Evaluating the aggregate structure in hot-mix asphalt using three ...

945

Evaluating the aggregate structure in hot-mix asphalt using three-dimensional computer modeling and particle packing simulations Naga Shashidhar and Kasthurirangan Gopalakrishnan

Abstract: In a hot-mix asphalt (HMA) pavement, the aggregate structure serves as a backbone and is primarily responsible for resisting pavement distresses. A sound aggregate structure implies optimal packing of aggregates providing both particle–particle contact and sufficient void space to fill in asphalt. In this paper, three-dimensional particle packing concepts are applied to the study of aggregate structure in HMA. A sequential deposition packing algorithm was used for packing typical aggregate gradations. The packing fraction and the distribution of particle–particle contacts in the simulated compact were studied. The packing simulation gave satisfactory results when aggregates above a certain minimum size were considered. Regression models were established to estimate the coordination number of any size aggregate in the compact. Such studies, in conjunction with the recent advances in X-ray computed tomography imaging techniques and discrete element modeling (DEM) simulations, have tremendous potential to help develop a deeper understanding of the HMA aggregate structure, develop and optimize the various parameters that describe the aggregate structure, and relate these parameters to the performance of pavements in a scientific way. Key words: packing, aggregate structure, computer simulation, aggregate–aggregate contact, pavement performance. Résumé : Dans une mélange à chaud de béton bitumineux (« hot-mix asphalt (HMA) »), l’agencement des granulats sert de squelette et est la principale source de la résistance des chaussées contre les dommages importants. Un sain agencement des granulats implique un tassement optimal des granulats, avec un contact particule à particule et suffisamment de vides pour les remplir de béton bitumineux. Le présent article applique les concepts tridimensionnels de tassement des particules à l’étude de l’agencement des granulats dans les HMA. Un algorithme de tassement avec déposition séquentielle a été utilisé pour le tassement des granulométries typiques des granulats. La fraction de tassement et la distribution des contacts interparticule dans le compactage simulé ont aussi été étudiées. La simulation de tassement a offert des résultats satisfaisants lorsque des granulats plus gros qu’une certaine dimension minimale ont été considérés. Des modèles de régression ont été établis pour estimer l’indice de coordination d’un granulat de toute dimension dans le bitume compacté. De telles études, en conjonction avec les plus récents avancements dans les techniques d’imagerie par tomographie aux rayons X assistée par ordinateur et les simulations par modélisation par éléments discrets (« discrete element modeling (DEM) »), présentent un potentiel énorme pour nous aider à mieux comprendre l’agencement des granulats HMA et le relier au rendement des bétons bitumineux de manière scientifique. Mots clés : tassement, agencement des granulats, simulation par ordinateur, contact granulat–granulat, comportement de la chaussée. [Traduit par la Rédaction]

Shashidhar and Gopalakrishnan

Introduction When a truck applies a load on pavement, the load is distributed and transmitted to the layers underneath. Thus, the function of the pavement is to transfer and distribute the load and provide a smooth surface for a pleasant ride.

954

A typical dense-graded hot-mix asphalt (HMA) pavement is made of 86% by volume of aggregates bound with about 10% by volume of asphalt and incorporates 4% air voids. The asphalt binder is a viscoelastic material that has a much lower ability to carry loads when compared to aggregates. Since the binder is viscoelastic, it tends to deform and flow under load. It is generally recognized that the coarse aggre-

Received 16 March 2005. Revision accepted 2 March 2006. Published on the NRC Research Press Web site at http://cjce.nrc.ca on 22 September 2006. N. Shashidhar.1 FHWA/Soil and Land Use Technology, Inc., 6300 Georgetown Pike, McLean, VA 22101, USA. K. Gopalakrishnan.2,3 Turner–Fairbanks Highway Research Center, 6300 Georgetown Pike, McLean, VA 22101, USA. Written discussion of this article is welcomed and will be received by the Editor until 31 December 2006. 1

Present address: Corning Inc., One Riverfront Plaza, HP-CB-08-7, Corning, NY 14831, USA. Corresponding author (e-mail: [email protected]). 3 Present address: Department of Civil Engineering, 498 Town Engineering Building, Iowa State University, Ames, IA 50011, USA. 2

Can. J. Civ. Eng. 33: 945–954 (2006)

doi:10.1139/L06-046

© 2006 NRC Canada

946

gates in HMA pavements form a skeleton that primarily supports the load. Shashidhar et al. (2000) showed that the stress is transmitted from particle to particle through the aggregate skeleton in HMA. Using photoelastic methods, it was shown that the loads were transmitted from aggregate to aggregate in a twodimensional representation of asphalt concrete. Appropriate sized glass disks (6.3 mm thick) were used to simulate coarse aggregates (i.e., aggregates larger than 4.35 mm in diameter). Fine aggregates and asphalt were used with these disks to represent coarse, fine, and stone matrix asphalt (SMA) gradations. These gradations were compacted in a two-dimensional assembly such that the glass disks were transparent to light. The stresses in the glass disks could be studied when this sample was stressed with a static load and viewed under polarized light. It was qualitatively shown that the stresses are conducted from particle to particle. It was also shown that different aggregate gradations give different aggregate structures and therefore different stress patterns. There is much key evidence that supports the significance of aggregate structure in HMA. The better performance of SMA is attributed to a better coarse aggregate skeleton in these mixes compared to that in the dense-graded HMA (Scherocman 1991; Brown et al. 1998). The Bailey method of gradation selection (Vavrik et al. 2001) reportedly produces an aggregate blend that is packed together in a systematic manner to form an aggregate skeleton. This method has been successfully used by the Illinois Department of Transportation (IDOT) to design and control mixes for their projects. Although the Bailey method is intuitive, there are several points that need clarification. It is generally considered that coarse aggregates form the aggregate skeleton, and it is unclear how these coarse aggregates are defined. In general, aggregates greater than the 4.75 mm sieve (No. 4) or the 2.36 mm sieve (No. 8) are considered as coarse aggregates. The Bailey method defines the cut-off for the coarse aggregate as the nominal maximum aggregate size multiplied by 0.22. Since the origins of these cut-off limits are uncertain, it is unclear if the aggregates below this cut-off will participate in aggregate skeleton. It is also uncertain what effect the aggregate gradation will have on this cut-off. Also, if the role of coarse aggregates is defined as skeleton forming, what would be the role of fine aggregates? Do they just fill the voids in the skeleton and make the mix less pervious? This is important because they increase the durability of the pavements in this role. Although the importance of aggregate structure (namely, coarse aggregate interlock) in HMA is widely recognized, there are no direct methods of estimating the degree of aggregate interlock. The recent developments in computer technology, image-analysis techniques, and nondestructive imaging (such as X-ray computed tomography) have demonstrated the potential to quantify the aggregate structure in HMA. These methods have been applied recently in studying the differences among different laboratory compaction methods, improving the simulation of laboratory compaction to field compaction, and predicting the permeability of HMA mixtures (Masad and Button 2004). The presence of aggregate interlock in SMA has been investigated using indirect methods (Brown and Mallick 1995;

Can. J. Civ. Eng. Vol. 33, 2006

Brown and Haddock 1996). At the National Center for Asphalt Technology (NCAT), Brown et al. (1998) investigated the ability of different compaction techniques (such as the Marshall hammer and the Superpave gyratory compactor) to determine whether stone-on-stone contact exists in the coarse aggregate fraction of SMA mixes. One of the methods currently used at the NCAT is to dry-rod the coarse aggregate and determine the voids in the coarse aggregate in the dry-rodded condition (VCADRC) (AASHTO 2000). It is assumed that stone-on-stone contact exists if voids in the coarse aggregate (VCA) in the mixture is equal to or less than VCADRC. It is not clear, however, to what extent the stone-on-stone contact exists. Aggregate interlock can be fully characterized if all the particle–particle contacts in a compact are counted, the average number of contacts per particle is calculated, and the stability of this structure is somehow established. If all the characteristics can be measured, it will be possible to study the effects of various parameters on aggregate interlock, such as aggregate gradation, angularity, and the influence of flat and elongated particles. The mix can then be optimized to give the most stable structure. If the aggregate characteristics can be optimized to give the most stable structure, the mix should become extremely resistant to rutting and the influence of the asphalt binder should be minimal. It is extremely difficult to measure the aggregate–aggregate contacts directly. Perhaps the nature of relationships between the aggregate characteristics and the aggregate interlock could be evaluated by other techniques, and then such relationships could be used to achieve the best aggregate skeleton. Physical methods such as the paint method (Bernal and Mason 1960) and optical techniques (Bernal et al. 1970) have been used to study the contacts between the particles. These methods are tedious, however, and do not have the resolution necessary to distinguish between contacts and near-contacts. Also, it is not known how these methods will perform if the aggregates vary in size. In areas such as ceramics and powder metallurgy, computer-simulation methods have proven to be effective and consistent in studying the contacts and their distributions in random packing structures (Nolan and Kavanagh 1993). The discrete element method (DEM) (Cundall and Strack 1979) has been used for modeling granular systems. The DEM treats particles as distinct interacting bodies. Interactions between particles are described by contact laws that define forces and moments created by relative motions of the particles. DEM offers a better opportunity to understand the micromechanical behavior of granular materials considering particulate assembly using microparameters such as average coordination number, induced anisotropy coefficients, and fabric tensors. Contact type is identified by contact properties such as normal stiffness, tangential stiffness, coefficient of friction, and adhesion between types of particles (Sitharam et al. 2004). Detailed microscopic information in a numerically simulated assembly of spheres can be used to trace the evolution of the microstructure and the contact forces during shear deformations. Numerical simulations using DEM on plane assemblies of discs by several researchers have been used to study such relationships (Cundall 1978; Cundall and Strack 1979; Rothenburg 1980; Bathrust 1985; Jean and Moreau 1992; Rothenburg and Bathurst 1992). In particulate media, © 2006 NRC Canada


each particle will be in contact with several of its neighbors, forming a group where each particle interacts with neighbors at the contact point and through contact forces (Sitharam et al. 2004). To extend the DEM techniques to real aggregates, one major step involves using a range of particle sizes. Zhong et al. (2000) developed methods to use this technique for accommodating particles with sizes spanning two decades. The determination of the initial structure of the granular assembly is an important input to this modeling. The typical technique to obtain this starting assembly is particle packing simulations. Although simulation methods have been used extensively in modeling the microstructure of Portland cement concrete (Bentz 1997), their application for studying asphalt concrete is relatively new. This paper demonstrates the potential benefits of using particle packing simulations to study the aggregate structure in HMA. The asphalt materials team at the Turner–Fairbanks Highway Research Center (TFHRC) is currently working to develop the scientific basis underlying the HMA mix design process and to establish criteria to predict pavement performance through a multidisciplinary initiative entitled Simulation, imaging and mechanics of asphalt pavement (SIMAP). Shashidhar et al. (2000) modeled and studied the stress patterns within unbound aggregates in HMA using DEM techniques. This paper focuses on applying particle packing simulation concepts for modeling the microstructure of HMA. The modeling is expected to start by simulating the packing of spherical particles with a specified gradation, and then add other factors such as surface roughness, presence of asphalt binder, and aggregate angularity. The model is expected to calculate characteristics of the compacted asphalt mixture, such as bulk density, voids content, void distribution, and average coordination number of each particle, and yield information useful to mixture design and particulate mechanics modeling of HMA.

Potential benefits of computer simulation Computer simulations could be used to study the aggregate skeleton. For instance, Gopalakrishnan et al. (2005) have successfully developed a method to estimate the degree of compaction in HMA by studying the aggregate structure from two-dimensional image analysis of HMA specimens and from results obtained from computer simulations. A twodimensional cross section of HMA was simulated in a computer using the mixture design information, and certain statistical parameters based on nearest-neighbor distance methods were computed. The simulation results were compared with the results obtained from the image analysis of actual HMA specimens, and both results compared favorably. In computer packing simulation, the gradation is input into the computer. The computer then packs the particles into a compact. An advantage of studying computersimulated models as opposed to physical models is that the geometrical properties of the packing structure can be accurately determined, as the particle coordinates are precisely known to the machine accuracy of the computer. Since the exact locations of these particles are known, the number of contacts, the coordination number of these contacts, and other characteristics of the compact can be estimated. It is

947

possible to use mathematical descriptors for particle shape and surface texture and introduce flat and elongated particles, and thereby study their effects on HMA properties. Also, the computer could be programmed to find the optimized gradation that would yield a mix with desired properties. In this loop, the computer would start with an initial gradation and calculate the characteristics of the compact. It could then vary the inputs and check the characteristics of the compact in a closed loop until the desired characteristics are achieved. Perhaps the most significant benefit is that computer simulations cost less and require less effort than conducting real experiments. It is admitted that it takes effort to develop the computer code and theory and validate the code, but, once developed, several simulations can be run in a short time. For instance, if the effect of different gradations on the density of HMA was to be evaluated, the aggregates have to be sieved and batched. They have to be heated, mixed with hot asphalt, compacted, and cooled. Then the specific gravity has to be measured by, say, the saturated surface-dry method (AASHTO 2005). The whole process could take as long as a day to test one sample. If, however, a computer simulation technique has been developed, the complete testing for a given gradation can be done in a few minutes. Furthermore, the variability in measured properties would be much higher than what could be obtained through computer simulation techniques. Sometimes the real differences between properties may not be detected by physical testing because the noise level is so high. In simulation, it is possible to achieve a much lower coefficient of variation. Multiple simulation runs can further reduce the coefficient of variation. If an anomalous result is obtained by computer simulation, the resultant compact can be examined to determine if there was anything unusual in the compact. In a physical sample, however, it is extremely difficult to pinpoint the cause of such anomalies unless techniques like X-ray computed tomography are available. An important benefit of modeling and simulations is that they increase our understanding of the mechanism behind certain phenomena. For instance, when the permeability of concrete was higher than it should have been, several tests were conducted. Scanning electron microscopy (SEM) of the concrete indicated the cement matrix to contain more air voids adjacent to the aggregate particles. Computer modeling of diffusion (Garboczi and Bentz 1992) was able to show that the increased air voids could indeed increase the diffusion rate. By knowing this information, low-permeability concrete could be designed.

Particle packing simulations Many types of computer simulations have been used in particle packing studies as reviewed by Barker (1994). One of the earliest and the simplest is the sequential deposition of spheres under unidirectional gravitational force (Vischer and Bolsterli 1972). In particle packing simulations, a particle system (e.g., powder) with a spread of particle sizes is closely approximated to a logarithm-normal or Gaussian distribution and is studied by computer simulated packing of spheres. The packing density or packing fraction is one of the most significant outcomes of such a modeling study. © 2006 NRC Canada

948

In addition to packing fraction, the distribution of the number of contacts (or coordination number) is considered to be an important characteristic of the simulated compact. A quantitative description of the contacts between aggregates becomes a necessity in granular materials because the load is always transferred as chains through contacts in granular materials (Dantu 1957). The coordination number indicates the number of particles in contact with a given particle. In most random packing studies, the distribution of coordination numbers has been found to be effective in characterizing the packing geometry and in studying the structural properties of particle packings (Powell 1980). As noted earlier, the physical methods for determining these contacts, which include the paint method (Bernal and Mason 1960) and the optical techniques (Bernal et al. 1970), do not have the resolution necessary to distinguish between contacts and near-contacts. To study these contacts and their distributions, computer simulations are more appropriate. The effect of size distributions on packing characteristics of the particles have been studied through physical experiments (Sohn and Moreland 1968; Dexter and Tanner 1972) and computer simulations (Nolan and Kavanagh 1993). These studies have typically used logarithm-normal or Gaussian particle-size distributions with systematic variation of the standard deviation (or range of particle sizes). These studies showed that the packing fraction increased as the standard deviation of the particlesize distribution increased. When computer simulations are used, a rather narrow particle-size range has to be used. For instance, Nolan and Kavanagh (1993) used a particle-size range of 0–20, whereas Powell (1980) used sizes from 0.0 to 1.0. The researchers who used theoretical models have used wider size distributions. For instance, Bierwagen and Saunders (1974) used particles of sizes ranging from 0.1 to 10.0, a two-decade range. Packing aggregate gradations for HMA range from 19.0 mm to less than 0.075 mm, a span of 2.5 decades. Packing granular materials of wide ranges of particle sizes has not been reported in the literature.

Special needs for hot-mix asphalt Applying particle packing simulations to design HMA has significant potential benefits and associated challenges. Previously, particle packing techniques have been successfully applied to cement concrete to maximize the amount of aggregates that can be put in the concrete. HMA differs from Portland cement concrete, however, in the functionality of its constituents. Portland cement is stronger and stiffer, with its modulus and strength being in the same order as those of the aggregates, which is not the same case with asphalt cement. In HMA, the primary means of withstanding a compressive load is by aggregate–aggregate contact, shear loads by aggregate interlock, and tensile stresses by the strength of asphalt cement. Thus, the aggregate structure becomes very important in HMA. The primary challenge in applying particle packing concepts to HMA stems from the aggregate size distribution or gradation. The aggregates used in HMA typically have a 19 mm maximum size, and about 5% of the aggregates (by weight) are finer than 0.075 mm, a span of 2.5 decades. In such cases, tens of millions of small particles have to be considered to have a representative number of large aggre-


gates in the simulated compact. At the present moment, such a simulation is not practical, and hence it becomes necessary to restrict the minimum aggregate size for packing, irrespective of the actual gradation. This will become clearer during the case study discussion.

Packing algorithm The packing algorithm used in this study is similar to that of Vischer and Bolsterli (1972) and Jullien and Meakin (1987). The spheres are added to a box sequentially under a unidirectional gravitational force, and each sphere is rolled to its stable position. The stable position is defined as that position where a sphere rests on three contacts. The spheres are dropped from random locations within the box. The algorithm includes five attempts for each sphere to find its lowest (minimum-energy) position, which was characterized by Vischer and Bolsterli as reproducing the effect of vibration on the settling sphere. The packing fraction or packing density, defined as the volume occupied by particles per unit volume of the compact, is calculated on a central core inside the box to avoid edge effects. Currently, the program can accommodate 25 different particle sizes. It is noted that the packing model is currently designed to pack aggregates in the form of solid spheres. The first step in packing simulation is to validate the packing program by comparing the results it produces with the data reported in the literature. To do this, two different studies were conducted. In the first study, 10 000 spheres 10 mm in diameter were packed in a box of dimensions 200 mm × 200 mm. This set-up had a box size to sphere diameter ratio of 20:1. To avoid edge effects, the outer 10 mm layer of the box was cut from all sides, and only the inside core was used for further analysis. Four independent simulations gave a packing fraction of 0.5939 and a standard deviation of 0.0011. In the second experiment, 80 000 spheres 3 mm in diameter were packed in a 120 mm × 120 mm box to yield a box size to sphere diameter ratio of 40:1. Four independent simulations yielded a packing fraction of 0.5940 with a standard deviation of 0.0006. By increasing the box size to sphere diameter ratio from 20:1 to 40:1, the mean packing fraction changed a little while halving the standard deviation. These results are comparable to the values reported in the literature, which are generally in the range of 0.58–0.60 for monosized spheres (Vischer and Bolsterli 1972; Powell 1980).

Case study In an effort to evaluate individual aggregate gradations in terms of construction characteristics and pavement performance, Goode and Lufsey (1962) developed an aggregate gradation chart on which all maximum-density gradations can be represented by straight-line plots. This chart can be especially useful in studying the influence of gradation on aggregate packing characteristics in terms of voids in mineral aggregate (VMA). By using a simple power-law aggregate distribution (eq. [1]), Goode and Lufsey showed that minimum VMA or maximum densities, irrespective of the nominal maximum aggregate size, always occur when slope k equals 0.45. This equation plots as a straight line on a © 2006 NRC Canada


logarithm–logarithm chart with a slope k. The power-law aggregate gradation equation used by Goode and Lufsey is as follows: [1]

949 Fig. 1. Aggregate gradations in a logarithm–logarithm chart (Goode and Lufsey 1962).

P = 100(d/dmax)k

where P is the total percent passing a given sieve size d, and dmax is the nominal maximum size of aggregate. The gradation equation was evaluated for different values of k (i.e., 0.31, 0.38, 0.45, 0.52, 0.59, and 0.66) by Goode and Lufsey (1962). The HMA specimens were molded in the laboratory using the gyratory method (US Army Corps of Engineers 1962), and the VMA values were determined. The gyratory compaction has a stronger resemblance to field compaction than does impact compaction. This means that the internal structure of specimens compacted with a gyratory compactor will show a closer resemblance to that resulting from actual after-construction roadway traffic (Ping et al. 2003). In the present study, the packing characteristics of the gradation equation, in terms of packing fraction and distribution of contacts, were evaluated for the different k values. The specific objectives were to study the dependence of particle–particle contacts on aggregate gradation and verify if the results from the Goode and Lufsey (1962) study could be obtained from computer simulation.

Experimental To pack the gradation, the number of particles retained in each sieve has to be calculated. A maximum size of 13.3 mm was considered throughout as according to the study by Goode and Lufsey (1962). The gradations considered in this study are shown on a logarithm–logarithm plot in Fig. 1. Each particle is considered to be spherical, with diameter equal to the average of the sieve (size) it is retained in and the next larger sieve. The total number of particles required for packing each size is approximated based on percent retained on each sieve interval and bulk specific gravity of the aggregate. Using this procedure, the total number of particles required to pack the entire gradation exceeds 12 million. In estimating the numbers, the intent was to have at least five particles of the maximum aggregate size. The packing was done in a box of size 25 mm × 25 mm. Packing 12 million particles needs considerable computing resources, and the current packing program could only pack a maximum of 80 000 particles. For this reason, the aggregate gradation had to be truncated. Three different cases were considered: (i) 70% of the gradation curve (70% retained and 30% passing), (ii) 75% of the gradation curve (75% retained and 25% passing), and (iii) 80% of the gradation curve (80% retained and 20% passing). These three cases would yield three different minimum particle sizes.

Discussion of results Packing fraction The packing fraction is a measure of the denseness of the packing and is a ratio of the volume of particles packed to the total volume. This is always a fraction less than unity. In terms of aggregate packing, packing fraction is the fraction of voids in a compacted sample. It is related to the VMA

such that a VMA of 14 is equivalent to a packing fraction of 0.86. The packing fraction results from the simulation runs are compared with the results obtained by Goode and Lufsey (1962) in Fig. 2. The simulation results do not agree well with those of Goode and Lufsey. In all three cases, the maximum packing fraction is achieved at different k values except for the 75% gradation case, where the maximum packing fraction occurs at k = 0.45, consistent with the findings of Goode and Lufsey. It is noted that the current packing model simulates aggregates as hard spheres and therefore there is a limitation on the maximum achievable packing fraction. The HMA specimens used in the Goode and Lufsey (1962) study were molded in the laboratory using the gyratory method (US Army Corps of Engineers 1962) with randomly shaped aggregates, and the VMA values were determined. Also, it is noted that all the aggregates could not be packed due to the current limitations of the packing model, and therefore the aggregate gradations had to be truncated. Thus, it is possible that a completely different packing structure evolved in considering the truncated particle gradation. To test the suitability of using the hard-sphere packing model to model crushed aggregates, however, a new case study was performed, in which the aggregate gradations consisting only of coarse aggregates (i.e., 0% passing 4.75 mm) were packed using the packing program. The results from the simulation runs were then compared with the results obtained by Brown et al. (1998), who dry-rodded the coarse aggregates in the laboratory and studied the VCADRC. Brown et al. studied the dry-rodded method in an effort to determine when stone-on-stone contact exists in the coarse aggregate fraction of SMA. Note that the packing fraction, in this case, would be equal to (100 – VCADRC)/100. The original gradations used by Brown et al. were used in the simulation runs. The simulation runs gave a packing fraction of 0.6, and Brown et al. reported a value of 0.608 (i.e., VCADRC = 39.2%). Brown et al. (1998) indicated that VCADRC for most aggregates is in the range of 37%–42% (i.e., packing fractions in the range of 0.58–0.63) and that different types of coarse aggregates having the same gradation will provide similar VCA values. To test this, nine different coarse aggregate © 2006 NRC Canada

950 Fig. 2. Comparison of laboratory packing fractions (PF = (100 – VMA)/100) obtained by Goode and Lufsey (1962) with those from computer simulations.

gradations from various original studies were considered (Brown and Mallick 1995; Brown et al. 1998; Abdulshafi et al. 1999). The packing fractions obtained from the simulation runs were in the range of 0.600–0.646 (Gopalakrishnan and Shashidhar 2000). These results are encouraging and indicate that the hard-sphere packing model performs satisfactorily when particles above a certain minimum size (in this case, 4.75 mm) are considered. More research is required to establish the critical minimum particle size, however. Distribution of contacts The transmission of force within particulate materials occurs only through the interparticle contacts, but each contact force is highly sensitive to the local arrangement of the surrounding particles and boundary loading conditions. Because of this strong dependence on particle arrangement, contact forces will usually be distributed in a complex, non-uniform manner, even when an isotropic and homogeneous assembly of particles is subjected to uniform loading (Antony et al. 2005). This complex behavior is revealed in photoelastic studies of two-dimensional discs reported by several investigators (e.g., Drescher and Jong 1972; Behringer et al. 1999; Hartley and Behringer 2003). These studies have revealed that even a uniformly applied load is transmitted by relatively rigid, heavily stressed chains of particles, which form a relatively sparse network of contacts, carrying greater than average normal contact forces. The distribution of normal contact forces within a medium falls into one of two categories, namely the “weak force chains” and “strong force chains.” The weak force chains are comprised of contacts that carry less than average contact force, and the strong force chains carry a greater than average force. Sliding occurs predominantly within the weak chains, and these contacts primarily contribute to the mean stress (Antony et al. 2005). In HMA, loads are mainly transmitted through the aggregate skeleton at a higher temperature to demonstrate its granular behavior but are transmitted through both aggregate contacts and binder at a lower temperature to demonstrate its continuum behavior. By the skeleton view, the average number of contacts for coarse aggregates, the contact normal distribution, the branch vector distribution, and the particle


orientation distribution are relevant; by the continuum view, mean solid path, the air void surface area, the damage tensor, and the local void volume fraction and its gradient are relevant (Wang et al. 2004). Using photoelastic methods and DEM simulations, Shashidhar et al. (2000) demonstrated that HMA behaves as a granular material (at relatively high temperatures, the binder is soft and has little restraint on the aggregates) and that applied loads are transmitted through force chains just as in unbound granular materials. By the aggregate skeleton view of HMA, some of the relevant parameters that describe the HMA skeleton include the average number of contacts per particle of coarse aggregates forming the skeleton, the contact normal distribution (number of contacts in different orientations), and the particle orientation distribution (number of particles with their particle orientations in certain ranges) (Wang et al. 2004). This paper focuses on studying the distribution of contacts for each particle size in the packing assembly. The approach taken here is to measure the distribution of contacts for any given particle size in the compact. The distribution of contacts was found to be Gaussian with a mean and standard deviation. Returning to the discussion of the original case study, Fig. 3 illustrates the Gaussian contact distribution curves (normalized by the total number of particles for a given size) when 70% of the gradation curve was packed with a k value of 0.52. Similar distributions were obtained for all other cases. Figure 3 shows that the distributions are narrower for smaller size particles and broader for larger size particles. This is expected because a larger particle can be surrounded by a range of small particles and therefore a wider range of contacts, whereas a smaller particle will fit into the void space produced by a few larger particles and therefore fewer contacts. Thus, the mean number of contacts increases as the particle size increases. All the curves are truncated at three contacts due the constraint in the packing algorithm, which seeks a minimum of three contacts before the particle is considered to be in a stable position. Also, one can observe that the data get noisier as the particle size increases because there are fewer larger particles in the packing assembly than smaller particles. Therefore, the smaller sizes have better statistics than larger particles. All these distributions were fit with a Gaussian distribution and the mean and standard deviations estimated. The fits were excellent, with R2 values greater than 0.99 for smaller particles. As the particle size increased, however, noise in the data caused poorer fits. The fits that had R2 values less than 0.90 were not considered for further analysis. The mean number of contacts is plotted in Fig. 4 as a function of particle size for different k values. Similar information is shown in Fig. 5 for the standard deviation of contacts. Upon normalizing the size of particles (d) by the minimum particle size (dmin), a single curve is obtained relating the mean number of contacts to (normalized) particle size, d/dmin, as shown in Fig. 6. A similar trend was obtained for standard deviation, as seen in Fig. 7. Similar results were obtained for all other cases, indicating the possibility of developing mathematical models relating the mean coordination number and contact spread (standard deviation) to particle size in a simulated compact, irrespective of aggregate gradation. The following regression models were established: © 2006 NRC Canada

Shashidhar and Gopalakrishnan Fig. 3. Distribution of contacts (70% gradation packed, k = 0.52). The different particle sizes (in mm) are given in the box along with the symbols used in the graph.

Fig. 4. Mean number of contacts versus particle size (d).

Fig. 5. Standard deviation of contacts versus particle size (d).

[2]

mean = A + B (d /d min) + C (d /d min) 2

[3]

standard deviation = D + E (d /d min) + F (d /d min) 2

where the regression coefficients A–F are given in Table 1 for selected k values. Using these regression models, it is

951 Fig. 6. Mean number of contacts versus normalized particle size (d/dmin).

Fig. 7. Standard deviation of contacts versus normalized particle size (d/dmin).

possible to estimate what the number of contacts would be if the gradation were packed down to smaller size particles. Due to limitations in computing capacity, the packing model could only pack to a minimum size of 0.6 mm for k = 0.52 aggregate gradation. Using the regression models for mean and standard deviation of contacts, however, it is possible to estimate the number of contacts in packing the entire gradation. For instance, if an aggregate gradation with a maximum size of 13.3 mm and a minimum size of 0.075 mm were packed at k = 0.52, a 2.36 mm particle would have a mean number of contacts of 914 and a standard deviation of 8.55. This is certainly a large number of contacts, and it is not known if all contacts carry equal importance or if some are more important than others from the viewpoint of load transmission. The mechanical properties of the packing are dependent on how many of these contacts actually transmit force when the packed assembly is stressed. Travers et al. (1986) showed through photoelastic experiments on a twodimensional packing of cylinders that only two or three contacts seemed to transmit load, even though there seemed to be six contacts on average. Therefore, it would be useful to formulate an appropriate criterion based on the coordination number to qualify each contact as active (or load-bearing) or © 2006 NRC Canada

952


Table 1. Regression coefficients for mean and standard deviation of contacts. Regression model coefficient

k = 0.38

k = 0.52

k = 0.59

A B C D E F

2.585017 0.640355 0.917728 –0.428106 1.238840 –0.058550

2.400044 0.725284 0.897473 –0.046100 0.861960 –0.023970

1.46225 1.58623 0.73191 –0.10004 0.88174 –0.03134

passive (or space-filling) in load transmission. This is discussed in more detail by Gopalakrishnan and Shashidhar (2006).

Future research Some of the important parameters of interest that are related to the study of contacts such as the effect of internal friction angle and the orientation of contact planes were not considered in this study due to the limited scope of this research. The angle of internal friction of loose granular materials is measured by their angle of repose, which is a unique material property of the grains (Abriak 1999; Vallejo and Mawby 2000). Samples with coarse sand grains have a higher friction angle compared with samples totally composed of fines. The simplest method of determining the angle of internal friction of mixtures is the direct shear test using a Casagrande shear box apparatus. A study of the orientation of contact planes can explain the inherent and induced anisotropies in the compact. For most granular materials the manner in which the sample was formed and consolidated influences the yield locus, since it introduces anisotropy, even in a material that is initially isotropic. Induced anisotropy is the material anisotropy introduced into the sample by the consolidation procedure, and inherent anisotropy stems from the shape of the particles and the way the bedding was formed (Molenda and Horabik 2004). Using DEM simulations, Antony et al. (2005) performed a detailed analysis on the force-transmission characteristics of granular materials at microscopic level and presented a connection between the directional orientation of force networks and the invariants of the macroscopic stress tensor: the nonsphere systems were able to build up a strongly anisotropic network of strong force-transmitting contacts. Aggregate particles in an HMA mixture have different shapes, surface texture and friction, and orientations, which make the aggregate contact more complicated. Chang and Meegoda (1999) proposed a micromechanical model for simulating HMA based on DEM which allowed for aggregate– binder–aggregate contact and aggregate–aggregate contact. Until now, a vast majority of research studies related to micromechanical modeling of HMA microstructure has been limited to two dimensions because of computational constraints (You and Buttlar 2004). As mentioned previously, Shashidhar et al. (2000) have shown that load transmission in HMA takes place in the form of force chains and that particle-to-particle contact exists in the formation of these chains. Significant questions still exist as to whether some contacts are more important

than others and about the function of fine aggregates in load transmission. A study of particle–particle contacts in the simulated compact will be useful in answering these questions. The current study is meant to lay the foundations for investigating this research area further. The knowledge of the relationship between the aggregate structure and the mechanical properties will enable one to engineer the pavements to have the appropriate aggregate structure to maximize performance. This approach is very similar to that used when developing materials with desired properties (designer materials), such as high thermal conductivity materials for electronic substrates or materials with high toughness. The ability to tailor and optimize material attributes, thus manufacturing designer materials, is increasingly being realized in structural systems (Chong and Garboczi 2002). By developing the relationship between aggregate structure and performance, potential approaches to change the aggregate structure to improve the performance could be developed. Masad and Button (2004) have summarized the experimental and analysis methods used in characterizing and quantifying the aggregate structure of HMA which include two-dimensional imaging techniques and three-dimensional nondestructive imaging techniques such as X-ray computed tomography (Shashidhar 1999). The nature and distribution of interparticle contacts in computer-simulated compacts with a wide particle size range, similar to that found in HMA, were discussed in this paper. The results have to be validated using physical experiments and discrete element techniques.

Summary and conclusions In this research, particle packing simulation concepts were applied to the study of aggregate structure in hot-mix asphalt (HMA). Previous research by the authors showed that the changes in HMA aggregate structure due to compaction could be quantified through two-dimensional image-analysis techniques and computer simulations. In this study, the aggregates were modeled as hard spheres, and some typical aggregate gradations used in HMA were packed using a computer program. The packing characteristics of the compact were studied in terms of packing fraction and distribution of contacts. A case study was presented to illustrate some of the significant outcomes resulting from the application of particle packing simulation concepts to the study of aggregate structure in HMA. Due to computational limitations, the entire gradation could not be packed. The packing fractions achieved using the simulations did not agree with the laboratory test results (in terms of voids in mineral aggregate from the compacted HMA specimens) obtained by previous researchers. A related case study to test the suitability of using the hard-sphere packing model to model packing of crushed aggregates showed satisfactory results when particles above a certain minimum size are considered. The coordination number of any size particle in the packing was found to be a Gaussian distribution whose mean and standard deviation can be estimated using regression models. Future research should focus on deriving an appropriate criterion based on the coordination number to qualify each © 2006 NRC Canada


contact as active (or load-bearing) or passive (or spacefilling) in load transmission. Future research efforts will also focus on applying suitable contact mechanics principles to introduce the asphalt binder (or mastic) at the particle–particle contacts and apply this model to virtual specimens reconstructed from nondestructive images of pavement core sections. It is recognized that the results of this study as presented here are limited in terms of practical utility. The authors believe, however, that the application of particle packing simulation concepts to the study of aggregate structure in HMA in conjunction with the recent advances in X-ray tomography imaging techniques and DEM simulations have tremendous potential to help us develop a deeper understanding of the HMA aggregate structure, develop and optimize the various parameters that describe the aggregate structure, and relate these parameters to the performance of pavements in a scientific way. This will provide the foundations to building more durable and long-lasting pavements.

Acknowledgments The authors gratefully acknowledge the efforts of Dr. Xibing Dou, Senior Research Engineer at the Turner–Fairbanks Highway Research Center, in developing the computer simulation packing program described in this paper. The second author wishes to acknowledge the National Highway Institute for financial assistance through the Eisenhower Graduate Research Fellowship.

References AASHTO. 2000. Standard method of test for bulk density (unit weight) and voids in aggregate. AASHTO Test Procedure T19M/T19, American Association of State Highway and Transportation Officials (AASHTO), Washington, D.C. AASHTO. 2005. Standard method of test for bulk specific gravity of compacted hot-mix asphalt using saturated surface-dry specimens. AASHTO Test Procedure T166, American Association of State Highway and Transportation Officials (AASHTO), Washington, D.C. Abdulshafi, O.A., Fitch, M.G., Kedzierski, B., and Powers, D.B. 1999. Laboratory optimization of asphalt concrete intermediate course mixes to improve flexible pavement performance. In Hotmix asphalt mixtures. Transportation Research Record 1681, pp. 69–75. Abriak, N.E. 1999. Shearing behavior of granular material in silo: microscopical approach. Powder Handling and Processing, 11(3): 301–304. Antony, S.J., Kuhn, M.R., Barton, D.C., and Bland, R. 2005. Strength and signature of force networks in axially compacted sphere and non-sphere granular media: micromechanical investigations. Journal of Physics D: Applied Physics, 38: 3944–3952. Barker, G.C. 1994. Granular matter. Springer-Verlag, New York. Bathrust, R.J. 1985. A study of stress and anisotropy in idealized granular assemblies. Ph.D. thesis, Department of Civil Engineering, Queen’s University, Kingston, Ont. Behringer, R.P., Howell, D., Lou, K., Sarath, T., and Veje, C. 1999. Predictability and granular materials. Physica D: Nonlinear Phenomena, 133(1–4): 1–17. Bentz, D.P. 1997. Three dimensional computer simulation of Portland cement hydration and microstructure development. Journal of the American Ceramics Society, 80(1): 3–21.

953 Bernal, J.D., and Mason, G. 1960. Coordination of randomly packed spheres. Nature (London), 188: 910–911. Bernal, J.D., Cherry, I.A., Finney, J.L., and Knight, K.R. 1970. An optical machine for measuring sphere coordinates in random packings. Journal of Physics E: Scientific Instruments, 3: 388– 390. Bierwagen, G.P., and Saunders, T.E. 1974. Studies of the effects of particle size distribution on the packing efficiency of particles. Powder Technology, 10: 111–119. Brown, E.R., and Haddock, J.E. 1996. Method to ensure stone-onstone contact in stone matrix asphalt paving mixtures. In Aggregates, filler, and neutralized materials in asphalt mixtures. Transportation Research Record 1583, pp. 11–18. Brown, E.R., and Mallick, R.B. 1995. Evaluation of stone-on-stone contact in stone-matrix asphalt. In Hot-mix asphalt design, testing, evaluation, and performance. Transportation Research Record 1492, pp. 208–219. Brown, E.R., Haddock, J.E., Crawford, C., Hughes, C.S., and Lynn, T.A. 1998. Designing stone matrix asphalt mixtures: Volume II(a) — Research results for Part 1 of Phase I. National Cooperative Highway Research Program Report NCHRP 9-8/2, Transportation Research Board, National Research Council (US), Washington, D.C. Chang, G.K., and Meegoda, J.N. 1999. Micromechanical simulation of hot-mix asphalt. ASCE Journal of Engineering Mechanics, 123(5): 495–503. Chong, K.P., and Garboczi, E.J. 2002. Smart and designer structural material systems. Progress in Structural Engineering and Materials, 4: 417–430. Cundall, P.A. 1978. BALL — a computer program to model granular media using the distinct element method. Technical Note TN-LN-13, Advance Technology Group, Dames and Moore, London, UK. Cundall, P.A., and Strack, O.D.L. 1979. A discrete numerical model for granular assemblies. Géotechnique, 29(1): 47–65. Dantu, P. 1957. Contribution a l’étude mecanique et geometrique des milieux pulverulents. In Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering, London, UK, 12–24 August 1957. Butterworths Scientific Publications, London, UK. Vol. 1, pp. 144–148. Dexter, A.R., and Tanner, D.W. 1972. Packing densities of mixtures of spheres with log-normal size distributions. Nature (London), Physical Science, 238: 31–32. Drescher, A., and Jong, D.J. 1972. Photoelastic verification of a mechanical model for the flow of a granular material. Journal of the Mechanics and Physics of Solids, 20(5): 337–351. Garboczi, E.J., and Bentz, D.P. 1992. Computer simulation of the diffusivity of cement-based materials. Journal of Materials Science, 27: 2083–2092. Goode, J.F., and Lufsey, L.A. 1962. A new graphical chart for evaluating aggregate gradations. Journal of the Association of Asphalt Paving Technologists, 31: 176–207. Gopalakrishnan, K., and Shashidhar, N. 2000. Quantifying aggregate structure in hot-mix asphalt using simulation and imaging techniques. Unpublished report, US Federal Highway Administration, Turner–Fairbanks Highway Research Center, McLean, Va. Gopalakrishnan, K., Shashidhar, N., and Zhong, X. 2005. Attempt at quantifying the degree of compaction in HMA using image analysis. In Advances in Pavement Engineering: Proceedings of the ASCE 2005 Geo-Frontiers Conference, Austin, Tex., 24–26 January 2005. Edited by E.M. Rathje. CD-ROM. American Society of Civil Engineers (ASCE), Geotechnical Special Publication 130, p. 155. © 2006 NRC Canada

954 Gopalakrishnan, K., and Shashidhar, N. 2006. Structural characteristics of three-dimensional random packing of aggregates with wide size distribution. International Journal of Information Technology, 3(3): 201–208. Hartley, R.R., and Behringer, R.P. 2003. Logarithmic rate dependence of force networks in sheared granular materials. Nature (London), 421(6926): 928–931. Jean, M., and Moreau, J.J. 1992. Unilaterality and dry friction in the dynamics of rigid body collections. In Proceedings of the Contact Mechanics International Symposium, Lausanne, Switzerland, 7–9 October 1992. Edited by A. Curnier. Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland. Jullien, R., and Meakin, P. 1987. Simple three-dimensional models for ballistic deposition with restructuring. Europhysics Letters, 4(12): 1385–1390. Masad, E., and Button, J. 2004. Implications of experimental measurements and analyses of the internal structure of hot-mix asphalt. In Bituminous paving mixtures 2004. Transportation Research Record 1891, pp. 212–220. Molenda, M., and Horabik, J. 2004. On applicability of a direct shear test for strength estimation of cereal grain. Particle and Particle Systems Characterization, 21(4): 310–315. Nolan, G.T., and Kavanagh, P.E. 1993. Computer simulation of random packings of spheres with log-normal distributions. Powder Technology, 76: 309–316. Ping, W.V., Leonard, M., and Yang, Z. 2003. Laboratory simulation of field compaction characteristics (phase I). Final Research Report FL/DOT/RMC/BB-890(F), Florida A&M University, Tallahassee, Fla. Powell, M.J. 1980. Computer-simulated random packing of spheres. Powder Technology, 25: 45–52. Rothenburg, L. 1980. Micromechanics of idealized granular systems. Ph.D. thesis, Department of Civil Engineering, Carleton University, Ottawa, Ont. Rothenburg, L., and Bathurst, R.J. 1992. Micromechanical features of granular assemblies with planar elliptical particles. Géotechnique, 42(1): 79–95. Scherocman, J.A. 1991. Stone mastic asphalt reduces rutting. Better Roads, 61(11): 26. Shashidhar, N. 1999. X-ray tomography of asphalt concrete. In Hot-mix asphalt mixtures. Transportation Research Record 1681, pp. 186–192. Shashidhar, N., Zhong, X., Shenoy, A., and Bastian, E., Jr. 2000. Investigating the role of aggregate structure in asphalt pavements. In Aggregates, Asphalt Concrete, Bases and Fines: Pro-

Can. J. Civ. Eng. Vol. 33, 2006 ceedings of the 8th Annual Symposium of the International Center for Aggregates Research (ICAR), Denver, Colo., 12–14 April 2000. International Center for Aggregates Research (ICAR), University of Texas at Austin, Austin, Tex. Sitharam, T.G., Dinesh, S.V., and Srinivasa Murthy, B.R. 2004. Critical state behaviour of granular materials using three dimensional discrete element modelling. In Granular materials: fundamentals and applications. Edited by S.J. Antony, W. Hoyle, and Y. Ding. The Royal Society of Chemistry, Athenaeum Press Ltd., Gateshead, UK. pp. 135–156. Sohn, H.N., and Moreland, C. 1968. The effect of particle size distribution on packing density. Canadian Journal of Chemical Engineering, 46: 162–167. Travers, T., Bideau, D., Troadec, J.P., and Messager, J.C. 1986. Uniaxial compression effects on 2D mixtures of ‘hard’ and ‘soft’ cylinders. Journal of Physics A, Mathematical, Nuclear and General, 19: L1033–L1038. US Army Corps of Engineers. 1962. Development of the gyratory testing machine and procedures for testing bituminous paving mixtures. US Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss. Vallejo, L.E., and Mawby, R. 2000. Porosity influence on the shear strength of granular material – clay mixtures. Engineering Geology, 58: 125–136. Vavrik, W.R., Pine, W.J., Huber, G., Carpenter, S.H., and Bailey, R. 2001. The Bailey method of gradation evaluation: the influence of aggregate gradation and packing characteristics on voids in the mineral aggregate. Journal of the Association of Asphalt Paving Technologists, 70: 132. Vischer, W.M., and Bolsterli, M. 1972. Random packing of equal and unequal spheres in two and three dimensions. Nature (London), 239: 504–507. Wang, L.B., Paul, H.S., Harman, T., and Angelo, J.D. 2004. Characterization of aggregates and asphalt concrete using X-ray computerized tomography: a state-of-the-art report. Journal of the Association of Asphalt Paving Technologists, 73: 467–500. You, Z., and Buttlar, W.G. 2004. Discrete element modeling to predict the modulus of asphalt concrete mixtures. ASCE Journal of Materials in Civil Engineering, 16(2): 140–146. Zhong, X., Shashidhar, N., and Chang, C.S. 2000. DEM stability for static analysis in aggregate packing. In Proceedings of the 14th ASCE Engineering Mechanics Conference, Austin, Tex., 21–24 May 2000. American Society of Civil Engineers (ASCE), Reston, Va.

© 2006 NRC Canada