Mitigation of Low-temperature Cracking in Asphalt Pavement by Selection of Material Stiffness

Abstract

A simple model dedicated to prevention or mitigation of low-temperature cracking in asphalt pavement is discussed in this paper. Having defined minimum temperature distribution in asphalt concrete pavement qualitative considerations on distribution of material stiffness are performed. As a result closed-form mathematical formulas allowing to reduce tensile stresses due to temperature drop are proposed in this approach.

Full text

Procedia Engineering 111 ( 2015 ) 748 – 755
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of organizing committee of the XXIV R-S-P seminar, Theoretical Foundation of Civil Engineering (24RSP)
doi: 10.1016/j.proeng.2015.07.141
ScienceDirect
Available online at www.sciencedirect.com
XXIV R-S-P Seminar, Theoretical Foundation of Civil Engineering (24RSP) (TFoCE 2015)
Mitigation of low-temperature cracking in asphalt pavement
by selection of material stiffness
Aleksander Szweda*, ,QH].DPLĔVNDa
aWarsaw University of Technology, Al. Armii Ludowej 16, Warszawa 00-637, Poland
Abstract
A simple model dedicated to prevention or mitigation of low-temperature cracking in asphalt pavement is discussed in this paper.
Having defined minimum temperature distribution in asphalt concrete pavement qualitative considerations on distribution of
material stiffness are performed. As a result closed-form mathematical formulas allowing to reduce tensile stresses due to
temperature drop are proposed in this approach.
© 2015 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of organizing committee of the XXIV R-S-P seminar, Theoretical Foundation of Civil
Engineering (24RSP)
Keywords: Pavement; Temperature; Stiffness; Thermal stress; Multi-layer half-space; Elasticity.
1. Introduction
Thermal load is one of the primary causes of pavement cracking [3,6,7,11,14]. Engineered layered system of a
pavement is subjected to cooling driven shrinkage which may lead to the formation of regularly spaced transverse
cracks [6,11]. When thermal stresses in a shrinking pavement reach the tensile strength of asphalt concrete (AC),
low-temperature induced cracks form across the width of a pavement. Such cracks usually form after an extreme
cooling event with temperature drop below
O
-20 C
and at least
O
-10 C
drop per one day of cooling event [3,6,7].
Formed cracks allow for the infiltration of water, which results in rapid deterioration of a pavement especially during
freeze-and-thaw cycles. Loss of continuity of a pavement structure yields in redistribution of stresses in adjacent to
* Corresponding author. Tel.: (+4822) 234 56 76, fax: (+4822) 825 88 99.
E-mail address: aszwed@il.pw.edu.pl
© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of organizing committee of the XXIV R-S-P seminar, Theoretical Foundation of Civil Engineering (24RSP)

749
Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
the crack zone by change from positive to negative bending of a pavement cross-section. This scenario of events
leads to progressive development of damage,loss of durability and affects the ride quality of a pavement.
Typically in the Central Europe one day temperature oscillation for awinter cooling conditions result in
O
10 -15 C
drop of the temperature [3]. Frequently distress in pavements is observed, since the generated thermal
stresses due to restraint reach the tensile strength of AC pavement.Prediction of the temperature distribution in
function of time in the pavement structure allows then to derive thermal stresses. The present paper describes a
simple method of estimating stresses due to temperature change during one-day cycle of typical cooling.Theory of
elasticity [12] is used for estimation of stresses in the layered pavement structure. In low temperatures AC exhibits
predominately elastic properties with reduced influence of viscosity phenomenon [1].Obtained stresses compared to
the tensile strength are then used to define maximum allowable stiffness of the selected material.
We analyze a homogeneous half-space as well as a typical three-layer pavement structure placed on a half-space
[5].First we consider one-dimensional transient problem of heat conduction in cooling program [2] to evaluate
temperature distribution across the structure.Obtained temperature field is treated as an input loading to an elasticity
problem.Objective of the study is to analyzestress field in a pavement subjected to thermal load and use this
information to formulate a criterion of choosing Young’s modulus for individual layer.Maximum tensile stress
criterion is used with prescribed uniaxial strength of the AC material.
2. Transient heat transfer in a half-space
2.1. Formulation and solution of the problem
We consider a three-dimensional problem, which can be reduced to a one-dimensional problem of heat
conduction for a half-space [2], as shown in Fig. 1. Initial temperature for the medium is assumed to be
O
0C
. The
boundary surface
0x
is subjected to the prescribed temperature load given by afunction of time

0
sinTt
Z
,
which was selected to represent change of temperature (during one day) at the pavement surface for the winter
conditions.
We formulate problem in the following way: find the temperature field

,Txt
satisfying the governing
equation, and boundary and initial conditions:
 
2
2
,,
1for 0, 0
Tx
t Txt xt
at
x
ww
!!
w
w
,
w
ac
O
U
,(1)

0
(0, ) sin for 0TtT t t
Z
!
,(2)
( ,0) 0 for 0Tx x !
,(3)
(,) 0 for 0
x
Txt t
of
o!
,(4)
where
O
-thermal conductivity, W/m/K;
U
-density, kg/m3;
w
c
-heat capacity, J/K/kg [2].
Fig. 1. The initial-boundary problem of half-space subjected to temperature.

750 Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
Solution of the problem may be achieved using the Laplace transform technique. Taking Laplace’s transform of
equations (1)-(4) with respect to time,next solving an ordinary differential equation with respect to
x
,and then
performing the inverse Laplace’s transform of the solution, we obtain the following result [2]:
 
 
2
0
3/2
sin -
,exp
04-
4-
t
Tx
Txt x d
at
at
ZW
WW
SW
§·
f
¨¸
¨¸
©¹
³
.(5)
Obtained formula (5) for the temperature as a function of time and depth is then used for estimation of induced
thermal stresses.
2.2. Analysis of temperature field in ahalf-space
For a rough estimate of the temperature distribution, the pavement and the substrate are treated as a homogeneous
half-space. Material coefficient
a
involved in (1) is ranging from
-7 1
4.5 10 s

u
to
-7 1
710s

u
, compare codes
[8,9,10]. For the calculations we chose the following values of other parameters:
0.85 W/m/K
O
,
3
1800 kg/m
U
,
900 J/kg/K
w
c
which results in
-7 1
5.25 10 sa

u
.
The temperature applied to the boundary is defined by function (2) in problem formulation.Assumed cooling
conditions during one day are following: minimum temperature amplitude
O
0
15 CT 
andfrequency coefficient
/ 86400 s
ZS
(period of 24 hours).The resulting temperature distribution in time for the selected points in the
half-space is shown in Fig. 2. In the top layer of the structure directly connected to the environment we observe the
rapidest changes of the temperature.Change of temperature inside the half-space getting slower (delayed in time and
dumped in value)when we select points deeper in the half-space. Distribution of the temperature is highly
dependent on the thermal diffusivity coefficient
a
in (1), which includes influence of the ratio of the conductivity
and the heat accumulation phenomenon [2,13].
Fig. 2. Evolution of temperature at selected depths.
Fig. 3. Temperature distribution across the medium at the selected instants.

751
Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
Distribution of the temperature in function of depth
x
for the selected time instants is shown in Fig. 3. In case of
cooling during the first 12 hours significant part of temperature distribution can be approximated by a linear
function [3,14]. At this stage of decrease of the temperature, critical change of temperature is governed by the
boundary loading in top layers of the half-space. In case of heating (i.e. returning of the temperature load to
O
0C
)
the distribution of temperature evolves to be more nonlinear. After about 6 hours of heating maximum of the
temperature drop moves into the lower layers of the half-space, but absolute value of the change is decreasing. For
high enough depth
x
,temperature aims at the initial value (3), and its change is insignificant during the process.
3. Stresses in a three-layer pavement subjected to atransient temperature field
3.1. Formulation and solution of the problem
An elastic three-layer pavement structure resting on an elastic half-space is subjected to temperature field

,Txt
, as shown in Fig. 4. We seek the components of stress tensor
ı
in the Cartesian coordinate system.Under
the applied thermal load and symmetry of the problem all functions are dependent on one spatial variable
x
and
time. Shear components of the stress tensor are zero, i.e.
0
xy xz yz
VVV
,which yields in null distortional strain
when Hooke’s law for isotropic material is assumed [12]. The only non–zero displacement is the vertical direction
x
which implies only one component of strain tensor
xx
H
to be non-zero.
The equilibrium equationsfor each layer and the substrate half-space are (only
x
variable is given in formulas):
1
d()
0
d
xx
x
x
V
,
2
d()
0
d
xx
x
x
V
,
3
d()
0
d
xx
x
x
V
,
d()
0
d
S
xx
x
x
V
,(6)
where 1
xx
V
,
2
xx
V
,
3
xx
V
are stress functions in the first,second and third layer, accordingly, and
S
xx
V
defines stress in
the substrate.Boundary and continuity conditions for the problem take the form:

1
00
xx
V
,
 
12
11xx xx
VG VG
,

23
12 12xx xx
VGG VGG
 
,

3
123 123
S
xx xx
VGGG VGGG
 
.(7)
Solving differential equations (6) with conditions (7) we obtain null functions for thefollowing stress components:
1
() 0
xx
x
V
,
2
() 0
xx
x
V
,
3
() 0
xx
x
V
,
() 0
S
xx
x
V
.(8)
Using the Hooke’s law for isotropic material with thermal strain component [12] and applying
0
yy zz
HH
(null
horizontal strain components) for all the layers we get:
 

(1 - )
() - ()
11-2 1-2
iii
ii
xx xx
EE
xxTx
QD
VH
QQ Q

,
 

() () - ()
11-2 1-2
iii
ii i
yy zz xx
EE
xx x Tx
QD
VV H
QQ Q

,(9)
where
1, 2, 3,iS
;
i
D
is the coefficient of thermalexpansion for i-th layer, i
Eis the elastic modulus for i-th layer
and
Q
is the Poisson’s ratio. Substituting (8) into (9) we get:

1
,(,)
1-
ii
xx
xt T xt
Q
HD
Q

,
(,) (,) - (,) - ( ,)
1- 1-
ii
ii i i
yy zz T
EE
xt xt T xt xt
VV D H
QQ
,(10)
where
(,)
i
T
xt
H
is free thermal strain in i-th layer. Having defined temperature distribution

,Txt
distribution of
lateral stresses (10) can be investigated.

752 Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
Fig. 4. Boundary value problem of the three-layer pavement resting on an elastic substrate.
3.2. Distribution of stresses in layered half-space due to temperature load
For the graphical representation of results (10) we consider the following typical data for perpetual pavement [4]:
1
5cm
G
,
2
10cm
G
,
38cm
G
,
1
9GPaE
or
112GPaE
,
2
14GPaE
,
312GPaE
,
2GPa
S
E
,
0.2
Q
.
The thermal expansion coefficient
D
varies significantly due to changes of temperature [6,13] but we assume it
constant (average) and for all layers equal to
-5 O
210 / Cu
[7,14].
Fig. 5. Horizontal stresses at selected points of layers for stiffness of the top layer: a)
1
9GPaE
; b)
1
12GPaE
.
Fig. 6. Distribution of stress across the layers for selected instants of time in case of stiffness
1
9GPaE
.

753
Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
For typical proportion of Young’s moduli for the layers, the extreme value of the lateral stress components occurs
in the second layer [1,3,14]. For high stiffness 1
E(comparable to the stiffness of the second layer) extreme value
can also be located in the first layer. The maximum stress in layer 1 (directly exposed to the environment) is reached
at the time in which the air temperature achieves its extreme value. In case of the two other layers the extreme
respond is delayed due to the insulation formed by the layer above, compare Fig. 5. Maximum difference between
extreme values of stress for layer 1 and 2 reaches
1MPa
for the assumed data.
The distributions of lateral stress along the
x
axis for the first and the third layer is approximately linear, see
graphs in Fig. 6. For the second layer distribution of lateral stress components differ significantly from a straight line
for the ending of considered loading process, i.e. for time instants
t
in the interval from 20 h to 24 h. It is caused by
transitional location of the second layer, placed between the first layer exposed to the environment and the third
layer where change of temperature is low (as
x
tends to infinity the temperature reaches the reference limit
O
0C
).
4. Distribution of stiffness in pavement structure
Result for the horizontal stresses (10) obtained in a closed mathematical form is a simple formula, which can be
used to define the design requirement. Generated tensile thermal stresses should be less than the tensile strength of
AC material used in each layer for any time instant of cooling:
(,)
ii
T
xt f
V
d
.(11)
In order to design stiffness distribution in pavement layers we can calculate the envelope of the temperature
distribution given by (5) andthen calculate the envelope of the lateral stresses using (10). Assuming generated
stresses to be equal to the tensile strength
T
f
across the structure we can derive the following relationship:
 
(,) -1- -1-
(,) (,)
ii
iTT
ii
T
ff
Ext Txt xt
QQ
DH
.(12)
Formula (12) defines maximum allowable stiffness of AC used in structural design of a pavement. Minimum of the
Young’s modulus calculated from (12) for the regarded layer can be used as an estimate of desired stiffness.
Obtained stiffness is dependent on the temperature distribution

,Txt
and material parameters:
i
D
,
i
T
f
,
Q
for
each AC layer (here constant Poisson’s ratio is assumed, but it can be different for each layer).
Distributions of stiffness at selected instants of time during cooling process with assumed tensile strength of
value
5MPa
i
T
f
,and
-5 O
210/ C
i
D
u
,
0.2
Q
are presented in Fig. 7. According to solutions (12) and (5) the
distribution of Young’s modulus given by formula (12) is an increasing function of
x
variable (depth).Selection of
the critical time instant (resulting in maximum tensile stress) for a point in layer can give optimum prediction of
allowable stiffness at that point.
Fig. 7. Distribution of stiffness in layers for selected instants of time.

754 Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
In order to propose more effective for use criterion for selection Young’s modulus for each layer due to the
temperature drop event we can draw qualitative conclusions from distributions shown in Fig. 5 and Fig. 6. Based on
the time and spatial distribution of the lateral stresses we observe that the extreme stress in layer 1 occurs at its
upper surface at a moment in which the extreme environment temperature is achieved
1
12ht
.The extreme tensile
stress in layer 2 is reached at its top surface about two hours after the extreme environment temperature drop is
reached
214ht
. Note that the extreme values of stresses for layers 1 and 2 are comparable in values. Additionally,
the extreme stress in layer 3occurs at its upper surface at
3
18ht
and it is approximately equal to half of the value
of extreme stresses for layers 1 and 2.
Taking into consideration above conclusions and using relationships (5) and (12) we obtain following
expressions for the Young’s moduli of each layer:
 
1
10
-1- -1-
(0, )
TT
ff
ETt T
QQ
DD
,

2
12
-1- (,)
T
f
ETt
QDG
,

3
123
-1- 2( ,)
T
f
ETt
QDGG

.(13)
In case of the third layer, half of the allowed stiffness was taken for the rough estimate, since for the perpetual
pavements fatigue criterion requires decreasing of the stiffness in the layer [4]. The same values of the tensile
strength and coefficient of thermal expansion were assumed in this study, but this assumption can be neglected in
general case.
Table 1. Maximum allowable Young’s moduli for pavement layers.
No.
1
G
[cm]
2
G
[cm]
3
G
[cm]
1
E
[GPa] 2
E[GPa] 3
E[GPa]
14 8 613.3 16.4 12.2
2412 613.3 16.4 14.8
3510 813.3 17.2 14.04
4512 913.3 17.2 15.6
5612 10 13.3 18.1 16.4
An exemplary thickness and stiffness for each layer calculated according to criterion (13) are given in Table 1.
The calculated values of stiffness change in a narrow range for the assumed typical conditions.Obtained stiffness of
material in layer 1 depends on thickness and stiffness of layer 2. Results of application of the criterion (13) as a
function of time are shown graphically in Fig. 8. Maximum stresses in the firstand the second layer are equal to the
assumed strength (11), while maximum stress in the third layer reaches 50% of the tensile strength.
Fig. 8. Stresses at the top and bottom of each layer for the case No. 3given in Table 1.

755
Aleksander Szwed and Inez Kamińska / Procedia Engineering 111 ( 2015 ) 748 – 755
Fig. 9. Envelope for critical temperature during cooling event and envelope of maximum allowed stiffness.
5. Conclusions
Typical one day temperature oscillation for a winter cooling conditions frequently result in transverse cracking of
AC pavement.To prevent or mitigate low-temperature cracking of pavements,a simple one-dimensional model was
analysed to find critical loading conditions. Due to temperature drop,tensile stresses develop in apavement
structure, and high possibility of cracking in top layers occurs.The proposed simple method of designing stiffness
distribution for the three-layer pavement result in simple formulas for evaluating Young’s moduli of the individual
layer.
In Fig. 9 the envelopes of the critical temperature and envelope of the allowable stiffness in inhomogeneous
pavement structure are presented. All other parameter involved in the solution are kept constant, but theirs variability
can be included into the model (especially coefficient of thermal expansion [6,7]). Criterion of the maximum tensile
stress for calculation of the stiffness was used. From the presented graph in Fig 9,it can be observed that the most
fragile to thermal cracking are top layers of the pavement structure. The obtained simple formulas (12) can be used
in the design practice to mitigate low-temperature cracking and enhance durability of pavements.Approximation of
the critical temperature distribution can be used to simplify analysis [3,14].
The proposed procedure for stiffness selection can be easily extended on the transversally isotropic material.
Application of this type of material can enhance reduction of stresses in the top layer of the pavement. For this
purpose reduction of stiffness in the horizontal direction is required. Additionally, time dependence of the material
parameters involved in the model should be included in further research [1,13].
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[5]Catalogue of typical structures of flexible and rigid pavements*HQHUDOQD'\UHNFMD'UyJ.UDMRZ\FKL$XWRVWUDG*GDĔVN
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[8] PN –(10DWHULDá\LZ\URE\EXGRZODQH–:áDĞFLZRĞFLFLHSOQR–ZLOJRWQRĞFLRZH–7DEHODU\F]QHZDUWRĞFLREOLF]HQLRZH
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