Search

A new nonlinear lattice model for an influence of the hydrogen concentration on the elastic constants of the lattice model of a material is developed. A weakly nonlinear long-wavelength continuum model is considered, and a model nonlinear equation for the dynamics of concentration of hydrogen is obtained asymptotically. The model predicts a decrease in the stiffness coefficient as well as a light local moving increase due to the localized nonlinear concentration wave propagation.

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 dense packing of particles are considered. Defects are modeled by removing or changing the mass of randomly selected lattice atoms. Based on the results of molecular-dynamics modeling, an empirical dependence of the thermal diffusivity on the density of defects is elaborated. It also turns out in a good agreement with experimental data.

It is shown that plane longitudinal nonlinear strain waves in a 2D graphene-type hexagonal lattice are described by a nonlinear double dispersion equation previously developed for the description of waves in an elastic rod. A procedure is developed to derive the governing equation as a continuum limit of the original lattice model. The lattice is described by an interaction of two sub-lattices and both translational and angular interactions between the lattice masses are taken into account.

The models of the wave description of the hydrogen concentration in materials are discussed and compared. Special attention is paid to the role of nonlinear effects.

It is shown that angular stiffness in the hexagonal lattice model plays a significant role in the geometrical nonlinear terms in the equations of the continuum limit. A geometrically nonlinear discrete model is formulated for the hexagonal lattice by considering the interaction of two sublattices. An asymptotic procedure is developed in order to obtain the nonlinear coupled equations of motion in the continuum limit of the discrete model. An interaction of longitudinal and shear plane strain waves is studied by using the solutions of the obtained equations.

A new nonlinear governing equation is obtained for the dynamics of hydrogen concentration. Numerical solution of the equation allows us to describe evolution of localized disturbance of the hydrogen concentration and the significant influence of nonlinearity on the shape of localized waves of the hydrogen concentration.

It is shown that highly nonlinear shear waves propagation in an elastic layer laying on the visco-elastic foundation subjected to the upper surface external loading may be modeled by the sine Gordon equation with the additional control term. The last has the form of the distributive feedback control function where normal loading provides desired target function. It allows us to justify the control model previously obtained for nonlinear localized waves of the sine Gordon equation.

The feedback control algorithm is applied to provide stable propagation of a two-step shock waves for nonlinear isothermal Euler equations despite the desired profile and velocity of the waves do not correspond to an analytical solution of the equations. Two cases are considered: transition to the two-step shock wave solution form the usual one-step wave and generation of a wave with a two-step front from an initially undisturbed velocity field. In both cases arising of two-step shock waves is obtained and an influence of the control algorithm coefficients on the shape of the waves is established.

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 properties of the dispersion relation for longitudinal and shear plane waves. Then we develop an asymptotic procedure to obtain continuum two-dimensional non-linear equations to study the transverse instability of weakly non-linear localized plane longitudinal and shear waves. We find that the additional interactions used in the model may affect the sign of the amplitude of the plane strain waves (existence of compression (minus sign) or tensile (plus sign) plane waves) and their transverse stability.

In this work the energy transfer in a one-dimensional harmonic crystal is investigated. In particular, a comparison between the discrete approach presented by Klein, Prigogine, and Hemmer with the continuum approach presented by Krivtsov is made. In the pioneering work of Klein and Prigogine the transfer of thermal energy is considered. In particular, an expression is obtained, which allows to calculate the thermal energy of each particle as a function of time. Later, Hemmer derived and used similar expressions to solve several particular problems in context of heat conduction. In the work of Krivtsov—in contrast to the discrete approach—a partial differential continuum equation is derived from the lattice dynamics of a 1D harmonic crystal. This so-called ballistic heat equation describes the propagation of heat at a finite speed in a continuous one-dimensional medium. The current work compares analyses based on the discrete equation of Klein, Prigogine, and Hemmer with those from the continuum-PDE-based one by Krivtsov. There is an important difference between the approaches. The continuum approach is derived from the dynamics of the crystal lattice, in which only kinetic degrees of freedom were excited and then thermal equilibration occurred. In contrast to that we consider in the discrete approach explicitly given equal kinetic and potential initial energies. Several exactly solvable initial problems are studied by using both methods. The problem of point perturbation shows a discrepancy in the results obtained in the framework of the continuous and discrete approaches. It is caused by the fact that the smoothness conditions of the initial perturbation is violated for the continuum approach. For other problems it is shown that at large spatial scales, where the one-dimensional crystal can be considered as a continuous medium, the discrete and the continuum relations converge. The asymptotic behavior of the difference between two aforementioned approaches is analyzed.

The feedback control algorithm is applied to provide stable propagation of the shock waves for nonlinear isothermal Euler equations even if the desired profile and velocity of the wave do not correspond to their analytical solution. Also the control algorithm helps to suppress scheme oscillations on the shock wave solutions, this is studied on an example of the Lax-Wendroff second-order scheme.

The dynamics of two-dimensional lattice structures is studied. The complexity of the structure includes non-neighboring interactions between the lattice masses, consideration of both translational and rotational interactions and also their nonlinear character (physical nonlinearity). The asymptotic procedures are developed to obtain the governing nonlinear equations of motion in the continuum limit. The equations obtained are studied both analytically and numerically. Of special interest are the propagation and transverse instability of the plane solitary strain waves. It is shown that the dynamics of longitudinal and shear waves is different in various two-dimensional lattices. The relationships for the elastic constants are obtained to characterize the type of the localized strain waves (tensile or compression), their transverse instability and possible auxetic behavior. Numerical solutions are obtained that describe unstable and stable dynamics of the plane longitudinal and shear waves.

The influence of internal hydrogen in metals is modeled using a bi-continual model. The internal medium is introduced in the bi-continuum for modeling the diffusion of mobile hydrogen. Study of the model equations was carried out under the assumption of a small fluctuation of the concentration of mobile hydrogen under the action of external mechanical load. The case of appearance of the nonlinear solitary waves exceeding the average concentration is considered. The conditions are obtained under which the appearance of a decaying solitary wave is possible.

A two-dimensional model is considered in the form of a phononic crystal having a rectangular lattice with elastically interacting ellipse-shaped particles possessing two translational and one rotational degrees of freedom. The linear differential-difference equations are obtained by the method of structural modeling to describe propagation of longitudinal, transverse and rotational waves in the medium. It is found analytically how the coefficients of the equations depend on the sizes of the particle and on the parameters of interactions between them. The dispersion properties of the model are analyzed. Existence of a backward wave is established. The threshold frequencies of acoustic and rotational waves in some crystalline materials with cubic symmetry are estimated.

The distributed feedback control is developed to support propagation of localized non-linear waves for the double sine-Gordon equation and the dispersive sine-Gordon equation previously obtained for the description of dynamic processes in complex crystalline lattices. The control allows the propagation of both bell-shaped and kink-shaped waves with permanent shape and velocity, with a functional form which does not correspond to the known analytical solutions of the equations. The results might be used for choosing external loading providing desired strain localization or variations in the internal structure of the lattice.

Localization of the waves of the sine-Gordon equation depends on the shape of the initial condition. It is shown how initially motionless Gaussian distribution may be modified to obtain propagation of localized waves in both directions. However, the resulting localized wave profile is described neither by an asymptotic envelope- wave solution to the sine-Gordon equation nor by its exact traveling breather solution. The distributed control algorithms are developed to achieve wave localization independent of the shape of the initial condition. It is shown that localization of the waves in both directions is achieved by means of a feedforward (nonfeedback) control. The waves are similar to the envelope wave solution. The feedback distributed algorithm is shown to provide both localized waves according to analytical solutions and their unidirectional propagation.

It is shown, that particular kink-shaped wave solutions to nonlinear nonintegrable equation may be employed to account for important features of the kink evolution observed in numerical solutions and to check the last solutions without computations. Thus, an exact traveling wave solution may predict boundary conditions suitable for the kink realization in numerics. A quasistationary asymptotic solution may detect an evidence of dispersion at the numerical kink shape, while an asymptotic solution obtained by the multiple scale method, describes variations in the kink amplitude, slope and velocity.

A new distributed speed-gradient feedback control algorithm for the sine–Gordon (SG) equation is proposed. It creates the antikink traveling wave mode for a broader class initial conditions compared to the uncontrolled system. In real physical problems it is difficult to provide consistent initial conditions for the second-order (in time) equations. Therefore for an uncontrolled system even small variations in the initial velocity relative to that of the exact antikink solution of the SG equation give rise to growing oscillations. The control algorithm allows one both to suppress defects and to obtain stable propagation of an antikink in the form of the exact traveling wave solution of the SG equation. In contrast to the existing algorithms the proposed algorithm does not require additional dissipative term for wave generation.

New governing equation is obtained for nonlinear modeling of dynamics of hydrogen concentration in alloys. An influence of the nonlinear terms on the dynamics of hydrogen concentration is studied. A particular case is found when the governing equation admits exact kink-shaped traveling wave solution. Numerical simualtion reveals different influence of the nonlinear terms arising in the equation due to different physical reasons. Qualitatively different effects such as arising of the tail behind the localized wave and formation of the counterpart wave are found. An influence of the polarity of the input on behavior of the localized wave of concentration is described.

A two-dimensional graphene lattice model with translational and angular interactions between the elements of two sublattices is considered. The nonlinearity is introduced via nonlinearly elastic translational forces between the masses in the lattice. The nonlinear strain behaviour is studied using the continuum limit of the lattice that results in the derivation of four coupled governing equations describing the dynamics of interacting sublattices. An asymptotic procedure is developed to account for weakly transversely perturbed nonlinear longitudinal plane waves giving rise to a two-dimensional nonlinear governing equation for longitudinal strains.

Evolution of an initial localized bell-shaped state for the sine-Gordon equation is considered. It is obtained numerically that variation in the parameters of the localized input gives rise to different propagating waves as time goes. The speed gradient feedback control method is employed to achieve unified wave profile weakly dependent on initial conditions. Two speed-gradient like algorithms are developed and compared. It is shown that the algorithm using coefficient at the second spatial derivative term in the sine-Gordon equation allows one to generate the same wave with prescribed energy from different initial states having different energies.

An asymptotic procedure is developed to obtain governing two-dimensional nonlinear equations as a result of the continuum limits of the original discrete hexagonal lattice model. Possible continualization is analyzed on the basis of linearized discrete equations. New weakly nonlinear continuum equations for plane and weakly transverse disturbed shear waves in a hexagonal lattice are obtained using different continuum limits. It is shown that nonlinear plane shear waves are described different from the model of isotropic continuum.

A two-dimensional discrete model for a hexagonal (closed-packed) lattice with elastically interacting round particles possessing two translational and one rotational degrees of freedom is considered. The linear differential-difference equations are obtained by the method of structural modeling to describe propagation of longitudinal, transverse and rotational waves in the medium. The dispersion properties of the model are analyzed. Existence of a backward wave is revealed. The numerical estimations of threshold frequencies of acoustic and rotational waves are given for some values of microstructure parameters.

The algorithm is developed to model two-dimensional dynamic processes in a nonlocal square lattice on the basis of the shift operators. The governing discrete equations are obtained for local and nonlocal models. Their dispersion analysis reveals important differences in the dispersion curve and in the sign of the group velocity caused by nonlocality. The continuum limit allows to examine possible auxetic behavior of the material described by the nonlocal discrete model.

Two kinds of nonlinearity are studied in the framework of a model of a discrete diatomic model of an acoustic metamaterial. It is shown that the continuum limit of a discrete model is similar to the equations obtained using a model of a reduced continuum with a microstructure. An asymptotic approach is developed to obtain a modulation nonlinear governing equation for dynamical processes in an acoustic metamaterial. New metamaterial features caused by nonlinearity are found.

Three different continuum limits for modeling non-linear plane waves in two-dimensional hexagonal lattice are obtained. New coupled non-linear continuum equations are obtained to study the interaction of a macro-strain wave and the waves caused by variations in an internal structure. New analytical solutions are obtained to describe localized non-linear strain waves. It is shown that the solutions are different from those of the 1D lattice model due to the inclusion of non-neighboring interactions in a lattice.

A previously developed strongly nonlinear continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. It suggested suitable expression for the forces in the discrete model; however, its continuum limit not only explains the nonlinear terms continuum model but also gives rise to some additional terms. It turns out that one of them supports bell-shaped localized variations in the diatomic material but suppresses kink-shaped variations.

The influence of an external loading on nonlinear dynamic processes in a layer having a di-atomic crystalline structure is studied. An asymptotic procedure is developed to transform the solution of the initial two-dimensional problem to the solution to the coupled one-dimensional nonlinear equations for longitudinal macrostrains and variations in the microstructure. It is shown that an external loading on the lateral surface of the layer can play the role of a feedback control supporting the stable propagation of localized strain waves. The weakly nonlinear case is considered to obtain a new single governing equation for longitudinal strains.

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch.

It is shown, how even particular traveling wave asymptotic solution may describe the defects on the shock wave profile caused by the dispersion features of the numerical scheme of the coupled nonlinear gas dynamics equations. For this purpose the coupled nonlinear partial differential equations or the so-called differential approximation of the scheme, are obtained, and a simplification of the method of differential approximation is suggested to obtain the desired asymptotic solution. The solution is used to study the roles of artificial viscosity and the refinement of the mesh for the suppression of the dispersion of the scheme.

A general form of a distributed feedback control algorithm based on the speed–gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions. This article is part of the themed issue ‘Horizons of cybernetical physics’.

Nonlinear strain wave propagation along the lamina of a periodic two-component composite was studied. A nonlinear model was developed to describe the strain dynamics. The model asymptotically satisfies the boundary conditions between the lamina, in contrast to previously developed models. Our model reduces an initial two-dimensional problem into a single one-dimensional nonlinear governing equation for longitudinal strains in the form of the Boussinesq equation. The width of the lamina may control the propagation of either compression or tensile localized strain waves, independent of the elastic constants of the materials of the composite.

The interaction of the components of composite solitary waves governed by nonlinear coupled equations is studied numerically. It is shown how predictions of the known exact traveling wave solutions may help in understanding and explaining the process of reshaping seen as head-on and take-over collisions of individual solitary waves. The most interesting results concern the switch in the sign or the periodic modulation of the amplitude of the solitary wave and the direction of its propagation due to collisions.

Generation of bell-shaped localized wave solutions to nonlinear coupled equations is studied numerically. It is shown how predictions of the known exact traveling wave solutions may help in understanding and explanation of arising of localized wave solutions of permanent shape and velocity in numerical solution. It is found that just coupling gives rise to changing of the sign of the amplitude of the localized bell-shaped wave.

A non-linear dynamic model is developed to account for material inhomogeneities in a growth plate in long bones. The governing equations are obtained to account for non-linear dispersive, viscoelastic and inhomogeneous features of the growth plate. The evolution of non-linear strain waves over the material inhomogeneities is obtained via the asymptotic solutions. It is shown that variations in the amplitude and the width of both the bell-shaped and kink-shaped waves reflect the position and the size of the inhomogeneity. This may be used for a detection of the growing plate features and in the development of the reaction–diffusion equation for the stimulus of the growth of long bones.

It is shown that for some seismic media both quadratic and cubic non-linearities should be taken into account in the governing equation for longitudinal waves. The new equation is obtained to account for non-linear surface waves in a medium surrounding a non-linearly elastic rod. Exact solutions of the equation allow us to describe simultaneous propagation of tensile and compressive localized strain waves. Various interactions between these waves give rise to both the multi-bump and “Mexican hat” localized wave structures closer to the surface waves recently observed in experiments.

It is found that simultaneous existence of compressive and tensile localized strain solitary waves in a rod can be described by the model equation containing both quadratic and cubic nonlinear terms. Only propagation of these waves is described by exact travelling wave solutions. However, numerical solution demonstrates that both kinds of these waves may be generated from an arbitrary input and interact each other keeping their shape and velocity. Moreover, one and the same input gives rise to a different number of these kinds of waves, and it is quadratic nonlinearity that determines it.

The control of the smoothness of an antikink profile of the Sine Gordon equation is considered. It is shown that variations in the initial conditions give rise to oscillations on the profile of the antikink. The control algorithm based on variation of one of the coefficient of the equation is developed to recover the smooth wave shape and the phase velocity of the antikink.

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).