Gennadi Mikhasev

On damping vibrations of three-layered beam containing magnetorheological elastomer

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.

Shell and Membrane Theories in Mechanics and Biology

On the theories of plates and shells at the nanoscale.- Homogenization approach in the theory of plates and shells.- Wavelet analysis of nonlinear mechanics of shells.- Solid mechanics modeling in ophthalmology.- Theory of shells and theory of curvilinear rods: A comparative analysis.- Initial-value problems in general asymptotic theory for thin walled elastic structures.- Second-order isotropic and anisotropic plate theories.- Actual problems of nanomechanics.- Analysis of free vibrations of multi-walled carbon nanotubes based on theories of orthotropic cylindrical shells and non-local elasticity.- A shell theory for CNTs of arbitrary chirality.- The application of shell model to biological membranes.- Studying free vibrations of totally reconstructed middle ear based on the plate theory and FEM simulation.- Concerning approaches to modeling and restoration of inhomogeneous initial stress fields in plates.- Some problems of equilibrium and stability of nonlinearly elastic circular membranes.- New approach for studying nonlinear dynamic response of thin plates in a viscoelastic medium.- Aspiration of a nonlinear elastic spherical membrane.- Movement modelling of soft microrobot of amoebalike type in the heterogeneous environment.- Identification of the elastic modulus of polymeric materials based on compression of thin-walled cylindrical specimens.- On gradient theories of plates and shells with applications to nanomechanics.- Nanoplate stability.- Asymptotic solutions for thin layers in articular contact.- Three-dimensional exact analysis of functionally graded and laminated piezoelectric plates.

Prediction of Eigenfrequencies of the Middle Ear Oscillating System After Tympanoplasty and Stapedotomy

The mathematical model of the reconstructed middle ear subjected to tympanoplasty and stapedotomy is proposed. The biomechanical system consists of a restored tympanic membrane made from cartilage and the total prosthesis replacing the malleus-incus-stapes chain. The reconstructed eardrum coupled to the circular prosthesis plate is modeled as an annular circular elastic plate. The prosthesis stem is considered as a rigid cylindrical rod that projects through a small perforation in the fixed footplate stapes into the vestibule of the inner ear. Resting in the perilymph, the piston-like stem is under static and dynamic forces acting from the cochlear liquid or/and additional spring element. The static forces assert stable position of the prosthesis but result in initial membrane stresses in the reconstructed eardrum. The eardrum motion is governed by the system of three differential equations written in terms of normal and tangential displacements taking into account the initial stresses. Free low-frequency vibrations of the prestressed biomechanical system are studied using two methods: when the initial stresses are small, a perturbation method is applied; if not, the finite element approach is utilized to predict the natural frequencies.

Soft Suppression of Traveling Localized Vibrations in Medium-Length Thin Sandwich-Like Cylindrical Shells Containing Magnetorheological Layers via Nonstationary Magnetic Field

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.

Mathematical Model for Analysis of Translational Displacements of Tooth Root

AbstractAnalytical 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 ligaments 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 pe...

Effect of Magnetic Field on Free and Forced Vibrations of Laminated Cylindrical Shells Containing Magnetorheological Elastomers

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.

Some Problems on Localized Vibrations and Waves in Thin Shells

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.

Shell Theory-Based Estimation of Local Elastic Characteristics of Biological Cells

The shell theory-based approach for the elastic analysis of biological cells is proposed. This approach introduces the estimation of the local Young’s modulus of the single cell on the basis of the shell theory and data of the atomic force microscopy (AFM). This method is applicable to evaluate the elastic properties of the cell membrane which stretched under action of the atomic force microscope indenter (AFMI). The cell is represented by a thin shallow spherical shell experiencing the concentrated outward force. The influence of cytoskeleton on the cell deformation is disregarded. Taking into account microscopic sizes of the cell and the indenter tip, the internal nanoscale parameter is introduced into the constitutive equations. On the basis of the experimental data and developed shell model we give a rough estimate of local Young’s modulus for the red blood cell.