Deformation of transtropic plates subject to mixed boundary conditions at their ends

Homogeneous solutions are constructed for three-dimensional static problems in the theory of elasticity for transversely isotropic plates whose top edges are free of stress and whose bottom edges are clamped. The roots of the transcendental equations are given. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 12–19, 1999     

Homogeneous solutions in three-dimensional problems of the theory of elasticity for transtropic plates with restrained ends

Homogeneous solutions of equations of the theory of elasticity for transtropic plates, on the flat faces of which the components of the displacement vector are zero, are obtained. The solution consists of the turbulent and potential states of the plate.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 10–13, 1988.     

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.     

Vibrations of transtropic plates in the case of mixed boundary conditions

Homogeneous solutions are obtained for three-dimensional dynamical problems in the theory of elasticity for transtropic plates with a stress-free upper edge and a rigidly clamped lower edge. The dispersion relations are studied. Donetsk State University. Translated from Teoreticheskaya i Prikadnaya Mekhanika, No. 29, pp. 52–62, 1999.     

The limit state of a layer compressed by rough plates

The possibility of presenting the solution of the problem of the compression of a three-dimensional layer by two rough plates, if use is made of Prandtl's assumption of a linear variation of the shear stresses over the thickness (not depending on the coordinates along the plates), is analysed. The case of an anisotropic material with a yield point obeying Hill's condition is also considered.     

The mixed boundary-value problem for a transtropic layer with a cavity

Using superposition of solutions and Fourier series we study the stressed state in a transversally isotropic layer with a cavity. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics of the body, and the nature of the load variation. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995 pp. 8–15.     

Equilibrium conditions for multilayered anisotropic viscoelastic plates in a complex state of stress

Systems of linear differential-difference equations describing the complex state of stress of multilayered viscoelastic and elastic plates are obtained.Moscow Institute of Electronic Machine Building; All-Union Scientific-Research Institute of Glass-Reinforced Plastics and Glass Fiber, Moscow Region. Translated from Mekhanika Polimerov, No. 2, pp. 281–288, March–April, 1973.     

On a method of studying the stress state of orthotropic shells and plates weakened by a hole with cracks extending to its edge

On the basis of the Timoshenko kinematic hypothesis for shells we give a formulation of the problem of studying the stress-strain state of orthotropic plates and shells weakened by a combined stress concentrator (a hole with two symmetric cracks extending to its edge). We propose a method of solving such problems on the basis of the finite-element method. To simulate the singularity of the stresses and displacements in a neighborhood of the tip of a crack we apply special finite elements with degenerate faces and nodes displaced by 1/4 the length of an edge. The stress intensity factors are found in terms of the displacements of the nodes of such elements. We give the results of computation of the concentration coefficients and the stress intensity factors for spherical and cylindrical shells loaded by internal pressure and for a cylindrical shell and a plate under the action of a distending load with various concentrators: a circular hole, an isolated crack, and a combined concentrator. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 48–54.     

The stress state of a plate with heterogeneous compression loaded by concentrated force and momentum

By the introduction of special modifications in the complex potentials of G. V. Kolosova and N. I. Muskhelishvili, using the method of linear conjugation and singular integral equations, the author solves a problem involving the stress-deformation state of a plate with heterogeneous inclusions, loaded with concentrated force and momentum.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 68–72, 1989.     

The stress state of thick plates under homogeneous boundary conditions of mixed type on the faces

We propose a method of using homogeneous solutions in the case of mixed boundary conditions on the faces of thick plates. We obtain the exact solution of the problem of the stress state of a lamina weakened by a cylindrical cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 3–8.     

Elastoplastic state of an isotropic Cylindrical pipe with closed ends

A series expansion technique is used to construct an analytic solution for the elastoplastic equilibrium of a pipe with closed ends subjected to a constant external pressure. Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 25–27, 1999.     

The mixed problem for a transtropic cylinder under local heating of the surfaces

By the method of homogeneous solutions we study the stressed state of a transversally isotropic cylinder under local heating of the lateral surfaces. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics, and the variation of the heating zone on one of the lateral surfaces of the cylinder. Five figures, 3 tables. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pages 3–10.     

The homology of a locally finite graph with ends

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension 1 captures precisely the topological cycle space of graphs but works in any dimension.     

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π₁(G^′) with G^′⊆G finite.     

The stress state of a plastic layer with a variable yield strength under a flat deformation

We study the stress state of a plastic layer with a variable yield strength in a strip under a flat deformation with a tensile load. We approximately calculate the first integrals of the system of plastic equilibrium equations, obtain an analog of the first Hencky theorem, and solve the conjugation problem for stresses on the contact boundary.     

Effect of the boundary conditions at the ends of laminated beams on their state of stress and strain

The necessity of taking into account the boundary conditions at the end faces of laminated structures is demonstrated with reference to a simple model.Mekhanika Polimerov, Vol. 4, No. 3, pp. 504–508, 1968     

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19 pp. 50–61, 1988.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.