Anton M. Krivtsov

Peter the Great St.Petersburg Polytechnic University | SPBSTU · Department of Theoretical Mechanics

Излагаются методы аналитического описания двух переходных процессов, реализующихся в гармонических (линейных) кристаллах со случайными начальными условиями: уравнения кинетической и потенциальной энергий и перераспределения энергии по степеням свободы. Для аналитического описания используется подход, основанный на анализе динамики ковариаций переме...

A thermal diode or rectifier is a system that transmits heat or energy in one direction better than in the opposite direction. We investigate the influence of the distribution of energy among wave numbers on the diode effect for the junction of two dissimilar harmonic chains. An analytical expression for the diode coefficient, characterizing the di...

Quasi-static growth of planar three-dimensional cracks in homogeneous and three-layer media is studied numerically using the particle dynamics method. It is shown that in the homogeneous medium cracks with different convex and nonconvex initial shapes tend to become circular (penny-shaped). The crack "forgets" its initial chape approximately when i...

We present analytical and numerical investigations of energy propagation in systems of massive particles that interact via harmonic (linear) forces. The particle motion is described by a scalar displacement, and the particles are arranged in a simple crystal lattice. For the systems under consideration we prove the conservation of the total energy...

We propose and examine a potential analogy between mass transfer (in space) and energy transfer (in solids). We adapt classical equations of matter dynamics to describe the dynamics of energy transfer. Such fundamental quantities as the effective mass, momentum, moment of inertia and other quantities typical for bodies of matter are introduced for...

In this paper, we propose an approach to define thermal conductivity for a purely ballistic transient heat conduction and study its size dependence for two-dimensional structures in circular geometry in order to use this dependence as a purely ballistic regime signature. Then, a review of various experimental techniques by which the thermal conduct...

In the paper, we deal with ballistic heat transport in a graphene lattice subjected to a point heat source. It is assumed that a graphene sheet is suspended under tension in a viscous gas. We use the model of a harmonic polyatomic (more exactly diatomic) lattice performing out-of-plane motions. The dynamics of the lattice is described by an infinit...

We present a review of the results in the field of discrete thermomechanics that have been achieved in the Institute for Problems in Mechanical Engineering RAS over the past decade. The focus is set on the novel approach for analytical description of non-equilibrium thermomechanical processes in crystalline solids. One, two, and three-dimensional p...

In the paper we deal with ballistic heat transport in a graphene lattice subjected to a point heat source. It is assumed that a graphene sheet is suspended under tension in a viscous gas. We use the model of a harmonic polyatomic (more exactly diatomic) lattice performing out-of-plane motions. The dynamics of the lattice is described by an infinite...

The kinetic theory is widely used in the description of thermal transport at the micro- and nanoscale. In the theory, it is assumed that heat is carried by quasi-particles, obeying the Boltzmann transport equation. These quasi-particles are sometimes associated with phonons. However, since phonons are not localized in physical space, they cannot pl...

The work is devoted to the description of unsteady thermal processes in low-dimensional structures. To obtain the relationship between the microscopic and macroscopic descriptions of solids, it is necessary to understand the heat transfer mechanism at the micro-level. At the latter, in contrast to the macro-level, analytical, numerical, and experim...

The equilibration of sinusoidally modulated distribution of the kinetic temperature is analyzed in the $\beta$-Fermi-Pasta-Ulam-Tsingou chain with different degrees of nonlinearity and for different wavelengths of temperature modulation. Two different types of initial conditions are used to show that either one gives the same result as the number o...

The equilibration of sinusoidally modulated distribution of the kinetic temperature is analyzed in the β-Fermi-Pasta-Ulam-Tsingou chain with different degrees of nonlinearity and for different wavelengths of temperature modulation. Two different types of initial conditions are used to show that either one gives the same result as the number of real...

The effect of discrete breathers (DBs) on macroscopic properties of the Fermi-Pasta-Ulam chain with symmetric and asymmetric potentials is investigated. The total to kinetic energy ratio (related to specific heat), stress (related to thermal expansion), and Young’s modulus are monitored during the development of modulational instability of the zone...

We study conversion of thermal energy to mechanical energy and vice versa in an α-Fermi-Pasta-Ulam-Tsingou (FPUT) chain with a 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 phenome...

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite s...

Рассматриваются нестационарные тепловые процессы в низкоразмерных структурах. Понимание теплопередачи на микроуровне необходимо для получения связи между микро- и макроскопическим описанием твердых тел. На макроскопическом уровне распространение тепла описывается законом Фурье. Однако на микроскопическом уровне аналитические, численные и эксперимен...

An asymptotic representation is obtained at large times for the thermal wavefront propagating in a one-dimensional harmonic crystal. The propagation of thermal waves from a localized thermal perturbation and the transition zone between regions with different temperatures is considered. An explicit solution is given for a number of the simplest form...

Modeling of the heat transfer in ideal crystal lattice with defects is performed for measuring the heat conductivity coefficient. A non-steady process in closed system is studied. The method is based on comparison of the results of molecular dynamics simulation and solution of the heat equation. Two-dimensional and three-dimensional structures with...

We study conversion of thermal energy to mechanical energy and vice versa in $\alpha$-FPU chain with spatially sinusoidal profile of initial temperature. We show analytically that thermal expansion and temperature oscillations, caused by quasiballistic heat transport, excite mechanical vibrations with growing amplitude. This new phenomenon is refer...

We study transient thermal process in infinite nonlinear crystal after initial deformation. Initially particles have random displacements and random velocities corresponding to a uniform initial temperature. After homogeneous deformation the crystal is in the nonequilibrium state. As a result of the transient process the crystal transfers to a new...

The features for the unsteady process of thermal equilibration (“the fast motions”) in a one-dimensional harmonic crystal lying in a viscous environment (e.g., a gas) are under investigation. It is assumed that initially the displacements of all the particles are zero and the particle velocities are random quantities with zero mean and a constant v...

For a number of crystals the existence of spatially localized nonlinear vibrational modes, called discrete breathers (DBs) or intrinsic localized modes (ILMs), has been demonstrated using molecular dynamics and in a few cases the first-principle simulations. High-resolution imaging of DBs is a challenging task due to their relatively short lifetime...

An influence of the second neighbor interaction on the process of heat propagation in a one‐dimensional crystal is studied. Previously developed model of the ballistic nature of the heat transfer is used. It is shown that the initial thermal perturbation evolves into two consecutive thermal wave fronts propagating with finite and different velociti...

The features for the unsteady process of thermal equilibration (the "fast motions") in a one-dimensional harmonic crystal with nearest-neighbor interactions lying in a viscous environment (e.g., a gas) are under investigation. It is assumed that initially the displacements of all the particles are zero and the particle velocities are random quantit...

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial t...

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite s...

We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear par...

The analytical model of unsteady ballistic heat transfer in a one-dimensional harmonic crystal is analyzed. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-rever...

This work presents the thermodynamical analysis of the ballistic heat equation from the viewpoint of two approaches: Classical Irreversible Thermodynamics (CIT) and Extended Irreversible Thermodynamics (EIT). A formula for calculation of the entropy within the framework of EIT for the ballistic heat equation is derived in this work. Entropy is calc...

An instant homogeneous thermal perturbation in the simplest model of the finite harmonic one-dimensional crystal is studied. Previously it was shown that for the same problem in the infinite crystal the kinetic temperature performs oscillations with decreasing amplitude described by the Bessel function of the first kind. In the present paper it is...

We consider dynamics of a one-dimensional harmonic chain with harmonic on-site potential subjected to kinematic and force loadings. Under kinematic loading, a particle in the chain is displaced according to sinusoidal law. Under force loading, a harmonic force is applied to a particle. Dependence of the total energy supplied to the chain on loading...

This review paper analyzes the entropy concept and the second law of thermodynamics in the context of one-dimensional media. For simplicity, only thermal processes are taken into account and mechanical motions are neglected. The relation between entropy and temperature and the constraints on the direction of the heat flux are discussed. A compariso...

The results of the computer simulation for the circumsolar gas-dust cloud evolution are presented. The particle dynamics method is used. We show that gas-dust clusters can be formed in ring-shaped structures of protoplanetary disks. It is demonstrated that the clusters are formed as a result of the counteracting of the self-gravitational force of t...

The work is devoted to description of unsteady thermal processes in low-dimensional materials. One-dimensional harmonic crystals with alternating masses and stiffnesses are considered. Analytical solution demonstrates the ballistic nature of heat propagation, which is confirmed by numerical simulations based on the particle dynamics method. It is s...

An analytical model of high frequency oscillations of the kinetic and potential energies in a one-dimensional harmonic crystal with a substrate potential is obtained by introducing the nonlocal energies [1]. A generalization of the kinetic temperature (nonlocal temperature) is adopted to derive a closed equation determining the heat propagation pro...

We consider a two-dimensional square lattice model extended by additional not closed neighboring interactions. We assume the elastic forces between the masses in the lattice to be nonlinearly dependent on the spring elongations. First, we use an analysis of the linearized discrete equations to reveal the influence of additional interactions on the...

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 ma...

An approach for analytical description of thermal processes in harmonic lattices is presented. We cover longitudinal and transverse vibrations of chains and out-ofplane 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 notio...

One-dimensional chain of pointwise particles harmonically coupled with nearest neighbors and placed in six-order polynomial on-site potentials is considered. Power of the energy source in the form of single ac driven particles is calculated numerically for different amplitudes $A$ and frequencies $\omega$ within the linear phonon band. The results...

One-dimensional chain of pointwise particles harmonically coupled with nearest neighbors and placed in six-order polynomial on-site potentials is considered. Power of the energy source in the form of single ac driven particles is calculated numerically for different amplitudes $A$ and frequencies $\omega$ within the linear phonon band. The results...

An approach for transition from discrete to continuum description of thermomechanical behavior of solids is discussed. The transition is carried out for several perfect anharmonic crystals with pair force interactions: one-dimensional crystal, quasi-one-dimensional crystal (a chain possessing longitudinal and transverse motions), two and three-dime...

Пособие посвящено описанию быстрых и медленных тепловых процессов в гармонических кристаллах. К быстрым относятся выравнивание кинетический и потенциальной энергий и перераспределение энергии по пространственным направлениям. К медленным - процесс распространения тепла. Изложен оригинальный подход, позволяющий описывать данные процессы аналитически...

We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear par...

Unsteady heat transfer in a harmonic chain is analyzed. Two types of thermal perturbations are considered: 1) initial instant temperature perturbation, 2) external heat supply. Closed equations describing the heat propagation are obtained and their analytical solution is constructed.

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 crystal...

We consider two transient thermal processes in uniformly heated harmonic crystals: (i) equalibra-tion of kinetic and potential energies and (ii) redistribution of the kinetic energy among the spatial directions. Equations describing these two processes in two-dimensional and three-dimensional crystals are derived. Analytical solutions of these equa...

The method of particle dynamics is used for both analytical and numerical investigation of tensor properties of the Mie–Grüneisen equation of state for two-dimensional solids with crystalline structure. It is demonstrated analytically that the Grüneisen function essentially depends on the ratio between the eigenvalues of the deformation temperature...

An approach for analytical description of unsteady heat transfer in infinite harmonic lattices is presented. Evolution of initial spatial distribution of kinetic temperature is investigated. Lattice dynamics equations are written in a form valid for one-dimensional chains and out-of-plane vibrations of two-dimensional lattices. The description of h...

In this work exact solutions for the equation that describes anomalous heat propagation in 1D harmonic lattices are obtained. Rectangular, triangular, and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the...

In this work exact solutions for the equation that describes anomalous heat propagation in 1D harmonic lattices are obtained. Rectangular, triangular, and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the...

В работе рассматриваются два высокочастотных тепловых процесса, происходящие при переходе гармонических кристаллов в состояние термодинамического равновесия: выравнивание кинетической и потенциальной энергий и перераспределение энергии по пространственным направлениям. Получено уравнение с детерминированными начальными условиями, описывающее оба пр...

Рассматриваются два переходных тепловых процесса, происходящие в однородно нагретых гармонических кристаллах: 1) выравнивание кинетической и потенциальной энергий; 2) перераспределение кинетической энергии по пространственным направлениям. Выведены уравнения, описывающие оба процесса в двухмерном и трехмерном случаях. Получены аналитические решения...

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...

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...

We consider relaxation to thermodynamic equilibrium in two- and three-dimensional harmonic crystals. Two processes are investigated: (i) equilibration of kinetic and potential energies and (ii) distribution of kinetic energy among spatial directions. Deterministic Cauchy problem describing both processes is formulated using the correlation analysis...

A one-dimensional harmonic crystal on an elastic substrate is considered as a stochastic system into which randomness is introduced through initial conditions. The use of the particle velocity and displacement covariances reduces the stochastic problem to a closed deterministic problem for statistical characteristics of particle pairs. An equation...

We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as graphite, diamond, fullerene, and carbon nanotubes. The interatomic bond is modeled by a hyperboloid–shape truss structure. The elastic characteristics of this bond are determined. Previous known structural...

Linear elastic deformation of the two-dimensional triangular lattice with multiple vacancies is considered. Closed-form analytical expressions for displacement field in the lattice with doubly periodic system of vacancies are derived. Effective elastic moduli are calculated. The results are compared with the ones obtained by molecular dynamics simu...

An analytical model of unsteady heat transfer in a one-dimensional harmonic crystal is presented. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed system of differential-difference equations determining unsteady thermal processes is derived. For an instantaneous heat perturbation a time-reversible therma...

An analytical approach for description of unsteady heat transfer in a one-dimensional harmonic crystal is presented. A covariance temperature is introduced. A constitutive law for the heat flux, an analogue of Fourier's law for the considered system is obtained. A reversible thermal wave equation for the kinetic temperature is derived and an integr...

Due to their excellent mechanical properties and extra high electroconductivity, suspended graphene sheets recently were proposed as perspective working elements of nanosystems. This work is devoted to derivation of natural frequencies of such sheets. Two different approaches are proposed. The first one is based on representation of the graphene sh...

A closed system of differential-difference equations describing thermal processes in one-dimensional harmonic crystals is obtained in the paper. An equation connecting the heat flow and the kinetic temperature is obtained as a solution of the system. The obtained law of heat conduction is different from Fourier's law and results in an equation that...

This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are ob...

The article presents results of numerical experiments performed to evaluate the effective rheological properties of a mixture of a fluid with solid particles. The numerical simulation of the Couette and Poiseuille flows shows that in the both cases, the effective viscosity and non-Newtonian properties of the suspension coincide to the accuracy of s...

Thermal expansion of a classical chain with pair interactions performing longitudinal and transverse vibrations is investigated. Corresponding equations of state are derived analytically using series expansions of pressure and thermal energy with respect to deformations of the bonds caused by thermal motion. In the first approximation the equation...

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....

The paper proposes a discrete mechanical model of monolayer graphene. A relation between parameters of the model and elastic characteristics of its equivalent continuum is derived by comparing the energy of small strains on micro-and macroscales. The relation allows one to determine the microscale interaction parameters from experimental data and,...

A study was conducted to consider and find an analytical solution to the problem of oscillations of the kinetic and potential energies in a one-dimensional crystal. An analytical solution for the linear interaction of particles, random initial velocities, and zero initial displacements was derived. It was shown that the time dependence of energies...

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 eq...

The problem of description of large inelastic deformations of solids is considered. On a simple discrete model it is shown that the classical concept of deformations used in continuum mechanics can exhibit serious difficulties due to reorganizations of the internal structure of materials. The way of construction of constitutive equations in continu...

A mechanical model of diatomic lattice which takes into account force and moment interactions is proposed. Relations between macroscale elastic moduli and microscale longitudinal and transverse stiffnesses of interatomic bonds are derived. In the framework of the proposed model, crystals with diamond and sphalerite lattices are considered. It is sh...

The advance in nanotechnology has lead to necessity to determine strength properties of crystal structures. Stability of a structure under finite deformations is closely connected with its strength. In this work stability of plane triangular (single atomic layer of FCC and HCP) and FCC lattices under finite strain is investigated. Analysis and mode...

The origin of the Moon remains an unsolved problem of the planetary science. Researchers engaged in celestial dynamics, geophysics, and geochemistry are still discussing various models of creation of our closest cosmic neighbour. The most popular scenario, the impact hypothesis involving a collision early in the Earth's history, has been substantia...

A mechanical model of diatomic crystal lattice with force interaction between atoms and angular interaction between bonds taken into account is proposed. Some relations between the macroscopic moduli of elasticity and the microparameters of the longitudinal rigidity of interatomic bonds and of the angular interaction rigidity are obtained for cryst...

The pair force interaction potential that allows one to describe a deviation from spherical symmetry, which is typical for hexagonal close-packed structures, is constructed using the “spherically symmetric” Mie potential that depends only on the interatomic distance. The parameters of the considered potential, which ensure the stability of hexagona...

The stability of the ideal two-dimensional triangular crystal lattice, which is a single atomic layer in the FCC and HCP structures was studied. Morse and Lennard-Jones potentials were used as the interaction law. An important distinction of Morse potential from Lennard-Jones potential is that during the compression of the material to the point the...

This paper reviews classical theories of coarse-graining and gives a short introduction to representative coarse-grained atomistic models that were developed based on structure reduction, an assumption of homogenous deformation, and field representation. The applicability and limitations of these coarse-grained models are analyzed on the basis of t...

In our paper, we suggest a model allowing study of the growth process of the double system (planet-satellite) as a result of the accumulation of scattered material from the common dust condensation. The model consists of two components—a computer component and an analytic component. In the course of the numerical experiment, the computer model allo...

The relations that describe the interactions of particles with rotational degrees of freedom in the study of mechanical properties of graphene are presented. Molecular-dynamic simulation of deformation and fracture of graphene upon tension is performed. During simulation, elastic, and strength characteristics of graphene are calculated. The propert...

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtainin...

The speed-gradient variational principle (SG-principle) for nonstationary nonequilib-rium systems is formulated and illustrated by an example. It is proposed to use the SG-principle to model transient (relaxation) dynamics for systems satisfying maximum entropy principle. Nonstationary processes generated with the method of dynamics of particles ar...

The elastic properties of diatomic crystals are considered. An approach is proposed that permits calculating the elastic characteristics of crystals by using the interatomic interaction parameters specified as many-particle potentials, i.e., potentials that take into account the effect of the environment on the diatomic interaction. The many-partic...

In this paper the effects of oblique impact loading of brittle rocks are investigated. The drilling process in hard rocks is simulated using particles dynamics (PD). The rock sample and impactor are described by particles with different bond strength. Impact is generated by applying a dynamical force to the impactor. The results are compared with t...