In recent years, in search of an alternative to the methodologies described above, the linear amplitude sweep test has been proposed to evaluate fatigue in asphalt cements. This method considers evaluating the sample with an increasing deformation stress to simulate the state of solicitations that are generated in a pavement.

The results of the linear amplitude scan are applicable to the Simplified Theory of Viscoelastic Continuous Damage (S-VECD), [5,6,8] which models the internal work of the material and quantifies the damage by means of microstructural changes in the sample [9, 10]. Based on the study carried out by Schapery and Park [11], the damage accumulation law is reached, represented by eq. (2).

Where S is the damage growth in the sample, W the work done, t the time, and α the sample integrity parameter. This theory is recently applied to studies to predict fatigue in both asphalts and asphalt mixtures [4,9].

The sample integrity parameter α was previously estimated from the curve of relaxation modulus (G*(t)) as a function of time. To obtain it, a conversion of the G* module from the frequency domain to the time domain has to be carried out. This conversion is not used regularly due to the complexity involved in its application. To simplify obtaining α, it is possible to quantify it by means of the slope of the frequency sweep test, in which the sample is considered without damage [5]. Finally, it is expressed as α=1/m, where m is the slope of the frequency sweep test, with which the sample without damage is characterized. On the other hand, the work done by the sample can be modeled according to eq. (3).

Where W is the work done expressed as energy dissipated, γ₀ the specific applied shear strain, G* the complex shear modulus and δ the phase angle. In this way, eq. (2) can be integrated numerically using eq. (3) in order to arrive at the expression for damage growth, shown in eq. (4).

Where Cj is the ratio between the modulus at each point and the initial modulus of the material without damage, γ₀ is the applied shear strain, and t is time.

To quantify the damage presented by the samples during the test, the C-S damage curves are plotted, which represent the integrity of the sample as a function of the accumulated damage. Said curve can be adjusted by means of eq. (5).

Where C₀ =1 and C₁ and C₂ are the fit coefficients of the model. Differentiating eq. (5) and substituting it into eq. (2), we can finally arrive at eq. (1), which in its extended form appears as eq. (6).

Where f is the test frequency (defined at 10 Hz), D_f is the failure criterion adopted, k = 1+ (1- C₂) α, and B = -2α. Based on the study carried out by Masad [12], it is considered that the specific strain provided by the asphalt represents between 12 and 90 times the specific strain of the asphalt mixture. Finally, Bahía sets this parameter at 50 times and proposes a specific shear deformation of 2,5 % for pavements greater than 10 cm and 5,0% for pavements less than 10 cm [13-15].

Several authors propose different failure criteria D_f [1,2,7,9,16] to define the fatigue limit of asphalts contemplating this test. The present work proposes to consider the two most used criteria to obtain the resulting equivalent axis curves. In the first place, the drop in the value of the C-S curve by 35 % from the initial value, data that unifies the position for the evaluated binders. The second corresponds to the peak value of the shear stress exerted on the sample evaluated in the C-S curve [15].

The asphalt binders used are a conventional asphalt classified by viscosity (CA-30) and three binders modified from the base asphalt with increasing rates of rubber from RTR (15%RTR, 20%RTR and 25%RTR). Table 1 shows the properties of the asphalts used.

Table 1

S.P.: Softening Point

Table 2

Source: Prepared by the author.

After aging the asphalts in the RTFO, the LAS test is carried out in the DSR. Through the previously performed analysis, the parameters of the model are calculated according to the S-VECD to evaluate the coefficient α of integrity of the material and the coefficients C1 and C2 of the damage curve. Table 2 shows the tests at 20 °C, 25 °C and 30 °C below.

In the first instance, an analysis of the alpha parameter is carried out, which indicates the integrity of the sample without damaging it. Said parameter strongly depends on the stiffness of the sample, increasing the value of α when the stiffness increases. Therefore, the value of α shows a first indication of fatigue resistance of the samples, increasing with the increase of RTR as can be seen in Fig. 4.

Figure 4 Source: Prepared by the author.

With the adjustment parameters found, the C-S curves of material integrity are plotted as a function of the accumulated damage. Figs. 5-7 show the curves for 20 °C, 25 °C and 30 °C, respectively. For the three temperatures, it can be seen in the graphs that the asphalts modified with RTR maintain a greater integrity of the material for damage intensities greater than 50. In the same way, a superior performance is evidenced when the percentage of incorporation of RTR increases, although in a less pronounced. It can also be shown that the asphalts with RTR have a 60 % material life based on the damage intensity limit value for the unmodified asphalt, for the evaluations at 20 °C.

Figure 5 Source: Prepared by the author.

Figure 6 Source: Prepared by the author.

Figure 7 Source: Prepared by the author.

From the developed C-S graphs, the models are assembled to estimate the fatigue life based on the maximum specific deformation of the material and the failure criterion adopted for its evaluation. In the first instance, these curves are evaluated with the 35 % reduction of the initial integrity to develop the model seen in eq. (1). As can be seen, in principle, for 20 °C there is a considerable benefit from the incorporation of RTR as shown. see in Figs. 8-10.

As previously mentioned, based on the tests carried out, the comparison between failure criteria is proposed to obtain the fatigue life curves. With the curves already exposed with a failure criterion of reduction of 35 % of the initial integrity, the fatigue life curves are formed with the criterion of the maximum shear stress peak, together with its corresponding value in the C-S curve. Fig. 11 shows comparatively the asphalts evaluated at 20 °C and with an applied deformation of 2,5 %.

The graph at 25 °C follows the same trend, however, a difference is evident between the different proportions of rubber incorporated. The fatigue phenomenon manifests itself at intermediate temperatures (approximately 10 °C - 30 °C).

Figure 8 Source: Prepared by the author.

Figure 9 Source: Prepared by the author.

Figure 10 Source: Prepared by the author.

Figure 11 Source: Prepared by the author.

Therefore, as it is tested at a higher temperature, improvements in performance are evidenced and, in turn, materials with a higher degree of modification present better performance. In this last graph you can see an improvement in the incorporation of 25 % compared to the other alternatives.

As can be seen, there is a significant difference between the proposed criteria, a gap that widens with the growth of RTR incorporation. On the other hand, the 35 % criterion, although it is conservative, and is not something minor when evaluating fatigue in pavements, is a fixed criterion for all types of asphalt. In this way, for asphalts modified with RTR, the life of the material is being underestimated, since it is considered to be in a state of failure, when its remaining life is considerable. The criterion of reduction of an initial value is equivalent to the criterion of the reduction of 50 % of the initial modulus to evaluate asphalt mixtures by means of the fatigue beam at four points. As previously mentioned, these criteria are being questioned due to the randomness when it comes to pinpointing a material failure point.

On the other hand, although the life data considering the peak shear stress are much higher than those evaluated by the 35% reduction, it is a criterion that reveals the variation of the failure point of the material and that overestimates the useful life of the material.

Figure 12 Source: Prepared by the author.

As can be seen in Fig. 12, despite obtaining a higher peak stress, the CA-30 sample reaches this value prematurely and falls sharply compared to asphalts with RTR. The modified asphalts show a more stable curve as the percentage of RTR increases. Said effect is contemplated in the C-S curves that evaluate the internal work carried out by the material. On the other hand, the displacement of the stress peaks in the samples translates into large displacements in the C-S curves. Finally, this has a considerable effect when evaluating the number of axes considered to be fatigued.