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The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed.

Free high-frequency longitudinal vibrations of an inhomogeneous nanosize rod are studied on the basis of the nonlocal theory of elasticity. The upper part of the spectrum with a wavelength comparable to the internal characteristic dimension of the nanorod is investigated. An integral-form equation with a Helmholtz kernel, containing both local and nonlocal phases, is used as the constitutive one. The original integrodifferential equation is reduced to a fourth-order differential equation with variable coefficients and a pair of additional boundary-value conditions is obtained. Using the WKB-method, we construct a solution of the boundary-value problem in the form of the superposition of a main solution and edge-effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model, providing an estimate of the upper part of the spectrum of eigenfrequencies.

Based on the equivalent single-layer linear theory for laminated shells, free and forced vibrations of thin cylindrical sandwich panels with magnetorheological core are studied. Five variants of available magnetorheological elastomers differing in their composition and physical properties are considered for smart viscoelastic core. Coupled differential equations in terms of displacements based on the generalized kinematic hypotheses of Timoshenko accounting for transverse shears with coefficients depending on the complex shear modulus for a smart core are used to govern vibrations of cylindrical panels. Assuming conditions of simple support for straight and curvilinear edges, solutions in the explicit form describing natural modes as well as an equation with respect to the required complex eigenfrequencies are found. To predict the shell response to an external harmonic force, the general solution of non-homogeneous governing equations is derived in the form of series in natural modes. To estimate damping capability of magnetorheological elastomers under consideration, the principle tunable parameters, the lowest natural frequencies and associated logarithmic decrements are calculated for the same panels with different magnetorheological elastomers under the action of a magnetic field of different intensities. Finally, the amplitude–frequency plots for magnetorheological elastomer-based panels of different opening angles with and without magnetic field are presented.

Pathological changes in the oscillating system of the middle ear can lead to a decrease of the susceptibility threshold of the auditory analyser to sound vibrations and, consequently, to partial or complete hearing loss. Cartilage implants are most often used for the reconstruction of the tympanic membrane, since they help to avoid complications after treatment. Evaluation of the dynamic characteristics (eigenmodes and eigenfrequencies) of the reconstructed middle ear is the most important problem for analysing the quality of operations that improve the auditory conductivity and develop further recommendations for optimal prosthetics. The aim of this study is to estimate the eigenfrequencies of the middle ear free oscillations after prosthetics on the basis of a mathematical model involving transverse vibrations of the cartilage graft and movement of the prosthesis connecting the reconstructed tympanic membrane and the base of the foot stapes plate. The values of natural frequencies are evaluated for different positions of the nodal lines, averaged geometrical parameters and elastic properties of the tympanic membrane, as well as the foot plate of stapes and prosthesis.

In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that, for a beam, a meaningful energy bounded solution of the purely nonlocal theory may still be defined as the limit solution of the two-phase nonlocal theory. For this, we consider the problem of free vibrations of a flexural beam under the two-phase theory of nonlocal elasticity with an exponential kernel, in the presence of rotational inertia. After recasting the integro-differential governing equation and the boundary conditions into purely differential form, a singularly perturbed problem is met that is associated with a pair of end boundary layers. A multi-parametric asymptotic solution in terms of size-effect and local fraction is presented for the eigenfrequencies as well as for the eigenforms for a variety of boundary conditions. It is found that, for simply supported end, the weakest boundary layer is formed and, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free end conditions bring a strong boundary layer and eigenfrequencies are heavily affected by rotational inertia, even for the lowest mode, in a manner opposite to that brought by nonlocality. Remarkably, all asymptotic solutions admit a well defined and energy bounded limit as the local fraction vanishes and the purely nonlocal model is retrieved. Therefore, we may define this limiting case as the proper solution of the purely nonlocal model for a beam. Finally, numerical results support the accuracy of the proposed asymptotic approach.

The problem on buckling of a thin laminated non-circular cylindrical shell under action of axial compressive forces non-uniformly distributed along edges is considered. It is assumed that some layers are made of a “soft” material so that the reduced (effective) shear modulus for the entire package is much less than the reduced Young's modulus. The differential equations based on the generalized hypotheses of Timoshenko and including the effect of transverse shears are used to predict the buckling of laminated cylinders regardless a number of layers and their mechanical properties. Using the asymptotic method, the buckling modes are constructed in the form of functions rapidly decaying far away from some generatrix at the reference surface. It is shown that accounting transverse shears strongly effect on the buckling modes and corresponding critical buckling forces. In particular, the preferable buckling form for a medium-length thin laminated cylinder with a low reduced shear modulus (as compared with the reduced Young's modulus) is found to be a system of small dents in the axial direction, whose amplitudes decay in the circumferential direction without oscillations; whereas the buckling of a shell with a relatively large reduced shear modulus may occur with formation of waves in both the axial and circumferential directions. As an example, the buckling of cylindrical sandwiches assembled from the ABS-plastic and magnetorheological elastomer with variable shear modulus under different levels of an applied magnetic field is examined

Governing equations based on the generalized kinematic hypotheses of Timoshenko and including the effect of transverse shears are used to predict the buckling of a medium-length thin laminated cylindrical shell under action of the external normal pressure. It is assumed that some of layers are made of a “soft” material so that the effective shear modulus turns out to be too less than the effective Young’s modulus for the laminate. Of all possible variants of boundary conditions, the boundary conditions corresponding to the simple support of edges with and without diaphragms in their planes are considered. For the case of the simply supported edges with diaphragms, the critical buckling pressure as well as the modes of buckling are found in an explicit form. If one of the edges is free from the diaphragm, the boundary-value problem is solved by using the asymptotic approach, a solution being constructed in the form of the superposition of functions describing the main stress state and the edge effect integrals. It is shown that the absence of the edge diaphragm accounts for the appearance of the edge transverse shears (non-classical edge effect integrals) whose decay rate is lower than that of the classical simple edge effect integrals. The effect of edge shears and diaphragms as well on both the critical buckling pressure and eigenform is studied for a laminated cylindrical shell with any number of layers regardless of materials used for laminae. As an example, the buckling of a cylindrical sandwich assembled from the ABS-plastic and magnetorheological elastomer under different levels of an applied magnetic field is examined.

Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral\ud models, along with the simplified differential model, are studied in case of free longitudinal vibrations\ud of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel\ud in the nonlocal integral models, the integro-differential equation corresponding to the two-phase\ud nonlocal model is reduced to a fourth order differential equation with additional boundary conditions\ud taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and\ud asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies\ud and associated modes found in the framework of the purely nonlocal model and its ”equivalent”\ud differential analogue are also compared. A detailed analysis of solutions suggests that the purely\ud nonlocal and differential models lead to ill-posed problems.

In this chapter, based on the equivalent single layer model for thin laminated members, natural modes and corresponding eigenfrequencies for laminated elastic beams plates and cylindrical shells are studied taking into account shears. At first, elastic vibrations of laminated beams are analyzed in Sect. 4.1, the emphasis being made on non-uniformly stressed beams contacting with an elastic inhomogeneous medium. Then, in Sect. 4.2, the eigenmodes and frequencies of elastic rectangular plates are analyzed for two variants of boundary conditions: if all edges are simply supported and have diaphragms preventing shears, the boundary-value problem is solved in the explicit form; and if one of edges is free of a diaphragm, the solution of a corresponding boundary-value problem is constructed in the form of the superposition of the main stress-strain state and the edge effect integrals accounting for the edge shears. Section 4.3 is devoted to vibrations of a circular cylindrical shell of an arbitrary length with constant geometrical and physical parameters. In Sect. 4.4, the localized natural modes for a medium-length laminated cylinder is investigated. And finally, Sect. 4.5 contains the problem on free localized vibrations of a laminated cylindrical shell under axial forces no-uniformly distributed in the circumferential direction. In the last two sections, natural modes are constructed by using the asymptotic method. In all problems, the effect of shears on the natural frequencies is analyzed. Examples on free vibrations of laminated cylinders and panels assembled from different materials are considered.

In this chapter, we consider thin-walled laminated beams, plates and shells containing layers made of viscoelastic smart materials (VSMs). Generally, from all variety of these materials, the magnetorheological elastomer MRE-1 with properties specified in Chapt. 2 will be used for damping layers or core. To compare the damping capabilities of this material with others, we will study also vibrations of thin-walled laminates assembled from other smart materials (MREs, MRFs and ERCs) described in Chapt. 2.

In this chapter, we study the elastic buckling of thin-walled elastic laminated structures. As a preliminary, the simplest problems on stability of laminated beams and plates are considered in Sect. 3.1. Then, using the derived in Chapt. 2 governing equations based on the equivalent single-layer model, some classes of problem on the buckling of thin elastic laminated cylindrical shells under different loading (external pressure, axial compression and torsion) are considered. In Sect. 3.2, the buckling of a medium-length laminated cylindrical shell under external pressure is investigated. As the special case, using the asymptotic Tovstik’s method, the localized buckling modes of a thin non-circular cylindrical shell with an oblique edge are studied. The problems on buckling of axially compressed laminated cylinders are considered in Sect. 3.3; a cylindrical shell under action of non-uniform axial forces is also examined. Finally, Sect. 3.4 is devoted to stability of laminated shells under axial torsion. In all cases, the influence of boundary conditions and transverse shears on the critical values of the buckling load parameter is analyzed. To verify the applied equivalent single-layer model, the finite-element analysis is performed for some of problems. We also show that the application of smart materials (i.e., magnetorheological elastomers) for assembling sandwiches or multi-layered thin cylinders allows to increase significantly the total stiffness of a structure and the critical buckling load as well.

Based on the equivalent single layer model for laminated shells, parametric vibrations of thin laminated non-circular cylindrical shells under non-uniform axial load periodically varying with time are studied. As the governing equations, the non-linear coupled differential equations written in terms of the displacement and stress functions accounting for transverse shears are used. It is assumed that the effective (reduced) shear modulus for an entire laminated package is much less than the reduced Young’s modulus. Using the asymptotic method of Tovstik in combination with the multiple scales method with respect to time, solutions of the governing equations are constructed in the form of functions which are exponentially decay far from some generatrix and growing with time in the case of parametric resonance. The system of two differential equations with periodic in time coefficients and accounting for shears is derived to determine the amplitude of parametric vibrations. The main regions of parametric instability taking into account transverse shears were found. An example of parametric vibrations of a sandwich cylinder with the magnetorheological core affected by a magnetic field is considered.

In this appendix, the definitions of symbols O, o, ~ and asymptotic expansions met in the book are shortly given.

We discuss the propagation of localized surface waves in the framework of the linear Gurtin–Murdoch surface elasticity and taking into account a roughness of a free boundary. We derive a boundary-value problem for anti-plane motions with curvilinear boundary and surface stresses. Using the asymptotic technique developed earlier, we obtain the form of a localized wave and analyze its amplitude evolution. As the main result we present the dependence of the wave amplitude on the roughness magnitude. The presented results could be used for non-destructive evaluation of the surface microstructure using surface waves-based devices. In particular, measuring the decay rate with the depth one can estimate roughness of a surface and appearance of new surface defects.

Buckling of short multi-walled carbon nanotubes (MWCNTs) under external radial pressure is studied on the base of a multiple-shell model. The modified Mushtari-Donell-Vlasov type equations taking into account the van der Waals (vdW) interaction forces between adjacent tubes are used as the governing ones. In contrast to a majority of available studies on buckling of MWCNTs, which consider only the simply supported boundary conditions, this paper based on the asymptotic approach allows for the study of the buckling behavior of MWCNTs with different variants of the boundary conditions at the tube edges. At first, the pre-buckling membrane hoop stress-resultants induced by radial pressure are determined for each wall. Then, introducing a small parameter defined as a thickness-to-radius ratio, the asymptotic solutions of the boundary value problem are constructed for different cases which depend on the outermost radius of a MWCNT. The relevance of the present approach is confirmed by good agreement between asymptotic estimates and exact values of the buckling radial pressure for simply supported double- and triple-walled nanotubes determined on the base of the accepted shell model. In addition, the validity of the asymptotic estimates is justified by comparing theirs with existing data obtained on the base of the available multiple-shell model taking into account the pressure dependence of the interlayer vdW forces. The influence of the outermost radius, aspect ratio and boundary conditions as well on the buckling radial pressure is analyzed in this study.

Based on modified Flugge equations and nonlocal elasticity theory, free axisymmetric oscillations of a long double-walled carbon nanotube embedded into an inhomogeneous elastic medium is studied. The ambient medium is simulated by the Winkler foundation. Van der Waals forces are introduced in order to take into account the interaction between the nanotube walls. Using Tovstik’s asymptotic method, eigenmodes are constructed in the form of functions that decay far from the line on the surface of the outer wall, on which the modulus of subgrade reaction has a local minimum. Eigenmodes and eigenfrequencies corresponding to the coand counterdirected wall motions are found. It has been found that introducing a nonlocality parameter into the model results in eigenmodes that are not inherent in macroscale shells. In particular, an increase in the stretching force leads first to greater localization of vibrations and increase in the amplitudes of tangential atomic oscillations and, second, to reduction in the frequencies in the case when the tube lies in a sufficiently stiff medium.

Understanding of viscoelastic response of a periodontal membrane under the action of short-term and long-term loadings is important for many orthodontic problems. A new analytic model describing behavior of the viscoelastic periodontal ligament after the tooth root translational displacement based on Maxwell approach is suggested. In the model, a tooth root and alveolar bone are assumed to be a rigid bodies. The system of differential equations for the plane-strain state of the viscoelastic periodontal ligament is used as the governing one. The boundary conditions corresponding to the initial small displacement of the root and fixed outer surface of the periodontal ligament in the dental alveolus are utilized. A solution is found numerically for fractional viscoelasticity model assuming that the stress relaxation in the periodontal ligament after the continuing displacement of the tooth root occurs approximately within five hours. The character of stress distribution in the ligament over time caused by the tooth root translational displacement is evaluated. Effect of Poisson’s ratio on the stresses in the viscoelastic periodontal ligament is considered. The obtained results can be used for simulation of the bone remodelling process during orthodontic treatment and for assessment of optimal conditions of the orthodontic load application.

Analytical modeling of stress-strain state of a periodontal ligament in the case of the translational displacement of a tooth root was carried out. The tooth root was assumed as a rigid body. The boundary conditions corresponding to the translational displacement of the root and fixed external surface of the periodontal ligament in the dental alveolus were considered. The system of differential equations describing the periodontal ligament’s plane-strain state induced by the translational motion of the tooth were used as the governing equations. An analytical solution was found for the governing equations in the explicit form. Comparative analysis of the concentrated force generated by the prescribed translational motion of the tooth root was performed using the obtained analytical solution and the model of an incompressible periodontal ligament in the form of a circular paraboloid and hyperboloid. The mathematical model developed in this paper can be used to analyze stresses and strains in the periodontal tissue during orthodontic movement.

Free and forced vibrations of thin medium-length laminated cylindrical shells and panels assembled from elastic materials and magnetorheological elastomer (MRE) embedded between elastic layers are studied. The equivalent single layer model based on the generalized kinematic hypotheses of Timoshenko is used for the dynamic simulation of laminated shells. The full system of differential equations taking into account transverse shears, written in terms of the generalized displacements, is used to study free vibrations of long sandwich cylindrical shells with the MRE cores. To predict free and forced vibrations of medium-length sandwich cylindrical shells and panels, the simplified equations in terms of the force and displacement functions are utilized. The influence of an external magnetic field on the natural frequencies and logarithmic decrement for the MRE-based sandwich cylindrical shells is analyzed. If an applied magnetic field is nonuniform in the direction perpendicular to the shell axis, the natural modes of the medium-length cylindrical sandwich with the homogeneous MRE core are found in the form of functions decreasing far away from the generatrix at which the real part of the complex shear modulus has a local minimum. The high emphasis is placed on forced vibrations and their suppressions with the help of a magnetic field. Damping of medium-length cylindrical panels with the MRE core subjected to an external vibrational load is studied. The influence of the MRE core thickness, the level of an external magnetic field and the instant time of its application on the damping rate of forced vibrations is examined in details.

Laminated cylindrical sandwich shells composed by embedding magnetorheological elastomers (MREs) between elastic layers is the subject of this investigation. Physical properties of the magnetorheological (MR) layers are assumed to be functions of the magnetic field induction and curvilinear coordinates. A system of differential equations with complex variable coefficients depending upon the magnetic field and based on both the assumptions of the generalized kinematic hypothesis for the whole sandwich and experimental data for MREs is used as the governing one. To analyze damping capabilities of adaptive materials, free vibrations of a three-layered circular cylinder containing MRE core layer are studied at different levels of the magnetic field. Using the asymptotic approach, eigenmodes of free vibrations of a laminated cylindrical shell with variable physical characteristics of MRE are constructed in the form of functions decaying far from the weakest plot on the shell structure. It has been shown that applying constant magnetic field may result in localization of eigenmodes corresponding to low-frequency spectrum of three-layered circular thin cylinder with embedded nonuniform MRE layer. Dependencies of natural frequencies, damping decrement and parameter characterizing the power of the eigenmode localization on the intensity of applied magnetic field are

Free axisymmetric vibrations of a single-walled carbon nanotube (SWCNT) embedded in a nonhomogeneous elastic matrix are studied on the base of the nonlocal continuum shell theory. The effect of the surrounding elastic medium are considered using the Winkler-type spring constant which is assumed to be variable along the tube axis. The tube may be prestressed by external tensile forces. The Flugge type shell equations, including the initial membrane hoop and axial stresses, are used as the governing ones. The constitutive equations are formulated by considering the small-scale effects. Using the asymptotic approach, the SWCNT eigenmodes are constructed in the form of functions decreasing rapidly away from some “weakest” line which is assumed to be far from the tube edges. This study shows that introducing the small length scale parameter into the tube model allows to take into account inclusions in the surrounding elastic matrix which may results in the strong localization of some eigenmodes. The dependence of the natural frequencies and the power of the localization of the corresponding modes upon the constant of non-locality, tensile stresses and the nanotube radius is analyzed.

Some problems on localized vibrations and waves in thin isotropic and laminated cylindrical shells are considered in this Chapter. To study vibrations of thin laminated shells, the equivalent single layer model for the whole packet of a sandwich is proposed. The basic goal of this paper is to demonstrate two asymptotic approaches for studying localized vibrations of thin shells. At first, the asymptotic method of Tovstik is applied to study free stationary vibrations localized in a neighbourhood of a fixed generatrix or parallel called the weakest one. As an interesting example, free localized vibrations of a laminated cylindrical shell containing polarized magnetorheological elastomer and affected by an external magnetic field are analyzed. Then the asymptotic method for investigation of running localized waves (wave packets) in thin shells is stated. The solution of governing equations is constructed in the form of a superposition of wave packets running in a thin non-circular prestressed cylinder in the circumferential direction. The influence of non-uniform stationary and dynamic pressures on running wave packets is briefly studied.

A medium length thin laminated cylindrical shell composed by embedding magnetorheological elastomers (MREs) between elastic layers is the subject of this investigation. Physical properties of MREs are assumed to be functions of the magnetic field induction. Differential equations with complex coefficients depending upon the magnetic field and based on experimental data for MREs are used as the governing ones. The shell is subjected to perturbations in their surface so that the initial displacements and velocities are localized in a neighborhood of some generatrix. The problem is to study the response of the MRE-based shell to the initial localized perturbations and the applied time-dependent magnetic field. The asymptotic solution of the initial boundary value problem for the governing equations is constructed by superimposing families of localized bending waves running in the circumferential direction. It is shown that applying the time-dependent magnetic field result in soft suppression of running waves.

In this article, we present the results of investigations of viscoelastic properties of magnetorheological elastomer containing carbonyl iron particles. Frequencies of natural vibrations of three-layered beam, supporting constructions of which are made from aluminum, and the inner layer—from magnetorheological elastomer—are calculated, and the dependence of vibrations on induction of the applied magnetic field is obtained. Nonstationary vibrations of the beam at pulse impact of magnetic field are found.

The process of wave propagation in an infinitely long non-homogeneous narrow medium (waveguide) bounded by a surface of revolution is considered. An asymptotic solution of the wave equation is constructed in the form of localized families of short waves (the wave packets) running in the longitudinal direction, the wave length being of the same order as the characteristic width of the waveguide. As a particular case, the found solution permits to study free oscillations of the medium near the cross-section having the maximum diameter. The effects of reflection of the traveling wave packets from some cross-sections and a localization of the wave processes in a neighborhood of the section with the maximum diameter are revealed.