Discrete Element Modeling of Asphalt Concrete ... - Springer Link

Journal of Wuhan University of

Technology-Mater. Sci. Ed.

Dec. 2011

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DOI 10.1007/s11595-011-0393-z

Discrete Element Modeling of Asphalt Concrete Cracking Using a User-defined Three-dimensional Micromechanical Approach CHEN Jun1, PAN Tongyan2, HUANG Xiaoming3 (1.College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China; 2. Department of Civil Engineering, Catholic University of America, Washington, DC 20064, USA; 3. School of Transportation, Southeast University, Nanjing 210096, China) Abstract: We established a user-defined micromechanical model using discrete element method (DEM) to investigate the cracking behavior of asphalt concrete (AC). Using the “Fish” language provided in the particle flow code in 3-Demensions (PFC3D), the air voids and mastics in asphalt concrete were realistically built as two distinct phases. With the irregular shape of individual aggregate particles modeled using a clump of spheres of different sizes, the three-dimensional (3D) discrete element model was able to account for aggregate gradation and fraction. Laboratory uniaxial complex modulus test and indirect tensile strength test were performed to obtain input material parameters for the numerical simulation. A set of the indirect tensile test were simulated to study the cracking behavior of AC at two levels of temperature, i e, 10 ℃ and 15 ℃. The predicted results of the numerical simulation were compared with laboratory experimental measurements. Results show that the 3D DEM model is able to predict accurately the fracture pattern of different asphalt mixtures. Based on the DEM model, the effects of air void content and aggregate volumetric fraction on the cracking behavior of asphalt concrete were evaluated. Key words: discrete element method; asphalt concrete; cracking behavior; three-dimensional simulation; micromechanics

1 Introduction Asphalt concrete (AC) cracking, thermal or fatigue, constitutes one of the major failure mechanisms of flexible pavements. Understanding the mechanical fundamentals behind the initiation and propagation of cracks is a critical step in developing accurate, mechanistic design procedures based upon pavement performance. Previous studies predicting AC cracking behavior were based primarily on experimental tests such as the beam bending tests and indirect tensile tests. However, cracking properties predicted from these tests come at the expense of conducting time-consuming and labor-costly test procedures. Micromechanical modeling, capable of reducing greatly the testing costs, has shown great potential in characterizing asphalt– aggregate mixtures for both material evaluation and structural design purposes. The major numerical method employed for characterizing AC behavior has been the finite ©Wuhan University of Technology and SpringerVerlag Berlin Heidelberg 2011 (Received: Dec. 29, 2010; Accepted: Apr. 10, 2011) CHEN Jun (陈俊): Ph D; E-mail: [email protected] Funded by the National High-tech Research and Development of China ('863' Program) (No. 2006AA11Z110)

element method (FEM), based on continuum-damage theory and/or fracture mechanics[1-3]. The FEM-based methods have been successful in capturing the stressstrain distribution within the asphalt mixtures and its effect on the stiffness anisotropy[4-6]. However, the current limitation with this approach is the convergence difficulties in modeling that frequently alters the three-dimensional (3D) aggregate contact geometry (aggregates coming in and out of contact and sliding during loading). Furthermore, the modeling of aggregate or mastic fracture during strength test simulations is very cumbersome based upon current finite element capabilities. Discontinuity-based numerical approaches for material modeling appeared in the late 1960s in more or less parallel developments of soil and concrete mechanics, developed via properly modifying existing continuum methods. One discontinuity-based method, i.e., the discrete/distinct element method (DEM), has received considerable attention in the past 30 years after it had been gradually developed by Cundall[7]. Pioneering works by Rothenburg et al[8] and later on by Chang and Meegoda[9] demonstrated the capability of DEM in effectively studying the interaction between idealized elastic aggregates. Using the Particle Flow

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Code in 2-Demensions (PFC2D), Buttlar, You, Abbas and Kim further developed two-dimensional (2D) image-based micromechanical models to predict asphalt mixture stiffness and fracture behavior [10-12]. DEM simulation of asphalt mixtures in these studies however were performed using mainly 2D image-based model. These 2D models are limited in capturing the interlock effect of aggregate particles in the 3D domain, therefore are normally inconsistent to laboratory results. The 2D image-based models are also highly dependent on image processing techniques. The main objective of this paper is to develop a user-defined 3D DEM model for investigating the cracking behavior of asphalt concrete at the micromechanical level. A commercial discrete element code-the particle flow code in 3-Demensions (PFC3D) was used. The irregular shape of the aggregate particle was modeled using a clump of spheres based on DEM. The 3D microstructure of asphalt mixture was constructed using a used-defined computer-based program implemented in PFC3D. Different from the previous studies using DEM, which were mainly image-based, the air voids, mastic and coarse aggregate shape, gradation and volumetric fraction in this micromechanical model all can be generated using a user-defined function in PFC3D. This capacity greatly facilitates the simulation of different mixture designs. The 3D DEM model was used to simulate the indirect tensile tests to obtain the cracking characteristics of AC. The simulation results were compared to experimental results to assess the validity of the DEM model. The effects of a comprehensive set of input material parameters, including the air void content, aggregate modulus, aggregate fraction, cohesive strength and adhesive strength, on the fracture behavior of asphalt mixture were investigated and discussed.

2 Experimental Aggregate shape has significant effects on the behavior of asphalt mixtures[13-15]. The 3D irregular shape of individual aggregate particles was first modeled using a clump of spheres in PFC3D. Coarse aggregates with a given gradation were then dumped into a cylinder frame built in PFC3D to construct an aggregate blend. Subsequently, hexagonal packing spherical elements (i e, each element has eight neighboring discrete elements in the three dimensions) were generated to fill out the cylinder. The spheres located in the aggregate subdomain were modeled as aggregate elements and others were treated as mastics. Air void was generated by randomly deleting some mastic discrete elements according to a pre-determined air void content. The 3D discrete element model of

asphalt mixture was prepared finally after removing the original clumps generated to model the aggregate irregular shape. 2.1 Generation of irregular-shaped 3D particles In this step, overlapping spheres were used to form clumps based on a procedure that is capable of controlling the sphericity and angularity of each clump. The procedure to prepare 3D irregular-shaped aggregate as developed by Lu and McDowell[16] was adopted in this study. Fig.1 illustrates the representative irregular-shaped aggregate particles of different sizes as was used in this study. It should be noted that the shape and angularity of the particles can be easily changed by setting different shape parameters when generating the clumps.

For each clump, the volume is determined through equation (1) as follows, (1) where, n and m give the numbers of spheres and overlaps in a clump, respectively; Vi the volume of the ith sphere in the clump; Vjoverlap denotes the volume of the jth overlap between the successively generated spheres in the clump and is formulated as equation (2), (2) where, R1 and R2 are the radii of the two interesting spheres respectively; h1 and h2 the heights of two caps respectively. With the radius of spheres and the heights of caps obtained in PFC3D, the volume and the mass of each clump can be easily determined when the sphere density is specified. 2.2 Generation of 3D coarse aggregate blend After being generated, the coarse aggregate particles blended in an arbitrary gradation were set in a cylinder with a height of 63.5 mm and a diameter of 100 mm, to generate the aggregate structure for the asphalt mixture. The cylinder was generated in PFC3D using three walls. A check for possible particle overlaps and particle generation outside the cylindrical space were performed, and the generated particles were accepted only when there were no overlaps among them and the particles were completely included

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inside the given space. In this study, coarse aggregate particles larger than 2.36 mm were placed into the cylinder to construct the aggregate assembly, with due consideration of the computing efficiency of DEM. Fig.2 shows a snapshot of the aggregate assembly consisting of aggregate clumps of different sizes in the cylinder, i e, the 13.2 mm-16 mm 80 g (3 clumps), 9.5 mm-13.2 mm 240 g (213 clumps), 4.75 mm-9.5 mm 320 g (612 clumps) and the 2.36 mm-4.75 mm 160 g (829 clumps). It is noteworthy that the mass of each aggregate and the entire aggregate assembly can be easily controlled using the user-defined function, which makes it very convenient to construct a 3D DEM model with different aggregate gradations and volumetric fractions.

2.3 3D discrete element modeling of aggregate-mastic system In PFC3D, it is not convenient to define directly the material properties for a multi-inclusion sample with irregular packing. However, with known material properties and regular packing geometries, it is possible to develop analytical expressions for modulus and strength. In order to obtain regular packing, a hexagonal particle arrangement was used for this study. Although there were other possible arrangements, such as random or square particle arrangements, the hexagonal arrangement was chosen with the following considerations: for the model under construction, it is not practical to assign heterogeneous bond properties to the random particle contacts; the theoretical Poisson’s ratio used for loading the square arrangement is zero in one of the principle directions of square arrangement.

Hexagonal packing spheres (63,222 spheres), each with a radius of 1 mm, were generated to fill out the

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cylinder in this step. Spheres located in the aggregate clumps were considered as aggregate elements and others were considered as asphalt mastics. The volume fraction of the aggregate phase was computed by dividing the number of aggregate elements (32,045 spheres) by the total number of elements, which gave a value of 50.1%. After deleting the clump generated in the first step, the 3D discrete element model of an aggregate-mastic system was developed. Fig. 3(a) shows the DEM model of the cylinder filled with hexagonal array of spheres, and Fig. 3(b) illustrates the developed 3D aggregate-mastic digital sample. 2.4 Air voids in 3D discrete models Two methods are available in PFC3D for generating air voids in the 3D discrete element model of digital samples. One is to set the mechanical parameters as zero for void elements in 3D models, and the other is to randomly remove some mastic discrete elements until the void volume percentage reaches the desired value. In this study, the second method was adopted and the air void contents were introduced at 0%, 3% and 6% to investigate on the effect of air voids on AC cracking behavior. The total numbers of void elements for the 3% and 6% air voids were 1,897 and 3,793, respectively. The air void distributions of the 3D models are shown in Fig.4. It should be noted that due to the complexity of air void distribution in field and laboratory samples [17], the air void distribution and the size of void were assumed to be uniform in this study.

2.5 Contact model and material properties There are four types of contacts in PFC3D that could be used to represent four different types of interaction within the digital sample of asphalt mixture: contacts within asphalt mastic, between mastic and aggregate, between adjacent aggregates, and within aggregates. Considering the viscoelastic behaviors of asphalt mastic, the Burger’s model was employed for the contacts within asphalt mastic and between mastic and aggregate. The spring elements with stiffness k n and ks were used for the contacts between adjacent aggregates and within aggregates to simulate the linear elastic behavior of aggregate. The Burger’s model parameters were determined by the method developed by Liu et al[18]. The two parameters of the Maxwell

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element (E1 and 1 ) are expressed as (3) (4) *

where, the dynamic modulus (E ) at the loading frequency  is determined through dynamic tests in a laboratory condition. Then, the parameters E2 and 2 in the Kelvin element can be solved by substituting E1 and 1 into equations (5) and (6) at the loading frequency  , (5)

(6) where,  is the phase angle of mastic. After determining the macroscopic parameters of the Burger’ s model, a conversion from macroscale properties to the microscale model input parameters (as shown in Fig.5) was needed using the following set of equations, (7) (8) (9) (10) (11) (12) (13) (14) where, L is the sum of two neighboring spheres’ radius, and  is the Poisson’s ratio and is set as 0.5 for asphalt mastic. In addition to the parameters used in Burger’s model, the stiffness kn and ks in spring element should also be obtained to characterize the elastic behaviors of aggregate element. For known macro-properties and regular packing geometries, it is feasible to develop analytical expressions for modulus and strength. The relationship between macro- and micro-stiffness properties offered in PFC3D was adopted in this study. (15) where, E is the apparent Young’s modulus, for aggregate phase the modulus is fixed at 50 GPa in the study, kn the input normal stiffness in PFC3D, and R

the radius of discrete element in the 3D models.

3 Results and Discussion 3.1 Laboratory measurements Laboratory uniaxial complex modulus tests and indirect tensile (IDT) tests were performed to determine the bulk material properties of mastics. The sand mastic had a nominal maximum aggregate size of 2.36 mm, which was obtained from the mixture’s aggregate gradation by eliminating all the aggregates bigger than 2.36 mm. The sand mastic had around 14% asphalt content by weight. The gradation of mixture is shown in Table 1. Young’s modulus of mastics was obtained at 10 ℃ and 15 ℃ with a 10 Hz loading frequency using a compressive uniaxial dynamic modulus test (ASTM D3497-79, 2003). Parameters in Burger’s model were obtained by solving equations (3)-(6) and listed in Table 2. The tensile strength of mastics was obtained following laboratory test procedures, with average values of 5.0 MPa and 2.1 MPa at 10 ℃ and 15 ℃, respectively. The asphalt mixture samples of a 100 mm diameter and a 63.5 mm height were compacted to a target 3% air void level by volume, with a nominal maximum aggregate size of 13.2 mm and 5.0%

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asphalt content by mass. The indirect tensile tests were performed at two temperature levels: 10 ℃ and 15 on the cylindrical samples. To compare experimental measurements with DEM predictions, the gradation of coarse aggregates (shown in Table 1) in the experimental mixture was the same as that simulated by the 3D model developed above. 3.2 DEM simulation of AC cracking Discrete element simulations of a set of indirect tensile tests were conducted using PFC3D to study the cracking behavior of AC. The mechanical compressive force was loaded to the top plate of 3D model with a constant displacement velocity of 50 mm/min, as shown in Fig. 6(b). The bottom plate of the sample was fixed in all directions. Figs. 6(a) and (c) present two cross-sections of the 3D model for better observation of the changing in microstructure and cracking patterns of the digital asphalt mixture sample during numerical testing. Throughout the simulation, the compressive loadings applied to the numerical sample and the horizontal tensile strains were monitored every ten calculation steps. The horizontal stress of the 3D model can be determined via equation (16), where  is the tensile stress of the numerical sample, h the thickness of the sample and was set to be 63.5mm, and P the vertical force applied to the sample. (16) Fig.7 illustrates the experimental and numerical results expressed in terms of horizontal tensile stress versus strain for the samples with the 16 mm nominal maximum aggregate size. Same loading rates were applied in the numerical simulation and experimental tests. The samples prepared in laboratory have the same gradation and air void content as the corresponding numerical 3D models. The results show that the global responses agree with experimental results. Due to the material complexity in experimental samples, the numerical results show little deviation from the experimental ones. Some reasons for the different results are: the spatial distribution of coarse aggregates

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and air voids in the 3D DEM model is not the same as those in the laboratory samples; theoretically, the crack path is not deterministic considering the stochastic nature of the stability in loading frame and control of loading pattern and values.

3.3 Data analysis and discussion Different from the previous studies that used image-based model, the volume parameters of the digital samples of asphalt mixture in this study can be controlled by user-defined functions when developing the 3D DEM model. The effects of coarse aggregate fraction, modulus, air void percentage, the strength of mastics and the strength between the mastics and aggregate on AC cracking behavior were investigated in the following sections. The air void was generated by randomly removing mastic elements as illustrated in Fig.4. The simulation of indirect tensile test was performed at 15 ℃. The applied stress and strain response of 3D

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models at different air void levels are plotted in Fig.8. It was found that with the increasing air void content from zero to 4% the tensile strength decreased from 1.4 MPa to 1.34 MPa. The tensile strength decreased from 1.34 MPa to 1.1 MPa as the air void increased from 3% to 6%. It is expected that the mixtures with low air void exhibit better resistance to fracture than that with high air void contents.

3D DEM models with air void level of 3% and different coarse aggregate volume (40%, 45% and 50%) were constructed by changing the mass of coarse aggregate included into the cylinder. The simulation of indirect tensile test was performed on the models at 15 ℃. The stress and strain responses are shown in Fig.9. It can be observed that increase in aggregate volume leads to higher tensile strength and larger deformation of asphalt mixture. When the aggregate volume increased from 40% to 50%, the mixture tensile strength and deformation was increased by 33% and 56% respectively. Cracking behavior of 3D models with different aggregate modulus: 25 GPa, 50 GPa and 75 GPa are shown in Fig.10. It is clearly that the mixture tensile strength and deformation did not change significantly with aggregate modulus. Moreover, compared to air voids and aggregate modulus, the aggregate volume has more significant effect on mixture fracture behaviors, especially on the tensile deformation. It is implied that the increase in coarse aggregate volume is among the most effective approaches to increase mixture’s resistance to tensile cracking.

Effects of cohesive strength of asphalt mastics and adhesive strength between asphalt mastic and aggregate were also investigated. Fig.11 shows the applied stress and strain responses with the adhesive strength of 2.5 MPa and different adhesive strength magnitudes. Increased cohesive strength can increase the tensile strength and deformation of mixtures to a certain extent. Similar plots were produced for the mixtures with the adhesive strength range from 2.5 MPa to 4.0 MPa and cohesive strength fixed at 2.5 MPa, as shown in Fig.12. When the adhesive strength increased, the fracture intensity and strain of mixture increased also.

Moreover, the area under the curve indicates the fracture toughness that equals the energy consumed for the initiation and propagation of cracks. It can be seen that increased cohesive strength can cause increase in fracture toughness as compared to adhesive strength, especially in process of the crack propagation. From this point of view, increase in cohesive strength has

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more effects on cracking resistance of asphalt mixture than the increase in adhesive strength does.

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Modeling of Asphalt Materials Using Finite Elements and Doublet Mechanics[J]. Mechanics of Materials, 2005,37(6): 641-662

4 Conclusions a) A 3D discrete element model of asphalt mixture was constructed, which takes into consideration the irregular shape, gradation, volumetric fraction of aggregate and the void contents using the “Fish” language provided in PFC3D. The user-defined 3D DEM sheds light on the cracking behavior of AC from the micromechanical perspective. b) The simulations of indirect tensile tests were performed to predict the fracture intensity and deformation at 15 ℃and 10 ℃. The favorable agreement between the 3D discrete element prediction and laboratory measurements indicates that the 3D discrete element model developed in this study is capable of simulating the cracking behavior of asphalt mixtures as a multi-inclusion composite. c) The effects of air void percentages, aggregate volume, the aggregate modulus, cohesive strength and adhesive strength on the cracking behavior were discussed based on the model developed in this paper. Compared to air voids and aggregate modulus, coarse aggregate volume fraction has more significant effect on mixture fracture behaviors, especially on the tensile deformation. Increasing in cohesive strength has more effects on cracking resistance of asphalt mixture than the increase in adhesive strength does.

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