An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

Solution of boundary-value problems of plates by the Vekua method for approximations <Emphasis Type="Italic">N</Emphasis> = 1 and <Emphasis Type="Italic">N</Emphasis> = 2

In this paper, the problem of bending for an isotropic plate with constant thickness 2h is considered. Problems of bending for infinite plates with a circular hole in which a rigid body is placed in the cases of approximations N = 1 and N = 2 of the Vekua theory are solved. We consider the case where the body is soldered. Problems of bending of circular rings are also solved. The results obtained are compared to the corresponding results obtained by plane classical bending theory. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

On construction of approximate solutions of equations of nonlinear and nonshallow shells

In the present paper, by means of Vekua's method, the system of differential equations for the nonlinear theory of nonshallow shells is obtained. Then the method of a small parameter is used for them and some basic boundary-value problems are solved. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

On the three-dimensional thermostressed state of rectangular plates under nonuniform heating

We consider three-dimensional boundary-value problems of the stationary theory of heat conduction and thermoelasticity for rectangular homogenous isotropic plates of arbitrary thickness. It is assumed that the temperature or heat flux density prescribed on the top and bottom surfaces admit a representation in the form of double trigonometric series. A closed-form analytic solution is obtained for the boundary-value problems of thermoelasticity in the case of plates with contacting edges along the lateral faces. Numerical computations are given for three types of boundary-value problems using the software package mathcad PLUS 6.0 for thin and thick plates. We construct the graphs of variation of the temperature, deflection, and normal stresses over the thickness of the plate. Three figures, 1 table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp 18–26.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

Constructing autonomous boundary-value linear systems from acausal input-output functions

Afrequency-domain realization theory is developed for the class of autonomous, but not necessarily stationary, boundary-value linear systems. It is shown that this realization problem, which consists of constructing autonomous boundary-value linear systems from prescribed input-output functions (weighting patterns), reduces to the factorization of several rational matrices in two variables having separable denominators. This factorization problem is examined and a method is given for constructing minimal factorizations for such rational matrices.     

A variational formulation and investigation of boundary-value problems of the non-linear theory of plates using an energy approach

A variational formulation of boundary-value problems of the non-linear dynamic theory of elasticity using the Hamilton functional is presented. The quasi-static boundary-value problem for thin plates is considered. The initial system of equations, in a two-dimenonal formulation, is represented in terms of generalized forces and displacements. The sufficient conditions for the existence and uniqueness of a weak solution are established.     

On a method of solving physically nonlinear problems in the bending of thick rectangular plates

We study the three-dimensional boundary-value problem of the physically nonlinear theory of elasticity for bending of a homogeneous isotropic thick plate by transverse forces. We assume that the intensity of the load is such that the connection between the stresses and small strains within the elastic limit can be described by nonlinear relations in the form of H. Kauderer. We develop an approximate analytic method that makes it possible to reduce the original nonlinear boundary-value problem to a recursive sequence of corresponding linear boundary-value problems. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, No. 40, 1997, pp. 30–35.     

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 51–56, 1986.     

Wave solutions of equations of Timoshenko type for the transverse vibrations of plates with parameters periodic on one coordinate

The hyperbolic system of equations that describes the vibrations of plates inhomogeneous along one rectangular coordinate in the context of the Timoshenko theory is presented in canonical hamiltonian form, assuming the solution is periodic on a second coordinate. In the case of periodic inhomogeneity we study the structure of the solutions of certain wave boundary-value problems for plates of this type using the general properties of periodic hamiltonian systems. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 105–111.     

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.     

Numerical solution of the boundary-value problem for a nonlinear diffusion equation in image processing

Diffusion filtering methods involve solving an initial boundary-value problem for the diffusion equation in which the initial condition is specified by a function representing the filtered image. The output of this filter is the solution u(x, y, t) of the initial boundary-value problem at some fixed time t = T. In a previous study we have proposed a new version of the diffusion filtering method that ensures improved noise removal due to inclusion of a dependence of the diffusion coefficient on local image intensity. The present study analyzes the resulting finite-difference method for the initial boundary-value problem, examines its numerical implementation, and analyzes its efficiency on prototype and real images. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 35–43, 2006.     

Three-dimensional mixed problems of thermoelasticity for isotropic plates

We obtain the homogeneous thermal solutions due to a temperature field for the three-dimensional thermoelastic problem for isotropic plates on whose plane faces homogeneous thermal and mixed mechanical conditions of flat face and diaphragm type are prescribed. This makes it possible to reduce the thermoelastic boundary-value problem to the corresponding elasticity problem. Translated fromTeoreticheskaya i Prikladnaya Mekhanika. No. 25, 1995, pp. 3–8.     

Inverse problems of reconstructing parameters of the Navier-Stokes system

An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem consisting of reconstruction of the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the results of approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For solution of the problem by this method, the concepts and constructions of open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the results of approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For solution of the problem by the dynamical method, the concepts and constructions of the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications and regularizations of the methods for solution of problems proposed that are based on one piece of a priori information or another about the desired solution and solvability conditions of the problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

Boundary-value problems of the asymmetric theory of elasticity for thin plates

Boundary-value problems of the three-dimensional asymmetric micropolar, moment theory of elasticity with free rotation are considered for thin plates. It is assumed that the total stress-strain state is the sum of the internal stress-strain state and the boundary layers, which are determined in an approximation using asymptotic analysis. Three different asymptotic forms are constructed for the three-dimensional boundary-value problem posed, depending on the values of dimensionless physical constants of the plate material. The initial approximation for the first asymptotic form leads to a theory of micropolar plates with free rotation, the initial approximation for the second asymptotic form leads to a theory of micropolar plates with constrained rotation, and the initial approximation for the third asymptotic form leads to a theory of micropolar plates with “small shear stiffness.” The corresponding micropolar boundary layers are constructed and studied. The regions of applicability of each of the theories of micropolar plates constructed are indicated.     

Boundary-Value Problems in a Cut Plane

We present a survey of the theory of boundary-value problems in the plane with curvilinear cuts. The problem of linear conjugation, the Riemann-Hilbert problem, and the Riemann-Hilbert-Poincare problem are considered in detail both in the classical setting and for a cut plane, with an emphasis on problems with shifts. The main focus is on the solvability conditions and index formulas in various function classes. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

The stressed state of glass disks during strength testing

By applying the improved theory of plates, which makes it possible to determine all components of the stress tensor, we study the stressed state of glass disks under axisymmetric bending. We obtain a closed-form solution of the corresponding boundary-value problem for systems of singularly perturbed equations. It is shown that the size of the zone of pure bending-the domain with uniformly stretched surface-differs from those known in the literature. The results obtained make it possible to determine the optimal geometric parameters of the punch-glass disk-support system in strength testing of the glass by the method of axisymmetric bending.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 73–79.     

The exponential characteristic of the Volterra integral operator with kernel depending on 2<Emphasis Type="Italic">N</Emphasis> variables

Exponential growth is estimated for integral operators with continuous periodic kernels of exponential type. The result is applied to the boundary-value problem for a general hyperbolic differential equation having periodic coefficients with Goursat boundary conditions. It is proved that the exponential characteristic for such a boundary-value problem has canonical form. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.