stress–strain data.

rutting depths of 408C dry, 408C saturated, and 408Cfreeze conditionings. To simulate the field situation,adequately the wet rut was applied as APA testing formand a pretreatment of 608C soaking for 4 h wereperformed for all samples. All mixture specimens weretested with APA where the temperature of air and waterbath was kept at 608C.

5. Results and analysis

5.1. Binders

GDSM is the most important parameter that is indi-rectly obtained from the dynamic stress sweeps testingof binders, and is calculated by means of the relationshipbetween the applied stresses and the resulting strains.Based on the data set of peak stresses and peak strainsin DSR testing, a typical distribution state of data pointsis plotted in Fig. 1. It is clearly shown that the distri-bution of peak stress–strain data is very similar to theconstitutive curve represented by the Box-Lucas modelw16x, which is shown in Fig. 2. The Box-Lucas modelis expressed as the following form,

xysa 1yb (1)Ž .

wherex is a variable,a andb are constants and 0-b-1, and ‘ysa’ is the asymptote equation of the Box-Lucas model.The Box-Lucas model was applied as shown in Fig.

3, where the equation is given as the following form,

gtsa 1yb (2)Ž .

wheret is the shear stress,g is the shear strain andg)0. a, b are constants and 0-b-1. Corresponding to Eq.

(1), the equationtsa is the asymptote equation of Eq.(2). The meaning of this asymptote equation is thatwhen the shear straing™`, the shear stress tendstowards a constant. Based on this and physical impli-cation in Eq.(2), the coefficienta was defined as shearstress constantt .0The Box-Lucas model was fitted well with all the

peak stress–strain data obtained from DSR tests. Thefitted results are summarized in Table 5.For asphalt binder materials, the shear modulus is an

important measure of the material’s stiffness and isdefined as the ratio of shear stress to shear strain. In thegeneralized stress–strain relation, the shear modulus isobtained from,

dtGs (3)

dg

whereG is the shear modulusw5x.Consequently, GDSM was obtained by taking the

first-order derivative of the variableg in Eq. (2). Theresult is given in Eq.(4) and the coefficienta isreplaced witht ,0

gG st9syt lnbb (4)d 0

403W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Table 5Results of the boxlucas model

Binder label No.a t0 b Fitted equationb R2

70�0%SBS 1 15 247.3 0.78666 ts1524.7(1y0.78666)g 0.996492 8731.0 0.68753 ts8731.0(1y0.68753)g 0.996093 8390.1 0.66608 ts8390.1(1y0.66608)g 0.99773

70�1%SBS 1 9100.5 0.66608 ts9100.5(1y0.66608)g 0.997522 9159.7 0.66046 ts9519.7(1y0.66046)g 0.998773 10 610.0 0.69223 ts10610.0(1y0.69223)g 0.99549

70�3%SBS 1 12 392.5 0.64219 ts12392.5(1y0.64219)g 0.997632 11 946.3 0.60277 ts11946.3(1y0.60277)g 0.995513 11 982.2 0.51365 ts11982.2(1y0.51365)g 0.99519

90�0%SBS 1 7124.0 0.76956 ts7124.0(1y0.76956)g 0.995792 7588.1 0.74923 ts7588.1(1y0.74923)g 0.997123 7584.7 0.76031 ts7584.7(1y0.76031)g 0.99574

90�2%SBS 1 9641.4 0.73620 ts9641.4(1y0.73620)g 0.996512 9950.2 0.72708 ts9950.2(1y0.72708)g 0.998663 10 347.5 0.75622 ts10347.5(1y0.75622)g 0.99801

90�5%SBS 1 10 625.2 0.61062 ts10625.2(1y0.61062)g 0.997322 10 828.1 0.62028 ts10828.1(1y0.62028)g 0.998673 10 099.8 0.59194 ts10099.8(1y0.59194)g 0.99745

90�8%SBS 1 16 592.4 0.56612 ts16592(1y0.56612)g 0.999352 16 656.3 0.57041 ts16656.3(1y0.57041)g 0.998023 16 583.0 0.46801 ts16583.0(1y0.46801)g 0.99774

Three specimens were prepared for each of seven binders.a

The constitutive equation istst (1yb) .b g0

In Eq. (4), G is the Generalized dynamic sheard

modulus.Once Eq.(2) is determined, Eq.(4) is determined

accordingly. In addition, if a level of the shear strain isdetermined,G then can be calculated from Eq.(4).d

Here, the strain level for calculating GDSM was deter-mined based on the assumption of small-strain theoryof elasticity and viscoelasticity, where the binder’s angu-lar rotation u was approximated by sinu, namely,sinufu w10x. It is widely accepted that for smalldeformation, the asphalt can be modeled as the gener-alized linear viscoelastic fluidw17x. As a viscoelasticmaterial, in the region of small strain, the stiffnessproperties of asphalt binders, including GDSM, areindependent of constitutive selection of stresses andstrains. Three strain levels of 5%, 10%, and 15% weretried. All the three strains were satisfied with theapproximation of sinufu when the binder specimens of25-mm diameter and 1-mm thickness were selected.Putting these three strain levels into Eq.(4), in whichthe parameterst andb have been determined by fitting0

the Box-Lucas model to the peak stress–strain data, theresults are shown in Table 6, and the equation corre-sponding to each lebel of binder is also listed in Table6. For each of the seven binders, the averaged values ofGDSM at three strain levels were calculated, respective-ly, based on the values of triplicate tests. The results ofthe averaged GDSM are given in Table 7 and thesequence is arranged in the ascending order of theaveraged GDSM.

Fig. 4 illustrates the relationship between GDSM andthe shear strains. It suggests that the values of theaveraged GDSM do not seem significantly different atthree strain levels of 5%, 10% and 15%. This phenom-enon was similar for all the seven binders used in thestudy. Statistical comparison of GDSM at three strainlevels of 5%, 10% and 15% was conducted. The Tukeytest was used in the comparison at the significance levelof as0.05 w18x. Results of the comparison are shownin Table 8. The results indicate that, at the 0.05 level,the differences between the means of GDSM at threestrain levels are not significant. So, it could be judgedthat the three strains for calculating GDSM of thebinders were so small that they were located in theregion of linear viscoelasticity, where the stiffness indi-cators of the viscoelastic materials are independent ofthe strains or stresses. In the study described as follows,the strain level of 10% was selected for obtaining GDSMof the binders.From Eq.(2), in which the coefficienta is replaced

with t , we obtain0

tgb s1y (5)

t0

Applying Eq. (5) to Eq. (4),

gG sy1524.7ln0.78666Ø0.78666 (6)d

For Eq.(6), if a value of the shear stress is determined,the portion ofy(t yt) in Eq. (6) would be a subtrac-0

404 W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Table 7Averaged values of GDSM

Binder label Averaged GDSM(Pa)

Strain at 5%a Strain at 10% Strain at 15%

90�0%SBS 2017.1 1989.5 1962.390�2%SBS 2960.1 2915.7 2872.070�0%SBS 3388.7 3332.1 3276.570�1%SBS 3725.6 3652.6 3581.090�5%SBS 5107.2 4981.5 4858.870�3%SBS 6328.5 6155.9 5988.190�8%SBS 10 129.8 9809.6 9499.7

The strain levels tried for calculating the Generalized dynamica

shear modulus.

Table 6Calculated results of GDSM

Binder label No. Equationa GDSM (Pa)

Strain at Strain at Strain at5%b 10% 15%

70�0%SBS 1 G sy1524.7 ln 0.78666 0.78666gd 3615.1 3572.0 3529.4

2 G sy8731.0 ln 0.68753 0.68753gd 3210.4 3150.8 3092.3

3 G sy8390.1 ln 0.66608 0.66608gd 3340.7 3273.5 3207.7

70�1%SBS 1 G sy9100.5 ln 0.66608 0.66608gd 3623.6 3550.7 3479.3

2 G sy9159.7 ln 0.66046 0.66046gd 3721.6 3645.2 3570.4

3 G sy10610.0 ln 0.69223 0.69223gd 3831.6 3761.8 3693.2

70�3%SBS 1 G sy12392.5 ln 0.64219 0.64219gd 5368.1 5250.5 5135.5

2 G sy11946.3 ln 0.60277 0.60277gd 5896.3 5748.9 5605.3

3 G sy11982.2 ln 0.51365 0.51365gd 7721.2 7468.2 7223.5

90�0%SBS 1 G sy7124.0 ln 0.76956 0.76956gd 1841.8 1817.8 1794.1

2 G sy7588.1 ln 0.74923 0.74923gd 2159.4 2128.4 2097.9

3 G sy7584.7 ln 0.76031 0.76031gd 2050.1 2022.2 1994.7

90�2%SBS 1 G sy9641.4 ln 0.73620 0.73620gd 2907.8 2863.7 2820.1

2 G sy9950.2 ln 0.72708 0.72708gd 3121.2 3071.8 3023.3

3 G sy10347.5 ln 0.75622 0.75622gd 2851.2 2811.7 2772.6

90�5%SBS 1 G sy10625 2 ln 0.61062 0.61062gd 5113.5 4988.9 4867.4

2 G sy10828.1 ln 0.62028 0.62028gd 5049.3 4930.2 4813.8

3 G sy10099.8 ln 0.59194 0.59194gd 5158.8 5025.3 4895.3

90�8%SBS 1 G sy16592.4 ln 0.56612 0.56612gd 9175.5 8918.1 8668.0

2 G sy16656.3 ln 0.57041 0.57041gd 9092.0 8840.4 8595.7

3 G sy16583.0 ln 0.46801 0.46801gd 12 121.9 11 670.3 11 235.6

The constitutive equation isG syt ln b b .a gd 0

The strain levels tried for calculating GDSM.b

tive constant and Eq.(6) can be expressed as:

G sy t yt lnb (7)Ž .d 0 a

wheret is the determined value of shear stress.a

According to Eq.(7), when the shear stress is deter-mined and kept constant att , G decreases with thea d

increase in the parameterb, where 0-b-1. From thededuction of the functional relationship betweenG andb

b, it is clearly shown that the parameterb in Eq. (2)has the influence on the Generalized dynamic shearmodulus.Experimentally, for each label of binders, the param-

eter b could be determined by fitting the Box-Lucasmodel to the peak stress–strain data collected in theDSR tests. Based on the results listed in Table 5, theaverage values of the parameterb are calculated for allthe seven binders and the results are shown in Table 9.The GDSM values at the strain level of 10% are shownin Table 9.From Table 9, a tendency could be seen clearly that

with the composition of the binders being changed, thevalue of parameterb varied accordingly. At the sametime, GDSM calculated at the strain level of 10%appeared to increase with the decrease in theb value.A one-way analysis of variance(ANOVA) was used todetermine whether the effect of the parameterb on

GDSM would be significantw18x, in which GDSM wasthe response variable and the parameterb was theinfluencing factor. The GDSM values were calculatedat the strain level of 10%.The results of the one-way ANOVA are shown in

Table 10. A a-level of 0.05 was selected. The mostimportant statistic value in the analysis of variance tableis P-value, which is the probability computed in theF-test to reject the null hypothesisw18x. In one-wayANOVA, P-value can determine the effect of the influ-encing factor. In brief, ifP-value is less than or equalto the selecteda-level, the effect of this factor is

405W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Fig. 4. Generalized dynamic shear modulus vs. small shear strains.

Table 8Means comparison for GDSM at three shear strain levels

Shear strain Mean(mm) Difference between Simultaneous confidence intervals Significantlevel (%) means(mm)

Lower limit Upper limitat 0.05 level

10a 4690.9714315 4576.90952 114.0619 y1804.23137 2032.35518 No5 4808.14762 y117.17619 y2035.46946 1801.11708 No15b 4576.90955 4808.14762 y231.2381 y2149.53137 1687.05518 No

The mean of GDSM at 10% strain minus the means of GDSM at 15%, 5% strain.a

The mean of GDSM at 15% strain minus the mean of GDSM at 5% strain.b

Table 9Average values of parameterb

Binder label Average value of parameterb GDSM at strain level of 10%(Pa)

90�0%SBS 0.7597 1989.590�2%SBS 0.7398 2915.770�0%SBS 0.7134 3332.170�1%SBS 0.6729 3652.690�5%SBS 0.6076 4981.570�3%SBS 0.5862 6155.990�8%SBS 0.5348 9809.6

Table 10Results of one-way analysis of variance

Variance source Degrees freedom Sum squares Mean square F ratio P-value

Parameterb 1 76997069.2 76997069.2 22.10174 5.13205E-4Error 12 41805064.4 3483755.36

significant. IfP-value is larger than the selecteda-level,the effect is not significant. From the results in Table10,P-values5.13205E-4, there is sufficient evidence toclaim that the effect of the influencing factor, namelyparameterb, on GDSM is significant at thea-level of0.05.It is proven from the above analysis, both theoretically

and experimentally that parameterb is directly correla-tive with the generalized dynamic shear modulus. The

type of asphalt binders decides the value of parameterb, which can be regarded as an attribute possessed bybinders themselves. Concerning an asphalt binder,parameterb is constant. In view of the close relationbetween the parameterb and GDSM, parameterb isdefined as theattenuation factor of GDSM. The mean-ing of the attenuation factor of GDSM is that with theincrease in parameterb, the generalized dynamic shearmodulus of binder decreases.

406 W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Fig. 5. Plot of average rutting depth vs. log generalized dynamic shearmodulus for mixture A.

Table 11Average rut depth for two mixtures

Label of mixture Average rut depth(mm)a Label of mixture Average rut depth(mm)

70�0%SBS-A 3.632 70�0%SBS-B 4.15170�1%SBS-A 2.217 70�1%SBS-B 3.75470�3%SBS-A 1.843 70�3%SBS-B 2.18990�0%SBS-A 4.944 90�0%SBS-B 5.18190�2%SBS-A 4.269 90�2%SBS-B 4.55890�5%SBS-A 2.586 90�5%SBS-B 3.22790�8%SBS-A 1.233 90�8%SBS-B 1.955

The overall average rut depth was calculated from the average rut depths of left, middle, right wheels.a

All parameters in Eqs.(2) and (4) have explicitlyphysical meanings. To demonstrate this point, Eq.(2)is rewritten in the following form,

gtst 1yAF (8)Ž .0

wheret is the shear stress constant,af is the attenuation0

factor of GDSM and 0-AF-1, t is the shear stress,gis the shear strain. Eq.(8) is called theconstitutiveshear stress–strain equation of asphalt binder. Corre-sponding to the changes in Eqs.(2) and (4) can bechanged as the following form,

gG syt lnAFAF (9)d 0

whereG is GDSM, t is the shear stress constant,afd 0

is the attenuation factor of GDSM and 0-AF-1, g isthe shear strain.

5.2. Mixtures

All samples of mixtures were tested by APA at 608C. The wet rut was applied as the APA testing. Theoverall average rutting depths of samples are calculatedas mentioned above, and the results are shown in Table11.

5.3. Rutting depth and generalized dynamic shearmodulus

Regression analysis was conducted to examine therelationship between average rutting depth of samples,which was obtained from the wet rut testing of APA at60 8C, and GDSM of binders, which was derived fromthe dynamic stress sweep testing of binders at 608Cand calculated at the strain level of 10% according toEq. (9).Regression analysis was conducted according to the

following equation:

RDsAqBlogGDSM (10)

where the independent variable is GDSM of binders,RD is the average rutting depth of mixture samples.

The result of the regression analysis for mixture A issummarized in Fig. 5, and each point on the graphrepresents one of the seven mixtures. There is a cleartrend for rutting depth to reduce as GDSM increases.The correlation between the rutting depth of mixturesamples and GDSM of binders is good withR value2

of 0.876, whereR is the correlation coefficient. The2

relationship is given by Eq.(11) and the regression lineis plotted in Fig. 5.

RDs23.93y5.792logGDSM (11)

For mixture B, a plot of average rutting depth againstlogarithm GDSM of binders is shown in Fig. 6. It issimilar to Mixture A in the trend that the average ruttingdepths of samples reduce with the increase in GDSM ofbinders. The correlation between the average ruttingdepth and GDSM is fairly high(R s0.948). The2

relationship is given by Eq.(12) and the regression lineis plotted on the graph.

RDs23.60y5.519logGDSM (12)

From the results of regression analysis, both for mixtureA and for mixture B, the close relationships were found

407W.G. Wong et al. / Construction and Building Materials 18 (2004) 399–408

Fig. 6. Plot of average rutting depth vs. log generalized dynamic shearmodulus for mixture B.

between the rutting depth of mixtures and GDSM ofbinders. Based on the correlation established in thisstudy, as a stiffness indicator of binders, it is appropriateto apply GDSM to the rutting resistance of polymer-modified asphalt.

6. Conclusions

The generalized dynamic shear modulus(GDSM)conveys three aspects of connotations. First, the notioncomes from the generalized stress–strain system wherethe differential quotient of shear stress to shear strain isdefined as the shear modulus. Second, the testing datafor developing the constitutive equation of shear stress–strain and GDSM are derived from the dynamic stress–strain test, where the shear stresses applied to the testspecimen vary with the time in the sinusoidal fashion.Third, as a measure of material’s shearing stiffness,GDSM reflects the permanent deformation resistance ofviscoelastic materials, including the asphalt binder.The constitutive equation of shear stress–strain can

be obtained by fitting the Box-Lucas model to the peakstress–strain data, which were obtained by applyingdynamic stress sweeps of DSR at 608C. The parametersin the constitutive equation have specific physical mean-ings from both functional deduction and experimentalanalysis.Using regression analysis, close correlations are estab-

lished between the average rutting depth of mixtures at60 8C and GDSM of binders calculated at the strainlevel of 10%. For two kinds of dense-grade mixturesused in this study, the values of correlation coefficientR are 0.876 and 0.948, respectively.2

The method to calculate GDSM presented in thisstudy is straightforward. In addition, the physical mean-ings of the expression used to depict it are explicit. Asa stiffness indicator, GDSM is appropriate for ranking

the rutting resistance of binders for both unmodified andSBS modified binders.

Acknowledgments

This research was supported by a grant of theResearch Committee of The Hong Kong PolytechnicUniversity (project account: PC57). Special thanks areexpressed to the reviewer for providing pertinent andavailable comments.

Appendix A: Notation

The following symbols are used in this paper:af: attenuation factor of generalized dynamic

shear modulus;G: shear modulus;G , GDSM: generalized dynamic shear modulus;d

RD: average rutting depth of mixture samples;g: shear strain;t: shear stress;t :0 shear stress constant;t :a the determined value of shear stress.

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