A cumulative damage model for predicting and assessing raveling in asphalt pavement using an energy dissipation approach

Abstract

Raveling is a common distress of asphalt pavements, defined as the removal of stones from the pavement surface. To predict and assess raveling quantitatively, a cumulative damage model based on an energy dissipation approach has been developed at the meso level. To construct the model, a new test method, the pendulum impact test, was employed to determine the fracture energy of the stone-mastic-stone meso-unit, while digital image analysis and dynamic shear rheometer test were used to acquire the strain rate of specimens and the rheology property of mastic, respectively. Analysis of the model reveals that when the material properties remain constant, the cumulative damage is directly correlated with loading time, loading amplitude, and loading frequency. Specifically, damage increases with superimposed linear and cosine variations over time. A higher stress amplitude results in a more rapidly increasing rate of damage, while a lower load frequency leads to more severe damage within the same loading time. Moreover, an example of the application of the model has been presented, showing that the model can be utilized to estimate failure life due to raveling. The model is able to offer a theoretical foundation for the design and maintenance of anti-raveling asphalt pavements.

Graphical abstract

Keywords

asphalt pavement / raveling / cumulative damage / dissipation energy theory

1 Introduction

Raveling is a common distress observed in asphalt pavements, defined as dislodging of stones from the pavement surface under traffic loads [1,2]. Kneepkens et al. [3] described raveling as a domino-like effect, where the initial loss of a stone causes adjacent stones to be removed more easily due to the lack of support from one direction, resulting in the expansion of the raveling site and other distresses like potholes and delamination. Raveling, along with the associated distresses, can especially affect the performance of asphalt pavements in open-graded friction courses, chip seal pavement, highway ramps, and airport pavement [4–7].

Currently, two main mechanisms have been proposed to explain the development of raveling. One suggests that the stone will be removed when the stone-mastic-stone meso-unit of pavement is subjected to stress that exceeds its strength [8,9]. However, failure in nonlinear materials such as asphalt mixtures is not instantaneous, as it is a damage-cumulative process [10]. The other proposed mechanism suggests that raveling is a fracture-related phenomenon and a degradation process due to dissipation of energy from the interaction of traffic and pavement, meaning stones will be lost from the pavement when the accumulated dissipated energy of the stone-mastic-stone meso-unit reaches the total energy required for fracture [11].

To better understand the phenomenon of raveling and prevent its development, several research efforts have been made. In terms of macro-level laboratory experiment methods, several asphalt mixture experiments have been proposed for evaluating resistance to raveling. The most commonly used test is the Cantabro loss method, which is also utilized to design the optimum asphalt content for porous asphalt pavements [12–14]. Additionally, researchers in Europe have proposed other test methods such as Aachener raveling tester, Darmstadt Scuffing Device, Rotating Surface Abrasion Test, Tribo Route Device, etc. [15,16]. These tests mainly use scuffing machines to simulate the repeated abrasive effect of vehicle tires on asphalt pavement, and the resistance to raveling is quantified by measuring the mass loss of specimens before and after the tests. Despite the widespread use of these test methods, studies have shown that results of these asphalt mixture tests at the macro level cannot establish a direct connection with actual road performance, making it difficult to provide an effective basis for material design [15,17]. Consequently, studies of raveling need to be conducted at a meso-level scale. In light of this, the binder bond strength (BBS) test has been used by many researchers to test the pull-off strength of the asphalt-stone interface Mishra and Singh [18], Asif and Ahmad [19] evaluated the factors influencing the asphalt-stone interfacial strength by BBS test and made some recommendations for pavement material design Guo et al. [20] compared the asphalt-stone interfacial property under dry and wet conditions. However, the load mode for the BBS test is tensile at a constant loading rate, quite different from the load mode of traffic, which is dynamic impact load. Therefore, results of the BBS test may differ significantly from the actual road performance.

Numerical simulations have also been employed to understand the process of raveling from a mesostructure perspective, and these have significantly improved the effectiveness and accuracy of such research. Researchers [21–24] analyzed the stress and strain response of the stone-mastic-stone meso-unit of chip seals and friction courses with the help of the finite element method and studied the effects of temperature, stone spread rates, and stone shape on the response stresses when subjected to traffic loads. However, these studies were mainly based on qualitative analysis of raveling susceptibility and lacked a criterion for quantitative assessment. To address this drawback, Manrique-Sanchez et al. [25] proposed a raveling sensitivity index R.I., calculated by the ratio of the dissipated energy after a wheel passes over the pavement to the total energy required for fracture at a stone-on-stone contact element. While this research made significant progress in the quantitative evaluation of raveling, it only considered a single vehicle load and failed to predict the effect of repeated traffic loads during the road service life. As raveling results from the accumulation of damage, it is very necessary to consider the effect of repeated traffic loads over a long-term period. In addition, the total fracture energy used to calculate the R.I. index was calculated using surface energy theory, which has been proven to be several orders of magnitude away from the actual fracture energy [26]. Given this, the effectiveness of this index needs to be further discussed. In summary, an effective approach to obtaining the fracture energy at stone-on-stone contact and a criterion to quantitatively evaluate and predict raveling over a long-term period is quite lacking.

The law of cumulative damage has been applied a lot in previous studies as a criterion for assessment of the fatigue life of materials. Alrayes et al. [27] used this approach to modeling the cyclic crack propagation in concrete, which provided a further understanding of crack growth and damage accumulation. Maggiore et al. [28] and Zhang and Oeser [29] employed the cumulative damage law to characterize the fatigue life and understand the fatigue behavior of asphalt mixtures. As mentioned above, raveling can be considered a degradation process of energy dissipation, indicating that this law is also adaptable for characterizing raveling.

To address the above issues, this paper aims to construct a cumulative damage model of raveling at a meso-level scale to provide a criterion for the material design of anti-raveling pavements. To achieve this, a new test method, the Pendulum Impact Test, was proposed to determine the actual energy required for fracture of the stone-mastic-stone meso-unit. In addition, a high-speed camera was utilized to capture the displacement rates of specimens during the test, which was used to form a relationship between strain rate and fracture energy. Furthermore, based on the assumption that the damage was due to the deformation and failure of the mastic between two stones, the dynamic shear rheometer (DSR) test was used to obtain the viscoelastic parameter of mastic, and this was incorporated in the developed model. The prediction model of raveling was constructed based on the energy dissipation approach, and the further behavior of the model was analyzed.

2 Cumulative damage based on energy dissipation theory

For viscoelastic materials, the total work of deformation per unit volume of material results can be given by Refs. [30–32]

(1)W=U_{\mathrm{s}}+U_{\mathrm{d}},

where U_{\mathrm{s}} is the stored energy per unit volume, which is contributed by its elastic property; U_{\mathrm{d}} is the dissipated energy per unit volume due to viscous effect. The relation of total input energy, storage energy, and dissipated energy for the whole process of damage is illustrated in Fig.1.

It has been shown in previous work that energy dissipation results from material damage, and the two are closely related [33]. Therefore, the cumulative damage factor used for describing the degree of cumulative damage can be defined as Jin et al. [34]

(2)D=U_{\mathrm{d}}(t)/U_{\mathrm{f}},

where U_{\mathrm{d}}(t) is the cumulated dissipated energy per unit volume after the action of the load for t seconds; U_{\mathrm{f}} is the total energy required for fracture per unit volume under the corresponding load level. When the load is periodic and the action time is a multiple of the period time, it can also be expressed as

(3)D=U_n/U_{\mathrm{f}},

where U_n is the cumulated dissipated energy per unit volume after the action of the load for n cycles.

The cumulative damage factor of the stone-mastic-stone meso-unit was taken as the assessment index for the cumulative damage model of raveling, with a higher value indicating more severe damage. When D\ge 1, the cumulated dissipated energy exceeds the total fracture energy indicating that structural failure has happened in the stone-mastic-stone meso-unit and the distress of raveling has occurred.

In this paper, U_{\mathrm{d}}(t), U_n, and U_{\mathrm{f}} at stone-on-stone contact were expressed by functions based on strain rate derived from the laboratory experiments described in the next section.

3 Materials and experimental methods

3.1 Materials

Styrene butadiene styrene (SBS) asphalt was used to prepare the mastic. Limestone mineral filler was added to the asphalt with a filler/asphalt ratio of 0.8. The properties of asphalt and limestone mineral filler are listed in Tab.1 and Tab.2, respectively. The rheological property of the SBS mastic was tested by a DSR, Malvern Kinexus Laboratory + The complex shear modulus G^* and phase angle δ of SBS mastic were fitted to the Continuous aggregate model (CAM) model [35], expressed as Eqs. (4) and (5):

(4)G^{\ast }=\frac{G_{\mathrm{\infty }}}{{\left[1+{\left(2\text{π}{f}_{\mathrm{c}}/{\alpha }_{\mathrm{T}}\omega \right)}^k\right]}^{m_{\mathrm{e}}/k}},

(5)\delta =\frac{90m_{\mathrm{e}}}{1+{\left(2\text{π}{f}_{\mathrm{c}}/\omega \right)}^k} ,

where \omega is the angular frequency (rad/s); f_{\mathrm{c}} is a location parameter where loss modulus equals storage modulus; {\alpha }_{\mathrm{T}} is horizontal shift factor at a given temperature T; G_{\mathrm{\infty }} is the glassy shear modulus when frequency tends to infinity; k and m_{\mathrm{e}} are dimensionless shape parameters. The horizontal shift factor in time-temperature superposition, {\alpha }_{\mathrm{T}}, can be derived according to the Williams−Landel−Ferry (WLF) formulation [36]:

(6)\log {\alpha }_{\mathrm{T}}( T)=- \frac{c_1\left(T-T_0\right)}{c_2+\left(T-T_0\right)} ,

where T_0 is the reference temperature, 30 °C is selected in this paper; c_1 and c_2 are constants.

The master curves of complex shear are illustrated in Fig.2, and the parameters are presented in Tab.3.

Four types of stones were used: Granite, diabase, limestone, and basalt. The stones were cut into cube shapes with a side length of (11 ± 0.5) mm, and surface polishing was performed to avoid variability caused by texture.

3.2 Pendulum impact test

3.2.1 Test instruments and specimens

A new experimental method, known as the Pendulum Impact Test, was used in this study to evaluate the fracture energy of stone-on-stone contact. This test was developed by the Institute of Road Research, South China University of Technology (Patent No. ZL201820282945.0) and is shown in Fig.3. The Pendulum Impact Tester consists of 1) impacting system; 2) temperature-controlling system; and 3) measuring system. The impacting system has a pendulum to apply an impact load and a base to hold the specimen in place. The temperature-controlling system is connected to the base to maintain a stable temperature throughout the experiment, ranging from −10 to 80 °C. The measuring system obtains the fracture energy absorbed in the fracture process

The diagram of the 4) specimen is also shown in Fig.3, which consists of two stones connected by mastic. During the test, the lower stone was firmly secured by the base, while the upper stone was impacted by the pendulum from a certain height. The specimen was damaged at the mastic between the two stones. The fracture energy was calculated by the difference in the height of the pendulum before and after impact, based on the energy conservation principle, which can be expressed as Eq. (7).

(7)E_{\mathrm{f}}=mgr\left(\alpha -{\alpha }^{\prime }\right),

where E_{\mathrm{f}} is the fracture energy of the specimen; m is the mass of the pendulum; g is the acceleration due to gravity; r is the radius of rotation of the pendulum; α and α' are the angles between the arm of the pendulum and the vertical plane where the center of the specimen is located, respectively before and after impact.

The details of specimen types and test plans are presented in Tab.4. All types of specimens were tested at −10, 0, 10, 20, 30, 40, 50, 60, 70, and 80 °C, respectively, and each type of specimen was tested three times.

3.2.2 Test specimen preparation

The specimen for the Pendulum Impact Test was composed of two identical stones bonded with mastic. The stones used in this study were polished to ensure consistent roughness. An adapted Spiral Micrometer equipped with a metal groove was used to adjust the thickness of the mastic between two stones during the fabrication process of specimens. As shown in Fig.4, two stones were put into the groove to form the specimen, and the total thickness of the two stones was initially measured. Subsequently, mastic was applied to the contact surface between the two stones, and the bonded stones were put into the groove again, and compressed until the desired thickness of 0.2 mm of mastic was achieved. Finally, the excess mastic was cut off after cooling.

3.3 Digital image method

A high-speed camera was utilized to capture the specimen destruction process during the Pendulum Impact Test at 2000 frames per second, as illustrated in Fig.5. During the test, the lower stone of the specimen was fixed, while the upper stone was impacted by the pendulum laterally. The videos of the fracture process of specimens were captured by the high-speed camera The positional information of the center of the upper stone at each frame of the videos was captured by an algorithm written in Python language, which could be used to calculate the horizontal deformation of the specimen. The deformation of the specimen was regarded as the deformation of the mastic between stones since the stones were considered rigid. Additionally, the time between each frame was known according to the setting of the high-speed camera. Consequently, the horizontal displacement rate could be calculated from the displacement and the time. The flowchart of the algorithm is shown in Fig.6. The shear strain rate of the specimen was then determined by dividing the displacement rate by the dimension of the specimen.

3.4 Frequency sweep test

Since fracture occurs in the mastic between the two stones, the rheology property of mastic is required to construct the raveling prediction model. To obtain the relaxation spectrum of mastic, frequency sweep tests were conducted using a DSR, Malvern Kinexus Laboratory +, produced by Malvern Analytical Company. The frequency tests ranged from 0.1 to 10 Hz and were conducted on each sample at six different temperatures (10, 20, 30, 40, 50, 60 °C) with a consistently small strain amplitude of 0.01% within the linear viscoelastic region. The generalized Maxwell model was then employed to determine the relaxation spectrum of mastic.

4 Test results

4.1 Fracture energy

The fracture energies of stone-mastic-stone specimens at different test temperatures and their average value are demonstrated in Fig.7.

It can be seen that there was little difference among the test results of specimens fabricated by different stones. To analyze the differences among the test data of four types of specimens (SG, SD, SL, and SB) at every test temperature, an Analysis of Variance (ANOVA) test was applied, and the results are shown in Tab.5. A significance level of 0.05 was applied, meaning that if the value of significance was greater than 0.05, there were considered to be no significant differences among the data of four types of specimens [37,38]. As demonstrated in Tab.5, at all test temperatures, the values of significance are greater than 0.05. Consequently, it can be concluded that the type of stone had no significant effect on fracture energy. Several previous studies also researched the impact of stone types on the asphalt-stone interface Chu et al. [39] studied the impact of stone surface anisotropy on the asphalt-stone interface by molecular dynamics simulation, and the results showed that the stones with alkaline molecular had a stronger connection with asphalt than the stones with acid molecular . However, according to the study results by Khasawneh et al. [40] the granite, an acidic rock, had a better bonding strength with asphalt than the marble, which is alkaline, contrary to the expected result. The potential reason is that the distribution of the stone composition is random and complex, leading to uncertainty in the composition of the stone surface that is in contact with asphalt. Fischer et al. [41] also found that the surface roughness and porosity play more important roles than the chemical properties in the asphalt-stone interface. Consequently, the effect of stones is a combination of diverse factors, with no significant regularity.

When discussing the effect of temperature, it is apparent from the curve of average value that the fracture energy was significantly affected by temperature, as asphalt is a viscoelastic material. Initially, the interface fracture energy rose with the rising temperature and attained a peak value at a specific temperature, approximately 40 °C for SBS mastic. Subsequently, the fracture energy decreased as the test temperature continued to climb. According to the previous study [42], the failure behavior of stone-mastic-stone meso unit is related to mastic-stone adhesion and mastic cohesion. To discuss the reason for the reversal of the trend at 40 °C, it is supposed that the required energy for adhesive failure is low at the low temperature and grows with temperature, while, in contrast, the energy for cohesive failure is high at low temperature and decreases with temperature, as shown in Fig.8. The fracture energy is mainly determined by the lower one of the above two kinds of energy, since the failure will occur first in relatively weak areas. Therefore, it is expected that an inflection point will appear at the intersection of the two curves, which is located at 40 °C for SBS mastic. Although it is hard to test the specific value of the energy for adhesive and cohesive failure at different temperatures with current laboratory equipment, this supposition is consistent with the change of the curve of fracture energy and the experiment phenomenon.

4.2 Shear strain rate

As previously mentioned, a high-speed camera was utilized to capture the displacement rate of specimens during the Pendulum Impact Test. After that, the shear strain rate of specimens was computed from the displacement rate and the dimension of the specimen in the load direction and is plotted in Fig.9.

The NOVA test was also applied to analyze the impact of stone types on the shear strain rate, as shown in Tab.6. The results indicated no significant variance in shear strain rate across the four different types of specimens (SG, SD, SL, and SB) at the same temperature, which was consistent with the findings of the fracture energy. It suggested that stone type had no significant effect on either shear strain rate or fracture energy. Accordingly, the four stone types were not distinguished in the subsequent model construction process.

The curve of average value in Fig.9 revealed that the variation of shear strain rate demonstrated an opposite trend to that of fracture energy, with respect to temperature. In particular, the shear strain rate decreased initially with increasing temperature before reaching a minimum value of approximately 40 °C, after which it increased gradually.

4.3 Relaxation spectrum of mastic

The relaxation shear modulus in the time domain for mastic can be expressed as follows:

(8)G(t)=G_{\mathrm{\infty }}+\sum _{i=1}^n{G}_i{\text{e}}^{-t/{\tau }_i},

where G_{\mathrm{\infty }}, G_i, and {\tau }_i are infinite relaxation shear modulus, Prony coefficients, and relaxation time, respectively. The shear storage modulus G^{\prime } and shear loss modulus G^{″} are expressed as follows [43]:

(9)G^{\prime }(\omega )=G_{\mathrm{\infty }}+ \sum _{i=1}^n\frac{G_i{\omega }^2{\tau }_i^2}{{\omega }^2{\tau }_i^2+1},

(10)G^{″}(\omega )=G_{\mathrm{\infty }}+ \sum _{i=1}^n\frac{G_i\omega {\tau }_i}{{\omega }^2{\tau }_i^2+1},

where \omega is the angular frequency.

Based on the data obtained from the frequency sweep test, the shear storage modulus G^{\prime } at 30 °C was fitted to Eq. (10) to obtain the Prony coefficients, which are presented in Tab.7. The relaxation shear modulus according to Eq. (9) is shown in Fig.10.

5 Construction of cumulative damage model of raveling

According to Eq. (3), the key point of building a cumulative damage model of raveling was to get the expressions for total energy required for fracture per unit volume (represented by U_{\mathrm{f}}) and cumulative dissipated energy per unit volume after n cycles of load (represented by U_{\mathrm{n}} in the following sections) at a stone-on-stone contact, which are discussed in the following Subsection 5.1.

5.1 Relation between U_{\mathrm{f}} and strain rate

The U_{\mathrm{f}} at stone-on-stone contact was calculated by dividing the fracture energy obtained from the Pendulum Impact Test by the mastic volume of the specimen, while the shear strain rate (represented by {\gamma }^{\ast }) was obtained using the digital image method as presented previously.

As illustrated in Fig.11, a strong correlation was observed between U_{\mathrm{f}} and {\gamma }^{\ast } for each specimen, which was fitted to Eq. (11) at different temperatures.

(11)U_{\mathrm{f}}=a\exp \left(b{\gamma }^{\ast }\right),

where a and b are the fitting parameters. The fitting parameters, R-squared (R²), and p-value of the parameters at different temperatures are presented in Tab.8.

The curves in Fig.11 present a goodness of fit, as all the values of R-squared (R²) in Tab.8 are not less than 0.89. The p-value of the parameters shown in Tab.8 represents the confidence level of the fitted parameters. It is considered that a p-value less than 0.05 means a confidence level greater than 95% in statistics [44]. It can be observed that all p-values are less than 0.05, demonstrating the model’s reliability.

5.2 Expression of U_n after n loading cycles

U_n was calculated by rheological parameters of mastic as follows. The relation between complex shear compliance and complex shear modulus is:

(12)J^{\ast }(i\omega )G^{\ast }(i\omega )=1,

Therefore, shear storage compliance J^{\prime } and shear loss compliance J^{″} can be expressed as

(13)J^{\prime }(\omega )=\frac{G^{\prime }(\omega )}{G^{\prime 2}(\omega )+G^{″2}(\omega )},

(14)J^{″}(\omega )=\frac{-G^{″}(\omega )}{G^{\prime 2}(\omega )+G^{″2}(\omega )} .

While periodic shear stress \tau (t)={\tau }_0\cos \omega t is applied, the real part of responding shear strain can be expressed as

(15)\begin{array}{rl}\gamma (t)=& \mathrm{R}\mathrm{e}\left[J^{\ast }(i\omega )\tau (t)\right]\\ \\ =& J^{\prime }\left(\overline{\tau }+{\tau }_0\cos \omega t\right)-J^{″}{\tau }_0\sin \omega t,\end{array}

Then, the shear strain rate can be obtained by differentiating the Eq. (15), giving

(16){\gamma }^{\ast }(t)=- {\tau }_0\omega \left(J^{\prime }\sin \omega t+J^{″}\cos \omega t\right),

Therefore, the input energy per unit volume can be calculated by

(17)U(t)={\int }_0^t\tau \text{(}\xi \text{)d}\gamma \text{(}\xi \text{)}= {\int }_0^t\tau \text{(}\xi \text{)}\frac{\text{d}\gamma \text{(}\xi \text{)}}{\text{d}\xi }\text{d}\xi =U_{\text{s}}+U_{\text{d}},

where U_{\mathrm{s}} is the storage energy and U_{\mathrm{d}} is the dissipated energy. U_{\mathrm{d}} can be expressed by

(18)U_{\mathrm{d}}( t)=- \frac{1}{4}{\tau }_0^2{J}^{″}(2 \omega t+\sin 2\omega t),

Thus, the dissipated energy after n cycles of load can be calculated by

(19)U_{\mathrm{n}}=U_{\mathrm{d}}\left(\frac{2n\text{π}}{\omega }\right)=-n\text{π}{\tau }_0^2{J}^{″}.

5.3 Function of cumulative damage factor

The average shear strain rate within a load cycle was taken as the shear strain rate of mastic between two stones, which was calculated by

(20)\begin{array}{rl}{\overline{\gamma }}^{\ast }=& \frac{1}{T}{\int }_0^T\left|{\gamma }^{\ast }(t)\right|\text{d}t\\ =& \frac{\omega }{\text{2π}}{\int }_0^{\frac{\text{2π}}{\omega }}\left|-{\tau }_0\omega \left(J^{\mathrm{\prime }}\text{sin}\omega t+J^{″}\text{cos}\omega t\right)\right|\text{d}t\\ =& \frac{2}{\text{π}}{\tau }_0\omega \sqrt{J^{\mathrm{\prime }2}+J^{\mathrm{\prime }\mathrm{\prime }2}},\end{array}

Therefore, according to Eq. (11), the total energy for fracture can be calculated as follows:

(21)U_f=a\exp \left[b\left(\frac{2}{\text{π}}{\tau }_0\omega \sqrt{J^{\prime 2}+J^{″2}}\right)\right],

Thus, by combining this with Eq. (18), the cumulative damage factor at any time can be expressed as follows:

(22)D(t)=\frac{U_{\mathrm{d}}(t)}{U_{\mathrm{f}}}=\frac{-{\displaystyle \frac{1}{4}}{\tau }_0^2{J}^{″}(2\omega t+\sin 2\omega t)}{a\exp \left[b\left({\displaystyle \frac{2}{\text{π}}}{\tau }_0\omega \sqrt{J^{\prime 2}+J^{″2}}\right)\right]} ,

To determine the cumulative damage factor after n cycles of load (D_n), the loading time t is assumed to be 2\text{π}n/\omega. The expression is as follows:

(23)D_n=\frac{U_n}{U_{\mathrm{f}}}=\frac{-n\text{π}{\tau }_0^2{J}^{″}}{a\exp \left[b\left({\displaystyle \frac{2}{\text{π}}}{\tau }_0\omega \sqrt{J^{\prime 2}+J^{″2}}\right)\right]} .

6 Discussion and application of cumulative damage model

6.1 Discussion of cumulative damage model

It can be seen from Eq. (23) that when the material properties are constant, the cumulative damage is mainly related to the loading frequency, loading time, and load amplitude. To study the variation law of the cumulative damage with the factors mentioned above, the cumulative model at 30 °C was taken as an example. The fitting parameters (a and b), shear storage compliance J^{\prime } and shear loss compliance J^{″} at 30 °C were brought into Eq. (23) to calculate the cumulative damage factor.

To discuss the variation law of cumulative damage with loading time and load amplitude, an angular frequency \omega of 1 rad/s was specified. The cumulative damage factor was then calculated for four different stress amplitudes, which were 200, 400, 600, and 800 kPa. As shown in Fig.12, all the curves displayed a linear variation in general and a cosine variation locally. It can be seen from Fig.12 that the higher the shear stress amplitude, the more pronounced the cosine fluctuation is. In addition, the cumulative damage factor can be seen to increase more rapidly with greater stress amplitude, resulting in a higher probability of raveling.

The relationship between cumulative damage and angular frequencies (\omega) was also studied. Load cycle n was taken as 10, while the shear stress amplitudes were once again taken as 200, 400, 600, and 800 kPa. When \omega = 0, cumulative damage factor D also equals 0 since the shear loss compliance J^{″} is 0. While \omega is infinitesimally close to zero, as depicted in Fig.13, the cumulated damage factor can be seen to approach infinity. It can be also seen from Fig.13 that as the loading angular frequency \omega increased, the value of D initially decreased rapidly and then gradually converged to a constant value. This suggested a lower load frequency would lead to more severe damage in the same loading time. However, when the load frequency was high enough, it had little impact on the cumulative damage.

6.2 Example of cumulative damage model application

According to existing research, a tire with a load of 17.8 kN and inflation pressure of 966 kPa will approximately generate a max shear stress of 474.3 kPa on asphalt pavement and a tire-road contact length of 15.3 cm [45,46]. If it is assumed that the vehicle speed is 80 km/h, and then the load angle frequency \omega is about 911.4 rad/s, which is calculated by Eqs. (25) and (26) [47], from which:

(24)t=L/\left(17.6V_{\mathrm{s}}\right),

(25)\omega =2\text{π}/t,

where L is tire-road contact length (in), V_{\mathrm{s}} is the operating speed (m/h), and t is the time of load (s).

The stone-mastic-stone meso-unit was assumed to be subject to the given load conditions. The cumulative damage curve, as calculated from Eq. (22), is plotted in Fig.14. It can be seen that the cumulative damage factor exhibited a linear increase with a superimposed sinusoidal oscillation, with a fitted slope of 5.826 × 10⁻⁵ s⁻¹. The cumulative damage after one loading cycle can be determined from Eq. (23) to be 4.02 × 10⁻⁷. Consequently, fracture failure is expected to occur in the stone-mastic-stone meso-unit after 17164 cycles of loading.

It is worth noting that in order to intuitively illustrate how the model works, the example above was calculated under an assumed load. In practical application, other stress analysis methods like the finite element method, should first be employed to obtain the stress response of the stone-mastic-stone meso-unit under the traffic load, after which the cumulative model can be used to predict the degree of the damage and the progression of raveling.

7 Conclusions and outlook

To quantitatively predict the occurrence and progression of raveling, this study has demonstrated the construction of a cumulative damage model of of the stone-mastic-stone meso unit using an energy dissipation approach and has analyzed the behavior of the model. The following conclusions can be drawn.

1) When the material properties are constant, the cumulative damage of the stone-mastic-stone meso unit is closely related to loading frequency, loading time, and load amplitude.

2) The cumulative damage of the stone-mastic-stone meso-unit displayed an increasing trend of superimposed linear and cosine variation with loading time, and the cosine fluctuation was more noticeable for higher load amplitude.

3) The cumulative damage factor increased more rapidly with greater stress amplitude, increasing the probability of raveling.

4) A lower load frequency led to more severe damage in the stone-mastic-stone meso-unit within the same loading time. However, while the load frequency was high enough, it had little effect on the cumulative damage.

5) The cumulative damage model of raveling can be utilized to estimate the failure life of raveling, and can offer a theoretical basis for predicting and accessing raveling.

Hopefully, this research can provide a foundation for the design and maintenance of anti-raveling asphalt pavements. It’s also noteworthy that the cumulative damage model in this paper was constructed for SBS asphalt, and models based on other types of asphalt can also be established using the proposed approach. Additionally, this is a preliminary model wherein the impacts of aging, water, and F/A ratio of mastic, etc., were not taken into account for the time being; these factors are expected to be studied in the future.

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