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.     

The three-dimensional problem for an anisotropic plate of arbitrary thickness

By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external forces. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 11–19.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

Dirichlet problem for the static elasticity equations outside curvilinear cracks on the plane

We study the Dirichlet problem for the static elasticity equations outside several curvilinear cracks on the plane. The existence and uniqueness of a classical solution are proved. We obtain an integral representation of the solution in the form of potentials whose densities are determined from a uniquely solvable system of Fredholm integral equations of the second kind. We analyze the singularities of the derivatives of the solution at the endpoints of open curves and obtain a closed-form solution of the problem for the case in which all cracks are located along segments of one and the same straight line.     

Inverse problems of the plane theory of elasticity : PMM vol. 38, n 6, 1974, pp. 963–979

In connection with the fact that failure of a structure ordinarily starts at sites of the most acute stress concentrations near cavitiies, it is of interest to determine the shape of the equally strong outlines of holes on which the technologically inevitable stress concentration would be least as compared with all other outlines.An effective exact solution of some inverse plane problems of the theory of elasticity concerning the determination of equally strong outlines of holes is proposed. A formulation of the problem is given first and the fundamental relationships are presented. Then the general problem for any number of holes in an infinite plane is reduced to a standard Dirichlet problem for the exterior of the same number of parallel slits on a parametric plane. An effective exact solution is found by this method for the case of one and two holes as well as for the case of periodic and doubly-periodic series of holes. The question of application of the solutions obtained to the theory of a minimum weight structure is considered.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

Triple trigonometric integral equations with an application

Closed form solution has been obtained for certain triple integral equations with trigonometric kernels which arise in the crack and punch problems in the linear theory of elasticity. As an application solution is obtained for a two dimensional punch problem.     

On the approximate solution of some integral equations of the theory of elasticity and mathematical physics PMM vol. 31, no. 6, 1967, pp. 1117–1131

In this paper the development of the method presented in [1] is carried out with application to two types of integral equations encountered in mathematical physics in the investigation of many mixed problems with circular separation line of boundary conditions and in the investigation of plane mixed problems.

The algorithm is given for reducing these integral equations to solution of equivalent infinite linear algebraic systems. It is proved that the resulting infinite systems are quasi completely regular for sufficiently large values of dimensionless parameter A which enters into the systems. It is shown that reduction (truncation) of infinite systems results in finite systems of linear algebraic equations with almost triangular matrices. The last circumstance simplifies considerably the solution of these finite systems after which the solution of initial integral equations is found from simple equations. For given accuracy of the approximate solution and decrease of parameter λ the number of equations in reduced systems increases.

As an example the solution is presented for the axisymmetric problem of a die acting on an elastic layer lying without friction on a rigid foundation.     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.     

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.     

Solution of three-dimensional problems of the theory of elasticity for a picewise homogeneous medium with constant poisson''s ratio : PMM vol. 43, no. 6, 1979, pp. 1122–1125

Singular integral equations of the theory of elasticity are studied for a piecewise homogeneous medium with the same Poisson's ratio. It is shown that a solution can be obtained using the method of successive approximations. Use of the potential method for the fundamental problems of the theory of elasticity leads to singular integral equations of second kind [1], In the case of the second internal and external problems, and of the first internal problem, the spectral properties of the integral operators allow the use of the method of successive approximations to obtain a solution.     

Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity

This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved.     

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.     

Periodic set of the interface cracks with contact zones

A closed form solution to the plane problem of the theory of elasticity for an infinite isotropic bimaterial space (plane) with a periodic set of the interface cracks with frictionless contact zones near its tips is obtained. By means of the complex function presentation the problem is reduced to the combined Dirichlet–Riemann boundary value problem for a sectionally–holomorphic function and solved exactly. The equations for the determination of the contact zone length as well as the closed form expressions for the stress intensity factors are carried out. The variation of the mentioned values with respect to the distance between the cracks is illustrated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A periodic system of parallel slits with contacting edges in a distended plate

We consider the problem of distention of an infinite plate containing a periodic system of parallel slits whose edges make contact in one of its face planes. We take account of the local bending of the plate in a neighborhood of the defects. On the basis of an approximate analytic solutions and a numerical solution of the singular integral equation of the problem we study the influence of the period of location of the defects on the size of the slit openings and the distribution of the reaction in the contacting edges. We compute the stress intensity factors and moments and determine the destructive load. We give a comparison of the results obtained with the known solution of the periodic problem for parallel slits with load-free edges. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 40–45.     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.     

Analysis of stresses in an infinite elastic body with a locally curved covered nanofiber

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for determining the stress distribution in nanocomposites with unidirectional locally curved covered nanofibers is used to investigate the normal stresses acting along nanofibers. The investigation is carried out for an infinite elastic body containing a single locally curved covered nanofiber in the case where there exists a bond covering cylinder of constant thickness between the nanofiber and the matrix material. It is assumed that the body is loaded at infinity by uniformly distributed normal forces in the fiber direction. Upon formulation and mathematical solution of the boundary value problem, the boundary form perturbation method is used. Numerical results for the stress distribution in the body and the influence of geometrical nonlinearity on this distribution are presented and interpreted.