Abstract：A deformation prediction model for the dynamic creep test is deduced based on the linear viscoelastic (LVE) theory. Then, the defect of the LVE deformation prediction model is analyzed by comparing the prediction of the LVE deformation model with the experimental data. To improve accuracy, a modification of the LVE deformation prediction model is made to simulate the nonlinear property of the deformation of asphalt mixtures, and it is verified by comparing its simulation results with the experimental data. The comparison results show that the LVE deformation prediction model cannot simulate the nonlinear property of the permanent deformation of asphalt mixtures, while the modified deformation prediction model can provide more precise simulations of the whole process of the deformation and the permanent deformation in the dynamic creep test. Thus, the proposed modification greatly improves the accuracy of the LVE deformation prediction model. The modified model can provide a better understanding of the rutting behavior of asphalt pavement.
Key words：deformation prediction; rutting; viscoelasticity; dynamic creep test
Rutting or permanent deformation can greatly reduce the service life of the pavement. In the analysis of permanent deformation, two main aspects are of particular interest: the laboratory test techniques to measure the resistance of asphalt mixtures to permanent deformation and the mechanics models of rutting prediction[1].
Various laboratory test methods are currently used to estimate the permanent deformation property of asphalt mixtures, such as the static creep test, dynamic modulus test, and dynamic creep test[2]. Compared with the dynamic creep test, the static creep test made a larger error due to its static loading condition[3]. The permanent deformation in the dynamic creep test had a good correlation with field performance[4]. Many researchers modified the dynamic creep test to simulate the real condition of road. Jiang et al.[5] proposed an optional multiple repeated load test to consider the axial load spectrum in real pavement. Li et al.[6] modified the dynamic creep test to simulate the layers condition and the temperature gradient of the actual pavement. Also, some studies[7-8] used a smaller platen to better simulate the actual confinements in pavement.
In terms of the mechanics models, the viscoelastic model is widely used in the deformation prediction of asphalt mixtures. The linear viscoelastic (LVE) model, though widely used[9-11], is incapable of long-term rutting prediction due to a lack of consideration of densification and material damage[12]. Thus, some nonlinear viscoelastic models were developed[13-15]. However, they were conducted by some numerical technologies and could not yield a function for deformation prediction. Though many models of the permanent deformation prediction of the dynamic creep test have been developed[13,16], research on the prediction of whole-process deformation (the deformation throughout the dynamic creep test) is rare[9].
The objective of this study is to improve the accuracy of deformation prediction of the dynamic creep test. Specifically, a LVE deformation prediction model for the dynamic creep test is first deduced. Then, a modification to the linear term of the LVE deformation prediction model is made to fit the nonlinear property of permanent deformation. Finally, the accuracy of the modified model in predicting the whole-process deformation and the permanent deformation is verified by comparing the predicted result with the experimental data.
The load used in the dynamic creep test is a repeated load consisting of a haversine loading period and a rest period, to simulate the traffic load on asphalt pavement. The function of the load form is represented as
n∈N
(1)
where σmax is the peak value of the haversine load; t0 is the haversine loading period; T is the duration of a load cycle, T=t0+td, and td is the rest period.
The creep compliance of the Burgers model, J(t), is represented as
(2)
where J(t) is the creep compliance at time, GPa-1; t is the test time, s; E1, E2, η1, η2 are the parameters of the Burgers model.
According to the Boltzmann superposition principle in the linear viscoelastic theory, the strain of asphalt mixtures can be represented by the convolution integral.
(3)
where ξ is the integration variable; ε(t) is the strain of the analyzed material.
Substituting Eq.(2) into Eq.(3), the strain for the 1st load cycle (0<t≤T) can be represented as
(4)
where
(5)
(6)
where ω=2π/t0.
Based on the Boltzmann superposition principle, the LVE deformation prediction model in the (n+1)-th load cycle (nT<t≤(n+1)T) can be deduced as
(7)
2.1 Material properties
The commonly used dense-grade asphalt mixture AC-13 was selected in this study. No.70 asphalt was used as the asphalt binder, the properties of which are listed in Tab.1. Optimal asphalt content was determined to be 5.0%. The aggregate gradation was designed as shown in Tab.2.
All specimens were fabricated by a superpave gyratory compactor (SGC). The cylindrical specimens were 100 mm in diameter and 100 mm in height.
Tab.1 Main technical indices of asphalt binder
BinderP25℃/0.1mmPITR&B/℃D15℃/cmγNo.70asphalt63.4-0.8847.6>1001.035Notes:P25℃isthepenetrationat25℃;PIisthepene-trationindex;TR&Bisthesofteningpoint;D15℃istheductilityat15℃;γistherelativedensity.
Tab.2 Aggregate gradation of AC-13 asphalt mixture
Sievesize/mm191613.29.54.752.36Passingpercentage/%10010092.777.446.632.1Sievesize/mm1.180.60.30.150.075Passingpercentage/%21.014.68.26.45.4
2.2 Dynamic creep test
The dynamic creep test was conducted by a universal testing machine. The repeated load described in Eq.(1) was applied. t0 and T were chosen as 0.45 and 6 s based on the intersection traffic survey. The temperature was maintained at 60 ℃. The vertical deformation of specimen was measured by linear variable differential transducers (LVDTs). The first test was conducted with a stress level of 0.7 MPa and 100 cycles of deformation were measured in the first test to verify the accuracy of the whole-process deformation prediction. Three other tests with different stress levels (0.6, 0.7 and 0.8 MPa) were conducted to verify the accuracy of the permanent deformation prediction.
The experimental data in the 1st cycle of the dynamic creep test is used to fit the Burgers parameters. The nonlinear deformation equation for the 1st cycle is deduced and shown in Eqs.(5) and (6). The parameters of the Burgers model are regressed and listed in Tab.3, with the coefficient of correlations R2 being equal to 0.986. Fig.1 shows that the fitted curve matches well with the experimental data. It can be concluded that the LVE deformation prediction model can precisely simulate the recoverable deformation in a single load cycle.
Fig.1 Experimental data and the fitted curve in the first load cycle
Tab.3 The parameters of Burgers model
CoefficientsE1/MPaE2/MPaη1/(MPa·s)η2/(MPa·s)R2Values449.259930.9991850.735555.6480.986
4.1 Defect of LVE deformation prediction model
Fig.2 compares the experimental results and the prediction of the LVE deformation prediction model within the first 60 s (10 cycles) in the dynamic creep test. However, the prediction error becomes significant as the number of loading cycles increases.
Fig.2 Experimental data in the first 10 loading cycles and the predicted deformation by the LVE model
The increase of permanent deformation in every load cycle (from the (n-1)-th to the n-th load cycle) is calculated as
Δεpd(n)=εpd(n)-εpd(n-1)=ε(nT)-ε((n-1)T)
(8)
where εpd(n) is the increase of permanent deformation from the (n-1)-th to the n-th load cycle; εpd(n) is the permanent deformation at the end of the n-th load cycle where t=nT, so εpd(n)=ε(nT).
Substituting Eq.(7) into Eq.(8), the following equation is obtained:
(9)
The first term of Eq.(9) is a constant term and the second one is an exponential term. Clearly, the exponential term decreases to 0 as the number of loading cycles n increases, so the value of εpd(n) will converge to the constant term. Tab.4 compares the exponential term with the constant term. The exponential term becomes smaller as the number of cycles increases. Therefore, the constant term is the main factor contributing to the permanent deformation. The permanent deformation in the LVE deformation prediction model can be calculated as
From Eq.(10), it is clear that the predicted permanent
Tab.4 Comparison between the constant term and the exponential term
CyclenumberConstanttermExponentialtermRatio11.22×10-42.84×10-84.29×10321.22×10-41.35×10-129.00×10731.22×10-46.43×10-171.89×101241.22×10-43.06×10-213.97×101651.22×10-41.46×10-258.35×1020Note:Ratiorepresentstheratioofthevalueofconstanttermtothevalueoftheexponentialterm.
deformation is a linear equation of the number of loading cycles n. Also, it is reasonable to conclude that, in the Burgers model, η1 is the parameter which has the most significant influence on the prediction of permanent deformation.
4.2 Modification of the LVE deformation prediction model
As mentioned above, the constant term in the expression of ε2(t) causes the linear increase of the predicted permanent deformation. A modification of the constant term should be made to fit the nonlinear property of the permanent deformation of the asphalt mixtures.
The constant term in ε2(t) is modified by an exponential function,
(11)
where C1 and C2 are two additional coefficients.
Two other factors are considered in this modification. First, the continuity of the model (when t=t0, 
(t0) should be equal to ε1(t0)) is ensured by this modification. It can be verified by substituting 
into Eq.(5) and Eq.(11).
Secondly, this modification also guarantees the compatibility of 
(t) with ε2(t). It can be verified, in the special case when C1=C2=0, (t) degrades to ε2(t). Therefore, (t) is a generalization of ε2(t), so 
is bound to have a better applicability than ε2(t).
Due to these reasons, the modified deformation prediction model can be acquired by replacing ε2(t) with 
The mathematical expression is
As discussed in Section 4.1, when simulating permanent deformation, the exponential term in ε2 is ignored, so the modified permanent deformation prediction model can be deduced and simplified.
P1n+P2(1-e-P3n)
(13)
where P1, P2, P3 are three interim coefficients used to simplify the nonlinear regression.
Using the experimental data in the first 10 loading cycles of the test, the additional coefficients of the modified model C1, C2 can be regressed based on the nonlinear equation in Eq.(12), where C1=2.512 1×10-3, C2=8.010 2×10-3, and R2=0.973.
The modified prediction model and the experimental data in 100 loading cycles are compared in Fig.3. The enlarged figure shows that even at the end of the test, the prediction matches well with the experimental data in each load cycle. Therefore, the modified model is believed to be capable of providing an accurate prediction for the whole process of the dynamic creep test.
Fig.3 Experimental data in the 100 loading cycles and the predicted deformation by the modified model
To verify the prediction of the permanent deformation, dynamic creep tests with three different stress levels of 0.6, 0.7, 0.8 MPa were conducted. The values of the regression coefficients in Eq.(13) are listed in Tab.5.
The prediction curve of the permanent deformation and the experimental data of the dynamic creep test are compared in Fig.4. The permanent deformation curve can be divided into three stages: 1) Decelerating in the primary
Tab.5 Regression coefficients of the modified permanent deformation prediction model 
Stress/MPaRegressioncoefficientsP1P2P3R20.61.12×10-68.90×10-31.74×10-30.99770.72.75×10-67.73×10-33.5×10-30.99630.83.59×10-66.94×10-34.87×10-30.9976
stage; 2) Increasing linearly in the secondary stage; 3) Accelerating in the tertiary stage [2]. It can be concluded that the modified model can provide an accurate prediction for the permanent deformation in the primary and secondary stage, but it cannot simulate the accelerating in the tertiary stage.
(a)
(b)
Fig.4 Permanent deformation at three different stress levels. (a) Permanent deformation at three different stress levels; (b) Three stages of the permanent deformation
1) The LVE deformation prediction model can precisely simulate the deformation in a single load cycle, but it cannot simulate the nonlinear property in permanent deformation.
2) The constant term of ε2(t) in the LVE deformation prediction model leads to the linear increase of the permanent deformation prediction, and it is this defect that needs to be modified in the LVE model.
3) In all the Burgers model parameters, η1 has the most significant influence on the prediction of permanent deformation.
4) The modified deformation prediction model can accurately predict the whole-process deformation in dynamic creep tests.
5) The modified deformation prediction model can accurately predict the permanent deformation in the primary and secondary stages, but it cannot simulate the permanent deformation in the tertiary stage, so further research needs to be conducted.
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References：
摘要：基于线性黏弹性理论(LVE)推导出沥青混料在动态蠕变实验中的变形预估模型.然后,将线性黏弹性变形预估模型和实验结果对比,分析说明了线性黏弹性预估模型的不足.最后,为了提高预估准确性,对线性黏弹性预估模型进行了修正,使其具有与沥青混合料变形特性相符的非线性特性,并用实验数据对修正模型进行了验证.结果表明,线性黏弹性变形预估模型无法模拟沥青混合料的永久变形的非线性特性,而修正变形预估模型可以准确地预测动态蠕变实验中变形的全过程以及永久变形.说明了所提出的修正方法可以有效地提高线性黏弹性变形预估模型的准确性,该修正模型可以为沥青路面的车辙预估提供指导.
关键词：变形预估;车辙;黏弹性力学;动态蠕变实验
中图分类号：U414
Foundation item：The National Natural Science Foundation of China (No.51378121).
Citation：：Yan Tianhao, Huang Xiaoming, Zhang Zhigang, et al. Modification of the linear viscoelastic deformation prediction model of asphalt mixture[J].Journal of Southeast University (English Edition),2017,33(1):86-90.
DOI：10.3969/j.issn.1003-7985.2017.01.014.
DOI：10.3969/j.issn.1003-7985.2017.01.014
Received 2016-07-03.
Biographies：Yan Tianhao (1991—), male, graduate; Huang Xiaoming (corresponding author), male, doctor, professor, huangxm@seu.edu.cn.