Materials and Structures
DOI 10.1617/s11527-007-9303-4
ORIGINAL ARTICLE
Micromechanical finite element framework for predicting
viscoelastic properties of asphalt mixtures
Qingli Dai Æ Zhanping You
Received: 4 December 2006 / Accepted: 27 August 2007
RILEM 2007
Abstract A micromechanical finite element (FE)
framework was developed to predict the viscoelastic
properties (complex modulus and creep stiffness) of
the asphalt mixtures. The two-dimensional (2D)
microstructure of an asphalt mixture was obtained
from the scanned image. In the mixture microstructure, irregular aggregates and sand mastic were
divided into different subdomains. The FE mesh
was generated within each aggregate and mastic
subdomain. The aggregate and mastic elements share
nodes on the aggregate boundaries for deformation
connectivity. Then the viscoelastic mastic with
specified properties was incorporated with elastic
aggregates to predict the viscoelastic properties of
asphalt mixtures. The viscoelastic sand mastic and
elastic aggregate properties were inputted into micromechanical FE models. The FE simulation was
conducted on a computational sample to predict
complex (dynamic) modulus and creep stiffness. The
complex modulus predictions have good correlations
Q. Dai
Department of Mechanical Engineering-Engineering
Mechanics, Michigan Technological University,
Houghton, MI 49931, USA
e-mail: [email protected]
Z. You (&)
Department of Civil and Environmental Engineering,
Michigan Technological University, Houghton, MI
49931, USA
e-mail: [email protected]
with laboratory uniaxial compression test under a
range of loading frequencies. The creep stiffness
prediction over a period of reduced time yields
favorable comparison with specimen test data. These
comparison results indicate that this micromechanical
model is capable of predicting the viscoelastic
mixture behavior based on ingredient properties.
Keywords Microstructure Micromechanical modeling Finite element method Asphalt mixture Viscoelasticity Complex modulus Creep stiffness
1 Introduction
Heterogeneous asphalt mixtures comprise of graded
aggregates bound with mastic (asphalt binder plus
fine aggregates and fines). For such materials, the
macro properties depend on the aggregate and mastic
microstructure. Important micro behaviors related to
mastic properties include: volume percentage, viscoelastic and viscoplastic responses, microcracking, and
bonding strength. The microstructural features of
aggregate include: mineralogy, elastic modulous,
size, shape, texture, and packing geometry. Chen
et al. [1] evaluated the internal structures of asphalt
mixtures constituting of four different percentages of
flat and elongated (F&E) aggregates, and studied how
the engineering properties of these mixtures change
in terms of rut depth, particle movement and
Materials and Structures
orientation, and strain. The experimental procedure
used in their study relied on the two-dimensional
(2D) images subjected to wheel loading. It was
concluded that the low percentages of F&E aggregates result in a stable internal structure that could
develop stone-on-stone contact and provide a better
interlocking mechanism [1].
Because of the heterogeneous nature of asphalt
mixtures, a micromechanical model for asphalt
concrete mixtures is needed to study and characterize their properties. Micromechanical models can
predict fundamental material properties based upon
the properties of the individual constituents such as
the mastic and aggregate. Micromechanical models
have tremendous potential benefits in the field of
asphalt technology, for reducing or eliminating
costly tests to characterize asphalt-aggregate mixtures and design purposes. Since these models allow
a more thorough examination of microstructural
material behavior, such as strain distribution within
the aggregate skeleton and asphalt matrix, they can
ultimately provide a powerful tool for optimizing
mixture design on the basis of mechanistic
performance.
The use of micromechanical models to predict
properties of asphalt mixtures and mastics has drawn
increasing attention over the past 10 years, and a
number of approaches have been investigated.
Asphalt mixture was investigated by non-interaction
particle micromechanics models without specified
geometry [2–6], as well as, with specified geometry
[7–10].
Discrete element method (DEM) was employed
on cemented particulate materials in recent years
[11–18]. A 2D micro-fabric discrete element model
(MDEM) concept was developed to predict the
stiffness of asphalt mixtures [19–21]. The MDEM
holds promise to balance the advantages of microstructural model resolution with all the benefits of
the discrete element approach including the simulation of aggregate cracking, debonding between
aggregates and mastic, and microcracking initiation
and propagation. However complex contact laws
need to be developed for predicting complex mastic
constitutive behavior. In addition, it is a challenge to
use circular element to simulate the irregular shaped
aggregate.
On the other hand, finite element modeling of
asphalt concrete microstructure potentially allows
accurately modeling of aggregate and mastic complex constitutive behaviors and microstructure
geometries. Research work has been conducted using
FE techniques [22–31]. In addition, some research
have been reported on the three-dimensional (3D)
microstructure of asphalt mixture [29, 32–34]. A
displacement discontinuity boundary element
approach was applied in the modeling of asphalt
mixtures [35]. An equivalent lattice network
approach was developed and applied, where the local
interaction between neighboring particles was modeled with a special frame-type FE [36–39]. A mixed
FE approach was developed to study asphalt mixtures
by using continuum elements for the effective asphalt
mastic and rigid body defined with rigid elements for
each aggregate [40]. A unified approach for the rateindependent and rate-dependent damage behavior
was developed using Schapery’s nonlinear viscoelastic model. Properties of the continuum elements were
specified through a user material subroutine within
the ABAQUS code and this allows linear and
damage-coupled viscoelastic constitutive behavior
of the mastic cement to be incorporated. In these
models, the microstructure of real asphalt materials
was simulated with idealized elliptical aggregates and
polygonal effective mastic zones [41]. Using image
processing and ellipse fitting methods, particle
dimensions and locations were determined from
digital photographs of the sample’s microstructure.
These models have been promising in predicting the
mixture behavior, however, the aggregate shape is
idealized as ellipse. Even though many research
studies have been conducted on micromechanical
modeling of asphalt mixtures, the irregular shape of
the aggregate in the mixture still has not been
successfully modeled to capture the micromechanical
aggregate-to-aggregate contact behavior.
2 Objectives
The objectives of this study are to (1) develop a
microstructure-based FE model for heterogeneous
asphalt mixture, and; (2) predict viscoelastic mixture
properties (e.g., complex modulus and creep stiffness) by using the FE model with the input of the
viscoelastic properties of sand mastic and elastic
modulus of the aggregates.
Materials and Structures
3 Scope
The asphalt mixture in this study is modeled as
irregular shaped aggregates bonded with sand mastic.
The microstructure of the mixture was obtained from
the 2D scanned image of a sawn asphalt concrete.
Therefore, the image of the asphalt mixture includes
the coarse aggregate and sand mastic. The sand
mastic is a mixture of fine sand and asphalt binder.
Comparing with the coarse aggregate, the sand
particles are so small that they can distribute in the
asphalt binder uniformly. Therefore, it is reasonable
treat the sand mastic as a homogonous composite.
The FE mesh was generated within each aggregate
and mastic subdomains by sharing nodes on the
aggregate boundaries. Viscoelastic properties of the
mastic elements are calibrated with the laboratory
experimental test data and inputted into a user
material subroutine within the ABAQUS code. Then
the viscoelastic mastic was combined with elastic
aggregates to predict the properties of asphalt mixtures. It should be noted that the air void in the
mixture is ignored in this 2D model due to the
limitation of smooth sawn image. Ongoing simulation work is conducted on X-ray scanned images with
the indication of air void distribution.
4 Microstructure of asphalt mixture
In this study, the 2D microstructure of asphalt
concrete was obtained by optically scanning
Fig. 1 Microstructure of an
asphalt mixture specimen
surface. (a) An original
image of the surface, (b)
The aggregate skeleton
(sieving size [1.18 mm)
smoothly sawn asphalt mixture specimens. A highresolution scanner was used to obtain grayscale
images from the sections. Image processing technique was used to process and analyze images.
Figure 1 demonstrates the image processing on an
asphalt specimen (with the nominal maximum aggregate size of 19 mm). Figure 1a shows an optical
scanning image of a mixture specimen with 86 mm
width and 106 mm height. A scanner with
1600 · 3200 Dots Per Inch (DPI) optical resolution
was used. After improving the image contrast, the
outlines of aggregates were converted into manysided polygons using a custom developed macro
program in Image Pro Plus to define the microstructure of asphalt mixture [15]. The average of the
polygon diameter was chosen as a threshold to
determine which aggregates would be ‘‘retained’’ on
a given sieve (i.e., the rest of the aggregates would be
‘‘passed’’ the given sieve), although some other
measurement parameters were also attempted in
analyzing the gradation of the aggregates [42].
The polygons used in the micromechanical model
were treated as coarse aggregates. Figure 1b is the
coarse aggregate retained on 1.18 mm sieve (i.e., No.
16), where the fine aggregates passing 1.18 mm were
filtered as sand mastic. Elastic properties of aggregates [20, 21, 42] were assigned to aggregate
subdomains, and viscoelastic mastic properties were
also evaluated with mastic creep tests. Finite element
simulation was conducted to predict the mixture
behavior by combining aggregate and mastic
properties.
Materials and Structures
5 Mastic viscoelastic model
each increment was developed in the following
format:
Generalized Maxwell model was widely used for
viscoelastic solids such as asphalt mixture. The
generalized Maxwell model was applied to simulate
the linear and damage-coupled viscoelastic behavior
of asphalt mixture [40]. The linear constitutive
behavior for this Maxwell-type model can be
expressed as a hereditary integral
Z t
deij ðsÞ
rij ¼ E1 eij þ
ds
ð1Þ
Et
ds
0
where Et is expressed with a Prony series
Et ¼
M
X
ðtsÞ
Em e qm ;
and
m¼1
qm ¼
gm
Em
ð2Þ
In these equations, E1 is the relaxed modulus, Et is
the transient modulus as a function of the time, Em,
gm and qm are the spring constant, dashpot viscosity
and relaxation time respectively for the mth Maxwell
element.
The reduced time (effective time) is defined by
using the time-temperature superposition principle as
Z t
1
nðtÞ ¼
ds
ð3Þ
a
T
0
where the term aT ¼ aT ðTðsÞÞ is a temperaturedependent time-scale shift factor (Fig. 2).
Three-dimensional behavior can be formulated
with uncoupled volumetric and deviatoric stress–
strain relations. A displacement-based incremental
FE modeling scheme with constant strain rate over
Dr ¼ K De þ DrR
ð4Þ
Where Dr and De are incremental stress and strain, K
is the incremental stiffness and DrR is the residue
stress vector.
The volumetric constitutive relationship is
expressed with the volumetric stress rkk and strain
ekk in the general form
Z n
dekk ðn0 Þ 0
rkk ðnÞ ¼ 3K1 ekk ðnÞ þ
3Kt ðn n0 Þ
dn
dn0
0
ð5Þ
relaxed bulk
where K1 ¼ E1 =3ð1 P2mÞ is the
ðnn0 Þ
q
m
modulus, Kt ðn n0 Þ ¼ M
K
e
is the tranm¼1 m
sient bulk modulus, and Km = Em/3(1–2m) is the bulk
constants for the spring in the mth Maxwell element.
The incremental formulation of the volumetric
behavior is obtained with constant volumetric strain
kk
rate Rkk ¼ De
Dn ;
"
Drkk ¼ 3 K1 þ
N
X
Km q m
m¼1
Dn
qDn
1e
#
m
Dekk þ DrRkk
ð6Þ
and the residual part DrRkk can be expressed in a
recursive relation with the history variable Sm,
DrRkk ¼
M
X
Dn
1 eqm Sm ðnn Þ;
and
m¼1
Dn
Dn
Sm ðnn Þ ¼ 3Km Rkk qm 1 eqm þ Sm ðnn1 Þeqm
ð7Þ
E∞
E1
E2
E3
η1
η2
η3
E M −1
η M −1
EM
ηM
For the initial increment,
theDn history variable Sm(n1)
q
m
equals to 3Km Rkk qm 1 e
and is similar to the
following formulations.
For the deviatoric behavior, the constitutive relationship is written using deviatoric stress
^ij ¼ rij 13 rkk Dij and strain ^eij ¼ eij 13 ekk Dij ;
r
^ij ðnÞ ¼ 2G1^eij ðnÞ þ
r
Z
0
Fig. 2 The generalized Maxwell viscoelastic model for the
sand mastic
n
2Gt ðn n0 Þ
d^eijðn0 Þ dn0
dn0
ð8Þ
where G1 ¼ E1 =2ð1
relaxed shear modðnn0 Þ
Pþ mÞ is the
q
m
ulus, Gt ðn n0 Þ ¼ M
G
e
is the transient
m¼1 m
shear modulus, and Gm = Em/2(1 + m) is the shear
constants for the spring in the mth Maxwell element.
Materials and Structures
The formulation of the deviatoric behavior is
obtained with constant deviatoric strain rate
D^e
R^ij ¼ Dnij ;
"
#
N
X
Dn
Gm qm 1 eqm D^eij þ D^
rRij
D^
rij ¼ 2 G1 þ
Dn
m¼1
ð9Þ
and the residual part D^
rRij can be expressed in the
recursive relation
D^
rRij ¼
N
X
Dn
1 eqm Sm ðnn Þ;
calculated and then the incremental 3D linear viscoelastic behavior was formulated as
3 2
32
3
2
K1 K2 K2 0
Dexx
0
0
Drxx
7
6 Dr 7 6 K K
6
0
0
0 7
1
2
76 Deyy 7
6 yy 7 6
7 6
76
7
6
6 Drzz 7 6 K1 0
0
0 76 Dezz 7
7¼6
76
7
6
7
6 Dr 7 6 6
K3 0
0 7
76 Dexy 7
6 xy 7 6
7 6
76
7
6
4 Dryz 5 4 K3 0 54 Deyz 5
Drxz
and
Dn
Dn
Sm ðnn Þ ¼ 2Gm R^ij qm 1 eqm þ Sm ðnn1 Þeqm
þ6
6
6
6
4
ð10Þ
Drxx ¼ 1=3Drkk þ D^
rxx
"
#
N
X Km q qDn
m
1 e m Dekk
¼ K1 þ
Dn
m¼1
"
#
N
X
Gm qm qDn
1 e m D^exx
þ 2 G1 þ
Dn
m¼1
þ 1=3DrRkk þ D^
rRxx
ð11Þ
where Dekk and Drkk are the incremental volumetric
strain and stress, D^exx and D^
rxx are the incremental
deviatoric strain and stress components, and DrRkk and
D^
rRxx are the recursive part of the volumetric and
deviatoric behavior given in Eqs. 6 and 9. Incremental stresses Dryy and Drzz are determined in the same
manner.
The incremental shear stress can be formulated by
using only the deviatoric behavior. For example,
Drxy ¼ D^
rxy
"
¼ 2 G1 þ
M
X
Gm q m
m¼1
Dn
1e
qDn
m
R 3
þ D^
rxx
þ D^
rRyy 7
7
7
R 7
þ D^
rzz 7
7
7
D^
rRxy
7
7
R
D^
ryz
5
K3
DrRkk
6 DrR
6 kk
6
6 DrR
6 kk
m¼1
The incremental normal stresses can be then
formulated by combining the volumetric and deviatoric behavior. For example,
2
Dexz
ð13Þ
D^
rRzx
where
"
#
N
X
Dn
Km qm K1 ¼ K1 þ
1 e q m
Dn
m¼1
"
#
N
X
4
Gm qm qDn
þ G1 þ
1e m
3
Dn
m¼1
"
#
N
X
Km qm qDn
K2 ¼ K1 þ
1e m
Dn
m¼1
"
#
N
X
2
Gm qm qDn
G1 þ
1e m
3
Dn
m¼1
"
#
N
X
Gm qm qDn
K3 ¼ 2 G1 þ
1e m
Dn
m¼1
ð14Þ
This viscoelastic model was defined in the ABAQUS
user material subroutine for mastic subdomains. A
displacement-based time-dependent FE analysis was
conducted by integrating elastic aggregate and viscoelastic mastic subdomains to predict the global
behavior of asphalt mixture.
#
rRxy
D^exy þ D^
ð12Þ
rxy are the incremental shear
where D^exy and D^
deviatoric strain and stress components, and the
recursive term D^
rRxy is also given in Eq. 9.
Once the incremental stress components are
developed, the incremental stiffness terms can be
6 Laboratory tests of aggregates, sand mastic, and
compacted asphalt mixture
The purpose of this section is to measure and
evaluate the material properties of sand mastic,
aggregate (rock), and compacted asphalt mixture
through laboratory tests. The uniaxial compression
laboratory tests of sand mastic and aggregate
Materials and Structures
(cylinder specimen) were conducted to provide
material input parameters for the FE models. The
goal of the mixture test is to provide a comparison
with the model prediction in order to validate the FE
model simulation.
The input parameters for the FE models include
not only the microstructure information but also the
material properties of the aggregate and mastic at
different loading conditions. In this study, a modulus
of 55.5 GPa [20, 21, 42] for the limestone was used
for different temperatures and loading frequencies.
The uniaxial compression creep test was conducted on the sand mastic and mixture samples under
different temperatures (0C, –10C, and –20C). The
mastic contains aggregates passing sieve 1.18 mm
and asphalt content is about 14% [20, 21, 42]. The
sand mastic was comprised of the portion of the
aggregate gradation finer than the 1.18 mm sieve
combined with the volume of binder normally used in
the entire asphalt concrete mixture. The creep
stiffness at different loading time and temperatures
were obtained from the inverse of creep compliance.
A regression fitting method was employed to evaluate
mastic viscoelastic properties with a generalized
Maxwell model at the reference temperature of –
20C, and the time shift factors were calculated for
0C and –10C. Master stiffness curves were generated for mastic and asphalt mixture from creep tests
[43]. The shifted creep stiffness and fitted master
curve for sand mastic at the reference temperature –
20C are shown as Fig. 3. The model for sand mastic
includes one spring and four Maxwell elements in
parallel. The viscoelastic parameters of the sand
mastic are: E1 ¼ 59:7 MPa; E1 = 5710.6 MPa,
Mastic Creep Stiifness (GPa)
1E+02
0 C Test Data
-10 C Test Data
-20 C Test Data
Fitted Model
1E+01
1E+00
1E-01
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
Reduced Time (sec)
Fig. 3 The shifted creep stiffness and fitted master curve for
the sand mastic
E2 = 2075.1 MPa,
s2 = 311.9 s,
s1 = 26.2 s,
E3 = 1449 MPa, s3 = 1678.8 s, E4 = 734.9 MPa,
and s4 = 19952.6 s. The relaxed and transient moduli
are determined from the master curve of mastic creep
stiffness. Uniaxial behavior was extended to multiaxial (3D) constitutive formulation with the uncoupled volumetric and deviatoric Eqs. 5 and 8. The
E
elastic bulk and shear moduli are K ¼ 3ð12t
Þ and
E
G ¼ 2ð1þtÞ : The terms K1 and G1 are the relaxed
bulk and shear moduli, and Km and Gm are the bulk
and shear constants for the spring in the mth Maxwell
element. The Poisson’s ratio m is assumed as 0.3 for
asphalt mixture. Therefore, the multi-axial properties
were connected with the uniaxial measurements.
For the gyratory compacted mixture specimens,
uniaxial compression tests were conducted to measure complex (dynamic) modulus and creep stiffness
with a number of mixture specimens. The complex
moduli were measured at different temperatures and
loading frequencies. The creep stiffnesses were
calibrated over a period of loading time at different
temperatures. This database was used to validate FE
model for prediction of mixture complex moduli and
creep stiffness at different loading conditions.
7 Development of micromechanical finite element
models for asphalt mixtures
As mentioned previously, the microstructure of the
asphalt mixture was divided into different aggregate
and mastic subdomains. The FE mesh was generated
within the subdomains of aggregates and mastic and
along the subdomain boundaries. Due to the very
irregular aggregate and complex mastic distribution,
the three-node triangle elements were used in the FE
mesh for the complex geometry. Figure 4 shows the
FE meshes in the aggregate and mastic subdomains of
a specimen surface. Finite elements in the neighboring subdomains share the nodes on irregular
boundaries as shown in Fig. 4, and therefore the
displacements of neighboring subdomains were connected through the shared nodes. For this 2D FE
mesh, plane stress elements with a solid section
thickness were applied for both aggregate and mastic
subdomains.
After the FE model has been developed, uniaxial
compression test was simulated. For the compression
simulation, the x- and y-displacements of the nodes
Materials and Structures
Fig. 4 The asphalt mixture
image and the three-node
triangle FEM meshes for
the aggregate and mastic
subdomains. (a) A scanned
image of the specimen
surface, (b) The finite
element meshes of the
specimen surface, (c)
Enlarged meshes for the
aggregates and mastic
on the bottom layer and the x-displacements of the
nodes on the top layer were constrained. The constant
or dynamic force loading was evenly divided and
imposed on nodes of the top layer. The generalized
Maxwell model parameters for mastic and the elastic
modulus of aggregates were inputted to the FE
model. Simulation was conducted to predict the
global viscoelastic properties of the asphalt mixture.
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.
Fig. 5 The compression
strain contour in a portion
of the digital specimen
One of the benefits of the micromechanical model
is to present the detailed stress and strain distributions
within the microstructure of the mixture sample.
Figure 5 shows the compression strain distribution
contour in a portion of the digital specimen under a
uniaxial constant compression force loading. The
aggregate skeletons are indicated in the figure with
the skeleton curves. The high compression strains
were generated in the vertical mastic gap between
neighboring aggregates. The computational results
show that the highest local strain is about eight times
of average strain for this portion of the computational
sample. Figure 6 shows the shear strain distribution
contours in the same portion of the digital specimen
Materials and Structures
Fig. 6 The shear strain
contour in the same portion
of the digital specimen
under a constant compression force loading. This
figure indicates that high shear strain zones are
distributed in horizontal narrow mastic gap along the
large-size aggregates. From our previous study, it was
found that under the mechanical loading, the cracks
initiate in the narrow gap between coarse aggregates
of asphalt mixture specimen. These strain distributions agree with the laboratory observations. The
strain contours help examine micro material behavior, such as strain intensities within the asphalt mastic
and aggregate phases. They also can provide useful
information in analyzing crack initiation and optimizing mixture design on the basis of mechanistic
performance for further study.
8 Complex modulus simulation and results
Sinusoidal cyclic loading was imposed to the simulation specimen for calculating the complex modulus
under the different loading frequencies (10 Hz, 5 Hz,
1 Hz and 0.1 Hz) as shown in Fig. 7. In the
simulation, the loading cycles were taken as 30 for
0.1 Hz, 50 for 1 Hz, 5 Hz, and 10 Hz. In these
figures, the constant cyclic curve is the imposed stress
load with the right-side axial scale, and the other
curve indicates the strain response with the left-side
axial scale. For better illustration, the final several
cycles were magnified in the right-side figures for
each frequency. The computation points are indicated in the right-side figures. The magnitude of the
dynamic modulus was calculated using the last ten
cycles for each frequency. As indicated in these
figures, the strain increases with the loading time for
each frequency. Comparing different frequency
responses, it was found that the strain value
deceases with the loading frequencies. It also
indicates the complex modulus value increases with
the loading frequencies due to decreasing relaxation
time.
Figure 8 shows the complex modulus comparisons
between FE simulation results and test data for
different loading frequencies at a test temperature of
–20C. As mentioned previously, the mastic master
curve was obtained at the reference temperature –
20C. Therefore the FE simulation with the input
mastic properties generated the mixture behavior at
this selected temperature. The comparisons indicate
that the simulation results are reasonable and applicable for complex modulus prediction although it
slightly under predicted the mixture modulus. The
differences between the predictions and measurements may cause by the following reasons: (1) the
aggregates sieved from the 2D image may reduce the
real aggregate content/percentage, and (2) the 2D
mixture microstructure model may underestimate the
Materials and Structures
0.35
12
0.30
10
0.25
8
0.20
6
0.15
4
0.10
2
0.05
0
0.00
3.0
4.0
0.35
12
0.30
10
0.25
8
0.20
6
0.15
4
0.10
2
0.05
0.00
0
4.6
5.0
4.7
Strain
0.40
12
0.35
0.30
10
0.25
0.20
8
6
0.15
0.10
4
2
0.05
0.00
0
0
2
4
6
8
Com pression S tress (GP a)
Stress
14
Com pression S train
Com pression S tress (GP a)
Stress
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
10
8
6
4
2
0
9.0
10
8
0.3
6
0.2
4
0.1
2
0
0.0
Stress
40
Straint
0.5
0.4
10
8
0.3
6
0.2
4
0.1
2
0.0
0
47
48
Stress
0.7
10
0.6
0.5
8
0.4
6
0.3
4
0.2
2
0.1
0
0.0
150
10
12
Strain
12
100
9.8
49
50
Loading time (s)
0.8
50
Stress
46
14
0
9.6
14
50
200
250
300
Loading Time (s)
Com pression Stress (GPa)
C om pr e s s ion Str e s s (GPa )
20
30
Loading time (s)
Com pression Stress (GPa)
0.4
Com pression S train
12
(d)
9.4
Strain
0.5
10
9.2
Loading Time (s)
14
0
Strain
12
10
Com pression S train
Com pression Stress (Gpa)
Stress
5.0
14
Loading Time (s)
(c)
4.9
Loading Time (s)
Loading Time (s)
(b)
4.8
Com pression Strain
2.0
14
Com pression Strain
1.0
Strain
Strain
14
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
12
10
8
6
4
2
0
260
270
280
290
Com pression Strain
0.0
Stress
Com pression Strain
Strain
Com pression S tress (GP a)
Stress
14
Com pression Strain
Com pression Stress (GPa)
(a)
300
Loading Time (s)
Fig. 7 The FEM simulation results under sinusoidal loading. (a) loading frequency = 10 Hz, (b) loading frequency = 5 Hz,
(c) loading frequency = 1 Hz, (d) loading frequency = 0.1 Hz
aggregate-aggregate contact or aggregate interlock
effects. With the development of 3D modeling and
the additional calibration of mastic and aggregates,
the model prediction perhaps will be closer to the test
data with more capacity to describe mixture complex
behavior.
In order to compare the FE prediction, the authors
also conducted the discrete element modeling to
predict the mixture complex modulus with the input
of the measured mastic complex moduli at different
loading frequencies and test temperatures, and aggregate modulus [19–21, 43]. It was found that the
Materials and Structures
100.0
Relaxation Modulus (GPa)
Mixture Complex Modulus (GPa) .
100
10
100.0
1000.0
10000.0
Reduced Time (s)
1
1
10.0
1.0
10.0
Test Data (-20C)
FEM Simulation
0.1
FEM
DEM (c=0.62), Calibrated
DEM (c=0.55), Calibrated
Test Data
10
Loading Frequency (Hz)
Fig. 9 Prediction from the FEM and the calibrated DEM
simulation, and the laboratory measurements of the asphalt
mixture creep stiffness for a reduced time up to 104 s
Fig. 8 The complex modulus comparison with the FEM
simulation and the test data at a temperature –20C
complex modulus predictions from the FE models
had a good agreement with the DEM and the lab
measurements.
9 Creep stiffness simulation and results
In this section, the creep stiffness simulation by using
the micromechanical FE model was discussed. In
addition, the predictions from the micromechanical
FE model, a microstructure based discrete element
model (DEM) [21] and a microstructure based FE
network model (FENM) were compared.
Micromechanical FE simulation was conducted
with a constant force loading condition to predict the
creep stiffness of asphalt mixture. When the constant
force was applied to the microstructure-based FE
model, the creep displacement responses were captured over a period of time. Then the macro creep
strain and creep stiffness of the asphalt mixture were
computed. The creep stiffness (reverse of the creep
compliance) varies with time. FE model simulation
was compared with the measurements of the master
curve of creep stiffness of the mixture across a
reduced time up to 104 s as shown in Fig. 9. The FE
model had a good comparison with test measurements, and slightly underpredicted the creep stiffness
(perhaps due to the limitation of 2D model).
In this case study, the authors consider that the
major contributions to creep deformation are viscoelastic properties. It also exits slight unrecoverable
deformation caused by the viscoplastic behavior for
real asphalt mixture. In the ongoing work, the authors
study the viscoplastic-viscoelastic behavior by
replacing the elastic spring with a viscoplastic-elastic
element in the presented model.
In order to compare the creep stiffness prediction
of a microstructure based discrete element model
(DEM) [21], the discrete element model predictions
with the coarse aggregate volume concentration
ratios (i.e, c = 0.55 and c = 0.62) were compared
with the master curve of the creep stiffness of the
mixture as shown in Fig. 9. A calibration method was
applied to reduce the possibility of under-counting
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 extra fine aggregate
particles (between 0.6 mm and 1.18 mm). Therefore,
the extra fine aggregate particles were part of the
original aggregate skeleton, and therefore the mixture’s coarse aggregate volume concentration ratio
increased. When comparing with the master curve of
the creep stiffness from discrete element models, the
model prediction was improved with the increasing
aggregate volume concentration ratio.
Finite element network model (FENM) using
elliptical aggregates was developed to study asphalt
mixture behavior including creep stiffness [41, 44,
45]. In the FENM model, the microstructure of
asphalt materials is simulated with idealized elliptical
aggregate and polygonal effective mastic zone.
FENM integrates viscoelastic mastic elements with
rigid elliptical aggregates to predict global mixture
behavior. Figure 10 shows the comparison among
mixture creep stiffness predictions from the FE and
idealized FENM simulation, and the laboratory
measurements. It was observed from the figure that
the FENM slightly over-predicted mixture creep
Materials and Structures
Relaxation Modulus (GPa)
100.0
FEM
FENM
Test Data
10.0
1.0
10.0
100.0
1000.0
10000.0
Reduced Time (s)
Fig. 10 Comparison among mixture creep stiffness prediction
from the FEM, and the idealized FENM simulation, and the
laboratory measurements
stiffness especially in the beginning of the loading
time. This occurs due to the model assumption of
rigid aggregates with infinity stiffness and idealized
particle and mastic zone shapes. The measurements
of the creep stiffness are between the FE and FENM
prediction bounds. Although some limitations within
2D micromechanical FE and FENM predictions, the
FE model predictions in general, are reasonable by
comparing the measurements of mixture creep tests.
10 Conclusions
The microstructure-based FE model was developed and
applied to predict viscoelastic properties (complex
modulus and creep stiffness) of heterogeneous asphalt
mixture. The 2D microstructure of asphalt mixture was
obtained by optically scanning the smoothly sawn
surface of asphalt specimens. In the microstructure,
aggregates and sand mastic were divided into different
subdomains. Finite element mesh was generated within
each aggregate or sand mastic subdomain. Therefore
the very irregular aggregate geometry and mastic
domains are modeled using a number of FEs. Then
the viscoelastic mastic with specified properties in an
ABAQUS user subroutine was combined with elastic
aggregates to predict the global viscoelastic properties
of asphalt mixtures.
An experimental program was developed to measure the properties of the aggregates, sand mastic, and
asphalt mixture for FEM simulation and validation.
The mastic viscoelastic properties and aggregate
elastic modulus were inputted for FE simulation.
The laboratory measurements of the complex
modulus and creep stiffness of the asphalt mixture
were used to compare the model predictions.
In general, micromechanical FE models provided
reasonable predictions of the complex modulus over a
range of frequencies, and creep stiffness across a
period of reduced loading time. In order to show the
different model predictions, comparisons have been
conducted among the FE model, a DEM and a FE
network model (FENM). FENM slightly overpredicted creep stiffness especially in the beginning of
loading time due to the model assumption of rigid
aggregates and idealized aggregate and mastic zone
shapes. Based upon the predictions from FE and DE
models, it was found that these models slightly
underpredicted mixture modulus and creep stiffness,
because of the limitation of aggregate-to-aggregate
contact and interlock effects in the 2D models. As
future modeling efforts are extended to three-dimensions, the prediction will be improved with larger
amount of inter-particle contacts and more measurements of aggregates and mastic.
Acknowledgement The authors acknowledge Dr. William
Buttlar’s assistance in the laboratory tests at the University of
Illinois at Urbana-Champaign.
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