Complex Modulus Determination of Asphalt Concrete Using Indirect Tension Test

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ABSTRACT

MOMEN, MOSTAFA HASSAN. Complex Modulus Determination of Asphalt Concrete

Using Indirect Tension Test. (Under the direction of Dr. Y. Richard Kim).

The purpose of this research is to present the results from an analytical/experimental

study on the dynamic modulus testing of hot mix asphalt (HMA) using the indirect tension

(IDT) mode. The analytical solution for dynamic modulus determination in IDT was

developed by Kim (14) using the theory of linear viscoelasticity. To verify the analytical

solution, temperature and frequency sweep tests were conducted on 24 asphalt mixtures

commonly used in North Carolina, using both axial compression and IDT test methods. In

doing so, a modified dynamic modulus test protocol is introduced that reduces the required

testing time by using more frequencies and fewer temperatures based on the

time-temperature superposition principle. A comparison of results from the axial compression and

IDT test methods shows that the dynamic modulus mastercurves and shift factors derived

from the two methods are in good agreement. It was also found that Poisson’s ratio is a weak

function of the loading frequency; its effect on the phase angle mastercurve is discussed.

After verification of the analytical solution, another study was conducted to evaluate the

effect of aggregate size on the variability of test results, where the coefficient of variation

(CV) was computed for each aggregate size and the results were compared. It was found that

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Complex Modulus Determination of Asphalt

Concrete Using Indirect Tension Test

BY

MOSTAFA MOMEN

A THESIS SUBMITTED TO THE GRADUATE FACULTY OF NORTH CAROLINA STATE UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

DEPARTMENT OF CIVIL ENGINEERING

RALEIGH

2004

APPROVED BY:

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DEDICATION

It is my honor to dedicate this thesis to my beloved wife for the support and love she

showed me throughout the two year journey of my graduate studies towards the Master of

Science degree. I would like also to dedicate this thesis to my father who encouraged me to

pursue a graduate degree and has been always there for me when I needed his guidance and

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BIOGRAPHY

Mostafa Momen was born in Urbana-Champaign, ILLINOIS where he lived with his

parents for three years while his father was pursuing a Doctor of Philosophy degree in civil

engineering. In1982, he moved to United Arab Emirates with his family where he lived for

one year and attended kinder garden. In 1983, he moved to Egypt with his family and

attended Manor House School for 11 years from which he graduated in June 1994. In

September of 1994 the author enrolled in the American University in Cairo majoring in

construction engineering and management, and received a Bachelor of Science Degree in

June 1999. He traveled in August 1999 to United Arab Emirates where he worked for a

consulting company as a structural engineer for two years, and then he worked with his father

as an arbitration engineer for one and a half years. In December 2002 he decided to go back

to the United States to pursue a graduate degree in civil engineering. He enrolled in North

Carolina State University where he worked as a graduate research assistant under the

supervision of Dr. Y. Richard Kim. This led to the research contained herein and to this

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ACKNOWLEDGEMENTS

First of all, I thank ALLAH for his guidance and support.

The author would like to thank the members of his advisory committee and all the

research group members for their help and guidance during his graduate studies. I also extend

a sincere appreciation for my advisor Dr. Y. Richard Kim for providing me the opportunity

to work with him as a research assistant and for his encouragement and technical assistance.

Special thanks to Mark King who helped me a great deal and he was also my partner and

dear friend throughout this whole process. I also would like to thank Dr. Ghassan Chehab,

Dr. Youngguk Seo, and Shane Underwood for their help to me in both my experimental and

analytical work.

Finally, my gratitude is extended to my parents, Hassan Momen and Faten Khalifa,

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LIST OF TABLES

Table 2.1 Coefficients for Poisson’s Ratio and Dynamic Modulus... 10

Table 2.2 Coefficients Used to Calculate Poisson’s Ratio and Creep Compliance... 13

Table 2.3 Coefficients in Eq. (2-42) ... 15

Table 3.1 Information on Asphalt Mixtures Tested in This Study ... 17

Table 3.2 Comparison of the NCHRP 1-37A and Modified Test Protocols ... 22

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LIST OF FIGURES

Figure 2.1 Schematic of the IDT Specimen Subjected to a Strip Load ... 6

Figure 3.1 (a) IDT Test Setup with SHRP LGD; (b) Surface-Mounted LVDTs... 20

Figure 3.2 Comparison of Mastercurves Between Three-Temperature (empty symbols) and Five-Temperature (filled symbols) Testing for S9.5C–Fine Mixture ... 23

Figure 4.1 Frequency Dependent Electronic Phase Angle for MTS ... 28

Figure 4.2 S 9.5C Fine Mixture Replicates Mastercurves ... 29

Figure 4.3 S 9.5C Fine Mixture Dynamic Modulus Mastercurve ... 30

Figure 4.4 Shift Factor as a Function of Temperature ... 30

Figure 4.5 Vertical and Horizontal Phase Angles Mastercurve... 32

Figure 4.6 Poisson’s Ratio Mastercurve ... 32

Figure 4.7 Poisson’s Ratio for 9.5 mm Mixtures... 33

Figure 5.1 Dynamic modulus master curves for: (a) S9.5A-Fine mixture; (b) S9.5B-Coarse mixture; (c) S9.5C-Fine mixture; (d) S9.5C-Coarse mixture... 37

Figure 5.2 Dynamic modulus master curves for: (a) S12.5C-Fine mixture; (b) S12.5D-Coarse mixture; (c) I19.0B-Fine mixture; (d) I19.0C-Coarse mixture... 38

Figure 5.3 Dynamic modulus master curves for: (a) I19.0D-Fine mixture; (b) I19.0D-Coarse mixture; (c) B25.0B-Fine mixture; (d) B25.0B-Coarse mixture... 39

Figure 5.4 S9.5A-Fine Mixture Phase angle mastercurves... 43

Figure 5.5 I19.0D-Fine mixture Phase Angle Mastercurves ... 44

Figure 6.1 Vertical Area of Interest ... 49

Figure 6.2 Horizontal Area of Interest... 49

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Figure 6.8 Point Strain Profile along Vertical Axis... 55

Figure 6.9 Point Strain Profile along Horizontal Axis... 56

Figure 6.10 Aggregate Orientation on the Surface of the Specimen ... 57

Figure 6.11 Vertical Strain Field Measurement along the Vertical Axis ... 58

Figure 6.12 Vertical Strain Field Measurement along the Horizontal Axis ... 58

Figure 6.13 Horizontal Strain Field Measurement along the Vertical Axis ... 60

Figure 6.14 Horizontal Strain Field Measurement along the Horizontal Axis ... 61

Figure 6.15 Point Strain Profile along Vertical Axis... 62

Figure 6.16 Point Strain Profile along Horizontal Axis... 62

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LIST OF SYMBOLS AND ABBREVIATIONS

• σ: Stress

• ε: Strain

• υ: Poisson’s Ratio

• _E* _{: Dynamic Modulus}

• φ: Phase Angle

• PG: Performance Grade

• NMSA: Nominal Maximum Size Aggregate

• LVDT: Linear Variable Differential Transformer

• LGD: Load Guide Device

• SHRP: Strategic Highway Research Program

• fR: Reduced Frequency

• DIC: Digital Image Correlation

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1. INTRODUCTION

1.1 Research Needs

The design methods adopted in the NCHRP 1-37A Design Guide (3) are based on

mechanistic-empirical principles in which the prediction of pavement responses and

performance must take into account fundamental properties of layer materials. Among these,

the most important property, but a relatively new concept to state highway agencies, is the

dynamic modulus of asphalt concrete. This property represents the temperature and

frequency (and, therefore, time) dependent stiffness characteristics of the material.

Recently, a significant amount of effort has been given to the development of a test

protocol to determine the dynamic modulus of HMA. This effort has resulted in a standard

test protocol that can be used for the NCHRP 1-37A Design Guide (3). This test protocol

calls for the use of axial compression testing for measuring the dynamic modulus.

One of the issues related to the dynamic modulus is its use in forensic studies and

pavement rehabilitation design. The current dynamic modulus protocol calls for the axial

compression testing of 100 mm diameter and 150 mm tall asphalt concrete specimens. It is

often impossible to obtain this size specimen from actual pavements. Given that a typical

asphalt layer thickness is less than a few inches and that coring is the most effective method

of obtaining specimens from actual pavements, the Indirect Tension (IDT) testing of cores

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differences between the axial compression and IDT tests raise questions about the

interchangeability of the dynamic modulus values obtained from the two test methods.

Two major differences between these two test methods are:

1. State of stress: Compression testing creates the uniaxial state of stress, whereas the

stress state in the IDT test is biaxial; and

2. The relationship between compaction direction and the direction in which the

stress-strain analysis is performed: In axial compression these two directions are the same,

whereas in IDT these two directions are perpendicular. The anisotropy that may exist

in the Superpave Gyratory Compactor specimen amplifies the effect of this

difference.

1.2 Research Objective and Significance

Recognizing the importance of the IDT dynamic modulus test for the pavement

industry, these issues were investigated by an analytical/experimental study at North Carolina

State University. The objective of this research is to document the findings from this study,

which may serve as an important foundation for developing a standard test protocol for the

IDT dynamic modulus test. The research presented herein was conducted as part of the

NCDOT research project HWY 2003-09 entitled “Typical Dynamic Modulus Values of

Asphalt Concrete Mixtures.”

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indirect tension testing on Superpave Gyratory compacted specimens simulates the

relationship between the compaction direction and critical tensile stress direction in the field

accurately. These advantages support the significance of using indirect tension specimens to

determine the dynamic modulus of asphalt concrete mixtures.

1.3 Thesis Organization

This thesus is composed of seven chapters. Chapter 1 presents the research needs and

the objectives to be achieved. Theoretical background and literature review about the indirect

tension test is presented in Chapter 2. Chapter 3 includes the description of the materials used

in the study, the specimen preparation methods, and the experimental program. Chapter 4

presents the laboratory testing results and analysis. Verification of test results is presented in

Chapter 5 by comparing IDT dynamic modulus and phase angle test results to those

determined from the axial compression mode and evaluating how different the results are

using statistical analysis. Chapter 6 presents the effect of aggregate size on the variability of

test results. Finally, Chapter 7 presents the conclusions of this research and suggestions for

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2. THEORETICAL BACKGROUND

2.1 Dynamic Modulus Determination in IDT

2.1.1 Linear Elastic Solution

Since Hondros (4) derived elastic solutions for the IDT test using the plane stress

assumption, the pavement community used those solutions until Roque and Buttlar (11, 12)

introduced correction factors that take into account the bulging effect of the specimen. Then,

Kim and Wen (15) introduced viscoelastic solutions for the IDT creep test using the theory of

linear viscoelasticity.

Unlike the uniaxial test, the stress and strain distributions in the IDT specimen are

biaxial in nature. This biaxial state of stress and strain can cause errors in determining the

material properties from the IDT test unless the derivation of the properties is carefully

handled. These errors may cause enough difference between the dynamic modulus

determined from the IDT test and that from the axial compression test that the criteria and

design methods based on the dynamic modulus determined from the axial compression test

cannot be used in forensic studies where the asphalt layer is not thick enough to yield 150

mm tall specimens to be used in axial compression testing.

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Uniaxial case: σ_y =E×ε_y or

E

y y

σ

ε = (2-1)

Biaxial case: _x 1 ( _x _y)

E σ νσ

ε = − (2-2)

where x and y denote the loading direction (i.e., the vertical direction) and the direction

perpendicular to the loading direction (i.e., the horizontal direction), respectively.

In the uniaxial case (i.e., the axial compression dynamic modulus test) in Eq. (2-1),

one can divide the axial stress (σy) by the axial strain (εy) to obtain the modulus. However, in

the biaxial case (i.e., the IDT dynamic modulus test) in Eq. (2-2), one cannot obtain the

modulus by dividing the horizontal stress (σx) by the horizontal strain (εx). Rather, the correct

way to determine the modulus of the material is to divide the biaxial stress (i.e., σx-νσy) by

the horizontal strain (εx). If the incorrect solution (i.e., σx/εx) is used to represent the modulus

of the material, it should not be considered the same as the modulus determined from the

axial test.

2.1.2 Linear Viscoelastic Solution

The linear viscoelastic solution for the complex modulus of HMA under the IDT

mode has been developed by Kim et al. (14) and is presented in this section. This solution

will be applied to the IDT test results to evaluate its accuracy against the dynamic modulus

determined by the axial compression test. A comparison between both elastic and

viscoelastic solutions in IDT will be presented.

Assuming the plane stress state, Hondros (4) developed the following expressions for

stresses and strains along the horizontal diameter of the IDT specimen subjected to a strip

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Y σy a R 2α P σx X

Figure 2.1 Schematic of the IDT specimen subjected to a strip load ) ( 1 y x x

E σ νσ

ε = − (2-3)

with

[

( ) ( )

]

2 tan / 1 / 1 tan / 2 cos / 2 1 2 sin ) / 1 ( 2 )

( ₂ ₂

2 2 1 4 4 2 2 2 2 x g x f ad P R x R x R x R x R x ad P x

x = −

           + − − + + − = − π α α α π

σ (2-4)

[

( ) ( )

]

2 tan / 1 / 1 tan / 2 cos / 2 1 2 sin ) / 1 ( 2 )

( ₂ ₂

2 2 1 4 4 2 2 2 2 x g x f ad P R x R x R x R x R x ad P x

y =− +

           + − + + + − − = − π α α α π

σ (2-5)

where

P = applied load

a = loading strip width, m

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ν = Poisson’s ratio

For the viscoelastic materials subjected to the sinusoidal load in a steady state, Eq.

(2-3) can be rewritten as:

) (

1

* x y

x

E σ νσ

ε = − (2-6)

where _E*_{is the complex modulus. It is often helpful to have}_E*_{in polar form,}

φ i

e E

E* = * ⋅ _(2-7)

where _E* _{is the dynamic modulus and}_φ_{is the phase angle calculated from the time lag}

between the load and the displacement. The response to the sinusoidal load applied in the

complex modulus test is the imaginary component of the response due to the complex load,

P, shown below:

) sin (cos

0

0e P wt i wt

P

P₌ iwt ₌ ₊ _(2-8)

where P0 and w are the amplitude and the angular frequency of the sinusoidal load used in

the complex modulus test, respectively.

Substituting Eqs. (2-4), (2-5), (2-7), and (2-8) into Eq. (2-6) results in:

[

(1 ) ( ) ( 1) ( )

]

2

) ,

( ( )

*

0 _e _f _x _g _x

ad E

P t

x iwt

x = + + −

− _ν _ν

π

ε φ _(2-9)

Integrate Eq. (2-9) over the gauge length to determine the horizontal displacement, U(t), and

obtain:       − + + = =

_∫

− − − − l l l l wt i l l

x e f x dx g x dx

ad E P dx t x t

U( ) ( , ) 2 ( ) (1 ) ( ) ( 1) ( )

*

0 ν ν

π

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where l is half of the gauge length. One may extract the response that only occurs due to the

sinusoidal input by taking an imaginary part of the total response. Therefore, the dynamic

modulus from the horizontal displacement, U(t), can be expressed as:

A t U ad wt P E ) ( ) sin( 2 ₀ * ⋅ − = π φ (2-11) where       − + + =

∫

− − l l l l dx x g dx x f

A (1 ν) ( ) (ν 1) ( ) (2-12)

with

( )

(

₂ 2₂ 2

)

₄ ₄

2 2 1 2 1 R / x cos R / x sin R / x x f + + − = α

α _and_(2-13)

( )

      + −

= − _tan_α

R / x R / x tan x

g ₂ ₂

2 2 1

1

1 _(2-14)

Similarly, the analogous expression for the dynamic modulus from the vertical

displacement, V(t), can be derived as:

B t V ad wt P E ) ( ) sin( 2 ₀ * ⋅ − = π φ (2-15) where       + − − =

_∫

− − l l l l dy y m dy y n

B (ν 1) ( ) (1 ν) ( ) (2-16)

with

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By equating Eqs. (2-11) and (2-15), one can obtain

) ( )

(t B U t V

A⋅ = ⋅ (2-19)

Then, one may derive the expression for Poisson’s ratio as follows:

) ( ) ( ) ( ) ( 2 2 1 1 t V t U t V t U γ β γ β ν + − − = (2-20) where

∫

− − − − = l l l l dy y m dy y

n( ) ( )

1 β ,

∫

− − − = l l l l dy y m dy y

n( ) ( )

2 β ,

∫

− − − = l l l l dx x g dx x

f( ) ( )

1

γ , and

∫

− − + = l l l l dx x g dx x

f( ) ( )

2

γ (2-21)

Combining Eqs. (2-11) and (2-15) yield a single form of the dynamic modulus, as

shown below: ) ( ) ( ) ( ) sin( ) ( ) sin( ₀ 0 * t U t V ad t BU wt P t AV wt P E ⋅ ⋅ − + − = π φ φ (2-22)

After substituting Eqs. (2-12) and (2-16) into Eq. (2-22), one can obtain:

) ( ) ( ) sin( 2 2 2 1 2 2 1 0 * t U t V ad wt P E β γ γ β γ β π φ − − −

= (2-23)

The vertical and horizontal displacements can be expressed in sine functions as follows:

) sin(

)

(t =V₀ wt−φ

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) sin(

)

(t =U₀ wt−φ

U (2-24b)

where V0 and U0 are the constant amplitudes of vertical and horizontal displacements,

respectively. Therefore, the final form of the dynamic modulus is:

0 2 0 2 1 2 2 1 0 * ₂ U V ad P E β γ γ β γ β π − − = (2-25)

Likewise, the expression for Poisson’s ratio can be simplified as:

0 2 0 2 0 1 0 1 V U V U γ β γ β ν + − −

= (2-26)

The coefficients, β1, β2, γ1, and γ2, in Eqs. (2-25) and (2-26), are calculated for

different specimen diameters and gauge lengths, and are presented in Table 2.1. Eqs. (2-25)

and (2-26) are based on the plane stress assumption. Kim and Wen (15) used the

three-dimensional finite element analysis to calculate the center strain in the IDT specimen and

concluded that the error due to the plane stress assumption is negligible.

Table 2.1 Coefficients for Poisson’s Ratio and Dynamic Modulus

Specimen

Diameter (mm) Gauge Length (mm) β1 β2 γ1 γ2

101.6 25.4 -0.0098 -0.0031 0.0029 0.0091

101.6 38.1 -0.0153 -0.0047 0.0040 0.0128

101.6 50.8 -0.0215 -0.0062 0.0047 0.0157

152.4 25.4 -0.0065 -0.0021 0.0020 0.0062

152.4 38.1 -0.0099 -0.0032 0.0029 0.0091

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2.2 Creep Compliance Calculation in IDT

Following Hondros’s analysis, Eqs. (2-3), (2-4), and (2-5) can be used to express

stresses and strains along the horizontal diameter of the specimen. Due to the viscoelastic

characteristics of asphalt concrete, strain is also time dependent. The elastic-viscoelastic

correspondance was used to solve the viscoelastic problem. The principle states that the

viscoelastic equations are similar to the elastic equations in the Laplace transformed domain

with the substitution of the Carson transformed modulus for the elastic modulus. Poisson’s

ratio is assumed to be constant. Thus, the transformed form of the stress-strain relationship is:

( )

ε_ ~ σ σ _ _ , x s E v x y

= 1_ − _ (2-27)

where

ε

_ ,

σ

__x, and

σ

__yare Laplace transformed strain and stresses while

E

~ is the Carson

transformed modulus. According to the theory of viscoelasticity, the relationship between the

relaxation modulus and creep compliance is:

E D~ × ~ =1 (2-28)

where

D

~ is the Carson transformed creep compliance. Substitute

E

~ with

D

~ , and Eq. (5-4)

becomes: ) ( ) , ( ~ _ _ _ y x v D s

x σ σ

ε = − (2-29)

Eqs. (2-4), (2-5), and (2-29) yield:

[

(1 ) ( ) ( 1) ( )

]

2 ) , ( ( ) _ ~ _ x g x f e ad P D s

x ₌ i wt− ₊_ν ₊ _ν ₋

π

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where P = PoH(t) for a creep test and

H(t) = 0, when t<0; 1, when t>0

Taking the inverse Laplace transform on both sides of Eq. (2-30) yields:

[

]

τ τ τ ν ν π ε d d dP t D x g x f ad t x t x = + + −

∫

−

0 ) ( ) ( ) 1 ( ) ( ) 1 ( 1 ) , ( (2-31)

Eq. (2-31) becomes:

[

(1 ) ( ) ( 1) ( )

]

) ( 2 ) ,

( 0 _D _t _f _x _g _x

ad P t x

x =_π +ν + ν −

ε (2-32)

The horizontal deformation U(t) is obtained by integrating the strain over the entire gauge

length:

∫

− = l l dx t x t

U( ) ε( , ) (2-33)

where l is half of the gauge length. Eqs. (2-32) and (2-33) yield:

      − + + =

_∫

− − l l l l dx x g dx x f ad E t D P t

U( ) 2 ( ) (1 ) ( ) ( 1) ( )

*

0 _ν _ν

π (2-34)

Rearranging Eq. (2-34) for creep compliance, D(t), yields:

A P t U t D o 2 ) ( )

( = (2-35)

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Similarly, the analogous expression for the creep compliance from the vertical

displacement, V(t), can be derived as:

B P t V t D o 2 ) ( )

( = (2-37)

where       + − − =

_∫

− − l l l l dy y m dy y n ad

B 1 (ν 1) ( ) (1 ν) ( )

π (2-38)

Eqs. (2-35) and (2-37) yields:

) ( ) ( ) ( ) ( 2 2 1 1 t V b t U a t V b t U a + + − =

ν (2-39)

[

( ) ( )

]

)

( cU t eV t

p d t

D =− + (2-40)

where a1, a2, b1, b2, c, and e are coefficients related to specimen diameter and gauge length

used to measure displacements. Note that Eq. (2-39) has the same form as Eq. (2-20). Values

of these coefficients for various specimen diameter and gauge length are presented in Table

2.2. One may observe that the coefficients in Table 2.1 and Table 2.2 are different but they

will yield the same Poisson’s ratio for any given specimen geometry.

Table 2.2 Coefficients Used to Calculate Poisson’s Ratio and Creep Compliance

25.4 3.385 1.081 1 3.122 0.7874 2.2783

50.8 4.58 1.316 1 3.341 0.4032 1.024

25.4 3.172 1.039 1 3.06 1.199 35.33

50.8 3.673 1.154 1 3.192 0.611 1.685

76.2 4.559 1.33 1 3.311 0.475 1.034

c e

a1 a2 b1 b2

(27)

It is note worthy that Eq. (2-40) is independent of Poisson’s ratio and, thus, is

material independent. This is due to the two-dimensional plane stress assumption used in the

derivation. Poisson’s ratio cannot be deleted from Eq. (2-40) if the viscoelastic solution was

based on a three-dimensional solution.

2.3 Determination of Center Point Strain from Horizontal Deformation

Knowing that the maximum tensile strain along the horizontal axis occurs at the

center point of the specimen and that the failure plane is along the vertical axis, the measured

deformation across a given gauge length must be converted into a center point strain, instead

of the average strain that is measured using the LVDT’s, in order to study the stress-strain

relationship of the element where the crack is initiated.

Using Eq. (2-32) and the theory of viscoelasticity, along the horizontal diameter of

the IDT specimen, the center point strain, εx=0, in a creep test is represented as follows:

[

(1 ) ( 0) ( 1) ( 0)

]

) ( 2 )

( 0

0 = + = + − =

= D t f x g x

ad P t

x _π ν ν

ε (2-41)

Eqs. (2-34) and (2-41) yield:

UA dv c

bv a U

x ₊ =

+ = =0

ε (2-42)

where a, b, c, and d are coefficients related to specimen diameter and gauge length used.

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Table 2.3 Coefficients in Eq. (2-42)

One can observe from Eq. (2-42) that the strain-deformation relationship is

independent of stress and dependent on Poisson’s ratio. Thus, Eq. (2-42) can be applied to

specimens subjected to any loading conditions. The dependency on Poisson’s ratio is due to

the plane stress assumption along the thickness of the specimen, however, the dependency is

insensitive to the change in Poisson’s ratio.

25.4 12.4 37.7 0.291 0.908

50.8 12.4 37.7 0.471 1.57

25.4 8.48 25.6 0.207 0.634

50.8 8.48 25.6 0.373 1.18

76.2 8.48 25.6 0.478 1.59

100

150 Specimen Diameter (mm)

Gauge

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3. SPECIMEN PREPARATION AND EXPERIMENTAL PROGRAM

3.1 Specimen Preparation

3.1.1 Asphalt Concrete Mixtures

Twenty four different North Carolina mixtures were used in this project to verify the

viscoelastic solutions for the dynamic modulus in IDT. All the aggregates used were granite

from six different sources in North Carolina. Four different nominal maximum aggregate

sizes were used: 9.5, 12.5, 19, and 25 mm. Three different PG grades were used (PG 64-22,

PG 70-22, and PG 76-22) from six different sources. Table 3.1 shows the details of the

asphalt concrete mixtures used.

The mixing and compaction temperatures for PG 64-22, PG 70-22, and PG 76-22

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Table 3.1 Information on Asphalt Mixtures Tested in This Study

Note: aS for surface mix, I for intermediate mix, and B for base mix

b_{Indicates the nominal maximum size of aggregate in mm (Superpave mix}

designation)

c_{Traffic volume indicator (A: lowest traffic, D: highest traffic)}

d_{Aggregate gradation (F: Fine, C: Coarse, 1: Fine)}

# Binder Mixture

Asphalt

content % Binder Source Aggregate Quarry

Aggregate

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3.1.2 Target Air Voids

The target air voids for all specimens tested in this project was 4%. A small study was

conducted using four mixtures with different NMSA to estimate the reduction in air voids

when the specimen is cut from 60 mm height to 38 mm. The study showed that in general the

reduction in air voids would be around 1% for 9.5 mm NMSA and it would increase by

approximately 0.5% increment as the NMSA increases to 12.5, 19, and 25 mm.

3.1.3 Maximum Specific Gravity Determination

The maximum specific gravity (Gmm) of all mixtures was measured according to

AASHTO TP34-94, commonly known as Rice Specific Gravity. Appropriate correction

factors were applied to correct for the specific gravity of water when the temperature of

water is different than 25°C. Three replicates were used to determine the Gmm of each

mixture by averaging the values obtained from the replicates. Maximum error allowed

between replicates was 0.011 according to the specification.

3.1.4 Specimen Fabrication

Three gyratory specimens for each mixture were prepared to 60 mm height and 150

mm diameter, and they were cut to 38 mm height. In order to achieve the target air void

(32)

3.2 Experimental Program

3.2.1 Testing System

Testing was performed using a closed-loop servo-hydraulic machine, manufactured

by MTS, using a 5 kip load cell, that is capable of applying load over a wide range of

frequencies ranging from 0.01 Hz to 25 Hz. The temperature control system of MTS is based

on the use of low pressure liquid nitrogen to condition the chamber where the specimen is

being tested. The temperature control system is able to achieve the required testing

temperatures ranging from -10°C to 35°C. An asphalt concrete dummy specimen with similar

geometry and a temperature probe (thermocouple) embedded in the middle of the specimen

was placed inside the chamber in order to monitor the actual temperature of the specimen

during testing.

The data acquisition system for MTS is fully computerized and is capable of

measuring and recording data from several channels simultaneously. Six channels were used

in IDT dynamic modulus testing: two for vertical displacements, two for horizontal

displacements, and one for load cell, one for actuator. Data acquisition programs were

prepared using Labview software for data collection and analysis.

The vertical and horizontal deformations were measured using linear variable

differential transformers (LVDTs). For IDT testing, two loose core type miniature XSB

LVDTs were mounted on each of the specimen faces using a 50.8 mm gauge length (2 inch),

as shown in Figure 3.1(b)

The Load Guide Device (LGD), developed from the Strategic Highway Research

(33)

Figure 3.1(a). From the NCHRP 1-28 study, it was found that when compared against other

loading devices with no column or four columns, the SHRP LGD with two guide columns

resulted in the least amount of “rocking” of the IDT specimen without causing significant

friction between the upper loading plate and guide columns under repetitive loading (16).

Figure 3.1 (a) IDT test setup with SHRP LGD; (b) Surface-Mounted LVDTs

(34)

3.2.2 Test Protocol

The NCHRP 1-37A draft protocol provides recommended testing temperatures and

frequencies, summarized in Table 3.2. It was recognized that this test protocol could be

simplified by reducing the number of testing temperatures and increasing the number of

testing frequencies without compromising the mastercurve construction. The result of this

action could mean that the required testing time to obtain a mastercurve would decrease

because of the reduced amount of temperature conditioning. This reduced testing time could

provide a significant benefit to practicing agencies. The test protocol proposed by the

NCHRP 1-37A project takes 11 to 12 hours to complete using the liquid nitrogen temperature

chamber. This amount of time is more than one working day. The major portion of this

period is consumed by temperature conditioning time. It would take longer than 12 hours if a

refrigeration type chamber is used. The modified test protocol, however, takes less than 8

hours and, therefore, allows the completion of one dynamic modulus test within one working

day.

The underlying principle that allows the modification of the original test protocol is

the concept of time-temperature superposition (or time-temperature equivalence); that is, the

same modulus value of a material can be obtained either at low test temperatures and long

times or at high test temperatures but short times. The material that exhibits this type of

behavior is called thermorheologically simple (TRS).

Since the mastercurve development combines the effect of testing frequency and

temperature into one variable, called reduced frequency, it is possible to modify the range of

frequency in each temperature to compensate for the modification in the number of test

(35)

and the decrease in the number of testing temperatures, as shown in Table 3.2 under the

modified test protocol. A small study was conducted to verify the similarity of test results

between the two test protocols by testing in uniaxial mode.

Table 3.2 Comparison of the NCHRP 1-37A and Modified Test Protocols

Test Protocol No. of Temps. Test Temperatures (°C) Testing Frequencies (Hz) NCHRP 9-19 5 54.4, 37.8, 21.1, 4.4, -10 25, 10, 5, 1, 0.5, 0.1

NCSU 3 35, 10, -10 25, 10, 5, 1, 0.5, 0.1, 0.05, 0.01

One key requirement in developing a dynamic modulus mastercurve is to have an

overlap of the dynamic modulus vs. frequency curves between adjacent testing temperatures.

Figure 3.2 illustrates mastercurve development based on the 1-37A protocol of five testing

temperatures and the modified protocol of three temperatures. To develop this curve, the

dynamic modulus vs. frequency plot at various temperatures is horizontally shifted along the

frequency axis in a logarithmic scale to form a single curve at a reference temperature. The

first step involved in the determination of shift factors is to determine the frequencies at the

reference temperature and the temperature in question that yield the same dynamic modulus

values. Since the horizontal shift is performed on the logarithmic scale, the shift factor is then

determined by calculating the ratio of the frequency at the reference temperature to the

frequency at the temperature in question. After the horizontal shift, the frequency at the

reference temperature is then called reduced frequency. Therefore, to find the frequencies at

two temperatures that yield the same dynamic modulus values, it is necessary for the data to

(36)

0 10000 20000 30000 40000

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency, Hz

IE

*I, M

P

a

-10C 4.4C 21.1C 37.8C 54.4C -10C 10C 35C

Figure 3.2 Comparison of mastercurves between three-temperature (empty symbols) and five-temperature (filled symbols) testing for S9.5C–Fine Mixture

The filled symbols in Figure 3.2 illustrate the overlap while using the 1-37A protocol.

It is noted that extra duplication of data exists at the lowest modulus or highest temperature

values. The empty symbols in Figure 3.2 show the overlap between adjacent temperatures

while using the three-temperature protocol. It can be seen from Figure 3.2 that, by adding

two low frequencies (0.05 and 0.01 Hz), sufficient overlap exists between temperatures. The

overlap is similar to that seen in five-temperature testing (filled symbols) between -10°C and

4.4°C or 4.4°C and 21.1°C. Some of the extra replication at the higher temperatures does not

exist in the three-temperature test data, which does not affect the quality of the overlap

(37)

Another important factor to consider in this comparison is the similarity between

dynamic modulus mastercurves from the two protocols. Each point in Figure 3.2 represents

the average of three replicates. When the average curves are compared, there is little

discrepancy between the protocols. This observation was verified for two different mixtures.

This finding suggests that one can use the three-temperature test protocol, which requires a

shorter testing time, and obtain the same dynamic modulus data as the five-temperature test

protocol would yield. All the IDT dynamic modulus tests in this project were conducted

according to the three-temperature test protocol.

The number of test replicates was three for each mixture type. In each frequency and

temperature combination of the three-temperature protocol shown in Table 3.2, the applied

load level was adjusted so that the resulting horizontal strain in the IDT specimen is between

(38)

4. DATA ANALYSIS AND RESULTS

4.1 Data Analysis

4.1.1 Software

LVDT and load cell measurements were collected using a National Instruments data

acquisition board and LabView software. The raw data were previewed before any analysis

was begun to ensure that the horizontal strain readings were within the target range (between

50 and 80 microstrains) and that all of the LVDTs read properly. Averaged deformations

were used to calculate the dynamic modulus and phase angle. In IDT, the vertical

deformations and horizontal deformations from two surfaces were averaged to determine the

deformation along each axis.

LabView software was used in determining the amplitudes of the sinusoidal load and

deformation histories. This program uses the Levenberg-Marquardt algorithm which is a

least squares approach to curve fitting. The last five cycles of data were analyzed and fitted

according to the following functional form:

) cos( )

(t =a+bt+c ωt+φ

f (4-1)

where f(t) is load or deformation time history;

a, b, and c are regression coefficients; φ is the phase angle; and

(39)

Coefficient c represents the amplitude of the sinusoidal waveform, and the dynamic modulus

is then calculated from the ratio of these coefficients from load and deformation histories.

The difference in the phase angles from the load and deformation analyses represents the

phase angle of the material.

The Labview software was programmed at North Carolina State University to utilize

the viscoelastic solution derivation in IDT to calculate the dynamic modulus and Poisson’s

ratio.

4.1.2 Mastercurve Construction

4.1.2.1 Dynamic Modulus

To construct a mastercurve the dynamic modulus vs. frequency curves at various

temperatures are horizontally shifted along the frequency axis in a logarithmic scale to form a

single curve at a reference temperature which was 10oC. The first step involved in the

determination of shift factors is to determine the frequencies at the reference temperature and

the temperature in question that yield the same dynamic modulus values. Since the horizontal

shift is performed on the logarithmic scale, the shift factor is then determined by calculating

the ratio of the frequency at the reference temperature to the frequency at the temperature in

question. After the horizontal shift, the frequency at the reference temperature is called

(40)

) (log

exp 1 1

*

R

f e d

b a

E Log

+ + +

= (4-2)

The mastercurve was constructed using the averaged dynamic modulus values from the three

replicates tested in each mixture. A detailed procedure is outlined in Appendix C to calculate

dynamic modulus for any of the mixtures tested in this study using the sigmoidal function

and the shift factor equation.

4.1.2.2 Phase Angle and Poisson’s Ratio

Phase angle is the time delay between stress and strain, and it is measured in degrees.

In order to determine the phase angle in IDT tests, the electronic and mechanical phase lags

due to machine compliance and electronic noise from the LVDT’s had to be determined first.

An aluminum specimen with same dimensions as IDT specimens was tested using the same

testing protocol used for dynamic modulus testing. Since aluminum is an elastic material, it

should exhibit no phase angle. The electronic phase angle values obtained from this test were

subtracted from those obtained from testing asphalt concrete specimens. It was found that

electronic phase angles were a function of frequency only and that temperature had a

negligible effect. Electronic phase angle values are presented in Figure 4.1. Then, the phase

angle mastercurves were developed using the shift factors determined from the construction

of the dynamic modulus mastercurves. In IDT two phase angle mastercurves were plotted,

the first is the “vertical phase” which is the phase angle between stress and vertical strain, the

second is the “horizontal phase” which is the phase angle between stress and horizontal

strain. It was discovered that the horizontal phase angles were always higher than the vertical

(41)

developed using the shift factors determined from the construction of the dynamic modulus

mastercurves.

0 5 10 15 20 25

0.001 0.01 0.1 1 10 100

Frequency

P

has

e A

ngl

e

Horizontal phase Vertical phase

Figure 4.1 Frequency dependent electronic phase angle for MTS

4.2 Results

In this section results from S 9.5C-Fine mixture is presented for demonstration

purposes, while the results for the remaining 23 mixtures could be found in Appendix A in a

graphical format. Also plotted in these figures are the axial compression data.

(42)

values from each replicate were averaged and a single mastercurve representing the mixture

was plotted in Figure 4.3.

4.2.2 Shift Factor

The reference temperature that was used as the basis for shifting the data was 10oC. A

quadratic equation was used to fit the three shift factor values obtained. Figure 4.4 shows a

shift factor plot against temperature.

0 5000 10000 15000 20000 25000 30000 35000

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency (Hz)

E*

(

M

P

a)

-10C Rep. 1 10C Rep. 1 35C Rep. 1 -10C Rep. 2 10C Rep. 2 35C Rep. 2 -10C Rep. 3 10C Rep. 3 35C Rep. 3

(43)

0 5000 10000 15000 20000 25000 30000 35000

0.000001 0.0001 0.01 1 100 10000 1000000

E*

(

M

P

a)

-10 C Averaged 10 C Averaged 35 C Averaged Fit

Figure 4.3 S 9.5C-Fine mixture dynamic modulus mastercurve

y = 0.00056x2_{- 0.15630x + 1.52179}

R2_{= 1.00000}

-4 -3 -2 -1 0 1 2 3 4

-15 -5 5 15 25 35

Log

S

hi

ft

F

ac

to

(44)

4.2.3 Phase Angle and Poisson’s Ratio

Horizontal and vertical phase angles were computed after subtracting the electronic

phase. The phase angle and Poisson’s ratio obtained from the three replicates were averaged

and plotted in Figures 4.5 and 4.6, respectively, against the reduced frequency used in

plotting the dynamic modulus mastercurve.

First to note from Figure 4.6 is that the average values of Poisson’s ratio at different

temperatures seem to be reasonable. Figures 4.7 to 4.10 present Poisson’s ratio values for all

24 mixtures tested. The plots were categorized according to the NMSA. Some of Poisson’s

ratio values at lower frequencies and 35°C exceeded the linear elastic limit of 0.5, indicating

that at these conditions specimens were damaged during the dynamic modulus test. It was

also observed that the amount of damage tends to increase as the NMSA increases, especially

(45)

0 5 10 15 20 25 30 35

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency

P

has

e A

ng

le

35 C Vert. 10 C Vert. -10 C Vert. 35 C Horiz. 10 C Horiz. -10 C Horiz.

Figure 4.5 Vertical and horizontal phase angles mastercurve

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.000001 0.0001 0.01 1 100 10000 1000000

P

oi

ss

on

's Ra

tio

(46)

0.0 0.2 0.4 0.6 0.8 1.0

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency P o is son' s R a tio

S 9.5C C S 9.5C F S 9.5A C S 9.5A F S 9.5B 1 S 9.5B C S 9.5B F

Figure 4.7 Poisson’s ratio for 9.5-mm mixtures

0.00 0.20 0.40 0.60 0.80 1.00

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency P oi sso n' s R at io

S 12.5D F S 12.5B C S 12.5B F S12.5C C S 12.5D C S 12.5C F

(47)

0.0 0.2 0.4 0.6 0.8 1.0

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency

P

oi

sso

n'

s R

at

io

I 19.0D F I 19.0B 1 I 19.0B C I 19.0B F I 19.0C C I 19.0C F I 19.0D C

Figure 4.9 Poisson’s ratio for 19.0-mm mixtures

0.0 0.2 0.4 0.6 0.8 1.0

0.000001 0.0001 0.01 1 100 10000 1000000

Reduced Frequency

P

o

is

son'

s R

a

tio

(48)

5. COMPARISON BETWEEN IDT AND AXIAL COMPRESSION TEST

RESULTS

Dynamic modulus is a fundamental material property that should remain the same

regardless of how the material is being tested, assuming that asphalt concrete is isotropic.

Thus, a comparison was conducted between results obtained from testing asphalt concrete in

IDT mode and that tested in uniaxial mode. A total of 12 mixtures will be represented

graphically, while the rest 12 mixtures are included in Appendix A. Statistical analysis was

conducted on the data from all 24 mixtures to evaluate how different IDT results are from

those on uniaxial compression tests.

5.1 Dynamic Modulus

5.1.1 Graphical Comparison

The IDT test data of the 12 mixtures were analyzed using the viscoelastic solutions

presented previously in the Literature Review section. The analysis of the axial compression

test data was performed in accordance with the NCHRP 1-37A protocol. The resulting

dynamic modulus mastercurves from these analyses are plotted in Figures 5.1 to 5.3 for the

12 mixtures. The data presented in these figures are the average of the three replicates. It can

be observed from these figures that the dynamic modulus mastercurves developed from the

IDT test using the biaxial LVE solution are generally in good agreement with those

(49)

factors obtained during the construction of mastercurves were essentially identical for the

axial compression and IDT tests.

(50)

Figure 5.1 Dynamic modulus master curves for: (a) S9.5A-Fine mixture; (b) S9.5B-Coarse mixture; (c) S9.5C-Fine mixture; (d) S9.5C-Coarse mixture (Uniaxial data from M. King, 2004)

37

100 1000 10000 100000