Effect of particle size ratio and contribution of particle size fractions on micromechanics of uniaxially compressed binary sphere mixtures
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DOI: 10.1007/s1003501707194
 Cite this article as:
 Wiącek, J., Parafiniuk, P. & Stasiak, M. Granular Matter (2017) 19: 34. doi:10.1007/s1003501707194
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Abstract
This paper is an extension of the recent work of Wiącek (Granul Matter 18:42, 2016), wherein geometrical parameters of binary granular mixtures with various particle size ratio and contribution of the particle size fractions were investigated. In this study, a micromechanics of binary mixtures with various ratio of the diameter of small and large spheres and contribution of small particles was analyzed using discrete element simulations of confined uniaxial compression. The study addressed contact normal orientation distributions, global and partial contact force distributions and pressure distribution in packings of frictional spheres. Additionally, the effect of particle size ratio and contribution of particle size fractions on energy dissipation in granular mixtures was investigated. The particle size ratio in binary packings was chosen to prevent small particles from percolating through bedding. The bimodality of mixtures was found to have a strong effect on distribution of contact normal orientation and distribution of normal contact forces in binary mixtures. Stress transfer in binary packing was also determined by both, particle size ratio and volume fraction of small particles. Dissipation of energy was higher in mixtures with higher particle size ratios and decreased with increasing contribution of small spheres in system.
Keywords
Micromechanics Discrete element method Binary granular material Particle size ratio1 Introduction
Granular materials play important role in many industries, especially in agriculture and food industry, pharmaceutical, cosmetics, metallurgy, and building industries. Materials are subjected to technological processes whose design requires advanced research into granular matter. Although a number of studies has been conducted on granular materials over last few decades, providing crucial knowledge on properties of particulate solids and valuable insight into nature of interactions between granules, many phenomena related to complex behavior of granular assemblies still remain unexplained. Therefore, structural and mechanical properties of materials which determine production, handling processes and processing of particulate solids remain a focus of intense research in physics, chemistry and environmental science.
Most particle packings involved in industrial and natural processes consists of the common property of polydispersity. The degree of particle size heterogeneity determines a rearrangement of particles and contact network in granular system. It provides different compaction characteristics of particulate materials and powders, which is very important in, inter alia, powder technology, powder metallurgy, ceramics, chemical industry and industry of pharmaceutical tablet manufacturing. Granular packings may be composed of one, two, three or more particle size fractions. The more number of particle size fractions is, the more difficult interpretation of effects observed in granular material is. Therefore, the indepth insight into a nature of simple particulate systems is required to understand behavior of more complex packings of nonuniformly sized grains. Binary mixtures represent the simplest case of polydisperse particulate systems. Up to date, a number of experimental [2, 3], theoretical [3, 4, 5] and numerical [1, 5, 6] studies have been conducted to investigate structural and mechanical properties of bimodal particulate beds. It was found that packing density was sensitive to friction [7, 8, 9], particle size ratio [1, 2, 6, 7, 9, 10, 11] and contribution of particle size fractions in grain assembly [1, 2, 3, 6, 8, 9, 11]. These factors were also found to determine geometric anisotropy, contact network and stress transmission in particulate system [9]. The studies conducted by McGeary [2] and Rassously [4] for binary mixtures with various particle size ratios have shown an increase in packing density of mixtures with volume fraction of small particles in packings increasing up to 60%. A further increase in contribution of small particles in mixture resulted in decrease in packing density of samples. Packing density was larger in samples with smaller ratio between diameter of small and large spheres. Wiącek [1] indicated that, packing density of binary mixtures with small to large particle size ratios larger than 0.7, did not reach maximum. It remained constant regardless on volume fraction of different fractions in mixtures. The experimental study conducted by Piston et al. [12] and DEM simulations prepared by Sánchez et al. [9] and by Wiącek [1] for bimodal mixtures indicated that both, particle size ratio (being a geometric factor) and contribution of particle size fractions in mixture (being a statistical factor) determined a partial coordination number. An increase in contribution of one of the size fraction in sample strongly affected number of contacts of different types and the larger spheres had more contacts with surrounding particles. Particle size ratio was found to have strong influence on anisotropy in the contact distribution and distribution of stress within the sample. Lade et al. [13] suggested that as the diameter ratio of small to large sphere decreased, the smaller granules better fitted within the pores between large particles resulting in reduced contribution of small grains in stress transfer. A numerical study conducted by Shire et al. [14] for bimodal packings of spheres have shown that small grains contributed approximately equally to stress transfer in packings with volume fraction of small particles not smaller than 30%. In packings comprising smaller number of small grains, a smaller contribution of these grains to stress transfer was observed, which decreased with decreasing particle size ratio. As the small particles are able to fit more efficiently in the voids, they are less likely to interact with the large spheres and participate in forming strong force chains.
The interactions between particles in granular packing are the sources of potential energy that propagates throughout the contact network. Part of that energy dissipates in the plastic deformation and due to work of frictional or adhesive forces in contacts [15, 16]. Interactions between grains in polydisperse granular packing are strongly affected by geometric and statistical factors. Therefore, dissipation of energy in particulate assembly is also determined by these factors [17, 18]. Twodimensional simulations of gravityfree Couette flow of binary granular systems, conducted by Karion and Hunt [18], indicated increased rate of energy dissipation in mixtures when the ratio of small to large granules decreased. Energy dissipated in the collisions was proportional to the mass of the colliding bodies, resulting in greater energy dissipation in systems composed of larger granules. DEM simulations of uniaxial compression of polydisperse sphere packings, conducted by Wiącek and Molenda [19], have shown that a loss energy increased as the degree of particle size heterogeneity increased. It is well known that efficiency of industrial processes involving handling of granular materials significantly depends on dissipation of energy in system. Although a dissipative nature of forces acting on interacting particles is nowadays wellknown, the search for method for reduction of energy loss during technological processes is still required. More insight is necessary to understand mechanisms of stress transfer and energy dissipation in nonuniformly sized particulate systems as well as to predict behavior of grain assembly under specific load conditions.
The review of literature shows that a large number of investigations involving binary granular mixtures addressed the ones with high difference between diameters of particles. Small grains percolated downwards through these beddings, which resulted in particlesize segregation affecting structural and mechanical properties of particulate assemblies. A reported project was devoted to the investigation of bimodal mixtures with relatively high particle size ratios, wherein percolation of small particles through larger ones did not take place. An influence of ratio between diameters of particles and contribution of granulometric fractions on mechanical properties of that kind of binary granular packings remains open question and is still insufficiently analyzed. Therefore, in this study, contact normal orientation distributions, global and partial contact force distributions and stress transfer in bimodal mixtures has been largely studied. Additionally, an effect of geometric and statistical factors on energy dissipation in particulate systems was investigated, which is of great interest in many industrial processes. As the experimental methods available do not provide complete information on interactions between particles inside the bedding, computational techniques find wide application. In the reported project, a Discrete Element Method was used to study mechanics of binary mixtures of spherical frictional particles.
2 Materials and methods
The numerical method applied in this paper is a Discrete Element Method (DEM). Discrete Element Method is a numerical technique for detailed investigation of mechanical behavior of granular systems, originally proposed by Cundall and Strack [20]. It allows for establishment of positions and velocities of particles in system through time integration of the ordinary differential equations formed for each individual particle on the basis of Newton’s second law of motion.
DEM input parameters
Parameter  Steel  

Poisson’s ratio  0.3  
Shear modulus (GPa)  77  
Density \((\hbox {kg}/\hbox {m}^{3})\)  7804  
Coefficient of restitution  Particle–particle  Particle–wall 
0.4  0.4  
Coefficient of static friction  Particle–particle  Particle–wall 
0.321  0.216  
Coefficient of rolling friction  Particle–particle  Particle–wall 
0.01  0.01 
Threedimensional simulations of the uniaxial confined compression test were performed by using EDEM software [21]. Samples were placed in a chamber of rectangular crosssection with rigid and frictional walls which did not deform under the applied load. The dimensions of sample were larger than 15 particle mean diameters which were adopted to be a representative elementary volume for polydisperse mixtures [22]. Granular material was compressed through the top platen that moved vertically with a constant velocity of 3 m/min until a maximum vertical pressure on the uppermost spheres of 100 kPa was adopted. The test procedure followed recommendations of the Eurocode 1 design standard [23]. Further details of contact model and generation procedure of sample are available in an earlier study by Wiącek [1].
3 Results and discussion
3.1 Packing microstructure
3.2 Force and stress distribution
Distribution of contacts in the granular packing strongly influences transmission of forces and spatial distribution of force chains [14, 17]. The studies of polydisperse packings, conducted by Voivret et al. [24] and Wiącek and Molenda [17] have shown that the strongest forces passed through the largest particles in system. Figure 6 presents an evolution of the average compressive force, defined as a sum of normal forces at contacts between particles, with volume fraction of small spheres in mixtures. The average compressive forces were normalized by mean compressive force in monodisperse sample comprising large particles. The forces exerted on spheres at contact points were found to be strictly related to the partial coordination numbers, presented by Wiącek [1]. The \(\bar{F}\) values and coordination numbers for different contacts followed the same paths with increasing contribution of small particles in mixtures. The average compressive forces exerted on small spheres by other particles were the largest in mixtures with g value of 0.8, which was strictly related to the largest number of contacts between these particles in mentioned samples [1]. The opposite tendency was observed for forces exerted on large spheres by small ones. The average compressive force exerted on large particles was higher in packings with smaller particle size ratio, wherein the largest average coordination number was observed [1]. Figure 6d shows that the average compressive force exerted on large spheres by the ones of the same size was slightly sensitive to the ratio of the diameter of small and large particles in binary mixture. The differences between \(\bar{F}\) values in mixtures with different particle size ratio were small and they lied within the range of scatter.
Regardless on the ratio of the diameter of small and large spheres and contribution of particle size fractions in binary mixtures, the largest compressive forces were exerted on the large spheres which confirms that mainly these particles contribute into stress transmission in granular bedding. Figure 7 shows the chains of compressive forces in compressed mixtures with \(g=0.4\) and volume fraction of small spheres \(f=0.2\) and \(f=0.8\), in xz plane. The black and white colors indicate the maximum and minimum values of forces. The chains of the largest forces passed through the largest particles in samples, which has been already observed by Wiącek and Molenda [17] in polydisperse sphere packings.
In the binary granular packings, small spheres fill partially the voids between large grains and carry reduced effective stress. They may percolate through the primary fabric produced by immobile large particles [25, 26]. Kenney and Lau [27] reported that the presence of primary fabric and loose small particles in particulate solid was an origin of internal instability of an assembly. The transfer of externally applied loads is not homogeneous in particulate bedding and stresses are transferred through limited number of particles. A number of studies have been performed over last few decades to measure the contribution of particle size fractions into stress transfer in granular materials [14, 28].
Figure 8 shows the evolution of global stress with volume fraction of small spheres in binary mixtures with different particle size ratios subjected to compressive load of 100 kPa. As the global stress in granular material is determined by its packing density, the p(f) relationships were similar to the ones between solid fraction and volume fraction of small particles in sample. In samples with particle size ratio of 0.4, the global stress increased twofold with volume fraction of small particles increasing form 0 to 0.6 due to increase in solid fraction of samples. A further increase in f value decreased the mean normal stress. An increase in the ratio of the diameter of small and large spheres from 0.4 to 0.6 resulted in decrease in global stress by 30% in packings with volume fraction of small particles of 0.6. A slight relationship between global stress and contribution of small spheres was observed in samples with \(g=0.8\), wherein slight changes in solid fraction and total compressive force with increasing f value were also observed.
3.3 Energy dissipation
4 Conclusions
The micromechanics of the binary granular mixtures subjected to compressive load was investigated, using the 3D DEM simulations. The micromechanical properties of granular packing strongly determine the internal response of material to externally applied loads and its macromechanical properties; however, they alone are also dependent on few factors. Therefore, in this study, the analysis of the effect of the geometric and statistical factors on the micromechanics of binary sphere packings was conducted. The geometric factor was the ratio of the diameter of small and large spheres in samples, while the statistical one was a volume fraction of small spheres in mixture. The particle size ratio in bidisperse samples was chosen larger than 0.4 to prevent small particles from percolating through bedding.
Numerical investigations of the binary granular systems showed a strong influence of both, particle size ratio and volume fraction of small particles on distribution of normal contact forces in binary mixtures. The analysis of the compressive force, defined as a sum of normal forces at contacts between particles, indicated that partial compressive force and partial coordination numbers followed the same paths with increasing contribution of small particles in mixtures. In this study, a strong relationship between anisotropy of the contact normal orientation in bidisperse packings and geometric and statistical factors was also indicated. An increase in global anisotropy was observed with increasing difference between the dimeters of spheres in samples. An increase in volume fraction of small spheres in packings also resulted in higher anisotropy of the contact normal orientation, however, an increase in anisotropy was smaller in mixtures with smaller particle size ratios.
The global pressure and solid fraction have followed the same paths with increasing contribution of small particles in mixtures. In samples with the particle size ratio of 0.6 and 0.4, solid fraction and global pressure reached maximum when the volume fraction of small spheres was 0.6, which was not observed in samples with larger ratio of the diameter of small and large spheres. The distribution of stress in the particulate bedding was not homogeneous and it strongly depended on the composition of the mixture. In packings with volume fraction of small particles of 20%, the contribution of small and large spheres into stress transfer was approximately equal. In mixtures with large number of small spheres and large particle size ratio, small spheres did not fit completely the voids between large ones and interacted with large grains to a lesser degree. Therefore, the stress in small spheres decreased as a volume fraction of small particles increased and ratio of the diameter of small and large spheres decreased. Both, the geometric and statistical factors were found to determine dissipation of energy in the examined packings. As the energy loss is a function of the radii of the interacting spheres, dissipation of energy decreased with decreasing particle size ratio and increasing volume fraction of small spheres in mixtures. These results indicate that the significant decrease in energy loss in particulate system may be achieved by selecting components of an appropriate size.
The bidisperse and multicomponent particulate packings are of great interest in many industrial processes and, therefore, an indepth insight into the nature of these systems is required. The results presented in this study provide the detailed knowledge of micromechanical properties of binary granular mixtures, which may also pave the way for better understanding behavior of more complex packings of nonuniformly sized grains.
Acknowledgements
This work was supported by the Ministry of Science and Higher Education of Poland [Grant Number 0625/IP2/2013/72].
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Conflict of interest
The authors declare that they have no conflict of interest.
Funding information
Funder Name  Grant Number  Funding Note 

Ministry of Science and Higher Education of Poland 

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