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.     

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.     

On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 156–160.     

Invariant integrals in a planar problem of elasticity theory for bodies with rigid inclusions and cracks

The article addresses a planar problem of elasticity theory for a body containing a rigid inclusion and a crack at the interface between the elastic matrix and the rigid inclusion. We show that the problem admits J- and M-invariant integrals. In particular, we construct an invariant integral of the Cherepanov-Rice type for rectilinear cracks.     

Algorithms of the asymptotic construction of linear two-dimensional thin shell theory and the st venant principle

A modified St Venant principle is formulated, governing the decay of the asymptotically dominant part of the stress-strain state due to a system of forces applied to an edge of a thin elastic body Four conditions for the satisfaction of the modified St Venant principle are derived and the possibility of using them to construct iterative processes for integrating the general equations of the theory of elasticity is established.     

Modeling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits

This paper is the second part of a work devoted to the modelling of thin elastic plates with small, periodically distributed piezoelectric inclusions. We consider the equations of linear elasticity coupled with the electrostatic equation, with various kinds of electric boundary conditions. We derive the corresponding effective models when the thickness a of the plate and the characteristic dimension ϵ of the inclusions tend together to zero, in the two following situations: first, when a ≃ ϵ, and second, when a ∕ ϵ tends to zero. Copyright © 2013 John Wiley & Sons, Ltd.     

The nonlocal theory of elasticity and its applications to the description of defects in solid bodies

The nonlocal theory of elasticity takes account of remote action forces between atoms. This causes the stresses to depend on the strains not only at an individual point under consideration, but at all points of the body. The stresses caused by defects in a nonlocally elastic medium have no nonphysical singularities, in contrast to the corresponding solutions obtained in the classical theory of elasticity. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 90–96.     

Application of singular integral equations in two-dimensional problems of the theory of elasticity for piecewise-uniform bodies with cracks

Using the method of singular integral equations we solve a two-dimensional problem of the theory of elasticity for an infinite plate containing an elastic inclusion of arbitrary configuration and a system of curvilinear incisions. The numerical solution is found by the method of mechanical quadratures for the case of an elliptic inclusion and a single polygonal crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 93–98.     

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

The nonsteady-state planar oscillations of a plate and oscillator on a two-layer base

We consider a plane dynamic contact problem for an inhomogeneous base of the following form: a soft layer on a rigid layer of an elastic half-plane. The layer is represented by a Winkler model, corresponding to the long-wave asymptotics of the equations of elasticity theory. The problem is reduced to a system of integro-differential equations that is solved numerically. We present the results of the computations of dynamic characteristics describing the oscillations of a rigid body and an oscillator on this base.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 64–68.     

Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.     

Some Boundary — Contact Problems of the Elasticity Theory with Mixed Boundary Conditions Outside the Contact Surface

n — Dimensional (n ≥ 2) boundary-contact problems of statics of the elasticity theory for homogeneous anisotropic media are investigated when the contact of two bounded domains occurs from the outside on some part of boundaries with mixed boundary conditions. Theorems on the existence and uniqueness of solutions of boundary-contact problems in Besov and Bessel potential spaces are obtained. The smoothness of solutions is studied in closed domains occupied by elastic media.     

Pseudo-Rigid Bodies: A Geometric Lagrangian Approach

The pseudo-rigid body model is viewed within the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians, the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on so(3)×so(3) and a second-order differential equation on the set of diagonal matrices with a positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas cloud and Riemann ellipsoids are presented.     

Solution of problems of the theory of elasticity for a half-space with planar boundary cracks

We propose a method of solving three-dimensional problems of the theory of elasticity for a half-space containing planar boundary cracks. The problem is reduced to a system of integro-differential equations for determining the functions that characterize the opening of the crack during deformation of the halfspace. The kernels of the equations, besides having poles, also have a fixed singularity at the points of intersection of the surface of the crack with the boundary of the half-space. The equations obtained are solved numerically for the case of cracks that are part of a circular region. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 58–63.     

Rotating elastic bodies in Einstein gravity

We prove that, given a stress‐free, axially symmetric elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary, axisymmetric solution to the Einstein equations coupled to the equations of relativistic elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency. © 2009 Wiley Periodicals, Inc.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.      

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.