It is well known that a modulationally unstable short-wavelength delocalized nonlinear vibrational mode (DNVM) can create chaotic discrete breathers (DBs) in a nonlinear lattice. A necessary condition for this is that the DNVM must have a frequency outside the phonon spectrum of the lattice. This phenomenon has been repeatedly analyzed for one- and two-dimensional lattices, and here it is studied for a bcc lattice with β-Fermi–Pasta–Ulam–Tsingou potential. Using the group-theoretical approach developed by Chechin and Sakhnenko, four DNVMs are found with a wave vector at the boundary of the first Brillouin zone and frequencies above the phonon spectrum. It is shown that the development of the modulational instability of all four DNVMs with amplitudes above a certain value leads to the formation of chaotic DBs, which is justified by calculating the energy localization parameter and the maximum particle energy. Chaotic DBs in the three-dimensional bcc lattice radiate their energy faster than in previously studied two-dimensional lattices. The results obtained describe one of the possible mechanisms of energy dissipation by a crystal lattice in a far-from-equilibrium system.

The paper presents the results of a study on the search for correct methods for measuring a high-value current pulse, which will be used to conduct research on the electroplastic effect. The electroplastic effect is the effect of electric current pulses on the plastic flow of metals. Electroplastic metal forming technology is a relatively new metal forming process that is energy efficient, environmentally friendly and versatile. In particular, it can be used to process metals or alloys that are difficult to process using conventional manufacturing processes. For the experimental study of the electroplastic effect, it became necessary to measure pulse currents of large magnitude, not only in amplitude, but also in the shape of the pulse. The pulsed current causes the formation of an alternating electromagnetic field near the conductors, so it can be measured with a Rogovsky current transformer. The results of the work present a schematic electrical diagram and a photograph with the appearance of an experimental installation for the study of the electroplastic effect. The results of measurements of the current value, the voltage drop on the sample and the dependence of the peak voltage values on the sample on the peak current value are shown. After making calculations and renormalising the data for the voltage drop on the sample according to the peak value of the current obtained on the transformer, the authors obtained the desired current values. The error of this method is estimated by calculating the total capacitance of capacitors, which does not exceed 2%.

The absorption of energy by a β-Fermi-Pasta–Ulam-Tsingu square lattice from harmonically driven rows of particles is studied numerically in a wide range of driving frequencies Ω and for relatively small driving amplitudes A. Two mechanisms of energy transfer in the lattice are described depending on the driving parameters Ω and A. When Ω is deep inside the phonon band, an extended wave is emitted from the driven row of particles. Interestingly, at the front, the emitted wave splits into wavepackets having fundamental frequency slightly larger than the driving frequency. When Ω is close to the upper edge of the phonon band ωmax but still within the band, the frequency of the wavepackets emitted by the driven row of particles is above the phonon band. Such wavepackets are actually discrete breathers (DBs). Thus, it is shown that DBs can contribute to energy transfer in the lattice not only when the driving frequency is outside the phonon band, as in classical supratransmission, but also when it is inside the phonon band. In classical supratransmission, the excitation of DBs can only be observed when the driving amplitude is above a threshold. This is also true for in-band driving, and the threshold driving amplitude decreases as the driving frequency approaches the phonon band edge.

This review examines recent research on chromium and chromium-based coatings for superalloys, focusing on their application methods and protective mechanisms in extreme environments. The study explores diffusion coatings, particularly the pack cementation method, and overlay coatings applied through thermal spraying techniques. The formation of protective Cr2O3 oxide scales and the role of chromium in enhancing oxidation and corrosion resistance are discussed. The review also addresses the challenges and limitations of chromium coatings, including the need for precise control of chromium content. Finally, it highlights the importance of selecting appropriate coating methods based on specific application requirements and environmental conditions.

В настоящее время множество канализационных очистных сооружений как по всей России, так и на территории Республики Саха (Якутия) находятся в процессе модернизации или даже реконструкции.Это происходит не только по причинам морального износа устаревшего оборудования и необходимости увеличения производительности в связи с внутренней миграцией населения, но и в какой-то степени из-за изменения культуры быта и поведения человека -появились загрязнения в виде крупных частиц -остатков средств личной гигиены.Исходя из вышеизложенного, модернизация технологического процесса биологической очистки сточных вод на примере канализационных очистных сооружений г.Мирный Республики Саха (Якутия) является актуальной задачей.Поскольку сточные воды, прошедшие очистку на канализационных очистных сооружениях г.Мирный, сбрасываются в р.Ирелях, они должны соответствовать показателям характеристик нормативно допустимого сброса в водные объекты очищенных сточных вод, которые согласно недавно внесенным изменениям (СП 32

The paper considers such nonlinear phenomena in condensed matter physics as Discrete Breezers (DB) and delocalised nonlinear vibrational modes (DNVM). DB are spatially localised vibrational modes of large amplitude that exist under conditions of nonlinearity of interatomic interactions and discreteness of the medium. The oscillation frequency of DB lies outside the phonon spectrum of low-amplitude crystal vibrations and does not resonate with phonons, i.e. it does not waste its energy on their excitation. DNVM are vibrational modes manifested in crystal lattices with translational symmetry, which exist for any oscillation amplitudes and regardless of the type of interaction between the elements of the system. In early works, the authors established a connection between DB and DNVM. A three-dimensional Body Centered Cubic (BCC) lattice with nearest and next-nearest interactions described by the β-Fermi-Pasta-Ulam-Tsingou (FPUT) interatomic potential is investigated. Properties of DNVM with the wave-vector on the boundary of the first Brillouin zone are analysed. DNVM are exact solutions to the equations of motion that can be found from the analysis of only the symmetry of the bcc lattice. Frequency response of DNVM for the case of soft- and hard-type anharmonicity is calculated. In the case of hard-type anharmonicity, four DNVM have frequencies bifurcating from the upper edge of the phonon spectrum and growing with the amplitude. By superimposing localisation functions on these DNVM, various DB were obtained, which were attributed to quasi-breezers. They are not single-frequency oscillatory modes with a finite lifetime and are formed due to overcoming the strength of the intersite potential. As a result of the study, six long-lived quasi-freezers were obtained based on four DNVM frequencies above the phonon band. The results of this study confirm the effectiveness of the search for long-lived quasi-freezers in complex lattices, starting with the analysis of DNVM. In the future, the obtained quasi-breeze solutions can be used as initial conditions for an iterative procedure for searching for exact DB. Thus, the presented work demonstrates a practical approach to the search for DB in high-dimensional lattices.

Body centered cubic (bcc) lattice with nearest and next-nearest interactions described by the β-FPUT interatomic potential is considered. Exact dynamical solutions in the form of zone-boundary delocalized nonlinear vibrational modes (DNVMs) are analyzed. 31 such solutions are revealed from the analysis of only the symmetry of the bcc lattice. Frequency response of DNVMs for the case of soft- and hard-type anharmonicity is calculated. In the case of hard-type anharmonicity, four DNVMs have frequencies bifurcating from the upper edge of the phonon spectrum and growing with the amplitude. Various quasi-discrete breathers are obtained by superimposing localizing functions upon these DNVMs. Our work demonstrates a practical approach to finding quasi-discrete breathers in higher-dimensional lattices. The obtained spatially localized long-lived vibrational modes inspire the search for various discrete breathers in bcc metals.

Numerical simulations are performed to investigate the formation of chaotic discrete breathers (DBs) in a bcc lattice using four zone-boundary modes with frequencies exceeding the crystal’s phonon spectrum. The study analyzes the impact of the stiffness of elastic bonds between first and second neighbors and identifies a specific range where the formation of chaotic DBs due to modulational instability of four vibrational modes is possible. The time evolution of the energy localization parameter and the maximum energy of all particles were monitored to control the formation of chaotic DBs. Their spontaneous nucleation was observed in a wide range of stiffness of elastic bonds and depends on the mode’s symmetry. Considering DBs in bcc metals, the paper focuses on the scenario where first-neighbor bonds are stiffer than second-neighbor bonds, as bond stiffness typically decreases with interatomic distance. In all four zone-boundary modes, formation of chaotic DB is observed under this condition.

The problem of finding various discrete breathers (DBs) in the β-Fermi-Pasta-Ulam-Tsingou simple cubic lattice is addressed. DBs are obtained by imposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) having frequencies above the phonon spectrum of the lattice. Among 27 DNVMs with the wave vector at the boundary of the first Brillouin zone there are three satisfying this condition. Seven robust DBs of different symmetries are found using this approach.

The molecular dynamics method was used to study the structure formation during austenite nanoparticles crystallization in the presence of carbon impurities. The paper describes the dependence of the melt cooling rate, particle size, concentration of carbon atoms in the particle on the resulting structure features during crystallization and temperature of the crystallization onset. Formation of the nanocrystalline structure of nanoparticles can be controlled by varying the cooling rate and introducing a carbon impurity: at a cooling rate above 1013 K/s in the model used, crystallization did not have time to occur; at a rate below 5·1012 K/s, the austenite particle crystallized to form a nanocrystalline structure. At the same time, with a decrease in the cooling rate, a decrease in the density of defects in the final structure was observed. At a rate of 5·1011 K/s or less, crystallization of carbon-free particles took place with the formation of low-energy grain boundaries (with a high density of conjugate nodes: special boundaries, twins). The crystallization temperature during cooling at a rate below 1012 K/s is inversely proportional to the particle diameter: as the particle size decreases, the proportion of free surface increases, which leads to a decrease in the probability of crystalline nuclei formation. In addition, the crystallization temperature increases with a decrease in the cooling rate. The introduction of a carbon impurity led to a decrease in the crystallization temperature of nanoparticles: in the presence of 10 at. %. As a percentage of carbon, it decreased by about 200 K for particles of different sizes. Carbon atoms often formed clusters consisting of several carbon atoms. Such clusters distorted the resulting crystal lattice of metal around them, preventing crystallization. In the presence of a carbon impurity, the final structure of the crystallized particles contained a higher density of grain boundaries and other defects. Carbon atoms, especially clusters of them, were fixed mainly at grain boundaries and triple joints.

This study examines the mechanism of nonlinear supratransmission (NST), which involves the transfer of disturbance to discrete media at frequencies not supported by the structure. A copper-graphene composite with 3 graphene layers inside copper matrix was considered. The investigation was carried out using atomistic modeling through molecular dynamics. Energy propagation in the crystal was carried out by harmonic influence according to the sinusoidal law. Obtained results shows that this composite has no forbidden zones, which indicates that the classical effect of nonlinear supratransmission cannot be manifested. Depending on the area of harmonic influence, different material behaviour can be observed, from smooth energy transfer deep into the crystal to its destruction.

The influence of the interface orientation on the intensity of dissolution of titanium in crystalline and amorphous aluminum is studied by molecular dynamics simulation. The following four orientations of the Ti–Al interface with respect to the Ti (hcp) and Al (fcc) lattices are considered: (1) (0001):(111), (2) (0001):(001), (3) ( 101̅0 ):(111), and (4) ( 101̅1 ):(001). The interface orientation is found to influence the intensity of dissolution of titanium in aluminum, which increases for the accepted designations in the order 1–2–3–4. An important phenomenon in this case turns out to be the formation of a thin (2–3 atomic planes thick) crystalline layer in aluminum, which repeats the crystal lattice of titanium, at the initial stage of dissolution. At a temperature below the melting point of aluminum, a grain boundary parallel to the interface forms behind this layer. At temperatures above the melting point of aluminum, this crystalline layer is preserved, but its thickness decreases gradually as the temperature increases. For aluminum in an amorphous state at temperatures below its melting point, the dissolution of titanium occurs at almost the same intensity as in the crystalline state of aluminum, which is explained by the formation of a similar crystalline layer in aluminum at the interface in all cases.

In recent decades, much interest has been shown in nonlinear lattice vibrations because crystalline materials are subjected to high-amplitude impacts in many fields of human activity. One of the effects of nonlinearity in discrete periodic structures is the possibility of existence of spatially localized high-amplitude vibrations, referred to as discrete breathers (DBs), or intrinsic localized modes. The problem of searching for DBs in nonlinear chains (i.e., one-dimensional crystals) can be solved in a fairly simple way, because the variety of possible DBs is small in this case. However, no general approaches to the search for DBs have been developed for high-dimension crystal lattices. Such an approach was derived based on the works by Chechin, Sakhnenko et al., who developed the theory of bushes of nonlinear normal modes, which (as applied to crystals) were later referred to as delocalized nonlinear vibrational modes (DNVMs). It has recently been noted that all known DBs can be obtained by superimposing localizing functions on DNVMs with a frequency beyond the phonon spectrum of the lattice. Since the Chechin and Sakhnenko theory makes it possible to find all possible DNVMs by considering the lattice symmetry, it has become possible to formulate the problem of determining all possible DBs in a given lattice. This approach has recently been applied with success to the search for DBs in a two-dimensional triangular lattice. The purpose of this study is to analyze and describe DBs in a two-dimensional square lattice obtained using a localizing function. As a result, new types of DBs of a square lattice are obtained, including one-dimensional DBs (i.e., those localized only in one of two orthogonal directions) and zero-dimensional DBs (i.e., those localized in two directions).

Нелинейные локализованные колебательные моды большой амплитуды с частотой колебаний за пределами фононного спектра называют дискретными бризерами (ДБ). Различают щелевые и т.н. ДБ с жёстким типом возбуждения. Первые возбуждаются в двухатомных кристаллах, например, в упорядоченных сплавах, атомы компонент которых значительно различаются по массе. Такие кристаллы имеют щель в плотности фононных состояний, внутри которой может находиться частота щелевого ДБ. Бризеры второго типа, как правило, имеют место в моноатомных кристаллах. Известно также, что они были получены и в некоторых двухатомных кристаллах. В данной работе изучается возможность возбуждения ДБ в кристалле Cr2Al. Хром-алюминиевые сплавы имеют большое практическое применение в электронагревательных устройствах большой мощности и промышленных электрических печах. Условия работы этих устройств создают предпосылки для возбуждения ДБ в кристаллической решётке сплавов, из которых изготавливают нагревательные элементы. Интерес к изучению возможности существования ДБ в указанном сплаве, связан с тем, что бризеры оказывают влияние на физические свойства материалов, могут снижать теплопроводность за счёт рассеяния фононов, а также способствуют возникновению и миграции дефектов. В работе была построена 3Dмодель ОЦК кристаллической решетки Cr2Al со сверхструктурой C11b. Рассчитаны коэффициенты потенциала Морзе, посредством которого осуществлялось взаимодействие между атомами.Рассмотрена возможность возбуждения ДБ в указанном модельном кристалле. Для поиска ДБ использовали методики, разработанные в работах Чечина Г.М., Сахненко В.П. с соавторами, а также Дмитриева С.В. с коллегами. Рассматривается также причина неустойчивости ДБ жёсткого типа возбуждения и их быстрого затухания, а также один из механизмов рассеяния энергии.

The paper considers the peculiarities of linear system simulation in the MatLab software package using the structural blocks of the Simulink application. Typical links of linear systems and their transfer functions are described. The simulation process is exemplified by the system of the object control by its rotation angle. A structural diagram is built for such a system, the mathematical model of which is made in the form of an aperiodic link of the first order. The parameters of the controllers forming the control signal are calculated. A detailed description of the model construction is provided in the Simulink application of the MatLab software package. As a result of simulation, the authors obtained the graphs during the study of the object control system: 1) under external impact without changing controller parameters; 2) under external impact when using a PID controller. The final model was built; it allows comparing the curves of the object rotation control signal in the system with the PD controller and the PID controller as well as displaying the transient curves of the object rotation angle.

The influence of discrete breathers (DBs) on the macroscopic properties of the Fermi–Pasta–Ulam chain with symmetric and asymmetric single-well potentials is investigated. The ratio of the total energy to the kinetic energy (which determines the specific heat) was monitored during the development of modulation instability of the short-wavelength vibrational mode with the wavenumber at the boundary of the Brillouin zone. Instability leads to the formation of chaotic DBs with subsequent transition to thermal equilibrium when DBs disappear due to the emission of energy in the form of low-amplitude phonons. The localization parameter, the number of DBs in the chain, and the average energy per one DB are given as functions of time. The number of DBs is close to the maximum at the point in time when the energy localization parameter reaches its maximum. It is found that DBs reduce the heat capacity for all considered parameters of the chain potential. This is due to the fact that the chain under consideration has a hard type of anharmonicity, at which the DB frequency increases with an increase in the amplitude. In the energy localization regime, the DB oscillation frequencies increase, which leads to an increase in the particle velocities and, accordingly, in their kinetic energy. An increase in the kinetic energy in the presence of DBs in the chain leads to a decrease in the ratio of total energy to kinetic energy, that is, to a decrease in the specific heat capacity. In chains with soft anharmonicity, the DB frequency decreases with an increase in the amplitude, and the opposite effect is obtained, i.e., DBs increase the heat capacity in this case. The obtained results can be useful for setting up experiments on the identification of discrete breathers in crystals by measuring their macroscopic properties.

All possible one-component delocalized nonlinear vibrational modes (DNVMs) in a square lattice are analyzed. DNVMs are obtained taking into account exclusively the symmetry of a square lattice, and therefore, they exist regardless of the type of interactions between particles. In this work, the interactions of the nearest and next nearest neighbors are described by the β -FPUT potential. For each DNVM, frequency, kinetic and potential energies are described as functions of amplitude. The mechanical stresses caused by DNVMs and the effect of DNVMs on the stiffness constants of the lattice are presented. DNVMs with higher vibration frequencies have a stronger effect on the mechanical properties of the lattice. Examples of analytical analysis of DNVMs are given. It is found that two of the sixteen one-component DNVMs can have frequencies above the phonon band in the entire range of their amplitudes. It is shown that these DNVMs can be used to construct discrete breathers by applying localizing functions. The modulational instability of these two DNVMs leads to the formation of chaotic DBs. The presented results contribute to a better understanding of the nonlinear dynamics of a square lattice by analyzing the properties of a class of delocalized exact solutions and demonstrating their connection with discrete breathers.

Standing and moving discrete breathers (or equally, intrinsic localized modes) in a square β-Fermi-Pasta-Ulam-Tsingou lattice are obtained by applying localizing functions to the delocalized nonlinear vibrational modes (DNVMs) found earlier by Ryabov and Chechin. The initial conditions used in our study do not correspond to exact spatially localized solutions, but make it possible to obtain long-lived quasibreathers. The approach employed in this work can easily be used to search for quasibreathers in three-dimensional crystal lattices, for which DNVMs with frequencies outside the phonon spectrum are known.