Micromechanics and Analogical Models for 
Forward and Inverse Problems in Asphalt 
Materials Low Temperature Characterization 
 
 
 
A THESIS 
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL 
OF THE UNIVERSITY OF MINNESOTA  
BY 
 
 
 
AUGUSTO CANNONE FALCHETTO 
 
 
 
 
 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS 
FOR THE DEGREE OF  
MASTER OF SCIENCE 
 
 
 
 
Mihai Marasteanu, Advisor 
 
August 2010  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
© Augusto Cannone Falchetto 2010 
 i 
ACKNOWLEDGEMENTS   
 
I would like to express my deepest gratitude to my advisor Dr. Mihai O. Marasteanu for 
his patience, kind support and guidance throughout the development of the thesis. I also 
want to thank the other committee members, Dr. Lev Khazanovich and Dr. Douglas M. 
Hawkins for their advice and contributions. I express also my appreciation to Dr. Hervé 
Di Benedetto for his precious advices. 
 
I would like to thank Mugurel I. Turos for the experimental work and my office mates, 
especially Ki Hoon Moon, for their support. 
  
I would also like to thank my cousins for they “far away” support and encouragement. 
Finally I am deeply thankful to my Mother and my Father for all they have done. 
 
 
 
 
 
 
 
 
 
 
 ii 
ABSTRACT  
The use of increased proportions of Reclaimed Asphalt Pavement (RAP) in the 
construction of asphalt pavements has become a top priority due to its economical and 
environmental benefits.  However, the blending process that occurs during mixing 
between the new virgin binder and the RAP aged binder is not well understood, and the 
question if total blending occurs or  other mechanisms take place that influence the 
effective properties of the mixture remains unanswered. For many years, various models 
have been developed and used to predict the composite asphalt mixture properties from 
the properties of the components. This type of approach is generally known as forward 
problem. More recently, researchers started to investigate the possibility of predicting 
binder properties from mixture properties (inverse problem).  
In this dissertation the inverse problem of obtaining asphalt binder properties from 
asphalt mixture properties at low temperature is investigated. First an extensive literature 
review of the models available is performed. Then the forward problem of predicting the 
asphalt mixture properties from asphalt binder properties is investigated using one semi-
empirical model, Hirsch, and one analogical model, Huet, and mixture creep stiffness 
data obtained with the Bending Beam Rheometer (BBR). Next, the same two models are 
applied to predict the binder properties from the mixture properties. Then, based on Huet 
model, expressions that relates the asphalt mixture stiffness to the asphalt binder stiffness 
and vice versa are obtained. 
 
 
 
 iii 
TABLE OF CONTENTS 
LIST OF TABLES ....................................................................................... vi 
LIST OF FIGURES .................................................................................... vii 
CHAPTER 1. INTRODUCTION ................................................................. 1 
Objective and Research Approach .............................................................................. 2 
Organization .................................................................................................................. 3 
CHAPTER 2. LITERATURE REVIEW .................................................... 4 
2.1. Asphalt Mixtures .................................................................................................... 4 
2.1.1. Linear Viscoelasticity ....................................................................................... 4 
2.1.2. Time-Temperature Superposition Principle (TTSP) ...................................... 5 
2.2 Composite Materials Models for Asphalt Mixture Characterization ................ 7 
2.2.1. Micromechanical Models ................................................................................ 7 
2.2.2. Higher Order Micromechanical Models ......................................................... 9 
2.2.3 Semi-empirical Model ..................................................................................... 13 
2.3. Analogical Models ................................................................................................ 15 
2.3.1. Discrete Spectrum Models ............................................................................. 15 
2.3.2. Continuous Spectrum Models ....................................................................... 19 
2.3.2.1. Parabolic Element ................................................................................... 19 
2.3.2.2. Huet Model............................................................................................... 21 
2.3.2.3. Huet-Sayegh Model .................................................................................. 22 
2.3.2.4. 2S2P1D Model ......................................................................................... 24 
2.4. Inverse Problem ................................................................................................... 28 
CHAPTER 3. MATERIALS AND TESTING ..........................................30 
3.1. Test Methods for low temperature characterization of asphalt binders and 
mixtures ....................................................................................................................... 30 
 iv 
3.1.1. Asphalt Binder Testing .................................................................................. 30 
3.1.1.1. Bending Beam Rheometer (BBR) ............................................................. 30 
3.1.1.2. Direct Tension (DT) Test ......................................................................... 31 
3.1.2. Asphalt Mixture Testing ................................................................................ 31 
3.1.2.1. Indirect Tensile Test (IDT) ....................................................................... 31 
3.1.2.2. Bending Beam Rheometer (BBR) test for Asphalt Mixtures .................... 32 
3.2. Materials and experimental work ...................................................................... 34 
3.2.1. Asphalt Binders .............................................................................................. 34 
3.2.2 Asphalt Mixtures ............................................................................................. 35 
CHAPTER 4. FORWARD PROBLEM IN LOW TEMPERATURE 
ASPHALT MIXTURE CHARACTERIZATION ....................................36 
4.1. Hirsch Model ........................................................................................................ 36 
4.2. Huet Model ........................................................................................................... 40 
CHAPTER 5. INVERSE PROBLEM IN LOW TEMPERATURE 
ASPHALT MIXTURE CHARACTERIZATION ....................................49 
5.1. Hirsch Model ........................................................................................................ 49 
5.2. Huet Model ........................................................................................................... 54 
CHAPTER 6. SUMMARY AND CONCLUSIONS .................................59 
6.1. Summary ............................................................................................................... 59 
6.2. Conclusions ........................................................................................................... 59 
REFERENCES ............................................................................................61 
APPENDIX A (CHAPTER 4) FORWARD PROBLEM IN LOW 
TEMPERATURE ASPHALT MIXTURE CHARACTERIZATION ...67 
A.1. Huet model fitting ............................................................................................... 67 
A.2. Huet model fitting parameters ........................................................................... 73 
 v 
$ĮUHJUHVVLRQSDUDPHWHUSORWV .............................................................................. 74 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 vi 
LIST OF TABLES 
Table 3.1. Asphalt binders and mixtures .......................................................................... 34 
Table 3.2. Mix design parameters ..................................................................................... 35 
Table 4.1. Parameters used in Hirsch model..................................................................... 37 
Table 4.2. Huet model parameters for four binder and corresponding granite mixtures .. 42 
7DEOHĮUHJUHVVLRQSDUDPHWHUIRUWKHIRXUELQGHUVPL[WXUHVJURXSV .......................... 44 
Table A.1. Asphalt binders and mixtures.......................................................................... 67 
Table A.2. Huet model parameters for PG-34 binder and corresponding granite mixtures
........................................................................................................................................... 73 
Table A.3. Huet model parameters for PG-34 binder and corresponding limestone 
mixtures............................................................................................................................. 73 
Table A.4. Huet model parameters for PG-28 binder and corresponding granite mixtures
........................................................................................................................................... 73 
Table A.5. Huet model parameters for PG-28 binder and corresponding limestone 
mixtures............................................................................................................................. 74 
7DEOH$ĮUHJUHVVLRQSDUDPHWHUIRUWKHIRXUELQGHUVPL[WXUHVJURXSV ......................... 76 
 
 
 
 
 
 
 
 
 
 vii 
LIST OF FIGURES 
Figure 2.1. Milton bounds for granite and limestone mixtures – Velasquez et al. (2010) 11 
Figure 2.2. Semi-empirical model proposed by Christensen et al. (2003) ....................... 13 
Figure 2.3. Maxwell model (a) and Kelvin-Voigt model (b) ............................................ 16 
Figure 2.4. Generalized Maxwell model ........................................................................... 17 
Figure 2.5. Generalized Kelvin-Voigt model ................................................................... 18 
Figure 2.6. Parabolic element ........................................................................................... 19 
Figure 2.7. Huet model – (Huet, 1963) ............................................................................. 21 
Figure 2.8. Huet-Sayegh model – (Sayegh, 1965) ............................................................ 23 
Figure 2.9. 2S2P1D model – (Olard et al., 2003) ............................................................. 25 
Figure 2.10. Binder to Mixture model scheme – (Di Benedetto et al., 2004) ................... 28 
Figure 3.1. Bending Beam Rheometer with thin asphalt mixture (Marasteanu et al., 2009)
........................................................................................................................................... 33 
Figure 3.2. Asphalt mixture beam preparation – (Marasteanu et al., 2009) ..................... 33 
Figure 4.1. Hirsch model for PG 58-34 M1 mixtures T=-24ºC ........................................ 37 
Figure 4.2. Hirsch model for PG 58-34 M1 mixtures T=-24ºC ........................................ 38 
Figure 4.3. Hirsch model for PG 58-28 U1 mixtures T=-18ºC ........................................ 38 
Figure 4.4. Hirsch model for PG 58-28 U2 mixtures T=-18ºC ........................................ 38 
Figure 4.5. Hirsch model for PG 64-34 M1 mixtures T=-24ºC ........................................ 39 
Figure 4.6. Hirsch model for PG 64-34 M2 mixtures T=-24ºC ........................................ 39 
Figure 4.7. Hirsch model for PG 64-28 U1 mixtures T=-18ºC ........................................ 39 
Figure 4.8. Hirsch model for PG 64-28 M1 mixtures T=-18ºC ........................................ 40 
Figure 4.9. Huet model for PG 58-34 M2 binder and granite mixture T=-24ºC .............. 41 
Figure 4.10. Log scale linear relationship EHWZHHQIJbinder DQGIJmix for the binders ........... 43 
and corresponding granite mixtures at PG-24 .................................................................. 43 
Figure 4.11. Huet model for PG 58-34 M1 mixtures T=-24ºC ......................................... 46 
Figure 4.12. Huet model for PG 58-34 M1 mixtures T=-24ºC ......................................... 46 
Figure 4.13. Huet model for PG 58-28 U1 mixtures T=-18ºC ......................................... 46 
Figure 4.14. Huet model for PG 58-28 U2 mixtures T=-18ºC ......................................... 47 
Figure 4.15. Huet model for PG 64-34 M1 mixtures T=-24ºC ......................................... 47 
 viii 
Figure 4.16. Huet model for PG 64-34 M2 mixtures T=-24ºC ......................................... 47 
Figure 4.17. Huet model for PG 64-28 U1 mixtures T=-18ºC ......................................... 48 
Figure 4.18. Huet model for PG 64-28 M1 mixtures T=-18ºC ......................................... 48 
Figure 5.1. Simplified mixture stiffness function for granite mixtures ............................ 50 
Figure 5.2. Simplified mixture stiffness function for limestone mixtures ........................ 50 
Figure 5.3. Hirsch model for PG 58-34 M1 mixtures T=-24ºC ........................................ 51 
Figure 5.4. Hirsch model for PG 58-34 M1 mixtures T=-24ºC ........................................ 51 
Figure 5.5. Hirsch model for PG 58-28 U1 mixtures T=-18ºC ........................................ 52 
Figure 5.6. Hirsch model for PG 58-28 U2 mixtures T=-18ºC ........................................ 52 
Figure 5.7. Hirsch model for PG 64-34 M1 mixtures T=-24ºC ........................................ 52 
Figure 5.8. Hirsch model for PG 64-34 M2 mixtures T=-24ºC ........................................ 53 
Figure 5.9. Hirsch model for PG 64-28 U1 mixtures T=-18ºC ........................................ 53 
Figure 5.10. Hirsch model for PG 64-28 M1 mixtures T=-18ºC ...................................... 53 
Figure 5.11. Huet model for PG 58-34 M1 mixtures T=-24ºC ......................................... 55 
Figure 5.12. Huet model for PG 58-34 M1 mixtures T=-24ºC ......................................... 56 
Figure 5.13. Huet model for PG 58-28 U1 mixtures T=-18ºC ......................................... 56 
Figure 5.14. Huet model for PG 58-28 U2 mixtures T=-18ºC ......................................... 56 
Figure 5.15. Huet model for PG 64-34 M1 mixtures T=-24ºC ......................................... 57 
Figure 5.16. Huet model for PG 64-34 M2 mixtures T=-24ºC ......................................... 57 
Figure 5.17. Huet model for PG 64-28 U1 mixtures T=-18ºC ......................................... 57 
Figure 5.18. Huet model for PG 64-28 M1 mixtures T=-18ºC ......................................... 58 
Figure A.1. Huet model for PG 58-34 M1 binder and granite mixture T=-24ºC ............. 67 
Figure A.2. Huet model for PG 58-34:M2 binder and granite mixture T=-24ºC ............. 68 
Figure A.3. Huet model for PG 58-28 U1 binder and granite mixture T=-18ºC .............. 68 
Figure A.4. Huet model for PG 58-28 U2 binder and granite mixture T=-18ºC .............. 68 
Figure A.5. Huet model for PG 64-34 M1 binder and granite mixture T=-24ºC ............. 69 
Figure A.6. Huet model for PG 64-34:M2 binder and granite mixture T=-24ºC ............. 69 
Figure A.7. Huet model for PG 64-28 U1 binder and granite mixture T=-18ºC .............. 69 
Figure A.8. Huet model for PG 64-28 M1 binder and granite mixture T=-18ºC ............. 70 
Figure A.9. Huet model for PG 58-34 M1 binder and limestone mixture T=-24ºC ......... 70 
 ix 
Figure A.10. Huet model for PG 58-34:M2 binder and limestone mixture T=-24ºC ....... 70 
Figure A.11. Huet model for PG 58-28 U1 binder and limestone mixture T=-18ºC........ 71 
Figure A.12. Huet model for PG 58-28 U2 binder and limestone mixture T=-18ºC........ 71 
Figure A.13. Huet model for PG 64-34 M1 binder and limestone mixture T=-24ºC ....... 71 
Figure A.14. Huet model for PG 64-34:M2 binder and limestone mixture T=-24ºC ....... 72 
Figure A.15. Huet model for PG 64-28 U1 binder and limestone mixture T=-18ºC........ 72 
Figure A.16. Huet model for PG 64-28 M1 binder and limestone mixture T=-18ºC ....... 72 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-34 
and corresponding granite mixtures .................................................................................. 74 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-34 
and corresponding limestone mixtures ............................................................................. 75 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-28 
and corresponding granite mixtures .................................................................................. 75 
Figure A.20 Log scale linear relationshLSEHWZHHQIJbinder DQGIJmix for the binders PG-28 
and corresponding limestone mixtures ............................................................................. 76 
 
 1 
Chapter 1. Introduction 
Asphalt mixtures used in pavement construction are complex heterogeneous material 
composed of aggregates, asphalt binder and air voids. Mastics are blends of asphalt 
binder and fine particles, typically passing sieve No. 200 (i.e., sizes finer than 75 
microns). The distribution of these three phases and their interaction define the 
mechanical properties of asphalt mixtures and contribute significantly to the performance 
and durability of flexible pavements. 
The models used to describe the properties of asphalt binders and asphalt mixtures 
range from very simple analogical models, such as the assembly of a spring and dashpot 
(Findley et al. 1989), to more complex analogical models that include parabolic elements 
(Huet, 1963; Sayegh, 1965;Olard, 2003; Olard et al., 2003; Di Benedetto et al. 2004), to  
much more complex micromechanical models in which the material is described 
according to the different phases that are present in the microstructure (Milton, 1981; 
Torquato, 1998).  
Asphalt binders are temperature-susceptible viscoelastic materials that have brittle 
behavior at low temperatures and are liquids at high temperatures. Asphalt binders are 
also subject to oxidative aging during pavement service life, that leads to dramatic 
changes in properties and cause higher brittleness at low temperatures and make binders 
more prone to cracking.   
In cold regions where extreme changes in temperature occur, the typical pavement 
distress is thermal cracking. It manifests as a series of almost regular spaced cracks that 
develop when then material strength limit is overcome. Low temperature properties of 
asphalt binders and mixtures are evaluated according to AASHTO standards. Bending 
 2 
Beam Rheometer (BBR) (AASHTO T 313-02 2006) is used to determine creep 
compliance of asphalt binder, and Direct Tension (DT) Test (AASHTO T 314-02 2002) is 
used to obtain binder failure stress and strain. For asphalt mixtures, Indirect Tension Test, 
IDT, (AASHTO T 322-03) is used to obtain both creep and strength. A much simpler 
method was recently developed that allows obtaining mixture creep compliance using the 
same BBR device used for testing binders (Marasteanu et al., 2009).  
Objective and Research Approach 
Most of the modeling efforts in asphalt materials characterization have focused on 
predicting the composite asphalt mixture properties from the properties of the 
components of the mixture (Forward Problem). However, due to economical and 
environmental issues, a different problem has emerged as more Reclaimed Asphalt 
Pavement (RAP) is used in pavement applications: do the new binder and the aged RAP 
binder blend completely or other mechanisms occur that influence the effective properties 
of the binder in the mixture? Since the chemical extraction and recovery process results 
in complete blending of the two, it becomes important to “extract” binder properties from 
mixture experimental data using various modeling techniques (Inverse Problem). 
The main objective of this thesis is to review and identify the most promising 
models that can be successfully used to predict asphalt mixture properties from properties 
of the components – Forward Problem (FP), and vice versa, to predict asphalt binder 
properties from mixture properties  - Inverse Problem (IP).  In this thesis only low 
temperature properties will be investigated, more specifically the creep compliance, since 
 3 
the use of increased amounts of RAP is most detrimental to the low temperature 
performance of asphalt pavements  
In order to accomplish this goal the following approach will be followed:  
x A review of micromechanics and analogical models in asphalt materials literature 
will be reviewed 
x Experimental work will be performed to obtain low temperature creep compliance 
data for asphalt binder and mixture specimens 
x Forward Problem will be evaluated for some of the reviewed models 
x The most promising FP models will be applied to  the Inverse Problem 
x The best model will be selected.  
Organization 
This thesis is divided into six chapters. Chapter 2 includes a general review on asphalt 
concrete characterization, micromechanics models, analogical models and a short 
description of the inverse problem. In Chapter 3 the materials and procedures used in the 
experimental phase are described. Chapter 4 and Chapter 5 present the analyses 
performed for the forward and inverse problems, respectively.  Chapter 6 contains a 
summary of this study, the conclusions, and recommendations for future research. At the 
end, one Appendix containing additional information on Chapter 4 is included. 
 
 
 
 4 
Chapter 2. Literature Review 
First, a brief introduction to linear viscoelasticity concepts and time temperature 
superposition principle is presented. Then, a review of current experimental test methods 
for asphalt materials characterization follows. The chapter concludes with a review of 
micromechanical models for heterogeneous materials, such as asphalt mixtures, as well 
as analogical models for viscoleastic materials, followed by an introduction to solving 
inverse problems. 
2.1. Asphalt Mixtures 
Asphalt mixtures can be classified as composite materials consisting of three phases: 
asphalt binder, aggregate, and air voids. A typical volumetric composition of this material 
is 5% of air voids, 20% of asphalt binder, and 75% of aggregate (NCAT, 2009). Similarly 
to other composite materials, the properties of asphalt mixtures are related to the 
properties of its components. The aggregate phase is considered linear elastic and the 
asphalt binder is considered viscoelastic, which results in a viscoelastic composite 
material with properties depending also temperature (Monismith and Secor, 1962).  
2.1.1. Linear Viscoelasticity 
Boltzmann’s superposition principle is generally used to express the constitutive 
relationship between stresses ı and strains İ for a linear viscoelastic non-aging material 
(Christensen, 1982; Findley et al., 1989) as: 
³ ww 
t
kl
ijklij dtEt
0
)()()( [[
[H[V                            [2.1] 
 5 
³ ww 
t
kl
ijklij dtDt
0
)()()( [[
[V[H             [2.2] 
Stress can be obtained from equation [2.1] knowing strain history and relaxation 
modulus E(t). Analogously, strain can be computed using equation [2.2] knowing stress 
history and creep compliance function D(t).  
The three functions, creep compliance, relaxation modulus, and complex 
modulus, can fully describe the behavior of linear viscoelastic materials. The functions 
are not independent of each other and various interconversion methods can be used to 
move from one function to another (Secor and Monismith, 1964; Mead 1994; Park and 
Kim, 1999; Park and Schapery, 1999; Marasteanu and Anderson, 2000).  For example, 
Hopkins and Hamming (1957) method has been used in many asphalt research papers to 
convert creep compliance to relaxation modulus; this method numerically solves the 
Volterra integral [2.3] assuming uniaxial state of stresses and isotropy. 
³  t dtDtEt
0
)()( [[             [2.3] 
According to different researchers, asphalt concrete can be assumed as linear 
viscoelastic at low temperatures (Lytton et al., 1993; Buttlar and Roque, 1994; Pellinen 
and Witczak, 2002).  
2.1.2. Time-Temperature Superposition Principle (TTSP) 
Time temperature superposition principle was first introduced by Leaderman (1943) who 
stated that temperature and time effects can be incorporated into the viscoelastic 
properties by a reduced time function ȟ: 
 6 
)(),( Ta
tTt
T
 [                       [2.4] 
where: 
t  time and 
aT (T)  shift factor function of temperature. 
Temperature and shift factor aT are related by the empirical expression proposed by 
Williams-Landel-Ferry (WLF) (Williams et al., 1955): 
S
S
T TTk
TTk
aLog 
 
2
1 )()(            [2.5] 
where: 
aT   shift factor function of temperature 
k1 and k2 material constants, 
TS  reference temperature, 
T  actual temperature. 
Alternatively, Arrhenius law can be used to determine shift factors for asphalt concrete 
(Anderson et al., 1991; Lytton et al., 1993; Marasteanu and Anderson, 1996; Di 
Benedetto et al., 2001; Pellinen and Witczak, 2002). By expressing creep compliance, 
relaxation modulus or complex modulus in term of reduced time ȟ and setting a reference 
temperature TS, it is possible to construct a “master curve” that represents the variation of 
these parameters over different temperature regimes. When developing performance 
prediction models (Wang et al., 2006; Di Benedetto et al., 2007; Masad et al., 2007), the 
above considerations and the need for reliable yet simple experimental methods to 
determine constitutive relations for asphalt mixtures are of utmost importance. 
 7 
2.2 Composite Materials Models for Asphalt Mixture Characterization 
2.2.1. Micromechanical Models  
At the macroscopic level composite materials are made of two or more phases. Generally 
one phase acts as a continuous matrix, while the others act as inclusion or reinforcement. 
The advantage of mixing two or more materials is given by the possibility of designing a 
new material with specific properties not achievable by a single phase. However, in order 
to predict the performance of the new composite both the properties of the constituents 
and high order microstructural information are needed. Several types of models, which 
provide solution for the estimation of the microstructural correlation functions for 
composite materials, are available in literature (Torquato, 2002). The effective response 
Keff of a composite material can be written as: 
 , ,if I eff iK Kȍ            [2.6] 
where Ki and iI  are the intrinsic properties of the ith phase and its corresponding 
volumetric fraction, and ȍ  is a parameter that gives the higher-order microstructural 
information.  
Asphalt mixtures can be classified as particulate composites that contain 
aggregate particles of various sizes and shapes randomly distributed in matrix of asphalt 
binder. In several research studies, asphalt mixtures are considered as two-phase 
materials (binder or mastic and aggregate) (Abbas et al., 2004; Papagiannakis et al., 
2002; Masad and Somadevan, 2002; Yue et al., 2003). Asphalt concrete was also 
evaluated as a three-phase material (large aggregate, small aggregate and mastic) with a 
two step method by Buttlar and Roque (1996), Wang et al. (2004) and Li and Metcalf 
(2005). 
 8 
For example the effective properties of asphalt concrete at low temperature were 
evaluated by Buttlar and Roque (1996) using classical micromechanical models. Due to 
the low order microstructural information (volume fractions) used in the models, the 
response was significantly underpredicted. 
The Generalized Self-Consistent Scheme (GSCS) model was implemented by 
Buttlar et al. (1999) to model asphalt mastic.  The predictions of the GSCS model was 
compared to the test results of specimens with different concentrations of particles and 
investigating three filler reinforcement regimes: volume filling, physiochemical effects, 
and particle interaction. The authors concluded that the physiochemical interaction 
between the particles and the binder is mainly responsible for the reinforcement effect of 
the particles on the mastics. The GSCS is a three phase model and represents a special 
case of the composite spheres model proposed by Hashin (Hashin and Shtrikman, 1963). 
This model consists of an infinite matrix of homogeneous material where a spherical 
inclusion is embedded in and does not take into account interaction between particles. 
Based on finite element simulations and mastic micromechanical modeling Masad and 
Somadevan (2002) found that the average strain in the asphalt mastic is three to five 
times higher than the strain in the mixture The authors also found that the mastic can be 
three to ten times stiffer than the binder. 
Hashin-Shtrikman (H-S) bounds [2.7] and [2.8] were used by Kim and Little 
(2004) to investigate the stiffening effect of two fillers. Based on experimental results and 
micromechanics analysis it was found that H-S model provides good prediction only for 
low volume fraction of the filler. 
 9 
11
1
12
2
1
43
31
GKKK
KK Lc

 I
I
     
22
2
21
1
2
43
31
GKKK
KKUc

 I
I
  [2.7] 
 
 
2
1
1 1 1
2 1 1 1 1
6 21
5 3 4
L
cG G K G
G G G K G
I
I   
     
 
1
2
2 2 2
1 2 2 2 2
6 21
5 3 4
U
cG G K G
G G G K G
I
I   
  [2.8] 
where:  
2121 GGKK  , 
L
cK ,
U
cK  lower and upper bound on bulk modulus, 
L
cG ,
U
cG  lower and upper bound on shear modulus, 
21,II   volume fractions of phase 1 and 2  21 1 II  , 
Buttlar and Dave (2005) developed blending charts for Recycled Asphalt 
Pavements (RAP) based on micromechanical modeling. Simple mixture laws ([2.9] and 
[2.10]), first order models and second order models ([2.7] and [2.8]) were used to obtain 
the effective properties of a two-phase material and to construct the charts. 
1 1 2 2effK K KI I   (arithmetic average)      [2.9] 
1
1 2
1 2
effK K K
I I § · ¨ ¸© ¹  (harmonic average)      [2.10] 
where: 
effK   effective properties of the material  
1K , 2K  properties of phase 1 and 2, 
21,II   volume fractions of phase 1 and 2  21 1 II  . 
2.2.2. Higher Order Micromechanical Models 
 10 
Zofka et al. (2006) compared different micromechanical models with finite element 
simulations and experimental results of asphalt concrete tested in 3-point bending at low 
temperatures. The finite element simulation was closely matched by the Milton (1981) 
bounds model. Milton’s simplified bounds are described in the following equations 
[2.11]: 
 
]
II
GK
KKKK Ue 43
3 22121

   
12
21
21
1314
114
1

»»
»»
»
¼
º
««
««
«
¬
ª

¸¸¹
·
¨¨©
§ 
 
]
II
GK
KK
K
K Le   [2.11] 
where: 
2
2
1
11
KKK
II   
2211 KKK II      
1
2
2
11
KKK
II   
1221 KKK II     
2
2
1
11
GGG
]]
]
   
2211 GGG ]]]   
21 1 ]]    
     22 1 3 12P u u Legendre polynomial   
0LOWRQ¶VQXPEHUȗ1 (geometry parameter) can be calculated with: 
 11 
³ ³³
' 'foo'
 
EE
E II]
1
1
2
)1(
3
21
01
)(),,(
2
9limlim duuP
rs
usrSdsdr      [2.12] 
where S3(1)(r,s,u) is the 3-point correlation function of the material.  Berryman (1985), 
Torquato (1991), and Zofka (2007) proposed approximate expressions IRUȗ1 as a function 
of the volume fractions. Velasquez (2009), Velasquez et al. (2010) also investigated this 
model to predict the experimental data obtained form 3-point bending test on asphalt 
mixtures at low temperature showing that Milton bounds are wide and poor predictors of 
the experimental results (Figure 2.1). 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
R
el
ax
at
io
n
 M
o
du
lu
s 
E,
 
G
Pa
Time, s
Experiment
Model bounds
T = -24°C   
 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
R
el
ax
at
io
n
 M
o
du
lu
s 
E,
 
G
Pa
Time, s
Experiment
Model bounds
T = -24°C   
 
Figure 2.1. Milton bounds for granite and limestone mixtures – Velasquez et al. 
(2010) 
 
Torquato (1998) proposed a three dimensional isotropic two-phase model ([2.13] and 
[2.14]). It presents the geometry parameters ] and K2 similar to [2.12], but including, the 
2-point correlation function as part of the integral. To ensure convergence of the integral, 
the 2-point correlation function is included in the integrand of [2.17] and [2.18].  The 
effective bulk and shear modulus can be expressed as:  
 12 
 
  2111
1
2
21
11
1
2
1
1
1
23
101
23
10
3
41
]NPINI
]NPINI
GK
G
GK
G
K
G
K
Ke

        [2.13] 
     
    °¿
°¾
½
°¯
°®
­
»¼
º«¬
ª

»¼
º«¬
ª


°¿
°¾
½
°¯
°®
­
»¼
º«¬
ª

»¼
º«¬
ª



 
212
11
11
121
2
11
11
2
21
11
1
2
212
11
11
121
2
11
11
2
21
11
1
2
11
11
1
2
325
2
3
623
21
2
325
2
3
623
2
26
891
]IKIP]INPPI
]IKIP]INPPI
GK
GKG
GK
GK
GK
G
GK
GKG
GK
GK
GK
G
GK
GK
G
Ge   
           [2.14] 
where: 
3
4 1
2
12
21 G
K
KK

 { NN          [2.15] 
 »¼
º«¬
ª


 {
11
11
12
12
21
26
89
GK
GKGG
GGPP                   [2.16] 
with phase one and two corresponding to the matrix and dispersions, respectively. The 
three point parameters ] and K2 are defined by the following integrals: 
³ ³³ f

f
»¼
º«¬
ª  
0
1
1
2
2
)2(
2
)2(
2)2(
3
021
2 )(
)()(),,(
2
9 duuPsSrSusrS
s
ds
r
dr
III]     [2.17] 
³ ³³ f

f
»¼
º«¬
ª  
0
1
1
4
2
)2(
2
)2(
2)2(
3
021
2
2 )(
)()(),,(
7
150
21
5 duuPsSrSusrS
s
ds
r
dr
III
]K               [2.18] 
where, P2 and P4 are the Legendre polynomials of order 2 and 4, respectively. The 
expressions for the Torquato model [2.13] and [2.14] were obtained by truncating an 
exact series expansion for the effective elastic stiffness tensor of two phase materials. For 
this model there is no assumption regarding the geometry of the microstructure but it 
 13 
requires statistical homogeneity (Torquato, 1998). Torquato model was applied by 
Velasquez (2009) and Velasquez et al. (2010) to the prediction of the asphalt mixture 
properties obtained form 3-point bending test. Torquato model resulted to be a poor 
predictor since the estimated relaxation did not match the experimental data. The high 
contrast between the stiffness of the phases and the inability to simulate contacts between 
particles are the main reasons why this model fails. 
2.2.3 Semi-empirical Model 
A semi-empirical model, based on Hirsch model (Hirsch, 1962) was proposed by 
Christensen et al. (2003) to estimate the extensional and shear dynamic modulus. The 
effective response is obtained assembling the elements of the mixture in parallel and in 
series (Figure 2.2).  
         
Figure 2.2. Semi-empirical model proposed by Christensen et al. (2003) 
 
The empirical factor Pc determines the amount of parallel or series elements in the 
mixtures. The general equation for this semi-empirical model is: 
Aggregate 
 
V
o
id
s 
 
A
sp
ha
lt 
bi
n
de
r 
 
A
gg
re
ga
te
 
Voids Asphalt binder 
 14 
> @     1211 »»¼
º
««¬
ª  
binderbinder
agg
agg
agg
binderbinderaggaggmix VE
V
E
V
PcVEVEPcE    [2.19] 
where: 
Emix  effective modulus of the mixture, 
Eagg, Vagg  modulus and volume fraction of the aggregate, 
Ebinder, Vbinder modulus and volume fraction of binder, and 
Pc   contact volume is an empirical factor defined as:  
1
1
2
0
P
binder
P
binder
VMA
EVFA
P
VMA
EVFA
P
Pc
¹¸
·
©¨
§ ˜
¹¸
·
©¨
§ ˜
         [2.20] 
where: 
VFA  voids filled with asphalt binder (%), 
VMA  voids between mineral aggregate (%),and 
P0, P1, P2 fitting parameters. 
Zofka et al. (2005) evaluated the use of Hirsch model (Hirsch, 1962) proposed by 
Christensen (Christensen et al., 2003) to predict the BBR mixture stiffness from the 
properties of the binder. The asphalt binders from the mixtures prepared in this study 
were extracted and tested in the BBR to obtain the stiffness and the m-values.  The 
experimentally determined binder stiffness values were input in equations [2.219] and 
[2.20] to predict the mixture stiffness based on the volumetric properties measured from 
the gyratory specimens. Since the predicted values were always higher than the measured 
stiffness values, equation [2.2193] was modified. The aggregate modulus, Eagg, which is 
equal to 4,200,000 psi, was replaced with a value of 2,750,000 psi based on these results 
 15 
and on numerical manipulation.  In order to improve the prediction of the laboratory 
mixture results, a further modification of the Hirsch model was proposed by Zofka (2007) 
introducing a new expression for the parameter Pc [2.21]: 
609.0ln1.0 ¹¸
·
©¨
§ 
a
EP binderc         [2.21] 
where: 
Ebinder  relaxation modulus of the binder in GPa, and 
a  constant equal to 1 GPa. 
The original Hirsch model consistently overpredicted the measured Emix values while 
with the new expression for Pc the model predicted measured Emix relatively well. Hirsch 
model was used by Velasquez (2009) and Velasquez et al. (2010) to estimate the asphalt 
mixture modulus obtained from BBR testing. It was found that Hirsch model predicts 
fairly well the relaxation modulus of the majority of the mixtures investigated. 
2.3. Analogical Models 
Different analogical models are available in literature. They may be very simple or much 
more complex and work on discrete or on continuous spectrum. The following 
paragraphs provide a short description of the most interesting models applied to asphalt 
concrete.  
2.3.1. Discrete Spectrum Models 
Dashpot and springs constitute the simplest analogical linear viscoelastic models (Ferry, 
1980; Findley, 1989). When spring and dashpot are assembled is series and in parallel 
Maxwell and Kelvin-Voigt model can be constructed respectively (Figure 2.3). 
 16 
 
 
(a) (b) 
Figure 2.3. Maxwell model (a) and Kelvin-Voigt model (b) 
 
Expressions [2.22], [2.23] provide the creep compliance D(t) and the relaxation function 
R(t) for the Maxwell model. 
K
t
E
tD  1)(           [2.22] 
W
t
EetR
 )(           [2.23] 
where: 
IJ  UHOD[DWLRQWLPHIJ Ș( 
The complex modulus for the Maxwell model is: 
222
222
*
1
)( KZ
ZKKZ
ZW
ZWZ 
  E
iEE
i
iEiE       [2.24] 
where: 
i  complex number (i2=-1) 
Expressions [2.25], [2.26] provide the creep function D(t) and the relaxation function R(t) 
for the Kelvin-Voigt model. 
¸¸¹
·
¨¨©
§  W
t
e
E
tD 11)(          [2.25] 
E 
Ș 
E Ș 
 17 
)()( tEtR KG          [2.26] 
where: 
IJ  relaxation time IJ Ș(, 
į  Dirac function. 
The complex modulus for the Kelvin-Voigt model is: 
ZKZ iEiE  )(*          [2.27] 
These two models are not able to describe the complex properties of asphalt 
material but can be used as basic components of more sophisticated models. A 
satisfactory description of the behavior of asphalt binder and concrete (Neifar and Di 
Benedetto, 2001) can be obtained combining two previous model into a Generalized 
Maxwell Model (n Maxwell elements in parallel plus one spring) or into a Generalized 
Kelvin-Voigt Model (n Kelvin-Voigt elements in series plus one spring and one linear 
dashpot) (Figure 2.4 and Figure 2.5). 
 
Figure 2.4. Generalized Maxwell model 
 
 
 
 
E0 E1 E2 
Ș1 Ș2 
Ei En 
Și Șn 
 18 
 
 
 
Figure 2.5. Generalized Kelvin-Voigt model 
 
In the case of discrete number of element and thus for a discrete spectrum the 
relaxation modulus for the Generalized Maxwell model can be expressed as (Ferry, 
1980): 
¦
 

f  
n
i
t
i
ieEEtR
1
)( W          [2.28] 
where: 
IJi  relaxation time of the ith  Maxwell element, 
Ei  spectral strength of the ith Maxwell element. 
The complex modulus presents the following expression: 
¦
 
f  
n
i i
i
i i
iEEiE
1
*
1
)( ZW
ZWZ         [2.29] 
Increasing the number of element without limit in the Maxwell model it is possible to 
obtain a continuous spectrum representation of the relaxation and complex modulus 
functions [2.30] and [2.31] respectively: 
³ f f f  )ln( )ln( ln)()( WW W WW deHEtR
t
       [2.30] 
Ș1 Ș2 Și Șn 
Ș0 E’ 
E1 E2 Ei En 
 19 
³ f f  
)ln(
)ln(
* ln
1
)()( WW WZW
ZWWZ d
i
iHiE        [2.31] 
where: 
H(IJ) dln(IJ) is the modulus associated with the relaxation time. 
For a discrete spectrum the relaxation modulus for the Generalized Kelvin-Voigt 
model can be expressed as: 
01
1111)( K
W ¸¸¹
·
¨¨©
§  
f 
¦ EeEtD
n
i
t
i
i
       [2.32] 
¸¸¹
·
¨¨©
§  ¦ f
n
i ii iEiE
iE ZKZKZ 0
* 111)(       [2.33] 
2.3.2. Continuous Spectrum Models 
The discrete numbers of elements included into the Generalized Maxwell or Kelvin-
Voigt models are not always enough to have a satisfactory representation of a complex 
linear viscoelastic material, even though the number of elements can be increased. More 
advanced analogical models with continuous spectrum were proposed by other authors 
and they can be represented by an infinite number of Kelvin-Voigt or Maxwell elements. 
2.3.2.1. Parabolic Element 
A parabolic element is an analogical model that can be schematized as in Figure 2.6. 
 
Figure 2.6. Parabolic element 
k
 
 20 
Creep function D(t) and complex modulus E* can be expressed as: 
k
t
tD ¹¸
·
©¨
§ WG)(           [2.34] 
)1(
)()(* * k
iiE
k
G
ZWZW          [2.35] 
where: 
i  complex number (i2=-1) 
E*  complex modulus, 
k  exponent, 
į  dimensionless constant, 
Ȧ  ʌ
IUHTXHQF\ 
IJ characteristic time varying with temperature accounting for the Time 
Temperature Superposition Principle (TTSP):  
 )()( 0 ST TTa WW   
aT shift factor at temperature T (can be determined from equation [2.5] 
WLF), 
IJ0 characteristic time determined at reference temperature TS 
ī  gamma function that can be expressed as: 
³f  * 0 1)( dtetn tn      
)()1( nnn * *       
n>0 or Re(n)>0 
t integration variable, 
n argument of the gamma function. 
 21 
2.3.2.2. Huet Model 
The Huet analogical model (Huet, 1963) is composed of two parabolic elements J1(t)=ath 
and J2(t)=btk plus a spring (stiffness E’) combined in series. (Figure 2.7) 
 
 
Figure 2.7.  Huet model – (Huet, 1963) 
 
The Huet model was proposed for binders and mixtures and presents a continuous 
spectrum that means it can be schematized by infinity of Kelvin-Voigt elements in series 
or Maxwell elements in parallel. The analytical expression of the Huet model for the 
creep compliance is: 
   
¸¸¹
·
¨¨©
§
** f )1(
/
)1(
/11)(
h
t
k
t
E
tD
hk WWG        [2.36] 
where: 
D(t)  creep compliance 
E’  glassy modulus, 
h, k  exponents such that 0<k<h<1, 
į  dimensionless constant, 
t  time, 
Nį
 
h
 
E’ 
 22 
ī  gamma function, 
IJ characteristic time varying with temperature accounting for the Time 
Temperature Superposition Principle (TTSP):  
 )()( 0 ST TTa WW   
aT shift factor at temperature T (can be determined from equation [2.5] 
WLF), 
IJ0 characteristic time determined at reference temperature TS. 
 An expression of the complex modulus for this model is also available [2.37], 
while there is no analytical formula for the relaxation function: 
hk ii
EiE 
f
 )()(1)(
*
ZWZWGZW        [2.37] 
where: 
i  complex number (i2=-1) 
E’  OLPLWRIWKHFRPSOH[PRGXOXVIRUȦIJĺ’*ODVV\PRGXOXV 
Ȧ  ʌ
IUHTXHQF\ 
A modified expression of the Huet model which includes a third term function of 
IJ and representing a dashpot in series was proposed by Maillard (Maillard, 2005). The 
modified model was applied to asphalt binder and a sensitivity study in the time domain 
was performed. 
2.3.2.3. Huet-Sayegh Model 
Since the Huet model does not perform well for mixes response at very low frequencies 
and/or high temperature, due to the inability to take into account the limiting value of the 
mixtures modulus related to the aggregate skeleton, Sayegh (Sayegh 1965) proposed a 
 23 
new expression [2.38] for the complex modulus introducing a spring in parallel into the 
Huet model (Figure 2.8).  
 
Figure 2.8. Huet-Sayegh model – (Sayegh, 1965) 
 
hk ii
EEEiE 
f

 )()(1)(
0
0
*
ZWZWGZW       [2.38] 
where: 
i  complex number (i2=-1) 
E’  limit of the complex modulus for ȦIJĺ’*ODVV\PRGXOXV), 
E0  limit of the complex modulus for ȦIJĺ 
h, k  exponents such that 0<k<h<1, 
į  dimensionless constant, 
IJ characteristic time varying with temperature accounting for the Time 
Temperature Superposition Principle (TTSP), and 
Ȧ  ʌ
IUHTXHQF\ 
Nį
 
h
 
E’- E0 
 
E0 
 24 
Six constants are required from this model (įNK(’, E0, and IJ0), one more than 
the Huet model. This model was applied by several authors (De La Roche, 1996; Neifar, 
1997; Olard – Di Benedetto, 2003; Bodin et al., 2004; Adams et al., 2006; Wistuba et al., 
2006; Chabot et al., 2010) with good results in the small strain domain for any range of 
frequencies and temperatures. It should be mentioned that this model presents some 
limitation when predicting binder modulus at very low frequencies where a parabolic 
element behavior is showed while a linear dashpot would be more appropriate. The 
model was also used by Neifar (Neifar et al., 2001) to calibrate a thermo-visco-plastic 
law named DBN law. This law allows describing with the same formalism different types 
of mixture behaviors according to the considered loading domain (Olard and Di 
Benedetto, 2005). A three dimensional extension of the DBN was also proposed by the 
same authors (Di Benedetto et al., 2007). It must be finally mentioned that there is no 
analytical expression for creep compliance in the time domain for this model. 
2.3.2.4. 2S2P1D Model 
An improved Huet-Sayegh model that takes into account the drawback for binder 
characterization was proposed by Di Benedetto and Olard et al. (Olard et al., 2003; 
Olard, 2003; Di Benedetto et al. 2004). This model is obtained by the Huet-Sayegh 
model and adding a linear dashpot in series with the two parabolic elements and the 
spring of rigidity E҄-E0
 
so that at low frequency it is equivalent to a linear dashpot in 
parallel with a spring of rigidity E0. The scheme of the model is shown in Figure 2.9 and 
the analytical expression of the complex modulus is given by [2.39]. 
  
 25 
 
Figure 2.9. 2S2P1D model – (Olard et al., 2003) 
 
1
0
0
*
)()()(1)( 
f

 ZEWZWZWGZW iii
EEEiE hk      [2.39] 
where: 
i  complex number (i2=-1) 
E’  limit of the complex modulus for ȦIJĺ’*ODVV\PRGXOXV 
E0  limit of the complex modulus for ȦIJĺ 
h, k  exponents such that 0<k<h<1, 
į  dimensionless constant, 
ȕ dimensionless parameter introduced to take into account the newtonian 
viscosity of the linear dashpot 
IJ characteristic time varying with temperature accounting for the Time 
Temperature Superposition Principle (TTSP), and 
Ȧ  ʌ
IUHTXHQF\ 
Nį
 
h
 
E’- E0 
 
E0 
Ș 
 26 
The seven constants (į ȕ N K (’, E0, and IJ0) required were determined with a 
minimization process from the experimental data at a reference temperature TS=10ºC for 
a series of binders and corresponding mixtures. Each mixture showed the same 
parameters į, k, h and ȕof the associated binder while only the static and glassy modulus 
(E0 and E’) and IJ0 seemed to be peculiar of the specific binder and mixtures. The values 
of E0 and E’ for the mixtures were in the range of 250 to 1050MPa and 41500 to 
45400MPa respectively. From the simple regression of the characteristic time of the 
mixture, IJ0mix, on the characteristic time of the corresponding binder, IJ0binder, at the 
reference temperature in log scale the authors found the following relationship: 
bindermix 00 10 WW D           [2.40] 
and applying the time temperature superposition principle: 
)(10)( TT bindermix WW D          [2.41] 
where: 
IJ0mix  characteristic time of mixture determined at reference temperature TS, 
IJ0binder  characteristic time of binder determined at reference temperature TS, 
IJmix  characteristic time of mixture at temperature T, 
IJbinder  characteristic time of binder at temperature T, 
Į  regression coefficient depending on mixture and aging. 
The value of Į was determined in the range 2.66 to 2.82 according to the different 
mixtures and binders investigated for a characteristic time evaluated at TS=10ºC (Olard et 
al., 2003; Olard, 2003; Di Benedetto et al. 2004). 
This model was used to calibrate the DBN model by Olard et al. (Olard et al., 
2005). Delaporte et al. (Delaporte et al., 2007) used the 2S2P1D model to investigate the 
 27 
linear viscoelastic properties of asphalt binder and mastics. The model gave good 
simulations for binders and mastics with and without aging. 
From the 2S2P1D model a relationship between the binder and the mix complex 
moduli (considering both the phase angle and the norm of the complex) was proposed 
[2.42] (Olard et al., 2003; Olard, 2003; Di Benedetto et al. 2004): 
> @
binderbinder
mixmix
binderbindermixmix EE
EEETEETE
0
0
0
*
0
* ),10(),( 
 
f
fZZ D    [2.42] 
where: 
E*mix  complex modulus of the mixture, 
E*binder  complex modulus of the binder, 
E’PL[  glassy modulus of the mixture, 
E0mix  static modulus of the mixture, 
E’ELQGHU  glassy modulus of the binder, 
E0binder  static modulus of the binder, 
T  temperature, 
Ȧ  ʌ
IUHTXHQF\ 
Į  regression coefficient depending on mixture and aging. 
The expression [2.42] is independent of the rheological model used to construct it and 
can be interpreted as a combination of transformation (Figure 2.10): 
- a negative translation of value E0_binder  along the real axis, 
- a homothetic expansion starting from the origin with a ratio of  
(E’BPL[ - E0_mix)/(E’Bbinder - E0_binder), 
- a positive translation of value E0_mix along the real axis. 
 28 
 
Figure 2.10. Binder to Mixture model scheme – (Di Benedetto et al., 2004) 
 
The expression [2.42] was also validated by Di Benedetto et al. (2004) for different 
mixtures and mastic then those used during formulation, providing promising results. 
2.4. Inverse Problem 
The prediction of a material property based on the measured (or observed) material 
response constitutes the objective of an inverse problem in mechanics. This process is 
called a parameter identification procedure. Two procedures for parameter identification 
for viscoelastic materials were proposed by Ohkami and Swoboda (1999). Both methods 
contain boundary control concept introduced by Ichikawa and Ohkami (1992).  
Amin et al (2002) developed a similar approach by combining FEM simulations 
with inverse scheme. The viscoelastic behavior was modeled by the authors using a 3-
parameter solid model (Maxwell model parallel with a spring). With a similar approach, 
FEM simulations combined with measured data was used by Bocciarelli et al. (2005) to 
construct objective function; this function was then minimized using trust-region 
approach.  
Kim and Kreider (2006) used numerical inversion for 2D problem for linear 
viscoelastic homogenous material with 3-7 parameters. Several potential problems with 
 29 
this scheme were detected. The solution might not be unique and might depend on the 
initial guess for optimization method and moreover there is no unique optimization 
approach that is suitable for all problem and material types.  
Zofka et al. (2005) obtained mixture stiffness (inverse of creep compliance) by 
performing Bending Beam Rheometer (BBR) tests on beams of asphalt mixture. The 
author used a modified Hirsch (Hirsch, 1962) model proposed by Christensen 
(Christensen et al., 2003) to “back-calculate” the asphalt binder stiffness and m-value. 
Since brute force was time consuming the original equation [2.19] was combined with an 
alternative procedure to the numerical minimization based on the observation that a 
simple function could be fitted to the mix stiffness versus binder stiffness data. Velasquez 
et al. (2010) using additional experimental data developed two expressions for the Pc 
parameter [2.20] and [2.21]. 
Zofka (2007) used an inverse scheme based on the Zevin’s method of iterative functions 
(Zevin, 1979; Arutyunyan and Zevin, 1988). The asphalt mixture is treated as a 2-phase 
composite material consisting of elastic aggregate particles of arbitrary shape and 
viscoelastic asphalt mastic. 
 30 
 Chapter 3. Materials and Testing 
3.1. Test Methods for low temperature characterization of asphalt binders and 
mixtures 
The following paragraphs provide a short description of the current test methods used to 
characterize the behavior of asphalt binders and mixtures at low temperature.  
3.1.1. Asphalt Binder Testing 
During the Strategic Highway Research Program (SHRP) two test methods were 
developed for the evaluation of the properties of asphalt binders at low temperatures: the 
Bending Beam Rheometer (BBR) and the Direct Tension (DT) test (Anderson et al., 
1994). 
3.1.1.1. Bending Beam Rheometer (BBR) 
The BBR is used to perform low-temperature creep tests on thin beams of asphalt binders 
conditioned at the desired temperature for one hour (AASHTO T 313-02 2006).  The 
asphalt beam (101.6x12.7x6.35mm) is tested in a three-point bending configuration. A 
constant load is applied instantaneously and maintained for all the duration of the test 
(240s) while the deflection at the mid span of the beam is continuously recorded. 
Correspondence principle and elastic solution for a simply supported beam are used to 
obtain the creep compliance. The creep stiffness, S(t), equal to the inverse of the creep 
compliance, D(t), is calculated as: 
)(4)()( 3
3
thb
lP
t
tS GH
V
˜˜˜
˜  
        
[3.1] 
where 
S(t)  flexural creep stiffness, function of time, 
 31 
ı   maximum bending stress in the beam, MPa, 
İW   bending strain (mm/mm), unction of time, 
P   constant load = 980±50mN , 
l   length of specimen (101.6mm),  
b  width of specimen (12.7mm), 
h  height of specimen (6.35mm),  
įW  deflection at the midspan of the beam at time t, and 
t  time. 
The m-value which is the slope of log stiffness versus log time curve is computed 
according to: 
 )log(
)(log)(
td
tSd
tm  
 
  
         
[3.2]                                 
Both stiffness and the m-value are used to determine the critical temperature. 
3.1.1.2. Direct Tension (DT) Test 
The Direct Tension (DT) is used to perform uniaxial tension tests at a constant strain rate 
of 1% per minute on dog-bone shaped specimens of asphalt binders until failure 
(AASHTO T 314-02 2002). The average stress and strain at failure are obtained from six 
replicates and for the same temperature for which creep stiffness and m value are 
measured on BBR. DT strength and thermal stress calculation can be used to calculate the 
critical cracking temperature for the specific binder (Bouldin et al., 2000). 
3.1.2. Asphalt Mixture Testing 
3.1.2.1. Indirect Tensile Test (IDT) 
 32 
Indirect Tensile test (IDT) (AASHTO T 322-03) is currently used to obtain creep 
compliance and strength of asphalt mixtures at low temperatures (Roque and Buttlar, 
1992; Buttlar and Roque, 1994; Zhang et al., 1997; Christensen, 1998; Roque et al., 
2002). During creep testing the cylindrical specimen is vertically loaded with a constant 
load resulting in an almost uniform tensile stress along the diameter of the sample. Four 
Linear Variable Differential Transducers (LDVT’s) are used to measure the vertical and 
horizontal displacement on both sides of the specimen for 1000±2.5s and from this the 
creep curve is obtained. Creep compliance D(t) is calculated according to elastic-
viscoelastic correspondence principle, elastic solutions for horizontal and vertical stresses 
and plane stress Hooke’s law. Strength test of asphalt mixtures can also be performed on 
IDT configuration when appropriate loading mode is applied to the specimen. 
3.1.2.2. Bending Beam Rheometer (BBR) test for Asphalt Mixtures 
Three-point bending test is currently the standard procedure used to determine creep 
compliance of asphalt binders at low temperatures (AASHTO T 313-02 2006). The device 
used to perform this test is the Bending Beam Rheometer (BBR) developed during the 
Strategic Highway Research Program (SHRP) (Bahia et al. 1992). In recent years, Zofka 
et al. (2005, 2006) and Zofka (2007) investigated the use of BBR to determine the creep 
compliance of asphalt concrete. Good agreement was found between the BBR and IDT 
testing procedures. Velasquez (2009) investigated the representative volume element 
when using BBR to test asphalt mixtures beams, and determined that a representative 
creep stiffness of asphalt concrete can be obtained from testing a minimum of three 
replicates of the thin mixture beams. A procedure similar to the one used for binders was 
proposed by Marasteanu et al., (2009) to test thin asphalt mixtures beams with BBR 
 33 
equipment.  A description of the beam preparation is detailed in the NCHRP 133 Final 
report (Marasteanu et al., 2009) and it include several cutting steps from the gyratory 
compacted cylinder through IDT specimens and finally to actual BBR beams. An 
example of a BBR asphalt mixture beam is shown in Figure 3.1, while Figure 3.2 
illustrates a scheme on how the beams are obtained from a cylindrical specimen. 
 
Figure 3.1. Bending Beam Rheometer with thin asphalt mixture (Marasteanu et al., 
2009) 
 
Step 1 
 
 
 
 
Step 2 
Step 3 Step 4 
 
Figure 3.2. Asphalt mixture beam preparation – (Marasteanu et al., 2009) 
 
Testing was performed according to AASHTO T 313-02 standard, using higher loads due 
to the higher stiffness of the mixtures. It was found that good results can be obtained 
 34 
using test loads of 1961 mN and 4413 mN at high (PG low temperature + 22Û& and 
intermediate low temperature levels (PG low temperature + 10Û&UHVSHFWLYHO\)RUWKH
lowest temperature level (PG low temperature - 2Û&WKHFUHHSVWLIIQHVVFDQEHSUHGLFWHG
from the data obtained at the higher two temperatures and from time-temperature 
superposition (Marasteanu et al., 2009). 
3.2. Materials and experimental work 
Two types of materials were used during the experimental phase of this study: eight 
asphalt binders and sixteen asphalt mixtures were tested using the Bending Beam 
Rheometer (BBR) (AASHTO T313-02). The testing was performed as part of NCHRP 
IDEA 133 project (Marasteanu et al., 2009). Table 3.1 presents the binder and mixture 
tested and the corresponding common temperature used during the models evaluation: 
Table 3.1. Asphalt binders and mixtures 
T(ºC) Binder Mixtures 
Granite (GR) Limestone (LM) 
-24 58-34:M1 58-34:M1:GR 58-34:M1:LM 
-24 58-34:M2 58-34:M2:GR 58-34:M2:LM 
-18 58-28:U1 58-28:U1:GR 58-28:U1:LM 
-18 58-28:U2 58-28:U2:GR 58-28:U2:LM 
-24 64-34:M1 64-34:M1:GR 64-34:M1:LM 
-24 64-34:M2 64-34:M2:GR 64-34:M2:LM 
-18 64-28:U1 64-28:U1:GR 64-28:U1:LM 
-18 64-28:M1 64-28:M1:GR 64-28:M1:LM 
 
3.2.1. Asphalt Binders 
Eight different asphalt binders with performance grades (PG) typical for cold climate 
were evaluated. The selected binders include both modified and unmodified (plain) 
binders as shown in Table 3.1. The short time aging that takes place during production 
 35 
and placing was simulated with the RTFOT procedure (AASHTO T 240) and only after 
that the binders were tested. BBR testing was performed according to AASHTO T 313-02 
standard using two replicates at each test temperature.  
3.2.2 Asphalt Mixtures 
Sixteen different asphalt mixtures were used in this study: the mixture were obtained 
combining the eight asphalt binders described in the previous paragraph with two 
aggregate types: granite and limestone (Table 3.1). Cylindrical specimens were prepared 
using gyratory compactor and volumetric properties were measured (Table 3.2). 
Table 3.2. Mix design parameters 
 Granite mixtures Limestone mixtures 
Optimum asphalt content [%] 6.0 6.9 
VMA [%] 16.3 16.2 
VFA [%] 75.9 75.0 
 
Specimens were cut into small beams that were tested in BBR. The asphalt 
mixtures beams were obtained from IDT specimens into whom the asphalt gyratory 
compacted cylinders were first cut and tested (Marasteanu et al., 2009). Prior to testing 
asphalt mixtures were short term aged according to current AASHTO specification 
(AASHTO R030-UL). For each mixture, eleven replicates (beams) were tested. However, 
some tests were discarded as outliers due to a few damaged beams during preparation, 
and the number of replicates considered in the analysis ranged from five to nine.    
 
 
 
 
 36 
Chapter 4. Forward Problem in Low Temperature Asphalt Mixture 
Characterization  
In this chapter, the BBR creep compliance binder data is used to predict the BBR creep 
compliance asphalt mixture and the predictions are compared with the mixture 
experimentally measured creep compliance. Based on literature review (Chapter 2), only 
the most promising models are selected. One semi empirical model, Hirsch model 
(Christensen et al., 2003), and one analogical model, Huet model (Huet, 1963) are 
hereafter evaluated.  
Since the mixture intermediate test temperature (I) matched the binder high test 
temperature (H), the comparison was performed only for this temperature to avoid any 
errors associated with time-temperature superposition shifting. 
4.1. Hirsch Model 
The asphalt concrete was modeled as a two-phase composite material assuming 
aggregates and asphalt binder as the two phases. Hirsch model equations [2.19], [2.20] 
and [2.21] were applied to the experimental data.  Two formulations of the Hirsch model 
were used according to results obtained in previous work (Zofka et al.,.2005; Zofka, 
2007; Velasquez, 2009; Marasteanu et al., 2009; Velasquez et al., 2010). In one study 
(Zofka et al., 2005) a value of aggregate modulus different from the original formulation, 
proposed by Christensen (2003), was used (Eagg=2750000psi – 19GPa instead of 
Eagg=4200000psi – 29GPa) with better fitting results. The alternative formulation of the 
Pc contact volume parameter [2.21] was used in other studies (Zofka, 2007; Velasquez, 
2009; Marasteanu et al., 2009; and Velasquez et al., 2010) in which the aggregate 
modulus was set to 25GPa and 30GPa for limestone and granite respectively. Table 4.1 
 37 
summarizes the parameters used for the models evaluation; G stands for granite and L for 
limestone. 
Table 4.1. Parameters used in Hirsch model 
Granite Limestone 
Hirsch-2 
Ea=2750000psi 
Pc expression [2.20] 
 
Hirsch-2 
Ea=2750000psi 
Pc expression [2.20] 
 
Hirsch-3G  
Ea=4351131psi 
Pc expression [2.21] 
Hirsch-3L 
Ea=3625942psi 
Pc expression [2.21] 
 
Figures 4.1 – 4.8 shows the plots of the creep stiffness S for both granite and limestone 
mixtures. 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.1. Hirsch model for PG 58-34 M1 mixtures T=-24ºC 
 
 
 
 38 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.2. Hirsch model for PG 58-34 M1 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.3. Hirsch model for PG 58-28 U1 mixtures T=-18ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.4. Hirsch model for PG 58-28 U2 mixtures T=-18ºC 
 
 39 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.5. Hirsch model for PG 64-34 M1 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.6. Hirsch model for PG 64-34 M2 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.7. Hirsch model for PG 64-28 U1 mixtures T=-18ºC 
 
 40 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 4.8. Hirsch model for PG 64-28 M1 mixtures T=-18ºC 
 
For the granite mixtures, the experimental curves are located between the two prediction 
curves: Hirsch-2 and Hirsch-3G. For the limestone mixtures, Hirsch-2 and Hirsch-3L 
result in similar prediction curves and both overestimate the experimental data except for 
the limestone mixture PG 58-28 U2 tested at T=18ºC. Overall, the Hirsch model seems to 
reasonably predict the creep stiffness of most mixtures investigated. 
4.2. Huet Model 
Among the analogical models reviewed in Chapter 2 the only model with continuous 
spectrum that presents an expression in the time domain for the creep compliance is the 
Huet model (Huet, 1963) [2.36]. 
This model does not have the additional dashpot in series and the spring in parallel that 
are present in the 2S2P1D model (Olard et al., 2003; Olard, 2003; Di Benedetto et al., 
2004) since the BBR experimental data is obtained at low temperatures or high 
frequencies. At low temperature and/or high frequency Huet model and 2S2P1D model 
give the same results. For higher temperatures and/or lower frequencies, the analysis 
would need to use 2S2P1D model. 
 41 
The five constants required by the model (į N K(’, and IJ) where determined 
through the minimization of the sum of the distances between the experimental creep 
compliance and that Huet model at n time points [4.1]. 
> @ ¹¸·©¨§ ¦ 
n
i
Huet tDtD
1
2exp )()(min         [4.1] 
where: 
Dexp(t)  experimental creep compliance, 
DHuet(t) model creep compliance. 
Figure 4.9 provides an example on how the model fit the experimental data for PG 58-34 
M1 modified asphalt binder and the corresponding asphalt mixture made with granite 
aggregate and tested on BBR at T=24ºC. The model fitting for the other binder-mixtures 
is presented in Appendix A. 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure 4.9. Huet model for PG 58-34 M2 binder and granite mixture T=-24ºC 
 
Visual inspection clearly indicates that Huet model can fit the experimental data 
very well for both asphalt binders and mixtures. This is true for all binders and mixtures 
evaluated. 
 42 
Table 4.2 shows the parameters of the model for four of the asphalt binders and 
the corresponding granite mixtures made with the same mix design and tested at T=-
24ºC. The parameters for the other binders and corresponding mixtures can be found in 
Appendix A. 
Table 4.2. Huet model parameters for four binder and corresponding granite 
mixtures 
Material į k h E’(MPa) /RJIJ 
Binder 
58-34:M1 2.42 0.18 0.60 3000 0.251 
58-34:M2 4.18 0.22 0.62 3000 0.497 
64-34:M1 3.50 0.21 0.64 3000 0.387 
64-34:M2 3.99 0.23 0.64 3000 0.328 
Mixtures 
58-34:M1:GR 2.42 0.18 0.60 28000 3.420 
58-34:M2:GR 4.18 0.22 0.62 30000 3.675 
64-34:M1:GR 3.50 0.21 0.64 30000 3.547 
64-34:M2:GR 3.99 0.23 0.64 29001 3.523 
 
It can be seen that the values for į, k, and h are the same for the binder and the 
corresponding mixture. This condition assures the validity of equation [2.42] when 
considering Huet model (in that case E0mix and E0binder are nil). In fact, Huet model is a 
simplified form of 2S2P1D model that gives the nearly the same results in the considered 
range of temperatures and frequencies. It can also be seen that the binders have similar 
values RI į į, k, and h, identical glassy modulus E’ (3000 MPa), and different 
characteristic time IJ. The same is true for the mixtures; in this case glassy modulus is in 
the 28000-30000 MPa range. The values of the characteristic time of mixtures were 
compared with those found by Huet (Huet, 1963) and reasonable agreement was found. 
Two expressions of the Huet model can be written: one for binder and one for mixture 
[4.2] and [4.3]:  
 43 
   
¸¸¹
·
¨¨©
§
** f )1(
/
)1(
/11)(
_
h
t
k
t
E
tD
h
binder
k
binder
binder
binder
WWG      [4.2] 
   
¸¸¹
·
¨¨©
§
** f )1(
/
)1(
/11)(
_
h
t
k
t
E
tD
h
mix
k
mix
mix
mix
WWG       [4.3] 
where 
Dbinder(t), Dmix(t) creep compliance of binder and mixture, 
E’BELQGHU, E’BPL[ glassy modulus of binder and mixture, 
IJbinderIJmix  characteristic time of binder and mixture. 
From the plot of the log(IJbinder) - log(IJmix) a linear correlation can be detected. Figure 4.10 
shows the relationship between the characteristic time of the binders PG 58-34 M1, PG 
58-34 M1, PG 64-34 M1 and  PG 64-34 M2 and those of the corresponding granite 
mixtures for a reference temperature T=24ºC in a log scale. Analogous linear relations 
were found for the other binders and mixtures evaluated (Appendix A).  
PG 58-34 M1
PG 58-34 M1
PG 64-34 M1
PG 64-34 M2
y = x + 3.17
R² = 0.98
3.40
3.45
3.50
3.55
3.60
3.65
3.70
0 0.1 0.2 0.3 0.4 0.5 0.6
Lo
g
1
0
(ʏ m
ix
) -
(T
=
2
4
º
C
)
Log10(ʏbinder)  - (T=24ºC)
 
Figure 4.10. Log scale linear relationship between IJbinder DQGIJmix for the binders  
and corresponding granite mixtures at PG-24 
 44 
Based on the strong linear correlation found (R2=0.98-0.99 for all the binders-mixtures), 
the following expression can be written to relate the characteristic time of the binders and 
corresponding mixtures with similar mix designs:  
bindermix WW D10          [4.4] 
where: 
IJbinder  characteristic time of binder, 
IJmix  characteristic time of mixture, 
Į  regression parameter which may depend on mix design. 
7KH Į SDUDPHWHU YDOXHV for all mixtures investigated are relatively similar and  range 
from 3.01 to 3.17, which reflects the fact that the mix designs were very similar mix even 
though they contain different type of aggregates (Table 4.3).  
7DEOHĮUHJUHVVLRQSDUDPHWHUIRU the four binders mixtures groups 
Mixtures PG -34 granite 
PG -34 
limestone 
PG -28 
granite 
PG -28 
Limestone 
Į 3.17 3.09 3.01 3.10 
Difference 0.08 0.09 
 
These results are very similar to the results reported by Olard and Di Benedetto in the 
case of the 2S2P1D model (Olard et al., 2003; Olard, 2003; Di Benedetto et al. 2004).  
Combining [4.2], [4.3] and [4.4] the following expressions between the creep 
compliance D(t) - creep stiffness S(t) of the mixture and the creep compliance D(t) - 
creep stiffness S(t) of the binder can be written respectively: 
mix
binder
bindermix E
E
tDtD
_
_)10/()(
f
f D        [4.5] 
 45 
binder
mix
bindermix E
E
tStS
_
_)10/()(
f
f D        [4.6] 
where: 
Dmix(t)  creep compliance of mixture, 
Dbinder(t) creep compliance of binder, 
Smix(t)  creep stiffness of mixture, 
Sbinder(t) creep stiffness of binder, 
E’BPL[  glassy modulus of mixture, 
E’BELQGHU glassy modulus of binder, 
Į  regression parameter which may depend on mix design, 
t  time 
These expressions are identical to equation [2.42]. The expressions are independent of the 
parameters used in the Huet model and thus, do not depend on the model they are built 
on.  
Expression [4.6] can be used to evaluate the forward problem for the asphalt 
binders and mixtures investigated. Figures 4.11-4.18 show examples of the Huet model 
predictions for granite and limestone mixtures at T =-24ºC and T=-18ºC.  
 
 46 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.11. Huet model for PG 58-34 M1 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.12. Huet model for PG 58-34 M1 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.13. Huet model for PG 58-28 U1 mixtures T=-18ºC 
 
 47 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.14. Huet model for PG 58-28 U2 mixtures T=-18ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.15. Huet model for PG 64-34 M1 mixtures T=-24ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.16. Huet model for PG 64-34 M2 mixtures T=-24ºC 
 
 48 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.17. Huet model for PG 64-28 U1 mixtures T=-18ºC 
 
Granite 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
S
ti
ff
n
e
ss
 S
 (G
P
a
)
Time (s)
Experimental
Huet
 
Figure 4.18. Huet model for PG 64-28 M1 mixtures T=-18ºC 
 
The Huet model predictions fit very well the experimental creep stiffness S(t) for 
all mixtures investigated and appears to represent better than Hirsch model the creep 
behavior of asphalt mixtures at low temperatures. 
 
 
 
 
 
 49 
Chapter 5. Inverse Problem in Low Temperature Asphalt Mixture 
Characterization 
Solving inverse problems is not trivial and may require some sophisticated procedures. 
Zofka (2007) found that Self-Consistent Model (SCM) (Yin et al., 2006) is not a good 
candidate for the inverse problem since it does not produce good prediction in the case of 
the forward model. Analogously, Milton (1981) and GSCS - Generalized Self-Consistent 
Scheme (Christensen 1979, Christensen and Lo 1979) – models present complicated 
expressions and require additional adjustment factors. This may results in bigger errors 
both in the case of forward and the potential inverse solutions.  
In the previous chapter, two models, Hirsch (Christensen et al., 2003) and Huet 
(Huet, 1963) based model were used to predict the mixture creep stiffness from the 
binder creep stiffness obtained from BBR testing (forward problem).In this chapter the 
two models are also evaluated if they can be used to back calculate asphalt binder creep 
stiffness Sbinder from the corresponding asphalt mixture creep stiffness Smix.  
5.1. Hirsch Model 
In section 4.1 of the previous chapter it was found that the Hirsch models in Table 4.1 
predict fairly well the stiffness of the majority of the mixtures investigated. 
In this dissertation the method proposed by Zofka et al. (2005) is used to 
investigate the inverse problem with the Hirsch model. First, based on the volumetric 
properties of the mixtures, plots of binder creep stiffness versus predicted mixture 
stiffness using modified equation [2.19] are generated for binder stiffness values between 
50 to 1000MPa (Figures 5.1-5.2). Then, a very simple function is fitted to the mix log 
stiffness versus binder log stiffness data, as shown in Figure 5.1 and 5.2:  
 50 
bEaE bindermix ˜ )ln(         [5.1] 
where a and b are regression parameters. 
Finally, the binder stiffness is simply calculated using equation [5.1] over the entire range 
of loading time. 
Hirsch-2 
y = 2.3345ln(x) - 2.6078
R² = 0.9988
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
A
sp
h
a
lt
 M
ix
tu
re
 S
ti
ff
n
e
ss
 (
G
P
a
)
Asphalt Binder Stiffness (MPa)
Simplified mixture 
stiffness function
 
Hirsch 3G 
y = 3.0922ln(x) - 4.5453
R² = 0.9978
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
A
sp
h
a
lt
 M
ix
tu
re
 S
ti
ff
n
e
ss
 (
G
P
a
)
Asphalt Binder Stiffness (MPa)
Simplified mixture 
stiffness function
 
Figure 5.1. Simplified mixture stiffness function for granite mixtures 
 
Hirsch-2 
y = 2.3377ln(x) - 2.6273
R² = 0.9988
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
A
sp
h
a
lt
 M
ix
tu
re
 S
ti
ff
n
e
ss
 (
G
P
a
)
Asphalt Binder Stiffness (MPa)
Simplified mixture 
stiffness function
 
Hirsch 3L 
y = 2.6595ln(x) - 4.1248
R² = 0.9974
0
2
4
6
8
10
12
14
16
18
20
0 200 400 600 800 1000
A
sp
h
a
lt
 M
ix
tu
re
 S
ti
ff
n
e
ss
 (
G
P
a
)
Asphalt Binder Stiffness (MPa)
Simplified mixture 
stiffness function
 
Figure 5.2. Simplified mixture stiffness function for limestone mixtures 
 
The parameters in Table 4.1 were introduced in the back calculation process along with 
the volumetric properties of the sixteen mixtures investigated (Table 3.2). The 
backcalculation algorithm was applied to the mixture data and binder creep stiffness was 
predicted and compared to the creep stiffness experimentally determined for the RTFOT 
 51 
binders used to prepare the corresponding mixtures.  Figures 5.3-5.10 present the model 
prediction of the asphalt binder creep stiffness for all the mixtures investigated: 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.3. Hirsch model for PG 58-34 M1 mixtures T=-24ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.4. Hirsch model for PG 58-34 M1 mixtures T=-24ºC 
 
 
 
 52 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.5. Hirsch model for PG 58-28 U1 mixtures T=-18ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.6. Hirsch model for PG 58-28 U2 mixtures T=-18ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.7. Hirsch model for PG 64-34 M1 mixtures T=-24ºC 
 
 53 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.8. Hirsch model for PG 64-34 M2 mixtures T=-24ºC 
 
Granite 
1
10
100
1000
10000
100000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.9. Hirsch model for PG 64-28 U1 mixtures T=-18ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3G
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Hirsch-2
Hirsch-3L
 
Figure 5.10. Hirsch model for PG 64-28 M1 mixtures T=-18ºC 
 
 54 
For both granite and limestone mixtures, the binder stiffness predictions, obtained 
with the Hirsch models, fit the experimentally determined binder creep stiffness poorly 
with very few exceptions. Thus the comparison of the predicted and experimentally 
determined data indicated less good agreement than what was found by Zofka et al. 
(2005).  It should be mentioned that in that previous work, the binders were chemically 
extracted from the mixture and then tested, while in this case the original binder was aged 
in the RTFOT and then tested. 
5.2. Huet Model 
In Chapter 4 an expression for predicting the creep stiffness of the mixture from the creep 
stiffness of the binder was found [4.6]. In this case, the approach is straight forward. 
First, from equation [4.4] the binder characteristic time is obtained from mixture 
characteristic time: 
mixbinder WW D 10          [5.2] 
where: 
IJbinder  characteristic time of binder, 
IJmix  characteristic time of mixture, 
Į  regression parameter which may depend on mix design. 
Then asphalt binder creep stiffness Sbinder can be easily predicted from the asphalt mixture 
creep stiffness Smix from equation [4.6]: 
mix
binder
mixbinder E
E
tStS
_
_)10/()(
f
f D        [5.3] 
where: 
 55 
Smix(t)  creep stiffness of mixture, 
Sbinder(t) creep stiffness of binder, 
E’BPL[  glassy modulus of mixture, 
E’BELQGHU glassy modulus of binder, 
Į  regression parameter which may depend on mix design, 
t  time. 
Expression [5.3] provides a formulation for the evaluation of the inverse problem and can 
be applied to the binders and mixtures investigated. The prediction uses Huet model and 
regression parameter Įpreviously determined and presented in Table 4.3. Figures 5.11-
5.18 present the predicted binder creep stiffness S(t) obtained using equation [5.3] for all 
the granite and limestone mixtures considered.  
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.11. Huet model for PG 58-34 M1 mixtures T=-24ºC 
 
 56 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.12. Huet model for PG 58-34 M1 mixtures T=-24ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.13. Huet model for PG 58-28 U1 mixtures T=-18ºC 
 
Granite 
 
Limestone 
 
Figure 5.14. Huet model for PG 58-28 U2 mixtures T=-18ºC 
 
 57 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.15. Huet model for PG 64-34 M1 mixtures T=-24ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
  (
M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.16. Huet model for PG 64-34 M2 mixtures T=-24ºC 
 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.17. Huet model for PG 64-28 U1 mixtures T=-18ºC 
 
 58 
Granite 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Limestone 
1
10
100
1000
10000
0.1 1 10 100 1000 10000
A
sp
h
a
lt
 B
in
d
e
r 
S
ti
ff
n
e
ss
 (M
P
a
)
Time (s)
Experimental
Huet
 
Figure 5.18. Huet model for PG 64-28 M1 mixtures T=-18ºC 
 
It is obvious that expression [5.3] predicts asphalt binder creep stiffness very well 
for all binders and mixtures investigated. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 59 
Chapter 6. Summary and Conclusions 
6.1. Summary 
In this dissertation, forward and inverse problems in asphalt mixtures with applications to 
low temperature mechanical properties were investigated based on experimental testing 
and modeling. 
The experimental part consisted of three-point bending creep tests performed with 
BBR on binders and mixtures beams (6.35 × 12.7 × 101.6 mm). Eight binders and sixteen 
asphalt mixtures were tested at low pavement service temperatures. In the analysis part,  
Hirsch semi empirical model and Huet analogical model were used with the experimtnal 
data to evaluate their performance to predict asphalt mixture creep stiffness from asphalt 
binder creep stiffness (forward problem) and, vice versa, to “extract” asphalt binder creep 
stiffness from asphalt mixture creep stiffness (inverse problem).   
6.2. Conclusions 
The main findings of this study can be summarized as follows: 
x In the case of the forward problem, Hirsch model predicted fairly well the creep 
stiffness of the majority of mixtures investigated. There is a small tendency to 
over predict the stiffness for the limestone mixtures.  
x In the case of the inverse problem,  Hirsch model predicted poorly asphalt binder 
creep stiffness from asphalt mixture creep stiffness 
x Huet model fitted very well the asphalt binder and asphalt mixture experimental 
data 
 60 
x A linear relationship was found between log characteristic times of asphalt 
binders and log characteristic times of the corresponding asphalt mixtures. 
x 7KHUHJUHVVLRQSDUDPHWHUĮZDVYHU\VLPLODUIRUDOODVSKDOWELQGHUVDQGPL[WXUHV
investigated.  The mixtures had similar mix designs, but were produced with 
different types of binders and two types of aggregates.  
x Huet model performed very well in both forward and inverse problems.  In 
particular, for the inverse problem of “extracting” asphalt binder creep stiffness 
from experimental mixture data, the model performed much better than Hirsch 
model.    
The analysis was performed on a set of mixtures with a similar mix design, and therefore, 
it is recommended that this approach is extended to asphalt binder and mixture 
combinations that include different mix designs, asphalt modifications, and aggregate 
types.  
The analysis should be extended to mixtures produced with high contents of RAP 
since Huet model has the potential of providing a much needed tool to improve the mix 
design of asphalt mixtures produced with large amounts of RAP. 
 
 
 61 
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 67 
Appendix A (Chapter 4) Forward Problem in Low Temperature 
Asphalt Mixture Characterization 
A.1. Huet model fitting 
The following plots (Figure A.1-A.16) show the Huet (Huet 1963) model fitting for all 
the asphalt binders and mixtures investigated (Table A.1). 
Table A.1. Asphalt binders and mixtures 
T(ºC) Binder Mixtures 
Granite (GR) Limestone (LM) 
-24 58-34:M1 58-34:M1:GR 58-34:M1:LM 
-24 58-34:M2 58-34:M2:GR 58-34:M2:LM 
-18 58-28:U1 58-28:U1:GR 58-28:U1:LM 
-18 58-28:U2 58-28:U2:GR 58-28:U2:LM 
-24 64-34:M1 64-34:M1:GR 64-34:M1:LM 
-24 64-34:M2 64-34:M2:GR 64-34:M2:LM 
-18 64-28:U1 64-28:U1:GR 64-28:U1:LM 
-18 64-28:M1 64-28:M1:GR 64-28:M1:LM 
 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0 200 400 600 800 1000
D
(1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.1. Huet model for PG 58-34 M1 binder and granite mixture T=-24ºC 
 
 
 
 68 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.2. Huet model for PG 58-34:M2 binder and granite mixture T=-24ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
0 200 400 600 800 1000
D
(1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.3. Huet model for PG 58-28 U1 binder and granite mixture T=-18ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0.00020
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.4. Huet model for PG 58-28 U2 binder and granite mixture T=-18ºC 
 
 69 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.5. Huet model for PG 64-34 M1 binder and granite mixture T=-24ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.6. Huet model for PG 64-34:M2 binder and granite mixture T=-24ºC 
 
Asphalt binder 
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.7. Huet model for PG 64-28 U1 binder and granite mixture T=-18ºC 
 
 70 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
J 
(1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.8. Huet model for PG 64-28 M1 binder and granite mixture T=-18ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0.00020
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.9. Huet model for PG 58-34 M1 binder and limestone mixture T=-24ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0.00020
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.10. Huet model for PG 58-34:M2 binder and limestone mixture T=-24ºC 
 
 71 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.11. Huet model for PG 58-28 U1 binder and limestone mixture T=-18ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.12. Huet model for PG 58-28 U2 binder and limestone mixture T=-18ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.13. Huet model for PG 64-34 M1 binder and limestone mixture T=-24ºC 
 
 72 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.14. Huet model for PG 64-34:M2 binder and limestone mixture T=-24ºC 
 
Asphalt binder 
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
0.00018
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.15. Huet model for PG 64-28 U1 binder and limestone mixture T=-18ºC 
 
Asphalt binder 
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
 
Asphalt mixture 
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
0 200 400 600 800 1000
D
 (
1
/M
P
a
)
Time (s)
Experimental
Huet
Figure A.16. Huet model for PG 64-28 M1 binder and limestone mixture T=-18ºC 
 
 73 
A.2. Huet model fitting parameters 
The following tables presents the Huet model fitting parameters for the four groups in 
which the sixteen mixtures were divided for the analysis. 
Table A.2. Huet model parameters for PG-34 binder and corresponding granite 
mixtures 
Material į k h E’(MPa) /RJIJ 
Binder 
58-34:M1 2.42 0.18 0.60 3000 0.251 
58-34:M2 4.18 0.22 0.62 3000 0.497 
64-34:M1 3.50 0.21 0.64 3000 0.387 
64-34:M2 3.99 0.23 0.64 3000 0.328 
Mixtures 
58-34:M1:GR 2.42 0.18 0.60 28000 3.420 
58-34:M2:GR 4.18 0.22 0.62 30000 3.675 
64-34:M1:GR 3.50 0.21 0.64 30000 3.547 
64-34:M2:GR 3.99 0.23 0.64 29001 3.523 
  
Table A.3. Huet model parameters for PG-34 binder and corresponding limestone 
mixtures 
Material į k h E’(MPa) /RJIJ 
Binder 
58-34:M1 2.42 0.18 0.60 3000 0.251 
58-34:M2 4.18 0.22 0.62 3000 0.497 
64-34:M1 3.50 0.21 0.64 3000 0.387 
64-34:M2 3.99 0.23 0.64 3000 0.328 
Mixtures 
58-34:M1:LM 2.42 0.18 0.60 22099 3.341 
58-34:M2:LM 4.18 0.22 0.62 27279 3.592 
64-34:M1:LM 3.50 0.21 0.64 27988 3.467 
64-34:M2:LM 3.99 0.23 0.64 23119 3.428 
 
Table A.4. Huet model parameters for PG-28 binder and corresponding granite 
mixtures 
Material į k h E’(MPa) /RJIJ 
Binder 
58-28:U1 4.33 0.22 0.58 3000 -0.032 
58-28:U2 4.23 0.25 0.64 3000 0.341 
64-28:U1 5.11 0.22 0.63 3000 0.855 
64-28:M1 2.38 0.20 0.51 3000 -0.504 
Mixtures 
58-28:U1:GR 4.33 0.22 0.58 33016 3.023 
58-28:U2:GR 4.23 0.25 0.64 30000 3.366 
64-28:U1:GR 5.11 0.22 0.63 34957 3.855 
64-28:M1:GR 2.38 0.20 0.51 38637 2.476 
  
 74 
Table A.5. Huet model parameters for PG-28 binder and corresponding limestone 
mixtures 
Material į k h E’(MPa) /RJIJ 
Binder 
58-28:U1 4.33 0.22 0.58 3000 -0.032 
58-28:U2 4.23 0.25 0.64 3000 0.341 
64-28:U1 5.11 0.22 0.63 3000 0.855 
64-28:M1 2.38 0.20 0.51 3000 -0.504 
Mixtures 
58-28:U1:LM 4.33 0.22 0.58 31347 3.059 
58-28:U2:LM 4.23 0.25 0.64 31163 3.426 
64-28:U1:LM 5.11 0.22 0.63 31278 3.985 
64-28:M1:LM 2.38 0.20 0.51 28217 2.616 
  
A.3. ĮUHJUHVVLRQSDUDPHWHUSORWV 
The following four plots presents the log scale linear relationship between the 
characteristic time of the binders and those of the corresponding mixtures. The regression 
SDUDPHWHUVĮIRUDOOWKHELQGHU-mixtures investigated are listed in Table A.6. 
PG 58-34 M1
PG 58-34 M1
PG 64-34 M1
PG 64-34 M2
y = x + 3.17
R² = 0.98
3.40
3.45
3.50
3.55
3.60
3.65
3.70
0 0.1 0.2 0.3 0.4 0.5 0.6
Lo
g
1
0
(ʏ m
ix
) -
(T
=
2
4
º
C
)
Log10(ʏbinder)  - (T=24ºC)
 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-
34 and corresponding granite mixtures 
 
 75 
PG 58-34 M1
PG 58-34 M1
PG 64-34 M1
PG 64-34 M2
y = x + 3.09
R² = 0.99
3.30
3.35
3.40
3.45
3.50
3.55
3.60
3.65
0 0.1 0.2 0.3 0.4 0.5 0.6
Lo
g
1
0
(ʏ m
ix
)  
-
(T
=
2
4
º
C
)
Log10(ʏbinder)  - (T=24ºC)
 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-
34 and corresponding limestone mixtures 
 
 
PG 58-28 U1
PG 58-28 U2
PG 64-28 U1
PG 64-28 M1 
y = x + 3.01
R² = 0.99
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Lo
g
1
0
(ʏ m
ix
) -
(T
=
1
8
º
C
)
Log10(ʏbinder) - (T=18ºC) 
 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-
28 and corresponding granite mixtures 
 
 
 76 
PG 58-28 U1
PG 58-28 U2
PG 64-28 U1
PG 64-28 M1 
y = x + 3.10
R² = 0.99
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Lo
g
1
0
(ʏ m
ix
) -
(T
=
1
8
º
C
)
Log10(ʏbinder) - (T=18ºC)
 
)LJXUH$/RJVFDOHOLQHDUUHODWLRQVKLSEHWZHHQIJbinder DQGIJmix for the binders PG-
28 and corresponding limestone mixtures 
 
7DEOH$ĮUHJUHVVLRQSDUDPHWHUIRUWKHIRXUELQGHUVPL[WXUHVJURXSV 
Mixtures PG -34 granite 
PG -34 
limestone 
PG -28 
granite 
PG -28 
Limestone 
Į 3.17 3.09 3.01 3.10 
Difference 0.08 0.09