Modeling the Primary and Secondary Regions of Creep Curves for SBS-Modified Asphalt Mixtures under Dry and Wet Conditions

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Modeling the Primary and the Secondary Regions of Creep Curves

for SBS Modified Asphalt Mixtures under Dry and Wet Conditions

Arash Saleh Ahari1, Seyed Arash Forough2, Ali Khodaii3, Fereidoon Moghadas Nejad4

1M.Sc. of Road & Transportation Engineering, Highway Division, Department of Civil and Environmental

Engineering, Amirkabir University of Technology, Tehran, Iran.

2PhD Candidate, Highway Division, Department of Civil and Environmental Engineering, Amirkabir University of

Technology, Tehran, Iran.

3Associate Professor, Department of Civil and Environmental Engineering, Amirkabir University of Technology,

Tehran, Iran.

4Associate Professor, Department of Civil and Environmental Engineering, Amirkabir University of Technology,

Tehran, Iran.

Abstract

This study was conducted to model the primary and the secondary regions of creep curves, derived from

dynamic creep tests, for dense graded polymer modified asphalt mixtures under different moisture

conditions. For this purpose, 96 Marshall specimens containing siliceous crushed stone aggregate with

85/100 penetration bitumen, modified with 4.5% of Styrene-Butadiene-Styrene (SBS) polymer modifier,

were fabricated and tested at four different loading frequencies, four different temperatures, and two

moisture conditions, dry and wet, with three replicate specimens for each experimental combination.

Statistical analyses were carried out on the test results and two approaches were proposed to model the

creep curves. In addition, a stepwise method was proposed by which it is possible to estimate the location

of the boundary point connecting the primary to the secondary regions of the creep curves. The proposed

models were compared to the previous ones, and demonstrated to be more accurate.

CE Database subject headings: Modeling; SBS Modified Asphalt mixture; Dynamic Creep Test; Creep

Curve; Primary Region; Secondary Region; Loading Frequency; Temperature; Moisture Condition.

Journal of Materials in Civil Engineering. Submitted January 9, 2013; accepted May 20, 2013; posted ahead of print May 22, 2013. doi:10.1061/(ASCE)MT.1943-5533.0000857

Copyright 2013 by the American Society of Civil Engineers

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Introduction

Background

Rutting in asphalt mixtures is one of the main distresses of flexible pavements defined as the

accumulation of permanent deformations in an asphalt mixture layer exposed to repeated traffic loading

especially at elevated temperatures (Kim, 2009). Generally, several testing methods have been developed

to evaluate the susceptibility and/or resistance of asphalt mixtures against the rutting phenomenon.

Among these testing methods, successfully used by various researchers, are Static/Dynamic Creep tests

(Khodaii and Mehrara, 2009; Mehrara and Khodaii, 2011), Wheel Track tests (Aschenbrener et al., 1993;

Izzo and Tahmoressi, 1999), Simple Shear tests (Sousa et al., 1994), and Indirect Tensile tests

(Christensen, 1998). All the mentioned testing methods can be used to evaluate the rutting susceptibility

and/or resistance of both unmodified and modified asphalt mixtures. However, the dynamic creep tests

can simulate the actual field situations more realistically than the other testing methods, especially for the

polymer modified asphalt mixtures (Mehrara and Khodaii, 2011).

During the dynamic creep tests on asphalt mixtures, several parameters are obtained by which the

accumulated permanent deformations due to the applied loads can be estimated. Figure 1 shows a

schematic explaining the relationship between the total plastic strains and the loading cycles in the

dynamic creep tests. As seen from this Figure, the accumulated permanent strain curve is divided into

three main regions, primary, secondary, and tertiary. The permanent strains accumulate rapidly, but

having a decreasing rate, within the primary region, until a constant accumulation rate at the beginning of

the secondary region. The boundary point connecting the secondary to the tertiary regions is defined as

the Flow Number after which the permanent strains accumulate rapidly with an increasing rate.

Generally, the creep curves derived from the dynamic creep tests are used to compare the resistance of

different asphalt mixtures against permanent deformations and rutting distress. For this purpose, it is

necessary to identify the locations of the boundary points connecting the primary to the secondary

regions, and the secondary to the tertiary regions. However, there is no general acceptance among various

researchers on how to identify these two boundary points on the creep curves. Furthermore, the rate of



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accumulated permanent strains on the creep curves is dependent upon several parameters among which

the loading frequency, temperature, and moisture condition are influential external parameters which

warrant further investigation.

Therefore, this study was undertaken to model the primary and the secondary regions of the creep curves

obtained from the dynamic creep tests on SBS modified asphalt mixtures at various loading frequencies,

different temperatures, and two moisture conditions, dry and wet, and to propose a stepwise method by

which the location of the boundary point connecting the primary to the secondary regions of the creep

curves can be identified.

Literature review

During the last four decades, several attempts have been made to model the permanent deformation of

asphalt mixtures, and different rutting models, e.g. power law models, VESYS model, Ohio state model,

AASHTO-2002 model, etc., have been proposed (Zhou et al., 2004; Zhou and Scullion, 2003; Monismith

and Ogawa, 1975). In addition, several methods have been used to characterize the creep curves obtained

from the dynamic creep tests. Polynomial fitting and statistical regression are among these methods

(Archilla et al., 2007; Biligiri et al., 2007).

Zhou et al. (2004) suggested that the Flow Number cannot be a suitable criterion for evaluating the

susceptibility and/or resistance of asphalt mixtures against permanent deformation and rutting distress.

Therefore, they proposed a three-stage model (one stage for each region of the creep curves) using a

simple algorithm having an iterative approach to identify the locations of the two boundary points. In

addition, they proposed a power function for the primary region, a linear function for the secondary

region, and an exponential function for the tertiary region of the creep curves as bellow:

, bP PSaN N N (1)

, and bP PS PS PS ST PS PSc N N N N N aN (2)

( ) 1 , and STf N NP ST ST ST PS ST PSd e N N c N N (3)

Where:



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P : Accumulated permanent strain,

N : Number of load repetitions,

PSN : Number of load repetitions corresponding to the starting point of the secondary region,

STN : Number of load repetitions corresponding to the starting point of the tertiary region,

PS : Plastic strain at the starting point of the secondary region,

ST : Plastic strain at the starting point of the tertiary region, and

a, b, c, d, f: Regression constants.

West et al. (2004) developed another three-stage model to characterize the creep curves from dynamic

creep tests. However, their proposed model cannot estimate the locations of the two boundary points on

the creep curves.

Mehrara and Khodaii (2009) evaluated the model proposed by Zhou et al. (2004) and developed an

improved model for the tertiary region of the creep curves of SBS modified asphalt mixtures. They

suggested that a second order parabolic function, Eq. (4), is more capable of accurately modeling the

tertiary region of the creep curves of these mixtures than the exponential function proposed by Zhou et al.

(2004).

2, and P ST ST ST ST ST PS ST PSd N N f N N N N c N N

(4)

Mirzahosseini et al. (2011) used computational methods to analyze permanent deformation of dense

graded asphalt mixtures. These researchers gathered a comprehensive laboratory database using the

dynamic creep tests on 270 asphalt mixture specimens, and suggested the following Eq. (5) for the Flow

Number:

/ / / 1Log Exp( )

2 / 5 2 5 2 / 4 / 4N

C S M F C S VMAF

C S VMA C S BP VMA M F VMA

(5)



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CS

: Weight percent of coarse aggregates to that of fine aggregates,

BP : Bitumen content (%),

VMA : Voids in mineral aggregates (%), and

MF

: Ratio of Marshall stability to flow.

Another study was carried out by Kalyoncuoglu and Tigdemir (2010) to develop an improved model for

the creep curves of SBS modified asphalt mixtures. These researchers proposed a logarithmic function to

accurately simulate the primary region of the creep curves derived from the dynamic creep tests on SBS

modified asphalt mixtures.

Materials and methods

The continuous aggregate gradation, having the nominal maximum size of 19mm, was used (Figure 2).

The aggregates, having the mechanical and physical properties presented in Tables 1 and 2, were siliceous

crushed stone obtained from a local quarry. In addition, 85/100 penetration pure bitumen modified with

4.5% of Styrene-Butadiene-Styrene (SBS) polymer modifier was used as binder. Marshall method was

utilized to fabricate 96 SBS modified asphalt mixture specimens having the same mix design

characteristics. For this purpose, all the specimens were mixed at the optimum bitumen content, 5.4%,

and compacted in a laboratory environment, applying 55 blows on each face. The 96 Marshall specimens

were used in a laboratory dynamic creep test program at four different loading frequencies, four different

temperatures, two moisture conditions, dry and wet, and three replicates. Table 3 shows the experimental

design used in this study for specimen fabrication.

The dynamic creep tests were carried out on the specimens using the Universal Testing Machine (UTM-

25) having the capability of applying up to 25kN loads. Generally, the modified asphalt mixtures are

expected to sustain the applied loads for much more loading cycles before failure than the unmodified

ones. Therefore, all the dynamic creep tests, due to the time limitations, were conducted up to the loading

cycle of 10000, and the tests were then terminated manually. Therefore, no creep curve entered the



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tertiary region. All the dynamic creep tests were conducted with a square wave loading pattern at the

stress level of 200kPa and a static pre-stress of 20kPa for 5 minutes as defined by the National

Cooperative Highway Research Program (NCHRP) Project 9-19 (NCHRP 465, 2002), four different

loading frequencies of 0.5, 1, 5, and 10Hz, four temperatures of 40, 20, 5, and -5°C, and three replicates.

It must be noted that because the water inside the wet specimens freezes at the temperature of -5°C, the

resulted ice acts as an additional factor resisting against the applied loads. It means the specimens'

components do not sustain the whole applied load and part of the load is carried by the ice inside the air

voids of the specimen, and the accumulated permanent strains may be unrealistic under such a condition.

Therefore, to model the creep curves of the wet specimens, it was decided to conduct the dynamic creep

tests at the three temperatures of 40, 20, and 5°C, and the temperature of -5°C was removed from the

experimental program of the wet specimens.

The loading times were 0.5, 0.1, 0.05, and 0.01 seconds; while the rest times between two successive

loading cycles were 1.5, 0.9, 0.15, and 0.09 seconds for the loading frequencies of 0.5, 1, 5, and 10Hz

respectively.

Before testing, all the wet specimens were first saturated in accordance with ASTM-D4867 (1995). An

asphalt mixture specimen having a saturation level of 55% to 80% was assumed to be acceptable and the

specimen could be directly used for the dynamic creep test under the wet condition. However, the

specimens, having the saturation level of less or more than the mentioned limit, should be saturated again

by the same method until satisfying the mentioned requirement. The wet specimens were also placed

inside a water vessel during the dynamic creep tests. Therefore, the surrounding water could enter in and

exit out of the specimens easily under the dynamic loading.

To accurately control the temperatures of both the specimen and environment, two thermometers, one

inside a dummy specimen and the other out of the specimen in the chamber, were used.

More details regarding the conducted dynamic creep tests on the SBS modified asphalt mixture specimens

can be found elsewhere (Khodaii et al., 2013).

Tests results and modeling



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The effects of different loading frequencies, various temperatures, and the two moisture conditions, dry

and wet, on the creep curves of SBS modified asphalt mixtures have been comprehensively investigated

in an earlier work (Khodaii et al., 2013). However, this paper aims to model the primary and secondary

regions of the creep curves for SBS modified asphalt mixtures under the mentioned treatments, and to

present a stepwise method by which the boundary point connecting the primary to the secondary regions

of the creep curves can be identified.

As mentioned earlier, the three-stage model proposed by Zhou et al. (2004), a power model for the

primary, a linear model for the secondary, and an exponential model for the tertiary regions, is the most

accurate model, among all the developed models in this field, to fit to the creep curves of unmodified

asphalt mixtures, and to identify the locations of the two boundary points on the creep curves. However,

Kalyoncuoglu and Tigdemir (2010) suggested that a logarithmic model more accurately simulate the

primary region of the creep curves of SBS modified asphalt mixtures than the power model proposed by

Zhou et al. (2004). Therefore, both the power and the logarithmic models were examined for the primary

region of the creep curves of all the specimens. In addition, Mehrara and Kodaii (2009) mentioned that a

second order parabolic function better models the tertiary region of the creep curves of SBS modified

asphalt mixtures than the exponential function proposed by Zhou et al. (2004). However, because all the

dynamic creep tests in this study were conducted up to the loading cycle of 10000, some specimens could

not enter the tertiary region, and remained in the secondary region. Therefore, it was unfortunately

impossible to model the tertiary region, and to identify the location of the boundary point connecting the

secondary to the tertiary regions of the creep curves for the specimens of this study.

In the course of the present study, the following method was used to fit the best model to the primary and

the secondary regions of the creep curves. The method was exemplified for the dry specimens at the

temperature of 40°C and the loading frequencies of 0.5 and 10Hz.

At first, the creep curves of all the specimens were plotted as the accumulated permanent strains versus

the loading cycles. It must be noted that all the creep curves were plotted using the average results of the

three replicates. It means that the accumulated permanent strains of the three replicate specimens were



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averaged for each single loading cycle, and then the averaged data points were used to plot the creep

curves. Figures 3 and 4 show the creep curves of the dry specimen at the temperature of 40°C and the

loading frequencies of 0.5 and 10Hz respectively. According to Zhou et al. (2004), a power model was

then fitted to each creep curve, and the model coefficients were determined using the statistical regression

analysis. Figures 3 and 4 also show the power models fitted as the trend lines to the creep curves of the

dry specimens at the temperature of 40°C and the loading frequencies of 0.5 and 10Hz respectively. It is

seen from Figure 3 that the power model having a high coefficient of determination, 0.973, was

acceptably fitted to the creep curve. However, it is evident from Figure 4 that despite the high coefficient

of determination, 0.957, the power model was unacceptably fitted to the creep curve for both the initial

and last loading cycles.

Afterwards, the accumulated permanent strain for each creep curve at the last loading cycle, N = 10000,

was calculated using the developed power model. For example, the accumulated permanent strains at the

loading cycle of 10000 for the dry specimens at the temperature of 40°C and both the loading frequencies

of 0.5 and 10Hz were calculated as below:

Loading Frequency of 0.5Hz: 0.078( ) 5757.748 10000 11810.075μsP Calculated

Loading Frequency of 10Hz: 0.214( ) 1202.281 10000 8629.904μsP Calculated

Then, the deviation error of the calculated accumulated permanent strains from the measured ones at the

last loading cycle, N = 10000, was determined for each creep curve using the following Eq. (6):

( ) ( )

( )

P Calculated P Measurede

P Measured

D

(6)

Where:

De: Deviation error of calculated accumulated permanent strains from measured ones at a certain loading

cycle,

( )P Calculated : Calculated accumulated permanent strain at a certain loading cycle (μs), and



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( )P Measured : Measured accumulated permanent strain at a certain loading cycle (μs).

Zhou et al., (2004) have considered a deviation error of less than 3% for the last loading cycle as an

acceptable accuracy for their model. It means that if the deviation error of the power model from the

measured accumulated permanent strain at the last loading cycle is less than 3%, the creep curve can be

assumed to be still within the primary region at the last loading cycle. However, if the deviation error at

the last loading cycle is larger than 3%, the creep curve can be assumed to be within the secondary region

at the last loading cycle. In the latter case, it is necessary to remove the last loading cycle, and to repeat

the above mentioned method until reaching a loading cycle with a deviation error of less than 3%. The

resulted loading cycle is defined as the loading cycle in which the boundary point between the primary

and secondary regions occurs. The deviation errors at the loading cycle of 10000 were calculated as

below for the dry specimens at the temperature of 40°C and the loading frequencies of 0.5 and 10Hz.

Loading Frequency of 0.5Hz: 11810.075 11841.710

100 0.27% 3%11841.710

P Calculated P Measured

eP Measured

D

Loading Frequency of 10Hz:

8629.904 8233.208

100 4.82% 3%8233.208


eP Measured

D

It is evident that the deviation error for the loading frequency of 0.5Hz, 0.27%, is less than 3%. It means

that the creep curve of the dry specimen at the loading frequency of 0.5Hz and the temperature of 40°C is

still within the primary region at the loading cycle of 10000. However, it is understood from Figure 3 that

the creep curve of this specimen has entered the secondary linear region before the loading cycle of

10000. On the other hand, the deviation error for the loading frequency of 10Hz, 4.82%, is larger than

3%. In other words, the creep curve of the dry specimen at the loading frequency of 10Hz and the

temperature of 40°C has entered the secondary linear region before the loading cycle of 10000. Therefore,

the above mentioned method was repeated again. However, no loading cycle was found with the



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deviation error of less than 3% as the boundary point connecting the primary to the secondary regions;

while it is clearly evident from Figure 4 that the creep curve of this specimen has entered the secondary

linear region. In addition, it is seen from Figure 4 that the power model could not fit well to the creep

curve for both the initial and the last loading cycles. Therefore, it can be concluded that neither the power

model nor the deviation error of 3%, which is to be checked at a single loading cycle, are good choices to

model the primary region of the creep curves of SBS modified asphalt mixtures. These problems were

realized to be more pronounced for the lower temperatures, 5, and -5°C, and the higher loading

frequencies, 5 and 10Hz. To eliminate these shortcomings, a logarithmic model with a deviation error of

1%, which must be simultaneously checked for all the loading cycles, was proposed during this study.

Figures 5 and 6 show the logarithmic models fitted to the creep curves of the dry specimens at the

temperature of 40°C and the loading frequencies of 0.5 and 10Hz respectively. It is evident from Figures

5 and 6 that the logarithmic models having the higher coefficients of determination, 0.993 and 0.998, than

those of the power models, 0.973 and 0.957, were acceptably fitted to the creep curves for all the loading

cycles of both the primary and the secondary regions.

Based on the above mentioned explanations, two different approaches may be taken in to consideration to

model the primary and the secondary regions of the creep curves. In the approach 1, the whole creep

curves shown on Figures 3 and 4 can be simultaneously modeled using the developed logarithmic

functions. Because the creep curves of Figures 3 and 4 obviously constitute of both the primary and the

secondary regions, it can be concluded that the logarithmic functions shown on Figures 5 and 6 have

modeled both the regions simultaneously. It means that there is only one region having no boundary point

on these creep curves. In other words, if the developed logarithmic functions are used to model the creep

curves, there would be no further need to separate these two regions from each other, and it is possible to

model the whole creep curves using the logarithmic functions simultaneously.

In order to check the developed logarithmic models for the creep curves of the dry specimens at the

temperature of 40°C and both the loading frequencies of 0.5 and 10Hz, the accumulated permanent strains

were calculated at all the loading cycles. For example, the accumulated permanent strains at the loading



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cycle of 10000 were calculated as below:

Loading Frequency of 0.5Hz: ( ) 788.991 Ln 10000 4518.586 11785.462μsP Calculated

Loading Frequency of 10Hz: ( ) 1169.961 Ln 10000 2515.093 8260.646μsP Calculated

Then, the deviation errors of the logarithmic models at all the loading cycles were calculated. For

example, the deviation errors at the loading cycle of 10000 were calculated as below for the dry

specimens at the temperature of 40°C and both the loading frequencies of 0.5 and 10Hz.

Loading Frequency of 0.5Hz: 11785.462 11841.710

100 0.47% 1%11841.710


eP Measured

D

Loading Frequency of 10Hz:

8260.646 8233.208

100 0.33% 1%8233.208


eP Measured

D

It is seen that both the deviation errors for the loading frequencies of 0.5 and 10Hz are less than 1%. It

means that the logarithmic models were fitted very well to both the primary and the secondary regions of

the creep curves of the dry specimens at the temperature of 40°C and the loading frequencies of 0.5 and

10Hz.

However, because the logarithmic functions, proposed in the approach 1, were used to model both the

primary and the secondary regions of the creep curves simultaneously, it was not possible to identify the

location of the boundary point connecting the two regions.

Figures 7 to 10 show the logarithmic models fitted simultaneously to both the primary and the secondary

regions of the creep curves for the dry specimens having all the selected loading frequencies at the

temperatures of 40, 20, 5, and -5°C respectively. In addition, Figures 11 to 13 show the same graphs for

the wet specimens as those shown on Figures 7 to 10 for the dry ones. It is clear from all these Figures

that the logarithmic models were fitted very well to all the creep curves of both the dry and wet specimens



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at all the selected loading frequencies and temperatures.

Table 4 presents the logarithmic model coefficients (a, b) and the coefficients of determination (R2)

resulted by the approach 1 for all the experimental combinations. It is understood from Table 4 that no

specific curve, polynomial, power, logarithmic, exponential, linear, etc., could be fitted well to the data

points to plot the model coefficients versus the loading frequencies or the temperatures. However, it is

evident that the model coefficients are only functions of the loading frequency and the temperature;

because all the specimens were fabricated under the same conditions and using the same mix design

characteristics. Therefore, if a dynamic creep test is carried out on a SBS modified asphalt mixture,

having the same mix design characteristics as for the mixtures used in this study, at a certain loading

frequency between 0.5 to 10Hz and a given temperature between -5 and 40°C, assuming a linear

relationship between the variations of every two loading frequencies or temperatures and the

corresponding model coefficients, it is possible to determine both the model coefficients of a and b via the

interpolation of the data presented in Table 4, and to model the primary and the secondary regions of the

creep curve simultaneously by the resulting logarithmic function. For example, the model coefficients of

a and b for a given SBS modified asphalt mixture specimen, the same as those used in this study, at the

loading frequency of 3Hz and the temperature of 10°C can be interpolated from Table 4 to be 673.374

and 667.578 respectively. Therefore, the primary and the secondary regions of the creep curve of this

specimen can be predicted and plotted with no need to conduct the related dynamic creep tests.

On the other hand, the approach 2, having a classical view point, considers the creep curves shown on

Figures 5 and 6 to constitute of both the primary and secondary regions separately. In this approach, a

stepwise method was proposed based on the following steps to model the two regions separately, and to

identify the location of the boundary point connecting the two regions:

Step 1: A loading cycle was visually selected among the initial loading cycles of the secondary linear

region on each creep curve. It must be noted that this loading cycle was not necessarily the boundary

point connecting the primary to the secondary regions,

Step 2: The loading cycles placed before the visually selected loading cycle were removed and a new



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creep curve, representing an approximation for the secondary linear region, was plotted using the

remaining loading cycles,

Step 3: A linear model was fitted to the approximate secondary linear region and the model coefficients

were determined using the statistical regression analysis,

Step 4: The accumulated permanent strains at all the loading cycles of the approximate secondary linear

region were calculated using the developed linear model coefficients,

Step 5: The deviation errors of the calculated accumulated permanent strains from the measured ones at

all the loading cycles of the approximate secondary linear region were determined using the Eq. (7),

Step 6: If all the deviation errors were simultaneously less than or equal to 1%, the linear model was

assumed to be a good representative for the secondary linear region. However, if at least one of the

deviation errors for a certain loading cycle was larger than 1%, the linear model, as an unsuitable

representative for the secondary linear region, was rejected. Then, the next loading cycles placed after the

visually selected loading cycle were removed one by one and all the steps 2 to 6 were repeated again until

reaching a loading cycle after which the deviation errors of all the loading cycles were simultaneously

less than or equal to 1%. Then, the linear model was selected as the final function governing the

secondary linear region of the creep curve,

Step 7: The logarithmic function resulted by the approach 1 was selected as the final function governing

the primary region of the creep curve,

Step 8: A set of simultaneous equations, having two equations and two unknowns, was established using

the two functions governing the primary and the secondary regions. Solving the simultaneous equations,

the accumulated permanent strain and the corresponding loading cycle in which the primary region

connects to the secondary region were identified.

The above mentioned stepwise method was used for the dry specimens at the temperature of 40°C and

both the loading frequencies of 0.5 and 10Hz, and it was realized that the loading cycles of 3495 and 6223

correspond to the boundary points connecting the primary to the secondary regions of the creep curves for

these two specimens respectively. As it is clear, the boundary point for the specimen tested at the lower



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loading frequency occurred at a lower loading cycle than that for the specimen tested at the higher loading

frequency. This observation was expected because of the higher resistance of the dry specimens at the

higher loading frequencies and vice versa as a direct result of the loading time. Permanent deformation is

function of many different factors among which loading time, load level, moisture condition, and

temperature are the most important external factors. Because the applied loading used in this study was

constant at 200kPa, the permanent strains of the dry specimens at a constant temperature were only a

function of the loading time. In addition, as the loading time increases at a constant loading cycle, the

permanent strain of the dry specimen increases too. On the other hand, as the resistance of a dry specimen

against the permanent strains decreases, the loading cycle corresponding to the boundary point connecting

the primary to the secondary regions of the creep curve, decreases too. Therefore, as the loading

frequency increases from 0.5 to 10Hz for the dry specimens at a constant temperature, the loading time

decreases; thus the loading cycle corresponding to the boundary point connecting the primary to the

secondary regions of the creep curve increases due to the higher resistance of the specimen against the

permanent strains.

For brevity, the logarithmic and the linear models fitted to the primary and the secondary regions of the

creep curves are shown on Figures 11 to 14 only for the dry specimens at the temperature of 40°C and the

loading frequencies of 0.5, 1, 5, and 10Hz respectively. It is evident from all these Figures that the

logarithmic and the linear models, having high coefficients of determination, were fitted very well to the

primary and the secondary regions of all the creep curves of the dry specimens at all the selected loading

frequencies and temperatures. It must be noted that the same results were obtained for the wet specimens.

Conclusion

This study was conducted to model the primary and the secondary regions of the creep curves derived

from the dynamic creep tests on SBS modified asphalt mixture specimens at different loading

frequencies, temperatures, and moisture conditions. For this purpose, two different approaches were

proposed. In the approach 1, both the primary and the secondary regions were simultaneously modeled

using a logarithmic function; while in the approach 2, having a classical point of view, the two regions



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were separately modeled via a logarithmic function, for the primary region, and a linear function, for the

secondary region. In addition, a stepwise method was presented by which the boundary point connecting

the primary to the secondary regions can be identified. In both the proposed approaches, the interpolation

technique can be utilized to model the primary and the secondary regions of the creep curves for a SBS

modified asphalt mixture, having the same mix design characteristics as for those used in this study, at a

certain loading frequency within the limit of 0.5 to 10Hz, and a given temperature between -5 to 40°C. It

must be noted that because all the specimens were fabricated using the same aggregate type, aggregate

gradation, bitumen type, and bitumen content, general application of the logarithmic models to asphalt

mixtures with aggregate types, aggregate gradations, bitumen types, and bitumen contents other than

those tested may yield inaccurate results. More research is needed to account for the effects of these

important variables.

References

Archilla, A. R., Diaz, L. G., and Carpenter, S. H. (2007). “Proposed method to determine the flow number

from laboratory axial repeated loading tests in bituminous mixtures.” Transportation research board

86th annual meeting, Transportation Research Board, Report Number 07-1901, Washington DC,

United States.

Aschenbrener, T., and Currier, G. (1993). “Influence of Testing Variables on the Results from the

Hamburg Wheel-Tracking Device.” Colorado Department of Transportation (CDOT), Report

Number CDOT-DTD-R-93-22, Denver, Colorado.

ASTM-D4867. (1995). “Standard Test Method for Effect of Moisture on Asphalt Concrete Paving

Mixtures.” American Society for Testing and Materials (ASTM), Annual Book of ASTM Standards.

Biligiri, K. P., Kaloush, K. E., Mamlouk, M. S., and Witczak, M. W. (2007). “Rational Modeling of

Tertiary Flow for Asphalt Mixtures.” Journal of Transportation Research Record., 2001, 63-72.

Christensen, D. W. (1998). “Analysis of Creep Data from Indirect Tension Test on Asphalt Concrete.”

Journal of the Association of Asphalt Paving Technologists, 67, 458–492.

Crockford, W. W. (2001). “Data Analysis—Load Controlled Dynamic Tests.” ShedWorks Inc., 2nd ed.



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Izzo, R. P., and Tahmoressi, M. (1999). “Testing Repeatability of the Hamburg Wheel-Tracking Device

and Replicating Wheel-Tracking Devices among Different Laboratories.” Journal of the Association

of Asphalt Paving Technologists, 68, 589-612.

Kalyoncuoglu, S.F., and Tigdemir, M. (2010). “A Model for Dynamic Creep Evaluation of SBS Modified

HMA Mixtures.” Journal of Construction and Building Materials, Science Direct, 25(2), 859-866.

Khodaii, A., Moghadas Nejad, F., Saleh Ahari, A., and Forough, S. A. (2013). “Investigating the Effects

of Loading Frequency and Temperature on Moisture Sensitivity of SBS Modified Asphalt Mixtures.”

Journal of Materials in Civil Engineering, Manuscript Number MTENG-1852, Under Review.

Khodaii, A., and Mehrara, A. (2009). “Evaluation of Permanent Deformation of Unmodified and SBS

Modified Asphalt Mixtures Using Dynamic Creep Test.” Journal of Construction and Building

Materials, Science Direct, 23(7), 2586–2592.

Kim, Y. R. (2009). “Modeling of asphalt concrete” ACSE Press McGraw-Hill Construction, United

States.

Mehrara, A., and Khodaii, A. (2011). “Evaluation of Asphalt Mixtures’ Moisture Sensitivity by Dynamic

Creep Test.” ASCE Journal of Materials in Civil Engineering, 23(2), 212-219.

Mirzahoseini, M. R., Aghaeifar, A., Alavi, A. H., and Gandomi, A. H. (2011). “Permanent Deformation

Analysis of Asphalt Mixtures using Soft Computing Techniques.” Journal of Expert Systems with

Application, 38(5), 6081-6100.

Monismith, C. L., Ogawa, N., and Freeme, C. R. (1975). “Permanent deformation characteristics of

subgrade soils due to repeated loading.” Journal of Transportation Research Record, 537, 1-17.

NCHRP 465. (2002). “Simple Performance test for Superpave Mix Design.”, National Cooperative

Highway Research Program (NCHRP), ARA, Inc. and ERES Consultants Division, Final Report,

Appendix C.

Sousa, J. B., Deacon, J. A., Weissman, S. L., Leahy, R. B., Harvey, J. T., Paulsen, G., Coplantz, J. S., and

Monismith, C. L. 1994. “Permanent Deformation Response of Asphalt Aggregate Mixes.” Strategic

Highway Research Program, National Research Council, Report SHRP-A-415, Washington DC, 437.



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West, R. C., Zhang, J., and Cooley, L. A. (2004). “Evaluation of the asphalt pavement analyzer for

moisture sensitivity testing.” National Center for Asphalt Technology (NCAT), Report Number 04-04.

Auburn University, Alabama.

Zhou, F., Scullion, T., and Sun, L. (2004). “Verification and Modeling of Three-Stage Permanent

Deformation Behavior of Asphalt Mixes.” ASCE, Journal of Transportation Engineering, 130(4),

486-494.

Zhou, F., and Scullion, T. (2003). “Preliminary field validation of simple performance tests for permanent

deformation.” Journal of Transportation Research Record, 1832, 209-216.



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Figure Captions List

Figure 1. Schematic showing the relationship between total plastic strains and loading cycles

Figure 2. Aggregate gradation used and the gradation limits

Figure 3. Creep curve and the fitted power model of the dry specimen under the temperature of 40°C and

the loading frequency of 0.5Hz

Figure 4. Creep curve and the fitted power model of the dry specimen under the temperature of 40°C and

the loading frequency of 10Hz

Figure 5. Creep curve and the fitted logarithmic model of the dry specimen under the temperature of

40°C and the loading frequency of 0.5Hz

Figure 6. Creep curve and the fitted logarithmic model of the dry specimen under the temperature of

40°C and the loading frequency of 10Hz

Figure 7. Logarithmic models fitted to whole the creep curves of the dry specimens under the

temperature of 40°C and all the loading frequencies






temperature of -5°C and all the loading frequencies

Figure 11. Logarithmic and linear models fitted to the primary and the secondary regions of the dry

specimen under the temperature of 40°C and the loading frequency of 0.5Hz


specimen under the temperature of 40°C and the loading frequency of 1Hz





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Secondary

Tertiary

Per

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Stra

in

Primary

P

Loading Cycles

Accepted Manuscript Not Copyedited



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60

80

100

assi

ng

Grad. Used

Grad. LimitsN

o. 2

00

No.

50

No.

8

No.

4

3/4

1/20

20

40

Per

cent

p

Sieve size (0.45 power)




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d.

= 5,757.748N 0.078R² 0 973

10000

12000

14000

rain

(μs

)

f=0.5, T=40 Power (f=0.5, T=40)

R² = 0.973

0

2000

4000

6000

8000

0 2000 4000 6000 8000 10000 12000

Per

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Loading Cycles




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d.

= 1,202.281N 0.214R² 0 9576000

700080009000

10000

rain

(μs

)

f=10, T=40 Power (f=10, T=40)

R² = 0.957

0100020003000400050006000

0 2000 4000 6000 8000 10000 12000

Per

man

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Loading Cycles




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d.

= 788.991Ln(N) + 4,518.586R² 0 993

10000

12000

14000

rain

(μs

)

f=0.5, T=40 Log. (f=0.5, T=40)

R² = 0.993

0

2000

4000

6000

8000

0 2000 4000 6000 8000 10000 12000

Per

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d.

= 1,169.961Ln(N) - 2,515.093R² 0 998

6000700080009000

rain

(μs

)

f=10, T=40 Log. (f=10, T=40)

R² = 0.998

010002000300040005000

0 2000 4000 6000 8000 10000 12000

Per

man

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Str

Loading Cycles




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R² = 0.993R² = 0.998R² = 0.99210000

12000

14000

ain

(μs)

Log. (f=0.5, T=40) Log. (f=1, T=40)

Log. (f=5, T=40) Log. (f=10, T=40)

R 0.992R² = 0.998

0

2000

4000

6000

8000

0 2000 4000 6000 8000 10000 12000

Per

man

ent

Stra

Loading Cycles




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R² = 0.989R² = 0.991R² = 0.983

700080009000

10000

ain

(μs)

Log. (f=0.5, T=20) Log. (f=1, T=20)Log. (f=5, T=20) Log. (f=10, T=20)

R² = 0.985

0100020003000400050006000

0 2000 4000 6000 8000 10000 12000

Per

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R² = 0.986

R² = 0.986R² = 0 9926000

700080009000

ain

(μs)

Log. (f=0.5, T=5) Log. (f=1, T=5)

Log. (f=5, T=5) Log. (f=10, T=5)

R² = 0.992

R² = 0.994

0100020003000400050006000

0 2000 4000 6000 8000 10000 12000

Per

man

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Stra

Loading Cycles




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R² = 0.992

R² = 0.9965000

6000

7000

ain

(μs)

Log. (f=0.5, T=-5) Log. (f=1, T=-5)

Log. (f=5, T=-5) Log. (f=10, T=-5)

R² = 0.986

R² = 0.9610

1000

2000

3000

4000

0 2000 4000 6000 8000 10000 12000

Per

man

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Loading Cycles




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d.

= 0.143N + 10,456.266R² = 0.993

Boundary Point at Loading Cycle of 3495

10000

12000

14000

rain

(μs

)

Linear (Secondary Region, f=0.5, T=40)

Log. (Primary Region, f=0.5, T=40)

= 788.991ln(N) + 4,518.586R² = 0.993

0

2000

4000

6000

8000

0 2000 4000 6000 8000 10000 12000

Per

man

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Str

Loading Cycles




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d.

= 0.165N + 9,704.376R² = 0.977


8000

10000

12000

ain

(μs)

Linear (Secondary Region, f=1, T=40)

Log. (Primary Region, f=1, T=40)

= 1,137.395ln(N) + 841.872R² = 0.998

0

2000

4000

6000

8000

0 2000 4000 6000 8000 10000 12000

Per

man

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tra

Loading Cycles




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d.

= 0.108N + 8,500.869R² = 0 983


8000

10000

12000

rain

(μs

)



R² = 0.983= 812.798ln(N) + 2,110.990

R² = 0.992

0

2000

4000

6000

0 2000 4000 6000 8000 10000 12000

Per

man

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Str

Loading Cycles




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d.

= 0.155N + 6,741.115R² = 0.988


6000

7000

8000

9000

rain

(μs

)



= 1,169.961ln(N) - 2,515.093R² = 0.992

0

1000

2000

3000

4000

5000

0 2000 4000 6000 8000 10000 12000

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Table 1. Mechanical properties of aggregatesProperty Value (%) StandardLos Angeles abrasion loss 25 ASTM C131Particles fractured in 1 face 87 ASTM D5821Particles fractured in 2 faces 93 ASTM D5821Aggregate coating 95 AASHTO T182Flakiness 10 BS – 812Sand equivalent 85 ASTM D2419Sodium Sulphate soundness 0.4 ASTM C88




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Table 2. Physical properties of aggregates Property Value Standard

Coarse Aggregates (retained on sieve #8) Bulk specific gravity (g/cm3) 2.325 AASHTO-T85 Apparent specific gravity (g/cm3) 2.502 AASHTO-T85 Water absorption (%) 1.60 AASHTO-T85

Fine Aggregates (passing sieve #8 and retained on sieve #200) Bulk specific gravity (g/cm3) 2.316 AASHTO-T84 Apparent specific gravity (g/cm3) 2.498 AASHTO-T84 Water absorption (%) 1.60 AASHTO-T84

Filler (passing sieve #200) Bulk specific gravity (g/cm3) 2.312 AASHTO-T100 Apparent specific gravity (g/cm3) 2.425 ASTM C128-04




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Table 3. Experimental design Experimental No. of Variable variables levels levels Loading frequency 4 0.5Hz, 1Hz, 5Hz, 10Hz Temperature 4 40°C, 20°C, 5°C, -5°C Moisture condition 2 Dry & Wet Replication 3 Dynamic creep tests




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Table 4. Logarithmic model coefficients and coefficients of determination for all the experimental combinations Mois. Freq. Temp. εp = a.Ln(N) + b R² Cond. (Hz) (°C) a b Dry 0.5 40 788.991 4518.586 0.993

20 559.657 3983.888 0.989 5 640.362 1955.689 0.986 -5 923.401 -2230.091 0.992 1 40 1137.395 841.872 0.998 20 777.507 1186.179 0.991 5 342.971 3252.168 0.986 -5 660.668 -757.971 0.996 5 40 812.798 2110.990 0.992 20 729.995 774.426 0.983 5 923.401 -2230.091 0.992 -5 224.319 -198.992 0.986 10 40 1169.961 -2515.093 0.998 20 503.781 1048.897 0.985 5 438.287 488.637 0.994 -5 62.271 -211.454 0.961

Wet 0.5 40 991.212 2452.949 0.983 20 599.019 3856.142 0.999 5 1009.488 -1089.058 0.999 1 40 1007.555 2527.488 0.999 20 611.460 4508.016 0.983 5 559.657 3483.888 0.989 5 40 861.264 4279.545 0.983 20 904.457 2814.996 0.991 5 616.344 3368.146 0.980 10 40 868.477 6421.988 0.963 20 503.781 7548.897 0.985 5 774.227 3352.065 0.991

Accepted Manuscript

Not Copyedited



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Documents

Modeling the Primary and Secondary Regions of Creep Curves for SBS-Modified Asphalt Mixtures under Dry and Wet Conditions