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2022 | Book

Recent Approaches in the Theory of Plates and Plate-Like Structures

Editors: Prof. Dr. Holm Altenbach, Prof. Dr. Svetlana Bauer, Prof. Dr. Victor A. Eremeyev, Prof. Gennadi I. Mikhasev, Prof. Dr. Nikita F. Morozov

Publisher: Springer International Publishing

Book Series : Advanced Structured Materials


About this book

This book presents the various approaches in establishment the basic equations of one- and two-dimensional structural elements. In addition, the boundaries of validity of the theories and the estimation of errors in approximate theories are given. Many contributions contain not only new theories, but also new applications, which makes the book interesting for researcher and graduate students.

Table of Contents

1. On one Class of Spatial Problems of Layered Plates and Applications in Seismology
The class of spatial nonclassical quasistatic and dynamic problems of the theory of elasticity for orthotropic layered plates was solved. We determined the stress-strain states (SSS) of Earth’s Lithospheric plates and blocks of Earth’s crust on the basis of data from inclinometers, strainmeters and other measuring instruments. Monitoring of changes in the stress-strain state of layered package with respect to the time makes it possible to trace the entire process of accumulation of critical deformation energy and the possibility of earthquakes occurrence.
Mher L. Aghalovyan, Lenser A. Aghalovyan
1. Asymmetric Buckling of Heterogeneous Annular Plates
Nonaxisymmetrical buckling of inhomogeneous annular plates subjected to normal pressure is studied. The effect of material heterogeneity and ratio of inner to outer radii on the buckling load is examined. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. For an annular plate with a small inner radius of a plate and Young’s modulus, decreasing towards the outer edge, the critical pressure for unsymmetrical buckling is sufficiently lower than for a plate with constant mechanical properties. For an annulus with large inner radius, the buckling pressure and the buckling mode number increases as the Young modulus decreases towards to outer edge of a plate.
Svetlana M. Bauer, Eva B. Voronkova
1. Bending Stiffness of Multilayer Plates with Alternating Soft and Hard Layers
The bending stiffness of a multilayer plate with alternating soft and hard layers is considered under the assumption that the deformation wavelength is substantially greater than the plate thickness. We discuss the approximate methods for determining the shear compliance required for replacing a multilayer plate with an equivalent single-layer Timoshenko–Reissner plate. A comparison is made with the exact solution of the three-dimensional problem of the theory of elasticity. The dependence of shear compliance on the ratio of Young’s moduli of layers and on their location is investigated.
Alexander K. Belyaev, Nikita F. Morozov, Petr E. Tovstik, Tatiana P. Tovstik
4. On the Bending of Multilayered Plates Considering Surface Viscoelasticity
We discuss the bending resistance of multilayered plates taking into account surface/interfacial viscoelasticity. Within the linear surface viscoelasticity we introduce the surface/interfacial stresses linearly dependent on the history of surface strains. In order to underline the surface viscoelasticity contribution to the bending response we restrict ourselves to the elastic behaviour in the bulk. Using the correspondence principle of the theory of viscoelasticity we present an e_ective bending relaxation function.
Victor A. Eremeyev, Violetta Konopi´nska-Zmysłowska
5. Buckling of a Ring-Stiffened Cylindrical Shell Under the External Pressure
In this paper, the problem of buckling of a thin elastic cylindrical shell supported by the rings of various stiffness is considered. The Rayleigh–Ritz method is used to obtain the problem’s analytical solution for the case of the simply supported edges of the shell. The parameters of the optimal distribution of the structure mass between the shell and the stiffening ribs, which is required to maximize the critical pressure, have been found. The solution of the problem of minimizing the mass of a structure at a given critical pressure is obtained. Here are considered the rings with zero eccentricity. The approximate analytical solutions are compared with the numerical solutions obtained by the finite element method.
Sergei B. Filippov, Grigory A. Nesterchuk
6. Free Vibration Corrugated Open Cylindrical Shells
The problem of the free vibrations of the thin circular corrugated open cylindrical shells are presented. The finite element method was used. The frequencies and forms of free vibrations of the in circular corrugated open cylindrical under conditions of different longitudinal bisecting and different boundary conditions shells are calculated. The dependence of the frequency of free vibrations on the variant of longitudinal bisecting is analyzed.
Alexander Ya. Grigorenko, Maksym Yu. Borysenko, Olena V. Boychuk, Natalia P. Boreiko
7. On a New Theory of the Cosserat Continuum with Applications in Electrodynamics
We consider an elastic Cosserat continuum of a special type. Next we suggest analogies between quantities characterizing the stress–strain state of the continuum and quantities characterizing electrodynamic processes. Taking into account the suggested analogies, we interpret equations describing the continuum as equations of electrodynamics. We identify parameters of our model by comparing the obtained equations with Maxwell’s equations. As a result, in the framework of our model, we obtain a set of differential equations that coincide with Maxwell’s first equation (the one with the displacement current term), the Gauß law for electric field, the charge conservation law, a modification of the Maxwell–Faraday equation and a modification of the Gauß law for magnetic field. We also obtain the Gauß law for gravitational field and introduce the concept of gravitating mass.
Elena A. Ivanova
8. Hierarchical Models of Conduction of Heat in Continua Contained in Prismatic Shell-like Domains
We construct hierarchical models for the heat conduction in standard and prismatic shell-like and rod-like 3D domains with non-Lipschitz boundary, in general.
George Jaiani
9. Dynamic Sliding Contact for a Thin Elastic Layer
The contribution is concerned with dynamics of a thin elastic layer, subject to sliding contact. Both one- and two-sided sliding contact are studied, revealing the presence of the fundamental vibration modes. First, mixed boundary conditions modelling two-sided sliding are addressed, allowing a factorisation of the dispersion relation. Then, the asymmetric problem of one-sided sliding contact is tackled, with mixed conditions along the contact surface and prescribed normal stress on the opposite face. Using symmetry, this problem is found to be related to that for a layer of a double thickness, with classical boundary conditions in terms of stresses. In this case, the fundamental mode of interest coincides with the zero-order Rayleigh-Lamb symmetric wave. A long-wave low-frequency perturbation scheme is implemented for the forced problem.
Julius Kaplunov, Danila A. Prikazchikov, Tomaž Savšek
10. Analytical Approach to the Derivation of the Stress Field of a Cylindrical Shell with a Circular Hole Under Axial Tension
A new analytical approach to the stress field problem of the cylindrical shell with a circular cutout under axial tension is proposed. Classical models because of an expansion into small parameter have narrow range of applicability and almost do not differ from Kirsch case for plate. The new approach opens up opportunities for the analytical study of the stress field. The idea is to decompose each basis function into a Fourier series by separating the variables, which allows us to obtain an infinite system of algebraic equations for finding coefficients. One of the important steps of the study is that the authors were able to prove which of the equations of the system is a linear combination of several others. Excluding it made it possible to create a reduced system for finding unknown coefficients. The proposed approach does not impose mathematical restrictions on the values of the main parameter that characterizes the cylindrical shell.
Stanislava V. Kashtanova, Alexey V. Rzhonsnitskiy
11. Analysis of Solutions for Elliptic Boundary Layer in Cylindrical Shells at Edge Shock Loading
This paper is devoted to analysis of the asymptotic solution behaviour for the elliptic boundary layer in cylindrical shells in small vicinities of the surface Rayleigh wave front under normal edge shock loading. The boundary layer is described by the elliptic equations along the thickness of shells and the hyperbolic equations which are defined boundary conditions on the faces. These boundary conditions on cylindrical faces characterise wave motion on them. The sought for solution is presented by the composite ones. The first one is theparticular solution, satisfeing only the boundary conditions on the shell edge. The boundary value problem for the second one is reduced to the problem for shock loading on the faces of the infinite cylindrical shell. This one is solved with the help of the Laplace transform on time and the Fourier transform on the longitudinal coordinate. Invertiation of the Laplace and Fourier transforms allows us represent the solution on the base of elementary function arctg of the complicated arguments. Analysis of this solution in a small quasifront vicinity by the asymptotic method defined properties of them at moving from the quasifront along the longitudinal coordinate.
Numerical calculations confirmed this quality analysis of the solution.
Irina V. Kirillova, Leonid Y. Kossovich
12. Dimension Reduction in the Plate with Tunnel Cuts
We carry out dimension reduction in the homogenization theory 3D periodicity cell problem for the plate with a unidirectional system of channel cuts. We demonstrate that the original 3D problem may be reduced to several 2D problems. The main attention is paid to the solution near the top and the bottom surfaces of the plate. Our numerical analysis indicates the existence of a new type of boundary layer at the upper and lower surfaces of the plate. We estimate the thickness of the found boundary layer. We also find a wrinkling effect on the top and bottom surfaces of the plate.
Alexander G. Kolpakov, Sergei I. Rakin
13. Topological Optimization of Multilayer Structural Elements of MEMS/NEMS Resonators with an Adhesive Layer Subjected to Mechanical Loads
The paper considers the problem of topological optimization of multilayer structural elements of MEMS/NEMS resonators with an adhesive layer under the action of mechanical loads. The purpose of this work is to obtain a design solution that is least susceptible to destruction due to an increase in the rigidity of the elements to be joined and, as a consequence, providing smoothing of stress peaks in the adhesive layer. To demonstrate the operation of the topological optimization algorithm for this class of problems, several examples are given that show significant improvements in the set target indicators. The problems were solved by the finite element method with the application of the sliding asymptotes method.
Anton V. Krysko, Jan Awrejcewicz, Pavel V. Dunchenkin, Maxim V. Zhigalov, Vadim A. Krysko
14. Forced Vibration Analysis of Laminated Piezoelectric Plates by a Strong Sampling Surfaces Formulation
This paper focuses on implementation of the sampling surfaces (SaS) method for the 3D vibration analysis of laminated piezoelectric plates. The SaS formulation is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface to introduce the displacements and electric potentials of these surfaces as basic plate variables. Such choice of unknowns allows the presentation of the laminated piezoelectric plate formulation in a very compact form. The feature of the proposed approach is that all SaS are located inside the layers at Chebyshev polynomial nodes that improves the convergence of the SaS method significantly. The use of outer surfaces and interfaces is avoided that makes possible to minimize uniformly the error due to Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of the equations of motion and the charge equation can be applied e_ciently to the obtaining of exact solutions for laminated piezoelectric plates, which asymptotically approach the 3D solutions of piezoelectricity as the number of SaS tends to infinity.
Gennady M. Kulikov, Svetlana V. Plotnikova
15. Asymptotic Analysis of Buckling of Layered Rectangular Plates Accounting for Boundary Conditions and Edge Effects Induced by Shears
Based on the equivalent single layer theory for laminated shells, buckling of layered rectangular plate under uniaxial compression with different variant of boundary conditions is studied. Equations in terms of the displacement, shear and stress functions, which take into account transverse shears inside the plate and near the edges with and without diaphragms, are used as the governing ones. Using the asymptotic approach, the buckling modes are constructed in the form of a superposition of the outer solution and the edge effect integrals induced by shears in the vicinity of the edges with or without diaphragms. Closed form relations for the critical buckling force accounting for shears are obtained for different variants of boundary conditions. It is detected that within one group of boundary conditions, the critical buckling forces can differ significantly depending on whether the edge is supplied with the diaphragm or not.
Gennadi Mikhasev, Rovshen Ataev
16. Semi-analytical Model for the Close-Range Stress Analysis of Transverse Cracks in Composite Plates
In this paper, an efficient method for the computation of the stress state in the vicinity of transverse cracks in symmetric fibre-reinforced composite laminated plates under tensile load is presented. To determine the stress field, the solution of the Classical Lamination Theory (CLT) of the uncracked structure is superimposed with a so-called “internal solution” which is based on a subdivision of the layers into an arbitrary number of numerical plies. The displacement field of the composite laminated plate is approximated by introducing a priori unknown interface displacement functions. By employing the principle of minimum total potential energy, the governing equations are obtained in a closed-form manner and yield a quadratic eigenvalue problem, which is solved numerically. In order to obtain a full description of the state variables, the underlying boundary conditions as well as the continuity conditions have to be utilized. Comparisons with two-dimensional finite element studies indicate that the semi-analytical method is able predict the stress field with similar accuracy while only using a fraction of the underlying computational effort.
Clemens Peiler, Andreas Kappel, Christian Mittelstedt
17. Shear Deformable Elastic Beam Models in Vibration and Sensitivity of Natural Frequencies to Warping Effects
A theory of shear deformable beams in vibration is formulated using a shear-warping theory whereby the cross section is allowed to warp according to a parametrically specified warping rule (parametric warping). A continuous family of beams is generated which is controlled by a warping parameter s ≥ 0 spanning from s = 0 (Timoshenko-Ehrenfest beam) to s = ∞ (Euler-Bernoulli beam) and intersecting the Levinson-Reddy model for s = 0:5. This enables one to express any response parameter as a function of s useful to describe the sensitivity of the beam’s behaviour to warping effects. The governing transverse displacement differential equation (DE) - of the fourth order in the case of no warping - is instead of the sixth order in the presence of warping effects, but remarkably the maximum order of time derivatives is still four. The vibration motion of the family’s general beam is characterized by two basic macroscopic space and time scales, which make it possible to ascertain that the terms of the governing DE with the fourth order time derivative are negligible with respect to the others. The simplified governing DE without fourth order time derivatives is applied to a beam case to derive the physically meaningful spectrum with warping effects and to assess the sensitivity of natural frequencies to the warping effects. Every frequency as a function of s exhibits a waved pattern featured by softening for 0 < s < sh (with smaller frequencies therein), by hardening for s > sh (with larger frequencies therein), sh varying with the vibration mode.
Castrenze Polizzotto, Isaac Elishakoff, Paolo Fuschi
18. Conceptual Approaches to Shells. Advances and Perspectives
The history of scientific discoveries in elasticity theory that preceded the creation of the theory of shells is described. The way of finding solutions and the logic of scientific discoveries are indicated, including the history of the question of the position of the neutral line of a loaded beam and the history of the derivation of the equations of elasticity theory. Some applied questions of analytical methods, including the method of asymptotic expansions, are discussed. The prospects and current use of methods of mechanics and the concept of shells in interdisciplinary research are described.
Oksana R. Polyakova, Tatiana P. Tovstik
19. Necessary Conditions for Energy Minimizers in a Cosserat Model of Fiber-Reinforced Elastic Solids
Pointwise necessary conditions for energy minimizers are derived in the context of a Cosserat model of fiber-reinforced elastic solids in which the Cosserat rotation field accounts for the kinematics of the embedded fibers, modelled as spatial Kirchho_ rods. The analysis requires careful consideration of constraints associated with the fact that fibers are convected by the continuum deformation field as material curves.
Milad Shirani, David J. Steigmann
20. Vibration Control of a Non-homogeneous Circular Thin Plate
Abstract Transverse vibrations of an inhomogeneous circular thin plate are studied. The plates, which geometric and physical parameters slightly differ from constant and depend only on the radial coordinate, are analyzed. After separation of variables the obtained homogeneous ordinary differential equations together with homogeneous boundary conditions form a regularly perturbed boundary eigenvalue problem. For frequencies of free vibrations of a plate, which thickness and/or Young’s modulus nonlinearly depend on the radial coordinate asymptotic formulas are obtained by means of the perturbation method. The effect of the small perturbation parameter on behavior of frequencies is analyzed under special conservation conditions: i) for a plate, the mass of which is fixed, if the thickness is variable, and ii) for a plate with the fixed average sti_ness, if Young’s modulus is variable. Asymptotic results for the lower vibration frequencies well agree with the results of finite element analysis with COMSOL Multiphysics 5.4.
Andrei L. Smirnov, Grigory P. Vasiliev
21. Modeling of an Inhomogeneous Circular Timoshenko Plate with an Elastically Supported Boundary
The deformation of a circular elastic plate of variable stiffness is studied in the present paper. The problem is considered in the framework of Timoshenko’s hypotheses with various conditions of support at the boundary, including the presence of elastic bonds. One of the applications of this model is the problem of modeling a lamina cribrosa sclerae (LC) of an eyeball. Timoshenko’s hypotheses are used in view of the necessity to take into account in modeling shear deformations of a LC. Elastic constraints in boundary conditions are characterized by two coefficients of subgrade resistance. An energy functional is presented with the use of the variational Lagrange principle for an inhomogeneous plate. It also takes into account the potential energy of bonds at the edge. Deflection of the plate and rotation angle of the normal are found using the Ritz method, which makes it possible to derive a solution based on the energy functional. The influence of the number of coordinate functions on the accuracy of the obtained solution is investigated. A comparison is made with the results obtained earlier in the framework of Kirchhoff’s plates.
Alexander O. Vatulyan, Olga A. Potetyunko, Ivan V. Bogachev
1. Effect of Distributed Dislocations on Large Deformations of Cylindrical Tube made of Micropolar Elastic Material
The generalized Lame problem for an elastic hollow circular cylinder at large deformations is considered. The cylinder is made of micropolar material and contains continuously distributed dislocations. The dislocation density tensor contains four nonzero components and describes the distribution of both screw and edge dislocations. Under the assumption that the material is isotropic, the problem is reduced to a system of nonlinear ordinary differential equations. For a special material model and an axisymmetric distribution of edge dislocations, an exact solution in an explicit analytical form is found.
Leonid M. Zubov, Evgeniya V. Goloveshkina
Recent Approaches in the Theory of Plates and Plate-Like Structures
Prof. Dr. Holm Altenbach
Prof. Dr. Svetlana Bauer
Prof. Dr. Victor A. Eremeyev
Prof. Gennadi I. Mikhasev
Prof. Dr. Nikita F. Morozov
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