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     

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.     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.     

Construction of the solutions of the equations of vibration of the classical theory of plates with parameters periodic on one coordinate

The system of classical equations of transverse vibrations of plates that are inhomogeneous on one coordinate and have exponential dependence of the solution on the other coordinate and time are presented in the canonical Hamiltonian form with suitably chosen canonical variables. For periodically varying parameters we use the general properties of periodic Hamiltonian systems to study the structure of the solutions of boundary-value problems for stationary vibrations of plates. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 109–113.     

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.     

Flexural vibrations of prestressed isotropic plates

We propose a method of solving problems of foreed vibrations of prestressed plates based on the perturbation method and making it possible to solve problems for both low and high frequencies. We describe a method of determining the ground natural frequency. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 75–78.     

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex N-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex N-fold integer minimization problems for which our approach provides polynomial time solution algorithms.     

Optimum control of the forced motions of systems with continuously distributed parameters : PMM vol. 38, n 6, 1974, pp. 1056–1062

We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].     

The thermomechanics of thermosensitive bodies

We deduce the equations of the generalized thermomechanics of thermosensitive uniform and piecewise uniform solid bodies, as well as thin plates and shells. We give methods of solving thermoelastic problems for thermosensitive bodies based on the application of the apparatus of distributions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 6–11.     

Asymptotic considerations for transverse bending of orthotropic sheardeformable plates

Departing from the known reduction of the sixth order linear theory problem of orthotropic shear deformable plates to two simultaneous third-order equations for the deflectionw and for a stress function variableJ we are concerned with the asymptotic decomposition ofw andJ into interior and edge zone solution contributions, and with the derivation of sequential boundary value problems for the leading terms of these two types of contributions, in generalization of known results for isotropic plates for which the decomposition is exact rather than asymptotic. Application of the general results to the solution of a specific problem is shown to be associated with a transverse shear stress resultant concentration factor which tends to infinity in the limit of absent transverse sheardeformability.

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Zusammenfassung Wir gehen aus von dem bekannten Resultat, dass die lineare Theorie sechster Ordnung von schubverformbaren orthotropischen elastischen Platten auf ein System von zwei Gleichungen dritter Ordnung für die Durchbiegungw und für eine SpannungsfunktionJ reduziert werden kann. Unsere Aufgabe ist die asymptotische Separation von Innenlösungsbeiträgen und Grenzschichtlösungsbeiträgen zuw undJ, und die Formulierung von rationellen Randwertproblemen für die führenden Glieder in der Entwicklung der Lösungsbeiträge, in Verallgemeinerung bekannter Resultate für dieisotropische Platte, für welche die asymptotische Separation in eine exakte ausartet. Die Anwendung unserer Ergebnisse auf ein Spezialproblem zeigt unter anderem die Existenz einer Spannungskonzentration, welche im Grenzfall der verschwindenden Schubverformbarkeit auf das Unendliche strebt.

     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach

We consider the motion of a collection of fluid loaded elastic plates, situated horizontally in an infinitely long channel. We use a new, unified approach to boundary value problems, introduced by A.S. Fokas in the late 1990s, and show the problem is equivalent to a system of one‐parameter integral equations. We give a detailed study of the linear problem, providing explicit solutions and well‐posedness results in terms of standard Sobolev spaces. We show that the associated Cauchy problem is completely determined by a matrix, which depends solely on the mean separation of the plates and the horizontal velocity of each of the driving fluids. This matrix corresponds to the infinitesimal generator of the C₀ ‐semigroup for the evolution equations in Fourier space. By analyzing the properties of this matrix, we classify necessary and sufficient conditions for which the problem is asymptotically stable.     

Flow of a micropolar fluid through two parallel porous boundaries

The present article contains the numerical solution for steady flow of a micropolar fluid between two porous plates using finite element method. The micropolar fluid fills the space inside the porous plates when the rate of suction at one boundary is equal to the rate of injection at the other boundary. The results for the fluid velocity and microrotation are graphically presented and the influence of micropolar fluid parameter K and parameter R is discussed. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Existence of solutions of vector variational inequalities and vector complementarity problems

In this paper, we consider vector variational inequality and vector F-complementarity problems in the setting of topological vector spaces. We extend the concept of upper sign continuity for vector-valued functions and provide some existence results for solutions of vector variational inequalities and vector F-complementarity problems. Moreover, the nonemptyness and compactness of solution sets of these problems are investigated under suitable assumptions. We use a version of Fan-KKM theorem and Dobrowolski’s fixed point theorem to establish our results. The results of this paper generalize and improve several results recently appeared in the literature.     

The method of volterra integral equations in contact problems for thin-walled structural elements

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 96–103. Original article submitted March 15, 1997.     

Error estimates for mixed finite element methods for nonlinear parabolic problems

We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart-Thomas spaces of index k, k ? 0. We obtain optimal order L²- and almost optimal order L^∞-error estimates for the finite element solution and order optimal L²-error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ? 1.     

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.     

Stability of multiply connected ribbed shells and plates in a magnetic field

We explain a method of studying electromagnetic loads, the stress-strain state, and the stability of multiply connected ribbed plates and shells made up of conducting materials and subject to the action of a variable magnetic field. We give the solutions of test problems. We study the influence of ribbing on the magnitude of the critical loads for various multiply connected plates.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 83–89.     

On pulsatile flow of a viscous fluid in a rotating channel

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK ²=ΩL ²/v which is reciprocal of Ekmann Number and frequency parameter σ=αL ²/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK ² the flow near the plates has a multiple boundary layer character.     

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.