Boundary-value problems of the asymmetric theory of elasticity for thin plates

Boundary-value problems of the three-dimensional asymmetric micropolar, moment theory of elasticity with free rotation are considered for thin plates. It is assumed that the total stress-strain state is the sum of the internal stress-strain state and the boundary layers, which are determined in an approximation using asymptotic analysis. Three different asymptotic forms are constructed for the three-dimensional boundary-value problem posed, depending on the values of dimensionless physical constants of the plate material. The initial approximation for the first asymptotic form leads to a theory of micropolar plates with free rotation, the initial approximation for the second asymptotic form leads to a theory of micropolar plates with constrained rotation, and the initial approximation for the third asymptotic form leads to a theory of micropolar plates with “small shear stiffness.” The corresponding micropolar boundary layers are constructed and studied. The regions of applicability of each of the theories of micropolar plates constructed are indicated.     

Equivalence of Two Sets of Transition Points Corresponding to Solutions with Interior Transition Layers

We establish the equivalence of two sets of transition points corresponding to solutions of singularly perturbed boundary-value problems with interior boundary layers. The first set appears in the formalism for constructing the asymptotics of the solution of a boundary-value problem and the second, in the direct scheme formalism for constructing the asymptotics of the solution of a variational problem.     

Internal and boundary calculations of thin elastic bodies

A method for the separate construction of the main stress-strain state (the internal calculation) and the boundary corrections (the boundary calculations) are discussed in the case of a linear static problem in the theory of shells and plates. It is assumed that the internal calculation is carried out using an iterative process based on the Kirchhoff-Love theory. The boundary calculation involves the construction of antiplane and plane boundary layers, that is, in the initial approximation they reduce to the solution of antiplane and plane problems in the theory of elasticity.

Investigation of the asymptotic behaviour of the boundary corrections shows that near a weakly clamped edge only the correction from the antiplane boundary layer is important and that near a fairly rigidly clamped edge only the correction from the plane boundary layer is important.

The advisability of the use of the shear theory of the bending of plates for investigating boundary elastic phenomena is discussed from the point of view of the results obtained. It is shown that, close to the free edge, its use is justified and is adequate for the method described in the paper both with regard to the numerical results and with regard to the nature of the mathematical apparatus. As a method for investigating boundary elastic phenomena, shear theories lose their meaning close to a fairly rigidly clamped edge since they only enable one to construct the minor part of the correction asymptotically.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.     

Surface and internal waves in a stratified layer of liquid and an analysis of the impedance boundary conditions

The surface and internal waves in a multilayered ideal liquid for specified displacements of the bottom are considered. The upper surface of the liquid is either free or displacements are specified on it. In the long-wave approximation, asymptotically accurate models of the waves propagating along the surface layers of an incompressible liquid are constructed both for intense stratification (the ratio of the densities of neighbouring layers are associated with a small parameter) and for weak stratification. The important case of this problem of the dynamic contact of rigid bodies through a layer of incompressible or compressible liquid is investigated. High-order impedance boundary conditions are constructed and the results of testing them using the exact solutions are presented.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Boundary stabilization of vibration of nonlocal micropolar elastic media

In this paper, we study the stabilization problem of vibration of linearized three-dimensional nonlocal micropolar elasticity. For this purpose, we need to demonstrate the well-posedness of the system of equations governing the vibration of three-dimensional nonlocal micropolar media for both forced (i.e. with boundary feedback) and unforced cases. We assume the non-homogeneous system of equations for the unforced (uncontrolled) case to establish the well-posedness. It should be pointed out that the well-posedness of the evolution equations in micropolar case has been studied by many authors; but, the well-posedness in the nonlocal micropolar is an open problem. Our tools in well-posedness analysis are the semigroup techniques. Afterwards, we pursue the stabilization problem and show that the vibration of the nonlocal micropolar elastic media will be eventually dissipated under boundary feedback actions consisting of stress and couple stress feedback laws. These control laws are simple, linear and can be easily implemented in practical applications. The stabilization proof is accomplished using Lyapunov stability and LaSalle’s invariant set theorems.     

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

Generalization of the boundary function method for solving boundary-value problems for bisingularly perturbed second-order differential equations

A bisingular boundary-value problem for an ordinary differential equation is considered. The asymptotics of the solution as the sum of an outer expansion and an analog of a number of functions of the boundary layer is constructed.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

Dual variational problems for boundary functionals of the linear theory of elasticity

Dual variational problem for use with the problem of minimization of the boundary functionals of three-dimensional theory of elasticity, is formulated using the method of orthogonal expansions at the boundary of the region constructed in /1/. Solutions of the initial and the dual problem obtained yield the estimates for the error of the approximate solutions of the boundary value problems of the theory of elasticity.     

A mixed boundary-value problem for the wave equation in a stratified medium for high-frequency oscillations

A boundary-value problem for the wave equation in a stratified medium with mixed boundary conditions on the boundary in the case of high oscillation frequencies is considered. The Helmholtz equation for a velocity function increasing monotonically with depth is investigated. The problem is reduced to an integral equation in the high-frequency approximation, and an explicitly smooth term of its asymptotic solution is constructed.     

A new version of boundary integral equations and their application to dynamic three-dimensional problems of the theory of elasticity

A version of boundary integral equations of the first kind in dynamic problems of the theory of elasticity is proposed, based on an investigation of the analytic properties of the Fourier transformant of the displacement vector, rather than on fundamental solutions. A system of three boundary integral equations of the first kind with Fredholm kernels is constructed, and the equivalence of the initial boundary-value problem on the vibrations of a bounded region and the system of boundary integral equations obtained is investigated. A version of the numerical realization, which combines the ideas of the classical method of boundary elements and the Tikhonov regularization method, is proposed. The results of numerical experiments are given.     

The equilibrium boundary element method in boundary-value problems of elasticity theory

An equilibrium boundary element method is proposed for solving boundary-value problems in the theory of elasticity, thermo-elasticity, the dynamical theory of elasticity, bar torsion calculations, and the bending of a plate. The idea is to use simultaneously the method of constructing bundles of functions which exactly satisfy the equilibrium equations, the boundary variational equations of mechanics, and the methods of discrete finite-element approximation. The variational method of constructing the resolving boundary equations ensures that the linear system is symmetric and easily coupled to the finite-element method. Since volume integrals are eliminated the dimensions of the problem are reduced by one, but, unlike the boundary element method, there is no need to know the fundamental solutions. The solution of some bar torsion and plate bending problems confirms the high numerical efficiency of the method.     

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.     

The Dirichlet problem in a domain with a slit

A basis of harmonic wavelets is constructed in an elliptic ring, and its approximation properties are investigated. The results are used to analyze the behavior of a boundary-value Dirichlet problem under the contraction of the inner boundary of the ring to a segment.     

Interpolating-orthogonal wavelet systems

A basis of harmonic wavelets is constructed in an elliptic ring, and its approximation properties are investigated. The results are used to analyze the behavior of a boundary-value Dirichlet problem under the contraction of the inner boundary of the ring to a segment.     

Numerical Simulation of Free Oscillations of Elastic Bodies with a Thin Coating

We propose an approach to the investigation of problems on free oscillations of elastic bodies with a thin coating. The method consists of applying a combined mathematical model which is based on the three-dimensional equations of elasticity theory in the domain of a body and on the two-dimensional equations of the theory of shells of the Timoshenko type in the domain of a thin coating. The systems of these equations are related by the conditions of conjugation on the surface of contact. For the numerical analysis of the eigenvalue problem, we used a scheme of the finite-element method constructed by using approximations of different dimensionality.