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Citation
Farrar, M. J., E. Y. Hajj, J. P. Planche, and M. Z. Alavi, 2013, A method to
estimate the thermal stress build-up in an asphalt mixture from a singlecooling event. Road Materials and Pavement Design, 14:sup1, 201-211.
As Published
http://dx.doi.org/10.1080/14680629.2013.774756
Publisher
Taylor & Francis
This Article
Author’s final manuscript
A method to estimate the thermal stress
build-up in an asphalt mixture from a single
cooling event
Michael J. Farrar* — Elie Y. Hajj** — Jean-Pascal Planche* —
Mohammad Zia Alavi**
*Western Research Institute
365 North 9th Street
Laramie, WY, 82072
[email protected]
[email protected]
**Department of Civil and Environmental Engineering
University of Nevada, Reno
Reno, NV, 89557
[email protected]
[email protected]
This paper suggests a method to estimate the thermal stress build-up in asphalt
from a single cooling event based primarily on the measured bitumen rheology. In order to
check the reasonableness of the calculation, first the Thermal Stress Restrained Specimen
Test (TSRST) was performed on a laboratory compacted, cylindrical asphalt specimen.
Concurrent to the TSRST test the thermal strain was measured from an unrestrained asphalt
specimen. As a result, the thermal stress build-up and coefficient of thermal expansion were
determined. The bitumen from the TSRST specimen was recovered and the bitumen low and
intermediate temperature rheological properties were determined using a dynamic shear
rheometer (DSR) technique (commonly referred to as 4mm DSR) that allows testing to -40°C
by way of a correction for instrument compliance. The estimated and measured TSRST
thermal stress build-up were compared and found remarkably similar. Also, the TSRST
thermal stress build-up was compared to the estimated thermal stress build-up using the
methodology in ASTM D6816-11, which includes an empirical Pavement Constant (PC), and
they were found to be significantly dissimilar suggesting that simply multiplying the binder
thermal stress by a PC (18 in this case) does not provide a particularly good estimate of the
mixture thermal stress build-up.
ABSTRACT.
KEYWORDS:
Thermal stress, Bitumen, Relaxation modulus, Thermal strain, TSRST, 4mm DSR.
EATA 2013, pages 1 to n
2
Farrar, Hajj, Planche and Alavi
1. Introduction
Thermal cracking is a very important damage event in an asphalt pavement’s life
cycle performance and is related to the thermal stress build-up in the pavement from
single or multi-cycle cooling events. Accurate prediction of thermal stress build-up
in asphalt is therefore essential in evaluating the low temperature cracking potential
of asphalt pavements. For example, ASTM D6816-11 requires the prediction of the
thermal stress build-up in the pavement in order to determine a critical cracking
temperature, Tcr. In the ASTM method, the bitumen creep compliance D(t), from the
Bending Beam Rheometer (BBR) test is interconverted to relaxation modulus, E(t),
and the thermal stress build-up in the bitumen to a selected cooling event is
determined using numerical methods to solve a convolution integral often referred to
as the Boltzmann hereditary integral.
The Boltzmann hereditary integral results from application of Boltzmann’s linear
superposition principle in which the effects of mechanical history are linearly
additive (Ferry, 1980). The thermal stress response σ(t) to an arbitrary thermal strain
history can be derived at any time, t, by applying the integral shown in equation 1.
𝑡𝑡
𝜎𝜎(𝑡𝑡) = ∫0 𝐸𝐸(𝑡𝑡 − 𝜏𝜏)
𝜕𝜕𝜕𝜕 (𝜏𝜏)
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[1]
where E(t) is the thermal relaxation modulus at time t, and ε(t) is the thermal strain
at time t.
The limits of integration are from time zero to t indicating that the material has
no strain history prior to time zero. Furthermore, the hereditary integral is applied
for material in the linear viscoelastic regime and the relaxation function E(t) is
entirely dependent on time and does not vary with the magnitude of applied strain.
In order to calculate the thermal stress build-up in the asphalt mix or pavement,
ASTM D6816-11 uses an empirical Pavement Constant (PC) proposed by Bouldin
et al. (2000) to convert the calculated thermal stress build-up in the bitumen to the
thermal stress in the asphalt pavement. It should be noted that this approach largely
depends on the magnitude of the PC value to predict thermal stress build-up in an
asphalt mixture.
Hence, the focus of this study is to present a more fundamental approach to
estimate the thermal stress build-up in an asphalt mixture from the thermal stress
build-up in bitumen without applying the empirical Pavement Constant. In this
study, the binder rheological properties at low temperature are determined using the
recently developed 4mm dynamic shear rheometer (DSR) technique rather than the
conventional BBR test. The 4mm DSR allows testing down to temperatures as low
as -40°C by application of a correction for instrument compliance (Sui et al., 2010).
Method of estimation of thermal stress build-up in pavement
3
2. Materials
The asphalt mixture used in this study consisted of a PG 64-22 unmodified
bitumen and an intermediate aggregate gradation with nominal maximum aggregate
size of 12.5mm. The asphalt mixture optimum binder content was 5.9% by total
weight of mixture designed for a medium traffic level, 3-10 million equivalent
single axle loads, following the Superpave volumetric mix design method,
AASHTO M323.
After mixing, the loose mixtures were subjected to short-term oven aging for
4 hours at 135±3°C. The mixtures were then compacted using a Superpave gyratory
compactor (SGC) to a target air void of 8±0.5%. Compacted samples were then
subjected to long-term oven aging for 5 days at 85°C according to AASHTO R30.
Specimens were long-term aged because thermal cracking is a long-term distress
mode. Two cylindrical specimens were cored from each SGC compacted sample for
testing in the thermal stress restrained specimen test (TSRST). After completion of
the testing, the bitumen was extracted and recovered from the mixture for further
testing.
2. Laboratory Testing
2.1. Thermal Stress Restrained Specimen Test (TSRST)
In the TSRST, an asphalt mixture is subjected to cooling below freezing from an
initial temperature of 5°C at a constant rate of 10°C per hour. Since the specimen is
restrained from contraction, increasing tensile stress is induced in the specimen by
decreasing the temperature until the specimen fractures. While attempts to measure
thermally induced stress in asphalt mixtures using the TSRST started in the mid1960’s, Arand (1987) working at the University of Braunschweig – Institute of
Technology, substantially improved the test system to insure that the stresses in the
specimen would not relax by continuously correcting the specimen length during the
test (Jung and Vincent, 1994).
The test samples consisted of 57 mm diameter by 140 mm height cylindrical
specimens cored from a large gyratory compacted sample perpendicular to the
direction of compaction, as shown in Figure 1.
The TSRST set up was also enhanced by adding a modular feature for measuring
the thermal strain in an unrestrained specimen concurrently with the stress
measurements from the restrained specimen. Figure 2 shows the details for the
utilized TSRST set up in this study.
The unrestrained specimen is placed on a frictionless roller stand to permit free
shrinkage or expansion of the specimen during thermal loading. The thermal
deformation measurements were obtained by two LVDTs attached to the two invar
4
Farrar, Hajj, Planche and Alavi
rods glued to the ends of the specimen. The invar rods extend out of the chamber to
maintain the LVDTs near ambient temperature while the temperature of chamber is
reduced. The unrestrained specimen is made up of two of the 57mm diameter
cylindrical specimens. The two specimens are chosen to have the same volumetric
measures and physical dimensions. The two cylinders are glued using a thin layer of
special epoxy to provide a uniform cross section along the length of the specimen.
The same epoxy is used to glue the TSRST specimen to the end platens and to glue
the invar rods to the ends of the unrestrained specimen. The epoxy was chosen to
have the same thermal contraction properties as common asphalt mixtures.
150 mm
140 mm
170 mm
57 mm
Figure 1. Side core TSRST cylinders obtained from a SGC specimen
Hydraulic Ram
Pedestals
LVDT
Environmental
Chamber
Temperature
Probe
Dummy
specimen
Restrained
Specimen
Unrestrained
Specimen
Invar Rod
Left
LVDT
Invar Rod
Frictionless
Roller Stand
Figure 2. Details of the TSRST setup
Thin Epoxy
Right
LVDT
Method of estimation of thermal stress build-up in pavement
5
2.2. Recovery of bitumen from the mixture
The bitumen extractions were completed by centrifuge method in accordance
with AASHTO T164A using a solvent of 85% toluene and 15% ethanol by volume.
The recovery process was completed utilizing a modified rotary evaporator
procedure (AASHTO T319). The recovery procedure was modified by increasing
the temperature, increasing duration, and applying higher vacuum conditions to
assure the satisfactory removal of the solvent, which was verified by gel permeation
chromatography (GPC). After completion of the bitumen extraction/recovery
process, the recovered binder was tested using the 4mm DSR technique.
2.3. Dynamic Shear Rheometry (DSR) using 4mm Diameter Parallel Platens
(4mm DSR)
The low and intermediate temperature (-30 to 45°C) rheological properties of the
recovered binder were measured with 4mm DSR. High temperature rheological
properties, typically measured with 25 mm diameter parallel plate geometry, were
not measured in this study.
The 4mm DSR test method is described elsewhere by Sui et al. (2010). The
method corrects for machine compliance and allows testing to as low as -40°C.
Dynamic shear moduli measurements were performed using a Malvern Kinexus
DSR. Frequency sweeps were typically performed at 15°C intervals over a
temperature range of -30 to 45°C and an angular frequency range of 0.1 to 100
rad/sec.
Storage and loss modulus master curves and estimated crossover moduli and
frequency were developed from frequency sweeps data using RHEATM software
designed by Abatech Consulting Engineers.
3. TSRST Results
The average thermal build-up stress and strain measurements from the restrained
and unrestrained specimens are shown in Figures3a and 3b, respectively. The test
was run until fracture occurred in the restrained specimen. As shown in Figure 3a,
the thermal build-up stress increased by decreasing temperature until -21.4°C at
which the asphalt mixture specimen fractured. The fracture stress was determined to
be 2.31MPa. Figure 3b shows the thermal strain measurements from the
unrestrained specimen. The slope of the strain thermal build-up represents the
coefficient of thermal expansion or contraction (referred to hereafter as CTE) for the
asphalt mixture. For the range of temperature applied for this mixture, the thermal
strain is considered to change linearly with temperature. Thus, the CTE was
determined to be 2.056×10-5 1/°C, which compares favourably with CTE values
6
Farrar, Hajj, Planche and Alavi
reported in the literature. This determined CTE was later on used in the Boltzmann
equation to predict the thermal stress build-up in the asphalt mixture.
Thermal Stress (MPa)
2.5
2.0
1.5
1.0
0.5
0.0
-24
-21
-18
-15 -12 -9
-6
Temperature (°C)
-3
0
3
Thermal Strain (mm/mm)
(a)
0.0006
y = -2.0564E-05x + 6.7321E-05
R² = 9.98
0.0005
0.0004
0.0003
0.0002
0.0001
0
-24
-21
-18
-15
-12 -9
-6
Temperature (°C)
-3
0
3
(b)
Figure 3. (a) Thermal build-up stress and (b) thermal build-up strain in asphalt
mixture
4. Method to calculate the thermal stress build-up in asphalt mixture from a
single cooling event based primarily on measured bitumen rheology
The first step is to generate the dynamic complex shear modulus G* and phase
angle δ for the recovered asphalt binder at the discrete time intervals selected. In this
study a time step of 72 seconds was selected. An initial temperature of 5°C and a
cooling rate of 10°C/hour were purposely applied to match the TSRST performed in
this study. However, an ending temperature of -35°C instead of -20°C was used to
carry the calculation out further. The total time of the cooling event was 4 hours, so
there were 200 discrete time intervals.
Method of estimation of thermal stress build-up in pavement
7
The Christensen–Anderson–Marasteanu (CAM) model (Marasteanu and
Anderson, 1999) was used to estimate G* and δ at each of the 200 time intervals and
corresponding temperatures. The CAM model is of the form:
𝜔𝜔
𝜈𝜈
𝐺𝐺 ∗ (𝜔𝜔) = 𝐺𝐺𝑔𝑔 �1 + � 𝑐𝑐 � �
𝜔𝜔
−𝑤𝑤
𝜈𝜈
[2]
The CAM model is similar to the CA model (Christensen and Anderson, 1992)
as it relates the frequency dependence of the complex modulus to the glassy
modulus (Gg), the crossover frequency (ωc) and the rheological index (R), but w is a
new parameter in the model that relates how fast or slow the phase angle converges
to the asymptotes at very low and high frequency.
The phase angle, δ, can be approximated as the first derivative of the log G*(ω)
with respect to log (ω). Rewriting equation 2 in terms of the log G*(ω) and
differentiating with respect the log (ω) derives the phase angle in terms of
frequency.
𝛿𝛿(𝜔𝜔) =
90𝑤𝑤
[3]
𝜔𝜔 𝜈𝜈
�1+� 𝑐𝑐 � �
𝜔𝜔
The crossover frequency at selected temperatures was determined using the
RheaTM software. The RHEATM software was used to generate G* and δ master
curves of the recovered binder at selected temperatures (-15 to 30°C in 15°C
intervals). The RHEA software uses the Williams-Landel-Ferry (WLF) (Williams et
al., 1955) equation to obtain an initial estimate of the horizontal shift, but the master
curve is refined using pair-wise shifting techniques (Rowe and Baumgardner, 2007).
The crossover frequency at any temperature beyond the selected temperatures can be
determined by plotting the log ωc versus temperature, which can be fit rather exactly
with a polynomial curve as shown in Figure 4. The change in crossover frequency
with temperature provides a convenient method for time temperature superposition,
and is a simple alternative to methods such as the WLF, Arrhenius form, and
Kaelble (Kaelble, 1985; Rowe and Sharrock, 2011) shift factors. The crossover
frequency can be determined from the polynomial equation shown in Figure 4 and
inserted into equations 2 and 3 allowing the estimation of G* and δ at each of the
200 time intervals.
The second step is to estimate the complex modulus of the asphalt mixture at
each time step. For this study the Al-Khateeb et al. (2006) model was used.
|𝐸𝐸 ∗ |𝑚𝑚 = 3 �
100 − 𝑉𝑉𝑉𝑉𝑉𝑉
100
��
(1+0.326|𝐺𝐺 ∗ |𝑏𝑏 )0.5
150 +0.0120 (|𝐺𝐺 ∗ |)0.5
� |𝐺𝐺 ∗ |𝑔𝑔
[4]
The model is essentially based on the binder complex modulus |G*|b as changes
in the percent voids in the mineral aggregate (VMA) only have a small effect. |G*|g
is the binder glassy modulus. The model is similar to the Hirsch model in that it is
8
Farrar, Hajj, Planche and Alavi
based on the law of mixtures for composite materials. Since the VMA has only a
small effect a value of 14 percent was assumed.
The RHEATM software and CAM model, which is programmed into the RHEA
software, were used to obtain a rough estimate of Gg of the recovered binder.
Accurate estimation of Gg even when low temperature rheology data are available,
as was the case in this study is a complex issue. In the present study, Gg was set to
1.0 GPa.
Figure 4. Temperature dependency of the log ωc
The third step is to determine the phase angle of the mix at each time step. Here
the Booij and Thoone (1982) approximation from Kramers-Kronig relations
(Kramers, 1927) connecting the real and imaginary parts of certain complex function
was used. The Booij and Thoone approximation can be written as:
𝜋𝜋
𝛿𝛿(𝜔𝜔) = �
2
𝑑𝑑 log |𝐺𝐺 ∗ (𝜔𝜔 )|
𝑑𝑑 log ⁡
(𝜔𝜔 )
�
[5]
The fourth step is to determine the mix relaxation modulus E(t) at each time step.
The mix storage modulus E'(ω) was determined by direct conversion of E* and δ.
(𝛿𝛿)
𝐸𝐸 ′ (𝜔𝜔) = 𝐸𝐸 ∗ (𝜔𝜔)𝑐𝑐𝑐𝑐𝑐𝑐⁡
[6]
𝐸𝐸(𝑡𝑡) ≈ 𝐸𝐸 ′ (𝜔𝜔)|𝜔𝜔 =2/𝜋𝜋𝜋𝜋
[7]
Interconversion of the storage modulus E'(ω) to E(t) was performed by the
approximate expression developed by Christensen (1982).
Method of estimation of thermal stress build-up in pavement
9
The fifth step is to numerical integrate the Boltzmann hereditary integral shown
in equation 1 from t = 0 to t = 14400 seconds to determine the stress at each time
step. The strain can be replaced by the thermal strain (αΔT) where α is the mix CTE
and ΔT is the change in temperature. The numerical integration is accomplished
using the trapezoidal rule.
The measured thermal stress build-up from the TSRST and the calculated
thermal stress are compared in Figure 5 and are remarkably similar, which suggests
the possible extent of physical hardening during the TSRST is similar to the extent
of physical hardening that occurs during the 4mm DSR test.
Figure 5. Comparison calculated and TSRST thermal stress build-up
5. ASTM D6816-11 Thermal stress build-up in the mix using the empirical
Pavement Constant (PC)
The ASTM D6816-11 method to determine the thermal stress build-up in the
mix involves first estimating the thermal stress build-up in the bitumen and then the
thermal stress in the asphalt by multiplying the bitumen thermal stress by the PC
factor. A PC = 18 was used for this analysis.
To convert the bitumen complex shear modulus G* to E* a Poisson’s ratio (ν) =
0.5 (at all frequencies) is assumed. However, it should be noted that ν is almost
certainly time dependent.
𝐸𝐸𝑏𝑏 = 𝐺𝐺𝑏𝑏 ∗ 2(1 + 𝜈𝜈)
[8]
The bitumen relaxation modulus Eb(t) at each time step was determined by first
determining the binder storage modulus Eb' by direct conversion of Eb* and δ per
equation 6 and then Eb(t) by applying equation 7.
10
Farrar, Hajj, Planche and Alavi
The next step is to numerically integrate the Boltzmann hereditary integral
shown in equation 1 to determine the stress at each time step. The strain can be
replaced by the thermal strain (αΔT) where α is the bitumen CTE and ΔT is the
change in temperature.
In a study by Marasteanu et al. (2007) bitumen CTE values below the glass
transition temperature ranged from 2.20 to 3.35 x 10-4/°C, while the CTE values
above the glass transition temperature ranged from 4.81 and 6.60 x 10-4/°C. Since
the CTE for the recovered bitumen used in this study was unknown, two CTE values
(1.00 and 3.00 x 10-4/°C) were selected representing a relatively broad range of
binder CTE values. The calculated asphalt thermal stress build-up using 1.00 and
3.00 x 10-4/°C CTE values are shown in Figure 6. Significant differences are
observed between the calculated thermal stress build-up according to ASTM D681611 method and the measured stress-build-up from the TSRST. This has important
implications in terms of estimating the critical cracking temperature (Tcr). The Tcr is
defined as the temperature at which the bitumen tensile strength, as measured in the
direct tension test, crosses the thermal stress curve. Substantial differences between
TSRST and PC based thermal stress curves would result in substantial differences in
Tcr. The differences also tend to confirm a study by Roy and Hesp (2001) where it
was found the pavement constant could vary between 3.4 and 16.7 based on
comparing the stress buildup in a restrained cooling test to the estimated stress
build-up using the pavement constant.
Figure 6. Comparison of thermal stress build-up: 1) TSRST, 2) Calc thermal stress
build-up this study, and 3) ASTM D6816 (4mm DSR data, PC = 18and two CTE’s
100 and 300 x 10-6/°C
Method of estimation of thermal stress build-up in pavement
11
6. Conclusions and future research
The following conclusions can be drawn based on the analysis performed in this
study:
– Since the calculated and the measured TSRST thermal stress development are
very similar the proposed method to calculate the thermal stress in the asphalt shows
considerable promise.
– The similar thermal stress development suggests the possible extent of physical
hardening during the TSRST is similar to the extent of physical hardening that
occurs during the 4mm DSR test.
– Using an empirical Pavement Constant to convert bitumen thermal stress
build-up to mixture thermal stress build-up is not an entirely satisfactory approach.
Of course, the work and calculations presented in this paper are limited to one
bitumen and asphalt, and additional research is needed to verify the proposed
method to calculate the thermal stress build-up from a single cooling event and
extend the application of the proposed method.
For example, the mechanistic-empirical pavement design guide (MEPDG) has
adopted a creep compliance parameter to characterize low-temperature behaviour of
bituminous materials and predict thermal cracking in the pavement. However, the
determination of the mixture creep compliance requires costly equipment, and
elaborate laboratory testing, and does not apply the MEPDG Global Aging System
(GAS). The GAS, which is an integral part of the MEPDG, is used to predict the
aged viscosity of the binder at any in-service time and depth. The bitumen viscosity
from the GAS is used in the asphalt dynamic modulus E* model to adjust the
modulus for the effects of oxidative aging.
It is suggested that the approach taken in this paper to calculate mixture
relaxation modulus could be applied to calculate mixture creep compliance by way
of interconversion of the MEPDG dynamic modulus. Further, by applying the GAS,
the mixture creep compliance could be determined at any in-service time and depth.
Finally, although the Al-Khateeb model appeared to perform adequately in this
analysis, it would be interesting to apply a more robust, fundamental rheological
model such as the 2S2P1D model developed by DI Benedetto et al. (2004), for
modeling the linear viscoelastic properties of both bituminous binders and mixtures.
Acknowledgements
The authors gratefully acknowledge the Federal Highway Administration, U.S.
Department of Transportation, for financial support of this project under contract
nos. DTFH61-07-D-00005 and DTFH61-07-H-00009. The RHEATM software
package, developed by Abatech Consulting Engineers, was used extensively during
12
Farrar, Hajj, Planche and Alavi
the data analysis phase of this study. Finally, the kind assistance of Mihai
Marasteanu in providing an example of the thermal stress build-up calculation for
bitumen greatly facilitated the development of thermal stress calculations presented
here for asphalt.
Disclaimer
This document is disseminated under the sponsorship of the Department of
Transportation in the interest of information exchange. The United States
Government assumes no liability for its contents or use thereof.
The contents of this report reflect the views of Western Research Institute and
the University of Nevada, Reno which are responsible for the facts and the accuracy
of the data presented herein. The contents do not necessarily reflect the official
views of the policy of the United States Department of Transportation. Mention of
specific brand names of equipment does not imply endorsement by the United States
Department of Transportation, Western Research Institute, or the University of
Nevada, Reno.
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