2.1 Asphalt Binder

A conventional asphalt binder with the following properties was used: penetration of 55/10 mm (^{ASTM D5 2013}); softening point at 50.2°C (^{ASTM D36 2014}); apparent viscosity of 4.12 poise at 135°C, 2.07 poise at 150°C and 0.85 poise at 175°C (^{ASTM D4402 2015}); temperature susceptibility index of -0.92 (Pfeiffer and Van Doormaal); and Performance Grade (PG) of 64-22 according to (Table 3) of the standard (^{AASHTO M320 2017}).

2.2 Mineral Aggregates

Mineral aggregates have a basaltic origin and the following properties: 100%/100% coarse aggregate angularity (^{ASTM D5821 2017}); 49.18% fine aggregate angularity (^{ASTM C1252 2017}); 9.59 flat and elongated particles (^{ASTM D4791 2019}), 61.2% clay clods (^{ASTM T176 2017}); 11.58% hardness (^{ASTM C131 2014}); 2.14 soundness (^{ASTM C88 2018}); absence of poor-quality particles (^{AASHTO T112 2017}); 2.953 g/cm³, 2.880 g/cm³, 0.849% of effective specific gravity, apparent specific gravity, and coarse aggregate absorption, respectively (^{ASTM C127 2015}); 2.974 g/cm³ of effective specific gravity of fine aggregate (^{DNER-ME 084 1995}) and 2.804 g/cm³ of effective specific gravity of the filler (^{DNER-ME 085 1994}).

2.3 Gradation Curve

The definition of the gradation curve was based on the Superpave method criteria for a nominal maximum size of 19 mm. Thus, the grading curve was made of 56.7% coarse aggregate (passing the –” sieve and retained on No. 4 sieve), 37.9 fine aggregate (passing through No. 4 sieve and retained on No. 200 sieve) and 5.4% filler (passing through No. 200 sieve and retained at the bottom). The filler composition was 3.9% stone dust and 1.5% hydrated lime. The grading composition is shown in (Figure 1).

Figure 1 Grading composition of the aggregate mixture 

2.4 Design of the Asphalt Mixture

The design of the asphalt mixture was carried out in the Superpave gyratory compactor for heavy traffic volume (N_initial= 9 rotations; N_project= 125 rotations; N_maximum= 205 rotations). The adopted design procedures followed the recommendations of the standards (^{AASHTO M323 2017}) and (^{AASHTO R35 2017}). The following parameters were established: rotation angle of 1.25°, vertical stress of 600 kPa and rotational speed of 30 rpm. The optimum binder content was based on the following criteria: air volume (Va) in N_initial > 11%, in N_project = 4% and in N_maximum > 2%; voids in mineral aggregate (VMA) ≥ 13%; voids filled with asphalt (VFA) between 65% and 75%; and filler/asphalt ratio between 0.8 and 1.6. The designed mixture was subjected to the modified Lottman, moisture-induced damage test (^{AASHTO T283 2018}).

The designed mixture had the following characteristics: binder content of 4.45%, Va of 4.01%, VMA of 14.12%, VFA of 71.57% and filler/asphalt ratio of 1.37, following all the Superpave method criteria. The results of the moisture-induced damage test showed a 5% reduction of the tensile strength, a value that is below the 30% limit set by the standard.

2.5 Production of Prismatic Specimens

Once the mixture was designed in the laboratory, the compaction of the asphalt mix plates was carried out, which produced prismatic specimens with the following dimensions: 38.1 cm x 6.35 cm x 5.08 cm. The plate compaction was done in the compactor of the LCPC (Laboratoire Central des Ponts et Chaussées), currently called IFSTTAR (Institut Français des Sciences et Technologies des Transports, de l’Aménagement et des Réseaux). The compaction process was based on the French standard (^{AFNOR NF P 98-250-2 1991}). Two plates of 60 cm x 40 cm x 9 cm were compacted; five prismatic specimens were cut out from each plate with a circular saw. The (Figure 2) shows the different stages of the specimens’ compaction and cutting process.

Figure 2 Specimen production: a) compaction; b) compacted plate; c) plate cutting; and d) prismatic specimens 

3.1 Determination of the Fatigue Strength of the Asphalt Mixture

This stage first determined the dynamic modulus of the asphalt mixture according to the recommendations of the standard (^{EN 12697-26 2018}) and, subsequently, the fatigue strength according to the standard (^{EN 12697-24 2018}); both tests were carried out in the four-point flexural fatigue machine. Eight specimens were selected out of ten, and from these eight, two were used for obtaining the dynamic modulus and six were used in the fatigue test. The dynamic modulus test applied a frequency of 10 Hz, temperature of 20°C, controlled deformation of 50 µm/m and continuous sinusoidal loading. The fatigue strength tests applied a frequency of 10 Hz, temperature of 20°C, continuous sinusoidal loading, controlled deformation and cracking criterion equal to 80% reduction of the initial modulus (considering that the initial modulus was established in the hundredth cycle), with the aim of enabling the application of different analysis approaches and failure criteria to the data obtained during the test.

3.2 Development of Fatigue Models

Following the obtainment of the fatigue test results, two fatigue cracking criteria were applied to the data: the conventional criterion and the Pronk & Hopman criterion. The conventional failure criterion defines the failure as the moment when the modulus of the material is reduced to 50% of its initial value. As for the Pronk & Hopman criterion, the dissipated energy rate is plotted based on the number of loading cycles; the failure point is equal to the number of cycles needed for the dissipated energy rate to deviate from a straight line tangent to the first section of the curve. According to the Pronk & Hopman criterion, the dissipated energy rate can be calculated with (Equation 1). The failure criteria are shown in (Figure 3).

(1)

Where: ER = the dissipated energy rate in cycle n = number of loading cycles; w_n = dissipated energy in cycle n; w_o = initial dissipated energy (50^th cycle); ε_o = initial deformation; σ_o = initial tension; δ_o = initial phase angle; ε_n = deformation for cycle n; σ_n = tension for cycle n; δ_n = phase angle for cycle n.

Figure 3 Representation of the conventional and Pronk & Hopman failure criteria 

After applying the failure criteria, fatigue models were developed for each criterion, with both the phenomenological and cumulative dissipated energy analysis approaches. In the phenomenological approach, the fatigue model was obtained by relating the microstrains to the number of cycles to failure, that is, by means of the Wöhler curve (Equation 2).

(2)

Where: N_f = number of cycles to failure; ε_t = tensile microstrain; k₁ and k₂ = constants.

Concerning the cumulative dissipated energy approach, the fatigue model was developed considering the relationship between the cumulative dissipated energy and the number of cycles, both at the time of fatigue cracking; the model is shown in (Equation 3).

(3)

Where: W_cu = cumulative dissipated energy in the crack; N_f = number of cycles to failure; Z y A = constants.

3.3 Fatigue Life Estimate of the Asphalt Wearing Course in the Pavement Structure

The final stage of the study estimated the fatigue life of the asphalt wearing course in a pavement structure, using the obtained fatigue models. The objective of this stage was to show how different analysis methods and failure criteria can affect the life cycle estimate of the asphalt wearing course subjected to dynamic loads.

With this purpose in mind, a numerical simulation using the ViscoRoute software (developed by IFSTTAR/France) was carried out, which considers the viscoelastic parameters of the asphalt wearing course. The software also includes the input data of loading speed and asphalt mixture temperature. The simulation considered a standard two-wheel axle of 8.2 tons moving at a speed of 72 km/h, a speed equivalent to a testing frequency of 10 Hz (^{Chabot et al. 2010}), with a distance of 32 cm between the wheels, load per wheel of 2050 kgf, circular contact area of 366 cm² between the wheel and the pavement, and contact pressure of 5.6 kgf/cm². A temperature of 20°C was considered for the asphalt wearing course, which was the same temperature used in the fatigue strength and dynamic modulus tests. The following characteristics were defined for the pavement structure: 18 cm asphalt wearing course, 18 cm base, 20 cm subbase, and infinite subgrade. The (Table 1) indicates the parameters used in the numerical simulation.

Table 1 Pavement structure characteristics 

The output data of the ViscoRoute software was the tensile microstrain in the lower face of the asphalt wearing course at 20°C. Subsequently, the fatigue models were used to confirm the number of cycles needed for the 8.2 ton axle to cause the fatigue cracking of the asphalt wearing course, considering the different analysis approaches and failure criteria.

With regard to the phenomenological approach, it was necessary to feed the models with the tensile microstrain obtained in the ViscoRoute software in order to estimate the fatigue cracking of the asphalt wearing course. As for the cumulative dissipated energy approach, the procedure was done differently and followed the four stages described below:

Stage 1: Development, based on fatigue strength laboratory results, of the mathematical model relating the initial dissipated energy (W_i) with the cumulative dissipated energy (W_cu) at the time of cracking (one model per each criterion).

Stage 2: Determination, through (Equation 4), of the initial dissipated energy (W_i) produced in the asphalt wearing course at 20°C by the passing of the standard two-wheel axle of 8.2 tons at a speed of 72 km/h (10 Hz).

(4)

Where: W_i = initial dissipated energy; ε_t = tensile microstrain obtained by the ViscoRoute software; |E*| = dynamic modulus of the asphalt mixture at 20°C and 10 Hz; δ_i = phase angle of the asphalt mixture at 20°C and 10 Hz. Where, |E*| and δi were obtained in the dynamic modulus test of the asphalt mixture, see (Table 1).

Stage 3: Prediction of the cumulative dissipated energy (Wcu) at the time of cracking of the asphalt wearing course, by using the initial dissipated energy (Wi) calculated in Stage 2 and the models developed in Stage 1.

Stage 4: Estimate of the number of cycles that the 8.2 ton axle should apply to the asphalt wearing course in order to cause fatigue failure. The estimate was possible by introducing the cumulative dissipated energy (W_cu), obtained in Stage 3, in the fatigue models of the cumulative dissipated energy approach.

4.1 Determination of the Fatigue Strength of the Asphalt Mixture

The four-point bending machine was used for the fatigue strength test, following the recommendations of the standard (EN 12697-24 2018). In total, six specimens were tested to failure. The (Table 2) indicates the number of cycles (N_f) needed to reach the assessed failure criterion and the respective reduction of the initial modulus.

Table 2 Number of cycles and reduction of the initial modulus fo the different criteria 

In (Table 2) it is possible to verify that all cycle numbers determined by the Pronk & Hopman criterion are more conservative than those determined by the conventional criterion. The Pronk & Hopman criterion shows, on average, a 59.2% reduction of the number of cycles needed to reach the fatigue failure, compared with the conventional criterion; that is, this criterion considers that the asphalt material is less resistant to the fatigue phenomenon. In relation to the reduction percentage from the initial modulus, the Pronk & Hopman criterion shows values below 50%, of approximately 37.8% (variance of 5.1%), which confirms the conservative nature of this criterion.

4.2 Development of Fatigue Models

The failure values obtained through the different analyzed criteria were used to build the material’s fatigue models. Two analysis approaches were evaluated: the phenomenological and the cumulative dissipated energy. Two fatigue equations were built for each approach, one for each analyzed failure criterion: conventional and Pronk & Hopman. The (Figure 4) and (Figure 5) show the resulting fatigue curves.

The (Table 3) shows the fatigue models obtained for each analysis approach and failure criterion.

Figure 4 Fatigue curve: phenomenological approach 

Figure 5 Fatigue curve: cumulative dissipated energy approach 

Table 3 Fatigue models 

Considering the phenomenological approach, both (Figure 4) and (Figure 5), as well as (Table 3), allow verifying that the Pronk & Hopman criterion produces a model that reflects a lower fatigue strength of the asphalt mixture, compared with the conventional criterion. This difference tends to decrease when the microstrain decreases and tends to increase when the microstrain increases; this behavior is characterized by the models’ inclination. In general, concerning the amplitude of the microstrains that pavements undergo in field testing, it is evident that the Pronk & Hopman criterion is more conservative than the conventional criterion in the phenomenological approach. When addressing the cumulative dissipated energy, the model generated by the Pronk & Hopman criterion shows a better performance in relation to failure for the same value of cumulative dissipated energy. Nevertheless, due to the models’ inclination, and for cumulative energy values higher than 840 MJ/m³, the conventional failure criterion tends to show a higher fatigue strength than that obtained with the Pronk & Hopman criterion. However, it is necessary to make a numerical simulation of the pavement structure in order to specifically and clearly evaluate the influence of the failure criteria and analysis methods on the life cycle estimate of the asphalt wearing course in dealing with fatigue.

4.3 Life Cycle Estimate in the Presence of Fatigue of the Asphalt Wearing Course in the Pavement Structure

In this stage, the numerical simulation helped to determine how many times the 8.2-ton axle (N_{8.2 ton}) should be applied to subject the asphalt layer to fatigue cracking; this number was determined for both analysis approaches and failure criteria. As a result of the numerical simulation, a tensile microstrain value of 70.7 µm/m was obtained in the lower face of the asphalt wearing course (20°C), caused by the passing of the standard two-wheel axle of 8.2 tons at 72 km/h (20 m/s). For this microstrain value, (Table 4) indicates the estimated life cycle of the asphalt wearing course in the presence of fatigue, according to the phenomenological approach for both failure criteria.

Table 4 Life Cycle of the Asphalt Wearing Course in the Presence of Fatigue, according to the phenomenological approach 

The results of (Table 4) allow observing that the life cycle of the asphalt wearing course, estimated according to the conventional failure criterion, is higher than that estimated by the Pronk & Hopman criterion. This could indicate that it is more conservative to consider the fatigue failure according to the dissipated energy criterion (Pronk & Hopman). In order to match the value estimated by the Pronk & Hopman criterion with the value estimated by the conventional criterion, the wearing course should be 2.29 cm thicker, which would reduce the tensile microstrain to 61.96 µm/m and equal the N_{8.2 ton} of both criteria. Regarding the Pronk & Hopman criterion, this thickness increase would correspond to a 12.72% increase above the 18-cm thickness.

With regard to the cumulative dissipated energy approach, and in order to apply the method, the mathematical model of initial dissipated energy (W_i) versus cumulative dissipated energy (W_cu) was developed for each failure criteria, based on fatigue strength laboratory results, as shown in (Figure 6).

Figure 6 Ratio of the initial dissipated energy and the cumulative dissipated energy for different failure criteria 

Next, and with the help of (Equation 4), the initial dissipated energy (W_i) caused by the passing of the 8.2-ton axle over the asphalt wearing course at 20°C, was calculated, considering a tensile microstrain of 70.7 µm/m, dynamic modulus of 7895 MPa and phase angle of 24.4°. The value of the initial dissipated energy (W_i = 0.051215 kJ/m³) allowed obtaining, for the models shown in (Figure 6), the value of the cumulative dissipated energy for the conventional criterion (W_cu = 443.7 MJ/m³) and the Pronk & Hopman criterion (W_cu = 263.7 MJ/m³); these values were used in the fatigue models of the energy approach. The (Table 5) indicates the estimated life cycle of the asphalt wearing course, in the presence of fatigue, according to the cumulative dissipated energy approach and for both failure criteria.

Table 5 Life Cycle, in the presence of fatigue, of the asphalt wearing course according to the cumulative dissipated energy approach 

According to (Table 5), and regarding the cumulative dissipated energy approach, it can be concluded that the Pronk & Hopman criterion is more conservative than the conventional criterion. In relation to the thickness of the asphalt wearing course, if the criteria’s estimated values are to match, the layer should be 20.62 cm thick when using the Pronk & Hopman criterion, which means an increase of 2.62 cm, or 14.56%, in relation to the layer thickness used for the conventional criterion.

Finally, the comparison between the phenomenological and the cumulative dissipated energy approach confirms that the fatigue cracking prediction for the asphalt wearing course is 26.10% and 22.39% higher with the energy approach than with the phenomenological approach, when comparing the conventional criterion with the Pronk & Hopman, respectively. These results reveal that the adopted failure approach and criterion will directly influence the design of the thickness of the asphalt wearing course in the pavement structure.