Components Of Displacement Vector Research Articles

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

Most refined bending models of medium-thick plates, which consider transverse shear and partial compression deformations, differ little. However, despite a significant increase in the order of the governing differential equations, the results obtained from their equations give mainly a small increase in accuracy compared to the existing theories. On the other hand, such an increase in the order of the constructed systems of differential equations requires a significant increase in the effort required to solve them, complicates their physical interpretation, and narrows the range of people who can use them, primarily engineers and designers. Therefore, developing a plate-bending model that incorporates all the above factors and is on par with previously applied theories regarding the complexity of the calculation equations remains relevant. For example, most of the applied theories that do not consider transverse compression cannot be used to solve problems of contact interaction with rigid and elastic dies and bases because it is impossible to satisfy the conditions at the contact boundary of the outer surface of the plate, as well as the boundary conditions at the edges of the plate. Therefore, to provide guaranteed accuracy of the results, some researchers of these problems have introduced such a concept as “energy consistency” between the functions of representation of the displacement vector components, their number, the order of equations, and the number of boundary conditions. The authors, based on the developed version of the model of orthotropic plates of medium thickness, investigate the problem of taking into account the so-called “energy consistency” effect of the bending model, depending on the order of the design equations and the number of boundary conditions, as well as its usefulness and disadvantages in practical calculations. The equations of equilibrium in displacements and expressions for stresses in terms of force and moment forces are recorded. For rectangular and circular plates of medium thickness, test problems are solved, and the numerical data are compared with those obtained using spatial problems of elasticity theory, as well as the refined Timoshenko and Reissner theories. An analysis of the obtained results is provided.

On the features of inhomogeneous residual stress identification using the digital speckle interferometry and the hole-drilling method

A phenomenological approach to the actual problem of determining the inhomogeneous residual stress-strain state in the components of high-tech engineering systems at the stages of their design and operation is presented. The approach is based on physical and mechanical methods of measuring displacements. Current physical models describe the physical regularities of the residual states attributed to changes in the structure by the interaction of defects and dislocations in the field of micro- and meso-stresses. At the same time, there are the problems of the transition to the macrolevel, the construction of multilevel models, and the conversion of these models in engineering practice. In the framework of phenomenological approaches, in the general case, the solution of this problem requires the solution of three-dimensional inverse problems of thermoelasticity. A well-known mechanical method for determining a uniform field of residual elastic stresses recommended by ASTM E837 is described. The method proposed earlier by one of the authors for determining an inhomogeneous (in the plane) field of residual elastic stresses is discussed. A method of the three-dimensional inhomogeneous residual elastic stress-strain state determination based on the experimental determination of the displacement vector components by the method of step-by-step point hole-drilling and data of digital speckle interferometry and digital image correlation is developed. The constitutive relations for the components of the displacement vector are written in the form of Volterra integral operators. The basic operator functions are the functions of four variables, i.e., the coordinates of the cylindrical system (r, θ, z) associated with the hole, and the hole depth h. A method for verification of the basic functions is presented. The problem is reduced to the determination of three displacement functions of three variables: hole radius r, h, and z. Numerical simulation of basic functions is carried out. The obtained results are consistent with the known experimental data and calculated values of the deformation on the surface depending on the depth of the hole according to the ASTM E837 Standard.

Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

Quadrilateral element in mixed FEM for analysis of thin shells of revolution

The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.

Investigation of the Stress-Strain State of a Rectangular Plate after a Temperature Shock

In this paper, the temperature shock phenomenon is considered. This phenomenon occurs during the operation of engineering structures on Earth and in outer space. A rectangular plate has been selected as a structural element exposed to temperature shock. It has a rigidly sealed edge and three free edges. A one-dimensional third initial boundary value problem of thermal conductivity was posed and solved to study the stress–strain state of the plate. Fourier’s law was used to solve this problem, taking into account the inertial term, since the temperature shock is a fairly fast-dynamic phenomenon. It was believed that all the thermophysical properties of the plate are constant and do not depend on its temperature. As a result, the temperature field of the plate was obtained after the temperature shock. This temperature field generates temperature stresses inside the plate, which lead to temperature deformations. To determine these deformations, the initial boundary value problem of thermoelasticity was posed and solved in this work. The Sophie Germain equation was used while solving this problem. To describe the plate, the theory of flexible plates was used, taking into account the stresses in the middle surface of the plate. Next, the accuracy of analytical solutions for the points displacement of a homogeneous plate subjected to a temperature shock was investigated. The temperature field was constructed using a numerical simulation. Functions of the displacement vector components were obtained using approximate analytical solutions. The accuracy of approximate analytical solutions for the components of the plate points deformation vector was estimated. The obtained results allow us to describe the stress–strain state of the plate after the temperature shock. The results of this work can be used in the design of engineering structures for both terrestrial and space purposes in terms of stability calculations and the implementation of deformation constraints.

Propagation of skew-symmetric unsteady shear waves from thick-walled shell in elastic space

Problems of propagation and diffraction of nonstationary waves in elastic bodies are of great theoretical and practical importance in such fields of science and technology as aircraft construction, shipbuilding, seismic exploration of minerals, seismic resistance of structures, and many others. The paper considers the problem of the propagation of skew-symmetric unsteady shear waves from a thick-walled spherical shell in elastic space. To solve the problem, the integral Laplace transforms in dimensionless time, and the method of incomplete separation of variables was used. In the image space, the problem is reduced to an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Formulas for the displacement vector components and the stress tensor are obtained. The transition to the originals is carried out using the theory of residues. Numerical experiments have been carried out, the results of which are presented in graphs. The obtained results of the work can be used in geophysics, seismology, and design organizations in the construction of structures and in the design of underground reservoirs.

A NUMERICAL ANALYSIS OF FORCED STEADY-STATE VIBRATIONS OF AN ELECTRO-VISCOELASTIC SYSTEM IN CASE OF A JOINT IMPACT OF ELECTRICAL AND MECHANICAL LOADS

In the current paper an estimation of a mechanical response of a structure on joint mechanical (force and kinematic) and electrical (voltage and current) impacts taking into account different options of their combinations was carried out. A mechanical response was esteemed basing on solution to the problem of natural vibrations for piecewise-homogeneous electro-viscoelastic bodies. A cantilever plate with piezoelectric element attached to its surface was chosen as an object under study. The plate was made of viscoelastic material the mechanical properties of which are described using complex dynamic moduli. A piezoelectric ceramic material was chosen as the one for piezoelectric elements. An amplitude of forced steady-state vibrations of a component of displacement vector that is perpendicular to the surface of the plate was chosen as a quantity that characterize a mechanical response. Variational equation of motion of the object under consideration is formulated based on relations of linear elasticity, linear viscoelasticity and Maxwell’s equations for a conducting media in a quasistatic approximation. Ranges of changing of values of excitation impacts for vibration modes taken into account were chosen on the basis of values of natural vibrations frequencies obtained from the solution to natural vibration problem for electro-viscoelastic bodies. Numerical implementation of the stated problem was performed in finite element software package ANSYS. Some characteristic peculiarities of a mechanical response depending on a type and magnitude of external impact were obtained as the result of the carried out numerical experiments. There was demonstrated that in case of joint impact of two loading factors there reveals an additional capability of control a mechanical response of an electro-viscoelastic system.

Analysis of recent Antarctic plate kinematics based on GNSS data

The research aims to study recent Antarctic plate kinematics based on Global Navigation Satellite System (GNSS) data. The research covers 60 continuous GNSS stations located on the Antarctic plate during 1994–2021. We determine the components of horizontal displacement vectors of these GNSS stations and construct the schematic map of their distribution. The results indicate that the direction of velocity vectors of horizontal displacements has a rotational nature in clockwise order. The analysis of deformation processes shows that a gradual decrease in the horizontal strain rates, maximum shear strain and area strain is taking place from west to east, in the direction of West Antarctica – the Transantarctic Mountains – East Antarctica. The absolute rotation pole of the Antarctic plate in the ITRF2014/IGS14 reference frame has been determined, and the obtained values are in good agreement with recent plate models.

ГЕОДИНАМІКА

The main goal is to determine and analyze the recent rotation poles of the main tectonic plates based on measurements of continuous GNSS stations for the period of 2002–2021. Using procedures based on the method of the least squares, we suggested an algorithm to determine recent rotation poles of tectonic plates on the basis of processing time series of daily solutions of continuous GNSS stations. The algorythm was implemented in the MathCAD software package. It uses, generalizes, and modernizes the approaches presented in previous studies. Structurally, this algorithm consists of five consecutive stages: transformation of data into an internal format; compliance check and time series filtering; determination of horizontal displacement rates; compliance check and filtering of specified velocities; determination of rotation poles. The algorithm involves the use of freely available time series of daily solutions of continuous GNSS stations, or any other data prepared in a similar format. The study has developed an algorithm to determine recent rotation poles of tectonic plates. It is based on processing time series of daily solutions of continuous GNSS stations. The algorithm was tested to define the recent rotation poles of the main tectonic plates. We determined the components of recent horizontal displacement vectors of 3169 continuous GNSS stations located on 7 large, 7 medium and 3 micro plates for the period of 2002-2021 in the ITRF2014/IGS14 reference frame. The accuracy of determining the component vectors of horizontal displacements is in the range of 0.9-6.4 mm and is on average 10–15% of the vector length. The research allowed us to construct a map scheme of the spatial distribution of the velocity field of recent horizontal movements of continuous GNSS stations. Recent rotation poles of the main tectonic plates were determined for the period 2002-2021 in ITRF2014/IGS14 reference frame. It was established that, in general, the obtained values of recent rotation poles correlate well with known models of tectonic plate movements. This confirms the correctness of the chosen method, as well as the reliability of the obtained results. Recent rotation poles of tectonic plates are the basis for modeling and analysis of global, regional and local geodynamic processes, so their accurate determination is an urgent and necessary task. GNSS data is an alternative, and recently, practically irreplaceable basis for determining such parameters. The rapid increase in the number of continuous GNSS stations, as well as the high quality of their measurements, contributes to improving the accuracy of determining the recent rotation poles of tectonic plates, but leads to the need for their constant recalculation and refinement. The presented algorithm and the obtained results can be used to develop new and refine existing models of tectonic plate movements and reference frames, as well as to forecast the movements of the Earth’s crust.

Algorithm of finite elemental SSS analysis of thin-walled technosphere objects based on a triangular discretion element with elastic-plastic deformation

The article describes an algorithm for the strength calculation of thin-walled objects of the technosphere, taking into account the plastic stage of the work of the material used, which is quite relevant with the increased requirements for material saving of designed and reconstructed objects. The finite element method is used as a tool for studying the stress-strain state of technosphere objects, taking into account the physical nonlinearity of the applied structural material. As a sampling element, a fragment of the middle surface of a thin-walled triangular technosphere object with main nodes at the vertices of the triangular element and additional nodes at the midpoints of the sides is used. The sought unknowns at the main nodes at the loading step are the increments of the displacement vector components and their first derivatives with respect to the curvilinear coordinates of the middle surface. At additional nodes, Lagrange multipliers act as unknowns, introduced to improve the consistency of the bin along the boundaries with adjacent elements. When deriving the plasticity matrix at the loading step, we used the hypothesis that the components of the stress increment tensor are proportional to the components of the strain increment tensor. The developed algorithm was implemented as a package of applied programs and verified by the example of calculating a rigidly restrained cylinder loaded with internal pressure.

Strength calculation algorithm for shell structures based on a four-node discretization element

The article describes a method for calculating shell structures taking into account transverse shear deformations during elastoplastic deformation. The geometric relationships between the increments of deformations in an arbitrary layer of the shell and the increments of the components of the displacement vector, the components of the vector of inclination angles of the normal are obtained on the basis of the hypothesis of S.P. Tymoshenko. The slope of the normal and the increment of the slope are performed from the original position of the normal. The physical relationships between the increments of the stress tensor components and the increments of the components of the deformation tensor at the loading step were obtained on the basis of the hypothesis that they were proportional, based on the deformation theory of plasticity. In the finite element implementation of the developed algorithm, a four-node fragment of the middle surface of the shell structure with nodes located at the vertices of the quadrangular fragment was used. The increments of the displacement vector components, their first derivatives with respect to curvilinear coordinates, as well as the increments of the components of the normal rotation angle vector were selected as the nodal variable parameters of the four-node finite element at the loading step. The size of the stiffness matrix of the four-node finite element was 44x44. The developed algorithm was tested on test problems for calculating a cylindrical shell using a duralumin alloy with a deformation diagram in the form of a two-link broken line as a material.

Element Differential Method for Solving Linear and Nonlinear Electromagnetic Problems

A novel numerical method named element differential method (EDM) is first presented to solve linear and nonlinear static electromagnetic problems. The main idea of this method is to use the direct differentiation formulation of the shape functions of Lagrange isoparametric elements to evaluate geometry and physical variables. A new collocation method is proposed to establish the system of equations, in which the governing equation is collocated at the internal nodes of the elements, the continuity conditions of tangential component of magnetic field intensity and normal component of electric displacement vector are collocated at interface nodes between elements, and Neumann boundary conditions are collocated at outer surface nodes. Compared with the finite element method, there is no need to use any variational or energy principles to establish the system of equations and there is no requirement for integral evaluation. The derived spatial derivatives can be directly substituted into the static electromagnetic governing equations and the boundary conditions to form the final system of algebraic equations. Three 2D numerical examples are used to verify the correctness and effectiveness of the presented method for solving linear and nonlinear static electromagnetic problems.

Energetic effects analysis of shear waves and their second harmonics in a waveguide with non-ideal component contact

In this article the energetic and kinematic effects that occur in the elastic shear wave and its second harmonics propagation are investigated. The waveguide consists of anisotropic elastic crystal layer of cubic system m3m class enclosed between crystal halfspaces of same anisotropy class. A slipping contact is assumed in the contact zone of waveguide parts. The research is based on a model of general geometrical end physical nonlinearity in dynamic deformation processes. It allows to use elastic potential with the quadratic and cubic deformation components and the deformations with nonlinear terms. The approach of nonlinear elastic wave characteristics expansion into rows of a small parameter is used. Due to this approach at the first stage it's necessary to solve the problem of finding the components of the localized shear wave displacement vector (the problem of the first approximation). In the second stage, using the obtained results of the first approximation problem, the representation of the components of the displacement vector for the second harmonics of the localized elastic wave is solved in analytical form (the problem of the second approximation). By using the obtained kinematic results, the energy effects can be evaluated in the form of a vector of the average for the period of power flow. Specific results of the study of the amplitude-frequency and energy characteristics of the localized shear type elastic waves in the considered waveguide structure were obtained using computer algebra methods.The calculations of cinematic and energetic characteristics (that in contrast to linear SH harmonic are P-SV type waves) have been carried out for NaCl layer and germanium halfspaces waveguide.

Advancing Ground-Based Radar Processing for Bridge Infrastructure Monitoring.

In this study, we further develop the processing of ground-based interferometric radar measurements for the application of bridge monitoring. Applying ground-based radar in such complex setups or long measurement durations requires advanced processing steps to receive accurate measurements. These steps involve removing external influences from the measurement and evaluating the measurement uncertainty during processing. External influences include disturbances caused by objects moving through the signal, static clutter from additional scatterers, and changes in atmospheric properties. After removing these influences, the line-of-sight displacement vectors, measured by multiple ground-based radars, are decomposed into three-dimensional displacement components. The advanced processing steps are applied exemplarily on measurements with two sensors at a prestressed concrete bridge near Coburg (Germany). The external influences are successfully removed, and two components of the three-dimensional displacement vector are determined. A measurement uncertainty of less than 0.1 mm is achieved for the discussed application.

ВИБРОСЕЙСМО ЗАЩИТА ЗДАНИЙ И СООРУЖЕНИЙ ОТ ДИНАМИЧЕСКОГО ВОЗДЕЙСТВИЯ ПРИРОДНОГО И ТЕХНОГЕННОГО ХАРАКТЕРА

In this article, the authors analyze results of their long-term researches on protection of buildings and structures against dynamic loads caused by the nature and/or human activities. They also give the grounds for necessity to provide vibration insulation of buildings, and show how to choose and calculate proper parameters for rubber vibroseismic insulators. Specifics of finite element method applied for static calculating the vibroseismic insulators is also described. In order to take into account weak compressibility of rubber, a moment finite element scheme was used, which assumes a triple approximation of the displacement vector components consisting of strain tensor and volume change function. Stress-strain state of the shock absorbers was determined for different standard sizes and diameters. The authors also describe two variants of calculation depending on the method of fixing the insulator’s ends. In the first case, the ends are vulcanized to the metal plates. In the second variant, the ends are free and can move in a horizontal plane. Influence of ratio of the shock absorber height and radius to the strain state of a structure was also analyzed. In order to determine factual levels of the soil and pile vibration (in two horizontal and one vertical directions), vibrodynamic studies were carried out. The obtained vibration signals were registered by the one-component vibration transducers 731A (vibration sensors) produced by the Wilcoxon Research company (the USA). Then, the vibration records were processed by the specialized program “Seismic Monitoring”. Based on the results of these studies, numerical calculations were performed in order to determine whether the predicted levels of the residential building vibrations are in compliance with the existing sanitary standards when exposed to real technogeneous loads. Vibrodynamic tests on vibration acceleration levels of the vibration-insulated reinforced concrete slabs and floors in residential building confirmed high effectiveness of the used vibroseismic insulation system with the rubber elements: the registered vibration acceleration levels in the residential building on all floors did not exceed acceptable levels set by the sanitary standards, and ensured comfortable living conditions under different dynamic impacts. The results of this work make it possible to design buildings with anti-seismic protection by using the designed rubber elements in accordance with the local conditions of the city of Almaty.

Modeling of the crystallization process of a polymer plate taking into account large deformations

The issues of modeling the thermomechanical behavior of a polymer material during cooling accompanied by crystallization are considered. The proposed approach is applied to the problem of deformation upon cooling of an infinite plate made of low-pressure polyethylene during crystallization. Mathematical formulations of thermal kinetic and thermomechanical boundary value problems are presented. The results of the numerical solution of a coupled heat-kinetic problem, including the non-stationary heat conduction equation and the crystallization kinetics equation obtained under the assumption that the characteristics of the material depend on temperature are given. When solving the boundary value problem of thermomechanics, the previously obtained phenomenological nonlinear constitutive relations are used, which continuously describe the behavior of a polymer material in a wide temperature range, including the range of phase transformations. The construction of constitutive relations was carried out using the Peng-Landel potential. A weak variational formulation of the boundary value problem, constructed using the Galerkin method, is presented. A linearization procedure is carried out, which makes it possible to reduce the solution of an initially nonlinear boundary value problem to solving a sequence of linear problems with relatively small increments of the displacement vector components. In this case, the linearization of geometric relations is performed by superimposing small increments of deformations on finite ones, linearization of the constitutive relations - expansion in a Taylor series with subsequent retention of linear terms under the assumption that the increment of the components of the strain tensor is small. An approach is demonstrated that naturally takes into account the small increments of temperature and structural deformations arising in the material. Some aspects of the numerical implementation of the created algorithm based on the finite element technology of constructing a discrete analogue of the problem under consideration are analyzed. The results of solving the problem under discussion in a linear formulation under the assumption of small deformations and in a nonlinear formulation are compared.

Strain-stress behavior of elastic layer in local loading

The paper studies the problem of a local loading of an elastic layer in 3D perspective. The solution of the boundary value problem subject to a concentrated force is constructed as a combination of two components. The first component is a classical solution of A. Lyav theory of elasticity, the second one is a solution proposed by I.М. Rapoport. The second component is distinctive in that it describes a point edge effect rapidly damping while moving off from the point of force application. This solution is built in a series form, namely, proper decompositions of the auxiliary nonself-adjoint differential operator. The convergence of series is ensured by a rapid growth of eigenvalues. Dying-away to zero at infinity is caused by the exponential law of Macdonald functions damping. The solution of the concentrated force action is used as a kernel to determine displacement vector components, tensors of deformations and strains in the problem of arbitrary local loading of an elastic layer. Eventually, analytical solution of the singular problem makes it possible to reasonably determine the strain-stress state in a local loading zone. Key words: strength, stress-strain behavior, theory of elasticity, differential equations, series, Bessel functions.

Volumetric discretization element with invariant displacements approximation in the SSS numerical analysis problems for the building structures

The article describes the sought quantities invariant approximation method in relation to the volumetric discretization element in the form of a parallelepiped with the main nodes located at its vertices. The displacement vector components and their partial derivatives of the first order were selected as the sought-for unknowns of the volume discretization element. An approximating expression when implementing the invariant approximation technique is written for the displacement vector of an element inner zone point and not for components of this vector isolated from each other. The approximating expressions obtained on the sought quantities’ invariant approximation basis contain the curvilinear coordinates system parameters used. The discretization volume element compatibility is improved due to the use of Lagrange multipliers in additional nodes located in the middle of the parallelepiped bases’ edges. The Lagrange correction factors ensure the displacement vectors normal components derivatives’ equality along the normals to the element base edges’ midpoints. The expanded stiffness matrix and the column of the volumetric discretization element nodal forces are formed by minimizing the conditional Lagrange functional.

Acoustic phonons in multilayer nitride-based AlN/GaN resonant tunneling structures

The study of physical processes associated with acoustic phonons in nitride-based nanosystems is of great importance for the effective operation of modern nanoscale devices. In this paper, a consistent theory of acoustic phonons arising in multilayer nitride-based semiconductor resonant tunneling structures, that can function as a separate cascade of a quantum cascade laser or detector is proposed. Using the physical and geometric parameters of a typical nanostructure, the spectrum of various types of acoustic phonons and the corresponding normalized components of the elastic displacement vector are calculated. It has been established that the spectrum of acoustic phonons of a multilayer nanostructure consists of two groups of the shear phonons dependencies and three groups of dependencies for a mixed spectrum of flexural and dilatational phonons. The dependencies of the acoustic phonons spectrum of the nanostructure and the components of the elastic displacement vector on its geometric parameters are studied. It has been established that for the components of the displacement vector u2 for shear phonons have a decrease in the absolute values of their maxima with increasing of energy level number. The components u1 and u3 of flexural and dilatational phonons behave respectively as symmetric and antisymmetric functions relatively the center of an separate selected layer of the nanostructure. The proposed theory can be further applied to study the interaction of electrons with acoustic phonons in multilayer resonant tunneling structures.

The Finite Element Desired Quantities Invariant Approximation Method in the Thin Shells Calculation Based on the Timoshenko Hypothesis

The displacement vector components vector (invariant) approximation implementation and the initial inclination angles by the hypothesis of S. P. Tymoshenko in obtaining the thin shell quadrangular finite element nodal forces stiffness matrix and the column is shown.