A periodic system of parallel slits with contacting edges in a distended plate

We consider the problem of distention of an infinite plate containing a periodic system of parallel slits whose edges make contact in one of its face planes. We take account of the local bending of the plate in a neighborhood of the defects. On the basis of an approximate analytic solutions and a numerical solution of the singular integral equation of the problem we study the influence of the period of location of the defects on the size of the slit openings and the distribution of the reaction in the contacting edges. We compute the stress intensity factors and moments and determine the destructive load. We give a comparison of the results obtained with the known solution of the periodic problem for parallel slits with load-free edges. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 40–45.     

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.     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

The stationary problem of thermoelasticity for a plate with a heated slit

We consider the stationary heat conduction and thermoelasticity problem for a plate that whose lateral surfaces are thermally insulated and which has a slit on whose edges a constant temperature is maintained. The heat conduction problem is solved using a modified logarithmic single-layer potential by reducing it to integral equations.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 50–54.     

A consistent solution of the problem of one-sided tension of an elastic plane with an astroidal hole

We consider the stress-strain state of an elastic plate with a hole in the form of an astroid under one-sided tension. The contact of edges of the hole near two opposite tips of the astroid is taken into account, which eliminates the contradiction with the classical solution concerning the overlapping of edges of the hole. We determined both the length of the regions of contact pressure and the distribution of contact stresses. For two contact-free tips of the hole, the value of the stress intensity factor is calculated.     

Generalized potentials and their application in solving problems of mathematical physics

We introduce the concept of potential for bounded bodies. We construct the potentials for a two-dimensional semi-infinite region. We solve the problems of determining the nonsteady state temperature fields in a semi-infinite plate with a straight-line cut situated perpendicularly to the boundary of the region under boundary conditions of the first or second kind on the boundary of the region and the edges of the cut.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 52–58.     

The max-cut problem and quadratic 0–1 optimization; polyhedral aspects,relaxations and bounds

Given a graphG, themaximum cut problem consists of finding the subsetS of vertices such that the number of edges having exactly one endpoint inS is as large as possible. In the weighted version of this problem there are given real weights on the edges ofG, and the objective is to maximize the sum of the weights of the edges having exactly one endpoint in the subsetS. In this paper, we consider the maximum cut problem and some related problems, likemaximum-2-satisfiability, weighted signed graph balancing. We describe the relation of these problems to the unconstrained quadratic 0–1 programming problem, and we survey the known methods for lower and upper bounds to this optimization problem. We also give the relation between the related polyhedra, and we describe some of the known and some new classes of facets for them.     

Model for Contact of Crack Boundaries in a Bending Plate

Closed cracks in a bending plate are studied in a two-dimensional formulation based on the Kirchhoff theory and the theory of the two-dimensional stressed state. A generalization of the end contact model is discussed which makes it possible formulate a geometrically linear contact bending problem for a plate with a rectilinear cut including the interaction of symmetric and antisymmetric deformation modes. A procedure for reducing the problem to a system of singular integral equations is described. An example of the calculations is given.     

Planar deformation of a plate with a one- or two-sided cover

We solve the problem of planar deformation of an isotropic plate fastened at one surface or symmetrically in its two surfaces to cover plates of a different material. As initial data we use the approximate relations of the two-dimensional formulation, which take account of the shear and the transverse compression of the plate. We study the dependence of the interlayer stresses on the ratio of the elastic moduli of the materials of the plate and the cover.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 99–102.     

The problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.     

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.     

Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.     

The heat-conduction problem for a shell with a system of diathermanous cuts

Using the two-dimensional Fourier transform and the elementary theory of distributions, we solve the heat-conduction problem for shells with a system of diathermanous cuts. We take account of heat exchange according to Newton's law on the lateral surfaces of the shells. For a spherical shell with two cuts of identical length we carry out numerical studies of the influence of the thermophysical properties of one cut on the jump in temperature of the adjacent cut. Three figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 86–89.     

Contact formulation of non-linear problems in the mechanics of shells with their end sections connected by a plane curvilinear rod

Starting from the consistent version of the geometrically non-linear equations of the theory of elasticity for small deformations and arbitrary displacements, a Timoshenko-type model that takes account of shear and compression deformations and also an extended variational Lagrange principle, an improved geometrically non-linear theory of static deformation is constructed for reinforced thin-walled structures with shell elements, the end sections of which are connected by a rod. It is based on the introduction into the treatment of contact forces and torques as unknowns on the lines joining the shells to the rods and it enables all classical and non-classical forms of loss of stability in structures of the class considered to be investigated. An analytical solution of the problem of the stability of a rectangular plate, that is under compression in one direction, supported by a hinge along two opposite edges and joined by a hinge with an elastic rod on one of the other two edges, is found using a simplified version of the linearized equations.     

Limit equilibrium of a closed transversally isotropic cylindrical shell with a nonthrough longitudinal crack

In the context of an analog of the Leonov-Panasyuk-Dagdeil model we consider the problem of limit equilibrium of a nonshallow transversally isotropic cylindrical shell weakened by a nonthrough surface longitudinal crack. Based on the equations that take account of the initial stresses, we reduce the problem to a system of two singular integral equations with unknown limits of integration. We carry out a numerical analysis of the dependence of the opening of the edges of the crack on the load and the geometric and physico-mechanical parameters of the shell. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 31–36.     

Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

Some exact representations for the three-dimensional thermoelastic problem

Some exact representations are obtained for the three-dimensional thermoelastic problem in a rectangular parallelepiped. We give an exact solution of the Dirichlet problem taking account of edges and angles. We use the Papkovich-Neiber representation.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 89–93.     

MAX-CUT has a randomized approximation scheme in dense graphs

A cut in a graph G = (V(G), E(G)) is the boundary δ(S) of some subset S η V(G) and the maximum cut problem for G is to find the maximum number of edges in a cut. Let MC(G) denote this maximum. For any given 0 < α < 1, ϵ > 0, and η, we give a randomized algorithm which runs in a polynomial time and which, when applied to any given graph G on n vertices with minimum degree ≥αn, outputs a cut δ(S) of G with $ P[|\delta(S)|\geq MC(G)(1-\epsilon)] \geq 1-2^{-n} $ We also show that the proposed method can be used to approximate MAXIMUM ACYCLIC SUBGRAPH in the unweighted case. © 1996 John Wiley & Sons, Inc.     

Fragile graphs with small independent cuts

A graph is called fragile if it has a vertex cut which is also an independent set. Chen and Yu proved that every graph with n vertices and at most 2n?4 edges is fragile, which was conjectured to be true by Caro. However, their proof does not give any information on the number of vertices in the independent cuts. The purpose of this paper is to investigate when a graph has a small independent cut. We show that if G is a graph on n vertices and at most (12n/7)?3 edges, then G contains an independent cut S with ∣S∣≤3. Upper bounds on the number of edges of a graph having an independent cut of size 1 or 2 are also obtained. We also show that for any positive integer k, there is a positive number ε such that there are infinitely many graphs G with n vertices and at most (2?ε)n edges, but G has no independent cut with less than k vertices. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 327–341, 2002     

Cut problems in graphs with a budget constraint

We study budgeted variants of classical cut problems: the Multiway Cut problem, the Multicut problem, and the k-Cut problem, and provide approximation algorithms for these problems. Specifically, for the budgeted multiway cut and the k-cut problems we provide constant factor approximation algorithms. We show that the budgeted multicut problem is at least as hard to approximate as the sparsest cut problem, and we provide a bi-criteria approximation algorithm for it.