Application of viscoelastic continuum damage approach to predict fatigue performance of Binzhou perpetual pavements

1.   Introduction

Over the past several decades, the type of pavement used most widely in China has been the semi-rigid asphalt pavement, which is characterized by a semi-rigid base layer, such as cement-stabilized aggregate and lime-fly ash treated soil, and overlays of asphalt concrete. This type of pavement structure is cost-effective and environmentally friendly as it is able to consume a considerable amount of industry waste (Cetin et al., 2010). In addition, the semi-rigid base provides stable support that is necessary for the high traffic volumes (and even overloading) typically seen in China. However, these semi-rigid materials are prone to fatigue cracking under repeated traffic loading. Once cracks initiate in the base, they propagate upward into the flexible asphalt layers, thereby leading to reflective cracking and finally the structural failure of the pavement. Hence, the design life of such pavements is usually limited to 10–15 years.

Inserting a stress-absorbing interlayer between the asphalt concrete and base materials has been one of the major remedies for reflective cracking. Nevertheless, it can only help to delay crack propagation even when high-quality material design and construction are guaranteed for the interlayer, because the base may crack as well. In the end, rehabilitation or even reconstruction is necessary for the whole pavement structure, including all the layers. With consideration of these unavoidable defects associated with the use of semi-rigid materials, another alternative, albeit expensive probably in the short term, is perpetual pavement. Perpetual pavements are typically designed to last 40–50 years without structural failure and only the surface asphalt layer requires rehabilitation or replacement when necessary. Therefore, in the long run, perpetual pavements could be a good choice that provides high cost-effectiveness and sustainability (Amini et al., 2012).

Traditionally, perpetual pavements have been designed by invoking the concept of limiting the critical pavement responses. Generally, the belief is that if the imposed traffic loads produce responses below certain threshold values, then structural damage will not accumulate. The critical pavement responses of interest are the vertical compressive strain at the top of the subgrade and the horizontal tensile strain at the bottom of the asphalt layers for structural rutting and bottom-up fatigue cracking, respectively (Behiry, 2012).

In order to limit rutting to the upper few inches of the pavements, an increase in the structure's thickness or in the stiffness of the materials is required so that the vertical load can be distributed more widely before reaching the subgrade. Experimental evaluation tools for material rutting resistance include the asphalt pavement analyzer, Hamburg wheel-tracking device, and heavy vehicle simulator, to list a few. To simulate the materials' macroscopic behavior and to further understand the underlying deformation mechanisms, readers are encouraged to follow the work by, for example, Cao and Kim, 2016, Choi, 2013, and Darabi et al. (2012).

On the fatigue side, one way to decrease the probability of bottom-up cracking is to increase the structural thickness as well, thereby reducing the maximum tensile strain at the bottom of the asphalt pavements. The bending beam test has been used conventionally in the laboratory to evaluate the fatigue resistance of asphalt mixtures and to investigate the concept of a fatigue endurance limit, which is the aforementioned threshold level of tensile strain. More advanced explorations regarding the concept of an endurance limit have been conducted by Underwood and Kim (2009) and Bhattacharjee et al. (2009) in the context of viscoelastic continuum damage (VECD) theory.

Despite their wide acceptance, the traditional design principles for perpetual pavements as mentioned above are subject to question. During its service life, a pavement undergoes complex traffic and environmental conditions, and thus, consideration of the effects of temperature, aging, and healing, should be included in both the material and structural design as they all lead to variations in the material properties. Clearly, to approve or disapprove a mix design by testing the material only under certain pre-specified conditions is not a sufficiently persuasive or rigorous approach.

This paper focuses on material characterization and pavement performance evaluation with regard to bottom-up fatigue cracking. A comprehensive analysis system is presented, and its applicability and versatility are demonstrated via simulations of the Binzhou perpetual pavement test sections in Shandong Province, China. The effects of temperature have been incorporated in this framework through the rigorous mechanistic modeling of the asphalt materials. As for other factors, such as aging and healing, continuous efforts are being made in the ongoing research and their effects will be taken into consideration in the future to complete the modeling framework.

2.   Overview of Binzhou test sections

Six different asphalt mixes were designed for the Binzhou project. The material designations and descriptions are listed in Table 1. Five different sections were constructed, and the structural layout is illustrated in Fig. 1. As can be observed from Fig. 1, Sections 1 Introduction, 2 Overview of Binzhou test sections, 3 Material modeling are categorized as full-depth asphalt pavements, which reflect the typical design of perpetual pavements in Europe and North America. In this project, Section 1 was designed using 70 με as the critical value for the tensile strain at the bottom of the pavements, whereas Sections 2 Overview of Binzhou test sections, 3 Material modeling were designed for 125 με. Section 5 was the semi-rigid asphalt pavement that is used widely in China, as mentioned previously. Section 4 had a design similar to that of Section 5 in which the thickness of the asphalt layers had been increased considerably by inserting a layer of large stone porous mixture (LSPM) right above the semi-rigid base. Note that for all the test sections, the top 30 cm of the subgrade was treated with lime in the field construction in order to increase the soil modulus.

Table  1.  Asphalt concrete materials used in Binzhou test sections.

Designation	Description
SMA	Stone matrix asphalt (PG 76-22, MAC modified)
Superpave-19	19 mm NMAS superpave (PG 76-22, MAC modified)
Superpave-25	25 mm NMAS superpave (PG 64-22)
LSPM	25 mm large stone porous mixture (PG 70-22, MAC modified)
F-1	12.5 mm NMAS superpave fatigue layer (PG 64-22)
F-2	12.5 mm NMAS superpave fatigue layer (PG 76-22, SBS modified)
Note: MAC means multigrade asphalt cement; SBS means styrene-butadiene-styrene.

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Figure  1.  Structural layout of test sections in Binzhou perpetual pavement project (unit: cm).

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As a relatively new type of asphalt mixture, LSPM is usually designed with a uniform gradation and large particle size, with the nominal maximum aggregate size (NMAS) typically greater than 25 mm, which is intended for an interlocking aggregate structure. Besides, a small amount of fine aggregate is also used to fill the voids to increase the material's stiffness and durability. The air void content is usually targeted at 13 percent to 18 percent, which gives the mixture excellent draining capability (Zhao and Huang, 2010). In addition, owing to its large particle size and high air void content, LSPM is able to function as a stress-absorbing interlayer and thus possesses the potential to reduce reflective cracking, according to numerical simulations and field observations conducted in China (Luo and Xu, 2007, Songgen et al., 2008, Yin, 2011).

3.   Material modeling

Strictly speaking, asphalt concrete is a viscoelastic–viscoplastic material, and which mechanism dominates the material's responses depends on the material's intrinsic properties, temperature, and loading rate. Because fatigue cracking usually occurs at moderate temperatures, and the structural layer of interest is the bottom asphalt layer that lies a few inches below the surface, it is reasonable to confine this study within the context of viscoelasticity. In the following sections, the classical linear viscoelasticity theory and continuum damage mechanics are reviewed briefly. Then, the viscoelastic continuum damage theory is applied to laboratory testing to characterize material stiffness and fatigue resistance.

3.1   Linear viscoelasticity

For a linear viscoelastic material, its mechanical response generally depends on the rate and history of the stress/strain input. According to the Boltzmann superposition principle, the response is usually expressed via a convolution integral of a unit response function, which describes the material's fading "memory" of the effects of the previous loading, and a term for the rate of input. In the case of a strain input, the stress response is expressed as follow

where σ is the stress response, ε is the strain input, $E(\cdot )$ is the relaxation modulus (i.e., unit response function), τ is the integration variable for time. Note that the input is initiated at time zero, and the lower limit of the integration is set as 0⁻ to accommodate the initial jump, if any, in the input.

For the purpose of numerical feasibility and convenience, the relaxation modulus function is usually represented via the Prony series as follows:

where M is the total number of Prony terms, E_∞, E_i, and ρ_i are the parameters to be determined using experimental data.

Before addressing the concept of damage in a viscoelastic body, Schapery's elastic-viscoelastic correspondence principle (Schapery, 1984) should be mentioned. The concept of pseudo strain is utilized and is defined as follows:

where ε^R denotes the pseudo strain, E_R is the reference modulus, typically set as unity.

The substitution of Eq. (3) into (1) yields

Eq. (4) resembles the constitutive relationship (i.e., Hooke's law) for the case of linear elasticity. Through the use of pseudo strain, viscoelastic problems can be reduced to elastic-like ones, which greatly facilitates analytical understanding and investigation. For instance, for a linear viscoelastic material that undergoes monotonic or cyclic loading without damage, its stress–strain curve would exhibit a nonlinear relationship or hysteresis loops due to strain energy dissipation. The stress-pseudo strain relationship, however, would be presented as a straight line (with the time/viscous effect being removed) with an inclination of 45° as determined by the reference modulus, E_R, according to Eq. (4). As discussed in the following section, this feature essentially helps to distinguish between the effects of viscosity and microcrack damage in the material responses.

3.2   Linear viscoelastic continuum damage theory

As aforementioned, an intact linear viscoelastic material demonstrates a linear relationship between the stress and pseudo strain during monotonic loading. When microcrack damage initiates, this relationship evolves into nonlinear regions, resulting in a reduction in the instantaneous secant modulus that is referred to as the secant pseudo stiffness. Analogous to elastic cases, this pseudo stiffness is used to characterize the material's deterioration due to damage. The material is treated as a continuum, with the microcracks assumed to be distributed uniformly within the body without coalescence. By ignoring the effects of anisotropy, nonlinear viscoelasticity, and plasticity, if any, the VECD framework allows for the hypothesis that any reduction in the pseudo stiffness is caused by the material's internal damage, exclusively:

where C is the secant pseudo stiffness, S is the internal variable that quantifies the material damage status.

The hypothesis is that between C and S there is a single-valued monotonically decreasing function, referred to as the damage characteristic relationship.

The pseudo stiffness value can be calculated directly using Eq. (5). Note that for the intact state that is free of damage, C- value is one, which is equal to the reference modulus, E_R, as obtained by substituting Eq. (4) into (5). Yet, the damage state variable usually can be obtained only through experimental investigation and theoretical hypotheses. In this case, Schapery's work potential theory, which is derived based upon thermodynamic principles, is employed to characterize the damage evolution process:

where α¹ is a material constant denoting the damage evolution rate, W^R is the pseudo strain energy density function, which is given as analogous to linear elastic cases as follows:

Initially developed for material constitutive modeling under monotonic loading conditions, the VECD theory and its algorithms later were extended and improved to accommodate cyclic loading scenarios (Daniel and Kim, 2002, Lee et al., 2000, Underwood et al., 2010, 2012), and it was then that the theory began to see its benefits in modeling the fatigue characteristics of asphalt materials.

In the laboratory, the C-S curve is usually determined by running cyclic direct tension fatigue tests in displacement-controlled mode in a closed-loop servo-hydraulic test system. Afterwards, an exponential function is typically used to fit the experimental curves for further manipulations:

where a and b are the fitting coefficients.

It is worth mentioning that the resulting damage characteristic relationship is a material intrinsic property that is independent of temperature, loading type (monotonic or cyclic), mode (stress, strain, or displacement-controlled), and other conditions such as loading amplitude and rate, as long as the viscoelasticity is maintained as the dominating mechanism in the material. It is also noted that the time–temperature superposition principle for the intact state is still applicable to the material despite the significant presence of damage (Chehab et al., 2002, Underwood et al., 2006), and therefore, the same time/frequency-temperature equivalence relationship can be used in the VECD framework.

3.3   Failure criterion

At this point, the damage characteristic relationship has been obtained. It prescribes the path that the damage evolution of each material point in the asphalt layers should be followed. However, the VECD theory does not provide a definition for the applicable region for the damage curves, because the fundamental assumption of the continuum would be violated as soon as microcracks start to coalesce and cause damage localization. Hence, a separate failure criterion is required to complete the VECD system.

Early explorations (Hou et al., 2010, Underwood et al., 2012) focused on the pseudo stiffness (C) by seeking a relationship to express the critical C-value at failure as a function of mixture type, i.e., whether the mixture contains reclaimed asphalt pavement (RAP) or not, the NMAS, and test conditions such as temperature and load frequency. Such criteria proved to be unreliable because of their high variability and dependence on the loading mode. However, the energy-based criterion developed by Sabouri and Kim (2014) presented a consistent and unified approach for failure characterization that had been verified successfully for both non-RAP and RAP materials (Norouzi and Kim, 2015, Norouzi et al., 2014). In the following, this model is reviewed in a slightly different way compared with its original presentation.

As shown in Fig. 2, attention is placed on the space of the stress and pseudo strain. To construct the damage characteristic curves, the values of C and S are computed in a cyclic manner according to the VECD theory (Underwood et al., 2010). After the stress peak in the first cycle, C begins to be redefined as the peak-to-peak pseudo stiffness immediately, designated as F. Because the pseudo strain energy dissipates slowly in the cyclic test, which is evidenced by the small area enclosed by the hysteresis loops before failure, F is deemed to be a natural substitute for C in cyclic analysis as a good indicator of material damage status or degree of integrity.

Figure  2.  Schematic representation of the characteristic dissipated pseudo strain energy.

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The material's deterioration is exhibited via a progressive decline in the F-value, which is accompanied by the loss in the material's ability to store the pseudo strain energy. For the specific stress and (pseudo) strain history in a cycle i, the amount of the maximum pseudo strain energy that can be stored if the material is intact is indicated by the triangular area, ADC, as shown in Fig. 2, where the AC line has a slope of unity, ${\epsilon }_{\mathrm{s}}^{\mathrm{R}}$ denotes the permanent pseudo strain, and ${\epsilon }_{0,\mathrm{ta}}^{\mathrm{R}}$ denotes the tensile amplitude of the pseudo strain in that cycle. The actual amount of maximum pseudo strain energy that can be stored in the material for this particular cycle is indicated by the area ADB.

The difference between the maximum pseudo strain energy for the intact material and that for the damaged material, indicated by ABC and denoted as $W_{\mathrm{C}}^{\mathrm{R}}$, is the pseudo strain energy that would have to dissipate in order to degrade from the intact state to the specific damage state for the material in that cycle. Therefore, in a given fatigue test, $W_{\mathrm{C}}^{\mathrm{R}}$ can be considered as another damage state variable, and in the original model presentation, it is referred to as the total released pseudo strain energy. However, it should be noted that this energy is not physically released or dissipated in a cycle during fatigue testing, and therefore, in this paper, it is referred to as the characteristic dissipated pseudo strain energy (CDPSE). This energy provides an effective and convenient approach to track the material's status by comparing the current damaged state with the virgin state in a cyclic manner. Lastly, it is also worth noting that, here, reverse loading and the potential healing effects are not considered.

According to Fig. 2, the CDPSE for cycle i is given as follow

When plotted with the cycle number until the cycle at failure, i.e., N_f, the CDPSE exhibits different profiles in terms of shape and magnitude depending on the test temperature and loading mode. However, it is found that the quantity G^R, as defined in Eq. (10), is consistently related to the fatigue life, N_f, regardless of the test temperature and loading mode.

where $\overline{W_{\mathrm{C}}^{\mathrm{R}}}$ is the averaged characteristic dissipated pseudo strain energy (ACDPSE) per cycle in a fatigue test, G^R is defined as the rate of change of the average released pseudo strain energy (per cycle) (Sabouri and Kim, 2014) throughout the test.

The relationship between G^R and N_f, which has been found from experimental data, is well expressed via a power function, or presented as a straight line in logarithmic scales:

where λ and κ are the fitting parameters.

Some inadequacy, however, was sensed regarding the physical significance, as cited above, assigned to G^R in the original model presentation. Basically, according to Eq. (10), G^R is simply equal to the accumulated CDPSE that is averaged twice over the entire fatigue life. It is difficult for the authors to relate the ratio of $\overline{W_{\mathrm{C}}^{\mathrm{R}}}$ to N_f with the rate of change of the ACDPSE. Instead, the ACDPSE itself has a well-defined physical meaning and may reasonably be related to the total amount of pseudo strain energy that should be dissipated in the intact material before reaching failure. Substituting Eq. (10) into (11) yields

The unified failure criterion expressed in the form of Eq. (12) is employed in this study.

4.   Experimental characterization

4.1   Dynamic modulus test

In order to identify the relaxation modulus function, the dynamic modulus test is often preferred over the relaxation test in the laboratory given the concerns related to machine compliance, capacity, and accuracy in observing instantaneous responses. For the dynamic modulus tests in this study, the plant-mixed loose mixtures acquired from the project site were compacted using a Servopac superpave gyratory compactor and then cored and cut to obtain cylindrical specimens 100 mm in diameter and 150 mm in height for each mix. All tests were conducted in accordance with the AASHTO TP62-10 testing protocol (AASHTO, 2010) using the asphalt mixture performance tester (AMPT). By virtue of the time–temperature superposition principle, the resulting modulus data points at different temperatures were then shifted horizontally in the space of the modulus versus frequency to construct a continuous and smooth mastercurve at a pre-specified reference temperature. The amount of shift for each temperature is termed as the shift factor. The dynamic modulus mastercurve is commonly expressed as a sigmoidal function as follows:

where |E*| is the dynamic modulus, f_r is the reduced frequency, f is the physical loading frequency, a_T is the shift factor at temperature T, δ, α, β, γ, α₁, α₂ and α₃ are parameters that can be identified in an optimization process in practice.

For the laboratory tests, a minimum of three replicates was employed, and the averaged dynamic modulus mastercurves were presented in Fig. 3, together with the phase angle mastercurves and the shift factor functions. A few observations can be made. The LSPM shows the lowest stiffness values at higher reduced frequencies (physically representing lower temperatures and/or higher loading frequencies) and relatively high stiffness values at lower reduced frequencies (physically representing higher temperatures and/or lower loading frequencies). These results are presumably due to the high air void content and the interlocking aggregate structure formed by the large particles, respectively. Conversely, the Superpave-25 mix shows the highest stiffness values at higher reduced frequencies and relatively low stiffness values at higher reduced frequencies. The F-1 and F-2 mixes are asphalt-rich mixtures designed primarily to mitigate fatigue and reflective cracking. They have the same aggregate structure design, but Fig. 3 shows that the mastercurve of the F-2 mix consistently lies above that of the F-1 mix due to the F-2 mix's use of SBS-modified binder that has a higher performance grade.

Figure  3.  Dynamic modulus test results for the asphalt mixes in Binzhou test sections. (a) Dynamic modulus master curves in arithmetic scales. (b) Dynamic modulus master curves in logarithmic scales. (c) Phase angle master curves. (d) Shift factor functions.

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With the dynamic modulus test results at hand, the coefficients of the relaxation modulus function, as shown in Eq. (2), can be calculated conveniently by following the algorithms described in Park and Schapery (1999). Basically, the dynamic modulus is converted to the relaxation modulus by solving a system of linear algebraic equations for the unknown Prony coefficients (E_i), while the time constants (τ_i) are specified a priori, usually a decade apart. The time–temperature shift factor function, Eq. (11), is applied to shift the obtained relaxation modulus function from the reference temperature to the actual test temperature for each fatigue test. Afterwards, the pseudo strain is calculated according to Eq. (3).

The fatigue resistance of asphalt mixtures is typically assessed by subjecting cylindrical specimens to direct tension cyclic tests in accordance with the AASHTO TP107-14 test protocol (AASHTO, 2014). As the practical difficulties are associated with other modes, these experiments are usually conducted in displacement-controlled mode to minimize the accumulation of plastic deformation. Each test is continued until the specimen is pulled apart. The material failure and fatigue life are determined as the time at which the calculated phase angle begins to drop (Reese, 1997). Obviously, this occurrence invalidates the extension of the VECD theory to include the material's status with the presence of macrocracks.

In earlier practice, the specimen geometry of 75 mm in diameter and 150 mm in height was used with a strain-measuring gauge length of 100 mm in the middle along the axis of the specimen. However, it had been observed that the chances were good that the specimen would fail near the ends beyond the gauge points. Such end failure could cause an inaccurate measure of the deformation, and more importantly, the true fatigue life could not be determined. Moreover, the damage characteristic curves generated from tests with end failure usually terminated at lower S-values, because higher levels of damage occurred at locations beyond the measurable range. Hence, for complete material characterization, middle failure is required to capture the material's macroscopic status from its intact state through failure. In order to raise the probability of middle failure, a comprehensive study of various geometric dimensions and gauge lengths was conducted by Lee et al. (2014) who recommended from finite element simulations and experimental investigations that the geometry of 100 mm in diameter and 130 mm in height should be used along with a 70 mm gauge length. Nevertheless, it is worthwhile to point out that experimental evidence also shows excellent overlap of the damage characteristic curves from both middle and end failures.

In this study, the above mentioned specimen geometry from Lee et al. (2014) was used for all the fatigue tests. For each mix, a minimum of two or three replicates was tested, depending on material availability. All tests ended with middle failure. In order to characterize the failure criterion, i.e., Eq. (12), three levels of fatigue life, N_f < 5000, 5000 < N_f < 25,000, and N_f > 25,000, were targeted by choosing the proper displacement amplitude based on trial tests or experience with similar mixes. All fatigue tests were performed at 19 ℃ and 10 Hz. The resulting damage characteristic curves were then fitted using Eq. (8), and the relationships are presented in Fig. 4.

Figure  4.  Damage characteristic curves for the asphalt mixes in Binzhou test sections.

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As previously stated, C is an indicator of material integrity while S quantifies the internal damage in the material. Fig. 4 shows that the material integrity of LSPM deteriorated quickly with damage, which is attributable to the high air void content and the open gradation of the mix. Conversely, for the F-2 mix, the material maintains its integrity even when it experiences a significant amount of damage, which can be explained by the high asphalt content and the use of SBS-modified binder. In addition, a general observation can be made by comparing all six mixes, which is that fatigue resistance usually can be enhanced by the use of smaller aggregate and modified asphalt binder. Nevertheless, a better material property does not always guarantee a better performance of the pavement, because the structural layout and the material properties in other layers should be also taken into consideration. Therefore, a fair and rigorous comparison of fatigue performance should only be possible by transferring the damage characteristic relationship obtained at the material level into a finite element structural analysis program, as is presented in the next section.

For each fatigue test, the ACDPSE is calculated via Eqs. (9) and (10), and is plotted with respect to fatigue life in Fig. 5. Besides, what is illustrated in the figure are the best fits for each material using Eq. (12). As can be observed from Fig. 5, when a given amount of energy is to be consumed, the LSPM fails during early cycles, whereas the F-2 mix is able to dissipate the energy at a slower pace so that the material can withstand much more cycles of loading before failure. The other four mixes demonstrate roughly the same fatigue resistance, with the F-1 and SMA mixes performing slightly better than the Superpave-19 and 25 mixes. Note that the line of the Superpave-25 mix lies immediately above the line for the Superpave-19 mix, indicating a marginal advantage in fatigue resistance of the Superpave-25 mix, which is contrary to the results shown in Fig. 4. This inconsistency is presumably related to the variability in the specimens, testings, and the processes of finding the turning point in the phase angle to determine the number of cycles to failure. After all, the ranking of fatigue resistance is expected to be the same using either the damage characteristic curves or the failure criterion lines. Nevertheless, the material fatigue performance should be finally assessed in the structural simulations.

Figure  5.  Relationship between ACDPSE and fatigue life for each mix in Binzhou test sections.

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5.   Fatigue performance predictions

5.1   The finite element package: LVECD program

A finite element structural analysis package has been developed and improved for pavement response and performance simulations along with the evolution of the VECD theory at NC State University (Cao et al., 2012, Park et al., 2014, Underwood et al., 2012). The latest version is called the layered viscoelastic pavement analysis for critical distresses (LVECD). In the following sections, the LVECD program inputs for the material properties, traffic, and climate data are explained, and some critical assumptions and simplifications are reviewed briefly. For details regarding the program, refer to the study by Eslaminia et al. (2012).

For this study, all of the asphalt materials were considered within the framework of linear VECD theory. The Prony coefficients, as shown in Eq. (2), were required for material stiffness, and the parameters used in Eqs. (8) and (11) were needed for predicting fatigue performance. The unbound materials were considered in the linear elastic domain. The elastic modulus was assumed as 1500 MPa for the lime fly-ash treated aggregate, 500 MPa for the lime fly-ash treated soil, 85 MPa for the 30 cm lime-stabilized subgrade, and 35 MPa for the natural subgrade. For all materials, Poisson's ratio was deemed as a constant irrespective of temperatures and loading conditions.

In the LVECD program, traffic is addressed in a number of ways. Either a wheel (single/dual), an axle (single/tandem/tridem), or a vehicle can be defined, with each tire having its own load and geometric characteristics if necessary. The traffic volume per day for the design lane is then given in terms of the total number of passes of the wheel, axle, or vehicle, whichever is defined in the previous step. In the present simulation, a single axle with dual tires is employed as the unit for traffic. To accommodate potential overloading, the axle load is set as 100 kN with tire pressure of 1.0 MPa. The number of passes of this traffic unit per day on the design lane is set at 10,000.

The temperature profile for the pavement structure is input on an hourly basis, and can be prepared using the climate information found in the enhanced integrated climate model (EICM). For simplicity, only one year temperature data are required for the LVECD program; this history is then repeated for the remaining years of the whole design life of the pavement. In this study, the climate information for the pavement site in Shandong Province was approximated by the averaged data over the state of Virginia in the United States, given the geographic similarities between the two regions.

In the LVECD program, adjacent pavement layers are assumed to be fully bound at the interface. The default setting for the meshing provides 11 and 101 nodes for the vertical and transverse directions, respectively, for each asphalt layer, which yields a total number of 1111 nodes. The mesh for the unbound elastic materials is relatively coarse. The responses of the stress, strain, and displacement at each node under the moving load of the traffic unit can be solved conveniently via Fourier transform techniques in both the time and space domains.

5.2   Simulations of fatigue performance and discussion

The fatigue performance of each section was predicted in a 15-year analysis period. As previously explained, the pseudo stiffness C is an indicator of material integrity, and thus, the C-contour can be utilized to examine and compare the performance of all the test sections, as presented in Fig. 6. Note that the control section, denoted as "Section Control" in Fig. 6(e), is a virtual section that is designed to be the same as Section 4 except for the use of the 15 cm Superpave-25 layer in place of the LSPM layer. The control section was added to the simulation only to make a parallel and thus a fair comparison with Section 4 in order to assess the potential of LSPM in resisting fatigue cracking. Also note that for all the contours, the maximum C-value is set at 0.8 instead of 1.0 with consideration of the inherent defects in virgin materials and certain adverse effects due to compaction and aging during pavement construction.

Figure  6.  C-contours of the Binzhou test sections after 15-year simulations. (a) C-contour for Section. (b) C-contour for Section. (c) C-contour for Section. (d) C-contour for Section. (e) C-contour for Section Control. (f) C-contour for Section.

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The fatigue performance of the test sections can be compared by examining the lowest C-value shown beside the colorbar for each section. The ranking among all the Binzhou sections from superior to inferior is Sections 1 Introduction, 2 Overview of Binzhou test sections, 3 Material modeling, Control, 4, and lastly Section 5. As expected, this sequence follows the general ranking of the thickness of the asphalt pavement. Section 1 is the thickest and thus demonstrated the best fatigue performance at the end of the 15-year simulation. The material damage is seen to concentrate in the layer of the Superpave-19 mix, as shown in Fig. 6(a). Owing to the use of SBS-modified binder in the bottom asphalt layer, Section 3 exhibits slightly better performance than Section 2. Another noticeable difference between these two sections is that the use of better materials in Section 3 also helps to shift the location of the damage concentration to an upper layer. In Section 2, the fatigue damage is concentrated at the bottom of the fourth layer, whereas in Section 3, the location of the major damage has shifted to the second layer. This finding is illuminating for perpetual pavement design that aims to avoid structural failure in the lower pavement layers.

For Sections 4 and Control, fatigue cracking initiated at the bottom of the asphalt pavement according to Fig. 6(d) and (e). By comparing the lowest C-values in these two sections, it appears that the utilization of the LSPM in the bottom asphalt layer would lead to a poorer design than the Superpave-25 mix in terms of fatigue cracking. Recall that in Figs. 4 and 5, at the material level the LSPM shows the least desired fatigue resistance due to its special aggregate gradation. Therefore, the above observation in the parallel structural comparison should be expected. However, Section 4 may actually perform better than the control section in the field because the placement of the LSPM immediately above the semi-rigid base may potentially reduce the distress caused by reflective cracking. Therefore, the previous conclusion drawn from the two sections is deemed valid only as far as bottom-up fatigue cracking is concerned. In the current LVECD framework, the mechanisms that underlie reflective cracking, which initiates in unbound materials, are not incorporated, and thus a truly fair comparison between Sections 4 and Control would require more advanced and comprehensive tools, and is beyond the scope of this study.

Section 5 is the conventional design of semi-rigid asphalt pavement and exhibits the worst fatigue performance at the end of 15 years. As shown in Fig. 6(f), damage is concentrated at the bottom of this thin asphalt pavement. The minimum C-value suggests that Section 5 may not be able to maintain its structural integrity for 15 years due to fatigue cracking, even if reflective cracking is not considered.

An alternative approach to fatigue performance is to investigate the percentage of material points with failure in the pavement cross-section. The failure percentage is calculated simply as the ratio of the number of material nodes with failure to the total number of finite element nodes in the cross-section. The evolution of the failure percentage with respect to the number of axle passes is plotted in Fig. 7 for each section.

Figure  7.  Evolution of failure percentages for Binzhou test sections after 15-year simulations.

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The ranking of fatigue performance according to Fig. 7 from superior to inferior is Sections 1, Control, 3, 2, 4 and 5, which is the same as for the C-based ranking except for the position of the control section. This inconsistency is attributed to the fact that a pavement with a low minimum C-value may possibly show a small area of material failure. In fact, a lower C-value may not necessarily suggest worse fatigue performance. Recall that the process of material deterioration follows the damage characteristic curve, and the C-contours shown in Fig. 6 are generated based on this principle without considering when failure would occur. Hence, the predicted C-values for the material nodes in the cross-section can decrease asymptotically to zero under repeated fatigue loading. Yet, beyond failure, the VECD theory is no longer applicable, which means that any further calculation of the pseudo stiffness is actually invalid for any material point that has already failed. Given these considerations, it is deemed that comparisons from the perspective of failure percentage are more reasonable for ranking different structural designs.

Again, it should be emphasized that all the simulations and conclusions presented here are conducted without involving the role of reflective cracking and other complexities, such as aging, healing, and shear failure. Judging purely from the pavements' fatigue performance, it is recommended that Sections 1 Introduction, 3 Material modeling can be used as ideal candidates for perpetual pavement design. All the other sections are ruled out because of either severe damage or the initiation of damage in the lower structural layers. Note that in Section 2 even though the fatigue damage concentrates in a lower layer as shown in Fig. 6(b), the predicted lowest C-value is as high as those in Sections 1 Introduction, 3 Material modeling, and therefore whether the design of Section 2 can be used as a candidate for perpetual pavement can not be concluded from the present analysis.

6.   Conclusions

This paper presents a systematic VECD framework for evaluating the fatigue performance of six (including one virtual control section) selected asphalt pavements at a pavement test site in Binzhou, China. Each asphalt mixture was treated as a linear viscoelastic continuum, and the constitutive modeling of the fatigue damage under repeated loading was conducted by identifying the damage characteristic curve for each mix. The VECD framework then was completed using the unified failure criterion that defined the applicable domain of the theories. The model parameters were determined through dynamic modulus testing and direct tension cyclic fatigue testing in the laboratory.

The obtained damage characteristic curves and failure criteria suggest a material level improvement in fatigue resistance with the use of a small aggregate particle size, dense aggregate gradation, and modified binder. All the material models were integrated in the finite element package, LVECD, which was used for the fatigue performance predictions of Binzhou perpetual pavement test sections. The 15-year fatigue simulation results indicate the recommendation that Sections 1 Introduction, 3 Material modeling should be used for perpetual pavement design.

The LVECD program provides a novel and comprehensive platform to understand and predict the long term fatigue performance of asphalt pavements. Despite the lack of field data in the current study, the material constitutive modeling and structural simulation results both conformed to the general engineering experience and expectations. Yet, for an accurate and fair evaluation of the program, a corresponding database of the field observations should be established and compared with the laboratory and simulation results. Moreover, regarding LVECD itself, on the material level, additional constitutive modeling should also be introduced, such as aging and healing, to accommodate a more realistic material property variation or deterioration in the field as far as bottom-up fatigue cracking is concerned.