Three-Dimensional Discrete Element Models for Asphalt ... - CiteSeerX

Three-Dimensional Discrete Element Models for Asphalt Mixtures Zhanping You, Ph.D., P.E.1; Sanjeev Adhikari2; and Qingli Dai, Ph.D.3 Abstract: The main objective of this paper is to develop three-dimensional 共3D兲 microstructure-based discrete element models of asphalt mixtures to study the dynamic modulus from the stress-strain response under compressive loads. The 3D microstructure of the asphalt mixture was obtained from a number of two-dimensional 共2D兲 images. In the 2D discrete element model, the aggregate and mastic were simulated with the captured aggregate and mastic images. The 3D models were reconstructed with a number of 2D models. This stress-strain response of the 3D model was computed under the loading cycles. The stress-strain response was used to predict the asphalt mixture’s stiffness 共modulus兲 by using the aggregate and mastic stiffness. The moduli of the 3D models were compared with the experimental measurements. It was found that the 3D discrete element models were able to predict the mixture moduli across a range of temperatures and loading frequencies. The 3D model prediction was found to be better than that of the 2D model. In addition, the effects of different air void percentages and aggregate moduli to the mixture moduli were investigated and discussed. DOI: 10.1061/共ASCE兲0733-9399共2008兲134:12共1053兲 CE Database subject headings: Micromechanics; Asphalts; Mixtures; Discrete elements; Two-dimensional models; Threedimensional models; Voids; Aggregates.

Introduction Asphalt mixture is a composite material of graded aggregates combined with asphalt binder and a certain amount of air voids. This multiphase material can be modeled with three phases, sand mastic, coarse aggregate, and air void phases. The mastic phase 共sand mastic兲 includes fine aggregate, sand, and fines which are embedded in a matrix of asphalt binder. One cubic inch of asphalt mixture has millions of mineral particles 共coarse and fine aggregates, sand, and fines兲, Therefore, it is necessary to define the particle size in mastic within a certain number, which is dependent on the computing capacity of the asphalt mixture model 共You and Buttlar 2004, 2005兲. In this paper, the mastic phase is considered as a uniform and homogeneous material, which in fact is composed of fines, fine aggregates smaller than 1.18 mm, and the asphalt binder. With the advanced modeling and computing capacity, the maximum aggregate size in a mastic can be even smaller than 0.075 mm 共No. 200 sieve兲 i.e., without fine aggregate 共Abbas et al. 2005; Buttlar et al. 1999兲. The asphalt mixture used in this study is a coarse Superpave Gyratory-compacted mix1 Donald and Rose Ann Tomasini Assistant Professor, Dept. of Civil and Environmental Engineering, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931-1295 共corresponding author兲. E-mail: [email protected] 2 Graduate Research Assistant, Dept. of Civil and Environmental Engineering, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931-1295. E-mail: [email protected] 3 Research Assistant Professor, Dept. of Mechanical EngineeringEngineering Mechanics, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931-1295. E-mail: [email protected] Note. Associate Editor: Ching S. Chang. Discussion open until May 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this paper was submitted for review and possible publication on December 7, 2006; approved on March 6, 2008. This paper is part of the Journal of Engineering Mechanics, Vol. 134, No. 12, December 1, 2008. ©ASCE, ISSN 0733-9399/2008/12-1053–1063/$25.00.

ture with a nominal maximum aggregate size of 19 mm. The asphalt specimen was scanned to obtain the microstructure of the aggregates and mastic. The combination of parallel twodimensional 共2D兲 images was used to capture the microstructure of three-dimensional 共3D兲 images. The micromechanical-based discrete element method 共DEM兲 was used to predict the modulus of the 2D and 3D models of the asphalt mixture. The objectives of this paper are to: 共1兲 develop 3D micromechanical DEMs to simulate the stress-strain response and predict the modulus of an asphalt mixture; 共2兲 compare the modulus prediction from both 2D and 3D discrete element models with the lab measurements; 共3兲 investigate the effect of different air void percentages on the asphalt mixture modulus; and 共4兲 study the parametric sensitivity of the aggregate moduli on the asphalt mixture moduli. Microstructure-based modeling techniques 共often referred to as micromechanical models兲 are recognized as important research areas in engineering and materials science. Progress in the development of a microstructure-based discrete element modeling technique will significantly advance the understanding of asphalt materials, resulting in safer, more secure, and longer-lasting asphalt pavement structures.

Historical Study and Current Methods of Micromechanical Models The modulus prediction of asphalt concrete is very difficult because it deals with the different sizes and shapes of aggregates bonded with time and temperature-dependent asphalt materials. The micromechanical model is a very useful method to predict the modulus of the asphalt concrete by providing the modulus properties of the constituent materials. There are a number of researchers who have used micromechanical models to predict the modulus of asphalt mixtures 共Christensen et al. 2003; Christensen and Lo 1979; Counto 1964; Hashin 1962, 1965; Hirsch 1962;

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1053

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Kerner 1956; Mori and Tanaka 1973兲. Hirsch 共1962兲 gave the basic model which used the parallel and series theory of the heterogeneous two-phase material. Hashin and Shtrikman 共1963兲 developed the composite arbitrary two-phase geometrical model based on the elastic theory. Hashin 共1965兲 introduced a model that has spherical particles embedded in the matrix. Mori and Tanaka 共1973兲 developed another model by using spherical elastic particles isotropically dispersed in an elastic matrix by applying effective bulk and shear moduli. Christensen and Lo 共1979兲 developed the composite sphere model which is embedded in an infinite medium of some unknown effective properties. The application of the micromechanical model to asphalt mixtures has been studied by many researchers in recent decades 共Buttlar and You 2001; Chang and Meegoda 1999; Dai and Sadd 2004; Dai et al. 2005; Guddati et al. 2002; Kim and Buttlar 2005; Papagiannakis et al. 2002; Park et al. 1999; Sadd et al. 2003, 2004; Wang et al. 2003兲. Micromechanical models of asphalt mixtures have been used to predict the asphalt mixtures physical properties based on the ingredients 共Li et al. 1999; Li and Metcalf 2005; Papagiannakis et al. 2002兲. Asphalt concrete was considered as a two-phase composite material of an aggregate in an asphalt matrix. The two-phase micromechanical model of the asphalt concrete was developed by using a 2D finite-element method 共FEM兲 or DEM. The strain distribution within asphalt binders in asphalt mixtures was reported by Kose et al. 共2000兲 and Masad et al. 共2002兲 by using X-ray computed tomography images. Based upon the progress of these researchers, a microstructure based FEM model of the asphalt mixture was developed by Dai et al. 共2006兲 and Dai and You 共2007兲. The multiphase asphalt mixture materials, including aggregates and mastic, were divided into different subdomains. Finite-element mesh was generated within each selected aggregate or mastic subdomain to predict the complex moduli of the asphalt mixture using uniaxial compression tests. The micromechanics-based DEM has received considerable attention in the past 20 years after it had been gradually developed by Cundall 共1971, 1987, 1990, 2000兲 and many others 共Chang and Chao 1991, 1994; Hart et al. 1988; Ng 2006; Ng and Changming 2001; Shimizu and Cundall 2001兲. Rothenburg et al. 共1992兲 developed a micromechanical model for asphalt concrete to investigate pavement rutting. An innovative feature of the model was the idea of intergranular interactions. Rothenburg’s model is useful for modeling asphalt concrete because it considers not only the aggregate-binder interaction, but also intergranular interactions in the presence of the binder. Chang and Meegoda 共1997兲 developed a model based on the discrete element method by modifying the TRUBAL program to simulate asphalt mixtures. These researchers developed the ASBAL model considering the viscoelastic elements of the asphalt mixture to represent the asphalt cement. Chang and Meegoda 共1999兲 then used the ASBAL program with Burgers’ viscoelastic parameters of springs and dashpots to study the temperature effect on the stress-strain behavior at higher temperatures. Collop et al. 共2004兲 applied a distinct element method 共similar to the DEM兲 to investigate the mechanical behavior of idealized asphalt materials. It was stated that the DEM can possibly be used to predict initial elastic, viscoelastic, and viscoplastic behavior of asphalt mixtures. Abbas et al. 共2005兲 used the DEM to simulate the dynamic mechanical behavior of asphalt mastics. The mastic used in the study contained the dispersions of aggregate fillers in an asphalt binder, and the fillers referred to are the fraction of mineral aggregates smaller than 75 ␮m. It was found that the DEM results captured

Table 1. Aggregate Gradation of Asphalt Mixture and Sand Mastic to Measure Dynamic Modulus Nominal maximum aggregate size of asphalt mixture 共percentage passing兲 Asphalt mixture Sieve size 共mm兲

Sand mastic

25 19 12.5 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075

100.0 — 98.9 — 77.5 — 68.4 — 56.2 — 46.2 — 31.7 100 20.8 65.6 12.6 39.7 8.3 26.2 5.6 17.7 Air void 共%兲 4.0 0a Asphalt content 共%兲 4.8 13.7 a Air voids were assumed to be zero for the sand mastic because of high asphalt content.

the stiffening behavior of asphalt mastics as a function of the volumetric concentration of mineral fillers. Recent studies 共Buttlar and Dave 2005; Buttlar and You 2001; You and Buttlar 2006; You and Dai 2007兲 have shown that the traditional micromechanical models using the composite spheres model and the arbitrary phase geometry in the 2D model do not adequately describe the complex microstructure of asphalt mixtures. These models predict 共over or under兲 the modulus of asphalt mixtures due to the inability to incorporate the aggregate interlock’s contribution to the overall response of the mixture. The 3D models are expected to have a better capability to predict the mixture’s aggregate interlock and improve the predictions. Therefore, 3D microstructure based modeling techniques for asphalt mixture materials are needed to improve the understanding of the fundamental properties of asphalt mixtures and pavements.

Laboratory Measurements The asphalt mixture used in this study was a 19 mm nominal maximum aggregate size mix, which contained Superpave performance grade 共PG兲 64-22 binder. The asphalt mixture had an asphalt content of 4.8% and was compacted in a Superpave gyratory compactor to a targeted air void level of 4%. The sand mastic was comprised of aggregate gradation finer than 1.18 mm and combined with the binder used in the entire asphalt mixture. The asphalt content in the sand mastic was 13.7%. The air void was assumed to be zero because of the high asphalt content in the sand mastic. Table 1 shows the aggregate gradation of asphalt mixture and sand mastic used in this study. The asphalt mixture was compacted using the gyratory compactor with a height of 160 mm and diameter of 150 mm. The sand mastic was compacted in a cylindrical mold with a height of 190 mm and diameter of 76 mm. The compacted asphalt mixture specimen had a diameter of 100 mm and height of 150 mm, and was obtained by coring and cutting the original specimen. The sand mastic specimen had a diameter of 75 mm and height of 150 mm and was obtained by trimming the ends of a 190 mm tall specimen.

1054 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

E* (GPa)

100

10 Mastic Measurements (0C) HMA Measurements (0C) Mastic Measurements (-10C) HMA Measurements (-10C) Mastic Measurements (-20C) HMA Measurements (-20C)

1 0.1

1

10

Loading Freq. (Hz)

a. Raw image of asphalt mixture

b. Image processing of aggregate

surface

>1.18mm

Fig. 2. Examples of image before and after image processing

Fig. 1. E* measurements for sand mastic and asphalt mixture across a range of temperatures 共0, −10, and −20° C兲 loading frequencies 共0.1, 1, 5, and 10 Hz兲

The compacted asphalt mixture specimen and sand mastic were used in measuring the dynamic 共complex兲 modulus, denoted E*, through uniaxial compressive testing. There were a total of five specimens and four sand mastic specimens that were used in finding the dynamic modulus test. The asphalt content in the sand mastic was around 13.7% by weight of the sand mixture, and therefore the sand mastic was flowable at high temperatures. The E* of both the sand mastic and asphalt mixture specimens were tested over a range of loading frequencies 共0.1, 1, 5, and 10 Hz兲 and test temperatures 共0°, −10, and −20° C兲. Fig. 1 shows the E* of sand mastic and asphalt mixture specimens. The dynamic modulus of the asphalt mixture and sand mastic specimens was also measured at 0.01 and 25 Hz, but the data were not included in the simulation. The E* of the sand mastic and modulus of aggregate were used as input parameters of the models. The aggregate blend used in this study was a mix of two or more types of geological rocks. A modulus of 55.5 GPa was used in the models for the aggregate. The lab measured E* of asphalt mixtures were used to compare the prediction results and to evaluate the modeling approach. In the E* tests, the total strain of the asphalt mixture and mastic were controlled in a linear viscoelastic range based upon a study by Airey et al. 共2004兲.

Discrete Element Models Used in This Study The DEM was used to simulate the complex behavior of the asphalt material by combining simple contact constitutive models with complex geometrical features. The particle flow code 共PFC兲 2D and 3D were used as discrete element codes in this study. The simpler contact models such as shear and normal stiffness, static and sliding friction, and interparticle cohesion can be employed in the DEM models. The stiffness model was used in this study. The contact stiffness was computed by assuming that the stiffness of the two contacting entities acts in series. The force-displacement law of the two elements in contact can be represented using different constitutive laws. The relationship between the average ¯ ij, in a volume, V, of the continuous media 共Cunstress tensor, ␴ dall 1971; Itasca Consulting Group 2004a兲 is defined by

¯ ij = ␴

1 V

冕

␴ijdV

共1兲

v

The stresses exist in the elements of aggregate and mastic for an asphalt mixture. Thus, the integral is replaced by a sum over the number of elements 共particles, N p兲 contained within the volume, V. The average stress tensor in a measurement circle is found by the following equation: ␴ij = −

1−n 兺NpV共p兲

F共c兲 兺 兺 兩x共c兲i − x共p兲i 兩n共c,p兲 j i

共2兲

The summations are taken over the centroids of the number of particles, N p, or discrete elements within the measurement circle and the Nc contacts of these spheres; n = porosity within the measurement circle; V共p兲 = volume of particle, 共p兲, equal to the area of 共c兲 particle times a unit thickness; x共p兲 i and xi = locations of a particle = unit normal vector centroid and its contact, respectively; n共c,p兲 i directed from a particle centroid to its contact location; and F共c兲 j = force acting at contact, 共c兲, arising from both the particles contact and parallel bonds. The strain-rate tensor of the discrete element method computed by a given measurement circle represents the best fit to the N p by measuring the relative velocity value, ˜V共p兲 i , of the particles at the centroids.

Preparation of 3D Discrete Element Models The 2D microstructure of the asphalt concrete mixture was first captured with a high-resolution scanner, manipulated using image processing techniques, and reconstructed into an assembly of discrete elements. 3D structure is constructed by using a combination of different layers of 2D images. The microstructure of the asphalt mixture was obtained by using scanned images of many slices as shown in Fig. 2. Image processing techniques were used to identify the aggregate skeleton from the image. For example, in this study an aggregate size larger than 1.18 mm was selected as the minimum aggregate size in the aggregate skeleton. Therefore, the mastic included all the aggregates finer than 1.18 mm. Fig. 2共a兲 shows the raw image of asphalt mixture before image processing and Fig. 2共b兲 shows the image of asphalt mixture after image processing. Air voids were not captured in the 2D images since the diamond saw may squeeze the asphalt into the air voids. In the modeling procedure, however, the air voids were virtually applied in the specimens.

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1055

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Model 1

Model 2

Model 4

Mastic

Aggregate Model 3

Model 4

a) 2D models

Model 3 Model 2 Model 1

b) 3D model Fig. 3. Illustration of reconstruction of 3D model from 2D models

The modulus measured in the laboratory was of 4% air void level. Therefore, it was necessary to introduce the air void level to compare the modulus of predictions and lab measurements. Each slice of the 2D image was then converted to 2D DEM models. A number of these models were used to form a 3D DEM model as described in the following steps. Step 1: 2D and 3D Models In this step, a rectangular shaped specimen with a height of 242 units 共81 mm兲 and width of 160 units 共53 mm兲 was generated from a collection of manipulated raw asphalt mixture images. The radius of each ball 共i.e., disk or discrete element兲 is 1 unit 共1 / 3 mm兲. There are a total of 9,680 balls in the 2D digital 共virtual兲 specimen. The volume of the aggregate was computed by dividing the number of aggregate elements by the total number of elements, which is 53% 共5,124 balls兲. Randomly generated air voids were used throughout the model. The air void was generated in the mastic element, keeping the aggregate constant. When the air voids were introduced at 4 and 7%, respectively, the total numbers of discrete elements were reduced accordingly. It should be noted that the air voids were not directly modeled with discrete elements in this paper. However, the discrete elements in the air void subdomain were removed from the DEM model. Fig. 3共a兲 shows the different images of 2D DEM models without air voids. When many of the 2D models are placed parallel together, a 3D model is reconstructed by using face-centered packing spheres 共i.e., each element has six neighboring discrete elements in the three dimensions兲. A rectangular prism shaped specimen with height of 81 mm, width of 53 mm, and depth of 32 mm was generated by using four different prepared 2D models. The depth of 32 mm prism was chosen to minimize the size of the sample and to reduce computational work. The radius of each sphere is 1 / 3 mm 共as a unit兲. There are a total of 464,640 spheres in this 3D asphalt mixture specimen without air voids.

The aggregate volume is 53% 共245,952 balls兲 of the whole specimen. Since the aggregate size was more than 1.18 mm, the minimum amount of balls used in the aggregate were two. Fig. 3共b兲 illustrates the generation of the 3D model using the four 2D models. The 2D plane was duplicated 12 times 共about 8 mm deep兲 because it was difficult to get a scanned image smaller than 8 mm deep without using X-ray computed tomography. The shear and normal strength bonds are assigned in the 3D model for the asphalt mixture. In order to simulate the air void effect on the mixture, artificial air voids were introduced into the model by randomly removing the mastic elements from the model. Fig. 4 illustrates the generation of three 3D models at different air void levels. Fig. 4共a兲 shows the overview of a 3D model at 0% air void level. In this study, the air voids were introduced at 4 and 7%, respectively. The air void levels of the 3D model are shown in Figs. 4共b and c兲. It should be noted that the air void distribution in the compacted specimen is not uniform for both field and lab specimens. The research team is currently working on the air void distribution in asphalt mixtures using the X-ray tomography technique. The total number of discrete elements and the aggregate volume concentration for 0, 4, and 7% air voids are listed in Table 2. It took a great deal of time to compare the 2D to the 3D DEM model as discussed in following sections. Step 2: Boundary Conditions of Discrete Element Models The force and displacement boundaries were generated in the rectangular and rectangular prism shaped asphalt mixture specimen. The biaxial compression method was employed in the 2D rectangular specimen and the triaxial compression method was employed in the 3D rectangular prism specimen. A constant confining stress environment was prepared with the four walls around the specimens in the biaxial compression simulation and six walls

1056 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

a) Overview of 3D model

b) Air void level of 4%

c) Air void level of 7%

Fig. 4. Illustration of 3D models at different air void levels

around the triaxial compression simulation. During the biaxial and triaxial test, the loading rate must be slow enough to ensure that the specimen remains in quasi-static equilibrium throughout the test, such that no inertial effects occur 共e.g., premature bond failure resulting from a stress wave passing through the specimen or platen loads in excess of their quasi-static equilibrium values兲. The adequacy of the loading rate can be verified by stopping the platens and observing that the platen load remains nearly constant. The walls were made longer than necessary to allow for the large strain during the test. These walls acted as boundary constraints. The top and bottom walls were moved at a specific velocity and their reaction forces were monitored or controlled by a servomechanism so as to maintain some prescribed reaction force. The axial compression force was evenly imposed on the top surface of the specimen with a constant loading speed. The stresses were computed by taking the average of the stresses acting on each set of opposing walls, whereas the stresses were computed by dividing the total force acting on the wall by the appropriate sample length. The strains on the x and y direction were computed by the ratio of change in length to the original length in the

respective axis. A detailed discussion of the biaxial and triaxial compression test can be found in many references 共Itasca Consulting Group 2004a,b兲. Step 3: Contact Model and Material Properties „Micro- and Macrostiffness and Strength Conversion… There are a number of contact models available. The constitutive model acting at a particular contact consists of three parts: a stiffness model, a slip model, and a bonding model 共Itasca Consulting Group 2004b兲. The stiffness model was used to predict the modulus of an asphalt mixture in the DEM model by applying compressive loads. The stiffness model provides a linear elastic relationship between the contact force and relative displacement between particles. The stiffness model is considered the same for tension and compression loading behavior. Usually, it is difficult to define the microproperties of a specimen to represent a real material. The input properties 共such as moduli and strengths兲 can be derived directly from measurements of laboratory specimens. However, in most of the discrete ele-

Table 2. Comparison of 3D and 2D Discrete Element Models for Asphalt Mixture

3D

2D

Number of discrete elements Aggregate volume to the entire specimen volume 共%兲 Time step 共s/cycles兲 Cycles Final axial strain 共%兲 Computation timea 共h兲 Ratio of computation time to the element number 共s/element兲

Number of discrete elements Time step 共s/cycles兲 Cycles Final axial strain 共%兲 Computation timea 共min兲 Computation time over element number 共s/element兲 a Using a PC with 3.2 GHz of CPU and 3.25 GB of RAM.

0%

4%

7%

464,640 53 7 ⫻ 10−5 20,700 0.07153 55 0.43

446,054 53 7 ⫻ 10−5 20,700 0.06973 49 0.40

432,115 53 7 ⫻ 10−5 20,700 0.06750 48 0.40

9,680 7 ⫻ 10−5 20,500 0.07153 13 0.081

9,293 7 ⫻ 10−5 20,500 0.06973 12 0.077

9,002 7 ⫻ 10−5 20,500 0.06750 11 0.073

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1057

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

⌬␴a =

⌬Fa 共2R兲2

共3兲

The increment in strain, ⌬␧a, and the relative normal displacement between two adjacent particles, ⌬u, can be expressed as Eq. 共4兲 ⌬␧a =

⌬u 2R

共4兲

Contact Number in 1000

600

⌬Fa = 共

兲⌬u = 共

1 2 Kn

兲共2R⌬␧a兲 = KnR⌬␧a

E=

⌬␴a Kn = ⌬␧a 4R

共6兲

Similarly, the expression of the apparent Young’s modulus, E, of the 2D assembly in terms of the input normal stiffness, Kn, can be written as 共Itasca Consulting Group 2004a兲 E=

⌬␴a Kn = ⌬␧a 2

共7兲

For the packing conditions, macro- and microstiffness properties are more complicated but can be found in a number of references. But for other packing methods, this equation is only useful to estimate the Young’s modulus 共Itasca Consulting Group 2004a,b兲. In the case of this study, since the closed-packing cubic arrays of spheres were used, it is easy to determine the relationship between the micro- and macrostiffness and strength. For example, when the macromodulus, E, was 1 GPa, the micro-normal stiffness, Kn, was computed as 4 GPa of the 3D DEM since the radius of the ball 共i.e., disk or discrete element兲 throughout the model was taken as 1 unit 共1 / 3 mm兲. Step 4: Simulation of Triaxial Compressive Test The aggregate and mastic domains were modeled separately and then combined. As discussed previously, the air voids were generated by randomly removing the mastic discrete elements so that the void volume percentage reached a desired value. After the boundary conditions were set up and the microparameters of the bonds and the stiffness were entered, the triaxial test simulations

300 200

0-1

1-10

10-100

100-200

200-600

Compressive Contact Force (MN)

a) Compressive contact force distribution in a model during simulation

250 Model at 0°C and 1Hz 200

Model at -20°C and 10Hz

150 100 50 0 0-1

1-10

10-100

Tensile Contact Force (MN)

共5兲

Therefore, from these equations, the exact expression of the apparent Young’s modulus E of the assembly in terms of the input normal stiffness, Kn, of the 3D DEM can be written as

Model at -20°C and 10Hz 400

0

This displacement gives rise to a normal force. The apparent normal force of the two adjacent particles, ⌬Fa, is the series of a combination of normal stiffness, Kn, and ⌬u for both particles 1 2 Kn

Model at 0°C and 1Hz

500

100

Contact Number in 1000

ment codes such as PFC 3D 共Itasca Consulting Group 2004b兲, “synthesis” material behavior from the microcomponents 共equivalent grains兲 makes up the material. In this case, an “inverse modeling” is used. The “inverse modeling” approach performs a number of simulations with assumed properties and compares the results with the desired response of the real intact material. When a match has been found, the corresponding set of properties may be used in the full simulation. However, for known macroproperties and regular packing geometries, it may be possible to develop analytical expressions for modulus and strength. Thornton 共1979兲 provided expressions for the strength of closed-packing cubic arrays of spheres. For a cubic array of spheres 共with six neighbors兲, when a radius, R, of the particles is considered, the relationship of the increment in axial force, ⌬Fa, and increment of axial stress, ⌬␴a, can be written as Eq. 共3兲 共Itasca Consulting Group 2004b兲

b) Tensile contact force distribution in a model during simulation

Fig. 5. Compressive contact forces, tensile contact forces, and displacement chains for 3D model with 4% air void

were conducted. The E* of the asphalt mastic over a range of loading frequencies 共0.1, 1, 5, and 10 Hz兲 and test temperatures 共0, −10, and −20° C兲 were used as input parameters of the asphalt mixture modulus. The aggregate modulus was fixed at 55.5 GPa 共You and Dai 2007兲. The asphalt mixture modulus of the specific loading frequency and temperature was predicted using the corresponding loading frequency and temperature of the sand mastic. During the simulation, the compressive and tensile contact force chains and the displacement chains were monitored to understand the load transfer in the aggregate-mastic-air microstructure. As an example, Figs. 5共a and b兲 show the magnitude distribution of the compressive and tensile forces of the 3D DEM model at 0 ° C temperature and 1 Hz loading frequency, in addition to −20° C temperature and 10 Hz loading frequency. The 4% air void level was used in this example. Two 3D models, namely Model A 共for 0 ° C temperature and 1 Hz loading frequency兲 and Model B 共for −20° C temperature and 10 Hz loading frequency兲, were compared herein. For the two 3D models, a total of 1,268,597 contacts were included. For Model A, there were 974,200 effective compressive force contacts and 314,076 effective tensile force contacts. For Model B, there were 996,636 effective compressive force contacts and 291,811 effective tensile force contacts. It should be noted that some contacts became inactive during a specific scenario in the microstructure and therefore they were not “effective.” When comparing compressive contact forces, it was found that Model B had a higher amount of contact-to-contact force 共200– 600 MN兲, and Model A had a higher amount of contact-to-contact force

1058 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Step 5: Interpretation of Results After the simulation, a challenging step is the interpretation of results. Because most discrete element codes such as the PFC3D model a nonlinear system of discrete particles, as it evolves in time, the interpretation of the results may be more difficult than with a conventional continuum code that produces a “solution” at the end of its calculation phase. In this study, the measurement spheres 共circles for 2D models兲, contact force histories, strain histories, and many other variables were traced to capture the macro- and micromechanical responses in the specimens. The modulus was computed from the plots of axial deviator stress history versus axial strain history as discussed in Step 4.

Results from 3D Discrete Element Models As discussed in the previous section, the 3D DEM model was simulated using a compression test, and the modulus was calculated from the stress-strain curve. The asphalt mixture modulus of the specific loading frequency and temperature was predicted by using the corresponding loading frequency and temperature of the sand mastic. A comparison of the predictions and measurements of the mixture modulus under a range of mastic modulus is shown in Fig. 6. The modulus of the 3D DEM model was predicted by using 0% 共or, no air void兲, 4 and 7% air void levels. When the mastic stiffness ranges of 6.19, 8.69, 10.07, 13.93, and 18.35 GPa 共measured at certain of temperatures and loading frequencies兲 were used in the 3D DEM model, mixture moduli were estimated as 15.95, 19.98, 21.95, 26.77, and 31.40 GPa, respectively, at the 0% air void level with the mixture measurements being 14.67, 17.67, 19.59, 24.13, and 30.94 GPa, respectively. It should be noted that the asphalt mixture tested in the lab had a 4% air void level. Therefore, the 3D DEM modulus prediction at 0% air void level overpredicts estimates the modulus measured in the lab, which was anticipated in lab measurements. Consequently, in

35 30

Mix Modulus (GPa)

共0 – 1 MN兲. When comparing tensile contact forces, it was found that Model B had a higher amount of contact-to-contact force 共10– 100 MN兲, and Model A had a higher amount of contact-tocontact force 共0 – 1 MN兲. The compressive and tensile contact forces become higher by decreasing temperatures and frequencies. Therefore, the modulus of an asphalt mixture increases with decreasing temperatures and frequencies. It was also observed that the aggregate skeleton carries most of the loads in the model and the aggregate did not carry much of the tensile force in the model. It is very important to compute and control the stress state in the specimen during the test. The specimen was loaded by specifying the velocities of the top and bottom walls. The stress was computed by taking the average wall forces divided by the appropriate areas. The strains in the radial and axial directions were computed using the current radius or specimen length and the original radius or specimen length. Throughout the loading process, the confining stress was kept constant by adjusting the radial wall velocity using a numerical servo mechanism. After the confining stress state had been reached, the specimen was tested by releasing the top and bottom platens from servo control, and specifying the velocities of the top and bottom platens. During each test, the servo control kept the confining stress constant at the requested value 共Itasca Consulting Group 2004b兲 and the velocities of the top and bottom platens created the compression test in that direction 共z, in this case兲.

25 20 15 Measurement 10

3D modeling 0% air void

5

3D modeling 4% air void 3D modeling 7% air void

0 0

5

10

15

20

Mastic Modulus (GPa)

Fig. 6. Comparing dynamic modulus 共E*兲 prediction of asphalt mixture using 3D DEM

order to closely simulate the mixture modulus, 4% air void is also studied in the 3D models. In addition, a 7% model was studied to compare the effect of air voids. It was found that the asphalt mixture prediction decreases with air void levels, which was anticipated. It was also shown that the 4% model prediction matched the asphalt mixture measurements well. At 4% air void level, the difference between the 3D model predictions and the measurement values was within 5%. When the large air void level 共7%兲 was introduced into the models, the predictions were lower than those of the 4% model and generated a larger difference with the asphalt mixture measurements.

Comparison of Predictions from 3D and 2D Discrete Element Models The 3D and 2D discrete element models are discussed herein to evaluate whether the 3D models yield a better prediction of the mixture modulus. Since the asphalt mixture was compacted to a 4% air void level, it is worthwhile to revisit the predictions and the measurements. The E* of the asphalt mastic over a range of loading frequencies 共0.1, 1, 5, and 10 Hz兲 and test temperatures 共0, −10, and −20° C兲 were measured and used as input parameters in the 3D discrete element models. Therefore, at each loading frequency and test temperature, the prediction of the asphalt mixture modulus was obtained from the triaxial compressive test simulations of the 3D DEM models. The asphalt mixture modulus predictions from the 3D models 共with 4% air void兲 and lab measurements are shown in Fig. 7, over a range of loading frequencies and test temperatures. It was found that for each temperature and frequency, the prediction is very close to the asphalt mixture modulus measurements. The paired t-test was performed to evaluate the mean difference between paired observations when the paired differences follow a normal distribution. The paired t-test was used to compare the asphalt mixture modulus predictions and lab measurements because there were 12 different points across a range of loading frequencies and test temperatures and each paired observation had paired differences. In this case, the paired observations were the predicted modulus and lab measurements at the same temperature and loading frequency. Table 3 shows the t-test results of asphalt mixture modulus predictions and lab measurements. Based upon the paired t-test result, it indicates that the confidence interval for the mean difference between the 3D prediction and the lab measurements included zero 共−0.39, 1.65兲, which suggests no significant difference between them under a 95% confidence interval.

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1059

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

100

10

Mixture E* (GPa)

Mixture E* (Gpa)

100

Measurements -20C Measurements -10C Measurements 0C 3D Prediction -20C 3D Prediction -10C 3D Prediction 0C 1

Measurements -20C Measurements -10C Measurements 0C 2D Prediction -20C 2D Prediction -10C 2D Prediction 0C

1

1 0.1

10

0.1

10

1

10

Loading Frequency (Hz)

Loading Frequency (Hz)

Fig. 7. Comparing modulus prediction using 3D DEM and laboratory measurements of asphalt mixture at 4% air void 共from different loading frequencies and test temperatures兲

Fig. 8. Comparing modulus prediction using 2D DEM and laboratory measurements of asphalt mixture at 4% air void 共from different loading frequencies and test temperatures兲

The p value 共p = 0.20兲 suggests that the data are consistent and the predictions and the measurements are similar. The 3D prediction data 共mean= 19.36兲 is very close to the measurement data 共mean= 19.99兲 over the four frequencies and three temperatures. The 2D discrete element models were also used to compare the asphalt mixture predictions and lab measurements. Even though the predictions were fairly good compared with the experimental measurements for some coarse mixtures, it was found that the 2D models underpredict estimates of the mixture modulus for this mixture. The mixture modulus predictions from the 2D models 共with 4% air void兲 and measurements are shown in Fig. 8 across a range of loading frequencies and test temperatures. The paired t-test results indicated that the confidence interval for the mean difference between the 2D prediction and the lab measurements did not include zero 共1.75, 3.80兲, which suggested that the difference between them was at a 95% confidence interval. The small p value 共p = 0.00兲 further suggests that the difference is significant. Specifically, the 2D prediction data 共mean= 17.21兲 is lower than the measurement data 共mean= 19.99兲 over the four frequencies and three temperatures. Therefore, the 3D models

yielded a better prediction over the four frequencies and three temperatures. In addition, comparison of the 2D and 3D models is listed in Table 2, where three air void levels are compared in terms of the number of discrete elements, aggregate volume to the entire specimen volume, time step, cycles, final axial strain, computation time, and the ratio of computation time to the element number. Both 2D and 3D models at 4% air void level were simulated by a time step of 7 ⫻ 10−5 s / cycles. There were a total of 20,700 cycles used for the 3D model and 20,500 cycles for the 2D model. The final strain level was 0.06973% for both 2D and 3D models. The computation time for the 3D model was above 49 h when using a PC with capacities of 3.2 GHz CPU and 3.25 GB RAM. For the 2D model, the computation time was about 12 min. The ratio of computation time to the discrete element number was 0.40 s/element for the 3D model, but 0.077 s/element for the 2D model. The 3D model simulation took a longer duration than the 2D model. It was found that the final axial strain was decreased with air void level, and computation time also decreased with air void level when the time step is the same.

Table 3. Paired t-Test for 3D and 2D Discrete Element Prediction for Asphalt Mixture 共4% Air Voids兲

Air Void Distribution in Discrete Element Models

Paired t-test and confidence interval: C1, C2 Paired t for C1-C2 N

Mean

12 19.99 C1 12 19.36 C2 Difference 12 0.63 95% CI for mean difference: 共−0.39, 1.65兲 t-test of mean difference= 0 共versus not= 0兲: t

SD

SE mean

6.05 4.84 1.61

1.75 1.40 0.46

value= 1.35 p value= 0.20

Paired t-test and confidence interval: C11, C2 Paired t for C11-C2 N

Mean

SD

SE mean

12 19.99 6.05 1.75 C11 12 17.21 4.80 1.39 C2 Difference 12 2.78 1.61 0.47 95% CI for mean difference: 共1.75, 3.80兲 t-test of mean difference= 0 共versus not= 0兲: t value= 5.97 p value= 0.00 Note: C1 = 3D prediction; C2 = lab measurements, C11= 2D prediction.

The air void distribution of 2D and 3D DEM models is investigated in this section. The air void distribution is very close to a uniform distribution 共such as cross sections yz1, yz2, yz3, and yz4兲 on the vertical cross section, as illustrated in Fig. 9. The air void level of 4% was generated by a random function in the yz plane. The modulus of the 2D and 3D models was distinguished at different air void levels. A comparison between the 2D and 3D model predictions is shown in Fig. 10, which has a mastic modulus of 10.07 GPa that was tested at −10° C temperature and 1 Hz frequency. It was found that the 3D models yielded a higher prediction than the 2D models. It also shows that the mixture modulus decreases with air void content. It was found that the 2D and 3D predictions were very close when the air void was 0%; but when the air void level became 4%, the mixture modulus decreased to 20% for 2D and 13% for 3D models. When the air void level was increased from 4 to 7%, the mixture modulus prediction decreased to 22% for the 2D and 12% for the 3D models. As mentioned previously, the 2D model underpredicted the asphalt mixture modulus because it does not account for the aggregate

1060 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Cross section yz1 Cross section yz2 Cross section yz3 Cross section yz4

Cross section xy z

x

y

Air void distribution in Air void distribution in

Air void distribution

Air void distribution in

cross section yz1

in cross section yz3

cross section yz4

cross section yz2

Fig. 9. Illustration of air void 共4%兲 distribution in 2D and 3D models

interactions. A comparison of the DEM model in Fig. 10 indicated that the 3D model simulation increased the predicted stiffness. The image analysis using the X-ray tomography shows the exact air void distributions for a given asphalt mixture specimen. The air void distributions using the X-ray tomography image are part of the ongoing research projects.

Mix Modulus (GPa)

25 20 15 10 3D 2D

5 0 0

2

4

6

Aggregate Modulus Sensitivity Study Using 3D Discrete Element Models In the study, the aggregate modulus was fixed at 55.5 GPa. However, based on the measurements of the limestone cylinder specimens, the modulus of the aggregate was found to be in the range of 30– 85 GPa 共You and Dai 2007兲. The aggregate modulus sensitivity study using the discrete element models was conducted at the following conditions: 共1兲 fixed asphalt mastic modulus; and 共2兲 no air voids in the model for both 3D and 2D models. The modulus prediction of the 2D and 3D models using aggregate sensitivity study is shown in Fig. 11. This observation is similar to the findings for other 2D models studied previously 共You et al. 2006; You and Dai 2007兲. It was found that the aggregate modulus has a significant effect on the asphalt mixture modulus. When the modulus of aggregate was increased from 30 to 80 GPa, the mixture modulus was increased to 39% using the 2D models and 46% using the 3D models. It should be noted that the 3D predictions yielded a slightly higher modulus over a range of aggregate modulus.

8

Air Void (%)

Summary and Conclusions

Fig. 10. Comparing of mixture modulus prediction of 3D and 2D under different air voids 共at 0, 4, and 7% air voids and mastic modulus fixed at 10.07 GPa兲

This study focused on: 共1兲 3D micromechanical discrete element models developed to simulate the stress-strain response and predict the modulus of an asphalt mixture; 共2兲 comparing the modu-

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1061

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

Mix Modulus (GPa)

30 25 20 15 10 3D 2D

5 0 0

50

100

150

Aggregate Modulus (GPa)

Fig. 11. Aggregate modulus sensitivity comparisons under 2D and 3D models 共0% air voids兲

lus prediction from both the 2D and 3D discrete element models with lab measurements; 共3兲 investigating the effect of different air void percentages to the mixture modulus; and 共4兲 studying the parametric sensitivity of the aggregate moduli on the asphalt mixture moduli. The microstructure of the asphalt mixture was obtained with a number of scanned 2D images. The 3D DEM model was then reconstructed from a number of 2D models. The stressstrain response of the 3D model was determined from compressive loading cycles. The input data to the 3D models included the aggregate and mastic stiffness as well as the microstructure of the aggregate skeleton and mastic distribution. The moduli of the 3D models were then compared with the experimental measured data. It was found that the 3D discrete element models were able to predict the mixture modulus over a range of temperatures and loading frequencies. In addition, the effect of different air void percentages and the aggregate modulus effect to the mixture modulus were investigated and discussed in this paper. The ongoing work at Michigan Technological University includes modeling of the 3D heterogeneous microstructure and advanced viscoelastic contact laws. The actual 3D microstructure, using X-ray tomography of several asphalt mixtures, is being studied to characterize the mixture properties. It is expected that the 3D models of the exact heterogeneous asphalt mixture will enhance the understanding of the mechanism of design and construction of asphalt pavement materials.

Acknowledgments This study was partially supported by the State of Michigan Research Excellence Fund. This material is also based in part upon work supported by the National Science Foundation under Grant No. 0701264. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the writer’s and do not necessarily reflect the views of the National Science Foundation.

References Abbas, A., Masad, E., Papagiannakis, T., and Shenoy, A. 共2005兲. “Modelling asphalt mastic stiffness using discrete element analysis and micromechanics-based models.” Int. J. Pavement Eng., 6共2兲, 137–146. Airey, G. D., Rahimzadeh, B., and Collop, A. C. 共2004兲. “Linear rheological behavior of bituminous paving materials.” J. Mater. Civ. Eng., 16共3兲, 212–220.

Buttlar, W. G., Bozkurt, D., Al-Khateeb, G. G., and Waldhoff, A. S. 共1999兲. “Understanding asphalt mastic behavior through micromechanics.” Transportation Research Record. 1681, Transportation Research Board, Washington, D.C., 157–169. Buttlar, W. G., and Dave, E. V. 共2005兲. “A micromechanics-based approach for determining presence and amount of recycled asphalt pavement material in asphalt concrete 共with discussion兲.” Asphalt Paving Technol., 74, 829–884. Buttlar, W. G., and You, Z. 共2001兲. “Discrete element modeling of asphalt concrete: Microfabric approach.” Transportation Research Record. 1757, Transportation Research Board, Washington, D.C., 111–118. Chang, C. S., and Chao, S.-J. 共1991兲. “Discrete element method for bearing capacity analysis.” Comput. Geotech., 12共4兲, 273–288. Chang, C. S., and Chao, S.-J. 共1994兲. “Discrete element analysis for active and passive pressure distribution on retaining wall.” Comput. Geotech., 16共4兲, 291–310. Chang, G. K., and Meegoda, J. N. 共1999兲. “Micro-mechanic model for temperature effects of hot mixture asphalt concrete.” J. Trans. Res. Record. 1687, Transportation Research Board, Washington, D.C., 95–103. Chang, K. G., and Meegoda, J. 共1997兲. “Micromechanical simulation of hot mix asphalt.” J. Eng. Mech., 123共5兲, 495–503. Christensen, D. W., Pellien, T., and Bonaquist, R. F. 共2003兲. “Hirsch models for estimating the modulus of asphalt concrete.” Asphalt Paving Technol., 72, 97–121. Christensen, R. M., and Lo, K. H. 共1979兲. “Solutions for effective shear properties in three phase sphere and cylinder models.” J. Mech. Phys. Solids, 27, 315–330. Collop, A. C., McDowell, G. R., and Lee, Y. 共2004兲. “Use of the distinct element method to model the deformation behavior of an idealized asphalt mixture.” Int. J. Pavement Eng., 5, 1–7. Counto, V. J. 共1964兲. “The effect of the elastic modulus of the aggregate on the elastic modulud, creep, and creep recovery of concrete.” Mag. Concrete Res., 62共48兲, 129–138. Cundall, P. A. 共1971兲. “A computer model for simulating progressive large scale movements in blocky rock systems.” Proc., Int. Symp. Rock Fracture, ISRM, Vol. II-8, Nancy, France, 129–136. Cundall, P. A. 共1987兲. Distinct element models of rock and soil structure, E. T. Brown, ed., George Allen and Unwin, London. Cundall, P. A. 共1990兲. “Numerical modelling of jointed and faulted rock.” Proc., Int. Conf. on Mechanics of Jointed and Faulted Rock, Mechanics of jointed and faulted rock, Balkema, Rotterdam, The Netherlands, 11–18. Cundall, P. A. 共2000兲. “A discontinuous future for numerical modelling in geomechanics?” Geotech. Eng., 149共1兲, 41–47. Dai, Q., and Sadd, M. H. 共2004兲. “Parametric model study of microstructure effects on damage behavior of asphalt samples.” Int. J. Pavement Eng., 5共1兲, 19–30. Dai, Q., Sadd, M. H., Parameswaran, V., and Shukla, A. 共2005兲. “Prediction of damage behaviors in asphalt materials using a micromechanical finite-element model and image analysis.” J. Eng. Mech., 131共7兲, 668–677. Dai, Q., Sadd, M. H., and You, Z. 共2006兲. “A micromechanical finite element model for viscoelastic creep and viscoelastic damage behavior of asphalt mixture.” Int. J. Numer. Analyt. Meth. Geomech., 30, 1135–1158. Dai, Q., and You, Z. 共2007兲. “Prediction of creep stiffness of asphalt mixture with micromechanical finite-element and discrete-element models.” J. Eng. Mech., 133共2兲, 163–173. Guddati, M. N., Feng, Z., and Kim, R. 共2002兲. “Toward a micromechanics-based procedure to characterize fatigue performance of asphalt concrete.” Transportation Research Record. 1789, Transportation Research Board, Washington, D.C., 121–128. Hart, R., Cundall, P. A., and Lemos, J. 共1988兲. “Formulation of a threedimensional distinct element model-part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 25共3兲, 117–125. Hashin, Z. 共1962兲. “The elastic moduli of heterogeneous materials.”

1062 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright

J. Appl. Mech., 29, 143–150. Hashin, Z. 共1965兲. “Viscoelastic behaviour of heterogeneous media.” J. Appl. Mech., 32共9兲, 630–636. Hashin, Z., and Shtrikman, S. 共1963兲. “A variational approach to the theory of the elastic behaviour of multiphase materials.” J. Mech. Phys. Solids, 11, 127–140. Hirsch, T. J. 共1962兲. “Modulus of elasticity of concrete affected by elastic moduli of cement paste matrix and aggregate.” J. Am. Concr. Inst., 59共3兲, 427–452. Itasca Consulting Group. 共2004a兲. PFC 2D Version 3.1, Minneapolis. Itasca Consulting Group. 共2004b兲. PFC 3D Version 3.1, Minneapolis. Kerner, E. H. 共1956兲. “The elastic and thermo-elastic properties of composite media.” Proc. Phys. Soc. London, Sect. B, 69, 808–813. Kim, H., and Buttlar, W. G. 共2005兲. “Micro mechanical fracture modeling of asphalt mixture using the discrete element method.” Electron. J. Assoc. Asph. Paving Technol.(AAPT), Vol. 74E 209–223. Kose, S., Guler, M., Bahia, H. U., and Masad, E. 共2000兲. “Distribution of strains within asphalt binders in HMA using image and finite element techniques.” Transportation Research Record. 1728, Transportation Research Board, Washington, D.C., 21–27. Li, G., Li, Y., Metcalf, J. B., and Pang, S.-S. 共1999兲. “Elastic modulus prediction of asphalt concrete.” J. Mater. Civ. Eng., 11共3兲, 236–241. Li, Y., and Metcalf, J. B. 共2005兲. “Two-step approach to prediction of asphalt concrete modulus from two-phase micromechanical models.” J. Mater. Civ. Eng., 17共4兲, 407–415. Masad, E., Jandhyala, V. K., Dasgupta, N., Somadevan, N., and Shashidhar, N. 共2002兲. “Characterization of air void distribution in asphalt mixes using x-ray computed tomography.” J. Mater. Civ. Eng., 14共2兲, 122–129. Mori, T., and Tanaka, K. 共1973兲. “Average stress in matrix and average elastic energy of materials with misfitting inclusions.” Acta Metall., 21, 571–574. Ng, T.-T. 共2006兲. “Input parameters of discrete element methods.” J. Eng. Mech., 132共7兲, 723–729. Ng, T. T., and Changming, W. 共2001兲. “Comparison of a 3-D DEM simulation with MRI data.” Int. J. Numer. Analyt. Meth. Geomech., 25共5兲, 497–507. Papagiannakis, A. T., Abbas, A., and Masad, E. 共2002兲. “Micromechanical analysis of viscoelastic properties of asphalt concretes.” Transportation Research Record. 1789, Transportation Research Board, Washington, D.C., 113–120. Park, S. W., Kim, Y. R., and Lee, H. J. 共1999兲. “Fracture toughness for microcracking in a viscoelastic particulate composite.” J. Eng. Mech.,

125共6兲, 722–725. Rothenburg, L., Bogobowicz, A., Hass, R., Jung, F. W., and Kennepohl, G. 共1992兲. “Micromechanical modelling of asphalt concrete in connection with pavement rutting problems.” Proc., 7th Int. Conf. on the Struct. Des. of Asphalt Pavements, Rd. Res. Lab, Univ. of Michigan, Ann Arbor, Mich., 476–498. Sadd, M. H., Dai, Q., and Parameswaran, V. 共2003兲. “Simulation of asphalt materials using finite element micromechanical model with damage mechanics.” Transportation Research Record. 1832, Transportation Research Board, Washington, D.C., 86–95. Sadd, M. H., Dai, Q., and Parameswaran, V. 共2004兲. “Microstructural simulation of asphalt materials: Modeling and experimental studies.” J. Mater. Civ. Eng., 16共2兲, 107–115. Shimizu, Y., and Cundall, P. A. 共2001兲. “Three-dimensional DEM simulations of bulk handling by screw conveyors.” J. Eng. Mech., 127共9兲, 864–872. Thornton, C. 共1979兲. “Conditions for failure of a face-centered cubic array of uniform rigid spheres.” Geotechnique, 29共4兲, 441–459. Wang, L. B., Myers, L. A., Mohammad, L. N., and Fu, Y. R. 共2003兲. “Micromechanics study on top-down cracking.” Transportation Research Record. 1853, Transportation Research Board Washington, D.C., 121–133. You, Z., and Buttlar, W. G. 共2004兲. “Discrete element modeling to predict the modulus of asphalt concrete mixtures.” J. Mater. Civ. Eng., 16共2兲, 140–146. You, Z., and Buttlar, W. G. 共2005兲. “Application of discrete element modeling techniques to predict the complex modulus of asphaltaggregate hollow cylinders subjected to internal pressure.” Transportation Research Record. 1929, Transportation Research Board, Washington, D.C., 218–226. You, Z., and Buttlar, W. G. 共2006兲. “Micromechanical modeling approach to predict compressive dynamic moduli of asphalt mixture using the distinct element method.” Transportation Research Record. 1970, Transportation, Research Board, Washington, D.C., 73–83. You, Z., Buttlar, W. G., and Dai, Q. 共2006兲. “Aggregate effect on asphalt mixture properties by modeling particle-to-particle interaction.” Proc., Sessions of the 15th U.S. National Congress of Theoretical and Applied Mechanics, American Society of Civil Engineers, L. Wang and E. Masad, eds., Boulder, Colo., 14–21. You, Z., and Dai, Q. 共2007兲. “A review of advances in micromechanical modeling of aggregate-aggregate interaction in asphalt mixture.” Can. J. Civ. Eng., 34共2兲, 239–252.

JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2008 / 1063

Downloaded 29 Jan 2009 to 141.219.24.206. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright