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Road Materials and Pavement Design
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Predicting thermal cracking of asphalt pavements from bitumen and mix properties
Bagdat Teltayev & Boris Radovskiy
To cite this article: Bagdat Teltayev & Boris Radovskiy (2017): Predicting thermal cracking of asphalt pavements from bitumen and mix properties, Road Materials and Pavement Design, DOI:
10.1080/14680629.2017.1350598
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https://doi.org/10.1080/14680629.2017.1350598
Predicting thermal cracking of asphalt pavements from bitumen and mix properties
Bagdat Teltayeva∗and Boris Radovskiyb
aKazakhstan Highway Research Institute, Almaty, Kazakhstan;bRadnat Consulting, Irvine, CA, USA
(Received 11 October 2016; accepted 24 June 2017)
The objective of this study was to develop approximate relationships for prediction of the rheological properties and tensile strength of asphalt concrete with conventional bitu- men binder to address the low-temperature cracking analysis. The parameters of modified Christensen–Anderson–Marasteanu model for relaxation modulus of bitumen were related with its penetration index and softening point by approximate formulas based on Van Der Poel’s nomograph for determining bitumen stiffness. Hot mix asphalt relaxation function has been described by the Christensen–Bonaquist model. Hot mix asphalt tensile strength as a function of loading time and temperature was related to the bitumen stiffness by equation based on Heukelom’s data and the Molenaar-Li empirical formula. The strength of hot mix asphalt has been also studied by performing tensile tests at constant strain rate and temper- atures. The results were analysed considering the calculated stress increase with temperature drop at various cooling rates. Critical temperature was estimated from comparison of thermal stress and tensile strength of asphalt concrete at constant stress rate.
Keywords: asphalt pavement; thermal stress; low-temperature cracking; relaxation modulus;
tensile strength; critical temperature
1. Introduction
Low-temperature cracking is one of the major causes of failure of asphalt pavements in cold weather climates, including much of the United States, Canada, Russia, Ukraine, Kazakhstan, and other countries at extreme northern and southern latitudes. Thermal cracking has been asso- ciated with the volumetric change in the asphalt concrete layer. When the temperature drops, the pavement tends to contract its volume but the thermal strains are unrealised. That causes the thermal stresses in pavement. They build up until they reach the strength of the material, leading to the formation of cracks to relieve these stresses.
Monismith, Secor, and Secor (1965) developed a theoretical calculation method for the ther- mally induced stress in asphalt pavement as in an infinite viscoelastic beam based on Humphreys and Martin (1963) solution. Boltzmann’s superposition principle and hereditary integral form of the constitutive equation for linear viscoelastic material was applied to relate time-dependent stresses and strains. This method is currently used for the estimation of critical cracking tem- perature by many researchers. Hills and Brien (1966) devised a simple pseudo-elastic method to compute the thermal stresses in asphalt pavement. The predicted stress was dependent on loading time equal to time interval of numerical integration over which the change of stress was computed. Fromm and Phang (1971) eliminated the dependency on the time interval of
*Corresponding author. Email:[email protected]
© 2017 Informa UK Limited, trading as Taylor & Francis Group
numerical integration by specifying loading times. However, the results of the stress compu- tations were still directly influenced by the loading time specified. Christison, Murray, and Anderson (1972) employed five different methods of stress computation, including the Moni- smith’s and Hills-Brien’s methods, and concluded that a potential of low-temperature cracking for a given pavement can be evaluated if the mix stiffness and strength characteristics are known as a function of temperature and time of loading. Buttlar and Roque (1994) developed the indi- rect tension test (IDT) to measure the creep compliance and tensile strength of asphalt mixtures at low temperatures. The IDT testing of cores is very appropriate for the evaluation of existing pavements. However, several differences between the axial tension and IDT tests raise ques- tions about the interchangeability of the creep compliance and tensile strength values obtained from the two test methods. Bouldin, Dongre, Rowe, Sharrock, and Anderson (2000) consid- ered the thermally induced stress in asphalt binder as in viscoelastic bar and reported that the midpoint of the binder’s glass transition is close to the pavement critical cracking temperature.
Based on the study of Bouldin et al. (2000), in AASHTO and ASTM specifications, MP1a-02 (AASHTO,2002), PP 42-07 (AASHTO,2007), and ASTM D 6816-02 (ASTM,2002), asphalt mixture thermal stresses were calculated from asphalt binder thermal stresses multiplied by an empirical pavement constant equal to 18. This was done to obtain a critical cracking temperature at which the mixture thermal stress curve and binder strength curve intersect (Moon, Marasteanu,
& Turos,2013). First, the binder tensile strength at a constant strain rate of 3%/min cannot be used as same as the mix tensile strength at different stress rates. Second, only one pavement constant is not sufficient to convert the binder properties to the mix properties. This method cannot reflect the difference in thermal stresses for mixtures prepared with the different aggre- gate source and gradation and the asphalt mixture volumetric (i.e. air void, binder quantity, etc.).
For instance, Roy and Hesp (2001) for limited set of binders estimated the variation of pavement
“constant” between 3.4 and 16.7. Hu, Zhou, and Walubita (2009), Prieto-Muñoz, Yin, and Buttlar (2013), and Dave, Buttlar, Leon, Behnia, and Paulino (2013) replaced the one-dimensional pave- ment model with the two-dimensional stress analysis of low-temperature cracking implemented within a viscoelastic finite element modelling framework. Farrar, Hajj, Planche, and Alavi (2013) combined the thermal stress restrained and unrestrained specimen tests of asphalt mixture. The measured restrained thermal stress development was very similar to the thermal stress build-up calculated from the Boltzmann’s hereditary integral. Relaxation modulus of asphalt mixture was determined by approximate conversion of complex modulus estimated from the model based on the law of mixtures for composite materials and from the measured complex modulus of the recovered asphalt binder (Farrar et al.,2013).
The low-temperature transverse cracking in asphalt pavements is the result of a combination of two distress mechanisms: single-event thermal cracking and thermal fatigue. The single- event thermal cracking as the significant contributor to transverse cracking of pavements was considered in this study. To better understand the ways to reduce low-temperature cracking, a mechanistic model is needed that includes the predicted rheological and fracture properties of asphalt mixture.
The objective of our study was to develop the relationships for prediction of the viscoelastic properties and tensile strength of asphalt concrete with conventional bitumen binder as a function of time and temperature and apply them to the low-temperature cracking problem.
2. Stiffness and relaxation modulus of bitumen
Stiffness modulus S(t)introduced by Van der Poel (1954) is the ratio of the constant applied uniaxial stress σ0 to the resulting uniaxial strainε(t)at timet. Based on the test results of 47 bitumens, Van der Poel developed a nomograph to determine the stiffness of asphalt cement as
a function of temperatureTand loading timet(or frequencyω) if the softening temperatureTs and the penetration indexPI are known. To calculate binder stiffness, it is convenient to use the BitProps program (free software) based on the scanned Van der Poel’s nomograph developed by G. Rowe and M. Sharrock (Abatech, Inc.).
Several researchers proposed the empirical equations for stiffness modulus of bitumen depend- ing on penetration and softening temperature based on Van der Poel’s data. Saal (1955) proposed the relationship between the bitumen stiffness and its penetration for load durationt=0.4 s and PI=0. Ullidz and Larsen (1984) proposed a simple equation for S(t)and limited its applica- bility to the duration of loading 0.01<t<0.1, range of penetration index −1<PI <1, and temperature range 10°C<T <70°C. Even in this narrow range of input parameters, the average coefficient of variation of stiffness calculated using the proposed equation from Van der Poel’s data is 40%. Shahin (1977) developed two regression equations based on Van der Poel’s nomo- graph to relate stiffness modulus with time, temperature, penetration index, and softening point:
one of them forS<1 MPa and another forS >1 MPa. The range of their applicability in terms of input parameters was not specified. Meanwhile, it is unknown what value of stiffness modulus will be and which of these formulas should be used. Moreover, often for the same set of input parameters, the first equation returnsS <1 MPa while the second givesS >1 MPa and it is not clear which of the results is correct. To our knowledge, the proposed formulas for the bitumen stiffness based on Van der Poel’s data were mostly developed using purely regression techniques without involving any rheological models.
In this study as a mathematical model for describing the stiffness of bitumen, Christensen and Anderson (1992) expression was used:
S(t)=Eg
1+ Egt
3η
β−1/β
, (1)
whereEgis the uniaxial glassy modulus of binder (MPa),ηis the steady-state viscosity (MPa·s), andβis constant.
The instantaneous value for longitudinal modulusEgwas obtained by extrapolation of values for stiffness modulus S(T,t) according to Van der Poel at low temperatureT and small load durationtfort→0. With the purpose of extrapolation, we used the model developed to describe the viscoelastic properties of amorphous glass forming polymers (Drozdov,2001) based on the theory of cooperative relaxation (Adam & Gibbs,1965). For the asphalt with certain penetration index, we selected the lowest temperature at which using the program BitProps the value of stiffnessS can be obtained fort=0.00005 s. This lowest temperature varied from −42° for PI = +2 to 9° forPI = −3. The stiffness modulusSwas obtained at that lowest temperature for a set of eight loading timest=0.00005, 0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005, and 0.01 s. Then using the Equations (18) and (37) of Drozdov (2001), the value of instantaneous modulus was found by fitting the Van der Poel’s data for stiffness modulus at eight loading times. It was concluded that for asphalts having the penetration index from −2 to +3, the average value of instantaneous longitudinal modulus equals toEg =2460 MPa with the standard deviation of 7%.
After fitting Equation (1) to Van der Poel data, the following equations were obtained:
β = 0.1794
1+0.2084PI −0.00524PI2, (2)
η=aT Ahrr(T)·η(Tr) (atT≤Ts−10),
η=aT WLF(T)·η(Tr) (atT>Ts−10), (3)
η(Tr)=0.00124
1+71 exp
−12(20−PI) 5(10+PI)
·exp
0.2011
0.11+0.0077PI
, (4)
whereTr is reference temperatureTr=(Ts−10)[°C], andaT(T)is time-temperature superpo- sition function:
aT Ahrr(T)=exp
11720·3(30+PI) 5(10+PI)
1
(T+273)− 1 (Ts+263)
, (5)
aT WLF(T)=exp
− 2.303(T−Ts+10) (0.11+0.0077PI)(114.5+T−Ts)
. (6)
For 1910 points in the range ofPI from −3 to +2,(T−Ts)from −45°C to 10°C, andtfrom 10−4to 104s, the standard deviation of stiffness calculated using Equation (1) from the Van der Poel’s data was 14.6% and the coefficient of determinationR2 =0.978.
By definition, stiffness modulus S(t)is the inverse of uniaxial creep compliance. A related function is the shear creep compliance:
J(t)= 1 Gg
1+
Ggt η
β1/β
, (7)
whereβis given by Equation (2) andGg = the glassy modulus in shear;Gg ≈Eg/3=820 MPa if the material is considered isotropic with Poisson’s ratioν=0.5, although it should be noted thatνis time dependent (Di Benedetto, Delaporte, & Sauzeat,2007).
There are different methods for converting the shear creep complianceJ(t)to relaxation modu- lusG(t). In this study, the modified Hopkins–Hamming algorithm (Tschoegl,1989) was selected, and relaxation modulus G(t) was approximated by Christensen, Anderson and Marasteanu (CAM) model (Marasteanu & Anderson,1999):
G(t)=Gg
1+ Ggt
η
b−k/b
, (8)
where bandkare constants. It can be shown that the parameterkis a function ofb from the following equation (Ferry,1980):
∞
0
G(t)dt=η, (9)
which can be treated as a definition of zero-shear viscosityη. After substituting Equations (8) to (9) and integration, it is easy to find that exactlyk=1+b. This leads to the following formula for relaxation modulus at shear:
G(t)=Gg
1+ Ggt
η
b−(1+1/b)
, (10)
where
b= 1
β +ln(π) ln(2) −2
−1
. (11)
The uniaxial relaxation modulus of binder can be found as Eb(t)≈3G(t), where its shear modulusG(t)is given by Equation (10).
Complex modulus of bitumen in shear was determined from creep compliance Equation (7) at a frequency equal to the inverse of the loading time using the Schwarzl and Struik (1968) method in the following form:
Gd(ω)= Gg (1+m(ω))
1+
Gg ηω
β−1/β
, δ(ω)=m(ω)π/2, (12)
where Gd(ω) is the norm of complex modulus,δ is phase angle,ω is frequency, (x)is the gamma function, and
m(ω)= (Gg/ηω)β
1+(Gg/ηω)β. (13)
For example, comparison of experimental results (Christensen & Anderson, 1992) at T=25°C versus calculated from Equation (12) values ofGd(ω)andδ(ω)is shown in Figure1 for SHRP core bitumen AAB-1 with pen 25=98 dmm, Ts=47.8°C,PI =0 (Mortazavi &
Moulthrop,1993). Tests were performed on dynamic shear rheometer with a range in frequency from 0.1 to 100 rad/s at temperatures −35, −25, −15, −5, 5, 15, 25, 35, 45, and 60°C. The data at all temperatures were then shifted with respect to time to construct the master curve at reference temperature 25°C (Christensen & Anderson,1992). Notice that the agreement between the calculated and experimental results for the norm of complex modulus and the phase angle is good.
The relaxation spectrum is a useful, fundamental way of characterising the time-dependent behaviour of asphalt binders (Christensen, Anderson, & Rowe,2016). According to Alfrey’s rule (Ferry,1980), relaxation spectrumH(τ)of binder is obtainable in first approximation as a negative slope of the relaxation modulus. Differentiating Equation (10) with respect to lntleads to
H(τ)=Gg(1+b) Ggτ
η b
1+ Ggτ
η
b−(2+1/b)
. (14)
Figure 1. An example of comparisons between the model predictions and experimental data for complex modulus and phase angle as a function of frequency: points – measured (Christensen & Anderson,1992), lines – calculated from Equation (12).
Figure 2. The relaxation spectrum for bitumen AAB-1 at 25°C: dashed line – from experimental data (Reproduced with permission from Christensen,1992); solid line – calculated from Equation (14).
Relaxation spectrum calculated in Christensen (1992, Figure 3.4) for the same SHRP core bitumen AAB-1 based on measurements of complex modulus is presented in Figure2together with calculated from Equation (14). The agreement between them is good. From Equation (14), it is easy to obtain the relaxation time corresponding to the maximum density of spectrum
τmax= η Gg
b 1+b
1/b
, (15)
and its maximum density
Hmax=Ggb b+1
2b+1
(2b+1)/b
. (16)
An acceptable estimate for the maximum density of relaxation spectrum of bitumen isHmax= Ggb/3. It follows that maximum density of spectrum is proportional to the parameterbof CAM model and is independent on temperature of bitumen.
One can see that the important rheological properties of bitumen such as stiffness, relax- ation modulus, complex modulus, and relaxation spectrum can be estimated from the pro- posed equations using the simplest standard properties of bitumen: penetration and softening temperature.
3. Stiffness and tensile strength of asphalt concrete
To analyse the temperature-induced stresses, the uniaxial relaxation modulus of asphalt concrete Emix(t)is needed. Christensen and Bonaquist (2015) recently improved their model:
Emix(t)=Pc[Eagg·(1−VMA)+Eb(t)·VFA·VMA], (17) Pc=0.006+0.994[1+exp(−(0.663+0.586·ln(VFA·Eb(t)/3)−12.9VMA
−0.17·ln(εs·106))]−1, (18)
whereEb(t)is relaxation modulus of binder (MPa),Eaggis elastic modulus of aggregate (MPa), VMA are voids in mineral aggregate (volume fraction), VFA are voids filled with asphalt (vol- ume fraction),εs=0.0001 is the standard target strain, andPcis the contact factor introduced in Christensen and Bonaquist (2015) and defined by Equation (18). The same model can be used for the stiffness modulus of asphalt concreteSmix(t)ifEb(t)is replaced by binder stiffnessS(t) defined according to Equation (1).
To estimate a critical temperature, the tensile strength of asphalt concrete as a function of temperature is needed. W. Heukelom has presented compelling evidence that the tensile strength of mix is related to the properties of bitumen contained herein (Heukelom,1966). He gave some data of tensile strength measurements of eight dense-graded mixes carried out at a variety of temperatures and speeds. Heukelom has showed that the relative tensile strength, that is, the tensile strength divided by its maximum value, can be represented by one curve for all mixes tested as a function of the stiffness of bitumen recovered (Heukelom,1966, Figure 22):
R(S)= Tensile Strength
Maximum Tensile StrengthOf Mix. (19)
The Heukelom curve can be approximated by equation
R= 0.774+0.039r+0.141r4.547
1+0.026r3.608·exp(1.245r), (20) wherer=log(Eg/S),Egis the uniaxial glassy modulus of binder (MPa),Sis the binder stiffness.
Molenaar and Li (2014) proposed an empirical equation to estimate the maximal tensile strength of asphalt concretePhas a function of mixture stiffness and the volumetric composition.
Six dense-graded mixes and one porous mixture with a bitumen of 10/20 to 70/100 penetra- tion and aggregate of maximum grain size from 4 to 32 mm were tested at the Delft University of Technology in a temperature range of 5–35°C at constant tensile strain rate from 0.0001 to 0.04/s. Regression analysis was performed to predict the maximum tensile strength of mixtures depending on mixture stiffnessSmix at 20°C and asphalt-void ratio (VFA). Molenaar and Li (2014) developed the following regression equation:
Ph =0.505·Smix0.308·VFA0.849. (21) The combination of Equation (20) based on the Heukelom curve for relative strength of mix with empirical Equation (21) for the maximal strength of mix (Molenaar & Li,2014) leads to the following expression for tensile strengthf as a function of temperatureTand the loading timet:
f =0.505·S0.308mix ·VFA0.849·0.774+0.039r+0.141r4.547
1+0.026r3.608·exp(1.245r). (22) whereSis the binder stiffness (MPa) defined in Equation (1) as a function of time and tempera- ture,Smixis the mix stiffness atT=20°C and at fixed loading time of 0.06 s, andf is the tensile strength under rectangular pulse loading of durationt.
Equation (22) relates the tensile strength of asphalt concretef with temperatureTand time to failuretfin terms of the binder creep stiffnessS(tf), as Heukelom (1966) suggested. At constant strain rateVε, the tensile strengthfεin the left-hand side of Equation (19) can be expressed as a product of time to failure, the strain rate, and the secant modulus at failureHmix(tf). The secant modulusHmix(t)is related to the relaxation modulusEmix(t)by the following equation (Smith,
1976):
Hmix(t)=1 t
t 0
Emix(t−τ)dτ. (23)
Thus using Equation (22), one can find the time to failuretfnumerically as a root of the following equation:
tfVεHmix(tf)=0.505·Smix0.308·VFA0.849· 0.774+0.039r+0.141r4.547
1+0.026r3.608·exp(1.245r), (24) wherer=log(Eg/S(tf)).
Then, the tensile strength of asphalt concretefεat constant strain rateVεcan be calculated as fε=tfVεHmix(tf). (25) The calculated tensile strength at constant strain rate was compared with test results of direct tensile measurements at Braunschweig Technical University (Stock & Arand,1993). The main features of testing machine developed at the University of Braunschweig are a very stiff frame, strain transducers for the measurement of the length of the specimen, and a step motor to apply strain to a specimen at a pre-determined rate. The samples were prismatic with a cross section of 40 mm by 40 mm. The specimen length was 160 mm. Seven asphalt concrete mixtures with the same content of different binders of 4.7% by weight of total mix were testedat constant strain rateVε=1·10−4/s (1 mm/min) and at temperatures 20, 5, −10, and −25°C. The test results for mix with unmodified bitumen B1 having the penetration indexPI= −0.692 and softening temperatureTs=49°C for the mix volumetric properties VMA=0.15 and VFA=0.733 are shown in Figure3together with tensile strength curve calculated from Equation (25). The cal- culated maximal strength and the general shape of curve agree with measured strength although the curve looks shifted along the temperature axis by approximately 4–5°C. Possible reasons of shifting will be discussed later.
In the Kazakhstan Highway Research Institute, the uniaxial tensile tests were performed on testing system TRAVIS (InfraTest GmbH), which is a version of device developed at the Uni- versity of Braunschweig (Figure4).This testing system includes a compact test frame integrated
Figure 3. Measured tensile strength values at constant strain rate (Stock & Arand,1993) and those calculated from Equations (24) and (25).
Figure 4. Machine used for the mechanical testing in the Kazakhstan Highway Research Institute.
in the temperature chamber. The load is applied to the specimen via heavy-duty screw jack and stepping motor. An electronic load cell is directly attached to the spindle. The equipment is con- trolled by a PC connected to the motor and the transducers. PC is used for measurement data logging, control of the test procedure, and the temperature test chamber.
Dense-graded asphalt mixture was prepared with the use of granite aggregate fractions of 5–10 mm (20%), 10–15 mm (13%), and 15–20 mm (10%) from the Novo-Alekseevsk rock pit (Almaty region); sand of fraction 0–5 mm (50%) from the plant Asphaltconcrete-1 (Almaty city) and activated filler (7%) from the Kordai rock pit (Zhambyl region). The air-blown bitumen was produced by the Pavlodar petrochemical plant from the crude oil of Western Siberia (Russia).
After short-time ageing, the bitumen penetration was 70 dmm (at 25°C), softening temperature Ts =48°C, andPI = −0.91.A mixture designated as K2 was prepared with 4.8% of bitumen by weight of aggregates. Mixture was compacted with the Cooper roller compactor (model CRT- RC2S) to the average void content of 3.6%. Rectangular specimens (50×50×160 mm3) were then sawed from these slabs and glued with the epoxy resin to the mounts. Specimens were tested with nominal deformation rate of 1 mm/min(Vε=1·10−4/s) . Tests were performed at temperatures 20, 10, 0, −10, −20, and −30°. The strength values were obtained from five tensile tests for all temperatures (Figure5). The average coefficient of variation was of 15.6%.
Figure5presents the comparison of the measured tensile strength of asphalt concrete samples and the strength of asphalt concrete predicted form Equations (24) and (25) forVε=1·10−4/s and the mix volumetric properties VMA=0.144, and VFA=0.75 (dashed curve).The mod- ulus of aggregate of bulk specific gravity Gsb=2.760 was estimated using the correlation recommended in Christensen and Bonaquist (2015) asEagg=7650G1.59sb =36, 000MPa.
In Figure5as before in Figure3, the calculated maximal strength and the shape of the predicted curve (dashed red) agree with measured tensile strength but the predicted curve is constantly shifted along the temperature axis to higher temperatures even more than in Figure3. This shift can be caused by various reasons. One of them might be the time-temperature superposition.
Obviously, the binder, regardless of the aggregates skeleton, drives the temperature dependency of a bituminous material (Di Benedetto et al.,2011). In this paper, the time-temperature superpo- sition function for the binderaT(T)in Equations (5) and (6) was determined from Van der Poel’s data. Meanwhile, Anderson et al. (1990) found that Van der Poel’s nomograph underpredicted the bitumen stiffness at long loading times and low temperatures. Although the time-temperature sensitivity of bitumen at low temperatures is an important issue and deserves further research
Figure 5. Measured and calculated tensile strength values: dashed curve – calculated from Equations (24) and (25) at constant strain rateVε=1·10−4/s, dotted curve – calculated from Equation (22) for average time to failuretf =40 s.
efforts, this reason does not explain the shift at temperaturesT>0°C on Figure5(the dashed curve).
Another possible reason of shift is the effect of machine compliance on specimen strain rates.
To facilitate data acquisition, the system TRAVIS was designed to extend the testing time at cold temperatures. As a result, because of machine compliance, the on-specimen strain rate was not constant, given that the cross-head displacement rate 1 mm/min remains constant throughout the test, due to the high stiffness of the material at low temperatures. Actual time to failure was between 40 and 50 s at temperatures 20, 10, 0, −10–20°C and around 20–25 s atT= −30°C.
To check the validity of this reason, we calculated the tensile strength of the same asphalt mix K2 at loading time 40 s from Equation (22) forPI = −0.91,Ts=48°C, VMA=0.144, and VFA=0.75 (dotted curve in Figure5). Shift of calculated curve to cold temperatures closer to measured strengths confirms the significant effect of machine compliance on test results. If a true constant-rate-of-deformation test is to be performed on asphalt concrete, feedback to loading unite must be from on-specimen LVDTs and not from the cross-head displacement as was the case in this experiment. This constant-strain loading requirement is difficult to achieve. More- over, although used by some researchers (Bouldin et al.,2000; Christison et al., 1972; Stock
& Arand, 1993), the advantages of constant strain-rate test are doubtful of its value for low- temperature cracking problem because the longitudinal strain in asphalt pavement induced by cooling is unrealised, that is, equals to zero up to appearance of the transverse crack. For the single-event low-temperature cracking analysis, the stress-controlled strength test looks more important than strain-controlled test.
At constant stress rate Vσ, time to failure equals the tensile strengthfσ over the stress rate tf =fσ/Vσ. In that case, the binder stiffness in the right-hand side of Equation (22) depends on
Figure 6. Tensile strength of asphalt concrete as a function of temperature calculated from Equation (26) at different stress rates.
tensile strength and the stress rate. The tensile strengthfσat constant stress rateVσcan be found numerically from Equation (26) as a function ofTandVσ:
fσ =0.505·S0.308mix ·VFA0.849·0.774+0.039r+0.141r4.547
1+0.026r3.608·exp(1.245r), (26) wherer=log(Eg/S(tf =fσ/Vσ)).
Figure6 depicts the tensile strength calculated from Equation (26) for the same as before asphalt mix K2 at different constant stress rates.
To illustrate, the mix strength at T= −2°C and Vσ =0.05MPa/s equals to 3.1 MPa (Figure6). In that case, time to failure is 62 s; stiffness of binder atT=20°C andt=0.06 s according to Equation (1) is S =6.02 MPa; stiffness of mix at T=20°C and t=0.06 s from Equation (17) equals toSmix=6040MPa. Parameter r is equal tor=log(2460/S(T=
−2◦C,t=62 s))=2.611, and the right-hand side of Equation (26) returns the tensile strength of 3.1 MPa. A 10-fold increase in stress rate shifts the strength vs. temperature curve to higher temperatures as much as around 7°C (Figure6).
4. Thermal stresses and critical temperature
Predicted relaxation modulusEmix and the stress rate dependent tensile strength fσ were used to estimate the critical single event cracking temperature. In this study, the critical cracking temperature(Tcr)is defined as the temperature at which the asphalt concrete tensile strength at constant stress rate crosses the thermal stress curve.
Low-temperature-induced stresses in asphalt pavement were calculated as in an infinite viscoelastic beam resting on frictionless rigid foundation:
σ (t)= − t
0
αmix(T(τ))Emix(ξ(t)−ξ(τ)) d
dτT(τ)
dτ. (27)
wheretis the present time (s),τ is the passed time (s),T(τ)is the temperature variation with time (°C),αmix is the asphalt mixture coefficient of thermal contraction (/°C),Emix is the relaxation modulus (MPa) andξ(t)is the reduced time:ξ(t)= 0tdτ/aT(T(τ)).
Coefficient of thermal contraction was assumed constantαmix=2.5·10−5/◦C. The proper- ties of binder and mix were taken as before: Ts =48°C, PI = −0.91, Eagg=36,000 MPa, VMA=0.144, VFA=0.75. The relaxation modulus of asphalt concrete was determined from Christensen–Bonaquist model – Equation (17).
A sinusoidal variation between the maximum and minimum temperatures during a day was assumed in the first example (Figure7).The temperature drops from 5°C to −35°C and returns to 5°C over 24-hour period. Thermal stress build-up was calculated according to Equation (27).
A tensile stress rate was estimated asVσ(t)=dσ(t)/dtand it varies from zero to 3.8·10−4MPa/s with the average of about 2·10−4MPa/s. Tensile strength of asphalt concrete fσ at Vσ =2× 10−4MPa/s according to Equation (26) as a function of T is shown on the same graph. The critical cracking temperature at which the thermal stress curve and asphalt concrete strength curve intersect isTcr= −33°C.
In the second example, the pavement temperature drops from 5°C to −35°C at different con- stant cooling rates. As expected, the stress accumulation rate was found dependent on cooling rate (Figure8). Cooling rate increase greatly increases the thermal stress rate. At the cooling rate of 2°C/h that is similar to the rates experienced by real pavements, the loading rate at the temperature close to −30°C is aroundVσ(t)=2·10−4MPa/s (Figure8). Figure9is a plot of stress according to Equation (27) at cooling rate of 2°C/h and of asphalt concrete strength ver- sus temperature from Equation (26) atVσ(t)=2·10−4MPa/s. The critical temperature is about Tcr= −34.5°C.
Low-temperature cracking of asphalt pavements is a widespread and costly problem in cold regions. Ability to predict the temperature-induced stress in asphalt pavement and asphalt con- crete strength starting from the properties of binder and mix should be very helpful for designing mixes that are more crack resistant.
Figure 7. Thermal stress and estimated fracture temperature for sinusoidal temperature variation during a day.
Figure 8. Thermal stress rate at different cooling rates.
Figure 9. Thermal stress and estimated fracture temperature at constant-rate cooling.
5. Conclusions
The following conclusions can be drawn based on the analysis performed in this study:
• The important rheological properties of bitumen, such as the creep compliance (Equation (7)), relaxation modulus (Equation (10)), complex modulus (Equation (12)), and relaxation
spectrum (Equation (14)), can be related with its simplest properties such as penetration and softening temperature based on experimental data and rheological models (CA model in this study) using the interrelationships of linear viscoelasticity rather than applying the purely regression techniques. It would be of interest to enlist a more comprehensive rhe- ological model such as the 2S2P1D model (Di Benedetto, Olard, Sauzeat, & Delaporte, 2004), for modelling the viscoelastic properties of both bitumen and mixture.
• The direct tensile strength of asphalt mixture as a function of temperature can be esti- mated depending on properties of binder and mix volumetric composition from empirical equations (22), (25), and (26). Evaluation of asphalt concrete strength needs a more comprehensive mechanically based approach to address a ductile-to-brittle transition at low temperatures, perhaps based on new theory of materials failure (Christensen, 2013).
• The feasibility of constant strain-rate test of asphalt mixture is doubtful of its value for low- temperature cracking problem because the longitudinal strain in asphalt pavement induced by cooling is unrealised up to appearance of the transverse crack. Moreover, the constant- strain loading requirement at various temperatures is difficult to achieve due to the testing machine compliance. For low-temperature cracking analysis, the strength test at constant rate of stress loading looks more valuable.
• The proposed method to determine the critical single-event cracking temperature as the temperature at which the thermal stress curve intersect the constant-rate strength curve leads to reasonable results.
Additional research is needed to verify the proposed method to calculate the thermal stress build-up from a single cooling event, to estimate the critical temperature and to extend the application of the proposed method.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by Road Committee of Ministry for Investments and Development of the Republic of Kazakhstan (Contract No. 36 dated 21.07.2016).
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