Prediction of Creep Stiffness of Asphalt Mixture
with Micromechanical Finite-Element
and Discrete-Element Models
Qingli Dai1 and Zhanping You2
Abstract: This study presents micromechanical finite-element 共FE兲 and discrete-element 共DE兲 models for the prediction of viscoelastic
creep stiffness of asphalt mixture. Asphalt mixture is composed of graded aggregates bound with mastic 共asphalt mixed with fines and fine
aggregates兲 and air voids. The two-dimensional 共2D兲 microstructure of asphalt mixture was obtained by optically scanning the smoothly
sawn surface of superpave gyratory compacted asphalt mixture specimens. For the FE method, the micromechanical model of asphalt
mixture uses an equivalent lattice network structure whereby interparticle load transfer is simulated through an effective asphalt mastic
zone. The ABAQUS FE model integrates a user material subroutine that combines continuum elements with viscoelastic properties for the
effective asphalt mastic and rigid body elements for each aggregate. An incremental FE algorithm was employed in an ABAQUS user
material model for the asphalt mastic to predict global viscoelastic behavior of asphalt mixture. In regard to the DE model, the outlines
of aggregates were converted into polygons based on a 2D scanned mixture microstructure. The polygons were then mapped onto a sheet
of uniformly sized disks, and the intrinsic and interface properties of the aggregates and mastic were assigned for the simulation. An
experimental program was developed to measure the properties of sand mastic for simulation inputs. The laboratory measurements of the
mixture creep stiffness were compared with FE and DE model predictions over a reduced time. The results indicated both methods were
applicable for mixture creep stiffness prediction.
DOI: 10.1061/共ASCE兲0733-9399共2007兲133:2共163兲
CE Database subject headings: Predictions; Creep; Stiffness; Finite element method; Discrete elements; Asphalt; Mixtures.
Introduction
Asphalt mixture, or hot mix asphalt 共HMA兲, is a well-known
composite material of graded aggregates bound with the mastic
共where mastic is a kind of fine mixture composed of asphalt
binder mixed with fines and fine aggregates兲 in asphalt pavement
construction. This multiphase material has different properties
from the original components—aggregate and mastic. The mastic
itself consists of fine aggregates, sand, and fines embedded in a
matrix of asphalt binder. The physical properties and performance
of HMA is governed by the properties of the aggregate 共shape,
surface texture, gradation, skeletal structure, modulus, etc.兲, properties of the asphalt binder 共grade, complex modulus, relaxation
characteristics, cohesion, etc.兲, and asphalt–aggregate interactions
共adhesion, absorption, physiochemical interactions, etc.兲. Aggre1
Research Assistant Professor, Dept. of Mechanical Engineering–
Engineering Mechanics, Michigan Technological Univ., 1400 Townsend
Dr., Houghton, MI 49931. E-mail: [email protected]
2
Donald and Rose Ann Tomasini Assistant Professor of Transportation
Engineering, Dept. of Civil and Environmental Engineering, Michigan
Technological Univ., 1400 Townsend Drive, Houghton, MI 49931.
E-mail: [email protected]
Note. Associate Editor: Ching S. Chang. Discussion open until July 1,
2007. Separate discussions must be submitted for individual papers. To
extend the closing date by one month, a written request must be filed with
the ASCE Managing Editor. The manuscript for this paper was submitted
for review and possible publication on June 1, 2005; approved on
September 13, 2006. This paper is part of the Journal of Engineering
Mechanics, Vol. 133, No. 2, February 1, 2007. ©ASCE, ISSN 07339399/2007/2-163–173/$25.00.
gate sizes vary depending on the different purpose of the asphalt
mixture, but typically, the sizes range from less than 0.075 mm
共75 ␮m兲 to as large as 37.5 mm. In addition, aggregate particles
in the mixture have different shapes, surface texture and friction,
and orientations, which make the aggregate contact more complicated. Asphalt binder is a material with very complex properties
that depends on temperature and loading time/frequency. Asphalt
mixture stiffness prediction is difficult when dealing with the different sizes of aggregate bonded with the asphalt binder. Therefore, the structure of asphalt mixture is very complex, which
makes properties 共such as stiffness and tensile strength兲 predictions very challenging. However, if some simplifying assumptions are employed, it is possible to construct a mathematical
model to arrive at approximate physical properties of asphalt concrete. For example, the approximation 共lower bound兲 presented
by Reuss 共1929兲 and Aboudi 共1991兲, Voigt approximation 共upper
bound兲 proposed by Voigt 共1889兲, model for elastic moduli of
heterogeneous materials 共Hashin 1962兲, Hirsch’s model 共Hirsch
1962兲, Counto’s model 共Counto 1964兲, Paul’s model 共Paul 1960兲,
the arbitrary phase geometry model 共Hashin and Shtrikman
1963兲, model by Mori and Tanaka 共1973兲, generalized selfconsistent scheme model 共Christensen and Lo 1979兲, and the
composite spheres model 共Hashin 1965兲 were available in the past
century. Most recently, Christensen et al. 共2003兲 reported a modified Hirsch’s model for estimating asphalt concrete complex dynamic modulus directly from binder modulus, voids in mineral
aggregate 共VMA兲, and voids filled with asphalt 共VFA兲.
A number of researchers have developed micromechanical
models to predict the fundamental properties of the mixture based
upon those of the phase constituents. For example, studies on
JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007 / 163
cemented particulate materials by Dvorkin et al. 共1994兲 and Zhu
et al. 共1996兲 provided information on the normal and tangential
load transfer between cemented particles. Chang et al. 共1999兲
derived an expression for elastic modulus of an assembly of
bonded granulates, based on the response of two particles that are
connected by an elastic binder. The expression considers the
particle/binder modulus, the particle size, the binder thickness, the
binder width, the assembly coordination number, the binder content, and the porosity. Contact mechanism based theory of viscoelastic Maxwell matrix composites was investigated by Zhu
共2001兲, and Zhu et al. 共2000, 2001兲. Zhu and Nodes 共2000兲 studied the effect of aggregate angularity and found that a higher
degree of aggregate angularity gives rise to a stiffer response of
asphalt pavement. This finding was consistent with the phenomenon of better interlocking interaction for coarse aggregates. In a
related study, Krishnan and Rao 共2000兲 used mixture theory to
explain air void reduction in asphalt materials under load.
Micromechanical modeling of HMA represents the possibility
of developing a practical tool to predict mixture properties. Buttlar and You 共2001兲 classified different micromechanical models
into several categories, which include noninteracting particles,
interacting particles, and the micromechanical finite- and discreteelement models. The following sections will introduce the micromechanical finite-element method 共FEM兲 and discrete-element
method 共DEM兲 that were applied for modeling complex asphalt
mixture.
In regard to FEM, Mishnaevsky and Schmauder 共2001兲 reported some recent advances in the numerical modeling of heterogeneous materials and concluded that the unit cell model and
real structure simulation allow the complex microstructure of materials to be taken into account. It was indicated that the FE simulation of the micromechanical behavior of multiphase materials
could be carried out with the multiphase finite-element 共MPE兲
approach. Li et al. 共1999兲 proposed a two-phase asphalt concrete
micromechanical model to predict the elastic modulus of HMA,
and asphalt concrete was assumed to be a two-phase composite
with aggregates dispersed in the asphalt matrix. Mohamed and
Hansen 共1999a兲 further developed a three-phase material model
for cement concrete, using truss finite elements at the microlevel.
The internal structure of the asphalt concrete was considered as a
three-phase material 共matrix, aggregate, and interfaces between
the matrix and the aggregate phases兲. The model considered the
randomness of the aggregate phase, as well as the probabilistic
nature of the properties of the three phases. Mohamed and Hansen
共1999b兲 applied the micromechanical model to predict the concrete response under shear and compressive loading. They recommended that three-dimensional analysis was needed to capture the
material responses quantitatively.
A number of other micromechanical finite-element models
have been used to study complex behavior of asphalt concrete.
Sepehr et al. 共1994兲 used an idealized finite-element microstructural model to analyze the behavior of an asphalt pavement
layer. To simulate the complex damage behavior in asphalt
mixture, Soares et al. 共2003兲 used cohesive zone elements to develop a micromechanical fracture model of asphalt materials.
Guddati et al. 共2002兲 recently presented a random truss lattice
model to simulate microdamage in asphalt concrete. Sadd et al.
共2004a,b兲 employed a microframe network model to investigate
the damage behavior of asphalt materials; this model used a special purpose finite element that incorporates the mechanical loadcarrying response between neighboring particles. For the elastic
response, Kose et al. 共2000兲 and Masad et al. 共2002兲 reported the
strain distribution within asphalt binders in asphalt concrete by
using a two-dimensional 共2D兲 finite element with the linear elastic
properties of the binder and aggregate. Wang 共2003兲 conducted
3D FE simulation by using x-ray tomography imaging technologies. In regard to viscoelastic properties, Buttlar and Roque
共1996兲 evaluated the potential use of micromechanical models to
predict the viscoelastic properties of asphalt mixtures and mastics.
Uddin 共1999兲 presented a micromechanical analysis method for
calculating the creep compliance of asphaltic mixes on a microscopic level using the laboratory viscoelastic characterization of
the binder and elastic material properties of the aggregates at a
given temperature.
The discrete-element method 共DEM兲 analyzes particulate systems by modeling the translational and rotational behavior of each
particle using Newton’s second law with appropriate interparticle
contact forces. Normally the scheme establishes an explicit, timestepping procedure to determine each of the particle motions.
DEM studies were conducted on granular materials or asphalt
mixture such as the work by Trent and Margolin 共1994兲, Buttlar
and You 共2001兲, You and Buttlar 共2004, 2005, 2006兲, Ullidtz
共2001兲, and Sadd et al. 共1997, 1998兲. Rothenburg et al. 共1992兲
developed a micromechanical discrete-element model for asphalt
concrete to investigate pavement rutting. Rothenburg et al.’s
model is useful for modeling asphalt concrete because it considers not only the aggregate–binder interaction but also intergranular interactions in the presence of the binder. Chang and
Meegoda 共1997兲 proposed a micromechanical model based upon
the discrete-element program ASBAL, which was developed
by modifying the TRUBAL program 共Cundall 1971, 1978兲 to
simulate hot mix asphalt 共HMA兲. The modified program allowed
aggregate–binder–aggregate contact and aggregate–aggregate
contact. An innovative feature in the model is the consideration
of both aggregate–binder–aggregate contact and aggregate–
aggregate contact. Zhong and Chang 共1997, 1999兲 adopted a
micromechanics approach to consider a contact law for the interparticle behavior of two particles connected by a binder. The
model was based upon the premises that the interparticle binder
initially contains microcracks, and the discrete-element method
was used to simulate the cracking behavior under uniaxial and
biaxial conditions.
The following sections present the micromechanical finiteelement 共FE兲 and discrete-element 共DE兲 modeling schemes
for the prediction of viscoelastic creep stiffness of asphalt
mixture. The 2D microstructure of asphalt mixture was obtained
by optically scanning the smoothly sawn surface of asphalt
mixture specimens. The FE microstructural model incorporates
an equivalent lattice network structure whereby intergranular
load transfer is simulated through an effective asphalt mastic
zone. The ABAQUS FE model combines the user-defined material subroutine with continuum elements for the effective asphalt
mastic and rigid body defined with rigid elements for each aggregate. An incremental FE algorithm was employed in an ABAQUS
user material subroutine for the mastic to predict global viscoelastic behavior of asphalt mixture. The FE simulation was conducted
on a computer generated sample from 2D scanned microstructure
with image processing and aggregate-fitting technologies. This
paper also discusses a cluster discrete-element 共DE兲 modeling
approach, which was applied to predict mixture creep stiffness.
For the DE model, the outlines of aggregates were converted into
polygons based on 2D scanned mixture microstructure. The polygons were mapped onto a sheet of uniformly sized disks, and the
intrinsic and interface properties of the aggregates and mastic
were then assigned for the simulation. An experimental program
was developed to measure the creep stiffness of the sand mastic
164 / JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007
Fig. 1. Asphalt modeling concept. 共a兲 Typical asphalt material; 共b兲
model aggregate system; and 共c兲 ABAQUS finite-element model.
for both FE and DE simulation. The laboratory measurements of
the creep stiffness of the asphalt mixture were employed to compare the predictions of the models.
Microstructural Finite-Element Model
of Asphalt Mixture
Microstructural Model
Asphalt mixture can be described as a multiphase material containing coarse aggregates, mastic cement 共including asphalt
binder and fine aggregates兲, and air voids. The load transfer between the aggregates plays a primary role in determining the load
carrying capacity and failure of such complex materials. In order
to develop a micromechanical model of this behavior, proper
simulation of the load transfer between the aggregates must be
accomplished. The aggregate material is normally much stiffer
than the mastic, and thus aggregates are taken as rigid particles.
On the other hand, the mastic cement is a compliant material with
elastic, inelastic, and time-dependent behaviors.
In general, asphalt mixture contains aggregate of highly irregular geometry as shown in Fig. 1共a兲. In this study, an approach
to simplify the irregular aggregate geometry is to allow variable
size and shape using an elliptical aggregate model as represented
in Fig. 1共b兲. In order to properly account for the load transfer
between aggregates, it is assumed that there is an effective mastic
zone between neighboring particles. It is through this zone that
the micromechanical load transfer occurs between each aggregate
pair. The finite-element model shown in Fig. 1共c兲 uses a rectangular strip to simulate the effective asphalt mastic zone, and incorporates the ABAQUS user material subroutine with continuum
elements to model the effective asphalt mastic behavior.
The ABAQUS finite-element scheme employed four-node
quadrilateral elements to mesh the effective cement material, and
a rigid body defined with two-node rigid elements sharing the
particle center to model each aggregate 关as shown in Fig. 2共a兲兴.
Fig. 2共b兲 shows an ABAQUS mesh in seven connecting particles
with four-node quadrilateral and two-node rigid elements. The
particle center is referred to as the master node of the aggregate
rigid body, and thus each rigid body element would have identical
translation and rotation as the aggregate. The rigid elements act to
link the mastic deformation with the aggregate rigid body motion
thereby establishing the micromechanical deformation behavior.
Properties of the quadrilateral elements are specified through a
user defined material subroutine within the ABAQUS code and
this allows incorporation of the following viscoelastic mastic cement models.
Fig. 2. ABAQUS modeling scheme. 共a兲 ABAQUS model for a
typical particle pair; 共b兲 ABAQUS mesh for seven connecting
particles.
Maxwell-Type Viscoelastic Model
The generalized Maxwell model was used to simulate the linear
and damage-coupled viscoelastic behavior of asphalt mixture by
Dai et al. 共2006a,b兲. The linear constitutive behavior for this
Maxwell-type model can be expressed as a hereditary integral
␴ = E ⬁␧ +
冕
t
Et
0
d␧共␶兲
d␶
d␶
共1兲
where Et is expressed with a Prony series
M
Et =
共t−␶兲
兺 E me − ␳
m=1
and ␳m =
m
␩m
Em
共2兲
In these equations, Et⫽transient modulus as a function of the
time; Em, ␩m, and ␳m⫽spring constant, dashpot viscosity, and relaxation time, respectively, for the mth Maxwell element.
The reduced time 共effective time兲 is defined by using the time–
temperature superposition principle as
␰共t兲 =
冕
t
0
1
d␶
␣T
共3兲
where the term ␣T = ␣T关T共␶兲兴⫽temperature-dependent time-scale
shift factor.
Three-dimensional behavior can be formulated with uncoupled
volumetric and deviatoric stress–strain relations. A displacement
JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007 / 165
冤 冥冤
⌬␴xx
K1
·
⌬␴yy
·
⌬␴zz
=
·
⌬␴xy
·
⌬␴yz
·
⌬␴xz
based incremental finite-element modeling scheme with constant
strain rate over each increment has been developed based on the
work by Zocher and Groves 共1997兲.
The volumetric constitutive relationship is expressed with the
volumetric stress ␴kk and strain ␧kk in the general form
␴kk共␰兲 = 3K⬁␧kk共␰兲 +
冕
␰
0
d␧kk共␰⬘兲
3Kt共␰ − ␰⬘兲
d␰⬘
d␰⬘
共4兲
M
Kme−共␰−␰⬘兲/␳m⫽transient
Kt共␰ − ␰⬘兲 = 兺m=1
冋
⌬␴kk = 3 K⬁ +
兺
m=1
册
K m␳ m
共1 − e−⌬␰/␳m兲 ⌬␧kk + ⌬␴Rkk
⌬␰
+
=
where
兺 − 共1 − e
−⌬␰/␳m
兲Sm共␰n兲,
Sm共␰n兲 = 3KmRkk␳m ⫻ 共1 − e
兲 + Sm共␰n−1兲e
冕
␰
2Gt共␰ − ␰⬘兲
0
−⌬␰/␳m
d␧ˆ ij共␰⬘兲
d␰⬘
d␰⬘
共7兲
N
Gme−共␰−␰⬘兲/␳m⫽transient shear modulus. The
where Gt共␰ − ␰⬘兲 = 兺m=1
formulation of the deviatoric behavior is obtained with constant
deviatoric strain rate R̂ij = ⌬␧ˆ ij / ⌬␰
冋
N
兺
m=1
册
G m␳ m
共1 − e−⌬␰/␳m兲 ⌬␧ˆ ij + ⌬␴ˆ Rij
⌬␰
共8兲
and the residual part ⌬␴ˆ Rij can be expressed in the recursive
relation
N
⌬␴ˆ Rij
=
冋
N
兺 − 共1 − e−⌬␰/␳ 兲Sm共␰n兲,
m
m=1
共9兲
and
冋
Sm共␰n兲 = 2GmR̂ij␳m共1 − e−⌬␰/␳m兲 + Sm共␰n−1兲e−⌬␰/␳m
The incremental normal stresses can be then formulated by combining the volumetric and deviatoric behavior, while the incremental shear stress can be formulated by using only the deviatoric
behavior. Once the incremental stress components are developed,
the incremental stiffness terms can be calculated and then the
incremental 3D linear viscoelastic behavior was formulated as
−
兺
m=1
K m␳ m
共1 − e−⌬␰/␳m兲
⌬␰
册
N
4
G m␳ m
共1 − e−⌬␰/␳m兲
G⬁ +
3
⌬␰
m=1
兺
N
K2 = K⬁ +
For the initial increment, the history variable Sm共␰1兲 equals
to 3KmRkk␳m共1 − e−⌬␰/␳m兲 and is similar to the following
formulations.
For the deviatoric behavior, the constitutive relationship
is written using deviatoric stress ␴ˆ ij = ␴ij − 1 / 3␴kk␦ij and strain
␧ˆ ij = ␧ij − 1 / 3␧kk␦ij
␴ˆ ij共␰兲 = 2G⬁␧ˆ ij共␰兲 +
冥冤 冥
⌬␧xx
⌬␧yy
⌬␧zz
⌬␧xy
⌬␧yz
⌬␧xz
共10兲
⌬␴ˆ Rxy
冋
共6兲
−⌬␰/␳m
K3
R
⌬␴Rkk + ⌬␴ˆ zz
+
and
K3
·
⌬␴Rkk + ⌬␴ˆ Rxx
K1 = K⬁ +
m=1
⌬␴ˆ ij = 2 G⬁ +
冤 冥
0
0
0
0
0
R
⌬␴ˆ zx
M
⌬␴Rkk
K3
·
·
0
0
0
0
⌬␴ˆ Ryz
共5兲
and the residual part ⌬␴Rkk can be expressed in a recursive relation
with the history variable Sm
0
0
0
K2
K2
K1
·
·
·
⌬␴Rkk + ⌬␴ˆ Ryy
bulk modulus. The
where
incremental formulation of the volumetric behavior is obtained
with constant volumetric strain rate Rkk = ⌬␧kk / ⌬␰
N
K2
K1
·
·
·
·
冋
冋
兺
m=1
K m␳ m
共1 − e−⌬␰/␳m兲
⌬␰
册
N
册
册
册
2
G m␳ m
共1 − e−⌬␰/␳m兲
G⬁ +
3
⌬␰
m=1
K3 = 2 G⬁ +
兺
N
兺
m=1
G m␳ m
共1 − e−⌬␰/␳m兲
⌬␰
共11兲
This viscoelastic model was defined in the ABAQUS user material subroutine, and then integrated with ABAQUS continuum
elements for mastic materials to implement a displacementbased time-dependent finite-element analysis and predict the global viscoelastic behavior of asphalt mixture.
Experimental Measurements of Mastic and Mixture
Creep Stiffness
An experimental program was developed to serve two main purposes: 共1兲 to measure the creep stiffness of the sand mastic, which
was a required FE and DE model input, and 共2兲 to obtain creep
stiffness of the HMA mixture, which was used to assess the FE
and DE model prediction. The aggregate creep stiffness was suitably estimated based upon mineralogical composition and limited
laboratory measurements using the uniaxial compression test.
The sand mastic mixture used in this study was composed of a
mixture of the fine aggregates in the mixtures passing the
2.36 mm sieve. For mastic and mixture testing, a single test mode
utilizing hollow cylinders was employed. The hollow cylinder
tensile 共HCT兲 test specimens had an outer diameter of 150 mm,
an inner diameter of 100 mm, and height of 115 mm, which were
used in the study by You 共2003兲 and You and Buttlar 共2005兲. The
creep stiffness of the sand mastic and mixture were tested up to
100 s of loading time and at three test temperatures 共0, −10, and
−20° C兲. The creep stiffness at different loading times and temperatures were obtained from the inverse of creep compliance. In
166 / JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007
Fig. 3. Shifted creep stiffness and fitted master curve for sand mastic
the creep compliance test, a constant internal pressure was applied to the inner wall of the HCT specimen for a 100 s time
period. The constant pressure was selected to make sure that the
maximum induced tensile strain value was 500 ␮␧ for a mixture
specimen and 1,500 ␮␧ for a sand mastic specimen, and was kept
under 25% of the estimated breaking pressure. After calculating
the stress and strain by measuring the applied pressure and induced volume change or strain directly by a strain gauge, creep
compliance was calculated as Eq. 共12兲, which was discussed in
the work by Buttlar et al. 共1999, 2004兲 and Al-Khateeb 共2001兲
␧t
␧t
D共t兲 =
=
␴t + ␯P 共F␴tc + ␯兲P
fitting method, respectively. The mastic master curve of creep
stiffness versus reduced time is obtained and plotted in logarithm
coordinates in Fig. 3. The evaluated Maxwell model with one
spring and four Maxwell elements for mastic materials are shown
in Fig. 4. The model elastic and transient moduli are given with
evaluated parameters as below
E⬁ = 59.7 MPa
Et = E1e−共t−␶兲/␳1 + E2e−共t−␶兲/␳2 + E3e−共t−␶兲/␳3 + E4e−共t−␶兲/␳4
= 5,710.6e−共t−␶兲/26.2 + 2,075.1e−共t−␶兲/311.9
共12兲
where ␧t and ␴t⫽simultaneous creep strain and stress;
␯⫽Poisson’s ratio; F␴tc⫽stress correction factor; and P⫽applied
constant internal pressure. The creep stiffness E共t兲 was calculated
as Eq. 共13兲.
E共t兲 = 1/D共t兲
Fig. 4. Evaluated viscoelastic Maxwell model with one spring and
four Maxwell elements for sand mastic
共13兲
Each specimen had two strain gauges and therefore each specimen had two creep strain readings. For each temperature, there
were three replicates for the mixture and two for the sand mastic.
It is proved that the low temperature creep stiffness measurements
from HCT are similar to those measured from indirect tension test
共IDT兲 according to the work done by Buttlar et al. 共1999, 2004兲
and Al-Khateeb 共2001兲. Creep stiffness results to compare the
uniaxial compression test and HCT test are not available at this
time. But You 共2003兲 conducted laboratory tests and found that
the measured value of mixture complex modulus is similar for the
uniaxial compression and HCT tests at small strain conditions
共same temperature and loading frequency兲. Similarly, mastic stiffness measurements from compression and HCT tests are also very
close. In addition, creep stiffness, or the reverse of creep compliance, is a fundamental material property. Therefore, the viscoelastic properties of mastic from different test methods are comparable. A regression fitting method was employed to evaluate
mastic viscoelastic properties with a generalized Maxwell model
and generate master stiffness curves for mastic and asphalt mixture from creep tests. The shift factors’ computation and master
curve generation procedures can be found by a number of documents such as Buttlar 共1995兲 and Buttlar and Roque 共1994兲.
The shifted creep stiffness versus reduced time for asphalt
mastic at the reference temperature −20° C is shown in Fig. 3.
The time–temperature shift factors ␣T共T兲 were calculated as 0.1
and 0.008 for the temperatures −10 and 0 ° C from the regression
+ 1,449.0e−共t−␶兲/1,678.8 + 734.9e−共t−␶兲/19,952.6
共14兲
Similarly, the shifted creep stiffness versus reduced time at
three temperatures and master stiffness curve 共reference temperature −20° C兲 for mixture are shown in Fig. 5. The time–
temperature shift factors ␣T共T兲 were calculated as 0.159 and
0.002 at the temperatures −10 and 0 ° C, respectively.
Finite-Element Prediction of Mixture Creep Stiffness
Micromechanical viscoelastic FE simulation was conducted on a
computer-generated image sample to predict the mixture creep
stiffness. The purpose of this study is to predict the mixture creep
Fig. 5. Shifted creep stiffness and fitted master curve for asphalt
mixture
JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007 / 167
Fig. 6. Uniaxial compression image model generation for finite-element simulation: 共a兲 smooth surface of asphalt specimen; 共b兲 elliptical fitted
aggregates; and 共c兲 image sample with aggregate and mastic
stiffness with the viscoelastic simulation on a computer-generated
image model. This numerical prediction with evaluated mastic
materials properties would be compared with creep test results
from laboratory asphalt specimens. Simulation samples were generated to capture the real microstructure of the asphalt mixture by
using image analysis procedures from the smoothly sawn surface
of laboratory compacted asphalt mixture specimens. These imaging techniques and computation model generation were described
in other work done by Dai 共2004兲 and Dai et al. 共2005兲.
Fig. 6 shows an image sample generation from a sectioned
surface of a HMA specimen. The specimen is a 19 mm 共nominal
maximum aggregate size兲 mixture produced in the work by You
共2003兲 and You and Buttlar 共2005兲. Its image was obtained from
a smoothly sawn surface by a high-resolution scanner. It was
preliminarily processed with imaging technology including sieving very fine aggregates. Fig. 6共a兲 shows a binary image of
the sectioned specimen, where this image specimen contains aggregates passed sieve 2.36 mm and its asphalt content is about
14%. In this specimen, the mastic portion includes aggregate finer
than the 2.36 mm sieve and asphalt binder. For the image processing, the boundary pixels of each sieved aggregate were extracted,
and the coordinates of these pixels were stored in an array for the
ellipse fitting. A least-squares, ellipse-fitting algorithm was incorporated to determine the “best” optimized ellipse to represent
each irregular aggregate geometry. The fitted ellipse was generated for each sieved aggregate with a developed MATLAB code
as shown in Fig. 6共b兲. Based on these fitted ellipse parameters
共center coordinates, size, and orientation兲, a computation model is
generated with maximally filled cementation between neighboring particles using MATLAB as shown in Fig. 6共c兲. For this compressive computational model, the porosity was approximately
2% and the aggregate percentage was about 62%. The air voids
are considered in the FE model, while the porosity is estimated
from the complex geometry of aggregates and asphalt mastic. A
displacement controlled compressive test 共virtual test兲 was conducted to capture the specimen response and compute the creep
stiffness of the mixture.
The FE creep simulation was conducted on the image model
as shown in Fig. 6共c兲. For the compression simulation, the x and
y displacements of the particles on the bottom layer and the x
displacements of the particles on the top layer were constrained.
The constant force loading was evenly divided and imposed on
particles of the top layer. The evaluated Maxwell model param-
eters for mastic were input for the model viscoelastic simulation.
In the simulation, axial strain was calculated by dividing the average vertical displacement of top particles with the initial height
of the undeformed specimen, and axial stress was obtained by
dividing the constant loading force on the top layer with the
specimen initial cross-section area.
The FE model simulation results for creep stiffness of the asphalt mixture were compared with the mixture stiffness master
curve over a reduced time period 共104 s兲 as shown in Fig. 7. The
comparison curves were plotted in linear and logarithm coordinates as shown in the figure. The average relative error between
model prediction and test data was approximately 11.7%. It is
observed from the figure that the mixture creep stiffness is overpredicted in the beginning of the loading time. This occurs due to
the 2D microstructural model assumption of rigid aggregates with
infinity stiffness. The other error is likely caused by the approximation of fitted elliptical aggregates. For the longer reduced time,
these mixture creep stiffness predictions are very close to the
master curve. Therefore, this proposed micromechanical FE
Fig. 7. Comparison of mixture creep stiffness from model prediction
and master curve 共measurements兲 versus reduced time
168 / JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007
Fig. 8. Mastic and aggregate elements in the MDEM model: 共a兲 Aggregate and mastic discrete elements 共with the enlargement of a piece of
aggregate A兲; 共b兲 the discrete element of the mastic 共with an enlargement of a portion of the mastic兲
model with a user material subroutine is capable of predicting the
global viscoelastic behavior of asphalt mixture.
Microstructural Discrete-Element Modeling
of Asphalt Mixture
The 2D microstructure of the asphalt mixture represented in the
DE model was obtained from the following image processing and
modeling procedures. First, the asphalt test specimens were
smoothly sawn, and the grayscale images of sectioned surfaces
were obtained with a high-resolution scanner. Second, by using
image processing techniques, the outlines of aggregates with a
certain size identification were converted into many sided
polygons. Third, each polygon 共aggregate particles兲 and its
associated mastic subdomain were modeled with a number of
uniformly sized disks or balls. Fourth, the disk or ball will be
assigned with required intrinsic and interface properties, and
then the DE model was used to conduct a certain amount of
simulation.
As discussed by You and Buttlar 共2004兲, the binder and remainder of the fine aggregate and fines were modeled as a homogeneous sand–asphalt mastic 共all aggregate finer than the
2.36 mm sieve兲. This dividing line between “coarse” aggregate
共modeled as aggregate兲 and “fine” aggregates placed in the homogeneous sand–asphalt mastic is an arbitrary decision, guided by
JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007 / 169
image resolution and available computing power as well as the
laboratory tests for mastic. The microstructure of asphalt mixture
was generated from the 2D image of a cross section of mixture
specimens. Fig. 8 shows a typical DE model for the aggregate and
mastic in a piece of asphalt mixture. Each aggregate particle is
modeled with a number of discrete elements. The mastic around
an aggregate particle is also modeled with many discrete elements. In the model shown in Fig. 8, the aggregates and mastic
were represented by 60,000 discrete elements, which were assigned a certain amount of microstiffness and microstrength by
examining the stiffness and strength 共cohesive and shear兲 of aggregates and asphalt mastic and the adhesion characteristics of the
aggregate-mastic. The normal and shear cohesive strength between two contacting elements was simulated using a simple contact bond model, which is applied at the contact point. When the
tensile and/or shear stress at a contact exceeds the strength of the
bond, the bond breaks and separation and/or frictional sliding can
occur. However, because the current analysis was geared toward
studying creep stiffness estimation at low strain levels, bond
breakage was suppressed by using very high bond strength value.
The developed DE models, considering the cohesion in aggregate,
cohesion in mastic, adhesion in aggregate–mastic interface and
aggregate–aggregate interaction, are superior to other available
micromechanical models. Numerical simulations such as the
uniaxial compression test and the HCT test were conducted to
obtain the creep stiffness of the mixture. In this study, a number
of images of slice of different mixture specimens were modeled
so that the prediction has less effect due to the possible microstructure variances. It is found that the effect is not significant
when a large number of images were modeled and simulated 共You
2003兲.
Asphalt mastic creep stiffnesses at different loading times and
test temperatures were used as input parameters in the DE models. Aggregate stiffness 共a typical value of 55.5 GPa for limestone兲 was used in the DE models. The specimen 共virtual mixture
or mixture DE model兲 responses including uniaxial stress and
strain were recorded in order to compute the mixture creep stiffness during compressive loading 共Itasca Consulting Group 2002兲.
Therefore, this DE approach was used to predict asphalt mixture
creep stiffness at different loading times and temperatures. It
should be noted that this study is to compare the mixture creep
stiffnesses from laboratory measurements and DE prediction at
several preselected temperatures and loading time. In addition,
the mastic creep stiffnesses measured from laboratory material
testing are the inputs for the DE simulation. The creep stiffness
measurements of viscoelastic mastic material vary with loading
times and temperatures. Therefore the mixture creep stiffness prediction using the mastic properties will also include the time–
temperature dependent viscoelastic properties. Thus the elastic–
viscoelastic correspondence principle will bridge the elastic
simulation and mixture viscoelastic response through the input of
viscoelastic mastic creep stiffness. The veracity of this approach
was indirectly validated by the favorable match between simulation results and predictions, as discussed by You and Buttlar
共2004, 2005兲.
The mixture creep stiffnesses at different loading times and
test temperatures were predicted 共or virtually measured兲 using
DEM simulation. For a typical nominal maximum aggregate size
of 19 mm mixture 共with coarse aggregate volume concentration
ratio of 0.55 computed from the DE models兲, it is observed that
the mixture creep stiffness is underpredicted at higher test temperatures and longer loading time. As shown in Fig. 9, the mixture creep stiffness is very close to the measurements when the
Fig. 9. DEM prediction 共calibrated and uncalibrated兲 and
measurements of mixture creep stiffness versus reduced time
reduced loading time is shorter, and the temperature is lower. A
calibration method was applied to reduce the possibility of undercounting the aggregate particles in the mixture model, since the
2D modeling approach may count insufficient aggregate–
aggregate contact or interlock. The calibration method was adding
some fine aggregate particles 共between 0.6 and 2.36 mm兲. Therefore, more fine aggregate particles were partial of the aggregate
skeleton, and thereby the mixture’s coarse aggregate volume concentration ratio increased to 0.62 in the DE. As shown in Fig. 9,
the mixture creep stiffness with a coarse aggregate volume concentration ratio of 0.62 increases along the reduced loading time.
In general, the mixture creep stiffness prediction at a lower reduced time is closer to the measurements. The difference may
come from the complex microstructure of the mixture or the possible limitation of the mastic measurements. As mentioned previously, the 2D mixture microstructure model may underpredict the
aggregate-aggregate contact or aggregate interlock effects. In the
followup study, the measurements of the mastic, aggregates, as
well as the mixture creep stiffness, will be further studied.
It should be noted that the air voids in HMA are ignored in this
model due to the limitation of the 2D image capture technique. A
detailed study of the DE model considering the air voids is underway at Michigan Technological University.
Summary of the Comparison of Finite-Element
and Discrete-Element Model Predictions
Fig. 10 show the comparison of the FE and DE predictions with
the test data across a reduced loading time 共up to 104 s兲. As
discussed in the previous section, the measurements are from
three test temperatures. As shown in Fig. 10, the FE prediction is
higher than DE. The main reason is that aggregates in the FE
model are assumed to be rigid while a typical value of aggregate
stiffness was used in the DE model. The figure shows that the
stiffness prediction from FEM intends to overpredict mixture
stiffness 共especially at lower reduced time兲. This could be caused
by the same reason: the FE model assumption of rigid aggregates
with infinite stiffness. Therefore, the overall mixture stiffness is
higher compared with the measurements. It is also observed that
the mixture creep stiffness with calibrated DEM is underpredicted
at higher test temperature and longer loading time. The difference
may be caused by a 2D microstructure modeling of mixture,
which underestimates the effects of aggregate–aggregate contacts
170 / JOURNAL OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2007
In general, both models give reasonable prediction of the creep
stiffness across the reduced loading time 共at three temperatures兲.
Based upon the predictions from FE and DE models, it is found
that the FE prediction is higher than DE 共calibrated DE prediction兲, because the aggregate particles in the FE model are assumed to be a rigid object, while a typical value of aggregate
stiffness was used in the DE model. The measurements of the
creep stiffness are between the FE and DE prediction bounds. It is
possible to calibrate both the DE and FE models further so that
the prediction represents the mixture properties better. As future
modeling efforts are extended to three dimensions, the prediction
will be improved with larger amounts of measurements of aggregate and mastic.
Fig. 10. Prediction 关finite-element method 共FEM兲 and discrete
element method 共DEM兲兴 and measurements of mixture creep stiffness
versus reduced time
or aggregate interlock. The measurements of the creep stiffness
are between the FE and DE prediction bounds. Although some
limitations within 2D micromechanical FE and DE models, the
model predictions in general, are reasonable by comparing the
measurements from three test temperatures.
Conclusions
The micromechanical finite-element 共FE兲 and discrete-element
共DE兲 models have been used to predict of viscoelastic creep
stiffness of asphalt mixture. For micromechanical FE and DE
model simulation, the 2D microstructure of asphalt mixture was
obtained by optically scanning the smoothly sawn surface of asphalt specimens. For the FE model, the microstructural model
incorporates an equivalent lattice network structure whereby intergranular load transfer is simulated through an effective asphalt
mastic zone. The ABAQUS FE model combines the user-defined
material subroutine with continuum elements for the effective asphalt mastic and rigid body defined with rigid elements for each
aggregate. An incremental FE algorithm was employed in an
ABAQUS user material subroutine for the mastic to predict global viscoelastic behavior of asphalt mixture. For the DE model,
the outlines of aggregates were converted into polygons based on
2D scanned mixture microstructure. The polygons were mapped
onto a sheet of uniformly sized disks, and the intrinsic and interface properties of the aggregates and mastic were then assigned
for the simulation.
An experimental program was developed to measure the creep
stiffness of the sand mastic for both FE and DE simulation. For
the FE models, a regression fitting method was employed to
evaluate mastic viscoelastic properties with a generalized Maxwell model and generate master stiffness curves for mastic and
asphalt mixture from creep tests. The evaluated mastic viscoelastic properties were input for FE simulation. The FE simulation
was conducted on a computer-generated sample obtained by applying image processing and aggregate-fitting technologies on a
2D scanned microstructure. For the DE models, the mixture viscoelastic responsess were achieved through the input of corresponding mastic creep stiffness at different loading times and
temperatures, where the correspondence principle is applicable.
The laboratory measurements of the creep stiffness of the asphalt mixture were used to compare the predictions of the models.
Acknowledgments
The writers acknowledge the discussions with Dr. William G.
Buttlar at the University of Illinois at Urbana–Champaign. The
writers also acknowledge the efforts and help of Dr. Buttlar and
Mr. Minkyum Kim in the preparation of the lab test at the University of Illinois at Urbana–Champaign.
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