A
PPENDIX
A
R
EVIEW OF
L
ITERATURE FOR
C
ONSTITUTIVE
M
ODELS IN
A
SPHALT
M
IXES
BY
:
T
HOMAS
W
EAVER
1
Numerical Modeling of Asphalt Concrete
Numerical models are being used increasingly in research
and practice
to predict the performance of
asphalt concrete pavement performance.
Numerical models able to account for many parameters that
impact asphalt concrete pavement behavior
are: discrete element models and constitutive models.
The discrete element
methodol
ogy models each individual particle within a material and uses contact
laws to govern the interaction of the particles. This procedure was initially developed by Cundall (1971).
Application of
the
discrete element method may lead to better understanding
of the physics governing
asphalt
concrete
behavior and
is
a valuable research tool. However, use of this method
is not
likely in
the near future by engineering practitioners to model asphalt concrete pavement performance.
Constitutive modeling is anothe
r method useful for modeling asphalt concrete behavior and models
material response on a more global level. This methodology is more developed than the discrete
element methodology and with more research, may be useful for engineering practitioners to ass
ess
long term asphalt concrete pavement performance.
The following
section
present
s
background information on constitutive models, in particular, models
that have been implemented in finite element programs
(Note, more work is required to complete this
lit
erature review)
.
1.1
Constitutive Models
Constitutive models are relationships correlating stress with strain. The simplest constitutive model is
Hooke’s law which states:
where
is stress, E is the modulus of elasticity, and
is strain.
More compl
ex models are required to
capture non

linear response of materials due to yielding, fracture, and other variables impacting
material behavior.
Two constitutive models that have been implemented in finite element programs are
the disturbed state concept (D
SC) constitutive model and an elastoviscoplastic model. These models are
described below.
1.1.1
Disturbed State Concept
The disturbed State concept (DSC) constitutive model was developed by Desai (2
001) and has been used
to model asphalt concrete pavement (Desa
i 2007).
This model is capable of including the effect of
elastic
ity
, plastic
ity
, creep, microcracking, and fracture under mechanical and environmental loading.
The DSC considers a material to exist in
two states,
a relatively intact state
(RI)
or a
microcracked
state
/fully adjusted (FA) state
. As the material is loaded, the material transforms from
RI
to
FA
. The DSC
relates these two states and the transformation from
RI
to the
FA
state through a disturbance value.
The disturbance value ranges fro
m 0 to 1 corresponding to relatively intact to significantly damaged
material, respectively.
The disturbance value controls the softening that occurs in a material after
yielding has occurred.
The relatively intact state is modeled
as an elastic

plastic
material
using a yield function
and elastic
material properties
.
The parameters used to define material response can be defined as function
s
of
temperature and loading rate.
Creep can also be included.
The fully adjusted state may be defined as a critic
al state where no volume changes occur during
yielding under a constant shear.
The disturbance value is obtained using the equation:
(
)
Where D is the disturbance,
D
is the
acculuative
deviatoric
platic
strain
, and the values D
u
, A, and
Z are
disturbance parameters.
1.1.1.1
Determination of DSC Paramters
The DSC model requires elastic parameters, plasticity parameters, creep parameters, disturbance
parameters, and thermal effects.
Not all parameters are needed if a certain aspect of the model is
neglected. The model parameters and methods for obtaining these parameters as provided by Desai
(2007) and are summarized below.
The elastic parameters are Young’s modulus, E, and Poisson’s ratio,
. The modulus value can be
obtained from a stress

strai
n curve
and is a function of mean pressure.
The hierarchical single yield surface developed by (Desai
et al.
1986) is used for the plasticity model.
Parameters needed to define this yield surface are
,
, n, a
1
,
1
, and
R
. The value of
is determined
from the plot of the ultimate envelope based on the asymptotic stress in the ultimate region
. The value
n is determine from the stress condition at which transition from compaction to dilation occurs
.
Hardening parameters a1 and h1 are det
ermined by computing the accumulated plastic strain from the
incremental stress

strain curve, and the cohesive intercept, c
, is used to find the value R.
Other parameters associated with creep, disturbance, and thermal effects are presented below.
Creep
p
arameters may be obtained from a plot of strain versus time. The disturbance parameters, A and Z, are
found from a stress vs. strain curve where softening occurs. The value D
u
is the residual response and
can be assumed to be 0.9.
Thermal effects can be
determined by plots of the parameter of interest with
temperature.
The number of parameters
and
effort
required
to obtain these parameters
will likely be cumbersome to
most engineers.
1.1.2
Elastoviscoplastic Model
A number of researchers have developed
elastoviscoplastic constitutive models for application to
asphalt concrete (Abdulshafi and Majidsadeh 1985, Scarpa
s
et al. 1997, Lu and Wright 1998, Seibi et al.
2001, Collop et al. 2003, and Masad et al. 2007). Differences in these models include the yie
ld surface
employed
(e.g. Drucker

Prager, HISS, Mohr

Coulomb)
and whether the model accounts for
microstructure behavior that influences anisotropy and damage. In general, these models decompose
the total strain rate into a viscoelastic strain rate and a
viscoplastic strain rate.
The constitutive model
proposed by Masad et al. (2007) is discussed in more detail below since this model accounts or m
aterial
anisotropy associated with the aggregate and damage associated with material cracking.
1.1.2.1
Anisotropy and
Damage
Material anisotropy is a result of the anisotropy associated with granular material in the asphalt
concrete mix. Masad et al. (2007) have proposed using image analysis of an asphalt concrete cross
section to determine the orientation of aggregates
within the material. Based on the orientation of the
aggregates, a vector magnitude
is computed. The vector magnitude D is used to modify the stresses
in the material computed using a viscoplastic model.
The concept
of modifying the effective stress in the material as a result of crack growth as proposed by
Kachanov (1958) has been implemented within the model proposed by Masad et al. (2007). When
computing the stress as implemented in the viscoplastic model, the eff
ective stress is multiplied by a
factor that increases the effective stress
in the material
. In this model, the variable controlling damage
ranges from 0 for undamaged to 1 for a completely damaged material.
1.1.2.2
Material Parameters
The material parameters req
uire
d
for the constitutive model proposed by Masad et al. (2007)
are listed
in Table 1 below.
Similar to the DSC model, many parameters are needed to define the material
behavior with an elastoviscoplastic model that can account for anisotropy and damage.
However, based
on results presented by Masad et al. (2007), some parameter
s
do not change significantly from one
material to another. For example, the anisotropy vector magnitude for granite and
general
grave
l
aggregate
s
were 28.6 and 26.1, respectively
. As long as typical aggregates are consistently utilized,
image analysis tests may be eliminated from the required testing sequence for determining
this
parameter
.
Correlations with other simplistic tests may also be possible for estimating appropriate
material parameters for this model.
Table
1
Material Parameters for Elastoviscoplastic Model (Masad 2007)
Parameter
Test
for Determining
Parameter
Anisotropy,
Image analysis
Viscoelastic stiffness, E
1
Uniaxial compression
Viscoelastic stiffness, E
2
Uniaxial compression
Poisson’s ratio,
Triaxial shear
Drucker

Prager friction angle,
Triaxial shear
Drucker

Prager cohesion,
0
Triaxial shear
Perzyna’s viscoplastic parameter,
Triaxial shear
Perzyna’s viscoplastic
parameter, N
Triaxial shear
Dilation Parameter,
Damage Parameter,
1
Triaxial shear
Damage Parameter,
1
Triaxial shear
Damage Parameter,
1
Triaxial shear
Hardening Parameter,
1
Triaxial shear
Hardening Parameter,
2
Triaxial shear
1.2
Summary
Numerical models have been developed for modeling asphalt concrete behavior.
Two types of
constitutive models were described above: the disturbed state concept and the elastoviscoplastic
models.
The challenge associated with using
constitutive models is
the number of parameters that must
be defined for using the model. Results of some laboratory testing by Masad et al. (2007) indicate that
it may be possible to reduce the amount of testing required to determine all of the material parameters
for the elas
toviscoplastic model. If the amount of laboratory testing can be reduced or more simple
tests can be substituted for some of the triaxial shear tests, this model may become useful for engineers
in assessing long

term asphalt concrete pavement performance.
2
References
Abdulshafi, A., and Majidzadeh, K. (1985). “Combo viscoelastic

plastic modeling and rutting of asphaltic
mixtures.”
Transportation Research Record 968
, Transportation Research Board, Washington, D.C.,
19

31.
Collop, C., Scarpas, A.T.,
Kasbergen, C., and de Bondt, A. (2003). “Development and finite element
implementation of a stress dependent elasto

visco

plastic constitutive model with damage for
asphalt.”
Transportation Research Record 1832
, Transportation Research Board, Washington, D
.C.,
96

104.
Desai, C.S., (2001).
Mechanics of materials and interfaces: The disturbed state concept
, CRC, Boca Raton,
FL.
Desai, C.S. (2007). “Unified DSC constitutive model for pavement materials with numerical
implementation,”
International Journal of G
eomechanics
, ASCE, Vol. 7, No. 2, 83
–
101.
Desai, C.S., Somasundaram, S., and Frantiziskonis, G. (1986). “A hierarchical approach for constitutive
modeling of geologic materials.”
Internation Journal for Numerical and Analytical Methods in
Geomechanics
, 1
0(3), 225

257.
Lu, Y., and Wright, P.J. (1988). “Numerical approach of viscoelastoplastic analysis for asphalt mixtures.”
Comput
ers &
Struct
ures
, 69, 139

157.
Masad, E., Dessouky, S., Little, D. (2007). “Development of an elastoviscoplastic microstructural

based
continuum model to predict permanent deformation in hot mix asphalt,”
International Journal of
Geomechanics
, ASCE, Vol. 7, No. 2, 119
–
130.
Scarpas, A., Al

Khoury, R., Van Gurp, C., and Erkens, S.M. (1997). “Finite element simulation of damage
deve
lopment in asphalt concrete pavements.”
Proc., 8
th
Int. Conf. on Asphalt Pavements
, Univ. of
Washington, Seattle, 673

692.
Seibi, A.C., Sharma, M.G., Ali, G.A., and Kenis, W.J. (2001). “Constitutive relations for asphalt concrete
under high rates of loadin
g.”
Transportation Research Record 1767
, Transportation Research Board,
Washington, D.C., 111

119.
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