Analysis of Fatigue and Fracture of Hot Mix Asphalt Mixtures

An accurate assessment of the fatigue life of hot mix asphalt (HMA) mixtures depends on the criteria used in the fatigue analysis. In the past, various studies have been conducted on crack initiation and crack propagation of the HMA mixtures. Most of these studies were focused on the beam samples with or without a sawed crack at the bottom. is paper presents and discusses two different fatigue life criteria for two-dimensional problems represented by cylindrical samples. One criterion is based on the rate of accumulation of the tensile horizontal plastic deformation (HPD) as a function of the number of load repetitions. e second criterion is based on fracture mechanics, stress intensity factor, and the rate of crack growth with respect to the number of load repetitions. It was found that, because of three-dimensional nature of the crack growth in cylindrical samples, the Paris’ law was violated. It is shown that the rate of crack growth criterion provides higher values of fatigue life relative to the rate of accumulation of HPD criterion. Although a trend could be established among the fatigue lives obtained by using the two criteria, it was found that the fatigue lives obtained from the rate of accumulation of HPDwere consistent and based on the actual measurement of HPD for HMAmixtures.


Introduction
e prediction of fatigue life of hot mix asphalt mixtures (HMA) is an important aspect of pavement design.Fatigue cracks are caused by repeated traffic loading and are typically initiated at the bottom of the HMA layer where the tensile stress and strain are the highest.With increasing number of load application, the cracks propagate to the surface where they appear as one or more longitudinal cracks, which will be connected by transverse cracking to form a pattern similar to an alligator hide.Many factors affect the fatigue life of HMA pavement such as the tensile strength of the asphalt binder, traffic load, construction practices, aggregate angularity and gradation, relative stiffness of the AC, and the base material and environmental conditions such as temperature and moisture.
In the past, many efforts have been made to estimate the fatigue life of laboratory compacted HMA mixtures.Such estimates are highly dependent on the criterion used.Hence, various criteria were developed and are reported in the literature [1][2][3][4][5][6][7][8][9][10][11][12].Monismith and Deacon (1969) [6] and Pell and Cooper (1975) [7] conducted displacementcontrolled trapezoidal fatigue test and proposed that for HMA mixtures the fatigue failure of the mixture is reached when the load value drops to a half of the initial load value.On the other hand for a stress-controlled fatigue test, the failure (i.e., fatigue life) was reached when the displacement value doubles the initial displacement value [11].e damage failure criterion was used based on the analysis of the stiffness modulus, , values along the number of load cycles.It was observed that the stiffness modulus decreases rapidly as the load application progresses.Aer this, a linear decrease of the stiffness modulus was reported.Finally, a rapid loss of stiffness modulus occurred which was related to crack development [9][10][11].Hence, the fatigue life was de�ned as the number of load cycles at which the rapid increase in stiffness modulus occurred.Similarly, for a displacement or stress controlled test, the change in the displacement or stress along the repeated loading was plotted.Based on this criterion, the HMA specimen failed when the displacement or stress values F 1: Stress intensity factor ( for a centrally cracked circular plate (aer [13]).
exhibited an accelerated increase while the loading process progressed [8,11,12].In this paper, a comparison between two different fatigue life criteria for two-dimensional problems represented by cylindrical samples is discussed and presented.One criterion is based on the rate of accumulation of the tensile horizontal plastic deformation (HPD) as a function of the number of load repetitions.e other criterion is based on fracture mechanics and takes into account stress intensity factor and the rate of crack growth with respect to the number of load repetitions.

Materials and Methods
Different types of HMA mixtures were constructed tested during this study.ese include the following.

AC� an� AC1� �trai��t an� Pol�mer�Mo�i�e� As��alt
(PMA) HMA Mixtures.e mix design for these mixtures was based on the standard Marshall Mix design procedure with 50 blows on each side of cylindrical samples.e samples were 10.16 cm in diameter and 6.35 cm thick.e Michigan Department of Transportation (MDOT) speci�cations of voids in mineral aggregates (VMA) and other criteria for the MDOT designated 4C HMA mixtures were followed.e polymers used for HMA modi�cation included styrenebutadiene-styrene (SBS), styrene-ethylene-butylene-styrene (SEBS), and styrene-butadiene-rubber (SBR).Detailed information regarding the mix design and polymer-modi�ed asphalt mixtures can be found in [5].
2.2.MDOT HMA Mixtures.ese mixtures were obtained from construction sites by MDOT and were compacted at the Michigan State University laboratory.Cylindrical samples of 150 mm diameter and 76 mm thick were compacted under the pressure of 600 KPa using Gyratory compactor.MDOT speci�cations of VMA and other criteria for the asphalt mixtures were followed for these mixtures.e compacted samples were cut into two 35 mm thick samples using watercooled diamond saw.For some samples, a slot (25 mm × 2.5 mm) was also made at the center of the sample along the vertical diameter.Samples with a slot in the center were tested under dynamic loading.
All samples were subjected to cyclic loads at 20 and 25 ∘ C using indirect tensile cyclic load tests.First, the sample was subjected to a sustained stress of 13 KPa.When the sample came to rest, a cyclic stress of 75 KPa was applied, and the deformation of the sample was measured in three directions using linear variable differential transducers (LVDTs).Each loading cycle consisted of 0.1 second load-unload time and 0.4 second of rest period.e resilient and total moduli of the mixtures were calculated using the resilient and total deformations, respectively [4].e sample characteristics and sample geometry are listed in Table 1.

Fatigue Life Criteria
e fatigue life of each test sample was analyzed using two fatigue life criteria.One criterion is based on the rate of accumulation of tensile horizontal plastic deformation (HPD) with respect to the number of load repetitions.e other is based on fracture mechanics, and it takes into account the rate of crack growth with respect to the number of load repetitions.Both criteria are discussed below.

e Rate of Accumulation of HPD Criterion
. is criterion was developed at Michigan State University, and it has been used in several research studies [5].In this criterion, it was hypothesized that, at certain number of load application, microcracks are initiated in the sample.With increasing number of load repetitions, they grow to macrocracks that can be visually detected.Once the microcracks are initiated and interconnected, the tensile horizontal plastic deformation starts to increase sharply, as a result of stress concentration at the crack tips.erefore, in this hypothesis, the fatigue life of a test sample is de�ned as the number of cycles at which the rate of accumulation of HPD increases.Based on the de�nition, the following procedure was established for the determination of the fatigue lives of the test samples.
(1) From the test results, the cumulative HPD (HPD)  is plotted as a function of the number of load repetition ().A polynomial equation is �tted to the test results such that where Δ  is the horizontal plastic deformation per load cycle.
(2) e rate of accumulation of HPD (RHPD)  is obtained by differentiating the above equation with respect to the number of load repetition () as shown below: (3) e normalized rate of accumulation of HPD (NRHPD)  at any load cycle is calculated as where (RHPD) 00 is the rate of accumulation of the HPD at 100th load cycle.
(4) e normalized rate of accumulation of HPD is then plotted on a semilogarithmic scale as a function of the number of load repetition ().
e slope of the curve decreases �rst, reaches a valley, and then starts to increase.e decrease in the normalized rate of accumulation of HPD at the beginning of the test is due to slight densi�cation and sample seating.However when cracks are initiated, the normalized rate of accumulation of HPD also increases.Hence, in this procedure, the fatigue life is de�ned as the number of load repetitions at which the rate of accumulation of HPD starts to increase.

e Rate of Crack Growth Criterion
. e asphalt concrete is a heterogeneous material consisting of asphalt cement and �ne and coarse aggregates.e fatigue cracks initiation in composite and heterogeneous material is related to the preexisting internal �aws.In HMA mixtures these internal �aws exist in the form of air voids, surface irregularities, and internal defects.ese internal �aws are randomly distributed in the HMA mixtures and control the geometry and propagation of fatigue cracks.e rate of crack growth criterion utilizes the concepts of fracture mechanics to relate the fatigue life to the rate of crack growth and stress intensity factor.With increasing number of load repetition the preexisting microcracks (internal �aws) at a given state of stress start to propagate and reach critical dimensions and pattern where the rate of crack propagation increases.e number of load repetitions at which the rate of crack growth increases is designated as the fatigue life of the HMA mixtures.e following two approaches were investigated to indirectly determine the crack length and growth.

Method of Compliance.
In this fatigue life criterion an indirect method was utilized to estimate the crack length at different load repetitions.e indirect method of crack length was utilized because it is not possible to get actual measurement of the crack length in a cylindrical sample subjected to indirect tensile cyclic loading.e indirect method is based on the fact that as the crack length increases from an initial �aw, the compliance ( of the sample also increases due to stress concentration in the vicinity of the crack tip.e compliance ( of a material subjected to load is de�ned as the reciprocal of modulus and is given by the following equation [14]: If the material behaves elastically, the increase in compliance is related to the crack length.e increase in the compliance as a result of crack growth is expressed by Irwin's equation as follows [13,15].
where  = compliance;  = Young's modulus;  = crack length;  = stress intensity factor,  = load per unit thickness;  = poisson ratio.is equation can also be written as e stress intensity factor can be obtained as follows [16]: where  is an applied stress,  is a diameter of the sample,  is the load per unit thickness, and  is a shape factor and is a function of  ratio as shown in Figure 1 [17].Substituting  in (3) and rearranging, the equation for  2  2 yields the following equation: If the initial compliance (  ) and the initial crack length (  ) are known, normalized compliance (NC) at any number of load application can be calculated from (6) as follows:  Equation (11) gives the relation between the compliance and the crack length.From indirect cyclic load test, the compliance of a sample can be calculated at different cycles by simply taking the inverse of the total modulus (.From this, the normalized compliance can be calculated by dividing all the data points with the compliance value at the 100th load cycle.Hence the crack length at a given load cycle can be obtained by using (11).Once the crack length is known, the stress intensity factor ( can be calculated from (7). e crack length as a function of number of cycles can also be calculated using another fracture mechanics technique that utilizes the vertical displacement of the sample.is technique provides a check for the crack length calculations using the compliance technique.e method of displacement is explained below.

Method of Displacement.
With the increase in number of load cycles the crack length and vertical displacement of the sample increase.e vertical displacement and crack length are related to each other by the following equation [18]: where Δ  = load point displacement with crack, Δ nocrack  = load point displacement with no crack present,  = crack length,  = radius of the sample,  = elastic modulus  = thickness of the sample, and  = applied load.
e load point de�ection with no crack present can be obtained from an appropriate analytical expression relating load and load point displacement to material properties and dimensions as Rearranging the above two equation will yield By using the numerical approach the crack length can be calculated from (14).Once the crack length is known, the  values for each load cycle can be calculated.e  and  material properties can be determined by plotting the  values and rate of crack growth in crack length on log-log scales and can be used for Paris law.

Results and Discussion
Figure 2 shows the effect of the number of load repetition ( on the modulus and normalized compliance (NC) of HMA mixture at test temperature of 20 ∘ C. It can be seen that the modulus values decrease and the normalized compliance increases with increasing number of load repetitions.A best-�t polynomial equation was obtained expressing the modulus as a function of the number of load repetition.e normalized compliance of the sample was then calculated by simply taking the inverse of the modulus and dividing the result by the compliance value at the 100th load repetition.e results were plotted as a function of the number of load repetitions and are shown in Figure 2. e normalized compliance increases as the number of cycle increases.Since the normalized compliance and the crack length are uniquely related, the crack length was calculated using (5) and plotted as a function of the number of load repetitions as shown in Figure 3.In order to validate the results the crack length was also calculated using the vertical displacement method (14) and plotted as a function of number of cycles (Figure 3).It can be seen that crack lengths calculated using both fracture mechanics methods are similar.e slight difference may be because of the best-�t curve used to �t the data.e examination of (11) and the data in Figures 2 and 3 indicate that the crack in HMA sample reaches its full length when the compliance is twice the initial value.is implies that when the modulus reaches its 50 percent value due to repeated loadings, the sample completely cracks.It should be noted that 50 percent reduction in modulus has also been used by various researchers as one of the fatigue criteria.
Similarly the measured cumulative HPD was plotted against the number of load repetition as shown in Figure 4. e normalized rates of both crack growth and cumulative HPD with respect to �rst data point (100th load repetition) were calculated and plotted against the logarithmic value of the number of load repetition as shown in Figure 5.It can be seen that the normalized rate of cumulative HPD decreases �rst, forms a valley, and then starts to increase.is is due to the initial densi�cation of the sample.Similarly, the normalized rate of crack growth decreases �rst, reaches a valley, and then increases.e decrease in the rate of crack growth is due to the crack being abstracted or shielded by large aggregates in the HMA mixtures.e crack had to de�ect and change its path and propagate around the boundary of the aggregates.is crack de�ection is referred to as crack shielding mechanism, which decreases the rate of crack growth.e crack tip shielding mechanisms have also been reported by Hertzberg [19] in polycrystalline alumina under cyclic loading.e crack growth rates were found to decrease with increasing crack length before arresting.When crack reached a critical dimension, it started to accelerate in a similar manner as that shown in Figure 5. Relative to the HMA mixtures, the number of load applications at which the crack starts to accelerate is considered the fatigue life of the HMA mixture.e stress intensity factor ( was also calculated, and the rate of crack propagation was plotted against the stress intensity factor as shown in Figure 6.Similar results for PMA mixtures without slot and straight MDOT HMA mixtures with slot are shown in Figures 7, 8, 9, 10, and 11. ese mixtures showed similar trend as that of the mixtures discussed above.e rate of crack growth decreases �rst and then starts to increase hence violating the Paris� law, which states that the rate of crack growth increases with the increase in the stress intensity factor and obeys the power law as given below: where  and  are material constants.e decrease in the rate of crack growth may be because there is no single crack, instead, there are many secondary cracks surrounding the main crack.Hence, the crack tip contains many microcracks embedded in the plastic zone.e rate of crack propagation and the path it follows mainly  depend on the energy balance at the crack tips.erefore, the energy due to the applied load is utilized by the main crack as well as the microcracks.Further, if there is a large amount of plastic deformation, the crack tips become blunt.Hence, the presence of microcracks and blunting mechanism at the crack tip reduces the effective stress intensity factor and the rate of crack growth [15,19].e test data was analyzed, and the fatigue lives of all samples were obtained using both criteria.Figures 12 and  13 show that the fatigue life obtained from the rate of crack growth criterion was higher than the rate of cumulative HPD criterion.Figure 14 shows the fatigue life calculated using both criteria.It was observed that fatigue life based on the rate of crack growth criterion yielded an average of 45 percent higher values than the rate of accumulation of HPD criterion.
e difference in the fatigue life obtained from the two criteria can be partly explained by the examination of the crack growth criterion.It is based on theory of elasticity with the assumptions that the material is homogeneous, isotropic elastic solid, and there are negligible changes in the material properties during testing.is is not the case for the HMA mixtures, which are heterogeneous mix of asphalt cement and �ne and coarse aggregates.Further, the criterion is based on two-dimensional crack growth.For cylindrical sample tested in indirect cyclic tensile loading, the crack propagates in three dimensions.Given that the crack initiates at the center of the sample, the crack will propagate along both the vertical and horizontal diameters as well as along the thickness of the sample as schematically shown in Figure 15.e crack length equation only calculates the crack length along the vertical diameter of the sample.is may result in higher fatigue life relative to the cumulative HPD criterion.On the other hand the rate of accumulation of HPD criterion of fatigue determination is based on direct measurement of HPD from the test sample.Since the fatigue of cylindrical sample is mainly due to the accumulation of HPD, the rate of accumulation of HPD is a true representation of fatigue crack growth.is criterion produced consistent and repeatable results for both straight and PMA mixtures [5].

Conclusions
In this paper, a comparison between two different fatigue life criteria for two-dimensional problems represented by cylindrical samples are discussed and presented.One criterion was based on the rate of accumulation of the tensile horizontal plastic deformation (HPD) as a function of the number of load repetitions.e other criterion was based on fracture mechanics and takes into account stress intensity factor and the rate of crack growth with respect to the number of load repetitions.Indirect tensile cyclic loading test was conducted on straight and PMA-modi�ed HMA mixtures with and without slot at the center of sample.It was found that the crack in HMA sample reaches its full length when the compliance is twice of its initial value.e rate of crack growth criterion showed higher fatigue life relative to the rate of accumulation of horizontal plastic deformation.is difference was attributed to the three-dimensional nature of crack propagation.Nevertheless, the rate of accumulation of HPD criterion produced consistent and repeatable results and is recommended to be used for the determination of laboratory fatigue life of HMA mixtures.

F 2 :F 3 :
A typical plot of the total modulus and normalized compliance versus the number.A typical plot of the crack length as a function of the number of load repetitions.

F 4 :F 5 :
A typical plot of cumulative HPD as a function of the number of load repetitions.A typical plot of normalized rate of the rate of the crack growth and cumulative HPD as a function of the number of load repetitions.

F 7 :F 8 :
Half-crack length and cumulative HPD as a function of the number of load repetitions for AC10-2% SBR mixture.Normalized rate of crack growth ( and rate of cumulative HPD ((CHPD)/) for AC10-2% SBR.

F 9 :
Half-crack length and cumulative HPD as a function of the number of load repetitions for MDOT AC mixture with slot at the center.

F 14 :
Comparison of the fatigue life between the rate of cumulative HPD and crack growth method.

F 15 :
ree-dimensional crack propagation model for centrally cracked cylindrical sample under indirect tensile cyclic loading.
T 1: Sample characteristics and sample geometry for fatigue testing.
Normalized rate of crack growth ( and rate of cumulative HPD ((CHPD)/) for MDOT AC mixtures.
F 11: - curve for AC10-2% SBR and MDOT AC mixtures with slot at the center.F 12: Comparison of the fatigue lives between the rate of cumulative HPD and crack growth criteria for AC straight and PMA mixtures.