Hirsh Model Asphalt a Apt 2003

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Christensen et al.

Hirsch Model for Estimating the Modulus of Asphalt Concrete

Donald W. Christensen 1, Jr., Terhi Pellien 1, and Ramon F. Bonaquist 2

Abstract

The purpose of this paper is to present a new, rational and effectivemodel for estimating the modulus of asphalt concrete using bindermodulus and volumetric composition. The model is based upon anexisting version of the law of mixtures, called the Hirsch model, whichcombines series and parallel elements of phases. In applying the Hirschmodel to asphalt concrete, the relative proportion of material in parallelarrangement, called the contact volume, is not constant but varies withtime and temperature. Several versions of the Hirsch model wereevaluated, included ones using mastic as the binder, and one in which theeffect of film thickness on asphalt binder modulus was incorporated intothe equation. The most effective model was the simplest, in which themodulus of the asphalt concrete is directly estimated from bindermodulus, VMA, and VFA. Models are presented for both dynamiccomplex shear modulus (|G*|), and dynamic complex extensionalmodulus (|E*|). Semi-empirical equations are also presented forestimating phase angle in shear loading and in extensional loading. The

proposed model was verified by comparing predicted modulus and phaseangles to values reported in the literature for a range of mixtures.

Key Words: Superpave, asphalt concrete, dynamic modulus, shearmodulus, models, law of mixtures.

Introduction

The purpose of this paper is to describe in detail the development anduse of a new model for predicting the modulus of asphalt concrete, whichis based upon an existing model for composite behavior called the Hirschmodel. This model was developed as part of NCHRP Projects 9-25 and9-31, the objectives of which are to evaluate existing Superpaverequirements for volumetric composition and modify them if warranted.The Hirsch model for hot mix asphalt concrete (HMAC) was developed

during the initial phases of these NCHRP projects to serve as a tool foranalyzing the effect of changes in air voids, voids in mineral aggregate(VMA) and other volumetric mix factors on the modulus of asphalt


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concrete and related mechanical properties. Although there are a numberof models for estimating the modulus of HMAC mixtures, these modelsare for the most part empirical, they rely on obsolete semi-empiricalasphalt binder rheological data, and do not adequately utilize volumetriccomposition in their formulation.

This paper presents the background needed to understand theconceptual development of the modified Hirsch model, and how thismodel has evolved to its current form. Several alternate formulations ofthe model are presented: one that potentially accounts for the stiffeningeffect of mineral filler within the mastic of the mixture; one that includesa term for the effect of film thickness; and a simple model based directlyon volumetric properties. These three models for predicting asphaltconcrete modulus were evaluated using a substantial set of data from theFederal Highway Administrations (FHWAs) Accelerated LoadingFacility (ALF) project, the MNRoad Project, and the WesTrack Project(Pellinen, 2001). These data were fit to the models using non-linear leastsquares. All three versions exhibited nearly identical accuracy, butmodel parameters relating to both mastic modulus and film thicknesseffects were not significant. Therefore, the simplest form of the Hirschmodel is recommended for general use by pavement engineers andresearchers in estimating modulus values of asphalt concrete mixtures.

This paper presents two versions of the Hirsch model for asphaltconcreteone for estimating the dynamic complex shear modulus |G*|,and one for estimating the dynamic complex modulus in extension, |E*|.Semi-empirical models are presented for estimating phase angles for

both loading methods based upon the Hirsch model. This paper includesfairly extensive verification of the Hirsch model, based upon a number ofcomparisons of predicted moduli values with measured values as

published in the literature. This paper is based upon appendices to

NCHRP Project 9-25 and Project 9-31 Interim Reports, but has beenedited for publication as a research paper (Christensen, 2001;Christensen, 2002).

Background

The proposed method for estimating HMAC modulus is based upon awidely used model for predicting properties of composite materials thelaw of mixtures. As generally formulated, the law of mixtures representsthe mechanical response of two separate phases in parallel (Nichols,

1976):

E c = v 1 E 1 + v 2 E 2 (1)


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Where E refers to the modulus, or some other material property, v refersto the volume fraction of a given phase, the subscript c refers to thecomposite, and the subscripts 1 and 2 refer to different phases present inthe composite. A less common form of the law of mixtures can bederived for phases in series (Nichols, 1976):

1/ E c = v 1 /E 1 + v 2 /E 2 (2)

Where the variables are as defined above. In the early 1960s, T. J.Hirsch developed a variation of the law of mixtures for modeling themechanical behavior of asphalt concrete (Hirsch, 1962). The Hirschmodel combines parallel and series arrangement of the phases (Nichols,1976):

1/ E c = v 1s /E 1 + v 2s /E 2 + (v 1p +v 2p )2 /(v1p E 1+v 2p E 2 ) (3)

Where v1s and v2s refer to the volume fractions of phases 1 and 2,respectively, in series arrangement, and v1p and v2p refer to the volumefractions of phases 1 and 2, respectively, in parallel arrangement. Anequivalent expression for the Hirsch model can be formulated if it isassumed that the relative proportions of phase 1 and phase 2 are the samein the series and parallel portion of the model:

( )

++

+=

22112

2

1

1 111

E v E v x

E v

E v

x E c

(4)

Where x is the ratio of phases in parallel arrangement to the total volume.When x = 1, the Hirsch model produces results identical to completely

parallel phases (Equation 1), whereas when x = 0, it represents a pure

series arrangement (Equation 2). This model is therefore quite flexible,and can be used to represent a wide range of composite behavior. Hirschfound that for several portland cement concrete mixtures, x had a valueof about 0.5. These three composite models are shown in Figure 1.


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V1 V2

V1

V2

V

VVV

1s

2s

1p 2p

(a) Parallel phases (b) Series phases (c) Hirsch model

Figure 1. Schematic representation of composite models for parallel,series, and Hirsch (combination) arrangement of phases.

Equation 3 is the basis for the proposed equation developed by theresearch team for predicting asphalt concrete modulus from bindermodulus and volumetric properties. Asphalt concrete tends to behavelike a series composite at high temperature, but more like a parallelcomposite at low temperature, and so the Hirsch model should beappropriate for estimating the modulus of asphalt concrete. However, forthis model to be useful in modeling the modulus of asphalt concrete, therelative proportions of the series and parallel phases must be time andtemperature dependent. The aggregate phase in the parallel portion ofthe model is important in characterizing the behavior and performance ofHMAC, as it represents that portion of the aggregate particles in intimatecontact with each other; this portion of the aggregate is therefore calledthe aggregate contact volume , Pc . In general, as the aggregate contactvolume increases, so will the modulus, strength, and resistance to

permanent deformation. High values of Pc indicate a very effectivestructure producing good strengths and stiffness, typical at lowtemperature. Low values of Pc tend to occur at high temperatures, andindicate mixtures with low strength and stiffness.

Various modifications of the Hirsch model were devised andevaluated in developing the final version. Four of the most important areshown in Figure 2. In many of the early formulations of the model, itwas assumed that the binder in asphalt concrete was the mastic (mineralfiller plus binder) rather than binder alone. In Figure 2(a), the bulk of theaggregate is in a series arrangement with a parallel combination of theaggregate contact volume and mastic. Better results were obtained withthe arrangement shown in 2(b), where the aggregate contact volume iscombined in parallel with a series arrangement of the bulk aggregate andmastic. In this figure, Vc refers to the aggregate contact volume, Va refers to the aggregate volume exclusive of the contact volume, Vm is the


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mastic volume, and Vv is the air void volume. For the configurationshown in 2(a), the appropriate equation for dynamic modulus would beas follows:

( )12'1'

++=

VmEm PcEa

Va

Ea

Va Ec (5)

Where: Ec = composite (asphalt concrete) modulusVa = volume fraction of aggregate, excluding contact volume

and mineral filler Ea = aggregate modulus Pc = aggregate contact volume, as volume fractionVm = volume fraction of mastic

Em = mastic modulus

Va

VmVc Vv

Va

VmVc

Vv

(a) Series Version (b) Parallel Version

Vc Vv Va,Vm

Vap VvpVmp

Vas

Vms Vvs

(c) Dispersed Version (d) Alternate Version

Figure 2. Schematic representation of four alternate versions of

modified Hirsch model.


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For the configuration shown in Figure 2(b), the equation for thecomposite modulus is given by the following equation:

( )1

2 '1

++=

Em

Vm

Ea

VaVcVcEa Ec (6)

Where the variables are as defined previously. The arrangement in2(b) can in fact be generalized by using an exponent m rather than

1 in the series portion of the model:

( ) ( )mmmm VmEm EaVaVcVcEa Ec 111 '1 ++= (7)

The exponent m can take any value from 1 to 1, though in this casemeaningful models would result only for values between 1 and about 0.2. For the case of perfect spheres of aggregates within the masticmatrix, m = -0.5; this arrangement is shown in Figure 2(c). Thus,Equation 7 represents a very flexible form of the Hirsch model.

In order to use the models represented in Figures 2(a), (b) and (c), themodulus of the mastic must be estimated. The NCHRP 9-25 researchteam has evaluated two approaches to calculating the modulus of themastic based upon the binder modulus and the volume fractions of binderand mineral filler. The first is a version of the Einstein equationdeveloped by Shashidhar and his associates (Shashidhar et al., 1999):

( ) Eb

Vf CVf AVf

Em++=

''11'1

(8)

Where: A = K E -1

K E = generalized Einstein coefficientVf = volume fraction filler in mastic = Vf/Vm C = (1- Vf max)/Vf ma x

2 Vf max = maximum volume fraction of filler in mastic

The Einstein equation is rational in form and appears to bereasonably accurate. However, the predictions are quite sensitiveto the maximum volume fraction of filler ( Vf max ). As the volumefraction of filler in the mastic approaches this maximum, the

predicted modulus of the mastic becomes extremely high; whenVf = Vf max , the predicted value of Em is infinite. In reality, the

binder would most likely never actually incorporate this amount offiller, as this represents a hypothetical maximum, at which the airvoid content of the mastic would be exactly zero. Instead, a certain


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amount of excess filler would occur throughout the mix as freefiller. However, this is difficult to account for within theframework of a relatively simple method for predicting mixturemodulus.

An alternative approach involves the use of the generalized law ofmixtures:

( ) nnn EbVb EaVf Em 1'' += (9)

Where:Vb = volume fraction binder in mastic = Vb/Vm

Eb = modulus of bindern = exponent with values from 1 to +1

For dispersed systems, n will range from slightly less than 0 to about 0.5. A preliminary evaluation based upon typical binder-filler systemshas indicated that the predictions of Equation 9 are of the samemagnitude as those of Equation 8 when n = -0.2. Equation 9, because ofits relative simplicity and robustness, is attractive for use in the variousmodels. Results using this equation compared favorably with those

produced using the Einstein equation. Therefore much of the analyses ofthe various forms of the Hirsch model therefore used Equation 9 forestimating mastic modulus.

Preliminary analyses showed that these three versions of the Hirschmodel did not exhibit good accuracy. The version of the model that wasfound to exhibit consistently good accuracy is shown in Figure 2(d).This alternate formulation of the Hirsch modelis very similar to theoriginal model. The only difference in this version is that the series and

parallel sub-units of the model are combined in parallel, rather than inseries. This in effect places more emphasis on the parallel sub-unit of themodel. This version of the Hirsch model produced the best results, andhas the additional advantages of being relatively simple and very similarto the original version of the model as formulated by Hirsch.Mathematically, it can be expressed using the following equation:

( ) ( )12

2 ''

' ++++

++=

VmsEmVvsVms

Ea sVa

VvsVms sVa

VmpEm pEaVa Ec

(10)

Where the variables are as defined previously, but the addition of thesubscripts p and s denote parallel and series phases, respectively (seefigure 2(d)). As with the standard version of the Hirsch model, this


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alternate formulation can be stated in somewhat simpler terms by usingthe contact volume Pc to represent the proportion of parallel to total

phase volume:

( ) ( ) ( )12'

1'

++++=VmEm

VvVm

Ea

Va PcVmEm EaVa Pc Ec (11)

Various types of functions where used to describe the contact factorfor use in Equation 11. Eventually, the research team found that thefollowing equation for the contact factor provided the best results:

1

1

'

'

2

0

P

P

VMA EmVFM

P

VMA EmVFM

P Pc

+

+

= (12)

Where VFM is the fraction of aggregate voids filled with mastic, VMA isvoids in the mineral aggregate, exclusive of mineral filler, and P 0, P 1,and P 2 are empirically determined constants. Equation 12 is in fact atype of logistic function, which produces a sigmoidal response in log-logspace typical of the behavior of many viscoelastic materials. The firstconstant in Equation 12, P 0, is directly related to the contact factor athigh temperatures and/or low frequencies; P 1 is an exponent related tothe rate of change of the contact factor with respect to the bindermodulus Em; P 2 is related to the location of the contact factor, which isdirectly related to the overall stiffness of the asphalt concrete mixture.

As will be discussed later in this paper, preliminary analysis of themastic version of the Hirsch model demonstrated that it was accurate, butalso seemed to indicate that it was not necessary to consider thestiffening effects of mastic. Instead, a simpler version of this model,which treats asphalt concrete as a simple three-phase system ofaggregate, asphalt binder, and air voids, seemed appropriate:

( ) ( ) ( )12

1

++++=VbEb

VvVb EaVa

PcVbEbVaEa Pc Ec

(13)Where Eb represents the binder modulus, and Vb represents the effective

binder volume. Note that the aggregate volumes in Equation 13 nowrepresent true aggregate volume including the volume of mineral filler.The corresponding equation for the contact factor is identical to Equation12, except that true VMA is substituted for VMA (voids plus bindervolume plus mineral filler volume).

A final modification of the Hirsch model was devised to determine iffilm thickness could be incorporated into the expression for modulus.The equation for this version of the Hirsch model is essentially identical


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to 13, but the effective binder modulus, Eb , is substituted for the true binder modulus:

( ) ( ) ( )12

'1'

++++=

VbEbVvVb

EaVa

PcVbEbVaEa Pc Ec

(14)The effective binder modulus is calculated by assuming that the

binder modulus at the aggregate surface is equivalent to the glassymodulus, but then decreases to the normal binder modulus value over acertain characteristic distance, t T , representing the transition zone

between the aggregate and binder. If the binder film thickness is calledt F , the equation for effective binder modulus is then given using thefollowing equation:

( ) Ebt E t t EbE t

EbT g T F

g F

+=' (15)

Where E g is the glassy modulus of asphalt cement binder, assumed to be1 GPa (145,000 lb/in 2).

The reader should keep in mind that the above functions for theHirsch model, though stated in terms of the extensional modulus E, can

be just as easily formulated in terms of the shear modulus G.Furthermore, the modulus values used in any of these equations can bedetermined from creep, stress relaxation, or dynamic modulus tests. TheHirsch model is a rational, though semi-empirical method of predictingasphalt concrete modulusthat is, its structure is logical and based uponthe law of mixtures, but its use in practice requires calibration withmeasured data. It should not be confused with a constitutive equation,which rigorously delineates the relationships among stress, strain, andmaterial properties (such as modulus) in 2 or 3 dimensions. The Hirschmodel can however be used to estimate modulus values that are used invarious such constitutive equations.

Data used in Refining the Hirsch Model

In order to evaluate the various versions of the Hirsch model, andrefine the most promising of these models it was necessary to establish adata base of modulus values for a wide range of mixtures. Such a dataset was created, based upon dynamic modulus measurements made atAdvanced Asphalt Technologies, LLC, (AAT) and at the Arizona StateUniversity (ASU) as part of NCHRP Project 9-19 (Pellinen, 2001). TheAAT data set consists of dynamic shear modulus and phase angle datacollected using the SST, using the frequency sweep tests. The ASU data


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set consists of dynamic compression modulus and phase angle data. Themixtures tested were the same for both data sets, and originated fromFHWAs Accelerated Loading Facility (ALF) project, the MNRoadProject, and the WesTrack Project; Pellinen has thoroughly summarizedthese projects and the materials used in both data sets (Pellinen, 2001).Inclusion of both shear and compression data allowed the research teamto develop versions of the Hirsch model for both cases, which is essentialsince the relationship between | G*| and | E *| for asphalt concrete mixturesis not at all straightforward nor well documented.

A summary of the database is given as Table 1. The databaseincludes results from testing on 18 mixtures using 8 different binders and5 different aggregate sizes and gradations. A total of 206 observationsare included in the dataset for each type of measurement (shear andcompression). A wide range of volumetric compositions is alsorepresented, although one shortcoming in this data is the lack of mixtureswith low air voids and low VFA. The data set is however extensive, andsuitable for initial development of the Hirsch model. It is possible thatlarger and more robust data sets could be used in the future to furtherrefine this method for estimating asphalt concrete modulus.

Method of Analysis

The general approach used in evaluating the various versions of theHirsch model, and refinement of the most promising was non-linear leastsquares (Chapra and Canale, 1988). Christensen has presented a detaileddescription of the use of this technique in analyzing IDT creep data(Christensen, 1998). In general terms, this procedure uses an iterative,

numerical procedure to calculate the values for parameters in a functionso that the sum of the square of the error terms is minimized. Graphicaltechniques were used to identify and eliminate several records that wereclearly outliers, and to determine if other potential predictor variablesexisted which were not included in the model. All of the outlierseliminated from the data were high-temperature measurements madeusing the SST at very low stress levels. These measurements wereconsidered to be unreliable, since the very low stress levels used weredifficult to measure and appeared to be quite noisy.

After refinement of the final versions of the Hirsch model, simplecurve fitting was used to develop a function for phase angle. Suchequations are sometimes needed for calculation of storage and loss

modulus values.


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Table 1. Summary of Database of Mixture Modulus Data

Factor ALF MNRoad WesTrack TotalDesign Method Marshall Marshall Superpave N/ABinders AC-5, 10, 20

SBS-modifiedPE-modified

AC-20120/150-

PEN

PG-64-22 8

Aggregate Sizesand Gradations

19-mm dense37.5-mm fine

9.5-mmfine

19-mm fine19-mm coarse

5

Mixtures 7 5 6 18Total DataPoints

78 59 69 206

For Complete DatabaseVoids, Vol. % 5.6 to 11.2VMA, Vol. % 13.7 to 21.6

VFA, % 38.7 to 68.0Temperatures, C

4, 21, 38

Frequencies 0.1 and 5|G*|, MPa 20.0 to 3,880|E*|, MPa 183 to 20,900Phase angle,degrees

8 to 61


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Results

The three most promising versions of the Hirsch model were allvariations of the alternate formulation shown graphically in Figure 2(d),and represented in its basic form in Equation 10. The three versions ofthis model are called here the mastic version (Equation 11), the simpleversion (Equation 13), and the transition zone version (Equations 14 and15). In all cases, Equation 12 was used to characterize the contact factor

Pc .

Model for |G*|

The results of the non-linear least squares analyses of these threemodels for dynamic complex shear modulus (| G*| as measured using theSST) are summarized in Table 2. The value for parameter P 0 could not

be determined reliably using the least-squares procedure, probably because the database did not include enough values at high temperaturesand low frequencies. However, it is important to include at least anestimate for this parameter, since it represents the limiting asphaltconcrete modulus at high temperatures and/or low frequencies.Comparing model predictions with published master curves for asphaltconcrete, it was estimated that an appropriate value for this constant isabout 3.

All three versions of the Hirsch model exhibited identical r 2 values of96.8 %, suggesting that the more complex versions of the model,intended to account for mineral-filler effects and the effect of filmthickness, are no more effective than the model treating asphalt concrete

as a simple three-phase system. For the mastic model, the law ofmixtures exponent for the mastic, n (Equation 9) has an unrealistic valueof 6.9, and a standard error of 281 %, indicating that this parameterdoes not contribute significantly to the accuracy of the model. For thetransition zone model, similar problems are observed in the estimate ofthe transition zone thickness; t T is estimated at only 0.03 microns, andhas a huge standard error of estimate of almost 4,000 %. It is thereforeconcluded that the most effective version of the Hirsch model is therelative simple version represented by Equation 13, which can be givenin terms of the complex shear modulus, | G*|, VFA, and VMA:


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Table 2. Summary of Least-Squares Estimation ofParameters for |G*| Hirsch Model.

Mastic ModelEquation(s): 11 & 12Parameter Estimate Std. Error

Ea , lb/in 2 635,000 7.2 %P 0 3 N/AP 1 0.678 2.4 %P 2 396 10.4 %n (Eqn. 9) 6.9 281%t T , microns N/A N/Ar 2 96.8 % N/A

Simple ModelEquation(s): 13 & 12Parameter Estimate Std. Error

Ea , lb/in 2 601,000 7.1 %P 0 3 N/AP 1 0.678 2.4 %P 2 396 6.9 %n (Eqn. 9) N/A N/At T , microns N/A N/Ar 2 96.8 % N/A

Transition Zone ModelEquation(s): 14, 15 & 12Parameter Estimate Std. Error

Ea , lb/in 2 601,000 7.2 %P 0 3 N/AP 1 0.678 2.4 %P 2 452 8.7 %n (Eqn. 9) N/A N/At T , microns 0.03 3,760 %r 2 96.8 % N/A


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( )

( )

1

*000,601

1001

1

000,10*1001000,601*

+

+

+=

binder

binder mi x

GVFA

VMAVMA

Pc

VMAVFAGVMA PcG

(16)

Where |G*| mix is the complex shear modulus for the mixture, and|G*| binder is the complex shear modulus for the binder, both in units oflb/in 2, and VFA and VMA are both given as percentages. The bindermodulus can either be determined experimentally using the dynamicshear rheometer (DSR) or similar device, or can be estimated from oneof several available mathematical models. It should be at the sametemperature and loading time selected for the mixture modulus, and inconsistent units. The contact factor is given by the following function:

678.0

678.0

*396

*3

+

+

=

VMA

GVFA

VMAGVFA

Pcbinder

binder

(17)

Figure 3 shows the | G*| values predicted using Equations 16 and 17versus measured values. There is generally good agreement.

1000

10000

100000

1000000

1000 10000 100000 1000000

Measured |G*|, psi

P r e

d i c t e d | G * |

, p s i

Figure 3. Predicted Versus Measured Shear Complex Modulus (r 2 =

96.8 %); Solid Line Represents Equality.


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Model for |E*|

Non-linear least squares analysis was also performed using the |E*|data gathered by Pellinen at Arizona State University as part of NCHRPProject 9-19 (Pellinen, 2001). In the initial calibration of the model for| E *|, performed during Phase I of NCHRP Project 9-25, the data set usedcoincided with that developed for the |G*| model. However, subsequentevaluation of the model showed that results at extreme high and lowtemperatures were not always accurate. Therefore, as part of Phase I of

NCHRP Project 9-31, an expanded data set was created, which includedadditional data at 9 and 54 C. This resulted in a somewhat greaterrange of values for both | E *| and phase angle in the data set.Furthermore, in the expanded data set replicate measurements wereaveraged, partly because of the greatly expanded size of the resultingdata set, but also to minimize variability due to experimental error. Itwas felt that this approach would provide a better estimate of theaccuracy of the model. The final values for the Hirsch model parametersfor dynamic complex modulus in extension were estimated to be asfollows:

Ea : 4,230,000 psi ( 6.5 %) P 0: 649 ( 9.0 %) P 1: 19.7 ( 20.7 %) P 2: .575 ( 3.0 %)

The r 2 value for the | E *| data (98.2 %) is slightly higher than for sheardata. Also, note the much higher value for Ea , which is expected ascompression moduli for most materials, including asphalt concrete, arealmost always much higher than shear moduli. The other parameters aresimilar to the values determined for the shear model. The predicted andmeasured values for | E* | are shown in Figure 4; residuals as a function of

predicted |E*| are shown in Figure 5. Applying appropriate rounding tothe coefficients listed above, the pertinent equation for compressionmodulus is as follows:

( )

( )1

*3000,200,4

10011

000,10*31001000,200,4*

++

+=

binder

binder mi x

GVFA

VMAVMA Pc

VMAVFAGVMA Pc E

(18)


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1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+04 1.0E+05 1.0E+06 1.0E+07

Measured |E*|, psi

P r e

d i c t e d | E * | , p s

i

Figure 4. Predicted and Measured values for Complex Modulus inCompression (r 2 = 98.2 %); Solid Line Represents Equality.

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

1.0E+04 1.0E+05 1.0E+06 1.0E+07

Predicted |E*|, psi

R e s

i d u a

l ( L o g

| E * | )

Figure 5. Residuals (Predicted-Measured) Values of Log |E*| forHirsch Model.

Note that the binder shear modulus in Equation 18 is multiplied by 3 as an estimate of the extensional modulus: | E*| binder 3|G*|binder. This is based on an assumption of incompressibility,that is, that Poissons ratio is 0.5. The following function is usedfor estimating the contact volume for use in conjunction with

Equation 18:


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58.0

58.0

*3650

*320

+

+

=

VMA

GVFA

VMA

GVFA

Pcbinder

binder

(19)

Prediction of Phase Angle

As discussed previously, in addition to a need for estimating modulusin shear and compression, there is also a need to estimate the phase anglefrom compositional data. For example, the fatigue models developedduring the Strategic Highway Research Program (SHRP) used the lossmodulus as a predictor variable, which is a function of both the complexmodulus and the phase angle (Taybali et al., 1995). The research teamfound a good empirical relationship between the log of the contact factor

Pc (Equation 12) and the phase angle. Plots showing these relationships,

which includes the empirically determined equations for phase angle as afunction of log ( Pc ), are shown in Figures 6 and 7. The followingfunction can be used to estimate phase angle using Pc as determinedfrom shear data (r 2 = 82.9 %):

( ) 6.9log39log5.9 2 += Pc Pc (20)

y = -9.4633x 2 - 38.834x + 9.6031

R2 = 0.8294

0

10

20

30

40

50

60

70

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Log (Pc)

P h a s e

A n g

l e ,

D e g

r e e s

Figure 6. Phase Angle as a Function of Log ( Pc ), Shear Data.


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y = -20.56x2 - 54.619x

R2 = 0.8906

0.00

10.00

20.00

30.00

40.00

50.00

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Log (Pc)

P h a s e

A n g

l e ,

D e g r e e s

Figure 7. Phase Angle as a Function of Log (Pc

), Compression Data.

The corresponding function for estimating from compression data isgiven by the following Equation (r 2 = 89 %):

( ) Pc Pc log55log21 2 = (21)

These relationships, though not highly accurate, are useful forestimating phase angle from asphalt concrete volumetric composition. Itshould be kept in mind that the measurement of phase angle is

particularly difficult, so the amount of scatter in Figures 6 and 7 is notsurprising. There are two significant differences in these plots. The

phase angle in shear (Figure 6) has a value of 9.6 when Pc is zero; thisindicates that in shear measurements, the phase angle is about 10 degreeseven at very low temperatures and high frequencies. For the extensionaldata, this is not true the phase angle is zero when Pc is zero. It is

possible that the non-zero phase angle when Pc is zero for shear data isan artifact, caused by friction in the horizontal bearing used in the SSTsystem (there are no bearings in the compression tests, outside of thoseinherent in the actuators). The other significant difference is that for thecompression data, there is a clear maximum in phase angle, whereas theshear data shows no maximum. In general, it appears that the phaseangle data in extension is somewhat more reasonable than thatdetermined in shear.


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Verification of the Model

In order to verify the final version of the Hirsch model in asindependent a manner as possible within the limited time availableduring the initial phases of NCHRP Projects 9-25 and 9-31, the modelsfor shear modulus (Equations 16, 17 and 20) were used to estimate shearmodulus for data as reported by Alavi and Monismith in research relatedto the original SHRP program (Alavi and Monismith, 1994). Thesemeasurements were made using a cylindrical shear test, a completelydifferent technique than that upon which the model was developed.Alavi and Monismith used one aggregate (19-mm nominal maximumsize) and one binder, but a range of asphalt and air void contents. Bindermodulus values were estimated using the model developed byChristensen and Anderson for SHRP binders (Christensen and Anderson,1992). Some volumetric information was estimated because of thelimited data reported by Alavi and Monismith. The predicted andmeasured values of shear modulus are shown in Figure 8; the agreementis good, though the Hirsch model appears to slightly under-predictmodulus values at higher levels. In Figure 9, predicted and measured

phase angles are plotted for these same data. The agreement here is notas good, especially at higher phase angles. In Alavi and Monismithsdata, the phase angle value, after peaking, decreases to very low values.In the SST shear data, on the other hand, the phase angle increases at adecreasing rate to a maximum value without any subsequent decrease.The research team believes this is an inherent difference in themeasurement methods. Phase angle predictions using the Hirsch modelare probably most accurate at low temperatures and/or high frequencies.

As a second verification of the model, the complex modulus in

compression (| E *|) was predicted using both the Hirsch model and usingAndrei and Witczaks equation, and compared to the measured valuesreported by Alavi and Monismith (Andrei Witczak, 1999; Alavi andMonismith, 1994). This comparison is shown graphically in Figure 10.The values for the Hirsch model are in excellent agreement, whereasAndrei and Witczaks equation slightly under-predicts at higher modulusvalues. Although this comparison is too limited to make broadgeneralizations, it suggests that the Hirsch model is in general agreementwith Andrei and Witczaks equation, and is at least as accurate. Hirschmodel predictions for | E *| are probably in better agreement with Alaviand Monismiths data, compared to the | G*| predictions, because theexperimental technique used by Alavi and Monismith for the shear

measurements was substantially different than that used by AAT inmaking the SST measurements. In contrast, Pellinens compression


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moduli were probably determined using methods giving resultscomparable to Alavi and Monismiths.

10

100

1000

10000

100000

10 100 1000 10000Predicted |G*|, MPa

M e a s u r e

d | G * |

, M P

Figure 8. Measured Complex Shear Modulus and Values PredictedUsing the Hirsch Model; Solid Line Represents Equality (data fromAlavi and Monismith, 1994).

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Predicted Phase Angle, Degrees

M e a s u r e

d P h a s e

A n g

l e ,

D e g r e e s

Figure 9. Measured Phase Angle s and Values Predicted Using theHirsch Model; Solid Line Represents Equality (data from Alavi andMonismith, 1994).


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100

1000

10000

100000

100 1000 10000 100000

Measured |E*|, MPa

P r e

d i c t e d | E * |

, M P a

Witczak ModelHirsch ModelEquality

Figure 10. Predicted and Measured |E*| Values; Solid LineRepresents Equality (data from Alavi and Monismith, 1994).

Another comparison useful for verification of the Hirsch model isshown in Figure 11, which is a master curve for mixture V0W1 fromAlavi and Monismiths study (Alavi and Monismith, 1994). This figureshows predicted and measured values for | E *| and phase angle as afunction of reduced frequency at 40 C. This figure confirms that thefrequency dependence and general shape of the functions for complexmodulus and phase angle as predicted by the Hirsch model arereasonable and in good agreement with experimental values. However, itappears that the phase angle predictions do not vary quite as stronglywith frequency as the measured values.

A more thorough verification of the Hirsch model was performedusing data from a recent sensitivity study by Witczak and his associates(Witczak et al., 2001). In this study, | E *| measurements were made on arange of mix variations based upon an Arizona Department ofTransportation mixture. This asphalt concrete used a 25-mm nominalmaximum aggregate size blend with an PG 64-22 asphalt. The basic mixdesign was varied using four different target air void levels (1.5, 4, 7 and10 %) and four binder content levels (3.9, 4.55, 5.2, and 5.9 %). Inaddition to measured | E *| values, modulus values were also predictedusing Witczaks equation. In Figure 12, | E *| values predicted using the

Hirsch model are compared to measured values reported by Witczak andhis team. The standard error in this case is 41 %, which is slightly betterthan the standard error for Witczaks model (45 %), but about double


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that for the experimental error for these data (20 %). Although not asgood as actual measurements, the accuracy of the model predictions is

probably suitable for many practical design and analysis applications.Estimated modulus values in fact are probably similar in reliability tomeasured values when only limited replicate tests can be performed, orwhen laboratory personnel are not experienced in making modulusmeasurements on asphalt concrete specimens.

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1000

10000

100000

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Reduced Frequency at 40 C, Hz

| E * |

, M P a

0

10

20

30

40

50

60

70

80

90

P h a s e

A n g

l e ,

d e g r e e s

Meas. |E*|Pred. |E*|Meas. PhasePred. Phase

Figure 11. Master Curve for Mixture V0W1 as Reported by Alaviand Monismith, and as Predicted Using the Hirsch Model.

A second comparison with these data is shown in Figure 13, which isa plot of modulus values predicted using the Hirsch model against values

determined using Witczaks equation. In this case, the agreement isextremely close. However, such close agreement between these modelswill not always occur; in this case, it is probably fortuitous and broughtabout by close agreement between the SHRP binder data used in theHirsch model and the routine data used in Witczaks equation. Given theclose agreement between these models, and the similar standard errorscompared to the measured |E*| values, this confirms the finding that theHirsch model and Witczaks equation are of similar accuracy and inreasonably close agreement. As pointed out earlier, the Hirsch model issimpler and more rational, and makes use of |G*| data as produced usingstandard Superpave binder tests..


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10

100

1000

10000

10 100 1000 10000

Measured |E*|, ksi

P r e

d i c t e d | E * | ,

k s

i

Figure 12. Predicted and Measured Dynamic Modulus Values UsingData from NCHRP 9-19 Sensitivity Study; Solid Line RepresentsEquality (Witczak et al., 2001).

10

100

1000

10000

10 100 1000 10000

Predicted |E*| (ksi), Witczak Model

P r e

d i c t e d | E * | ( k

s i )

, H i r s c

h M o

d e

l

Figure 13. Dynamic Modulus Values Predicted Using the HirschModel and Using Witczaks Equation, NCHRP 9-19 SensitivityData; Solid Line Represents Equality (Witczak et al., 2001).


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A final verification of the Hirsch model was done using bending beam data and mixture creep compliance data published as part of theSHRP project (Lytton et al., 1993). Mixture creep compliance valueswere determined on a range of field cores using the indirect tensile creeptests. Using reported binder creep modulus values as determined usingthe bending beam rheometer, and reported volumetric composition, theHirsch model was used to predict creep modulus values, which werecompared to values reported by Lytton and associates. The results of thiscomparison are shown in Figure 14. The Hirsch model appears to under-

predict the modulus values at high stiffness levels, but at this point themixture creep modulus is very high, in the range of 2 to 6 million lb/in 2 ,and approaching the glassy limit. At values below about 2 million lb/in 2,the predicted values are in reasonably good agreement with the measuredvalues. Comparison of Figure 14 with Figure 12 show that thesecomparisons are in agreement; the under-prediction seen with the Hirschmodel at very high modulus values is probably a function of theestimated constant glassy modulus value (4,200,000 lb/in 2). From a

practical perspective, once an asphalt concrete mixture is exhibitingmodulus values in this range, it will be extremely stiff and brittle, andwould be probably subject to high levels of thermal cracking and fatiguecracking if used in a pavement in this condition. This discrepancyshould therefore not be considered a significant problem.

100

1000

10000

100 1000 10000

Measured Creep Modulus, ksi

P r e

d i c t e d C r e e p

M o

d u

l u s ,

k s

i 5 seconds100 secondsequality

Figure 14. Predicted Creep Modulus Values Compared withMeasured Values, Using SHRP A-005 Low-Temperature Data; Solid

Line Represents Equality (Lytton et al., 1993).


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Comparison of |E*| and |G*| Values

A final important comparison can be made between the measuredmoduli in shear and compression for the entire data set used to developthe Hirsch model. Recall that the shear moduli were measured at AATusing the SST frequency sweep procedure, while the compression moduliwere determined entirely independently at Arizona State University,though the same materials were used. This allows a direct comparison ofindependently determined modulus values. A plot of | G*| versus | E *| forthese data is shown in Figure 15. Two important observations can bemade concerning this figure. First, the R 2 value for this relationship,92.9 %, is actually lower than the values for the predicted and measuredmoduli for both shear and compression data. This indicates that there isa large amount of noise in the modulus measurements. Because of therelatively poor precision of modulus determinations on asphalt concrete,it is possible that modulus predictions using the Hirsch model (or othersimilarly accurate procedure) might be nearly as accurate as independentmodulus measurements made on the same mixture. Thus for many

practical applications, using modulus values predicted using an accuratemodel is probably just as effective as using measured values, and ofcourse much less time consuming and expensive.

y = 0.0603x 1.0887

R2 = 0.9291

1000

10000

100000

1000000

10000 100000 1000000 10000000

Measured |E*|, psi

M e a s u r e

d | G * |

, p

|G*| = |E*|/2.8

Figure 15. Comparison of Measured Modulus in Shear (|G*|) andCompression (|E*|), for ALF, MNRoad, and WesTrack Data.


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A second important observation concerning Figure 15 is therelationship between | G*| and | E *|. Paving engineers might be temptedto try to convert compression moduli to shear moduli using the simplerelationship G = E /(1+ ), where is Poissons ratio. If a value of 0.4 isassumed for , this suggests that | G*| | E *|/2.8. The line representingthis equation is included on Figure 15, and clearly shows that thisapproximation is not at all accurate. Although the SST is certainly a farfrom ideal test system, the agreement between SST-based predictionsand Alavi and Monismiths data (Figure 5) suggests that the relationshipshown in Figure 15 is for the most part real. The inaccuracy of simpleconversions between shear and compression moduli is probably due toseveral factors, including non-linearity. However, non-linearity isgenerally not significant at low temperatures and high frequencies, but asseen in Figure 15, the simple conversion is not accurate even at highmodulus values. This discrepancy is probably caused in large part byanisotropy in the mechanical behavior of asphalt concrete. Otherresearchers have found similar discrepancies between HMAC modulusvalues made using different loading geometries (Taybali et al., 1994)Therefore, engineers should use caution in selecting modulus test data or

predicted modulus values for use in pavement design and analysis. Forexample, Bonnaures fatigue equations were developed on the basis offlexural modulus data; for best accuracy, the modulus values used inconjunction with his fatigue equation should therefore be values based onflexural measurements, or if these are not available, extensional data.Engineers should not rely on conversions between shear, compression,and flexural data based upon linear elastic theory and assumptions ofhomogeneity and anisotropy. Although not as elegant, empiricallydetermined relationships such as that illustrated here and reported byother researchers are likely to be more reliable.

Conclusions

A relatively simple version of the Hirsch model for composite behavior has been developed for estimating the complex modulusand phase angle of asphalt concrete in shear and compression.

The Hirsch model is both simpler and more rational than existingmodels for predicting modulus, and requires as input only asphaltconcrete volumetric composition and SHRP binder data, and so isideal for use in examining the relationships among HMACvolumetrics, modulus, and related aspects of pavement performance


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The model appears to be in good agreement with Andrei andWitczaks model, and is of similar accuracy.

Modulus values predicted using the Hirsch model are potentiallynearly as accurate as measured modulus values. For many pavementdesign and analysis procedures, predicted modulus values can beeffectively used, and can be determined much more quickly andcheaply. Using predicted modulus values should be consideredwhen reliable measurements by experienced lab personnel are notavailable.

Relationship among modulus values determined using different testmethods are complex and cannot be accurately predicted usingsimple linear elastic theory and assumptions of homogeneity andanisotropy. Engineers should be careful to select the appropriatemodulus value for their intended purpose, whether that value ismeasured or predicted using one of the available models.

References

Alavi, S. H., and C. L. Monismith, Time and Temperature DependentProperties of Asphalt Concrete Mixes Tested as Hollow Cylinders andSubjected to Dynamic Axial and Shear Loads, Journal of the

Association of Asphalt Paving Technologists , Vol. 63, 1994, pp. 152-175.

Andrei, D., Witczak, M.W., and Mirza, W., Development of a Revised Predictive Model for the Dynamic (Complex) Modulus of Asphalt Mixtures , NCHRP 1-37A Inter Team Technical Report, University ofMaryland, March 1999.

Bonnaure, F. P., A. H. J. J. Huibers, and A. Boonders, A LaboratoryInvestigation of the Influence of Rest Periods on the FatigueCharacteristics of Bituminous Mixes , Proceedings, the Association of

Asphalt Paving Technologists , Vol. 51, 1980, p. 104.

Bonnaure, F., G. Gest, G. Gravois, and P. Uge, A New Method ofPredicting the Stiffness of Asphalt Paving Mixtures, Proceedings, the

Association of Asphalt Paving Technologists , Vol. 46, 1977, p. 64.

Chapra, S. C., and R. P. Canale, Numerical Methods for Engineers, NewYork: McGraw-Hill, Inc., 1988, 812 pp.


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Christensen, D. W., Analysis of Creep Data from Indirect Tension Teston Asphalt Concrete, Journal of the Association of Asphalt PavingTechnologists , Vol. 67, 1998, pp. 458-489.

Christensen, D. W., NCHRP Project 9-25: Requirements for Voids in Mineral Aggregate for Superpave Mixtures, Interim Report to the National Cooperative Highway Research Program , Advanced AsphaltTechnologies, LLC, September 2001.

Christensen, D. W., and D. A. Anderson, Interpretation of DynamicMechanical Test Data for Paving Grade Asphalt, Journal of the

Association of Asphalt Paving Technologists , Vol. 61, 1992, pp. 67-98

Hirsch, T. J., Proceedings of the American Concrete Institute, Vol. 59,1962, p. 427.

Lytton, R. L., J. Uzan, E. G. Fernando, R. Roque, D. Hiltunen, and S. M.Stoffels, Development and Validation of Performance Prediction Modelsand Specifications for Asphalt Binders and Paving Mixes , Report SHRP-A-357, Washington, D.C.: Strategic Highway Research Program, 1993.

Nichols, R., Composite Construction Materials Handbook, EnglewoodCliffs, N. J.: Prentice-Hall, Inc., 1976, pp. 259-262.

Pellinen, T. K., Investigation of the Use of Dynamic Modulus as anIndicator of Hot-Mix Asphalt Performance, A Dissertation Presented inPartial Fulfillment of the Requirements for the Degree Doctor ofPhilosophy, Arizona State University, May 2001, 803 pp.

Shashidhar, N., S. P. Needham, B. H. Chollar, and P. Romero,Prediction of the Performance of Mineral Fillers in SMA, Journal ofthe Association of Asphalt Paving Technologists , Vol. 68, 1999, p. 222.

Tayebali, A. A., J. A. Deacon, and C. L. Monismith, Development andEvaluation of Surrogate Fatigue Models for SHRP A-003A AbridgedMix Design Procedure, Journal of the Association of Asphalt PavingTechnologists , Vol. 64, 1995, pp. 340-364.

Witczak, M. W., M. Bari, and M. M. Quayum, Sensitivity of Simple Performance Test Dynamic Modulus |E*|, NCHRP 9-19 Subtask C4b Report , Tempe, AZ: Arizona State University, Department of Civil and

Environmental Engineering, December 2001.

Hirsh Model Asphalt a Apt 2003

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