Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

The theory of micropolar thin elastic shells

A boundary-value problem of the three-dimensional micropolar, asymmetric, moment theory of elasticity with free rotation is investigated in the case of a thin shell. It is assumed that the general stress-strain state (SSS) is comprised of an internal SSS and boundary layers. An asymptotic method of integrating a three-dimensional boundary-value problem of the micropolar theory of elasticity with free rotation is used for their approximate determination. Three different asymptotics are constructed for this problem, depending on the values of the dimensionless physical parameters. The initial approximation for the first asymptotics leads to the theory of micropolar shells with free rotation, the approximation for the second leads to the theory of micropolar shells with constrained rotation and the approximation for the third asymptotics leads to the so-called theory of micropolar shells “with a small shear stiffness”. Micropolar boundary layers are constructed. The problem of the matching of the internal problem and the boundary-layer solutions is investigated. The two-dimensional boundary conditions for the above-mentioned theories of micropolar shells are determined.     

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 2: Methods for approximate solutions

This work considers the methods for solving approximately the four types of boundary equations arising when the third initial boundary value problem of the theory of elasticity is solved with the help of retarded elastic potentials. The convergence of these methods is proved.     

Mixed problems of axisymmetric elasticity theory for some bodies of revolution

We consider the problem of axisymmetric elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on the cavity surface and the main mixed problem of axisymmetric elasticity theory for a hyperboloidal layer formed by the two surfaces of a two-cavity hyperboloid of revolution symmetrical about the plane z = O. The problems are solved by the method of p-analytical functions. The solution of the first problem is reduced to solving a Fredholm integral equation of the second kind. We investigate the behavior of the normal stress near the boundary lines. The solution of the second problem is reduced to solving a system of two Fredholm integral equations of the second kind. Existence and uniqueness of the solution is proved for this system.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 88–101, 1989.     

Dynamic Reaction of a Composite Anisotropic Space with Tunnel Holes to the Action of Concentrated Shear Forces

An antiplane stationary dynamic problem of elasticity theory for a two-component anisotropic (orthotropic) space is considered, where one component of the pair is weakened by tunnel holes of arbitrary cross section. The space is subjected to the action of time-dependent harmonic shear forces concentrated along some line and shear stresses applied to the surfaces of the holes. To solve the problem, the fundamental solution for a composite space without holes is preliminarily constructed. Using integral representations for displacements in the composite space with holes, the boundary problem of elasticity theory is reduced to an integral equation of second kind, which is solved numerically by the method of quadratures.     

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 method of analytic functions of several complex variables in temperature problems of the theory of elasticity

Solving a temperature problem of the theory of elasticity with a known thermoelastic potential is reduced to finding scalar- and vector-valued analytic functions of two complex variables that satisfy the boundary condition and are solutions of the basic and adjoint problems of elasticity theory respectively. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol, 40, No. 1, pp. 45–48.     

Stress concentration control in the problem of plane elasticity theory

In engineering practice, one of the important problems is the problem of finding full-strength contours which permits to control stress concentration at the hole boundary. The article addresses the mixed problem of plane elasticity theory for doubly-connected domain with partially unknown boundary conditions. In the presented work the stress state of the given body and full-strength contours were defined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations

The existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations. The arguments are based only upon the positivity of the Green's functions and the Krasnosel'skii fixed point theorem. As an example, a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity.     

Analytical solution to a generalized spectral problem in a method of recalculating boundary conditions for the biharmonic equation

An iterative algorithm with an efficient preconditioner for the numerical solution to an elasticity problem in the approximation of plate theory with mixed boundary conditions is proposed and substantiated. Exact constants of energy equivalence for optimization of the iterative method are obtained. Inversion of the preconditioner is equivalent to the double inversion of a discrete analog of the Laplace operator with Dirichlet boundary conditions.     

Inverse Boundary Value Problem of the Plane Theory of Elasticity

There is given the formulation and the solution of the inverse boundary value problem of the plane theory of elasticity in this paper. The solution is reduced to finding the Taylor coefficients for the function which maps the unit dick onto the unknown domain.     

The boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of the boundaries

The existence and uniqueness of solutions of the boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of their boundaries are investigated in the Besov and Bessel potential classes using the methods of the potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

Asymptotic solution of the first boundary-value problem of the theory of elasticity of the forced vibrations of an isotropic strip

The first boundary-value problem of the theory of elasticity of the forced vibrations of an isotropic strip is solved by an asymptotic method. The asymptotic form of the components of the stress tensor and the displacement vector, which differ in principle from the asymptotic form in the corresponding static problem, is established. All the required quantities in the inner problem are determined and the conditions for resonance to occur are established. The solution in the dynamic boundary layer is constructed and the fundamental (inner) and boundary solutions are matched.     

The formulation of linearized boundary integral equations of the anisotropic theory of elasticity and their application in geometrical inverse problems

An elastic bounded anisotropic solid with an elastic inclusion is considered. An oscillating source acts on part of the boundary of the solid and excites oscillations in it. Zero displacements are specified on the other part of the solid and zero forces on the remaining part. A variation in the shape of the surface of the solid and of the inclusion of continuous curvature is introduced and the problem of the theory of elasticity with respect to this variation is linearized. An algorithm for constructing integral representations for such linearized problems is described. The limiting properties of the linearized operators are investigated and special boundary integral equations of the anisotropic theory of elasticity are formulated, which relate the variations of the boundary strain and stress fields with the variations in the shape of the boundary surface. Examples are given of applications of these equations in geometrical inverse problems in which it is required to establish the unknown part of the body boundary or the shape of an elastic inclusion on the basis of information on the wave field on the part of the body surface accessible for observation.     

Method of successive approximations in the nonlinear theory of viscoelasticity

A method of successive approximations, a generalization of the Il'yushin method of elastic solutions, is proposed for solving problems of the nonlinear theory of elasticity in which the stress-strain relation is given in the form of a time operator Frechet-differentiable in a neighborhood of zero. The nonlinear relaxation kernels are found from the given nonlinear creep kernels for the principal quadratic theory of elasticity. These relations make it possible to formulate the boundary value problem for this theory. By way of illustration the problem of the pressure exerted on a space by a sphere is examined within the framework of the developed theory. The question of the convergence of the method is discussed in relation to the quadratic theory of visco-elasticity.Presented at the Third All-Union Conference on Theoretical and Applied Mechanics, Moscow (January, 1968).Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 236–242, March–April, 1969.     

Solution of the fundamental problems of the theory of elasticity for a half-plane with holes and cracks

We obtain the general solution of the fundamental problems of the theory of elasticity for an isotropic half-plane with a finite number of arbitrarily situated elliptic holes whose boundaries may intersect or form rectilinear cuts or boundaries of curvilinear holes. On the rectilinear boundary the first problem and the second or mixed problem of the theory of elasticity are defined. We use general expressions obtained previously by the author for the complex potentials generated by solving the problem of linear coupling for cuts in a multiconnected region, conformal mappings, and the method of least squares. The problem is reduced to solving a system of linear algebraic equations. The results of numerical experiments are given for a half-plane with a crack in the case of the first fundamental problem and the action of various loads. Two figures, two tables. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 157–171.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

State of stress of nonuniformly heated plates loaded along the boundary surfaces : PMM vol. 43. no. 6, 1979, pp. 1065–1072

The general solution of ati elasticity theory problem for a constant thickness plate is constructed under the condition that a force and a nonuniformly heated plate are applied normally to the boundary planes. The solution is obtained as a result of applying the M.E. Vashchenko-Zakharchenko expansion formulas to the infinitely high-order differential equations obtained by A.I. Lur'e by a symbolic method [1,2], by a separate analysis of the symmetric and antisymmetric elasticity theory problems relative to the middle plane: 1) for constant temperature and given forces on the boundary planes; 2) for a given nonuniform heating and no forces. Simple formulas are presented to determine the state of stress in the case of a slowly varying external load and temperature of the unbounded plate. For a bounded plate the general solution for no forces on the boundary planes and heating resulted in the A.I. Lur'e solution [1].