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Streszczenie:
The analytical models of two spherical grains contact interactions, typical for several classes of slip and sliding regimes in the experimental testing, are proposed. They analyze the cases for coupling or decoupling the frictional microslip and sliding displacements during the kinematically induced sphere translation along a straight trajectory or the force-induced motion from the initially activated contact zone under constant vertical loading. In the slip mode, the evolution of sphere center horizontal displacement obeys the Mindlin-Deresiewicz theory rules either for the force or kinematically induced transverse motions of the sphere. In the frictional sliding mode, it is demonstrated that for the kinematically induced transverse motion of the sphere, the contact tractions are fully governed by the coupled evolution of slip and sliding displacements. When the account for contact slip velocity and the rate of contact plane rotation is made, then the coupling of slip and sliding modes theoretically results in a simple scaling multiplier imposed on the overlap resulted from the sliding mode. It generates a driving force fluctuation and affects the evolution of contact tractions. For transverse sliding of the sphere under constant vertical load and driving force, the contact tractions are essentially governed by the conditions of static equilibrium and are independent of the displacements generated in the slip mode. In this case, the slip displacement provides only the additive term to the sliding displacement of the sphere center, not affecting contact tractions.

Słowa kluczowe:
sphere–sphere contact, frictional slip and sliding, Coulomb friction, displacement and mixed force-displacement control, reciprocal motion

Streszczenie:
The present paper is aimed at developing the analytical description of the interaction of two contacting spheres for several classes of slip and sliding trajectories, typical in the experimental testing. The analysis accounts for memory effects in the slip regime and configurational effects in the sliding regime, expressed in terms of an active loading surface and memory surfaces within the space of contact forces. Analytical relations for contact response are derived for linear and piecewise-linear motion trajectories of the sphere. The problem of multiple contact interaction of the sphere moving over the regularly packed granular bed is also considered analytically. It is demonstrated that the dual contact activation-separation processes occur within the combined slip–sliding modes, essentially affecting the distribution of contact tractions. The results obtained are relevant for the class of contact problems requiring analysis of interaction of slip and sliding displacements, in particular in testing grain contact interaction aimed at specification of elastic, frictional and wear parameters.

Słowa kluczowe:
Sphere–sphere contact, Memory rules, Slip and sliding, Displacement and force control, Monotonic and reciprocal sliding, Friction, Granular bed

Streszczenie:
The analytical modelling of coupled slip and sliding contact response of two elastic spheres is presented for the kinematically imposed sphere centre relative motion trajectories. One sphere is assumed as a fixed, the other translating along a specified trajectory and remaining in contact condition. Two cases are considered, the first is corresponding to a linear trajectory with the contact engagement in the combined slip-sliding mode, the other is related to the contact initiation by normal loading and subsequent motion along an inclined linear trajectory. The formulae and diagrams of the evolution of driving force along the sliding path in terms of main contact geometry parameters were analytically specified. Further extensions and applications of the analysis can be envisaged in the creation of the translation controlled apparatus for the measurements of friction and restitution coefficients for the pair of spherical grains.

Słowa kluczowe:
soil, spherical grains contact interaction, slip and finite sliding modes, monotonic or reciprocal sliding, coefficients of friction and restitution

Streszczenie:
In Part I of the present study, the static analysis for the sliding of two identical spheres under displacement and force control was carried out. For linear and circular sliding trajectories, the contact traction evolution was analytically specified for both monotonic and reciprocal sliding regimes. Similarly, for the specified gravity loading, the driving force evolution and the sliding path were also determined. In the present Part II of the analysis, the dynamic response for the same sliding modes is presented. The contact traction and velocity evolutions are considered in detail. The analytical formulae are proposed for prediction of the tangential restitution coefficient, critical velocity, and time of contact for the displacement and load-controlled motions. The effects of loading and reloading in reciprocal sliding are also considered with account for the slip and sliding regimes. The generated results have practical aspects and can be implemented in modeling of the asperity and rough surface interaction, wear analysis and also in the development of the numerical discrete element method.

Streszczenie:
A simplified analytical model of tangential contact engagement, sliding and separation of two elastic, identical spheres is developed assuming the kinematically induced sphere motion trajectory or load controlled sliding motion. The evaluation of driving force during contact sliding motion is determined for both monotonic and reciprocal sliding motion. The analytical formulae and diagrams of driving force versus sliding path are specified for linear and circular paths. The sliding trajectories are also determined for the load controlled programs. The results presented can be applied in the experimental testing of frictional response of contacting bodies, in a wear study of rough surfaces or in the contact interaction analysis of granular material during flow. The results can also be relevant for the development of the discrete element method widely applied in simulation of granular material flow, where the sliding regime conditions prevail in grain contact interaction.

Streszczenie:
A simplified analytical modelling of tangential contact interaction of two elastic spheres was presented accounting for the kinematically induced sphere trajectory or a load controlled sliding process. The evolution of driving force in combined slip and finite sliding interaction was considered for both monotonic and reciprocal sliding motion. The analytical formulae and diagrams for the evolution of driving force along the sliding path in terms of the main contact geometry parameters were specified. The sliding trajectories and contact tractions were also determined. The analytical formulae were briefly presented for prediction of the tangential restitution coefficient and time of contact. The results presented can be applied in the experimental testing of frictional response of contacting bodies, wear studies of rough surfaces or in the contact interaction analysis of granular material during flow. The results can also be relevant for development of the discrete element method widely applied in simulation of granular material flow, where the sliding regime conditions prevail in intergranular contact interactions.

Słowa kluczowe:
sphere-sphere contact, slip and finite sliding, displacement and force control, monotonic and reciprocal sliding, traction and velocity evolution, coefficient of restitution, time period of contact interaction

Streszczenie:
The present paper provides both experimental and DEM analyses of the filling and discharge of pea grains from a 3D flat-bottomed bin. In the DEM model, the fixed mean values of the experimentally determined single particle data, such as the particle density, Young's modulus, Poisson's ratio as well as the sliding and rolling friction coefficients were incorporated to analyse their effects on the macroscale indicators, such as the wall pressure, discharge velocities and material outflow parameters. The effect of rolling friction was studied based on the experimentally measured single particle rolling friction coefficient. This analysis is aimed at the quantitative prediction of flow parameters as related to the identification of material parameters.

Słowa kluczowe:
Visco-elastic granular material, Discrete element method, Rolling friction, Wall pressure, Material velocity, Outflow mass, Outflow rate, Biosystems, Pea grains

Streszczenie:
The present paper provides a discrete element method (DEM) analysis of the filling and discharge processes of granular material in a 3D flat-bottomed bin. A granular aggregate of nearly round particles (20,400 pea grains, 7.2–7.8 mm in diameter) is considered. The numerical results are compared with the experimental data. The DEM analysis provides an accurate prediction of wall stress distribution and the outflow rate of discharge throughout the bottom orifice. The stress distribution developed within the granular material after filling and during the discharge phase is considered, and the transition from the active to passive stress state is discussed. This analysis aims to quantitatively predict the flow parameters related to the careful identification of the material parameters. The investigation presented may be useful for the ongoing development of DEM.

Słowa kluczowe:
Visco-elastic granular material, Discrete element method, Wall pressure, Granular material stress, Micro-properties, Bulk properties

Streszczenie:
By using the discrete element method (DEM) a comparison and observations on material flow patterns in plane-wedged, space-wedged, and flat-bottomed hopper were accounted for. Numerical results obtained by combining data of individual particles, statistical processing of particle assemblies and evaluation of the field variables provided the essential characteristics for different regimes of the discharge flow (within steady or unsteady state of flow) and the differences in differently shaped hoppers due to different microscopic inter-particle friction. For validation of the performed simulations, velocity patterns developed in three-dimensional flat-bottomed hopper containing 20,400 pea grains were also analysed. To represent the continuum by avoiding the local effects produced by the individual grains, the simulation results were focused on the mean velocity distributions with data smoothening. The effect of rolling resistance on granular material flow was also considered.

Słowa kluczowe:
DEM, Mass flow rate, Velocity distribution, Plane-wedged, Space-wedged, Flat-bottomed hoppers

Streszczenie:
The friction effect in filling and unsteady/steady discharge of three-dimensional wedge-shaped hopper was considered numerically by the discrete element method. The presented model is based on contact mechanics of a single spherical particle involving Hooke's law of spring interaction, static and dynamic friction as well as viscous damping forces. The hopper filling process en masse was performed while the modeling of uncontrolled and controlled discharge was simulated. The evolution of the kinetic energy of filling, the distribution of stresses within granular material and stresses acting on the walls, as well as the change in fields of material porosity were investigated at the stages of filling and discharge of the hopper. The variation of flow velocities, the discharged mass fractions as well as discharge fraction rates were analyzed. The friction effect was analyzed by varying inter-particle friction coefficient over the range of 0–0.6. A comparison of the numerical results with available continuum-based predictions was also presented.

Słowa kluczowe:
Visco-elastic granular material, 3D wedge-shaped hopper, Discrete element method, Wall pressure, Granular stresses, Fraction discharged, Bulk porosity

Streszczenie:
The paper presents the application of the discrete element method to modelling of granular material filling and discharge in 3D flat-bottomed hopper. A mathematical model, as well as the developed software code, operates with spherical visco-elastic non-cohesive frictional particles. The evolution of granular flow, internal forces and densification (rarefaction) are characterized by macroscopic parameters such as the discharge rates, the porosity fields and the wall pressures, as well as by microscopic evaluations in terms of coordination number, velocity patter ns and inter-particle contact forces. It was shown that qualitative characterization of flow may be done even by relatively rough models with small number of particles, which required to be increased, however, for more precise description of the localize d phenomena. Unsatisfactory evaluation of the stress peak during discharge is p resented as an illustrative example. The main focus of the paper is the analysis of particle friction effect and the consistency of micro and macro-phenomena in the time-dependent flow process.

Słowa kluczowe:
discrete element method, spherical particles, visco-elastic granular material, three-dimensional flat bottomed hopper

Streszczenie:
Multiobjective approach for optimization of the discharge flow parameters of viscoelastic frictional granular material in a wedge-shaped hopper with fixed outlet orifice is considered. The discharge time and the discharge flow rate are taken as optimality criteria, while the discharge mass of material and the shape of the hopper are considered as design variables. The discrete concept and numerical discrete element method (DEM) are applied to the analysis of granular flow. The filling problem considered as numerical generation of the initial conditions for discharge flow was initially solved. The presented DEM model was validated by examining the wall pressures and comparing them with the results of classical macroscopic pressure prediction. Variation of optimality criteria and Pareto set is demonstrated by the numerical results. Sensitivity of optimality criteria to interparticle friction is also investigated.

Słowa kluczowe:
Viscoelastic granular material, Hopper, Discrete element method, Optimization of flow parameters