An inverse approach to determine complex modulus gradient of fieldaged asphalt mixtures
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Abstract
This study develops a new mechanicalbased method to determine the complex modulus and modulus gradient of fieldaged asphalt mixtures using the direct tension test. Due to the nonuniform aging nature of the field cores, the mechanical responses must be measured at different depths. Meanwhile, the monotonic load is not applied at the neutral axis of the field core specimen due to the modulus gradient, the tensile part of the strain is used and should be separated from the measurement because of the eccentric loading. The modulus gradient parameters, the location of the neutral axis, and the stress distribution are first obtained using the elastic formulas for a series of loading times. Then the complex modulus is determined using the Laplace transform and the elastic–viscoelastic correspondence principle. An inverse approach and iteration are then proposed by using the pseudo strain to accurately calculate the modulus gradient parameters after the relaxation modulus and reference modulus are determined.
Keywords
Field cores Modulus gradient Eccentric loading Corresponding principles Pseudo strain1 Introduction
The dynamic modulus of asphalt mixtures is a material property and one of the most important inputs in the Pavement MechanisticEmpirical (ME) Design [1]. It is also used as an indicator for either the level of aging or damage of the asphalt mixtures. Due to its importance, the dynamic modulus has been widely used and well determined for the laboratorymixedlaboratorycompacted (LMLC) asphalt mixtures. In general, the modulus of the unaged LMLC asphalt mixtures is affected by some factors such as binder type and content, aggregate type and gradation, and mix design. These factors can be well controlled in the laboratory and the field construction. However, when considering the fieldaged asphalt mixtures, the effects of field aging process and nonuniform air void distribution cannot be ignored. On the other hand, the properties of fieldaged asphalt mixtures can provide valuable information regarding the pavement condition since they can be used to make maintenance decisions and performance predictions.
In general, the fieldaged asphalt mixtures become stiffer after a longterm aging period, which is similar with the LMLC mixtures under the longterm aging in the laboratory. In addition to the longterm aging, there is another unique aging feature for the field cores: nonuniform aging in the pavement depth. It is known that the surface of the asphalt layer suffers from the solar radiation and oxidative aging more than deeper layers, as the oxygen needs time to diffuse through the interconnected air voids into the pavement structure from the surface of the pavement. Thus less carbonyl area is formed at deeper layers due to the less volume of oxygen and contact area. As a result, the modulus at the surface is higher than the other layers, and finally a modulus gradient is developed.
In order to take into account the field aging of asphalt mixtures in the Pavement ME Design, considerable research efforts have been made to either simulate or analyze the field aging in the laboratory, in both binder level and mixture level, or extract binders using the solvent from the field cores then determine the complex shear modulus and phase angle of the aged binders [2]. One of the widely used methods is the AASHTO R30 aging procedure [3]. It has been questioned to be too moderate, which cannot be used to reflect the aging of the asphalt mixtures in the field. Meanwhile, in order to simulate aging in a rational way, it is suggested that for different types of asphalt mixtures such as unmodified and modified mixtures, different aging protocols need to be developed [4]. In addition, the complicated nonuniform aging is even more difficult to be simulated. For the binder extraction method, the viscosity is determined for the extracted aged binders at different pavement depths from the field cores, the viscosity gradient with pavement depth and aging time can also be obtained [5]. However, there is one main problem with this method: some effects such as air void distribution, aggregate gradation, binder absorption, and aggregatebinder interaction on the modulus of the mixtures are not considered [6, 7].
As a result, it is preferred to obtain the material properties of the field cores directly. It is known that conducting the mechanical tests on the field cores remains difficult mainly due to the geometry compared to the LMLC mixtures specimens. The typical issue is the required dimension for a cylindrical specimen. The thickness of field cores normally ranges from 26 to 100 mm (1 to 4 inches) and even smaller for the overlays, which is insufficient to be used in the standard dynamic modulus test. To overcome this issue, recent studies determine that the thickness for rectangular and cylindrical field core specimens can be as thin as 26 mm (1 inch) for both dynamic modulus test and damage test, and the test results are in the same ranges with those for the standard dimension specimen [8, 9]. This valuable conclusion provides a guide for dealing with those with small specimen geometries such as field cores. In these studies, the tested specimens are obtained from different depths of one original field core, which are used to reflect the modulus distribution along the pavement depth. The traditional uniaxial tension–compression test is then conducted at different temperatures and frequencies to obtain the dynamic modulus master curves of field cores at different depths. However, it should be noted that the aging has been found to be most severe in the top surface especially the top 13 mm (0.5 inch) [5, 10]. The methods mentioned above are actually to measure the average modulus of each field core specimen, which may not be able to capture this gradient feature at top 13 mm.
Under this circumstance, this study presents a new mechanical method to determine the complex modulus and modulus gradient of field cores using the direct tension test. The direct tension test is adopted because of the three key advantages: (1) it is simpler to conduct and only takes less than 1 min for a given temperature; (2) it causes no damage to the specimen if the strain limitation is carefully controlled; and (3) the tensile modulus is determined instead of compressive modulus. It has been shown that the tensile modulus and compressive modulus of asphalt mixtures are different in both the magnitudes and phase angles [11]. However, most tests are conducted in the compression mode [1] and flexion mode [12]. The tensile modulus is necessary, especially for the characterization of various types of cracking in the asphalt pavements. In this study, an inverse approach is proposed to accurately determine the complex modulus and modulus gradient at different temperatures using the elastic theory, pseudo strain concept and elastic–viscoelastic correspondence principle.
This paper is organized as follows. The next section describes the information and preparation for both field core specimens and LMLC specimens. The test protocol to determine the complex modulus and modulus gradient is also discussed. The following section provides detailed derivations and results of the complex modulus and modulus gradient using the inverse approach. The last section summarizes the findings and future work.
2 Direct tension test to measure modulus gradient
 1.
The materials for testing, containing asphalt field cores and LMLC mixtures;
 2.
The configuration and procedure of the direct tension test with a nondestructive monotonically increasing load; and
 3.
The characteristics of mechanical responses of field cores as well as their comparisons with those of LMLC mixtures.
2.1 Asphalt field cores and LMLC mixtures
The asphalt field cores used in this study include one type of hot mix asphalt (HMA). They are collected from a field project near the Austin Bergstrom airport in Texas. The field asphalt mixtures are fabricated with a PG 7022 asphalt binder and Texas limestone aggregates. The binder content is 5.2%, the nominal maximum aggregate size is 10 mm (3/8 inch). The detailed mix design and the aggregate gradation can be found in this report [13]. The cores are taken at the center of two lanes of a HMA section at 8 months and 22 months after construction. It is reasonable to assume that the collected cores are not damaged by traffic within this aging period when they are in the field.
In order to demonstrate the features of field cores, laboratory HMA specimens are also fabricated. The parallel tests are performed between the field and LMLC specimens to demonstrate the differences in the measured data. Two air void contents for the laboratory specimens are chosen. The tested LMLC specimens are obtained only from the center of the compacted cylinder samples for the purpose of having uniform air void distributions through their thicknesses.
Field cores and laboratory fabricated mixtures specimens tested in direct tension test
Material type  Air void content (%)  Field aging time (month)  Thickness (mm) 

HMA field cores  6.6  8  38 
5.8  8  51  
5.5  22  51  
5.3  22  38  
LMLC HMA  6.3  N/A  38 
5.2  N/A  38 
2.2 Direct tension test
The direct tension test is conducted using the Material Test System (MTS) shown in Fig. 2b. A nondestructive monotonically increasing load is applied on the rectangular specimens at 10, 20 and 30 °C at a ramp rate of 0.020 mm/min, respectively. This MTS is an electrohydraulic servo machine. It includes a load cell, a temperature chamber, and is connected to a desktop for reading, saving and analyzing the test results including the load and strains. The MTS is also equipped of ball joints. To keep the specimens intact, the maximum tensile strain is set below 100 microstrains as suggested in the literature [14, 15, 16]. This type of LVDT can measure 100 microstrains accurately. It takes approximately 2 h to change the temperature of the specimens from one to another, and it takes approximately 8 h to finish the entire set of the tests for three temperatures. A new set of specimens are put in the temperature chamber overnight to reach the temperature equilibrium and recover the temperature loss due to opening the chamber for unloading and removing the previous specimens.
2.3 Mechanical responses of field cores and LMLC mixtures
Figure 5 presents the measured vertical and horizontal strains of the field core specimen calculated from the readings of the deformations of one vertical and one horizontal LVDTs attached at the top. The vertical deformations are recorded by the four vertical LVDTs attached at the top, center and bottom, whereas the horizontal deformations are recorded by the two horizontal LVDTs attached on the top and bottom. Note that the vertical strains at the center of the specimen are calculated by averaging the readings from the two LVDTs attached on the two center sides. It is shown that as the tensile load increases, the vertical strain increases whereas the horizontal strain decreases.
Figures 6a and 6b compare the induced vertical strains obtained from the corresponding vertical deformation data for the field core specimen and LMLC specimen, respectively. Under the similar loading, the measured vertical strains in the field and laboratory specimens are obviously different. The three measured vertical strains in the field core specimen (Fig. 6a) have different magnitudes at the three locations, which are closely related to the modulus at each depth. However, the three measured strains for the LMLC specimen are almost identical from Fig. 6b. It is known that the LMLC specimen has an almost uniform modulus across the thickness. Therefore, the difference between three measured strains of the LMLC specimen is minimal. The measured strains at the top, center and bottom for the field core specimens are different, which is due to the nonuniform modulus distribution in the field cores. In general, the strain at the top is smallest and the strain at the bottom is largest, which reflects the modulus distribution. Figure 6c illustrates that the strain is smaller and increases slower for the field specimen with a longer aging time, which shows the longterm aging effect on the mechanical response.
3 Derivations of complex modulus gradient
 1.
How to determine the parameters of the modulus gradient of a field core specimen; and
 2.
How to convert the measured data and parameters of modulus gradient to its corresponding viscoelastic property: complex modulus.
3.1 Inverse application of viscoelasticelastic correspondence principle
 I.
In the first iteration:
 1.
Use the measured tensile strain of an undamaged field core specimen as the seed value for the pseudo strain. In other words, temporarily, there is an elastic relationship between the measured stress and the measured strain;
 2.
Utilize the elastic theory along with the measured load/strains to determine the modulus gradient parameters. The values of n and k are determined and checked for their dependence on loading time and frequency;
 3.
Convert the functions of the measured load/strains and modulus gradient parameters using the Laplace transform to calculate the corresponding viscoelastic property: complex modulus;
 4.
Calculate the relaxation modulus then the reference modulus using the calculated complex modulus; and
 5.
Calculate the pseudo strain by the reference modulus.
 1.
 II.
In the second iteration: replace the measured strain with the calculated pseudo strain as the seed value after the first iteration and repeat steps 2–5.
 III.
In the following iterations (normally 3–5):
 1.
Replace the pseudo strain in the previous iteration with the newest one and repeat steps 2–5; and
 2.
Stop the iteration when the pseudo strain is stable. Then the modulus gradient parameters converge, the complex modulus and pseudo strain will not change.
 1.
In the following subsections, the major steps are elaborated in sequence and the final equations of the complex moduli of different depths of the field cores are presented.
3.2 Determination of modulus gradient parameters
Using the pseudo strain to determine the modulus gradient parameters contains three steps discussed below.
Step 1
Decomposition of vertical strains in field core specimens
Step 2
Formulation of value and location of the load in field core specimens.
For the case of the LMLC specimen, the pseudo strains are the same at different locations and k equals to 1. Thus \(\overline{Z}\) reduces in \(\frac{d}{2}\) in Eq. (11), which is the centerline of the specimen. However, for a field specimen, \(\overline{Z}\) is always smaller than \(\frac{d}{2}\) given that k is larger than 1.
Step 3
Solve for n and k in the modulus gradient model.
Results of calculations of modulus gradient of a field core specimen (8 months aged at 30 °C) from direct tension test
Iteration  Loading time (s)  Pseudo strain (με)  Load (N)  a  b  c  n  k  n _{0}  k _{0} 

1st  5–15  6.12  30.41  1.26  0.73  0.85  2.96  2.44  2.54  2.16 
16–25  20.42  165.6  1.28  0.72  0.86  3.73  2.82  
26–35  34.71  268.34  1.30  0.72  0.87  3.89  3.09  
2nd  5–15  3.67  30.41  1.23  0.70  0.84  2.91  2.40  2.53  2.12 
16–25  12.15  165.6  1.26  0.71  0.85  3.68  2.79  
26–35  20.3  268.34  1.29  0.71  0.85  3.86  3.06  
3rd  5–15  3.53  30.41  1.22  0.68  0.82  2.90  2.39  2.52  2.11 
16–25  11.79  165.6  1.25  0.70  0.84  3.67  2.76  
26–35  19.15  268.34  1.27  0.70  0.84  3.85  3.05 
It should be noted that the exponent n and the ratio k are the material properties since they are the two parameters in the modulus gradient equation and must be included in the application of the correspondence principle which transforms an elastic equation into the Laplace transform of a viscoelastic equation. With each iteration, both n and k are determined to be timedependent, as seen in Table 2, which shows their final converged values. They both increase slightly with loading time and decreases slightly with frequency.
3.3 Determination of complex modulus using approximated n and k
After obtaining the modulus gradient parameters, the next step is to convert the elastic property to the corresponding viscoelastic property using the Laplace transform. The procedure is given below in sequence.
 Monotonic tensile load \(P(t)\):$$P(t) = a_{\text{P}} (1  {\text{e}}^{{  b_{P} t}} )$$(12)
 Tensile portions of the strains at the top and bottom of the field core specimen:$$\varepsilon_{\rm 0t} (t) = a_{0} (1  {\text{e}}^{{  b_{0} t}} )$$(13)$$\varepsilon_{\text{dt}} \left( t \right) = a_{\text{d}} \left( {1  {\text{e}}^{{  b_{\text{d}} t}} } \right)$$(14)
 Modulus gradient parameters n and k:$$n = n_{0} {\text{e}}^{{b_{n} t}}$$(15)$$k = k_{0} {\text{e}}^{{b_{k} t}}$$(16)
3.3.1 Determination of complex modulus using complex n and k
It should be mentioned that the calculations of the complex modulus above are based on the approximated results of Laplace transform of n and k by Eqs. (22) and (23). This generates a dynamic modulus gradient, but results in an issue that the phase angles at the top, center, and bottom are the same according to Eqs. (27), (30), (31), and (33). As a matter of fact, the phase angle should also have a gradient along the pavement depth. However, the derivations and computations become too complicated when using the accurate results of Laplace transform of n and k. In this study, the approximation method to calculate the complex moduli is adopted. The derivations and expressions of the complex moduli with complex n and k are presented in the Appendix, which also provides the phase angle gradient accurately.
3.3.2 Determination of relaxation modulus, reference modulus and modulus gradient
4 Conclusions and future work

The strains at different depths of the field core specimens are different, which is related to the modulus gradient, however, the strains for the LMLC mixtures are almost identical.

Due to the nature of the modulus gradient, the strains should be decomposed into tensile and bending portions from the elastic theory. The tensile portion is used and the two aging parameters n and k in the modulus gradient equation for different loading times and the modulus gradient can be obtained.

Using the Laplace transform and correspondence principle, the elastic forms can be further converted into the viscoelastic forms, which is used to determine the dynamic modulus.

An inverse approach with an iteration process for field cores is proposed using the pseudo strain concept. The relaxation modulus and reference modulus are determined to calculate the pseudo strain. Since the measured strain (i.e., viscoelastic strain) is not appropriate to be used in the elastic formulas, pseudo strain should be calculated to determine the accurate results of n and k, and dynamic modulus.
In a continuation of this paper, the dynamic modulus, viscoelastic Poisson’s ratio and the corresponding phase angles will be determined to obtain a full characterization of the viscoelastic properties of asphalt field cores. The viscoelastic properties of the field core specimens are elaborated and show the timedependency, nonuniform aging dependency and the longterm aging dependency. It is worth noting that the air void distribution of field specimens also has an influence on the dynamic modulus at different depths, and it should be taken into account carefully. In addition, the properties of warm mix asphalt mixtures (WMA) at the same aging condition are compared with the HMA dynamic modulus and Poisson’s ratio.
Notes
Funding
This study was funded by the National Cooperative Highway Research Program (NCHRP) Under Project NCHRP 0152: A MechanisticEmpirical Model for TopDown Cracking of Asphalt Pavement Layers.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
 1.AASHTO (2008) Mechanisticempirical pavement design guide. A manual of practice, interim edition. American Association of State Highway and Transportation Officials: WashingtonGoogle Scholar
 2.AlAzri NA, Jung SH, Lunsford KM, Ferry A, Bullin JA, Davison RR, Glover CJ (2006) Binder oxidative aging in Texas pavements: hardening Rates, hardening Susceptibilities, and the impact of pavement depth. Transp Res Board 0361–1981Google Scholar
 3.AASHTO (2006) Standard specification for mixture conditioning of hot mix asphalt (HMA). AASHTO R3002: WashingtonGoogle Scholar
 4.Braham AF, Buttlar WG, Clyne TR, Marasteanu MO, Turos MI (2009) The effect of longterm laboratory aging on asphalt concrete fracture energy. In: Asphalt Paving Technology: Association of Asphalt Paving TechnologistsProceedings of the Technical SessionsGoogle Scholar
 5.Farrar M, Harnsberger P, Thomas K, Wiser W (2006) Evaluation of oxidation in asphalt pavements test sections after four year of service. Paper presented at the International Conference on Perpetual Pavement, Western Research InstituteGoogle Scholar
 6.Luo R, Lytton RL (2012) Selective absorption of asphalt binder by limestone aggregates in asphalt mixtures. J Mater Civil Eng 25(2):219–226CrossRefGoogle Scholar
 7.Underwood BS, Kim YR (2012) Microstructural association model for upscaling prediction of asphalt concrete dynamic modulus. J Mater Civil Eng 25(9):1153–1161CrossRefGoogle Scholar
 8.Kutay M, Gibson N, Youtcheff J, Dongré R (2009) Use of small samples to predict fatigue lives of field cores: newly developed formulation based on viscoelastic continuum damage theory. J Transp Res Board 2127:90–97CrossRefGoogle Scholar
 9.Park H, Kim YR (2013) Investigation into topdown cracking of asphalt pavements in North Carolina. Transp Res Board 2368:45–55CrossRefGoogle Scholar
 10.NCHRP 137A (2004) Guide for mechanisticempirical design of new and rehabilitated pavement structures. National cooperative highway research program, transportation research board, national research council, WashingtonGoogle Scholar
 11.Luo X, Luo R, Lytton RL (2013) Characterization of asphalt mixtures using controlledstrain repeated direct tension test. J Mater Civil Eng 25(2):194–207CrossRefGoogle Scholar
 12.NF EN 1269726 (2004) Bituminous mixtures: test methods for hot mix asphalt—Part 26: StiffnessGoogle Scholar
 13.Glover CJ, Liu G, Rose AA, Tong Y, Gu F, Ling M, Arambula E, Estakhri C, Lytton RL (2014) Evaluation of binder aging and its influence in aging of hot mix asphalt concrete: Technical Report. Publication FHWA/TX14/066131. Texas A&M Transportation Institute, College StationGoogle Scholar
 14.Levenberg E, Uzan J (2004) Trixial small strain viscoelasticviscoplastic modeling of asphalt aggregate mixes. Mech Time Depend Mater 365–384Google Scholar
 15.Luo R, Lytton RL (2009) Characterization of the tensile viscoelastic properties of an undamaged asphalt mixture. J Transp Eng 136(3):173–180CrossRefGoogle Scholar
 16.Koohi Y, Lawrence JJ, Luo R, Lytton RL (2012) Complex stiffness gradient estimation of fieldaged asphalt concrete layers using the direct tension test. J Mater Civil Eng 24(7):832–841CrossRefGoogle Scholar
 17.Schapery RA (1994) Correspondence principle and a generated J integral for large deformation and fracture analysis of viscoelastic media. Int J Fract 25(3):195–223CrossRefGoogle Scholar
 18.Zhang Y, Luo R, Lytton RL (2012) Characterizing permanent deformation and fracture of asphalt mixtures by using compressive dynamic modulus tests. J Mater Civil Eng 24(7):898–906CrossRefGoogle Scholar
 19.Luo X, Luo R, Lytton RL (2015) Mechanistic modeling of healing in asphalt mixtures using internal stress. Int J Solids Struct 60:35–47CrossRefGoogle Scholar
 20.Luo X, Zhang Y, Lytton RL (2016) Implementation of pseudo Jintegral based Paris’ law for fatigue cracking in asphalt mixtures and pavements. Mater Struct 49(9):3713–3732CrossRefGoogle Scholar
 21.Findley WN, Lai JS, Onaran K (1989) Creep and relaxation of nonlinear viscoelastic materials: with an introduction to linear viscoelasticity. Dover Publications, New YorkzbMATHGoogle Scholar
 22.Schapery RA (1961) Two simple approximate methods of Laplace transform inversion for viscoelastic stress analysis. Guggenheim Aeronautical Laboratory, California Institute of Technology, PasadenaGoogle Scholar
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