The stressed state of an orthotropic plate with a two-sided external groove under pure bending

We use the Fourier method with curvilinear coordinates to obtain a closed-form solution of the problem of the stressed state of an infinite orthotropic plate bounded by the two branches of a hyperbola under pure bending. We determine the stresses in the plate, in particular, at its narrowest point. Three figures. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika No. 27, 1997, pp. 72–81.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

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.     

An analysis of the thermal stresses in glass elements of electrovacuum devices in the neighborhood of metallic inclusions

On the basis of the improved theory of plates, which makes it possible to determine all the components of the stress tensor, we study the thermally stressed state of a finite round plate with a concentric inclusion of a different material. We obtain the exact analytic solution of the corresponding boundary-value problem for the system of singularly perturbed equations, valid for any ratios between the diameters of the plate and the inclusion. In the contact zones the stresses differ significantly (both quantitatively and qualitatively) from those predicted by the classical plate theories. The results obtained make it possible to draw a number of conclusions that are useful for estimating the strength of electrovacuum devices.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 1–6.     

The stressed state in a plate caused by the distortion tensor

The improved plate theory, which makes it possible to determine all the components of the stress tensor, is generalized to the case when the stressed state of a plate is caused by the distortion tensor. The equations obtained can be useful in the study of residual stresses in thinwalled elements, in applications of the method of distortion in the mechanics of failure and elsewhere.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 53–57.     

Algorithms for ordering unorganized points along parametrized curves

Letn points be given on several arcs of curve. The equations of these curves are unknown. We give two algorithms, based on geometrical criteria, that allow to determine an order on these points: the order given by the curvilinear abscissa. The computational complexity of these algorithms is inO(n logn) and the space complexity inO(n).     

Displacement of a rigid rod in a viscous medium

The spatial displacement of an absolutely rigid vertical rod in a viscous medium is considered; the rod is hinged at the upper end to a platform moving on the surface of the viscous medium over a specified curvilinear trajectory. A load is attached to the lower end of the rod. Using a variational Lagrangian equation, a nonlinear system of ordinary differential equations in terms of the angles of rod rotation determining the position of points of this rod at any time is obtained. As an example, the problem is solved for conditions of acceleration, uniform motion, and deceleration of the reference point moving over a trajectory consisting of rectilinear sections and portions of circles. The differential equations obtained may be used in determining the position of rod-type elements suspended in a viscous medium.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 108–112, 1990.     

Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.     

The stressed state caused by a residual strain field in plates with stress concentrators

We obtain an analytic solution of the problem of determining the stressed state caused by a given residual strain field (while taking account of three-dimensional effects) in a round plate with a concentric foreign inclusion. We study the influence of the geometric parameters of the given system, the nonlinearity of the distribution of the given residual strains over the thickness of the plate, and a possible jump in distortion at the surface of contact on the stressed state of the system. We discover an internal boundary-layer effect that is significant in the case when the given residual strain field is strongly gradient.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 60–66.     

On a method of solving two-dimensional problems of the linear theory of viscoelasticity

We describe the basic propositions of the linear theory of viscoelasticity. We give transformation formulas for the resolvent integral operators of viscoelasticity with an arbitrary analytic kernel of difference type. The method of computing the irrational operator functions is illustrated by determining the real parameters of the two-dimensional stressed state of an orthotropic plate. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 86–96.     

Antiplane deformation of an isotropic massif with a curvilinear cavity

We solve the problem of antiplane deformation for an infinite isotropic massif with a curvilinear cavity in which a harmonic shear wave is propagating. The solution of the problem, which is based on the application of the theory of functions of a complex variable, is reduced to finding unknown constants from a system of linear algebraic equations. Numerical results are given for a circular cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 115–119.     

The stressed state of a three-dimensional plate with rigid inclusions

We study the stressed state of an isotropic plate with two rigid round inclusions stretched by uniform forces at infinity in the three-dimensional formulation. We give the results of numerical investigation. We exhibit the error of the corresponding applied theory. Four tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 34–42.     

Elastic equilibrium of an anisotropic plate of arbitrary thickness with an elliptic cavity

We study the stressed state of an anisotropic plate of arbitrary thickness weakened by a cylindrical elliptic cavity and subject to forces that are independent of the thickness coordinate. The results of numerical computations are described. Four figures. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 76–80.     

A method of determining the functional dependence between stress and deformation under isothermal loading of a reinforced plate

We propose an approximate method for specifying the functional dependence among the stress, deformation, temperature, and time under arbitrary laws of loading and heating of a reinforced plate in a uniaxially stressed state. The method is justified experimentally in tests of the uniaxial extension of models under isothermic and nonisothermic loading with stress and temperature varying at a constant rate.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 75–77.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Diffraction of nonstationary waves at curvilinear cavities in orthotropic bedrock

We propose a method of solving the two-dimensional dynamic problem for an elastic orthotropic body. On the basis of the fundamental solutions constructed for the equations of motion in displacements in the Laplace transform space with respect to time and the boundary element method, solving the boundary problem is reduced to solving a finite system of linear algebraic equations. We carry out an investigation of the stressed state of the bedrock near a tunnel cavity of circular cross-section. Two figures. Bibliography: littes. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 96–102 1991.     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

Bending of a thin orthotropic bar with an elliptic cut

To study the stressed state of an infinite orthotropic bar with an elliptic slit we use the solutions obtained by integrating the three-dimensional equations of the theory of elasticity. We give a comparison of the results of numerical computations with those obtained in the applied theory. One table. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 35–40.     

Parallel Implementation of Stability Analysis of Difference Schemes with MATHEMATICA

We consider a parallel algorithm for investigating the stability of the schemes of the finite-difference and finite-volume methods that approximate the two-dimensional Euler equations of compressible fluid on a curvilinear grid. The algorithm is implemented with the aid of the computer algebra system Mathematica 3.0. We apply a two-level parallelization process. At the first level, the symbolic computation of the amplification matrix is parallelized by a parallel computation of the matrix rows on different processors. At the second level, the values of the coordinates of points of the stability-region boundary are computed numerically. For the communication between the workstations, we apply a special program, LaunchSlave, which uses the MathLink communication protocol. Examples of application of the proposed parallel symbolic/numerical algorithm are presented. Bibliography: 15 titles.     

The dynamic stressed state of a rectangular composite plate rigidly fixed along its edges

Using the generalized dynamic theory of bending of plates, which takes account of the compliance of the material to transverse shear strains, we obtain an approximate solution of the problem of the dynamic stressed state of a composite plate with rigidly fixed edges subject to an impact load. We exhibit the contribution of the physico-mechanical and geometric parameters of the plate to the magnitude of the computed stresses and their time variation for different coefficients of internal dissipation of mechanical energy.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 129–133.