Effect of particle size ratio and contribution of particle size fractions on micromechanics of uniaxially compressed binary sphere mixtures

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 in-depth insight into a nature of simple particulate systems is required to understand behavior of more complex packings of non-uniformly 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]. Two-dimensional simulations of gravity-free 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 well-known, 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 non-uniformly 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 particle-size 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.

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.

Three-dimensional simulations of the uniaxial confined compression test were performed by using EDEM software [21]. Samples were placed in a chamber of rectangular cross-section 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].

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 in-depth 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 non-uniformly sized grains.