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.     

Solving some problems of elasticity theory by the integral equation method for a holomorphic vector

Under study are some problems of elasticity theory with nonclassical boundary value conditions. We assume that the load and displacement vectors are given on a part of the boundary, while on the other parts of the boundary, the load vector or the displacement vector may be given separately, and no conditions are imposed on the remaining part of the surface (of some nonzero measure).We consider the questions of uniqueness for the solutions to these problems. Solving the nonclassical problems is reduced to a system of singular integral equations for a holomorphic vector.     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type

This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.     

Two boundary value problems for a strongly anisotropic inhomogeneous elastic ring

The second boundary value problem (displacements are given on the boundary) and the improper mixed problem for a cylindrically orthotropic ring are studied. It is assumed that the coefficients of elasticity are continuously differentiable functions of the coordinates and depend on a small parameter in a specific manner. The form of the dependence of the coefficients on the small parameter is selected in such a way that in the case of constant coefficients it describes bonding of the ring by two families of very rigid fibers located along the radius vectors and concentric circles, where the stiffness of the fiber families is of identical order. Consequently, the coefficients of elasticity are represented in the form of products of constants which will later be called provisionally the “stiffnesses”, and functions of the coordinates. It is assumed that the stiffnesses in the radial and circumferential directions are equal and exceed and shear stiffness considerably. The asymptotic form of the solution of the boundary value problems under consideration is constructed when the ratio between the shear stiffness and the stiffness in the radial direction is used as the small parameter. In the case of the second boundary value problem the limit boundary value problem is described by a hyperbolic system of equations and is not solvable uniquely, since one of the families of characteristics is parallel to the boundary. When constructing the asymptotic form the necessity arises to average the coefficients of elasticity with respect to the circumferential coordinate. In this respect, there is an analogy with the results obtained in /1/ where the boundary value problem was studied for a second-order elliptic equation.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ?. The limit problem as ? → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.     

Vibrations of a thick orthotropic disk with inextensible face coatings

We propose a theoretical method of studying the three-dimensional dynamic problems of the theory of elasticity for plates on whose planar faces mixed homogeneous boundary conditions are imposed on the normal stresses and displacements in the plane of the faces. The method is based on the theory of dynamic homogeneous solutions of exponential type. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, pp. 108–114.     

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.     

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.     

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.     

Application of the pseudo-spectral method to 2D eigenvalue problems in elasticity

A pseudo-spectral approach to 2D vibrational problems arising in linear elasticity is considerede using differentiation matrices. The governing partial differential equations and associated boundary conditions on regular domains can be translated into matrix eigenvalue problems. Accurate results are obtained to the precision expected in spectral-type methods. However, we show that it is necessary to apply an additional “pole” condition to deal with ther=0 coordinate singularity arising in the case of a 2D disc.     

A variational method for the solution of biharmonic problems for a rectangular domain

We develop a variational method for the solution of biharmonic problems for a rectangular domain where, at one pair of its opposite sides, the unknown function and its normal derivative take zero values, and, at the other pair, certain inhomogeneous conditions are valid. The cases of semiinfinite and finite domain are considered. The method is based on the minimization of a quadratic functional determining the deviation of the solution from the given inhomogeneous conditions in the norm of L ₂. To solve this variational problem, we apply the expansion of the solution in the systems of complex biharmonic functions (the so-called Papkovich homogeneous solutions), which satisfy identically the given homogeneous conditions at the pair of opposite sides of the rectangle. This representation of the solution is somewhat different from that proposed earlier [V. F. Chekurin, “A variational method for the solution of direct and inverse problems of the theory of elasticity for a semiinfinite strip,” Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, No. 2, 58–70 (1999)]. We consider several variants of inhomogeneous boundary conditions arising in the problems of the two-dimensional theory of elasticity. Finally, we give an example of applying the proposed method for the determination of stress distributions in a rectangular area one of whose sides is rigidly fastened and the opposite one is subjected to the action of normal forces. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 88–98, January–March, 2008.     

Transient boundary element formulation for linear poroelasticity

The transient boundary integral equations for linear, isotropic poroelasticity are derived from a reciprocal theorem. Green's functions needed in the integral equations are found by a variable transformation technique originally proposed by Biot. This technique seperates the displacement field into an undrained part satisfying the Navier equation of elasticity and an irrotational part governed by a diffusion equation. Fundamental expressions of an instantaneous point force and a fluid dilatation are obtained in two and three dimensions. The resultant transient integral equations can be numerically implemented in a boundary element procedure for the solution of boundary value problems in poroelasticity.     

Localization near the corner point of the principal eigenfunction of the Dirichlet problem in a domain with thin edging

We establish that the principal eigenfunction of the Dirichlet problem in a domain with a thin heavy edging admits localization near the corner point of opening angle α > π. The edging amounts to a boundary strip of small width ɛ with the density function ɛ ^−2−m , m > 0, while it is O(1) in the remaining part of the domain. We derive the result by analyzing the essential and discrete spectra of an auxiliary problem in an infinite angle without the small parameter. We state several open questions about the structure of spectra of both problems.     

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.     

Global stress regularity of convex and some nonconvex variational problems

We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).      

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.