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.     

On generalized formulation of the equilibrium problem of an elastic strip

A “nonenergetic” formulation of the boundary value problems of statics of an elastic strip based on the principle of admissible displacements, is studied. The formulation makes possible, in particular, the study of problems concerning the strips of infinite energy, while retaining the external form of the “energetic” formulation /1–3/, and produces unique solvability of the problem under weaker restrictions imposed on the external loads. Such a formulation is also possible for other problems of the theory of elasticity.     

A Variational Approach for the Mixed Problem in the Elastostatics of Bodies with Dipolar Structure

In this study, we address the mixed initial boundary value problem in the elastostatics of dipolar bodies. Using the equilibrium equations, we build the operator of dipolar elasticity and prove that this operator is positively defined even in the general case of an elastic inhomogeneous and anisotropic dipolar solid. This helps us to prove the existence of a generalized solution for first boundary value problem and also the uniqueness of the solution. Moreover, relying on this property of the operator of dipolar elasticity to be positively defined, we can apply the known variational method proposed by Mikhlin.     

Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type

We study the solvability of a boundary value problem for a system of nonlinear second-order partial differential equations under given boundary conditions, which describes the equilibrium of elastic shallow shells with hinged edges in the framework of the Timoshenko shear model. The study method implies the reduction of the original system of equations to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.     

On one contact problem of plane elasticity theory with partially unknown boundary

Applications of elastic plates weakened with full-strength holes are of great interest in several mechanical constructions (building practice, in mechanical engineering, shipbuilding, aircraft construction, etc). It's proven that in case of infinite domains the minimum of tangential normal stresses (tangential normal moments) maximal values will be obtained on such contours, where these values maintain constant(the full strength holes). The solvability of these problems allow to control stress optimal distribution at the hole boundary via appropriate hole shape selection. The paper addresses a problem of plane elasticity theory for a doubly connected domain S on the plane z = x + iy, which external boundary is an isosceles trapezoid boundary; the internal boundary is required full-strength hole including the origin of coordinates. In the provided work the unknown full-strength contour and stressed state of the body were determined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of stresses and deformation properties of rock masses which is based on the solution of an inverse problem from the measurement data of the free surface displacement

A method is developed of interpreting the measurement data of the variations of displacements and deformations on the free surfaces caused by carrying out mining works. It allows us to quantitatively estimate the horizontal components of the natural stress fields and elastic characteristics of the constructive elements of the excavation technologies of solid mineral deposits. The method is based on the solution of inverse problems for the system of elasticity equations in the domains of an arbitrary shape. The structure of the proposed target functions is studied; the optimal values are determined of the weight coefficients that provide solvability of the inverse problem as well as the limit level of error in the input data of various types is estimated.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

A nonlinear elasticity model of macromolecular conformational change induced by electrostatic forces

In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson-Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existence of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small. The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods.     

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 1. Existence theorems

This work deals with four variants of boundary equations appearing if the third main initial boundary value problem is solved with the help of elastic retarded potentials. The solvability of these equations is proved, and the smoothness of their solutions researched.     

Boundary Equations in Dynamic Problems of the Theory of Elasticity

We study various nonstationary boundary equations that arise when the mixed problems of the theory of elasticity and thermoelasticity are solved by means of dynamic surface potentials. In the review, the results obtained are presented on their unique solvability in the scales of function spaces of the Sobolev type.     

On the existence of optimal shapes in contact problems

The optimal shape design of a two-dimensional elastic body on a rigid frictionless foundation is analyzed. The problem is to find the boundary part of the body where the unilateral boundary conditions are assumed, in such a way that the total energy of the system in the equilibrium state will be minimized. The solvability of the problem is proved.     

Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.     

工程中一类拼接问题的复变方法

讨论工程中一类含边裂缝弹性材料补强的拼接问题 .根据平面弹性复变方法 ,问题归结为一类解析函数的边值问题 ,通过有效的分析方法和积分变换 ,进一步将问题简化为一类奇异积分方程 ,证明了方程解的存在唯一 ,并对方程解的简化进行了研究 ,得到了弹性材料体内应力分布的封闭形式解 ,并导出一直裂缝情况裂缝尖端应力强度因子的表达式     

On the solvability of a boundary value problem for a fourth-order equation on a graph

We consider a boundary value problem for a fourth-order equation on a graph modeling elastic deformations of a plane rod system with conditions of rigid connection at the vertices. Conditions for the unique solvability are stated. We also present sufficient conditions for the problem to be degenerate.     

GLOBAL SOLVABILITY OF NONLINEAR WAVE EQUATION WITH A VISCOELASTIC BOUNDARY CONDITION

The paper deals with the global solvability of nonlinear wave equation with a viscoelastic boundary condition. The problem is a mathematical mode for nonlinare one-dimensional motion of an elastic bar connected with a viscoelastic element at one end of the bar. Under some physically reasonable assumptions, the boundary condition is dissipative and the existence of global smooth solution of the problem is proved for small data.     

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile

In this paper we consider a general mathematical model for the collision between the free-fall hammer of a pile-driver and an elastic pile whose ends are furnished with a bearing. When the free-fall hammer collides with the pile, the displacement of a cross-sectional area of the pile is the weak solution of an initial-boundary value problem involving a linear wave equation with memory boundary conditions. We generalize this problem into a nonlinear one with more general boundary conditions. Then we obtain the unique solvability and the regularity of the weak solution of this nonlinear problem. The unique solvability is shortly discussed in regard to the Galerkin method. The regularity result is obtained by a combination of a fixed-point technique and an energy method, and the convenience of this procedure is also pointed out.     

A rigid inclusion in the contact problem for elastic plates

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.     

Optimal design of an elastic beam with a unilateral elastic foundation: Semicoercive state problem

A design optimization problem for an elastic beam with a unilateral elastic foundation is analyzed. Euler-Bernoulli’s model for the beam and Winkler’s model for the foundation are considered. The state problem is represented by a nonlinear semicoercive problem of 4th order with mixed boundary conditions. The thickness of the beam and the stiffness of the foundation are optimized with respect to a cost functional. We establish solvability conditions for the state problem and study the existence of a solution to the optimization problem.     

A Boundary Integral Formulation for Poroelastic Materials

Wave propagation in porous media is an important topic, e.g. in geomechanics or the oil-industry. We formulate a linear system of coupled partial differential equations based on Biot's theory with the solid displacements and the pore pressure as the primary unknowns. To solve this system of coupled partial differential equations in a semi-infinite homogeneous domain the BEM (Boundary element method) is especially suitable. Starting from a representation formula a system of two boundary integral equations is derived. These boundary integral equations are used to solve related boundary value problems via a direct approach. Coercivity of the resulting bilinear form is shown, from which unique solvability of the variational formulation follows from injectivity. Using these results we derive the unique solvability of the related boundary integral equations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and uniqueness of weak solutions for a thin plate with elastic boundary conditions

Robin-type problems are studied for thin elastic plates with transverse shear deformation. These problems are reduced to analogous ones for the corresponding homogeneous equilibrium equation, whose solutions are then represented as single and double layer potentials. The unique solvability of the systems of boundary integral equations yielded by this procedure is discussed in Sobolev spaces.