04-christensen1-hirsch model for estimating the modulus of asphalt-dm

8/13/2019 04-Christensen1-Hirsch Model for Estimating the Modulus of Asphalt-DM

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Hirsch Model for Estimating the Modulus of Asphalt

Concrete

Donald W. Christensen1, Jr., Terhi Pellinen

2, and Ramon F.

Bonaquist

3

Abstract

The purpose of this paper is to present a new, rational andeffective model for estimating the modulus of asphalt concrete

using binder modulus and volumetric composition. The model is

based upon an existing version of the law of mixtures, called theHirsch model, which combines series and parallel elements of

phases. In applying the Hirsch model to asphalt concrete, the

relative proportion of material in parallel arrangement, called the

contact volume, is not constant but varies with time andtemperature. Several versions of the Hirsch model were evaluated,

included ones using mastic as the binder, and one in which the

effect of film thickness on asphalt binder modulus wasincorporated into the equation. The most effective model was the

simplest, in which the modulus of the asphalt concrete is directly

estimated from binder modulus, VMA, and VFA. Models arepresented for both dynamic complex shear modulus (|G*|), and

dynamic complex extensional modulus (|E*|). Semi-empirical

equations are also presented for estimating phase angle in shear

loading and in extensional loading. The proposed model was

verified by comparing predicted modulus and phase angles tovalues reported in the literature for a range of mixtures.

Key Words: Superpave, asphalt concrete, dynamic modulus,

shear modulus, models, law of mixtures.

Introduction

The purpose of this paper is to describe in detail the

development and use of a new model for predicting the modulus of

asphalt concrete, which is based upon an existing model for

1Senior Engineer and 3Chief Operating Officer, Advanced Asphalt Technologies LLC, Sterling VA2Assistant Professor, Purdue University, West Lafayette IN

The oral presentation was made by Dr. Christensen


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literature. This paper is based upon appendices to NCHRP Project

9-25 and Project 9-31 Interim Reports, but has been edited for

publication as a research paper (2, 3).

Background

The proposed method for estimating HMAC modulus is based

upon a widely used model for predicting properties of composite

materialsthe law of mixtures. As generally formulated, the lawof mixtures represents the mechanical response of two separate

phases in parallel (4):

Ec= v1E1+ v2E2 (1)

WhereErefers to the modulus, or some other material property, v

refers to the volume fraction of a given phase, the subscript crefersto the composite, and the subscripts 1 and 2 refer to different

phases present in the composite. A less common form of the lawof mixtures can be derived for phases in series (4):

1/Ec= v1/E1+ v2/E2 (2)

Where the variables are as defined above. In the early 1960s, T. J.

Hirsch developed a variation of the law of mixtures for modelingthe mechanical behavior of asphalt concrete (5). The Hirsch model

combines parallel and series arrangement of the phases (4):

1/Ec= v1s/E1+ v2s/E2+ (v1p+v2p)2/(v1pE1+v2pE2) (3)

Where v1sand v2srefer to the volume fractions of phases 1 and 2,

respectively, in series arrangement, and v1pand v2prefer to the

volume fractions of phases 1 and 2, respectively, in parallel

arrangement. An equivalent expression for the Hirsch model canbe formulated if it is assumed that the relative proportions of phase

1 and phase 2 are the same in the series and parallel portion of the

model:

( )

+

+

+=

22112

2

1

1 111

EvEvx

E

v

E

vx

Ec (4)


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Wherexis the ratio of phases in parallel arrangement to the total

volume. Whenx= 1, the Hirsch model produces results identical

to completely parallel phases (Equation 1), whereas whenx= 0, itrepresents a pure series arrangement (Equation 2). This model is

therefore quite flexible, and can be used to represent a wide rangeof composite behavior. Hirsch found that for several portlandcement concrete mixtures,xhad a value of about 0.5. These three

composite models are shown in Figure 1.

V1 V2

V1

V2

V

V

VV

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

the research team for predicting asphalt concrete modulus from

binder modulus and volumetric properties. Asphalt concrete tendsto behave like a series composite at high temperature, but more

like a parallel composite at low temperature, and so the Hirschmodel should be appropriate for estimating the modulus of asphalt

concrete. However, for this model to be useful in modeling the

modulus of asphalt concrete, the relative proportions of the seriesand parallel phases must be time and temperature dependent. The

aggregate phase in the parallel portion of the model is important in

characterizing the behavior and performance of HMAC, as itrepresents that portion of the aggregate particles in intimate contact

with each other; this portion of the aggregate is therefore called the

aggregate contact volume,Pc. In general, as the aggregate contactvolume increases, so will the modulus, strength, and resistance to

permanent deformation. High values ofPcindicate a very


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effective structure producing good strengths and stiffness, typical

at low temperature. Low values ofPctend to occur at high

temperatures, and indicate mixtures with low strength andstiffness.

Various modifications of the Hirsch model were devised andevaluated in developing the final version. Four of the mostimportant are shown in Figure 2. In many of the early

formulations of the model, it was assumed that the binder in

asphalt concrete was the mastic (mineral filler plus binder) ratherthan binder alone. In Figure 2(a), the bulk of the aggregate is in a

series arrangement with a parallel combination of the aggregate

contact volume and mastic. Better results were obtained with thearrangement shown in 2(b), where the aggregate contact volume is

combined in parallel with a series arrangement of the bulk

aggregate and mastic. In this figure, Vcrefers to the aggregate

contact volume, Va refers to the aggregate volume exclusive ofthe contact volume, Vmis the mastic volume, and Vvis the air void

volume. For the configuration shown in 2(a), the appropriate

equation for dynamic modulus would be as follows:

( )1

2'1'

+

+=

VmEmPcEa

Va

Ea

VaEc (5)

Where:

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

volume and mineral filler

Ea = aggregate modulusPc = aggregate contact volume, as volume fraction

Vm = volume fraction of masticEm = mastic modulus

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

VaVcVcEaEc (6)


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Va

VmVc Vv

Va

VmVc

Vv

(a) Series Version (b) Parallel Version

VcVv Va,

VmVap VvpVmp

Vas

Vms Vvs

(c) Dispersed Version (d) Alternate Version

Figure 2. Schematic representation of four alternate versions

of modified Hirsch model.

Where the variables are as defined previously. The arrangement in

2(b) can in fact be generalized by using an exponent mrather than

1 in the series portion of the model:

( ) ( ) mmmm VmEmEaVaVcVcEaEc 111 '1 ++= (7)The exponent mcan take any value from 1 to 1, though in this

case meaningful models would result only for values between 1and about 0.2. For the case of perfect spheres of aggregates

within the mastic matrix, m= -0.5; this arrangement is shown in

Figure 2(c). Thus, Equation 7 represents a very flexible form ofthe Hirsch model.

In order to use the models represented in Figures 2(a), (b) and

(c), the modulus of the mastic must be estimated. The NCHRP 9-

25 research team has evaluated two approaches to calculating themodulus of the mastic based upon the binder modulus and the

volume fractions of binder and mineral filler. The first is a version


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of the Einstein equation developed by Shashidhar and his

associates (6)

( )Eb

VfCVf

AVfEm

+

+=

''11

'1 (8)

Where:A = KE-1

KE = generalized Einstein coefficient

Vf = volume fraction filler in mastic = Vf/Vm

C = (1-Vfmax)/Vfmax2

Vfmax = maximum volume fraction of filler in mastic

The Einstein equation is rational in form and appears to bereasonably accurate. However, the predictions are quite sensitive

to the maximum volume fraction of filler (Vfmax). As the volume

fraction of filler in the mastic approaches this maximum, the

predicted modulus of the mastic becomes extremely high; whenVf= Vfmax, the predicted value ofEmis infinite. In reality, the

binder would most likely never actually incorporate this amount of

filler, as this represents a hypothetical maximum, at which the airvoid content of the mastic would be exactly zero. Instead, a certain

amount of excess filler would occur throughout the mix as free

filler. However, this is difficult to account for within theframework of a relatively simple method for predicting mixture

modulus.

An alternative approach involves the use of the generalized lawof mixtures:

( ) nnn EbVbEaVfEm 1'' += (9)

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

Eb = modulus of binder

n = exponent with values from 1 to +1

For dispersed systems, nwill range from slightly less than 0 toabout 0.5. A preliminary evaluation based upon typical binder-

filler systems has indicated that the predictions of Equation 9 are

of the same magnitude as those of Equation 8 when n= -0.2.Equation 9, because of its relative simplicity and robustness, is

attractive for use in the various models. Results using this


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equation compared favorably with those produced using the

Einstein equation. Therefore much of the analyses of the variousforms of the Hirsch model therefore used Equation 9 for estimating

mastic modulus.

Preliminary analyses showed that these three versions of theHirsch model did not exhibit good accuracy. The version of themodel that was found to exhibit consistently good accuracy is

shown in Figure 2(d). This alternate formulation of the Hirsch

model is very similar to the original model. The only difference inthis version is that the series and parallel sub-units of the model are

combined in parallel, rather than in series. This in effect places

more emphasis on the parallel sub-unit of the model. This versionof the Hirsch model produced the best results, and has the

additional advantages of being relatively simple and very similar to

the original version of the model as formulated by Hirsch.

Mathematically, it can be expressed using the following equation:

( ) ( )

12

2 ''

'

++++

++=

VmsEm

VvsVms

Ea

sVaVvsVmssVa

VmpEmpEaVaEc

(10)

Where the variables are as defined previously, but the addition of

the subscriptspandsdenote parallel and series phases,

respectively (see Figure 2(d)). As with the standard version of the

Hirsch model, this alternate formulation can be stated in somewhatsimpler terms by using the contact volumePcto represent the

proportion of parallel to total phase volume:

( ) ( ) ( )

12

'1'

++++=

VmEm

VvVm

Ea

VaPcVmEmEaVaPcEc (11)

Various types of functions where used to describe the

contact factor for use in Equation 11. Eventually, the researchteam found that the following equation for the contact factor

provided the best results:


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1

1

'

'

2

0

P

P

VMA

EmVFMP

VMA

EmVFMP

Pc

+

+

= (12)

Where VFMis the fraction of aggregate voids filled with mastic,VMA is voids in the mineral aggregate, exclusive of mineral filler,

andP0,P1, andP2are empirically determined constants. Equation

12 is in fact a type of logistic function, which produces a sigmoidalresponse in log-log space typical of the behavior of many

viscoelastic materials. The first constant in Equation 12,P0, is

directly related to the contact factor at high temperatures and/or

low frequencies;P1is an exponent related to the rate of change ofthe contact factor with respect to the binder modulus Em;P2is

related to the location of the contact factor, which is directly

related to the overall stiffness of the asphalt concrete mixture.As will be discussed later in this paper, preliminary analysis of

the mastic version of the Hirsch model demonstrated that it was

accurate, but also seemed to indicate that it was not necessary toconsider the stiffening effects of mastic. Instead, a simpler version

of this model, which treats asphalt concrete as a simple three-phase

system of aggregate, asphalt binder, and air voids, seemed

appropriate:

( ) ( ) ( )

12

1

++++=

VbEb

VvVb

Ea

VaPcVbEbVaEaPcEc

(13)WhereEbrepresents the binder modulus, and Vbrepresents the

effective binder volume. Note that the aggregate volumes in

Equation 13 now represent true aggregate volume including the

volume of mineral filler. The corresponding equation for thecontact factor is identical to Equation 12, except that true VMAis

substituted for VMA (voids plus binder volume plus mineral filler

volume).A final modification of the Hirsch model was devised to

determine if film thickness could be incorporated into theexpression for modulus. The equation for this version of the

Hirsch model is essentially identical to Equation 13, but theeffective binder modulus,Eb, is substituted for the true bindermodulus:


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( ) ( ) ( )

12

'1'

++++=

VbEb

VvVb

Ea

VaPcVbEbVaEaPcEc

(14)

The effective binder modulus is calculated by assuming thatthe binder modulus at the aggregate surface is equivalent to the

glassy modulus, but then decreases to the normal binder modulusvalue over a certain characteristic distance, tT, representing the

transition zone between the aggregate and binder. If the binder

film thickness is called tF, the equation for effective bindermodulus is then given using the following equation:

( ) EbtEtt

EbEtEb

TgTF

gF

+=' (15)

WhereEgis the glassy modulus of asphalt cement binder, assumedto be 1 GPa (145,000psi).

The reader should keep in mind that the above functions for the

Hirsch 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 be determined from creep, stress relaxation, or dynamicmodulus tests. The Hirsch model is a rational, though semi-

empirical method of predicting asphalt concrete modulusthat is,

its structure is logical and based upon the law of mixtures, but itsuse in practice requires calibration with measured data. It should

not be confused with a constitutive equation, which rigorously

delineates the relationships among stress, strain, and material

properties (such as modulus) in two or three dimensions. TheHirsch model can however be used to estimate modulus values that

are used in various such constitutive equations.

Data used in Refining the Hirsch Model

In order to evaluate the various versions of the Hirsch model,and refine the most promising of these models it was necessary to

establish a database of modulus values for a wide range of

mixtures. Such a data set was created, based upon dynamic

modulus measurements made at Advanced Asphalt Technologies,

LLC, (AAT) and at the Arizona State University (ASU) as part ofNCHRP Project 9-19 (4). The AAT data set consists of dynamic


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shear modulus and phase angle data collected using the SST, using

the frequency sweep tests. The ASU data set consists of dynamiccompression modulus and phase angle data. The mixtures tested

were the same for both data sets, and originated from FHWAs

Accelerated Loading Facility (ALF) project, the MnROADProject, and the WesTrack Project; Pellinen has thoroughlysummarized these projects and the materials used in both data sets

(1). Inclusion of both shear and compression data allowed the

research team to develop versions of the Hirsch model for bothcases, which is essential since the relationship between |G*| and

|E*| for asphalt concrete mixtures is not at all straightforward nor

well documented.A summary of the database is given as Table 1. The database

includes results from testing on 18 mixtures using eight different

binders and 5 different aggregate sizes and gradations. A total of

206 observations are included in the dataset for each type ofmeasurement (shear and compression). A wide range of

volumetric compositions is also represented, although one

shortcoming in this data is the lack of mixtures with low air voidsand low VFA. The data set is however extensive, and suitable for

initial development of the Hirsch model. It is possible that larger

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 the Hirsch model, and refinement of the most promising wasnon-linear least squares (7). Christensen has presented a detailed

description of the use of this technique in analyzing IDT creep data

(8). In general terms, this procedure uses an iterative, numerical

procedure to calculate the values for parameters in a function sothat the sum of the square of the error terms is minimized.

Graphical techniques were used to identify and eliminate several

records that were clearly outliers, and to determine if otherpotential predictor variables existed which were not included in the

model. All of the outliers eliminated from the data were high-

temperature measurements made using the SST at very low stresslevels. These measurements were considered to be unreliable,

since the very low stress levels used were difficult to measure and

appeared to be quite noisy.


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Figure 2(d), and represented in its basic form in Equation 10. The

three versions of this model are called here the mastic version(Equation 11), the simple version (Equation 13), and the transition

zone version (Equations 14 and 15). In all cases, Equation 12 was

used to characterize the contact factorPc.Model for |G*|The results of the non-linear least squaresanalyses of these three models for dynamic complex shear

modulus (|G*| as measured using the SST) are summarized in

Table 2. The value for parameterP0could not be determinedreliably using the least-squares procedure, probably because the

database did not include enough values at high temperatures and

low frequencies. However, it is important to include at least anestimate for this parameter, since it represents the limiting asphalt

concrete modulus at high temperatures and/or low frequencies.

Comparing model predictions with published master curves for

asphalt concrete, it was estimated that an appropriate value for thisconstant is about 3.

All three versions of the Hirsch model exhibited identical r2

values of 96.8 percent, suggesting that the more complex versionsof the model, intended to account for mineral-filler effects and the

effect of film thickness, are no more effective than the model

treating asphalt concrete as a simple three-phase system. For themastic model, the law of mixtures exponent for the mastic, n

(Equation 9) has an unrealistic value of 6.9, and a standard error

of 281 percent, indicating that this parameter does not contributesignificantly to the accuracy of the model. For the transition zone

model, similar problems are observed in the estimate of thetransition zone thickness; tTis estimated at only 0.03 microns, and

has a huge standard error of estimate of almost 4,000 percent. It is

therefore concluded that the most effective version of the Hirsch

model is the relative simple version represented by Equation 13,which can be given in terms of the complex shear modulus, |G*|,

VFA, and VMA:


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Table 2. Summary of Least-Squares Estimation of Parameters

for |G*| Hirsch Model.

Mastic Model

Equation(s

):

11 & 12

Parameter Estimat

e

Std.

Error

Ea, lb/in2 635,000 7.2 %

P0 3 N/A

P1 0.678 2.4 %

P2 396 10.4 %

n(Eqn. 9) 6.9 281%

tT, microns N/A N/A

r2 96.8 % N/A

Simple Model

Equation(s

):13 & 12

Parameter Estimat

e

Std.

Error

Ea, lb/in2 601,000 7.1 %

P0 3 N/A

P1 0.678 2.4 %

P2 396 6.9 %

n(Eqn. 9) N/A N/A

tT, microns N/A N/A

r2 96.8 % N/A

Transition Zone Model

Equation(s

):14, 15 & 12

Parameter Estimate Std.

Error

Ea, lb/in2 601,000 7.2 %

P0 3 N/A

P1 0.678 2.4 %

P2 452 8.7 %

n(Eqn. 9) N/A N/A

tT, microns 0.03 3,760 %

r2 96.8 % N/A


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( )

( )

1

*000,601

1001

1

000,10*1001000,601*

+

+

+=

binder

bindermix

GVFA

VMAVMA

Pc

VMAVFAGVMAPcG

(16)

Where |G*|mixis the complex shear modulus for the mixture, and

|G*|binderis the complex shear modulus for the binder, both in unitsof lb/in

2, and VFA and VMA are both given as percentages. The

binder modulus can either be determined experimentally using the

dynamic shear rheometer (DSR) or similar device, or can beestimated from one of several available mathematical models. It

should be at the same temperature and loading time selected for the

mixture modulus, and in consistent units. The contact factor isgiven by the following function:

678.0

678.0

*396

*3

+

+

=

VMA

GVFA

VMA

GVFA

Pc

binder

binder

(17)

Figure 3 shows the |G*| values predicted using Equations 16

and 17 versus measured values. There is generally good

agreement.

Model for |E*|Non-linear least squares analysis was alsoperformed using the |E*| data gathered by Pellinen at Arizona State

University as part of NCHRP Project 9-19 (1). In the initial

calibration of the model for |E*|, performed during Phase I ofNCHRP Project 9-25, the data set used coincided with that

developed for the |G*| model. However, subsequent evaluation 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

included additional data at 9 and 54C. This resulted in asomewhat greater range of values for both |E*| and phase angle in

the data set. Furthermore, in the expanded data set replicatemeasurements were averaged, partly because of the greatlyexpanded size of the resulting data set, but also to minimize


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1000

10000

100000

1000000

1000 10000 100000 1000000

Measured |G*|, psi

Predicted|G*|,psi

Figure 3. Predicted Versus Measured Shear Complex

Modulus (r2= 96.8 percent); Solid Line Represents Equality.

variability due to experimental error. It was felt that this approachwould provide a better estimate of the accuracy of the model. The

final values for the Hirsch model parameters for dynamic complex

modulus in extension were estimated to be as follows:

Ea: 4,230,000 psi (6.5 %) P0: 649 (9.0 %) P1: 19.7 (20.7 %) P2: .575 (3.0 %)

The r2value for the |E*| data (98.2 %) is slightly higher thanfor shear data. Also, note the much higher value forEa, which is

expected as compression moduli for most materials, includingasphalt concrete, are almost always much higher than shear

moduli. The other parameters are similar to the values determined

for the shear model. The predicted and measured values for |E*|

are shown in Figure 4; residuals as a function of predicted |E*| areshown in Figure 5. Applying appropriate rounding to the

coefficients listed above, the pertinent equation for compression

modulus is as follows:


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( )

( )

1

*3000,200,4

10011

000,10*31001000,200,4*

+

+

+=

binder

bindermix

GVFA

VMAVMAPc

VMAVFAGVMAPcE

(18)

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

Predicted|E*|,psi

Figure 4. Predicted and Measured values for Complex

Modulus in Compression (r2= 98.2 percent); Solid Line

Represents Equality.

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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-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

Residual(Log|

E*|)

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

for Hirsch Model.

58.0

58.0

*3650

*320

+

+

=

VMA

GVFA

VMA

GVFA

Pc

binder

binder

(19)

Prediction of Phase Angle As discussed previously, in addition

to a need for estimating modulus in shear and compression, there is

also a need to estimate the phase angle from compositional data.

For example, the fatigue models developed during the StrategicHighway Research Program (SHRP) used the loss modulus as a

predictor variable, which is a function of both the complexmodulus and the phase angle (9). The research team found a good

empirical relationship between the log of the contact factorPc

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

relationships, which includes the empirically determined equationsfor phase angle as a function of log (Pc), are shown in Figures 6

and 7. The following function can be used to estimate phase angle

usingPcas determined from shear data (r2= 82.9 %):

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


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y = -9.4633x2- 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)

PhaseAngle,

Degrees

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

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)

Pha

seAngle,Degrees

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

Data.

The corresponding function for estimating from compression

data is given by the following Equation (r2

= 89 percent):


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( ) PcPc log55log21 2 = (21)

These relationships, though not highly accurate, are useful for

estimating phase angle from asphalt concrete volumetric

composition. It should be kept in mind that the measurement ofphase angle is particularly difficult, so the amount of scatter in

Figures 6 and 7 is not surprising. There are two significantdifferences in these plots. The phase angle in shear (Figure 6) has

a value of 9.6 whenPcis zero; this indicates that in shear

measurements, the phase angle is about 10 degrees even at verylow temperatures and high frequencies. For the extensional data,

this is not truethe phase angle is zero whenPcis zero. It ispossible that the non-zero phase angle whenPcis zero for shear

data is an artifact, caused by friction in the horizontal bearing usedin the SST system (there are no bearings in the compression tests,

outside of those inherent in the actuators). The other significantdifference is that for the compression data, there is a clear

maximum in phase angle, whereas the shear data shows nomaximum. In general, it appears that the phase angle data in

extension are somewhat more reasonable than that determined in

shear.

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 available

during the initial phases of NCHRP Projects 9-25 and 9-31, the

models for shear modulus (Equations 16, 17 and 20) were used toestimate shear modulus for data as reported by Alavi andMonismith in research related to the original SHRP program (10).

These measurements were made using a cylindrical shear test, a

completely different technique than that upon which the model wasdeveloped. Alavi and Monismith used one aggregate (19-mm

nominal maximum size) and one binder, but a range of asphalt and

air void contents. Binder modulus values were estimated using themodel developed by Christensen and Anderson for SHRP binders

(11). Some volumetric information was estimated because of the

limited data reported by Alavi and Monismith. The predicted and

measured values of shear modulus are shown in Figure 8; theagreement is good, though the Hirsch model appears to slightly

under-predict modulus values at higher levels. In Figure 9,


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predicted and measured phase angles are plotted for these same

data. The agreement here is not as good, especially at higher phaseangles. In Alavi and Monismiths data, 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 a decreasing rate to amaximum value without any subsequent decrease. The researchteam believes this is an inherent difference in the measurement

methods. Phase angle predictions using the Hirsch model are

probably most accurate at low temperatures and/or highfrequencies.

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

compression (|E*|) was predicted using both the Hirsch model andusing Andrei and Witczaks equation, and compared to the

measured values reported by Alavi and Monismith (12, ,10). This

comparison is shown graphically in Figure 10. The values for the

Hirsch model are in excellent agreement, whereas Andrei andWitczaks equation slightly under-predicts at higher modulus

values. Although this comparison is too limited to make broad

generalizations, it suggests that the Hirsch model is in generalagreement with Andrei and Witczaks equation, and is at least as

accurate. Hirsch model predictions for |E*| are probably in better

agreement with Alavi and Monismiths data, compared to the |G*|predictions, because the experimental technique used by Alavi and

Monismith for the shear measurements was substantially different

than that used by AAT in making the SST measurements. Incontrast, Pellinens compression moduli were probably determined

using methods giving results comparable to Alavi andMonismiths.

Another comparison useful for verification of the Hirsch model

is shown in Figure 11, which is a master curve for mixture V0W1from Alavi and Monismiths study (10). This figure shows

predicted and measured values for |E*| and phase angle as a

function of reduced frequency at 40C. This figure confirms that

the frequency dependence and general shape of the functions forcomplex modulus and phase angle as predicted by the Hirsch

model are reasonable and in good agreement with experimental

values. However, it appears that the phase angle predictions do notvary quite as strongly with frequency as the measured values.


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10

100

1000

10000

100000

10 100 1000 10000

Predicted |G*|, MPa

Measured|G*|,MPa

Figure 8. Measured Complex Shear Modulus and ValuesPredicted Using the Hirsch Model; Solid Line Represents

Equality (data from Refereence 10).

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Predicted Phase Angle, Degrees

Measur

edPhaseAngle,

Degrees

Figure 9. Measured Phase Angle s and Values Predicted Using

the Hirsch Model; Solid Line Represents Equality (data from

Reference 10).


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100

1000

10000

100000

100 1000 10000 100000

Measured |E*|, MPa

Predicted|E*|,

MP

a

Witczak Model

Hirsch Model

Equality

Figure 10. Predicted and Measured |E*| Values; Solid LineRepresents Equality (data from Reference 10).

A more thorough verification of the Hirsch model wasperformed using data from a recent sensitivity study by Witczak

and his associates (13). In this study, |E*| measurements were

made on a range of mix variations based upon an ArizonaDepartment of Transportation mixture. This asphalt concrete used

a 25-mm nominal maximum aggregate size blend with an PG 64-

22 asphalt. The basic mix design was varied using four different

target air void levels (1.5, 4, 7 and 10 percent) and four binder

content levels (3.9, 4.55, 5.2, and 5.9 percent). In addition tomeasured |E*| values, modulus values were also predicted using

Witczaks equation. In Figure 12, |E*| values predicted using theHirsch model are compared to measured values reported by

Witczak and his team. The standard error in this case is 41 %,

which is slightly better than the standard error for Witczaks model(45 percent), but about double that for the experimental error for

these data (20 percent). Although not as good 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

to measured values when only limited replicate tests can beperformed, or when laboratory personnel are not experienced inmaking modulus measurements on asphalt concrete specimens.


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10

100

1000

10000

10 100 1000 10000

Measured |E*|, ksi

Predicted|E*|,

ksi

Figure 12. Predicted and Measured Dynamic Modulus Values

Using Data from NCHRP 9-19 Sensitivity Study; Solid Line

Represents Equality (13).

10

100

1000

10000

10 100 1000 10000

Predicted |E*| (ksi), Witczak Model

Predicted|E*|(ksi),HirschModel

Figure 13. Dynamic Modulus Values Predicted Using the

Hirsch Model and Using Witczaks Equation, NCHRP 9-19

Sensitivity Data; Solid Line Represents Equality (13).

A final verification of the Hirsch model was done usingbending beam data and mixture creep compliance data published

as part of the SHRP project (14). Mixture creep compliance values


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were determined on a range of field cores using the indirect tensile

creep tests. Using reported binder creep modulus values asdetermined using the bending beam rheometer, and reported

volumetric composition, the Hirsch model was used to predict

creep modulus values, which were compared to values reported byLytton and associates. The results of this comparison are shown inFigure 14. The Hirsch model appears to under-predict the modulus

values at high stiffness levels, but at this point the mixture creep

modulus is very high, in the range of 2 to 6 millionpsi, andapproaching the glassy limit. At values below about 2 millionpsi,

the predicted values are in reasonably good agreement with the

measured values. Comparison of Figure 14 with Figure 12 showthat these comparisons are in agreement; the under-prediction seen

with the Hirsch model at very high modulus values is probably a

function of the estimated constant glassy modulus value

(4,200,000psi). From a practical perspective, once an asphaltconcrete mixture is exhibiting modulus values in this range, it will

be extremely stiff and brittle, and would be probably subject to

high levels of thermal cracking and fatigue cracking if used in apavement in this condition. This discrepancy should therefore not

be considered a significant problem.

100

1000

10000

100 1000 10000

Measured Creep Modulus, ksi

PredictedCreepModulus,

ksi 5 seconds

100 seconds

equality

Figure 14. Predicted Creep Modulus Values Compared with

Measured Values, Using SHRP A-005 Low-Temperature Data;Solid Line Represents Equality (14).


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

A final important comparison can be made between themeasured moduli in shear and compression for the entire data set

used to develop the Hirsch model. Recall that the shear moduli

were measured at AAT using the SST frequency sweep procedure,while the compression moduli were determined entirelyindependently at Arizona State University, though the same

materials were used. This allows a direct comparison of

independently determined modulus values. A plot of |G*| versus|E*| for these data is shown in Figure 15. Two important

observations can be made concerning this figure. First, the R2

value for this relationship, 92.9 %, is actually lower than the valuesfor the predicted and measured moduli for both shear and

compression data. This indicates that there is a large amount of

noise in the modulus measurements. Because of the relatively

poor precision of modulus determinations on asphalt concrete, it ispossible that modulus predictions using the Hirsch model (or other

similarly accurate procedure) might be nearly as accurate as

independent modulus measurements made on the same mixture.Thus for many practical applications, using modulus values

predicted using an accurate model is probably just as effective as

using measured values, and of course much less time consumingand expensive.

A second important observation concerning Figure 15 is the

relationship between |G*| and |E*|. Paving engineers might betempted to try to convert compression moduli to shear moduli

using the simple relationship G=E/(1+), where is Poissonsratio. If a value of 0.4 is assumed for , this suggests that |G*| |E*|/2.8. The line representing this equation is included on Figure

15, and clearly shows that this approximation is not at all accurate.

Although the SST is certainly a far from ideal test system, theagreement between SST-based predictions and Alavi and

Monismiths data (Figure 5) suggests that the relationship shown

in Figure 15 is for the most part real. The inaccuracy of simple

conversions between shear and compression moduli is probablydue to several factors, including non-linearity. However, non-

linearity is generally not significant at low temperatures and high

frequencies, but as seen in Figure 15, the simple conversion is notaccurate even at high modulus values. This discrepancy is

probably caused in large part by anisotropy in the mechanical


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y = 0.0603x1.0887

R2= 0.9291

1000

10000

100000

1000000

10000 100000 1000000 10000000

Measured |E*|, psi

Measured|G*|,psi

|G*| = | E*|/2.8

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

Data.

behavior of asphalt concrete. Other researchers have found similardiscrepancies between HMAC modulus values made using

different loading geometries (9) Therefore, engineers should use

caution in selecting modulus test data or predicted modulus valuesfor use in pavement design and analysis. For example, Bonnaures

fatigue equations were developed on the basis of flexural modulus

data (15, 16); for best accuracy, the modulus values used in

conjunction with his fatigue equation should therefore be valuesbased on flexural measurements, or if these are not available,

extensional data. Engineers should not rely on conversions

between shear, compression, and flexural data based upon linearelastic theory and assumptions of homogeneity and anisotropy.

Although not as elegant, empirically determined relationships such

as that illustrated here and reported by other researchers are likelyto be more reliable.

Conclusions

A relatively simple version of the Hirsch model forcomposite behavior has been developed for estimating the


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complex modulus and phase angle of asphalt concrete in

shear and compression.

The Hirsch model is both simpler and more rational thanexisting models for predicting modulus, and requires as

input only asphalt concrete volumetric composition andSHRP binder data, and so is ideal for use in examining therelationships among HMAC volumetrics, modulus, and

related aspects of pavement performance

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

Modulus values predicted using the Hirsch model arepotentially nearly as accurate as measured modulus values.For many pavement design and analysis procedures,

predicted modulus values can be effectively used, and can

be determined much more quickly and cheaply. Using

predicted modulus values should be considered whenreliable measurements by experienced lab personnel are not

available.

The relationship among modulus values determined usingdifferent test methods are complex and cannot be

accurately predicted using simple linear elastic theory and

assumptions of homogeneity and anisotropy. Engineers

should be careful to select the appropriate modulus valuefor their intended purpose, whether that value is measured

or predicted using one of the available models.

References

1. T. K. Pellinen,Investigation of the Use of Dynamic Modulus asan Indicator of Hot-Mix Asphalt Performance, A Dissertation

Presented in Partial Fulfillment of the Requirements for the

Degree Doctor of Philosophy, Arizona State University, May

2001, 803 pp.2. D. W. Christensen,NCHRP Project 9-25: Requirements for

Voids in Mineral Aggregate for Superpave Mixtures, Interim

Report to the National Cooperative Highway ResearchProgram, Advanced Asphalt Technologies, LLC, September

2001.3. D. W. Christensen,NCHRP Project 9-31: Air Void

Requirements for Superpave Mix Design, Interim Report to the


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National Cooperative Highway Research Program, Advanced

Asphalt Technologies, LLC, April 2002.4. R. Nichols, Composite Construction Materials Handbook,

Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1976, pp. 259-262.

5. T. J. Hirsch,Proceedings of the American Concrete Institute,Vol. 59, 1962, p. 427.6. N. Shashidhar, S. P. Needham, B. H. Chollar, and P. Romero,

Prediction of the Performance of Mineral Fillers in SMA,

Journal of the Association of Asphalt Paving Technologists,Vol. 68, 1999, p. 222.

7. S. C. Chapra and R. P. Canale, Numerical Methods for

Engineers, New York: McGraw-Hill, Inc., 1988, 812 pp.8. D. W. Christensen, Analysis of Creep Data from Indirect

Tension Test on Asphalt Concrete,Journal of the Association

of Asphalt Paving Technologists, Vol. 67, 1998, pp. 458-489.

9. A. A. Tayebali, J. A. Deacon, and C. L. Monismith,Development and Evaluation of Surrogate Fatigue Models for

SHRP A-003A Abridged Mix Design Procedure,Journal of

the Association of Asphalt Paving Technologists, Vol. 64,1995, pp. 340-364.

10. S. H. Alavi and C. L. Monismith, Time and Temperature

Dependent Properties of Asphalt Concrete Mixes Tested asHollow Cylinders and Subjected to Dynamic Axial and Shear

Loads,Journal of the Association of Asphalt Paving

Technologists, Vol. 63, 1994, pp. 152-175.11. D. W. Christensen and D. A. Anderson, Interpretation of

Dynamic Mechanical Test Data for Paving Grade Asphalt,Journal of the Association of Asphalt Paving Technologists,

Vol. 61, 1992, pp. 67-98

12. D. Andrei, Witczak, M.W., and Mirza, W., Development of a

Revised Predictive Model for the Dynamic (Complex) Modulusof Asphalt Mixtures, NCHRP 1-37A Inter Team Technical

Report, University of Maryland, March 1999.

13. M. W. Witczak, M. Bari, and M. M. Quayum, Sensitivity of

Simple Performance Test Dynamic Modulus |E*|, NCHRP 9-19Subtask C4b Report, Tempe, AZ: Arizona State University,

Department of Civil and Environmental Engineering,December 2001.

14. R. L. Lytton, J. Uzan, E. G. Fernando, R. Roque, D. Hiltunen,

and S. M. Stoffels,Development and Validation of


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Performance Prediction Models and Specifications for AsphaltBinders and Paving Mixes, Report SHRP-A-357, Washington,D.C.: Strategic Highway Research Program, 1993.

15. F. P. Bonnaure, A. H. J. J. Huibers, and A. Boonders, A

Laboratory Investigation of the Influence of Rest Periods onthe Fatigue Characteristics of Bituminous Mixes,Proceedings, the Association of Asphalt Paving Technologists,

Vol. 51, 1980, p. 104.

16. F. Bonnaure, G. Gest, G. Gravois, and P. Uge, A New Methodof Predicting the Stiffness of Asphalt Paving Mixtures,

Proceedings, the Association of Asphalt Paving Technologists,

Vol. 46, 1977, p. 64.

04-christensen1-hirsch model for estimating the modulus of asphalt-dm

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