Vitaly A. Kuzkin

Vector-based model of elastic bonds for simulation of granular solids.

A model (further referred to as the V model) for the simulation of granular solids, such as rocks, ceramics, concrete, nanocomposites, and agglomerates, composed of bonded particles (rigid bodies), is proposed. It is assumed that the bonds, usually representing some additional gluelike material connecting particles, cause both forces and torques acting on the particles. Vectors rigidly connected with the particles are used to describe the deformation of a single bond. The expression for potential energy of the bond and corresponding expressions for forces and torques are derived. Formulas connecting parameters of the model with longitudinal, shear, bending, and torsional stiffnesses of the bond are obtained. It is shown that the model makes it possible to describe any values of the bond stiffnesses exactly; that is, the model is applicable for the bonds with arbitrary length/thickness ratio. Two different calibration procedures depending on bond length/thickness ratio are proposed. It is shown that parameters of the model can be chosen so that under small deformations the bond is equivalent to either a Bernoulli-Euler beam or a Timoshenko beam or short cylinder connecting particles. Simple analytical expressions, relating parameters of the V model with geometrical and mechanical characteristics of the bond, are derived. Two simple examples of computer simulation of thin granular structures using the V model are given.

Material frame representation of equivalent stress tensor for discrete solids

In this paper, we derive expressions for equivalent Cauchy and Piola stress tensors that can be applied to discrete solids and are exact for the case of homogeneous deformation. The main principles used for this derivation are material frame formulation, long wave approximation and decomposition of particle motion into continuum and thermal parts. Equivalent Cauchy and Piola stress tensors for discrete solids are expressed in terms of averaged interparticle distances and forces. No assumptions about interparticle forces are used in the derivation, thereby ensuring our expressions are valid irrespective of the choice of interatomic potential used to model the discrete solid. The derived expressions are used for calculation of the local Cauchy stress in several test problems. The results are compared with prediction of the classical continuum definition (force per unit area) as well as existing discrete formulations (Hardy, Lucy, and Heinz-Paul-Binder stress tensors). It is shown that in the case of homogeneous deformations and finite temperatures the proposed expression leads to the same values of stresses as classical continuum definition. Hardy and Lucy stress tensors give the same result only if the stress is averaged over a sufficiently large volume. Thus, given the lack of sensitivity to averaging volume size, the derived expressions can be used as benchmarks for calculation of stresses in discrete solids.

Computer simulation of effective viscosity of fluid-proppant mixture used in hydraulic fracturing

The paper presents results of numerical experiments performed to evaluate the effective viscosity of a fluid-proppant mixture, used in hydraulic fracturing. The results, obtained by two complimenting methods (the particle dynamics and the smoothed particle hydrodynamics), coincide to the accuracy of standard deviation. They provide an analytical equation for the dependence of effective viscosity on the proppant concentration, needed for numerical simulation of the hydraulic fracture propagation.

Interatomic force in systems with multibody interactions

The system of particles interacting via multibody interatomic potential of general form is considered. Possible variants of partition of the total force acting on a single particle into pair contributions are discussed. Two definitions for the force acting between a pair of particles are compared. The forces coincide only if the particles interact via pair or embedded-atom potentials. However in literature both definitions are used in order to determine Cauchy stress tensor. A simplest example of the linear pure shear of perfect square lattice is analyzed. It is shown that, Hardy’s definition for the stress tensor gives different results depending on the radius of localization function. The differences strongly depend on the way of the force definition.

High-frequency thermal processes in harmonic crystals

We consider two high frequency thermal processes in uniformly heated harmonic crystals relaxing towards equilibrium: (i) equilibration of kinetic and potential energies and (ii) redistribution of energy among spatial directions. Equation describing these processes with deterministic initial conditions is derived. Solution of the equation shows that characteristic time of these processes is of the order of ten periods of atomic vibrations. After that time the system practically reaches the stationary state. It is shown analytically that in harmonic crystals temperature tensor is not isotropic even in the stationary state. As an example, harmonic triangular lattice is considered. Simple formula relating the stationary value of the temperature tensor and initial conditions is derived. The function describing equilibration of kinetic and potential energies is obtained. It is shown that the difference between the energies (Lagrangian) oscillates around zero. Amplitude of these oscillations decays inversely proportional to time. Analytical results are in a good agreement with numerical simulations.

Jump detection using fuzzy logic

Jump detection and measurement is of particular interest in a wide range of sports, including snowboarding, skiing, skateboarding, wakeboarding, motorcycling, biking, gymnastics, and the high jump, among others. However, determining jump duration and height is often difficult and requires expert knowledge or visual analysis either in real-time or using video. Recent advances in low-cost MEMS inertial sensors enable a data-driven approach to jump detection and measurement. Today, inertial and GPS sensors attached to an athlete or to his or her equipment, e.g. snowboard, skateboard, or skis, can collect data during sporting activities. In these real life applications, effects such as vibration, sensor noise and bias, and various athletic maneuvers make jump detection difficult even using multiple sensors. This paper presents a fuzzy logic-based algorithm for jump detection in sport using accelerometer data. Fuzzy logic facilitates conversion of human intuition and vague linguistic descriptions of jumps to algorithmic form. The fuzzy algorithm described here was applied to snowboarding and ski jumping data, and successfully detected 92% of snowboarding jumps identified visually (rejecting 8% of jumps identified visually), with only 8% of detected jumps being false positives. In ski jumping, it successfully detected 100% of jumps identified visually, with no false positives. The fuzzy algorithm presented here has successfully been applied to automate jump detection in ski and snowboarding on a large scale, and as the basis of the AlpineReplay ski and snowboarding smartphone app, has identified 6370971 jumps from August 2011 through June 2014.

Enhanced vector-based model for elastic bonds in solids

A model (further referred to as the enhanced vector-based model or EVM) for elastic bonds in solids, composed of bonded particles is presented. The model can be applied for a description of elastic deformation of rocks, ceramics, concrete, nanocomposites, aerogels and other materials with structural elements interacting via forces and torques. A material is represented as a set of particles (rigid bodies) connected by elastic bonds. Vectors rigidly connected with particles are used for description of particles orientations. Simple expression for potential energy of a bond is proposed. Corresponding forces and torques are calculated. Parameters of the potential are related to longitudinal, transverse (shear), bending, and torsional stiffnesses of the bond. It is shown that fitting parameters of the potential allows one to satisfy any values of stiffnesses. Therefore, the model is applicable to bonds with arbitrary length/thickness ratio. Bond stiffnesses are expressed in terms of geometrical and elastic properties of the bonds using three models: Bernoulli-Euler beam, Timoshenko beam, and short elastic cylinder. An approach for validation of numerical implementation of the model is presented. Validation is carried out by a comparison of numerical and analytical solutions of four test problems for a pair of bonded particles. Benchmark expressions for forces and torques in the case of pure tension/compression, shear, bending and torsion of a single bond are derived. This approach allows one to minimize the time required for a numerical implementation of the model. Keywords: granular solid, elastic bond, torque interactions, V-model, discrete element method, distinct element method, particle dynamics.