Three-dimensional Problem Of Elasticity Theory Research Articles

Variation-Difference Method of Calculation of Layered Rubber-Metal Vibration Isolators Used for Protection of Reinforced Concrete Buildings from Anthropogenic Vibration

In the modern construction complex of the city of Moscow for the protection of buildings and structures from man-made vibration arising from the movement of trains of rail transportation (subway trains, railroad lines and trams). To protect buildings and structures from anthropogenic vibration arising from the movement of rail transport trains (subway trains, railroad lines and streetcars), Moscow uses layered rubber-metal vibration isolators [1]. Most often, to determine their static and dynamic characteristics, the finite element method (FEM) is used, which makes it possible to determine all components of the stress-strain state and frequencies of free oscillations in the loaded state practically for any structural forms of isolators. However, for the most popular software packages that implement FEM, the problem of optimizing the structural shape of the vibration isolator still requires significant time expenditures for multiple changes of the finite element mesh, repeated setting of boundary conditions and implementation of a series of calculations. Only some of the software packages implementing the FEM solve optimization problems of the shape of the product being calculated, most often it is related to foreign software products with universal functionality. Variation-difference method (VDM) is the closest to the finite element method (FEM) in terms of its computational capabilities. It is possible to create program modules that repeatedly and automatically solve three-dimensional problems of elasticity theory taking into account the changed geometry of the vibration isolator: the dimensions of the product, the location of perforations within the rubber layers, as well as the thickness of the rubber layer and other parameters important for obtaining an effective technical solution for vibration isolation of buildings. Further, the article describes the method of implementation of the variational-difference method (VDM) as applied to the solution of the problem of determining the components of the stress-strain state inside a three-dimensional layered vibration isolator with perforations of different sizes having different locations relative to the contour of the vibration isolator, i.e., the solution of the problem of optimizing the three-dimensional shape of the vibration isolator is given.

Numerical analysis of stresses near the vertices of a single and several radial spatial cracks

In problems of the theory of elasticity, there are singular solutions associated with the presence of infinite values of stresses at separate points (or lines), which are called singular. The great interest due to practical importance is those singular points that define the vertices of plane or spatial cracks. The algorithm for the numerical analysis of singular solutions in two-dimensional and three-dimensional problems of elasticity theory is considered. The numerical algorithm is based on extracting from the finite element solution the power law dependence of stresses near the singular points of the considered region, where singular solutions are possible. Based on the found power-law dependences, a conclusion about the absence or presence of a stress singularity and its nature is made. This algorithm is verified for two-dimensional and three-dimensional problems of the theory of elasticity by comparing the stress singularity exponents found by the proposed numerical method and obtained from known analytical and numerical solutions. The considered algorithm is applicable for variants in which the stress behavior in the vicinity of singular points is described by one power law dependence. One of the purposes of the article is to study the stress behavior near the common vertex of several radial spatial cracks in an isotropic material. In this case, the evaluation of stress behavior in the vicinity of the singular points is based on the construction of a correct stress field pattern by the finite element method when approaching the singular points. The results of numerical studies have shown that the highest stress level near the common vertex is observed for configurations with the equal angles between radial cracks.

Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres

In this study, three-dimensional problem of the theory of elasticity (3DPTE) for radially inhomogeneous (INH) transversally-isotropic thin hollow spheres is investigated using the asymptotic integration method. The basic relations and equilibrium equations for radially inhomogeneous transversally-isotropic thin hollow spheres are formed and inhomogeneous solutions (INHSs) and homogeneous solutions (HSs) are constructed. The built solutions completely reveal the qualitative structure of a three-dimensional stress-strain state of radially inhomogeneous transversely-isotropic spheres of small thickness and serve as an effective apparatus for solving boundary value problems, the basis for evaluating existing applied theories and for creating new, more refined applied theories. Asymptotic formulas are obtained that allow the calculation of the three-dimensional stress-strain state of spheres. New solution groups (boundary layer solutions) have been found that are absent in applied theories. The behavior of homogeneous solutions in the inner parts of the sphere and in the vicinity of conical sections has been studied, when the thin-walled parameter of the sphere tends to zero. The nature of the constructed homogeneous solutions is clarified. On the basis of the theoretical analysis, three types of the stress-strain state (SSS) in the radially inhomogeneous transversally-isotropic hollow spheres (RINHTIHSs) are considered: a penetrating stress state, a simple edge effect, and a boundary layer. Finally numerical calculations are made and the influences of inhomogeneity on the stress distributions are investigated.

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. A notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. A closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.

Forced vibrations of orthotropic shells when there is viscous resistance

Forced vibrations of orthotropic shells when there is viscous resistance are considered, when two versions of the spatial boundary conditions are given on the upper face of the shell, and the displacement vector is given on the lower face. The solution of the corresponding dynamic equations of the three-dimensional problem of elasticity theory is obtained by an asymptotic method. The amplitudes of the forced vibrations are determined and it is established that the viscous resistance causes the amplitudes of the forced vibrations to increase in the range of values of natural vibrations, but they remain finite. Boundary-layer type functions are obtained, and characteristic equations for determining the rate of decay of boundary vibrations in the direction from the side surface into the shell are established.

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

Abstract The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.

Methods of analysing ropes. The extension–torsion method

Two new approaches are used for calculating the stress–strain state of a rope and its stiffnesses. The first approach relies on the theory of fibrous composites and Saint-Venant's solution for a cylinder with helical anisotropy. The second approach is based on the solution by the finite element method of the three-dimensional problem of elasticity theory for a solid inhomogeneous cylinder formed by a finite number of elastic fibres arranged in helical lines and connected by a weak filler (in the sense that its Young's modulus is several orders of magnitude less than the Young's modulus of the fibre). The behaviour of the stiffness when the modulus of elasticity of the filler tends to zero is analysed, and the results of the limiting transition are discussed. The numerical results obtained are compared with calculations by other well-known applied theories.

An algorithm for solving three-dimensional problem of elasticity theory for transversally isotropic medium

An algorithm for solving a boundary-value problem of deforming structures made of composite materials is considered. The polynomial solutions of the equation system for the linear theory of elasticity for the anisotropic medium in displacements are used in this algorithm.

On the generalized Mindlin problem

The classical Mindlin paper [1] published in 1936 consists of first and second sections concerning concentrated forces perpendicular and parallel to the boundary of the elastic half-space, respectively. In paper [2], the fundamental solutions obtained by Lord Kelvin and Mindlin (the first section of [1]) are generalized for the case of a three-dimensional elastic wedge where a concentrated force is perpendicular to its edge under different types of boundary conditions at its sides. Below, we present expressions for three Papkovich–Neuber harmonic functions for a wedge that has unstressed sides and where a concentrated force parallel to its edge acts in its middle half-plane. When the opening angle of the wedge corresponds to a halfspace, the expressions for elastic displacements and stresses coincide with formulas from the second section of [1]. Displacements at the wedge edge are calculated as well. The solutions obtained by Boussinesq and Cherutti for an elastic wedge with one side subjected to normal and tangential loads and the other side free of stress are generalized in [3]. The problems of a threedimensional wedge are solved by presenting a harmonic function as the complex Fourier–Kontorovich– Lebedev integral and by reducing the three-dimensional problem of elasticity theory to the Hilbert boundary value problem generalized in the sense of Vekua [4–6].

Minimal requirements on the smoothness of data preserving accuracy of a one-dimensional model of rods

We consider cylindrical rods whose elastic properties smoothly vary along the axis, i.e., the fast change of mechanic characteristics is admitted only in cross directions. In the general case, asymptotically exact estimates for the difference of a solution to the three-dimensional problem of elasticity theory in a narrow cylinder and an approximate solution constructed within the framework of a one-dimensional model of a rod age given. As was shown, under wakened smoothness requirements on the right-hand sides of the problem, the accuracy of approximation becomes lower. Bibliography: 21 titles.

Application of functions of a complex variable to certain three-dimensional problems of elasticity theory

Methods of the theory of functions of a complex variable are applied to problems of the deformation of thin plates of constant or variable thickness considered in three dimensions. To that end, a third complex potential is added to the two complex Kolosov-Muskhelishvili potentials. The components of the displacement vector and the stress tensor are represented in terms of these three complex potentials. The formulations if the problems, characteristic for cases in which complex variables are used in problems of elasticity theory, are investigated.

Solution of three-dimensional problems of elasticity theory using new formulas for harmonic polynomials

Approximate solutions of three-dimensional problems of elasticity theory are sought in the form of linear combinations of vector functions each of which satisfies a differential equation. The linear-combination coefficients are found by energy minimization of the difference between exact and approximate solutions. This can be realized in the first and second basic problems. Simple recursion relations and differentiation formulas for similar harmonic polynomials are obtained. The above-mentioned vector functions are constructed using these formulas and the Trefftz representation. The problem of a truncated pyramid is considered.

Acceleration of the convergence of the method of homogeneous solutions in three-dimensional problems of the theory of plates

A method is proposed for the regularization of the calculation process in investigations of homogeneous solutions of three-dimensional problems of elasticity theory by the method of homogeneous solutions. A qualitative investigation is performed of a three-dimensional compression—tension problem with mixed boundary conditions. Questions are examined of the a priori finding of limit values of the unknowns in an infinite system of equations, of the behavior of the coefficients and the series convergence on the boundary as a function of properties of the functions.

Numerical solution of some three-dimensional problems of elasticity theory by Vekua's method

We consider the solution of the problem of elastic equilibrium of a three-dimensional orthotropic plate in the absence of displacements on the end surfaces under the action of forces applied to the lateral surfaces. The solution of the original problem by Vekua's method is reduced to the solution of a recursive sequence of two-dimensional problems. A numerical solution of these problems is obtained by computer using the finite-difference method. The effect of the number of Legendre polynomials on the accuracy with which the boundary conditions are satisfied is investigated.

Solution of three-dimensional problems of elasticity theory with the aid of re-expansion formulas

Solution of three-dimensional problems of elasticity theory with the aid of re-expansion formulas

Investigations of the Kurpsai dam on the basis of a numerical solution of the three-dimensional problem of elasticity theory

Investigations of the Kurpsai dam on the basis of a numerical solution of the three-dimensional problem of elasticity theory

Construction of a working model in the finite-element method for solution of three-dimensional problems of elasticity theory

Construction of a working model in the finite-element method for solution of three-dimensional problems of elasticity theory

Asymptotic behavior of the solution of elasticity theory problem for shell of positive curvature and small thickness

The state of stress and strain of a shell of positive curvature with one edge subjected to the effect of a sufficiently smooth load applied to the endface surface is studied. The case is investigated when the shell thickness is slight. It is proved that the shell state of stress consists of three parts: 1) the internal state of stress that does not possess the property of decay and encloses all domains of the shell body, 2) the slowly decaying state of stress (simple edge effect of shells), and 3) the rapidly decaying state of stress of boundary-layer type. Asymptotic expansions are presented for the components of states of stress and strain of the types 1), 2) and 3). Boundary conditions are formulated for each part of the solution constructed. A system of “two-dimensional” equations of the refined applied theory of shells is obtained on the basis of the solution of a three-dimensional problem of elasticity theory.

On the problem of a three-dimensional crack in an an isotropic elastic medium

A three-dimensional problem of elasticity theory for a homogeneous and anisotropic medium containing an isolated crack is considered. As a result of using a known analogy between a crack and a Somigliana dislocation, the problem is reduced to the solution of an elliptic pseudo-differential equation in the density of the dislocation moments simulating the crack. The properties of the operator in this equation are investigated. A regular representation of this operator is obtained in the class of sufficiently smooth functions. The possibility of applying the proposed regularization to a numerical solution of the problem is discussed. The structure of the tensor stress intensity factor on the smooth outline of an arbitrary crack in an anisotropic medium is analyzed.