Mechanics and Control of Solids and Structures

In this paper, we consider the mathematical methods which can be applied for the solution of two mechanical problems which have varying in space and time parameters, and some of them are described by the hyperbolic equations. The first problem is an interaction between the deformation waves and the diffusion process in a rigid solid. The main effect accompanying the transport of impurity in the material is a reduction of its initial rigidity parameters. Unlike the existing works, in our paper, an analytical approach is applied to the analysis. Considering the interaction of impurity waves and deformation waves, we obtained a number of solutions which can be used to develop acoustic diagnostics of defects in a material. For the one-dimensional case, analysis is given by taking into account the influence of a changing stress state of the environment. The second problem is a mathematical modeling of the thermomechanical processes in soils. We consider a plane thermal state of the soil surrounding the gas-main pipeline.

The problem of the generalized Kapitsa pendulum on the stability of the vertical position of the rod under the vertical vibration of the support was studied in various settings. A vertical deformable rod with a free upper end and clamped or simply supported lower end under the action of harmonic or stationary random vibrations of the support is considered. We model the rod as a system with several degree of freedom. The conditions for stability of the upper vertical position of the pendulum are found. Both bending and longitudinal vibrations of the bar are taken into account. We found the attraction basin of the stable vertical position.

Experiments show that the coefficient of shear viscosity calculated by formulas of the classical theory of viscous fluid loses the sense of material constant when the thickness of the liquid flow becomes small enough. It becomes an effective quantity that changes significantly with decreasing flow thickness and takes a finite limit value on the wall. The limit (named the boundary viscosity) can be considered as the empirical material characteristic of given liquid–solid surface pair. This and many other facts indicate that the classical theory fails near a solid surface, that most real liquids (including usual water) form near solid surfaces a specific thin (\(\sim \)some \(\upmu \)m and less) “subboundary” layer where new physical mechanisms of liquid flow become important. Modern experiments also show that the next specific ultrathin (\(\sim \)0.1 \(\upmu \)m and less) layer is formed under subboundary layer. It consists of very orderly packed molecules of the liquid and flow does not take place in this “solid-like” layer. These fundamental effects have to be taken into account in the modern theory of fluid. It is especially important for the analysis of such problems as filtration, lubrication, flow of suspensions, polymer solutions, polar, and other real liquids. We show here that this aim can be achieved by combination of the Eringen-Aero theory of micropolar fluid and new generalized boundary conditions that take explicitly into account the existence of boundary viscosity and solid-like layer.

This paper presents an overview of recent studies on the development of the theory of vibration processes and devices and on their industrial implementation. The work under consideration was completed by employees of the Laboratory of Vibrational Mechanics or with their participation. The fundamental principles of this work were established earlier due to the discovery and framing of the theory of self-synchronization of rotating bodies and the theory of vibrational displacement and to the development of new analytical approaches to studying the effect of vibration on nonlinear systems and media (vibrational mechanics and vibrational rheology). The work covered in this article provided a significant generalization and further development of these approaches. Their application allowed solving a large number of urgent applied problems in various fields of knowledge. As a significant recent achievement, a number of nonlinear vibration effects with a great potential for practical application have been discovered and studied. Moreover, new highly efficient vibration machines for the processing of natural and technogenic materials (crushers, screens, separators) and related laboratory and test vibration equipment are being designed, with the development of respective calculation methods.

A brief review of the research activity provided during the last three decades in the Laboratory of Mechanics of Nanomaterials and Theory of Defects at the Institute for Problems in Mechanical Engineering of Russian Academy of Sciences in the field of micromechanics of strength and plasticity in nanostructured materials is presented. It covers the works aimed at explanation and theoretical description of the following features in mechanical behavior of these materials: deviations from the classical Hall-Petch law, homo- and heterogeneous nucleation of dislocations, grain boundary sliding and mechanisms of its accommodation, rotational deformation, deformation twinning, deformation-induced grain growth and refinement, and interaction between deformation and fracture processes. Some most important and interesting results are discussed and compared with available data of experimental studies and computer simulations.

Solutions of dynamic equations of plane deformation for a nonlinear model of complex crystal lattice are obtained. Crystalline media of cubic symmetry are considered. Macrofield vector \(\mathbf {U} \) (acoustic mode) and microfield vector \(\mathbf {u} \) (optical mode) describe the deformation of the medium in the nonlinear model. In the case of plane deformation, the vectors \(\mathbf {U}\) and \(\mathbf {u}\) can be found from a system of four related nonlinear equations. A complex representation of the general solution of macrofield equations is given. Macrostress tensor \(\sigma _{ij} \) and vector \(\mathbf {U} \) are expressed through a function Q(t, x, y) , which is a dynamic analogue of the Airy function. A general solution of the microfield equations was found.

This article presents some results of our work on improving the original version of the artificial bases method in the following areas. Determination of linear wear resistance based on the formation of a hole with a simple and precise tool is given. Justification of simplified algorithms for evaluating the volumetric wear resistance in testing materials is done. The “block-on-ring” method for evaluating the volumetric wear resistance taking into account the level of hardness of the material is applied.

The problem of active vibration suppression of the distributed elastic system is considered in the example of a slender metal beam undergoing bending vibrations. Control systems include piezoelectric sensors and actuators. Three different strategies for vibration suppression are considered: local, modal and shape control strategy. The local approach means that each feedback loop includes only one sensor–actuator pair placed at specific location on the beam, while the modal strategy implies that each feedback loop corresponds to a specific vibration mode of the object. The shape control method is based on the compensation of known distribution of the external excitation using only one feedback loop with all available sensors and actuators. First, experimental results are obtained for the local and the modal control systems using the same two sensor–actuator pairs, and then the transfer functions in feedback loops for these systems are improved as the result of numerical modeling. After that, the modal method is compared numerically with the shape control strategy. The results show that the modal method is the most effective if it is needed to suppress several vibration modes of the object.

The paper is devoted to development of the speed-gradient method in the IPME RAS in 1986–2021 and its applications to the problems of mechanical engineering.

The influence of a stress concentrator on a chemical reaction front propagation in a solid is investigated for the reaction between diffusing and deformable solid constituents. A kinetic equation is used in a form of the dependence of the reaction front velocity on the normal component of the chemical affinity tensor which in turn depends on the mechanical stresses. A plane problem for a linear-elastic body with a cylindrical hole as a concentrator is studied using numerical FEM-simulations. An analytical solution of the axially symmetric problem is used for the verifications of the numerical procedure. Then reaction front propagation in the vicinity of the hole is studied for all-round and uniaxial external loadings.

The chapter describes the diffusion process in a composite material with non-uniformly distributed isolated spheroidal pores. The pores are assumed to present in the material initially or to form during the mass transport. The influence of the segregation effect, shape of pores, and its orientation scatter on the distribution of impurity is discussed. Various constitutive equations for the diffusion flux are compared. Consideration of Fick’s first law allows to account for effective diffusion properties of material with microstructure. Consideration of the constitutive equation introducing chemical potential allows to account for effective diffusion and effective elastic properties of a porous material.

A novel method of instability and stability of equilibrium points of autonomous dynamical systems using a flow and divergence of the vector field is proposed. A relation between the method of Lyapunov functions, Gauss (Ostrogradsky) and Chetaev theorems with the divergence conditions is established. The generalizations of Bendixon and Bendixon–Dulac theorems about a lack of periodic solutions in arbitrary order systems are considered. The state feedback control law design is proposed based on new divergence conditions. Examples illustrate the efficiency of the proposed method and the comparison with some existing ones.

Recently, there has been a tendency to use parallel structures in the design of intelligent robots. In particular, such structures are used in the smart electromechanical systems (SEMS) proposed by the IEMS laboratory. This is one of the variants of cyber physical systems (CPhS). Cyber physical systems the ability to integrate computing, communication, and storage of information, monitoring, and control of the physical world objects. The main tasks in the field of theory and practice CPhS are to ensure the efficiency, reliability, and safety of functioning in real time. It is important to keep in mind that the behavior of the system is based on making decisions based on information received from the sensors of the Central nervous system (CNS) about the environment and its own state. The task of making a decision about the behavior of SEMS in a group interaction of several SEMS is much more complicated, since in this case additional information about the planned behavior of other members of the group is necessary.

A brief review of solutions of boundary-value problems in the theory of elasticity for defects in nanoscale and nanocomposite solids, which were found during the last three decades in the Laboratory of Mechanics of Nanomaterials and Theory of Defects at the Institute for Problems in Mechanical Engineering of Russian Academy of Sciences, is presented. It covers the elastic behavior of dislocations, disclinations, and inclusions near free surfaces and interfaces in such inhomogeneous nanostructures as composite nanolayers, core-shell nanowires and nanoparticles, quantum dots and wires in subsurface layers, etc. Some relevant works dealing with application of the solutions found to the theoretical models describing the nucleation and development of different defect structures in the process of crystal growth and misfit stress relaxation in advanced composite nanostructures which are highly promising for use in modern electronics, optoelectronics and photonics, are also briefly reviewed.

A diffusion semi-Markov process on a bounded interval range of values with unattainable boundaries is considered. It is supposed that the process does not have any infinite-value interval of stop. So unattainability of the boundaries is provided only because of changing the coefficient of shift. A limit theorem of the alternating renewal process is used to derive a limit distribution of this diffusion semi-Markov process.

Dynamic fracture of a one-dimensional chain of identical linear oscillators (masses connected by springs) is regarded in the work. The considered system consists of arbitrary but finite number of links and the first mass is supposed to be fixed. Two types of load are discussed: free oscillations of the initially uniformly stretched chain and loading the chain with a short deformation pulse. Both problems are solved analytically for an arbitrary number of links. The obtained solutions are investigated, and a dynamic fracture effect related to the discreetness of the system is discussed: a deformation wave travelling through the chain is distorted and some links may be subjected to critical deformation. The obtained solutions for the chain are compared to the solutions of analogous problems stated for an elastic rod—a continuum counterpart of the considered discrete system. It is shown that the discussed fracture effect is not observed in a continuous system.

In the frame of the concept of vibrational mechanics, a rotating multi-mass mechanism with high-frequency periodic or stochastic excitation is considered, and the general equation for the averaged motion of such mechanisms is obtained (the equation of vibrational mechanics). The equation of vibrational mechanics has the same structure as the original Lagrange equation without excitation, but has a modified inertial coefficient and dissipative function, depending on the intensity of the excitation. As a result of this difference, the equilibrium position and eigenfrequency depend on the intensity of high-frequency excitation. As an example of modification of the global behaviour of a rotation mechanism, a centrifugal analogue of the Stephenson–Kapitsa pendulum is considered. The theory is applied also to the analysis of the centrifugal pendulum absorber with a complex kinematics. The sensitivity of the pendulum order deviation from the nominal order due to high-frequency excitation is studied. It is shown that the own rotation angle of the pendulum as a function of its position has a significant impact on this sensitivity.

The review provides a short list of the latest scientific achievements of the Laboratory of Structural and Phase Transformations in Condensed Matter, Institute for Problems in Mechanical Engineering (IPME RAS). We present the results on the developed growth method of thin silicon carbide films on silicon by coordinated atomic substitution, the properties of the obtained SiC/Si layers, and the prospects for their use as substrates for the growth of thin films of various materials. Special attention is paid to the experimental results on the deposition on SiC/Si substrates of various promising wide-bandgap semiconductors, such as aluminum nitride AlN, gallium nitride GaN, and their solid solutions. Data on the growth of the A^IIB^VI semiconductor compounds are presented. The possibility of using SiC/Si substrates for deposition of nanocrystals is discussed. A number of theoretical results devoted to the description of the growth mechanisms of these semiconductors are outlined.

An eigenvalue problem for a pair of harmonic functions is considered; it contains a spectral parameter in the Steklov condition on a part of the boundary. This problem concerns eigenvalues of fluid’s free oscillations in a vertical, cylindrical container; its cross-section is arbitrary, whereas a porous medium occupies a bottom-adjacent layer. Assuming the fluid to be inviscid, incompressible and heavy, properties of its eigensolutions are investigated.

In this chapter, the issues of global stability, bifurcations, and emergence of nontrivial limiting dynamic regimes in systems described by differential equations with discontinuous right-hand sides are considered within the framework of the theory of hidden oscillations. Such systems are important in the problems of mechanics, engineering, and control, and arise both a priori and as a result of idealization of some characteristics included in real physical systems. Determining the boundaries of global stability, scenarios of its violation, as well as identifying all arising limiting oscillations are the key challenges in the design of real systems based on mathematical modeling. While the self-excitation of oscillations can be effectively investigated numerically, the identification of hidden oscillations requires special analytical and numerical methods. The analysis of hidden oscillations is necessary to determine the exact boundaries of global stability, to estimate the gap between the necessary and sufficient conditions of global stability, and their convergence. This work presents a number of theoretical results and engineering problems in which hidden oscillations (their absence or presence and location) play an important role.

Two kinds of aluminum alloy, 1561 and 1565 alloys, were tested within impact velocity range of 241.9–744.8 m/s in two schemes of shock loading: (i) under uniaxial strain conditions and (ii) in high-velocity penetration. Combination of load regimes allows a formation of multiscale structure to be retraced. Formation of mesoscale-1 in form of micro-shears of 3–10 mkm is found to be identical for both kinds of alloy. As for the mesoscale-2 (50–150 mkm), the formation of dynamic structures for two kinds of alloy is of different nature. In 1565 alloy the transition from mesoscale-1 to mesoscale-2 occurs in form of structural instability whereas in 1561 alloy this transition happens gradually. In 1561 alloy the mesoscale-2 structural elements are elongated plaints and ellipsoids whereas in 1565 alloy the mesoscale-2 structures are the fault cells at the boundary of penetration cavern. Affect of transition from mesoscale-1 to mesoscale-2 in both aluminum alloys turns out to opposite: in 1561 alloy the transition on to mesoscale-2 decreases the resistance to high-velocity penetration whereas in 1565 alloy the formation of mesoscale-2 structures increases the resistance to penetration. Numerical simulation of impact aluminum mm-size projectile in same aluminum target with speeds \(\sim \)300 m/s in the moment \(\sim \)240 nanoseconds after the beginning of interaction at mesoscale-2 shown turbulization of particle motion of the environment at their movement in close proximity to target axis of gravity. Numerical researches demonstrated that transition of material to the structural and unstable state has the local and kinetic nature of impact damage material.

Molecular dynamics (MD) simulations of equilibrium structures and flows of polar water, nonpolar argon and methane, and mixtures of water and methane confined by single-walled carbon nanotubes (SWCNTs) with different cross sections have been performed. The results of these simulations show that equilibrium structures and flows of all confined fluids significantly depend not only on the shape of the SWCNT’s rectangular cross sections but also on the types of liquids inside SWCNTs. The cross sections of equilibrium structures of all confined fluids resemble replicas of cross sections of corresponding SWCNTs. In addition, the equilibrium structures formed by nonpolar argon atoms are most spatially ordered whereas nonpolar water molecules form the least spatially ordered equilibrium structures. It has been found that, for nonpolar methane flows through SWCNTs with rectangular cross sections, there are critical values \(f_{xc}\) of the external driving force below which an average flow velocity is equal to nearly zero and above \(f_{xc}\) methane molecules can flow. It has been also found that, for a sufficiently large value of the external driving force, the liquid argon flow through SWCNT with rectangular cross section with the ratio between its sides 1:4 demonstrates the ballistic frictionless regime. MD simulations of equilibrium structures and Couette flows of polar water molecules and nonpolar argon atoms between bounding carbon substrates disposed at the distance \(h = 1.5\) nm from each other have been also performed. Two symmetric configurations when both substrates have similar, either graphene-like crystalline or amorphous structures, and one asymmetric configuration consisted of one substrate with graphene-like crystalline structure and another substrate with amorphous structure have been considered. It has been found that, in all configurations under consideration, the Couette flow velocity profiles depend strongly on polarity of fluid particles confined between bounding substrates. In has been also found that, for substrates with different structures, the Couette flows depend strongly on which of the substrate is moving and which is fixed.

Presently in rules for fatigue assessment of steel and steel welded structures in different technologies subjected to intensive alternating service loading, the stress-life (S-N) criteria are recommended in several versions of approaches. The criteria and approaches are addressed at assessment of fatigue properties of structures; however, the procedures are accompanied with a series of approximations and uncertainties. The nature of drawbacks of the S-N criteria and approaches is commented and feasible means of improvement of the fatigue criteria evaluation and applications in fatigue assessment procedures are proposed.

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 perfect crystals with arbitrary harmonic and weakly anharmonic interactions are considered. The discussed topics cover three major areas: transition to thermal equilibrium, ballistic heat transfer, and thermoelasticity. The analysis reveals and elucidates such phenomena as thermal waves, heat flow from “cold” to “hot”, the existence of several kinetic temperatures, thermal echo, and ballistic resonance.

The chapter provides an overview of the results of studying the effect of hydrogen in mixtures with gases on strength, ductility, fatigue crack growth rate, and fracture morphology of the most commonly used pipeline steels X70, X80. The main methods of testing susceptibility of pipeline steels to hydrogen following the standards are briefly discussed. The results obtained by various authors show that there is a strong influence of partial hydrogen in mixtures with gases. Fatigue crack growth rate increases many times, the fracture morphology changes, and a quasi-cleavage fracture mode is observed. At the same time, tensile strength and yield strength of smooth tensile specimens made of the base metal practically do not change. This, in turn, can lead to an incorrect interpretation of the results of testing.

The actual problem of estimation the indicators of reliability, survival and risk based on real operating data, characterized by different conditions and loads, is considered. A new formulation of the physical principle of reliability is proposed, and a dynamic model of reliability, analysis of survival, and risk taking into account variable loads in the form of a system of differential equations is built. A load is applied to the input of a dynamic system, and a function of the probability of failure-free functioning of the system is formed at its output. The conditions for the equivalence of dynamic models are investigated. In the presence of self-similarity of damage accumulation processes, the general dynamic model is reduced to an equivalent simplified basic dynamic model. Methods for estimating the parameters of this model based on the operating time to failure at various loading histories have been developed. To solve the problem, the maximum likelihood method was used. The results of experimental verification of the dynamic model based on the results of testing LEDs for reliability under constant and variable loading are presented. The constructed model was used to calculate a test variable load and compared with experimental data. The comparison results confirm the effectiveness of the method of dynamic models of survival and reliability under variable loading and accelerated testing. The results of the work can be used in the theory of the reliability of systems with variable loads, in the analysis of survival, the theory of accelerated and forced tests, in the construction of models of damage accumulation, and technical diagnostics.

The structural parameter of composites with dispersed filler of nano- and micro-dimensions is proposed. The derivation of the physical wear model from the empirical wear formula is considered in order to predict the wear resistance of composites at the stage of their design. The developed model allows to estimate wear resistance of composites depending on filler concentration, size and peculiarities of filler particles distribution. The validity of the wear model is shown by the results of comparing the calculated results with the experimental curves.

The purpose of this chapter is to show some routes in describing the mechanism responsible for the formation of the temperature difference at the boundaries of the microfluidic hybrid aligned nematic (HAN) channel, initially equal to zero, if one sets up the stationary hydrodynamic flow or under the effect of an externally applied shear stress (SS) to the bounding surfaces. Calculations based on the nonlinear extension of the classical Ericksen–Leslie theory, supplemented by thermomechanical correction of the SS \(\sigma _{zx}\) and Rayleigh dissipation function, with accounting the entropy balance equation, show that due to the coupling among the \(\sigma _{zx}\), the gradients of the temperature \(\nabla T\) and the director \(\hat{\mathbf {n}}\) fields in the HAN channel the horizontal nematic flow \({\mathbf {v}}\) is excited. The direction and magnitude of \({\mathbf {v}}\) is influenced by both the heat flux \({\mathbf {q}}\) across the HAN channel and the strength of the \(\sigma _{zx}\).

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 conductivity is measured is presented. Finally, the thermal conductivity of graphene in purely diffusive regime is measured for one fixed sample size using Raman thermometry. The result of the proposed theoretical approach is a linear dependence on the sample size in the case of purely ballistic thermal conductivity. An outcome of an experimental study of graphene in a purely diffusive regime and the presented review of experimental methods are the basis for an extension of further experimental studies to the anomalous heat conduction regimes.

The problem of pulse-induced and acoustic cavitation of degassed water is considered within the framework of the incubation time approach. The analytical model to describe a dependency of the cavitation threshold on the pulse duration is developed and applied to experimental data in order to evaluate the main model parameter as the incubation time. The Sign-Perturbed Sums (SPS) method is used to get its estimation in the form of a confident interval. Obtained values of the incubation time are used to describe the dependency of the acoustic cavitation threshold on the ultrasonic frequency.