Structural model for the dynamic buckling of a column under constant rate compression

Dynamic buckling behavior of a column (rod, beam) under constant rate compression is considered. The buckling is caused by prescribed motion of column ends toward each other with constant velocity. Simple model with one degree of freedom simulating static and dynamic buckling of a column is derived. In the case of small initial disturbances the model yields simple analytical dependencies between the main parameters of the problem: critical force, compression rate, and initial disturbance. It is shown that the time required for buckling is inversely proportional to cubic root of compression velocity and logarithmically depends on the initial disturbance. Analytical expression for critical buckling force as a function of compression velocity is derived. It is shown that in a range of compression rates typical for laboratory experiments the dependence is accurately approximated by a power law with exponent close to 2/3. Theoretical findings are supported by available results of laboratory experiments. Keywords: dynamic buckling, Hoff problem, column, Airy equation, Euler force.Comment: 8 pages, 3 figure

Unsteady ballistic heat transport in infinite harmonic crystals

We study thermal processes in infinite harmonic crystals having a unit cell with arbitrary number of particles. Initially particles have zero displacements and random velocities, corresponding to some initial temperature profile. Our main goal is to calculate spatial distribution of kinetic temperatures, corresponding to degrees of freedom of the unit cell, at any moment in time. An approximate expression for the temperatures is derived from solution of lattice dynamics equations. It is shown that the temperatures are represented as a sum of two terms. The first term describes high-frequency oscillations of the temperatures caused by local transition to thermal equilibrium at short times. The second term describes slow changes of the temperature profile caused by ballistic heat transport. It is shown, in particular, that local values of temperatures, corresponding to degrees of freedom of the unit cell, are generally different even if their initial values are equal. Analytical findings are supported by results of numerical solution of lattice dynamics equations for diatomic chain and graphene lattice. Presented theory may serve for description of unsteady ballistic heat transport in real crystals with low concentration of defects. In particular, solution of the problem with sinusoidal temperature profile can be used for proper interpretation of experimental data obtained by the transient thermal grating technique. Keywords: ballistic heat transport; heat transfer; harmonic crystal; harmonic approximation; polyatomic crystal lattice; complex lattice; kinetic temperature; transient processes; temperature matrix; unsteady heat transport.Comment: 34 pages; 11 figures. arXiv admin note: text overlap with arXiv:1808.0050

On angular momentum balance for particle systems with periodic boundary conditions

The well-known issue with the absence of conservation of angular momentum in classical particle systems with periodic boundary conditions is addressed. It is shown that conventional theory based on Noether's theorem fails to explain the simplest possible example, notably jumps of angular momentum in the case of single particle moving in a periodic cell. It is suggested to consider the periodic cell as an open system, exchanging mass, momentum, angular momentum, and energy with surrounding cells. Then the behavior of the cell is described by balance laws rather than conservation laws. It is shown using the law of angular momentum balance that the variation of the angular momentum in systems with periodic boundary conditions is a consequence of (i) the non-zero flux of angular momentum through the boundaries and (ii) torque acting on the cell due to the interactions between particles in the cell with images in the neighboring cells. Two simple examples demonstrating both phenomena are presented.Comment: Keywords: angular momentum, periodic boundary conditions, open systems, balance laws, conservation laws, Noether theore

Approach to thermal equilibrium in harmonic crystals with polyatomic lattice

We study transient thermal processes in infinite harmonic crystals with complex (polyatomic) lattice. Initially particles have zero displacements and random velocities such that distribution of temperature is spatially uniform. Initial kinetic and potential energies are different and therefore the system is far from thermal equilibrium. Time evolution of kinetic temperatures, corresponding to different degrees of freedom of the unit cell, is investigated. It is shown that the temperatures oscillate in time and tend to generally different equilibrium values. The oscillations are caused by two physical processes: equilibration of kinetic and potential energies and redistribution of temperature among degrees of freedom of the unit cell. An exact formula describing these oscillations is obtained. At large times, a crystal approaches thermal equilibrium, i.e. a state in which the temperatures are constant in time. A relation between equilibrium values of the temperatures and initial conditions is derived. This relation is refereed to as the non-equipartition theorem. For illustration, transient thermal processes in a diatomic chain and graphene lattice are considered. Analytical results are supported by numerical solution of lattice dynamics equations. ${\bf Keywords}$: thermal equilibrium; stationary state; approach to equilibrium; polyatomic lattice; complex lattice; kinetic temperature; harmonic crystal; transient processes; equipartition theorem; temperature matrix.Comment: 29 pages; 11 figure

Discrete and Continuum Thermomechanics

In the present chapter, we discuss an approach for transition from discrete to continuum description of thermomechanical behavior of solids. The transition is carried out for several anharmonic systems: one-dimensional crystal, quasi-one-dimensional crystal (a chain possessing longitudinal and transversal motions), two- and tree-dimensional crystals with simple lattice. Macroscopic balance equations are derived from equations of motion for particles. Macroscopic parameters, such as stress, heat flux, deformation, thermal energy, etc., are represented via parameters of the discrete system. Closed form equations of state relating thermal pressure, thermal energy and specific volume are derived. Description of the heat transfer in harmonic approximation is discussed. Unsteady ballistic heat transfer in a harmonic one-dimensional crystal is considered. The heat transfer equation for this system is rigorously derived.Comment: 22 page

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.Comment: 4 pages; 2 figure

The maximum force in a column under constant speed compression

Dynamic buckling of an elastic column under compression at constant speed is investigated assuming the first-mode buckling. Two cases are considered: (i) an imperfect column (Hoff's statement), and (ii) a perfect column having an initial lateral deflection. The range of parameters, where the maximum load supported by a column exceeds Euler static force is determined. In this range, the maximum load is represented as a function of the compression rate, slenderness ratio, and imperfection/initial deflection. Considering the results we answer the following question: "How slowly the column should be compressed in order to measure static load-bearing capacity?" This question is important for the proper setup of laboratory experiments and computer simulations of buckling. Additionally, it is shown that the behavior of a perfect column having an initial deflection differ significantlys form the behavior of an imperfect column. In particular, the dependence of the maximum force on the compression rate is non-monotonic. The analytical results are supported by numerical simulations and available experimental data.Comment: 11 pages, 4 figure

Fast and slow thermal processes in harmonic scalar lattices

An approach for analytical description of thermal processes in harmonic lattices is presented. We cover longitudinal and transverse vibrations of chains and out-of-plane vibrations of two-dimensional lattices with interactions of an arbitrary number of neighbors. Motion of each particle is governed by a single scalar equation and therefore the notion "scalar lattice" is used. Evolution of initial temperature field in an infinite lattice is investigated. An exact equation describing the evolution is derived. Continualization of this equation with respect to spatial coordinates is carried out. The resulting continuum equation is solved analytically. The solution shows that the kinetic temperature is represented as the sum of two terms, one describing short time behavior, the other large time behavior. At short times, the temperature performs high-frequency oscillations caused by redistribution of energy among kinetic and potential forms (fast process). Characteristic time of this process is of order of ten periods of atomic vibrations. At large times, changes of the temperature are caused by ballistic heat transfer (slow process). The temperature field is represented as a superposition of waves having the shape of initial temperature distribution and propagating with group velocities dependent on the wave vector. Expressions describing fast and slow processes are invariant with respect to substitution $t$ by $-t$. However examples considered in the paper demonstrate that these processes are irreversible. Numerical simulations show that presented theory describes the evolution of temperature field at short and large time scales with high accuracy.Comment: 26 pages, 7 figure

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.Comment: Key words: suspension, proppant transport, hydraulic fracture, effective properties, viscosity, particle dynamics, smoothed particle hydrodynamic

Ballistic resonance and thermalization in Fermi-Pasta-Ulam-Tsingou chain at finite temperature

We study conversion of thermal energy to mechanical energy and vice versa in $\alpha$-Fermi-Pasta-Ulam-Tsingou~(FPUT) chain with spatially sinusoidal profile of initial temperature. We show analytically that coupling between macroscopic dynamics and quasiballistic heat transport gives rise to mechanical vibrations with growing amplitude. This new phenomenon is referred to as "ballistic resonance". At large times, these mechanical vibrations decay monotonically, and therefore the well-known FPUT recurrence paradox occurring at zero temperature is eliminated at finite temperatures.Comment: 6 pages; 5 figure