Solution of two-dimensional problems of elasticity and thermoelasticity for a rectangular region

We propose a separation-of-variables method for the biharmonic equation and construct a complete system of orthogonal functions for constructing exact solutions in the form of non-periodic trigonometric series for two-dimensional problems of the theory of elasticity and thermoelasticity for a rectangular region. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 19–25.     

Relaxation of variational problems in two-dimensional nonlinear elasticity

Consider a plate occupying in a reference configuration a bounded open set Ω ⊂ ℝ ² , and let be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:

Homogenization of two-dimensional elasticity problems with very stiff coefficients

In this paper we study the asymptotic behaviour of a sequence of two-dimensional linear elasticity problems with equicoercive elasticity tensors. Assuming the sequence of tensors is bounded in L¹, we obtain a compactness result extending to the elasticity the div-curl approach of [M. Briane, J. Casado-Díaz, Two-dimensional div-curl results. Application to the lack of nonlocal effects in homogenization, Comm. Partial Differential Equations 32 (2007) 935-969] for the conduction. In the periodic case this compactness result is refined replacing the L¹-boundedness by a less restrictive condition involving the oscillations period. We also build a sequence of isotropic elasticity problems with L¹-unbounded Lamé's coefficients, which converges to a second gradient limit problem. This loss of compactness shows a gap in the limit behaviour between the very stiff problems of elasticity and those of conduction. Indeed, in the conduction case a compactness result was proved in [M. Briane, J. Casado-Díaz, Asymptotic behaviour of equicoercive diffusion energies in dimension two, Calc. Var. Partial Differential Equations 29 (4) (2007) 455-479] without assuming any bound from above for the conductivities.     

Some equilibrium finite element methods for two-dimensional elasticity problems

Summary We consider some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL ₂-estimates of orderO(h ²) for both stresses and displacements.     

Concerning existence theorems for some problems of elasticity theory

We prove an existence theorem for the statistical elasticity theory equation for a homogeneous incompressible medium and its extension to the second and third boundary value problem case. We demonstrate, in the case of the first, second, and third problems that, as the solution of the elasticity theory equation with Lamé constants and converges to the solutions of the respective equations for incompressible material. An existence theorem in the rectangle is demonstrated for the third boundary value problem inw _q ² .Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 599–609, April, 1975.     

Solution techniques for some allocation problems

This paper presents methods for solving allocation problems that can be stated as convex knapsack problems with generalized upper bounds. Such bounds may express upper limits on the total amount allocated to each of several subsets of activities. In addition our model arises as a subproblem in more complex mathematical programs. We therefore emphasize efficient procedures to recover optimality when minor changes in the parameters occur from one problem instance to the next. These considerations lead us to propose novel data structures for such problems. Also, we introduce an approximation method to solve certain equations, which arise during the procedures.     

Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems

The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.     

Construction of fundamental solutions of two-dimensional problems of the nonstationary theory of elasticity

We propose a method of constructing the images of the fundamental solutions in the space of the Laplace transform with respect to time, leading to simple formulas. The method is illustrated using three dynamical problems: planar deformation for an anisotropic body; flexural vibrations of an anisotropic plate; and vibrations of a shallow isotropic shell of arbitrary Gaussian curvature. Quadrature formulas are given for computing the values of the fundamental solutions. We give a new interpretation and a new method of computing the values of the special functions used in the construction of singular solutions in problems of the static theory of shells. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 86–92.     

On some representations of solutions of dynamical problems in elasticity theory

The divergence and rotation of a translation vector field, which are solutions of wave equations, are used as auxiliary potentials for numerical solution of dynamical problems in elasticity theory. In a particular case, the problems of longitudinal translation are tested in two variants of numerical algorithms with their programming realization, some suggestions are given regarding the choice of the optimal algorithm for achieving highly precise calculations with a small number of discretization points.Translated from Dinamicheskie Sistemy, No. 9, pp. 27–33, 1990.     

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 two-dimensional quasistatic problems of thermoelasticity for multilayered bodies

We propose a new method of solving two-dimensional quasistatic problems of thermoelasticity for isotropic multilayered bodies. The method is based on the applicaiton of generalized functions. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 49–52.     

Numerical solution of some three-dimensional problems of elasticity theory by Vekua's method

We consider the solution of the problem of elastic equilibrium of a three-dimensional orthotropic plate in the absence of displacements on the end surfaces under the action of forces applied to the lateral surfaces. The solution of the original problem by Vekua's method is reduced to the solution of a recursive sequence of two-dimensional problems. A numerical solution of these problems is obtained by computer using the finite-difference method. The effect of the number of Legendre polynomials on the accuracy with which the boundary conditions are satisfied is investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 77–84, 1986.     

Solution of one-dimensional problems of elasticity and thermoelasticity in stresses for a cylinder

We propose a method of solving problems of elasticity and thermoelasticity in stresses, making it possible to give a simpler construction of the solutions of such one- dimensional problems for a cylinder, compared with the known method of solving in the displacements. We obtain a relation between the components of the stress tensor and the integral conditions of equilibrium and continuity that has important applications in the solution of inverse problems of thermoelasticity.Translated fromMatematichni Metodi ta Fiziko-Mechanichni Polya, Vol. 40, No. 3, 1997, pp. 103–107.     

Solution of the fundamental problems of elasticity theory for a multiconnected anisotropic half-plane

By analytic continuation of generalized complex potentials to upper half-planes we reduce the boundary conditions on a rectilinear boundary to problems of linear coupling for cuts of a multiconnected extended plane. By solving the latter problems we obtain general representations of the complex potentials in the case of a multiconnected anisotropic half-plane for different types of boundary conditions on intervals of the rectilinear boundary. As particular cases we give expressions for the complex potentials in the cases of action of external forces on the rectilinear boundary and dies both with and without friction. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 44–63.     

Solution of some inverse problems in nonequilibrium sorption dynamics

The paper considers inverse problems of sorption dynamics that determine the sorption isotherm from the output curve. Uniqueness theorems are proved for these inverse problems. Numerical solution methods are constructed. Some applications to model problems are described.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 5–15, 1986.     

Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems

Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.

     

Accuracy of solution of some boundary-value problems of elasticity theory by finite-difference methods

A system of nonlinear equations associated with finite-difference solution of problems of mathematical physics is considered. A so-called bounding linear system is constructed for the given nonlinear system. The solution accuracy of the bounding system is estimated by the magnitude of the discrepancy vector, which provides an estimate of the solution accuracy of the nonlinear system for the worst-case values of the nonlinear coefficients. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 24–28, 1986.     

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.Presented at the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Timoshenko Institute of Mechanics, Ukranian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 465–472, July–August, 2000.