Comparison of conventional, polymer, and rubber asphalt mixtures using viscoelastic continuum damage...

For Peer Review O

nly

Comparison of Conventional, Polymer, and Rubber Asphalt Mixtures Using Viscoelastic Continuum Damage Model

Journal: Road Materials and Pavement Design

Manuscript ID: RMPD-13-06-04.R2

Manuscript Type: Original Scientific Paper

Keywords: Fatigue, rubber-modified, polymer-modified, dynamic modulus, viscoelastic

continuum damage

URL: http://mc.manuscriptcentral.com/rmpd

Road Materials and Pavement Design

For Peer Review O

nly

Comparison of Conventional, Polymer, and Rubber Asphalt Mixtures

Using Viscoelastic Continuum Damage Model

Abstract. In this study, a laboratory experimental program was conducted to

compare the material properties and fatigue performance characteristics for

reference, polymer-modified and rubber-modified gap graded mixtures. These

mixtures were placed on E18 highway between the interchanges Järva Krog and

Bergshamra in the Stockholm area of Sweden. The advanced material

characterization tests included: dynamic (complex) modulus for stiffness

evaluation and the uniaxial tension-compression for fatigue assessment. The

data was used to compare the performance of the rubber-modified gap graded

mixture to the reference and the polymer-modified gap mixtures using the

viscoelastic continuum damage (VECD) approach. Different researchers have

successfully applied the VECD model to describe the fatigue behavior of asphalt

concrete mixtures. The damage characteristic (C-S) curves were established for

each of the three mixtures. The fatigue behavior for the three mixtures was

ranked based on the C-S curve results and the rubber-modified mixture showed

the best fatigue damage resistance followed by the polymer-modified mixture

and the reference mixture. The VECD approach provides a more comprehensive

analysis to evaluate fatigue resistance compared to tradition fatigue evaluation

using a number of cycles at a given stiffness reduction.

KEYWORDS: Fatigue, rubber-modified, polymer-modified, dynamic modulus,

viscoelastic continuum damage

Page 1 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

2

1. Introduction

Load-associated fatigue cracking is considered to be one of the most significant

distress modes in flexible pavements besides thermal cracking and rutting. The action

of repeated loading, caused by traffic induced tensile and shear stresses in the bound

layers, will eventually lead to a loss in the structural integrity of a stabilized layer

material. Fatigue cracking is a progressive distress and can be distinguished into three

different stages. An early stage of fatigue cracking consists of intermittent longitudinal

wheel path cracks. An intermediate stage of fatigue cracking called alligator cracking

because the cracking pattern resembles an alligator’s skin. In some extreme cases, the

final stage of fatigue cracking is disintegration when potholes form.

Different test methodologies have been developed over the past few decades

for measuring the fatigue behavior of asphalt concrete mixtures. One of the most

popular fatigue testing methods is the flexural beam fatigue to measure the fatigue life

of a compacted asphalt beam subjected to a repeated flexural bending. The AASHTO

T-321 is the standard procedure for the beam fatigue test (American Association of

State Highway and Transportation Officials [AASHTO], 2008). Other fatigue tests

have been developed such as such as diametral test (Roque & Buttlar, 1992), cantilever

rotating beam test (Pell, & Hanson, 1973), trapezoidal fatigue test (Van Dijk, 1975),

direct tension, or tension-compression (Raithby, & Ramshaw, 1972). The prediction

quality of the fatigue life using any of these test methods will depend on how exact the

method simulates the condition of loading, support, stress state and environment.

Moreover, selecting any of these test methods can be influenced by the availability and

Page 2 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

3

cost of the equipment, in addition to ease of use (Tangella, Craus, Deacon, &

Monismith, 1990).

In general, there are two main approaches that can be utilized to characterize

the fatigue behavior of asphalt concrete mixtures: phenomenological and mechanistic.

The phenomenological approach usually relates the stress or strains in the Hot Mix

Asphalt (HMA) layer to the number of load repetitions that cause failure (Strategic

Highway Research Program [SHRP], 1994). A mechanistic approach is inherently

more complex than the phenomenological one but it is more widely accepted because

it uses material properties based on stress-strain relationships. The mechanistic

approach can be implemented through dissipated energy (Carpenter, & Shen, 2005;

Van Dijk, & Visser, 1977), fracture mechanics (Jacobs, Hopman & Molenaar, 1996;

Majidzadeh, Kauffmann, & Ramsamooj, 1971), or continuum damage mechanics

(Chehab, Kim, Schapery, Witczak, & Bonaquist, 2002; Christensen, & Bonaquist,

2005; Daniel, & Kim, 2002; Kim, Lee, & Little, 1997a; Kim, & Little, 1990; Mun,

Chehab, & Kim, 2007; Kutay, Gibson, & Youtcheff, 2008; Park, Kim, & Schapery,

1996; Underwood, Kim, & Guddati, 2010; Zhang, Sabouri, Guaddati,&Kim, 2013).

A Continuum Damage Mechanics Approach (CDM) was developed through

research efforts at North Carolina State University and Texas A&M University. This

approach utilizes the viscoelastic correspondence principle and Work Potential Theory

(WPT) described by Schapery (1984) to remove viscous effects in monitoring changes

in pseudo-stiffness in repeated uniaxial tensile tests. Therefore, physical variables were

replaced by pseudo variables based on the extended elastic-viscoelastic

Page 3 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

4

correspondence principle to transform a viscoelastic (linear and/or nonlinear) problem

to an elastic case. In 1990-1991, Schapery (1990a, 1991b) developed a series of

damage models for elastic and viscoelastic media based on thermodynamics of

irreversible process and work potential theories with internal state variable to describe

evolution of micro-structural changes.

The Swedish Transport Administration, Trafikverket, has aggressively utilized

asphalt-rubber mixtures on highways within Sweden to mitigate pavement distresses

such as fatigue cracking (Kaloush, Biligiri, Zeiada, Rodezno, & Souliman, 2008).

However, the majority of the rubber-modified pavement sections have been tested and

evaluated mainly for noise and rolling resistance (Nordgren, & Preinfalk, 2009). To

date, adequate information regarding fatigue behavior of the Swedish rubber-modified

mixtures pertinent to its regional climatic conditions is not available.

In 2009, Arizona State University (ASU) and Trafikverket engaged in a joint

effort to understand the fundamental materials properties of the different gap graded,

unmodified and modified mixtures placed on the E18 highway in the Stockholm area

of Sweden (Kaloush et al., 2008; Kaloush, et al., 2010). As part of this project,

advanced mixture material characterization tests were performed that included rutting

evaluation, fatigue and thermal cracking evaluation, and crack propagation

phenomenon assessment. The test results and analysis of the advanced characterization

tests are presented in another paper submitted to the 2012 asphalt rubber conference

(Kaloush, et al., 2012). In 2012, the uniaxial tension-compression test was added to

Page 4 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

5

further evaluate fatigue damage utilizing the viscoelastic and continuum damage

model.

This paper presents results of the fatigue evaluation of the reference, polymer-

modified and asphalt rubber, gap-graded asphalt mixtures placed on the E18 highway

in the Stockholm area of Sweden. The uniaxial tension compression test was used

along with the simplified viscoelastic continuum damage model (S-VECD) to evaluate

resistance to fatigue damage. Substantial work has been carried out elsewhere for the

development of this model, and it is not the purpose of this paper to simply recreate

what has been done elsewhere (Chehab, Kim, Schapery, Witczak, & Bonaquist, 2002;

Daniel, & Kim, 2002; Kim, Lee, & Little, 1997a; Kim, & Little, 1990; Park, Kim, &

Schapery, 1996; Underwood, Kim, & Guddati, 2006; Underwood, Kim, & Guddati,

2010). Here, and for the first time, this approach is used to quantitatively compare

three alternative mixtures (asphalt rubber, polymer modified asphalt, and a

conventional gap graded mix) that are considered for placement on an in-service

roadway. In addition, the work presented in this paper includes an experimental

method involving the use of on-specimen, strain controlled fatigue loading, which has

heretofore not been used with respect to the S-VECD formulation

2. Objective

The main objective of this study was to compare fatigue behavior of gap-graded

mixtures; reference, polymer-modified and rubber-modified placed in the Stockholm

area of Sweden using the viscoelastic and continuum damage model.

3. VECD Model Background

Page 5 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

6

Continuum damage theories ignore specific micro-scale behaviours and instead

characterize a material using macro-scale observations. The VECD model consists of

three concepts:

(1) The elastic-viscoelastic correspondence principle;

(2) The continuum damage mechanics-based work potential theory; and

(3) The temperature-time superposition principle.

Schapery (1984) proposed the extended elastic-viscoelastic correspondence

principle (CP) which can be applicable to both linear and nonlinear viscoelastic

materials (Schapery, 1984). Schapery suggested that constitutive equations for certain

viscoelastic media are identical to those for the elastic cases, but stresses and strains

are not necessarily physical quantities in the viscoelastic body. Instead, they are pseudo

variables in the form of convolution integrals. The uniaxial pseudo strain (εR) is

defined according to Equation 1.

ττε

τε dd

dtE

E

t

R

R )(1

0

−= ∫ (1)

where:

ER = reference modulus;

E(t) = relaxation modulus and creep compliance, respectively;

t = elapsed time from specimen fabrication and time of interest;

τ = time when loading began; and

ε = measured strain.

Page 6 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

7

Schapery (1990) applied the method of thermodynamics of irreversible

processes and the observed phenomenon of path independence of work in damage-

inducing processes to develop the work potential theory to describe the mechanical

behavior of elastic composite materials with growing damage (Schapery, 1990). The

theory is general enough to allow for strong nonlinearities and to describe a variety of

mechanisms including micro- and macro-crack growth in monolithic and composite

materials. Three fundamental elements comprise the work potential theory: the pseudo

strain energy density function (Equation 2), the stress-pseudo strain relationship

(Equation 3) and the damage evolution law (Equation 4).

( ),R RW f Sε= (2)

R

R

Wσ

ε∂

=∂ (3)

RdS W

dt S

α ∂

= − ∂ (4)

The work potential theory specifies an internal state variable (S) to quantify

damage, which is defined as any microstructure changes that result in stiffness

reduction. Kim, Lee, & Little (1997b) characterized the growing damage for a

controlled-strain testing mode through the following constitutive equations (Equations

5-6) (Kim, Lee, &Little, 1997b):

2))((2

1 RR SCW ε= (5)

RSIC εσ )(= (6)

Page 7 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

8

Where I is the initial pseudo stiffness, and C is the normalized pseudo stiffness

via dividing the pseudo stiffness by I. Daniel, & Kim (2002) developed a simplified

numerical model to calculate S from measured data as a function of time shown in

Equation 7 (Daniel, & Kim, 2002).

)1(

1

1

)1(

1

2

1 )())((2

)( ααα

ε +−

+

= − −

−−= ∑ ii

N

i

R

ii ttCCI

tS

(7)

Chehab et al. (2002) and Underwood et al., (2006) verified that the time-

temperature superposition (t-TS) principle at high levels of damage is equally

significant (Chehab, Kim, Schapery, Witczak, & Bonaquist, 2002; Underwood, Kim,

& Guddati, 2006). Based on this validation, Equation 7 can be modified to produce

Equation 8.

)1(

1

1

)1(

1

2

1 )())((2

)( ααα

ξξεξ +−

+

= − −

−−= ∑ ii

N

i

R

ii CCI

S

(8)

Where ξ is the reduced time. Equation 7 or 8 can also be written in the following form:

)1(

1)1(2

11 ))((2

αα

α

ξε ++

−+ ∆

−−+= i

R

iiii CCI

SS

(9)

Underwood et al. (2010) developed a simplified VECD modelling technique

based on the analysis of cyclic data. This method allows for the prediction of the

fatigue life of asphalt concrete at various strain–stress amplitudes under different

temperatures using the dynamic modulus master curve and the cyclic fatigue data from

a single temperature and single stress or strain amplitude (Christensen, & Bonaquist,

2005). The proposed S-function takes on the form shown in Equation 10.

)1(

1

1

)1(

1)1(

2

11 )()())((2

αααα

ξε +++

−+ ∆

−−+= KCCDMR

SS i

R

NNNN

(10)

Page 8 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

9

where:

DMR = Dynamic Modulus Ratio = |E*|fp / |E*|LVE; |E*|fp is the fingerprint

modulus, and |E*|LVE is the linear viscoelastic modulus,

∆ξi = change in the average reduced time between analysis cycles,

K1 = developed functional parameter to account for the analysis of cyclic

data.

Also as part of this simplification the definition of normalized pseudo stiffness was

found to be dependent upon the DMR value as indicated by Equation 11.

DMRC

R

N

N

N ×=

εσ

(11)

The parameter α is believed to be a material property. It was recommended to

correlate α to the slope, m in the central part of the dynamic modulus master curve for

the log E(t)-log(t) relationship where α = 1/m for the stress-controlled tests and α =

1/m + 1 for the cross-head strain tests (Kim, Lee, & Little, 1997b). The C-S curve is a

unique relationship for each mixture where all the different curves for tests conducted

at different strain levels, temperatures, stress or strain-controlled, and monotonic or

dynamic are supposed to collapse on only one curve named the damage characteristic

(C-S) curve. In the analysis presented in this paper the characterization process was

carried out using cyclic repeated on-specimen strain controlled experiments. The test

method is described in Section 5 whereas the characterization process is briefly

outlined in Section 6. The C-S relationship can be also fitted to an analytical form

represented by Equation 12, where C1 and C2 are regression coefficients (Lee, 1996):

2)(1)( 1

CSCSC −= (12)

Page 9 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

10

4. Description of the Project, Mixtures, and Specimen Preparation

The test sections constructed as part of this project include: a reference mixture (ABS

16 70/100), polymer-modified mixture (ABS 16 Nypol 50/100-75) and a rubber-

modified mixture (Gap 16). Figure 1 presents the section of E18 where the project was

constructed along with a schematic test section layout. The polymer modified mixture

contained 3-6% polymer and the rubber-modified mixture contained approximately

20% ground tire rubber. Base bitumen was Pen 70/100 and all mixture designs were

accomplished using the Marshall method. Air voids in the field for all three mixtures

were approximately 3% (Kaloush et al., 2010). Table 1 and Table 2 display the mixture

characteristics and gradation, respectively.

The three variants of asphalt gap-graded mixtures and the associated binders

were sampled from the project sites during construction. At the ASU laboratories,

cylindrical gyratory samples were compacted for both dynamic modulus and uniaxial

tension-compression fatigue test. Two different specimen geometries were

manufactured for each test. For the dynamic modulus test, gyratory plugs were

compacted into 150 mm (6 inches) diameter and 170 mm (6.7 inches) tall specimens.

Then, one 100 mm (4 inches) diameter sample was cored from each gyratory plug. The

sample ends were sawn to arrive at typical test specimens of 150 mm in height. For

uniaxial tension-compression fatigue test, the compaction height was increased to 180

mm (7.1 inches) and the final specimen dimensions were 150 mm (6 inches) height

and 75 mm (3 inches) in diameter. Figure 2 shows cored specimens for the uniaxial

tension-compression fatigue test. The main reason to increase the compaction height

Page 10 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

11

was to allow for larger end cuts to produce a more homogeneous air void distribution

which increases the chances to have a middle failure in the uniaxial fatigue test.

During compaction of the uniaxial tension-compression test, a large difference

in compaction effort to achieve the target air void level was noticed between mixture

types. The reference mixture required approximately 500 gyrations followed by the

polymer-modified (~160 gyrations) and the rubber-modified (~12 gyrations). This

trend is reasonable given the higher binder content of the rubber-modified mixture.

However, the lower compaction effort required for the rubber-modified mixtures may

influence fatigue behavior as discussed in a latter section of this paper.

5. Test Methods

5.1 Dynamic Modulus Test

The dynamic modulus (|E*|) test, per AASHTO TP 62-07 (2007) was performed in the

laboratory at five temperatures -10, 4.4, 21.1, 37.8, 54.4°C (14, 40, 70, 100, and

130°F) and six load frequencies: 25, 10, 5, 1, 0.5 and 0.1 Hz. The stress levels were

varied with the frequency to keep the specimen response within a linear viscoelastic

limit (recoverable microstrain below 150 microstrain). The test parameter values;

dynamic modulus and phase angle, were measured at different temperatures and

frequencies. The average dynamic modulus and phase angle values were summarized

based on three replicates for each mixture. Figure 3 shows a typical instrumented test

specimens and the applied wave shape.

Page 11 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

12

5.2 Uniaxial Tension-Compression Test

The first step prior to running the test included gluing end plates to the specimen using

the jig shown in Figure 4. The applied glue is Devcon plastic steel 5 minutes epoxy

putty. The test specimen was then instrumented with three LVDTs to monitor material

response. The uniaxial tension-compression fatigue test is conducted to evaluate the

fatigue damage of the Swedish gap-graded mixtures using the viscoelastic and

continuum damage model. A servo hydraulic testing machine was used to load the

specimens under an on-specimen strain-control mode of loading. A dynamic sinusoidal

strain (continuous wave) was applied. The test software is capable of achieving and

maintaining the target on-specimen strain based on the outputs from the three LVDTs

by dynamically changing the actuator strain level to solve the machine compliance

issue. New software was developed for Arizona State University by IPC (Industrial

Process Control) company and is designated as UST032-v1.01b S-VECD fatigue test.

The uniaxial tension-compression fatigue tests were conducted using two specimens

from each mixture type where two different strain levels (low and high) were applied

to each specimen. The uniaxial tension-compression fatigue test was run until the

specimen reached 50% of its initial modulus. For a regular fatigue test, the initial

modulus or stiffness is measured at cycle number 50. In this particular study, the initial

modulus was measured at cycle number 100 as to allow enough time for the software

to reach the target on-specimen strain. At each loading cycle, the software calculates

the modulus and the phase angle plus the stress and the strain values from the actuator

and the three LVDTs.

Page 12 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

13

6. Test Results

6.1 Dynamic Modulus Test

The |E*| master curves were constructed for the three gap-graded mixtures. The shift

factors at different temperatures were first computed from the master curve of the

storage modulus [E′=|E*| cos(φ), where φ is the phase angle in degrees] then the same

values were used to construct the |E*| and φ master curves. The main reason for this

approach is that the storage modulus considers both |E*| and φ, which completely

describe the material behavior. In general, the shift factor function was modelled as a

2nd

order polynomial (Equation 13), while the |E*| data at various temperatures were

shifted with regard to a reference temperature of 21.1°C (70°F) with respect to

frequency until the data merge into single smooth pattern that can be mathematically

modelled by a sigmoidal function (Equation 14).

32

2

1log ααα ++= TTaT (13)

)(log

11

*log

rfe

E

γβ

αδ

+++=

(14)

where:

aT = shift factor for temperature, T;

α1, α2, α3 = shift factor function coefficients determined by optimization;

fr = reduced frequency of loading;

δ = minimum value of log |E*|;

δ+α = maximum value of log |E*|; and

Page 13 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

14

β, γ = parameters describing the shape of the sigmoidal function.

Figure 5 and Figure 6 present the dynamic modulus and phase angle

mastercurves for the reference, polymer-modified and rubber-modified mixtures,

respectively. For both dynamic modulus and phase angle, the coefficients for the shift

factor function and sigmoidal mastercurve are given in Table 3. At low test

temperatures, the polymer- modified mixture expressed the highest stiffness followed

by the reference and rubber-modified mixtures. In comparison, at low temperatures,

the rubber-modified mixture had the highest stiffness followed by reference and

polymer-modified. The rubber-modified mixture exhibited the lowest stiffness

compared to the other two mixtures at 21°C (70°F) which is the test temperature used

for the uniaxial tension-compression test.

It can be observed from Figure 6 that the phase angle increases with decreasing

reduced frequency till a certain point where it starts to decrease. This can be explained

that at low temperature and high loading frequency, the asphalt binder dominates the

behavior of asphalt mixtures and the mixture is more elastic resulting in a reduced

phase angle. By increasing temperature or decreasing loading frequency, the asphalt

mixture becomes more viscous as the binder becomes softer and thus, phase angle

increases. This trend is observed until a point when the asphalt binder becomes very

soft and the aggregates dominate the behavior of asphalt mixture. At this point, the

asphalt mixture exhibits more elastic behavior resulting in a decrease in phase angle.

Page 14 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

15

6.2 Uniaxial Tension-Compression Test

During this experiment a cylindrical asphalt concrete undergoes a controlled on-

specimen strain cyclic loading until failure point. The applied stress and on-specimen

axial strains are measured. These values are used to calculate pseudo strain, Equation

1, and pseudo secant modulus, Equation 11, internal state damage parameter, Equation

10, and finally to construct the damage characteristic curve and determine the

coefficients of the damage function shown in Equation 12.

Two specimens from each mixture type were tested at different strain levels:

300 and 250 microstrains on the reference-gap mixture, 300 and 400 microstrains on

the rubber-modified mixture and the polymer-modified mixture. However, it is

important to note that one specimen from each of the three mixtures were tested under

the same target on-specimen strain value (300 microstrain). The results are shown in

Table 4.

From Table 3, it appears that the reference mixture has the highest modulus

value compare to the modified mixture types. At 300 microstrain levels, both polymer-

modified mixture and the rubber-modified mixture, undergo higher load cycles before

failure compared reference-gap mixture. This appears to be reasonable since for strain-

controlled test, the lower the modulus the higher the fatigue life.

Moreover, the polymer-modified mixture appears to have slightly higher

number of cycles until failure compared to the rubber-modified mixture; however it

was expected that the rubber-modified mixtures would show relatively higher fatigue

life. This might be due to the fact that the rubber-modified mixture required much less

Page 15 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

16

compaction effort (gyrations) compared to the polymer-modified mixture to reach the

target air void level. This might decrease the fatigue resistance of the rubber-modified

below what was expected. Based on that, the mixture design for the rubber-modified

mixture might need to be modified to allow more voids in mineral aggregate (VMA) to

accommodate the excess amount of the asphalt rubber.

7. Viscoelastic Material Properties

The viscoelastic material properties were estimated through the relaxation modulus

calculation. The relaxation modulus values were calculated for each mixture type using

the exact inter-conversion method (Park, & Schapery, 1999). This method is based

upon the Fourier transformed relationship between storage modulus, E', and relaxation

modulus, E(t), Equation 15.

( )

=′ ∫∞

− dtetEjE tj

0

Re ωω (15)

Where j is the imaginary unit and Re{x} reflects the real component of the transform.

In Prony form, the relaxation modulus is given by Equation 16.

m

tN

m

mEEtEρ−

=∞ ∑+= exp)(

1

(16)

where:

E(t) = the relaxation modulus as a function of time, t, (kPa);

E∞ = the long-time equilibrium modulus (kPa);

Em = the modulus of Prony term number m (kPa);

ρm = the relaxation time of Prony term m (s); and

Page 16 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

17

N = the number of Prony terms used.

Substituting Equation 16 into Equation 15 leads to the Prony representation of E',

Equation 17.

∑=

∞ ++=′

N

m m

mmEEE

122

22

1)(

ρωρω

ω (17)

Since Equation 17 is derived based on the Prony representation of E(t), Equation 16,

the Em values in both are equivalent. Thus, if the terms can be found from E′ data then

they may be directly substituted into Equation 16 to predict the relaxation modulus.

The E′ values can be fit to the Prony representation using the collocation method.

Initially, relaxation times are assumed in one decade intervals over a range that

approximates the reduced times/frequencies evaluated in the dynamic modulus test

(approximately 2 x 10-8

to 2 x 108 seconds). Then, Equation 17 is manipulated to

separate the Prony term modulus from the summation term, Equation 18, where E∞ can

be estimated from the sigmoidal function of the |E*| since E∞ is the same for either |E*|

or E′ (e.g., E∞ is the long-time elastic response of the mixture). Finally, the Prony term

moduli are solved for by using inverse matrix operations as shown in Equation 19.

{ } { }m

m

m EEE

+=−′ ∞

122

22

ρωρω

(18)

{ } { }∞

−

−′

+= EEE

m

mm

1

22

22

1ρωρω

(19)

Table 5 summarizes the three asphalt mixtures’ Prony series coefficients and relaxation

times. Note that in this project the interconversion process has not been directly

Page 17 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

18

validated with comparisons to experimentally determined relaxation modulus.

However, the main components of this process has been used validated elsewhere

(Park, & Kim, 2001) and successfully used in time-domain constitutive modelling of

asphalt concrete under varying temperature, rate, and frequencies of loading

(Underwood, Kim, & Guddati, 2006; Underwood, Kim, & Guddati, 2010; Daniel, &

Kim 2002; Schapery, 1984) for different term values required for relaxation modulus.

8. Damage Characteristic Curve

The construction of C-S curves in this paper followed the most updated procedure

developed by Underwood et al. (2010) to calculate the internal state damage, S as

shown in Equation 10. The pseudo stiffness or modulus was calculated simply by

calculating the pseudo strain from cyclic data divided by the applied stress

(Underwood, Kim, & Guddati, 2010). Then the pseudo stiffness was normalized via

dividing the pseudo stiffness at each cycle by the initial pseudo stiffness at the first

cycle. For each mixture, two C-S curves were established from the two tested

specimens. Then a single model using Equation 12 was fitted through the collapsed

two curves to represent the model C-S curve for the mixture. Figure 7 demonstrates an

example of the C-S curve for the polymer-modified mixture while Figure 8 shows the

C-S curves for the three mixtures together. The coefficients of these C-S curves are

shown in Figure 8 and are also given in Table 3.

From Figure 8 it can be observed that the most favourably positioned damage

characteristic curves are obtained from the reference-gap and polymer-modified

mixtures. A more favourable damage characteristic curve is the one that has the

Page 18 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

19

greatest damage level for a given pseudo stiffness. This condition is more favourable

because, as seen in Equations 2 through 5, it means that the rate of damage growth for

a given pseudo energy input will be less and thus the incremental loss in pseudo

stiffness will also be less. However, one cannot, or should not, use this curve alone to

judge the fatigue resistance of the three mixtures.

While the damage characteristic curve constitutes an important component of

the overall mixture fatigue resistance, it is not the only characteristic that should be

considered. In addition, one must also consider the material’s resistance to

deformation, i.e., how much pseudo energy will be generated for any given strain

input. The resistance to deformation can vary as much or more between materials than

their differences in damage resistance. An example of how this interplay between

resistance to deformation and resistance to damage may affect the fatigue life can be

observed by considering the following hypothetical case. Suppose that there are two

mixtures, and that the first (Mixture 1) has low damage sensitivity, but the second

(Mixture 2) has high damage sensitivity (i.e., the C-S curve for Mixture 1 is positioned

above the C-S curve for Mixture 2). Ignoring the modulus effect, one would

immediately conclude that Mixture 1 is the better material with regards to fatigue

resistance. However, now suppose that the modulus for Mixture 1 is greater than that

of Mixture 2 by a factor of 4, and further suppose that both materials are subjected to

controlled strain loading at the same input level. According to the damage theory

(Equations 2 through 5), under these loading conditions Mixture 1 would generate 16

times more pseudo energy than would Mixture 2. Such an increase in pseudo energy

Page 19 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

20

may very easily overcome the differences in C-S curves and result in overall poorer

fatigue performance from Mixture 1.

It can also be theoretically shown (Park, 1994) that S is an internal state

variable that relates to a physical change within the material (microcrack formation,

internal dislocations, etc.), but does not necessarily give direct quantification of that

change. Consider the case where the physical transformation responsible for any

observed softening is microcracking. The S variable will increase as those microcracks

grow or multiply, but one cannot convert the value of S to, the number of microcracks,

cumulative crack area, etc. without some additional theoretical assumptions. This

constraint means that across different materials, the same S value does not necessarily

represent the same internal physical change. The practical implication of this

theoretical constraint is simply that in order to gain useful information on fatigue

cracking one must perform simulated predictions of the fatigue life at specific

conditions of interest. Underwood, Baek, & Kim (2012) derived the formulas for

predicting the material response to fully reversed constant stress and constant strain

loadings (Equation 20 and 21 respectively) and verified these formulations using

independent laboratory experiments (Hou, Underwood, & Kim, 2010). Here, these

equations are used to predict the fatigue performance of the three study mixtures at the

temperatures of 5, 20, and 27°C (41, 68, and 80°F) and the results are summarized in

Figure 9 and Figure 10.

( ) ( )( ) ( ) ( ) ( )

2 13

2

2 1 2 0. 1

2

1 *

C

r failure

failure

pp LVE

f SN

C C C E K

α αα

ααα α ε

− +

= − +

(20)

Page 20 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

21

( )

( )( )( )

2

2

2

ˆ23 1

2 10 1 20, 1

ˆ ˆ1* 2 *

ˆˆ ˆ

failure

C

S

r

failure C

pp

C Sf E

N dSC C SK

α

αα

ασ

−

−

=

∫ (21)

where:

Nfailure = predicted cycle number of cycles to failure;

fr = reduced frequency for the condition being simulated;

|E*| = dynamic modulus for the condition being simulated;

ε0,pp = peak-to-peak strain level for simulation;

σ0,pp = peak-to-peak stress level for simulation;

Sfailure = damage level at failure; and

other variables are given by Equation 22 and Equation 23.

21

ˆ

*

failure

failure

SS

Eα

α += (22)

( ) 1221

11 11ˆ *

C

C C Eα

α += (23)

Note that although the stress controlled function has been used to generate the

data in Figure 10, the simulation results are plotted against the initial strain instead of

the input stress level based on the typical convention (Tangella, Craus, Deacon, &

Monismith, 1990; SHRP, 1994). From these two figures it is observed that overall the

asphalt rubber mixture is expected to yield a longer laboratory fatigue life. The results

also show the complexities involved with fatigue assessment as the asphalt rubber and

polymer modified mixtures are expected to show similar fatigue resistance at 5°C

(41°F), but different results at 20°C and 27°C (68 and 80°F) particularly at small strain

Page 21 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

22

amplitudes. This behavior can be related to the complex interaction between modulus

and damage resistance that was mentioned earlier. The same trend holds in the case of

controlled stress testing when the mixtures are compared at consistent initial strain

levels. None of the results directly conflict the basic findings from the laboratory

experiment, which concluded that the asphalt rubber and polymer modified mixtures

have similar fatigue resistance, and that the reference-gap mixture has worse fatigue

resistance. One interesting finding is that the fatigue life predicted through the

continuum damage analysis using the uniaxial fatigue test showed similar results and

trends, when compared to the fatigue life measured by the beam fatigue test (Kaloush

et al., 2010). This shows the power of the continuum damage analysis to predict the

fatigue life relationships using only limited amount of tests.

One caveat in this analysis is that the controlled stress simulations are shown

with respect to initial strain level and not with respect to the input stress condition.

Since the reference mixture is stiffer than either the polymer modified or rubber

modified mixtures, when comparisons are made at the same stress level the reference

mixture would appear to outperform the other two. However, convention based on

historical correlations (Tangella, Craus, Deacon, & Monismith, 1990) suggests that

more useful insight on in-service fatigue performance is gained by examining

controlled stress laboratory tests in the manner performed herein. It should also be

mentioned that laboratory fatigue tests of the type simulated with the VECD model do

not generally take explicit consideration of differences in the fracture characteristics of

the materials. However, these characteristics may have important implications in

Page 22 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

23

rigorously relating laboratory fatigue tests to pavement cracking performance. Since

the rubber-modified asphalt mixture has been shown to outperform either of the other

two study mixtures in this regard, it is expected that the differences identified through

this investigation likely underestimate the overall net benefits of asphalt rubber in

mitigating or reducing the cracking phenomenon in asphalt concrete pavements.

9. Conclusions

This paper presented research performed to compare properties and fatigue

performance characteristics for reference, polymer-modified and rubber-modified gap

graded mixtures placed on the E18 highway in the Stockholm area of Sweden. The

advanced material characterization tests included: dynamic (complex) modulus for

stiffness evaluation and the uniaxial tension-compression for fatigue assessment. The

data were used to compare the performance of the rubber-modified gap graded mixture

to the reference and the polymer-modified gap mixtures using the viscoelastic

continuum damage (VECD) approach.

Dynamic modulus test results indicate that, at low test temperatures, the

polymer- modified mixture expressed the highest stiffness followed by the reference

and rubber-modified mixtures. In comparison, at low temperatures, the rubber-

modified mixture had the highest stiffness followed by reference and polymer-

modified. The rubber-modified mixture exhibited the lowest stiffness compared to the

other two mixtures at 21°C (70°F) which is the test temperature used for the uniaxial

tension-compression test.

Page 23 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

24

According to the uniaxial tension-compression fatigue test (300 µɛ), both the

polymer-modified mixture and rubber-modified mixtures, undergo higher load cycles

before failure compared to the reference-gap mixture. This appears to be reasonable

since for strain-controlled test, the lower the modulus the higher the fatigue life.

Moreover, the polymer-modified mixture appears to have slightly higher number of

cycles till failure compared to the rubber-modified mixture. However it was expected

that the rubber-modified mixture would show relatively higher fatigue life and this

discrepancy may be the result of differences in compaction effort.

Damage characteristic (C-S) curves were formulated and used along with the

measured linear viscoelastic characteristics to predict the fatigue performance of the

three study mixtures over a range of temperatures. This data clearly showed the

benefits of the rubber modified mixture in terms of laboratory fatigue resistance. This

VECD analysis presents a more powerful tool to evaluate fatigue resistance. Instead

of only looking at the number of cycles at a certain stiffness reduction, this method

considers internal state damage. Results indicate that VECD is a much more

comprehensive approach to access resistance to fatigue damage than the traditional

number of cycles at a certain stiffness reduction. The rubber-modified mixture exhibits

the greatest fatigue resistance followed by the polymer-modified and reference.

Finally, results obtained from this study will be compared to actual field

performance of the test sections once this information becomes available from

Trafikverket. At this time, ranking of mixtures from lab tests will be compared to field

ranking to validate the results of the VECD approach.

Page 24 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

25

10. References

American Association of State Highway and Transportation Officials T 321, (2008).

Standard method of test for determining the fatigue life of compacted hot-mix asphalt

(HMA) subjected to repeated flexural bending, AASHTO, Washington D.C.

American Association of State Highway and Transportation Officials TP62-07. (2007).

Standard method of test for determining dynamic modulus of hot-mix asphalt concrete

mixtures, AASHTO, Washington D.C.

Carpenter, S.H., & Shen, S. (2005). Application of the dissipated energy concept in

fatigue endurance limit testing. Transportation Research Record, Journal of the

Transportation Research Board, 1929, 165-173.

Chehab, G.R., Kim, Y.R., Schapery, R.A., Witczak, M.W. & Bonaquist, R. (2002).

Time-temperature superposition principle for asphalt concrete mixtures with growing

damage in tension state. Journal of the Association of Asphalt Paving Technologists,

72, 559-593.

Christensen, D.W. & Bonaquist, R. (2005). Practical application of continuum damage

theory to fatigue phenomena in asphalt concrete mixtures. Journal of the Association

of Asphalt Paving Technologists, 74, 963-1002.

Daniel, J.S. & Kim, Y.R. (2002). Development of a simplified fatigue test and analysis

procedure using a viscoelastic continuum damage model. Journal of the Association of

Asphalt Paving Technologists, 72, 619-650.

Hou, E. T., Underwood, B.S. & Kim, Y.R. (2010). Fatigue performance prediction of

North Carolina mixtures using the simplified viscoelastic continuum damage model,

Journal of the Association of Asphalt Paving Technologists, 79, 35-80.

Jacobs, M.M.J., Hopman, P.C., & Molenaar, A.A.A. (1996). Application of fracture

mechanics principles to analyze cracking in asphalt concrete. Journal of the

Association of Asphalt Paving Technologists, 65, 1-39.

Kaloush, K.E., Biligiri, K.P., et al. (2010). Laboratory evaluation of rubber & polymer

modified bituminous mixtures constructed in Stockholm (E18 Highway between the

Järva Krog & Bergshamra Interchanges), Arizona State University.

Kaloush, K.E., Biligiri, K.P., Nordgren, T., Zeiada, W.A. et al. (2012). Laboratory

evaluation of asphalt-rubber gap graded mixtures constructed on Stockholm highway

in Sweden, Accepted for Presentation - Asphalt Rubber 2012, Munich, Germany.

Page 25 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

26

Kaloush, K.E., Biligiri, K.P., Zeiada, W.A., Rodezno, M.C. & Souliman, M.I. (2008).

Laboratory pavement performance evaluation of Swedish gap graded asphalt concrete

mixtures–Malmo E-06 Highway, Arizona State University.

Kim, Y.R. & Little, D. (1990). One-dimensional constitutive modeling of asphalt

concrete. Journal of Engineering Mechanics, 116 (4), 751-772.

Kim, Y.R. Lee, H.J. & Little, D. (1997a). Fatigue characterization of asphalt concrete

using viscoelasticity and continuum damage theory. Journal of the Association of

Asphalt Paving Technologists, 66, 520-558.

Kim, Y.R., Lee, H.J., & Little, D. N. (1997b). Fatigue characterization of asphalt

concrete Using visco-elasticity and continuum damage theory. Journal of the

Association of Asphalt Paving Technologists, 66, 520-569.

Kutay, M.E., Gibson, N. & Youtcheff, J. (2008). Conventional and viscoelastic

continuum damage (VECD)-based fatigue analysis of polymer modified asphalt

pavements. Journal of the Association of Asphalt Paving Technologists, 77, 395-434.

Lee, H-J. & Kim, Y.R. (1998). Viscoelastic constitutive model for asphalt concrete

under cyclic loading. Journal of Engineering Mechanics, 124(1), 32-40.

Lee, H.J. (1996). Viscoelastic constitutive modeling of asphalt concrete using

viscoelasticity and continuum damage theory (Doctoral Dissertation). North Carolina

State University, Raleigh.

Majidzadeh, K., Kauffmann, E.M., & Ramsamooj, D.V. (1971). Application of

fracture mechanics in the analysis of pavement fatigue. Proceedings of the Association

of Asphalt Paving Technologists, 40, 227-246.

Mun, S., Chehab, G.R., Kim, Y.R. (2007). Determination of Time-Domain Viscoelastic

Functions Using Optimized Interconversion Techniques. International Journal of Road

Materials and Pavement Design, 8(2), 351-365.

Nordgren, T. & Preinfalk, L. (2009). Asphalt rubber - a new concept for asphalt

pavements in Sweden? Progress report - February 2009, Swedish Transport

Administration, Gothenburg, Sweden.

Park, S.W. & Kim, Y.R. (2001). Fitting Prony-Series Viscoelastic Models with Power-

Law Presmoothing, Journal of Materials in Civil Engineering, 13(1), 26-32.

Page 26 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

27

Park, S.W. & Schapery, R.A. (1999). Methods of interconversion between linear

viscoelastic material functions, part I - a numerical method based on Prony series,

International Journal of Solids and Structures, 36, 1653-1675.

Park, S.W. (1994). Development of a nonlinear thermo-viscoelastic constitutive

equation for particulate composites with growing damage (Doctoral Dissertation). The

University of Texas, Austin.

Park, S.W., Kim, Y.R. & Schapery, R.A. A. (1996). A viscoelastic continuum damage

model and its application to uniaxial behavior of asphalt concrete. Mechanics of

Materials, 24, 241-255.

Pell, P.S. & Hanson, J.M. (1973). Behavior of bituminous road base materials under

repeated loading, Proceedings of Association of Asphalt Paving Technologists, 201-

229.

Raithby, K.D. & Ramshaw, J.T. (1972). Effect of secondary compaction on the fatigue

performance of a hot-rolled asphalt, TRRL-LR 471, Crowthorne, England.

Roque, R. & Buttlar, W.G. (1992). The development of a measurement and analysis

system to accurately determine asphalt concrete properties using the indirect tensile

mode. Journal of the Association of Asphalt Paving Technologists, 61, 304-333.

Schapery R.A. (1991). Analysis of damage growth in particulate composite using a

work potential, Composites, Part B, Engineering, 1(3), 167–182.

Schapery, R.A. (1984). Correspondence principles and a generalized J integral for

large deformation and fracture analysis of viscoelastic media. International Journal of

Fracture, 25, 195-223.

Schapery, R.A. (1990). A theory of mechanical behavior of elastic media with growing

damage and other changes in structure. Journal of Mechanics and Physics of Solids,

38, 215-253.

Strategic Highway Research Program. (1994). SHRP-A-404 - Fatigue characteristics

of bitumen and bituminous mixes, University of California, Berkeley, National

Research Council, Washington, D.C.

Tangella, S.R., Craus, J., Deacon, J.A., & Monismith, C.L. (1990). Summary report of

fatigue response of asphalt mixtures, Technical Memorandum No. TM-UCB-A-003A-

89-3M, prepared for SHRP Project A-003A, Institute of Transportation Studies,

University of California, Berkeley.

Page 27 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

28

Underwood, B.S., Kim, Y.R. & Guddati, M.N. (2010). Improved calculation method of

damage parameter in viscoelastic continuum damage model, International Journal of

Pavement Engineering, 11(6), 459-476.

Underwood, B.S., Baek, C.M. & Kim, Y.R. (2012). Use of simplified viscoelastic

continuum damage model as an asphalt concrete fatigue analysis platform.

Transportation Research Record, Journal of the Transportation Research Board, (In

Press)

Underwood, B.S., Kim, Y.R., & Guddati, M.N. (2006). Characterization and

performance prediction of ALF mixtures using a viscoelastoplastic continuum damage

model. Journal of the Association of Asphalt Paving Technologists, 75, 577-636.

Van Dijk, W. (1975). Practical fatigue characterization of bituminous mixes. Journal

of the Association of Asphalt Paving Technologists, 44, 38.

Van Dijk, W., & Visser, W. (1977). The energy approach to fatigue for pavement

design. Journal of the Association of Asphalt Paving Technologists, 46, 1-40.

Zhang, J., Sabouri, M., Guddati, M., & Kim, Y.R. (2013). Development of a Failure

Criterion for Asphalt Mixtures under Fatigue Loading. International Journal of Road

Materials and Pavement Design, 14(2), 1-15.

Page 28 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

29

Table 1. Field mixture characteristics, Stockholm test section

Mix Binder

Content (%)

Air Voids

(%)

Max. Theoretical

Density

(Gmm)

Reference ABS 16 70/100 5.9 2.6 2.4642

Polymer ABS 16 Nypol 50/100-75 5.9 2.6 2.4558

Rubber GAP 16 8.7 2.4 2.3588

Table 2. Average aggregate gradations, Stockholm Highway.

Gradation

(% Passing by mass

of each sieve)

Sieve Size (mm) Reference-

Gap

Polymer-

Modified

Rubber-

Modified

22.4 100 100 100

16 98 98 98

11.2 65 65 68

8 38 38 44

4 23 23 24

2 21 21 22

0.063 10.5 10.5 7.5

Table 3. Summary of Model Coefficients.

Mix Type |E*| Mastercurve Shift Function Damage Model

δ α β γ α1 α2 α3 C1 C2 α Reference-

Gap 5.466 2.453 0.234 0.538 0.00003 -0.1272 2.673 0.0013 0.516 3.93

Polymer-

Modified 5.664 2.134 0.230 0.604 -0.00043 -0.0952 2.441 0.0034 0.435 4.09

Rubber-

Modified 4.941 3.124 0.222 0.331 0.00055 -0.1541 3.553 0.0043 0.421 4.85

Table 4. Fingerprint modulus (FP) and uniaxial fatigue test results.

Mixture

Type

Specimen

ID

Air

Voids %

Strain

Level

µs

FP

Modulus

MPa

Machine

Compliance

Factor

Initial

Stiffness

MPa

Initial

φ

Cycles

to

Failure

Reference-

Gap

SWC03 3.65 250 10,978 6.78 8,609 25.3 131,830

SWC02 3.82 300 10,310 6.45 7,435 28.6 11,030

Polymer-

Modified

SWP05 3.55 300 8,479 5.08 6,707 21.8 138,570

SWP06 3.65 400 8,913 5.43 6,214 24.1 28,620

Rubber-

Modified

SWR04 2.81 300 6,710 4.45 5,296 24.9 126,380

SWR06 3.11 400 7,123 4.63 5,134 25.5 27,200

Page 29 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

30

Table 5. Prony series parameters for the Swedish gap-graded mixtures.

Mixture

Type Reference-Gap Polymer-Modified Rubber-Modified

Terms ρm (s) Em (kPa) ρm (s) Em (kPa) ρm (s) Em (kPa)

1 2.00E+08 3,392 2.00E+08 6,109 2.00E+08 17,871

2 2.00E+07 3,234 2.00E+07 5,343 2.00E+07 9,742

3 2.00E+06 8,322 2.00E+06 13,514 2.00E+06 23,904

4 2.00E+05 18,416 2.00E+05 28,827 2.00E+05 43,334

5 2.00E+04 42,175 2.00E+04 63,873 2.00E+04 84,288

6 2.00E+03 99,953 2.00E+03 146,238 2.00E+03 170,776

7 2.00E+02 252,445 2.00E+02 354,928 2.00E+02 366,546

8 2.00E+01 692,406 2.00E+01 922,105 2.00E+01 822,717

9 2.00E+00 1,941,157 2.00E+00 2,404,968 2.00E+00 1,855,538

10 2.00E-01 4,361,535 2.00E-01 5,096,523 2.00E-01 3,106,013

11 2.00E-02 7,667,979 2.00E-02 8,613,774 2.00E-02 5,314,349

12 2.00E-03 11,958,953 2.00E-03 11,660,148 2.00E-03 8,198,165

13 2.00E-04 14,272,413 2.00E-04 12,125,054 2.00E-04 11,277,606

14 2.00E-05 14,052,870 2.00E-05 10,344,635 2.00E-05 13,795,241

15 2.00E-06 11,862,006 2.00E-06 7,665,242 2.00E-06 15,127,488

16 2.00E-07 8,988,787 2.00E-07 5,184,031 2.00E-07 15,055,265

17 2.00E-08 6,568,947 2.00E-08 3,423,696 2.00E-08 14,509,193

Page 30 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

31

List of Figures

Figure 1. Project location and schematic of test section layout (Kaloush et al., 2010).

Figure 2. Specimen manufacturing for uniaxial tension-compression fatigue test.

Figure 3. Specimen instrumentation and applied wave shape.

Figure 4. Specimen gluing via gluing jig.

Figure 5. Dynamic modulus master curves of the Swedish gap-goaded mixtures.

Figure 6. Phase angle master curves of the Swedish gap-goaded mixtures.

Figure 7. An example of S-C curve fitting model for the polymer-modified mixture.

Figure 8. Comparison of damage characteristic curves for study mixtures.

Figure 9. Simulation results for controlled strain test at 10 Hz loading frequency and

at; (a) 5°C, (b) 20°C, and (c) 27°C.

Figure 10. Simulation results for controlled stress test at 10 Hz loading frequency and

at; (a) 5°C, (b) 20°C, and (c) 27°C.

Page 31 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Paper Title: “Comparison of Conventional, Polymer, and Rubber Asphalt

Mixtures Using Viscoelastic Continuum Damage Model”

Journal: IJRMPD

Reviewer(s)' Comments to Author:

Reviewer: 1

Comments:

The paper reads very well and addresses an important issue. However, from the

reviewer’s point of view the paper doesn’t provide sufficient background

information to fully understand the concept. To improve on this, the reviewer

suggests the following:

Comment: In section 3 the VECD Model is described in some detail, however, it

is not clear how this was used in the research. In particular, no example is given

dealing with the application of this model and no section is included determining

the parameters of this model from real test results.

Response: Equations 1-9 are provided as background. Equations 10 and 11 are

the most important equations with respect to the current paper. The use of these

equations is more fully understood in subsequent sections of the paper, but the

authors understand that in its current form the reader could be confused as to how

this section ties into the remainder of the paper. Thus we have included the

following statement in the last paragraph to Section 3:

“In the analysis presented in this paper the characterization process was

carried out using cyclic repeated on-specimen strain controlled

experiments. The test method is described in Section 5 whereas the

characterization process is briefly outlined in Section 6.”

In addition we have also modified the first paragraph in Section 6.2:

“During this experiment a cylindrical asphalt concrete undergoes a

controlled on-specimen strain cyclic loading until failure point. The

applied stress and on-specimen axial strains are measured. These values

are used to calculate pseudo strain, Equation (1), and pseudo secant

modulus, Equation (11), internal state damage parameter, Equation 10,

and finally to construct the damage characteristic curve and determine

the coefficients of the damage function shown in Equation (12).”

Page 32 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

To clearly demonstrate that the model parameters are determined for each

mixture, the model equations (Equation 12) with the optimized coefficients are

have also been added to Figure 8.

Comment: In section 6.1 the dynamic (complex) modulus is dealt with. Equation

(12) is used to capture the |E*| master-curve. The authors state that they have

determined the shift factors as well as the beta and gamma parameters of the

model from the test results obtained for the three gap-graded mixtures

investigated in the paper. However, the parameters and the shift factor are not

included and no information is provided regarding the R2.

Response: The authors apologize for the oversight and have added the

mastercurve and shift factor coefficients to a new table, which became Table 4.

Table 1. Summary of Model Coefficients.

Mix Type |E*| Mastercurve Shift Function Damage Model

δ α β γ α1 α2 α3 C1 C2 α Reference-

Gap 5.466 2.453 0.234 0.538 0.00003 -0.1272 2.673 0.0013 0.516 3.93

Polymer-

Modified 5.664 2.134 0.230 0.604 -0.00043 -0.0952 2.441 0.0034 0.435 4.09

Rubber-

Modified 4.941 3.124 0.222 0.331 0.00055 -0.1541 3.553 0.0043 0.421 4.85

Page 33 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Comment: Section 6 “Viscoelastic Material Properties” should actually read

Section 7 “Viscoelastic Material Properties”. In this section the authors introduce

an approach developed by Park & Schapery that may be used to convert from the

frequency to time domain. This approach is very useful since it provides the

theoretical background to back calculate relaxation times as well as the relaxation

modulus from cyclic experiments. However, apart from the equation, not too

much detail is provided. The parameters of the prony series function used are

given in Table 4. The authors should show how well the actual relaxation

modulus fits the one determined with the Park & Schapery approach.

Response: The authors have corrected the typographical error. The referenced

section now reads “7. Viscoelastic Material Properties”. Within the length

limitations of the paper it is not possible to provide all of the details regarding the

interconversion methodology. However, the following was added in Section 7.

These additions also address the reviewer’s comments regarding matching with

actual relaxation data.

“The viscoelastic material properties were estimated through the relaxation

modulus calculation. The relaxation modulus values were calculated for each

mixture type using the exact inter-conversion method (Park & Schapery, 1999).

This method is based upon the Fourier transformed relationship between storage

modulus, E', and relaxation modulus, E(t), Equation 15.

( )

=′ ∫∞

− dtetEjE tj

0

Re ωω (15)

Where j is the imaginary unit and Re reflects the real component of the transform.

In Prony form, the relaxation modulus is given by Equation (16).

m

tN

m

mEEtEρ−

=∞ ∑+= exp)(

1

(16)

where:

E(t) = the relaxation modulus as a function of time, t, (kPa).

E∞ = the long-time equilibrium modulus (kPa);

Em = the modulus of Prony term number m (kPa);

ρm = the relaxation time of Prony term m (s); and

N = the number of Prony terms used.

Substituting Equation (16) into Equation (15) leads to the Prony representation of

E', Equation (17).

Page 34 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

∑=

∞ ++=′

N

m m

mmEEE

122

22

1)(

ρωρω

ω (17)

Since Equation (17) is derived based on the Prony representation of E(t),

Equation (16), the Em values in both are equivalent. Thus, if the terms can be

found from E′ data then they may be directly substituted into Equation (16) to

predict the relaxation modulus. The E′ values can be fit to the Prony

representation using the collocation method. Initially, relaxation times are

assumed in one decade intervals over a range that approximates the reduced

times/frequencies evaluated in the dynamic dynamic modulus test (approximately

2 x 10-8

to 2 x 108 seconds). Then, Equation (17) is manipulated to separate the

Prony term modulus from the summation term, Equation (18), where E∞ can be

estimated from the sigmoidal function of the |E*| since E∞ is the same for either

|E*| or E′ (e.g., E∞ is the long-time elastic response of the mixture). Finally, the

Prony term moduli are solved for by using inverse matrix operations as shown in

Equation 19.

{ } { }m

m

m EEE

+=−′ ∞

122

22

ρωρω

(18)

{ } { }∞

−

−′

+= EEE

m

mm

1

22

22

1ρωρω

(19)

Error! Reference source not found. summarizes the three asphalt

mixtures’ Prony series coefficients and relaxation times. Note that in this project

the interconversion process has not been directly validated with comparisons to

experimentally determined relaxation modulus. However, the main components of

this process has been used validated elsewhere (Park and Kim, 2001) and

successfully used in time-domain constitutive modelling of asphalt concrete under

varying temperature, rate, and frequencies of loading (Underwood et al., 2006;

Underwood et al., 2010; Daniel and Kim 2002; Schapery, 1984) for different term

values required for relaxation modulus.”

Comment: Section 7 “Damage Characteristic Curve” should actually read

Section 8 “Damage Characteristic Curve”. Again, only theory, no parameters are

provided and no example is presented.

Response: The authors have updated the section title to read “8. Damage

Characteristic Curve”. Also, as described in our response to this reviewer’s first

comment we have added the values for the coefficients into the paper in both

Figure 8 and Table 4.

Page 35 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Comment: Section 8 “Conclusions” should actually read Section 9

“Conclusions”.

Response: The authors have updated the section title to read “9. Conclusions”.

Comment: Page 5, 2nd sentence “This paper presents…”. The word “on” is

missing in that sentence.

Response: Corrected

Comment: Page 10 last sentence “The dynamic modulus …”. The word “test” is

missing in that sentence.

Response: Corrected

Page 36 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Reviewer: 2

Comment: I think technical parts of the paper are OK, but am quite concerned

with its scientific significance as a journal article. The authors simply retried what

NCSU teams has done so far to compare three mixtures. Comparison of mixtures

based on a methodology that has been published in a number of similar work

would not be valid for publication. The study seems better fit to a research report

rather than a journal article. In addition, the authors compared fatigue

performance with only one sample per case, which is not reasonable. Fatigue of

asphalt concrete is quite random, and one specimen would not show any valid

observations and/or conclusions. With these two significant drawbacks, I would

decline the paper for publication.

Response: Thank you for the useful comment. We understand how the original

version of the paper did not emphasize the scholarly or practical contributions of

the current paper. The authors do not intend for the scientific contribution of this

work to simply be the verification of the approach developed at NCSU. The

significance of this study is the application of the model to evaluate three

alternative mixes that are not well differentiated by current assessment methods,

but for which it is known that fatigue performance is different. To better explain

the significance of the work the following has been included:

“Substantial work has been carried out elsewhere for the development of

this model, and it is not the purpose of this paper to simply recreate what

has been done elsewhere (Chehab, Kim, Schapery, Witczak & Bonaquist,

2002; Daniel & Kim, 2002; Kim, Lee, Little, 1997a; Kim & Little, 1990;

Park, Kim & Schapery, 1996;Underwood, Kim, and Guddati, 2006;

Underwood, Kim & Guddati, 2010). Here, and for the first time, this

approach is used to quantitatively compare three alternative mixtures

(asphalt rubber, polymer modified asphalt, and a conventional gap

graded mix) that are considered for placement on an in-service roadway.

In addition, the work presented in this paper includes an experimental

method involving the use of on-specimen, strain controlled fatigue

loading, which has heretofore not been used with respect to the S-VECD

formulation.”

As explained in the new writing this is the first time that the S-VECD model has

been applied to evaluate asphalt mixtures that are being considered as alternatives.

The majority of the on-going work with the NCSU model focuses on the

application of this model as a means to generate data that will then be used in

pavement analysis tools (see the recent efforts concerning the HMA Performance

Related Specifications work). Other publications have examined it’s use for

moisture damage and/or oxidative aging assessment. The closest NCSU work to

Page 37 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

this current paper is the Underwood et al. 2006 assessment, which used the VECD

model formulation as applied to FHWA ALF test mixture. The work here is

differentiated by the 2006 publication in the application of the S-VECD model

(2006 used VECD, which is similar but distinct) and characterization of mixes

using an on-specimen strain controlled test (2006 used constant rate tension tests

and no follow-up work has used on-specimen strain controlled testing). Although

it is probably not a strong individual case for the merit of this paper, the mixtures

evaluated here are also the very first Gap graded mixtures evaluated with the S-

VECD model.

Comment: In addition, the authors compared fatigue performance with only one

sample per case, which is not reasonable. Fatigue of asphalt concrete is quite

random, and one specimen would not show any valid observations and/or

conclusions.

Response: The reviewer is correct that drawing conclusions based on a single

replicate test may have issues in assessment of fatigue performance. Variability in

fatigue tests can come from many different places related to both the inherent

specimen variability and also to the variability in setting-up and conducting the

test. However, the strength of the damage model approach is that it apparently

identifies an underlying and more fundamental characteristic of the material that

is less affected by experimental variability. The theory suggests that fatigue is

governed by a strain level, temperature, and mode of loading independent

function (the damage characteristic relationship). That a theory based on this

supposition can explain fatigue growth in asphalt concrete has been shown to be

valid in multiple papers from the NCSU group and associated researchers. The

authors acknowledge that it is debatable whether this truly has a physical

significance, e.g., whether it truly identifies the fundamental physical process.

The same argument could be made for many modelling methods used with asphalt

concrete, but keeping philosophical arguments aside, multiple experimental and

analytical studies have shown that this single damage curve exists regardless of

the test condition considered (within the realm of conditions that lead to damage

dominant behaviours). Therefore, comparisons of the damage characteristic

curves from tests at different conditions demonstrate at a more fundamental level

the repeatability of the experiments. So, while only a single experiment was

conducted at the respective strain levels the authors have confidence in the

validity of their observations because of the similarity in damage characteristic

curves for the two strain levels tested. The agreement shown in Figure 7 is typical

and better than that observed in many investigations referenced in this paper

(Underwood et al. 2010 and Kutay et al. 2008).

Page 38 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Reviewer: 3

Comment: It is relatively well written paper.

Response: Thank you

Comment: In some cases, however, the sentences regarding the contents of a

Table or Figure appear earlier than the table or figure number is cited. Please cite

the figure or table number first before mentioning the contents.

Response: The paper was revised accordingly.

Comment: In Figure 5 and Figure 6, no temperature information are given.

Response: The effect of temperature and the frequency is included in the

calculation of the reduced frequency (horizontal axis). So, all the dynamic

modulus and phase angle values in Figures 5 and 6 respectively are function of

the reduced frequency which are related to test temperature and frequency. It is

added into the paper that the shifting was done with respect to 21.1°C (70°F)

reference temperature.

Page 39 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 1. Project location and schematic of test section layout (Kaloush et al., 2010). 341x144mm (72 x 72 DPI)

Page 40 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 2. Specimen manufacturing for uniaxial tension-compression fatigue test. 301x119mm (72 x 72 DPI)

Page 41 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 3. Specimen instrumentation and applied wave shape.

337x129mm (72 x 72 DPI)

Page 42 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 4. Specimen gluing via gluing jig.

223x231mm (72 x 72 DPI)

Page 43 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 5. Dynamic modulus master curves of the Swedish gap-goaded mixtures. 313x179mm (96 x 96 DPI)

Page 44 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 6. Phase angle master curves of the Swedish gap-goaded mixtures. 362x184mm (72 x 72 DPI)

Page 45 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 7. An example of S-C curve fitting model for the polymer-modified mixture. 329x172mm (72 x 72 DPI)

Page 46 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 8. Comparison of damage characteristic curves for study mixtures. 293x179mm (96 x 96 DPI)

Page 47 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 9. Simulation results for controlled strain test at 10 Hz loading frequency and at; (a) 5 °C, (b) 20 °C, and (c) 27 °C.

367x236mm (72 x 72 DPI)

Page 48 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

For Peer Review O

nly

Figure 10. Simulation results for controlled stress test at 10 Hz loading frequency and at; (a) 5 °C, (b) 20 °C, and (c) 27 °C.

367x234mm (72 x 72 DPI)

Page 49 of 49



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

Date post:	20-Nov-2023
Category:	Documents
Upload:	sharjah
View:	1 times
Download:	0 times

Comparison of conventional, polymer, and rubber asphalt mixtures using viscoelastic continuum damage...

Documents