Aplicación de enfoques de análisis y criterios de rotura en ensayos de fatiga y su influencia en la predicción de la vida útil de la capa de rodadura asfáltica en la estructura del pavimento Use of analysis approaches and failure criteria on fatigue tests and their influence to estimate the lif

The obtaining of fatigue models of asphalt mixtures, are part of the procedures of advanced characterization of the material and are used for the design of pavements. However, there are several approaches to analysis and failure criteria that can be used. These approaches and criteria may not produce the same models of fatigue, resulting in different thicknesses of the asphalt surface in the pavement structure. This study presents the results of the influence of two approaches of analysis (phenomenological and accumulated dissipated energy) and two criteria of rupture (conventional and Pronk & Hopman) in obtaining fatigue models of the asphalt mixture, and its repercussion in the estimation of useful life of the asphalt surface in the pavement structure. The results show that the phenomenological approach is more conservative than the accumulated dissipated energy, as well as the Pronk & Hopman criterion in relation to the conventional criterion, that is, the type of approach and the criterion of rupture adopted will directly affect the final thickness of the projected asphalt surface.


Introduction
In general, asphalt pavements do not suffer abrupt or sudden cracks, but rather cumulative functional and structural deteriorations when subjected to dynamic loading. Therefore, it is important to study the origin of these pathological manifestations in order to improve the pavement's mechanical and functional performance (Bernucci et al., 2010).
The fatigue phenomenon is one of these deterioration mechanisms, which consists in subjecting a specific material to cyclic loads with low levels of stress and strain (Khalid, 2000). The repeated and alternate bending in the asphalt wearing course, caused by the dynamic loading of vehicle traffic, produces tensile and compressive stresses in the material, thereby fostering the nucleation and growth of microcracking on the inside that compromise the life cycle as the number of load applications increases (Melo, 2014).
In this context, the development of asphalt mixture fatigue models allows predicting the performance of the asphalt wearing course in the pavement structure, in the presence of fatigue cracking; consequently, it offers a more reliable design of the necessary thickness of the layer, based on the traffic volume and the project horizon that the highway should have (Melo, 2014).
However, different tests can be used to define the fatigue strength of an asphalt mixture, which allow adopting different analysis approaches and failure criteria. Some of these tests are: the four-point bending test, two-point bending test, uniaxial tension and compression testing, indirect tensile strength test, etc. (Zeiada, 2012). With regard to analysis approaches and failure criteria, most standards (AASHTO T321, 2014); (EN 12697-24, 2018) specify the phenomenological approach and the conventional failure criterion of 50% reduction of the material's initial modulus. Nevertheless, this analysis approach and this failure criterion, although widely used in the design of fatigue models, do not provide a detailed description of the material's deterioration mechanisms (Sybilski and Bankowski, 2007). Therefore, different researchers (Ameri et al., 2017); (Shadman and Ziari, 2017); (Pasetto and Baldo, 2017); (Dondi et al., 2013) are applying more mechanistic failure approaches and criteria, among which we find those based on dissipated energy. This paper reports the results of a study addressing the influence of applying different analysis approaches and failure criteria on the asphalt layer's fatigue life and the effect on the estimated life cycle of the asphalt wearing course in the pavement structure.

Materials
In order to carry out this study, an asphalt mixture was designed at the laboratory, and ten prismatic specimens were obtained (38.1 cm x 6.35 cm x 5.08 cm) for undertaking the fatigue tests and the complex modulus in the four-point bending test. The following materials were used for the design and production of the mixture: asphalt binder, mineral aggregates and dolomitic hydrated lime CH-1, classified as Type II according to (AASHTO M303, 2014).

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 ¾" 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).

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  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.

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.

Methods
Once the specimens were obtained, the methods used in the present study followed the structure defined below:

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.

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).
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.
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).
Where: N f = number of cycles to failure; ε t = tensile microstrain; k 1 and k 2 = 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).
Where: W cu = cumulative dissipated energy in the crack; N f = number of cycles to failure; Z y A = constants.

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 cm 2 between the wheel and the pavement, and contact pressure of 5.6 kgf/cm 2 . A temperature of 20°C was considered for the asphalt wearing course, which was the same temperature used in the fatigue strength and dynamic  (Table 1) indicates the parameters used in the numerical simulation.
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:

= * (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 δ i were obtained in the dynamic modulus test of the asphalt mixture, see (Table 1).

Stage 3: Prediction of the cumulative dissipated energy (W cu ) at the time of cracking of the asphalt wearing course, by using the initial dissipated energy (W i ) 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.

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.   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/m 3 , 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.

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   The results of (  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.

Conclusions
The studies undertaken in this research confirmed that the analysis approaches and failure criteria have an influence on the life cycle prediction of the wearing course in the pavement structure, in dealing with fatigue.
It was demonstrated that the Pronk & Hopman failure criterion leads to determine a shorter life cycle of the wearing course when compared with the conventional criterion, in both studied approaches. This allows concluding that the application of the Pronk & Hopman criterion is more conservative, while resulting in thicker wearing courses. With regard to the performance of the wearing course in reaction to the two analysis approaches, it was demonstrated that the cumulative dissipated energy approach predicts a longer life cycle, in both failure criteria. Consequently, it is concluded that the energy approach is less conservative than the phenomenological one.
When the cumulative dissipated energy approach is used for estimating the life cycle of the wearing course, in the presence of fatigue, the results are approximately 24% higher than those determined by the phenomenological approach. This means that, when designing the wearing course thickness, the energy approach will present smaller thicknesses than those determined by the phenomenological approach.

Acknowledgements
The authors wish to thank the Rede Temática do Asfalto/Petrobras for the investment made in the Pavement Laboratory of the Federal University of Santa Catarina and to the University itself for the support and encouragement to academic researches.