Evaluation of different methods for the estimation of the bitumen

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The main disadvantage of the previously mentioned approaches is that ... One paving grade bitumen (35/50) and a polymer modified bitumen (PMB 45/80-.

Evaluation of different methods for the estimation of the bitumen fatigue life with DSR testing Andrรฉ Pereira1, Rui Micaelo2, Luรญs Quaresma3 and Maria Teresa Cidade4 Department of Civil Engineering1,2,3, Department of Materials and CENIMAT/I3N4, Universidade Nova de Lisboa, Caparica, PORTUGAL, 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected]

Abstract Asphalt fatigue cracking is one of the phenomena that contribute most to degradation of road pavements and it may initiate within the bitumen or at the bitumen-aggregate interface. The cohesive cracking resistance can be evaluated with bitumen testing. Commonly, the Dynamic Shear Rheometer (DSR) is used for bitumen testing. This paper presents an evaluation of different methods proposed in literature for the estimation of the bitumen fatigue life. A neat and a polymer modified bitumen (PMB) were tested with time sweep tests (continuous and discontinuous loading) and with incremental load amplitude (linear amplitude sweep test). The results are analysed with the traditional approach (N f,50 corresponding to 50% initial modulus reduction) and other methodologies, namely the Ratio of Dissipated Energy Change (RDEC) and the Viscoelastic Continuum Damage (VECD) approach. The results obtained showed, as expected, that the PMB has a higher resistance to fatigue than the neat bitumen. The test conditions and the method used to evaluate the fatigue resistance lead to significant differences in the estimated bitumen fatigue life. The plateau value (RDEC) shows very good correlation with Nf,50 obtained from constant strain amplitude tests, regardless of the type of bitumen or test conditions. The fatigue life parameters obtained from the linear amplitude sweep test is very sensitive to the analysis method. Healing during non-loading periods has a large effect on the PMB fatigue life while no effect in the neat bitumen fatigue life for small to intermediate rest periods.

Keywords Fatigue life, DSR, VECD, healing.

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Introduction

Asphalt fatigue cracking due to repeated traffic loading is one of the phenomena that contribute most to degradation of road pavements. This failure mode is con-

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sidered in all pavement design guides. The fatigue resistance law, which relates the asphalt fatigue life with the loading conditions, is usually obtained from lab tests and then calibrated to field based on the local road network pavements data. Even that asphalt fatigue tests have been developed and widely used in the last decades, it is known that these tests are time consuming and relatively expensive. On the other hand, fatigue properties of mixtures are strongly related with those of binders and, in this context, fatigue testing of bituminous binders is an important task with considerable research potential (Partl et al. 2013; Botella et al. 2012). In fact, cracking may initiate within the bitumen (cohesive cracking) or at the bitumen-aggregate interface. The cohesive cracking resistance can be evaluated with bitumen testing and, commonly, the Dynamic Shear Rheometer is used for that purpose. Furthermore, this equipment can also be used to evaluate the socalled self-healing capacity of bitumen. This phenomenon can be defined as the self-recovery capability of bituminous materials under certain loading and/or environmental conditions, especially during rest time, and it can have a significant effect on the fatigue life of those materials (Shen et al. 2010). An adequate characterization of the bitumen fatigue resistance and healing potential will help with the design and selection of bitumens, which may contribute to the increase of road pavementsโ€™ service life.

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Estimation of fatigue life

There are several methods and failure criteria proposed in literature to evaluate the fatigue resistance of bituminous binders. By carrying out several tests it is possible to obtain a fatigue law of the material by fitting a power-law model, as: ๐‘๐‘“ = ๐ด โˆ™ ๐‘‹ ๐ต (1) where, X represents the cyclic loading amplitude (stress or strain amplitude), Nf is the number of cycles to failure and, A and B are material dependent constants. Focusing on strain-controlled fatigue tests, traditional fatigue analysis defines failure as the point at which the materialโ€™s complex modulus value reduces to 50% of its initial value and the corresponding number of cycles is typically denoted as Nf,50. Although arbitrary and controversial, this failure criterion has been widely used by many researchers (Soenen and Eckmann 2000; Lu et al. 2003; Soenen et al. 2004; Shen et al. 2010; Santagata et al. 2013). An alternative approach is based on the dissipated energy (DE) concept. When sustaining cyclic fatigue loading, the viscoelastic materials, like bitumen, exhibit different paths for the loading and unloading cycle and creates hysteresis loops (Shen et al. 2010). The area inside of the loop is the DE, which can be calculated with: (2) ๐ท๐ธ๐‘› = ๐œ‹ โˆ™ ๐œ๐‘› โˆ™ ๐›พ๐‘› โˆ™ ๐‘ ๐‘–๐‘› ๐›ฟ๐‘› where, ๐ท๐ธ๐‘› is the dissipated energy at cycle n and, ๐œ๐‘› , ๐›พ๐‘› and ๐›ฟ๐‘› are, respectively, the stress amplitude, the strain amplitude and the phase angle at cycle n.

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However, not all dissipated energy is responsible for damage propagation. Only the relative amount of energy dissipation coming from each additional load cycle, while excluding the energy dissipated through passive behaviours such as plastic DE, viscoelastic damping and thermal energy, will produce further damage (Shen et al. 2010; Sutharsan 2010; Zhang et al. 2013). Consequently, it was developed the concept of Ratio of Dissipated Energy Change (RDEC), which for a test in controlled strain mode is: ๐ท๐ธ๐‘Ž โˆ’ ๐ท๐ธ๐‘ ๐‘…๐ท๐ธ๐ถ๐‘ = (3) ๐ท๐ธ๐‘Ž (๐‘ โˆ’ ๐‘Ž) where, a and b are the initial and final number of cycles of the interval (usually 100 cycles) used for the calculation of RDEC. The RDEC approach has been proved to be a good approach for the characterization of asphalt materials fatigue and healing properties (Sutharsan 2010). During a fatigue test under a strain-controlled loading mode three different phases of RDEC variation are usually identified. First there is a fast increase of the RDEC value, then a phase with an almost constant value and finally a sharp decrease before the test is terminated. The RDEC value in the second stage is named as the Plateau Value (PV). Shen et al. (2010) found a unique relation between PV and Nf,50 (power-law model) for the complete set of bitumen type and test conditions. A higher magnitude of the PV indicates higher incremental damage energy and a shorter fatigue life. Thus, the effect of healing (self-recovery capacity under certain loading and/or environmental conditions) can be quantified as the reduction in the PV value due do the application of rest periods in the cyclic load test (Shen and Carpenter 2007; Shen et al. 2010; Sutharsan 2010; Van den bergh 2011). The main disadvantage of the previously mentioned approaches is that several tests must be performed at various strain amplitude levels. The duration of these tests is undefined before the test and it is known that some binders can take many hours to show enough degradation to accurately assess their fatigue properties. The Linear Amplitude Sweep (LAS) test was proposed to overcome this challenge employing a cyclic loading at systematically increasing loading amplitudes to accelerate damage and, as such, it can be seen as an accelerated procedure to characterize the fatigue resistance (Johnson 2010). LAS test results are analyzed using the viscoelastic continuum damage (VECD) approach to predict the bitumen fatigue life. The VECD is based on Schaperyโ€™s theory of work potential to model damage growth (Hintz et al. 2011, Willis et al. 2012). According to this theory, for a viscoelastic material, work is related to damage by: ๐‘‘๐ท ๐œ•๐‘Š ๐›ผ (4) = (โˆ’ ) ๐‘‘๐‘ก ๐œ•๐ท where, W is the work performed, D is the damage intensity and ฮฑ is a material constant that is related to the rate at which damage progresses. The parameter ฮฑ can be taken as 1+1/m or 1/m (there is no consensus in the literature as to which definition should be used). Nevertheless, Johnson (2010) defined ฮฑ as 1+1/m and demonstrated that the slope, m, of a log-log plot of storage modulus (Gโ€™) versus

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angular frequency (w) can be used to calculate ฮฑ. Thus, the LAS test protocol also includes a frequency sweep test at very low strain amplitude of 0.1% to obtain the undamaged material properties and to allow the calculation of m and ฮฑ. The amplitude sweep can be run directly after the low strain frequency sweep as no damage is induced during this stage (Johnson 2010; Hintz et al. 2011). The work done by Johnson (2010) resulted in the development of AASHTO TP101-12 (MARC 2014). According to this standard, the damage accumulation in the specimen during the strain sweep test can be calculated with: ๐‘

๐›ผ

1

๐ท(๐‘ก) โ‰… โˆ‘[๐œ‹ โˆ™ ๐›พ0 2 (๐ถ๐‘›โˆ’1 โˆ’ ๐ถ๐‘› )]1+๐›ผ (๐‘ก๐‘› โˆ’ ๐‘ก๐‘›โˆ’1 )1+๐›ผ

(5)

๐‘›=1

where, C(t) is equal to |G*|sin ฮด at time t divided by the initial โ€œundamagedโ€ value of |G*|sin ฮด, ฮณ0 is the applied strain for a given data point (%), |G*| is the complex shear modulus (MPa), ๐›ฟ is the phase angle and t is the test time (sec). The C(t)D(t) curve can be fitted with a power-law model: (6) ๐ถ(๐‘ก) = ๐ถ0 โˆ’ ๐ถ1 [๐ท(๐‘ก)]๐ถ2 where, C0 is the initial value of C(t) (C0=1), and C1 and C2 are curve-fit coefficients. The parameters A and B of the fatigue law can then be calculated with (MARC 2014): 1+(1โˆ’๐ถ2 )๐›ผ

๐‘“(๐ท๐‘“ ) ๐ด= [1 + (1 โˆ’ ๐ถ2 )๐›ผ](๐œ‹ โˆ™ ๐ถ1 โˆ™ ๐ถ2 )๐›ผ

(7)

๐ต = โˆ’2๐›ผ (8) where, f is the loading frequency (Hz), Df is the damage accumulation at failure, which is defined as the D(t) value that corresponds to a 35 percent reduction in C(t) and calculated with: 0.35 1/๐ถ2 (9) ๐ท๐‘“ = ( ) ๐ถ1

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Experimental

3.1 Materials One paving grade bitumen (35/50) and a polymer modified bitumen (PMB 45/8065) were selected for this study. PMB binder is modified with 2-4% of SBS polymer using an undisclosed procedure by the supplier. Both binders were tested without considering any aging effect. The conventional properties of bitumens are given in Table 1. The bitumens were rheologically characterized using a Bohlin Gemini HRnano rotational rheometer. For dynamic tests, this equipment is equivalent to a Dynamic Shear Rheometer (DSR). Frequency sweeps were performed in

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constant strain loading mode at temperatures from 15 to 75 ยบC and frequencies between 0.1 and 10 Hz. The tests were performed with the 8 mm (2 mm gap) and 25 mm (1 mm gap) plate-plate set-up. The results for the complex modulus (G*) and the phase angle (ฮด) showed, as expected, higher phase angle values (lower elasticity), at given stiffness level, for the paving grade bitumen in comparison with the PMB. Complete information about the rheological characterization can be found in Pereira (2014). Table 1 Conventional properties of bitumens Bitumen

Method

Property

35/50

PMB 45/80-65

EN 1426

Penetration (0,1 mm)

43.0

51.0

EN 1427

Softening point (ยบC)

51.0

70.4

3.2 Fatigue tests All fatigue tests were performed under controlled strain conditions, at a frequency of 10 Hz and at a temperature of 15ยบC. It was used the parallel plate setup with the 8-mm spindle and a 2-mm gap between the plates. Two types of fatigue tests were performed: time sweep tests at constant strain and linear amplitude sweep tests (LAS). The time sweep tests were performed without rest periods (continuous tests) and with rest periods (intermittent tests) to evaluate the effect of healing on fatigue life. The continuous tests were conducted at three strain levels: 1.2%, 1.6% and 2%. The intermittent tests were carried out at the single strain level of 1.2% and rest periods of 4 and 8 seconds were introduced after every 10 seconds of loading. For the neat bitumen it was also considered a rest period of 16 seconds. To limit the total testing time, all time sweep tests were carried out until a 50% reduction of the initial complex modulus was attained. Furthermore, the ratio between loading and rest duration was selected so that the duration of the tests did not exceed 14 hours. The time sweep tests results were analyzed considering the traditional failure criterion (Nf,50) and the RDEC approach. The LAS tests were performed based on the work done by Johnson (2010) and AASHTO TP 101-12 protocol (MARC 2014). Thus, before the fatigue test it was performed a frequency sweep at the low strain level of 0.1%, and over a range of frequencies from 0.1 to 30 Hz. Then, it was performed a strain sweep at 10 Hz. The test procedure consists of applying an initial 100 cycles at 0.1% strain to determine the undamaged linear viscoelastic properties of the bitumens and each subsequent load step consists of 100 cycles at a rate of increase of 1% applied strain per step for 19 steps, beginning at 1% and ending at 19% applied strain. The LAS tests results were analyzed using the VECD approach.

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Two repetition tests were performed for repeatability analysis purpose.

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Results and Discussions

4.1 Time sweep tests Figure 1 illustrates the complex modulus during the continuous fatigue tests for both bitumens. As shown, the higher the level of deformation, the shorter the number of cycles until there is a 50% reduction of the initial complex modulus and therefore, a shorter fatigue life. It is also noticed that for the same temperature (15ยบC), the neat bitumen has a higher initial complex modulus and, consequently, for the same strain level, the 35/50 bitumen is subject to higher shear stress.

G* (Pa)

3.E+07 35/50 (ฮณ=1,2%) 35/50 (ฮณ=1,6%) 35/50 (ฮณ=2%) PMB 45/80-65 (ฮณ=1,2%) PMB 45/80-65 (ฮณ=1,6%) PMB 45/80-65 (ฮณ=2%)

2.E+07

1.E+07

0.E+00 0

50,000

100,000 Loading cycles

150,000

Fig. 1 G* evolution for the 35/50 bitumen with different strain levels

The PMB shows a reduction of the complex modulus from the beginning of the test while the neat bitumen has a constant value during the first 2,000 cycles. Then, the 35/50 shows a much faster reduction of G* with the loading cycles than the PMB. Figure 2 presents the Nf,50 results and the fatigue laws fitted to data. As expected, the fatigue life of the neat bitumen is much lower than of the PMB. However, the increase of the strain level has a larger effect on the PMBโ€™ fatigue life. For the largest strain levels tested, the initial decrease in complex modulus counts for an important share of the 50% reduction of the initial complex modulus though it may not be necessarily due to fatigue. Some aspects such as the internal heating or the edge effects due to lower initial stiffness of the PMB may have caused the unexpected reduction of the initial complex modulus. Hence, the traditional failure criterion may not be the most appropriate to evaluate the fatigue resistance of the PMB, especially for higher strain levels.

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The effect of rest periods on the fatigue life defined by the traditional failure criterion is presented in Figure 3. 1.E+06 35/50

Nf,50 = 169418ฮณ-3.094 Rยฒ = 0.97

PMB 45/80-65

Nf,50

1.E+05

1.E+04 Nf,50 = 17144x-1.774 Rยฒ = 0.99 1.E+03 1

10

Strain, ฮณ (%)

Fig. 2 Traditional Nf,50 โ€“ strain relationship for bitumenโ€™s fatigue testing results

The effect of rest periods on the fatigue life is much more evident for the PMB than for the 35/50 bitumen. With the introduction of rest periods, the fatigue life of the PMB is significantly longer, showing a sharp increase with the duration of the rest period. Therefore, it can be inferred that the PMB has a larger self-healing capacity. Instead, in the case of the neat bitumen, the rest periods of 4 and 8 seconds lead to small reduction in the fatigue life, being comparable with the no rest situation. Nevertheless, for a rest period of 16 seconds there is an increment of the fatigue life and therefore it can be assumed that with sufficiently long rest periods, the neat bitumen also has self-healing capacity. 250,000

20,000

200,000

15,000

150,000

10,000

100,000

5,000

50,000

Nf,50

25,000

0

0

4 8 Rest period (sec)

16

0

0

4

8

Rest period (sec)

8 Fig. 3 Effect of rest periods on fatigue life

It should be emphasized that the PMB has a lower initial stiffness than the neat bitumen, meaning lower shear stress for the same strain level. So, the damage accumulated by the PMB tends to be smaller and the self-healing process might have developed more efficiently. The fact of the PMB showing a lower stiffness at 15ยบC should be seen as an important advantage and that undoubtedly contributes to the fatigue resistance during the test. To evaluate the fatigue resistance using the concept of RDEC, it was first calculated the dissipated energy in accordance with the Eq. (2). Then, it was calculated the RDEC in accordance with Eq. (3). However, as mentioned by Shen and Carpenter (2007) and Shen et al. (2010), due to testing noise, the raw DE data points are not directly useable since it can cause ambiguity for RDEC and PV calculation. Thus, in order to obtain a representative PV from fatigue testing data and minimize the error due do testing data variation, a linear regression was fitted to the DE-LC curve, and the slope of the curve was obtained. In order to ensure that the curve segment used for regression is in the plateau stage where the variation rate of DE is almost constant, the adjustment was done with respect to the following conditions: (i) for the 35/50 bitumen, the adjustment was made from cycle 2000 until the cycle in which there was a 50% reduction of the initial complex modulus; (ii) for the PMB, the adjustment was made from cycle 15000 until the cycle in which there was a 50% reduction of the initial complex modulus. These considerations were applied to all measurements (including repetitions of the same test), regardless of the strain level and the type of test (continuous or intermittent). Taking into account the linear regressions that were obtained and defining the PV as the RDEC value at the 50% stiffness reduction failure point (N f,50), it was possible to determine the PV for both binders tested and for the different test conditions. The results are plotted versus fatigue life (Nf,50) in Figure 4. Each data point represents a specific combination of bitumen and test conditions, with 6 points referring to the 35/50 and 5 points to the PMB.

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1.E-03

PV = 4.02Nf,50-1.19 Rยฒ = 0.98

PV

1.E-04

1.E-05

1.E-06 1.E+03

1.E+04

1.E+05

1.E+06

Nf,50 Fig. 4 PV versus Nf,50 plot for all tested binders

As shown, no matter the binder type (35/50 or PMB 45/80-65), the strain level (1.2%, 1.6% or 2%), the test type (continuous or intermittent) and the rest period duration (4s, 8s or 16s), all data points follow a unique PV-Nf,50 line with an Rsquare value of 0.98. This is in accordance with the conclusions of Shen et al. (2010). It is also noted that for higher strain levels, the PV tends to increase. This was expected because for higher strain levels, the variation of dissipated energy between each loading cycle is more pronounced and the fatigue resistance is lower. Furthermore, it is possible to confirm that the rest periods have a beneficial effect, particularly in the case of PMB 45/80-65, since they lead to a significant reduction in PV. For all test conditions that were used, the PMB 45/80-65 presents lower PV values than the 35/50 bitumen.

4.2. Linear amplitude sweep tests Following the protocol for the LAS test, the test begins with a frequency sweep test for the determination of the undamaged material properties. Due to the ambiguity in literature about the calculation of ฮฑ, it was determined as 1+(1/m), according to Johnson (2010), and as 1/m in accordance with a recent draft proposal for AASHTO TP-101-12 (MARC 2014). Table 2 shows the m and ฮฑ values obtained for the two bitumens. Figure 4 shows the variation of C(t) with D(t) obtained from the tests with the two different ฮฑ values. The influence of ฮฑ is so small that cannot be detected in the figure. The PMB shows a more gradual decrease of the integrity parameter C(t), which means that it is able to accumulate much more damage than the neat bitumen. Table 2 presents the fatigue laws obtained from the LAS test, which were calculated following the procedure described in section 2. The value of A is very sensitive to the value of ฮฑ, being approximately 15 and 25 times higher with

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ฮฑ=1+(1/m) for the 35/50 and the PMB, respectively. The differences in B values are justified by the variable B being a single function of ฮฑ (Eq. 10). For the same ฮฑ equation there are not significant differences between the bitumens. Figure 5 compares fatigue life predicted by the fatigue laws obtained from the time sweep tests (Nf,50) and from the LAS tests (Nf,LAS). It is concluded that the two methodologies provide very different laws regardless of ฮฑ equation selected for LAS. The fatigue laws from the time sweep test determine, in general, longer periods to failure for both bitumens. The A values derived from the time sweep tests are much lower than the ones obtained from the LAS test with ฮฑ=1+(1/m) while when ฮฑ is 1/m it is significantly higher for the 35/50 and lower for the PMB. Also, the B values obtained from the time sweep tests are significantly different between bitumens while the ones derived from the LAS are very similar. Johnson (2010) recommended the value of 35% in C(t) reduction for the determination of the fatigue life based on the comparison of time sweep and LAS tests. The results obtained in this study show that more tests are needed, with different materials and test conditions, to establish an adequate fatigue failure criteria. Table 1 LAS test results Bitumen

m

ฮฑ = 1+(1/m) ฮฑ = 1/m 1.297 0.771 35/50 2.297 1.437 PMB 45/80-65 0.696 2.437

C1 0.088 0.100 0.085 0.103

C2 0.510 0.481 0.461 0.421

Df 15.981 13.410 21.359 18.481

A 6485.5 102569.5 26042.0 613634.8

B -2,593 -4,593 -2,875 -4,875

1 35/50 (data) 35/50 (fit) PMB 45/80-65 (data) PMB 45/80-65 (fit)

C(t)

0.8 0.6 0.4 0.2 0 0

40

Fig. 4 Plot of C(t) versus D(t)

80

D(t)

120

160

200

11

1.E+06

1.E+05

Nf,50

1.E+04

1.E+03

1.E+02

PMB 45/80-65 LAS: ฮฑ=1/m PBM 45/80-65 LAS: ฮฑ=1/(1+m) 35/50 LAS: ฮฑ=1/m 35/50 LAS:ฮฑ=1/(1+m)

1.E+01 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Nf,LAS Fig. 5 Comparison of different fatigue lawsโ€™ predictions

5. Conclusions and Recommendations This paper presents an evaluation of different methods proposed in literature for the estimation of the bitumen fatigue life. A neat bitumen and a polymer (SBS) modified bitumen were tested with continuous and discontinuous loading (time sweep tests) and with incremental load amplitude (linear amplitude sweep tests). The results of time sweep tests were analysed with the traditional approach (Nf,50 50% initial modulus reduction) and using the Ratio of Dissipated Energy Change approach. The results of the linear amplitude sweep tests were analysed with the Viscoelastic Continuum Damage approach. The results obtained showed, as expected, that the PMB has a higher resistance to fatigue than the neat bitumen. However, it was found that the traditional failure criterion can significantly underestimate the true fatigue resistance of the PMB, particularly for higher strain levels. Healing during non-loading periods has a large effect on the PMB fatigue life while no effect in the neat bitumen fatigue life for small to intermediate rest periods. Consequently, it was possible to infer that the PMB shows a higher healing potential, at least for these testing conditions. By using the concept of RDEC and evaluating the fatigue resistance with the Plateau Value (PV), it was possible to confirm that regardless of the bitumen type, strain level, test type (continuous or intermittent) and rest period duration, all data points follow a unique PV-Nf,50 line.

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The fatigue life obtained from the linear amplitude sweep tests and using the VECD analysis shows that, for the same strain level and for the same definition of ฮฑ, the PMB always shows a higher fatigue resistance than the neat bitumen. The fatigue law obtained with this method is very sensitive to the value of ฮฑ considered. However, there are significant differences between the fatigue laws obtained with the two different methodologies. Such differences can be attributed to the fact that the testing procedures are different, the failure criteria being arbitrary and the VECD model formulation and its suitability for bitumen still needs further investigation. New specifications for the fatigue resistance and self-healing capacity assessment testing, as the AASHTO TP-101-12 (MARC 2014), are valuable improvements to current bituminous materialsโ€™ performance testing. It is required to continue research both in the methodologies used for the test analysis and in the lab testing.

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13 Soenen H, Eckmann B (2000) Fatigue Testing of Bituminous Binders with a Dynamic Shear Rheometer. Paper presented at the second Eurasphalt and Eurobitume Congress, Vol. 1, pp. 827-834, Barcelona, Spain, 2000. Sutharsan T (2010) Quantification of cohesive healing of asphalt binder based on dissipated energy analysis. MSc Thesis, Washington State University, Washington, D.C., USA. Van den bergh W (2011) The Effect of Ageing on the Fatigue and Healing Properties of Bituminous Mortars. PhD Thesis, Delft University of Technology, Delft, The Netherlands. Willis J, Turner P, Julian G, Taylor A, Tran N, Padula F (2012) Effects of changing virgin binder grade and content on rap mixtures properties. NCAT Report Nยบ 12-03, National Center of Asphalt Technology, Auburn University, Auburn, Alabama, USA. Zhang J, Sabouri M, Guddati M-N, Kim R (2013) Development of a failure criterion for asphalt mixtures under fatigue loading. Road Materials and Pavement Design, 14(2):1-15.

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