Optimization of the stressed state by a distribution of elastic parameters in elastic bodies

For designing an isotropic inhomogeneous body having a variable elastic modulus and a prescribed shape under the influence of an external force load we give a formulation and solution of the problem of determining the design that is optimal with respect to stress. The problem of optimal design reduces to a certain problem in the theory of elasticity for a non-linearelastic material. As an example we consider the problem of optimal design of an inhomogeneous cylinder. Four figures. Bibliography: 9 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 78–82.     

Antiplane strain of a multiconnected anisotropic body with longitudinal cavities and plane cracks

Applying the theory of generalized complex potentials and the method of least squares, we solve the problem of the stressed state of a multiconnected anisotropic body under an antiplane strain. The problem is reduced to a system of linear algebraic equations in the unknown constants that occur in the required functions. By numerical studies we exhibit the influence of the elastic and geometric characteristics on the stress distribution and the variation of the stress intensity factors in a cylinder with one or two cracks and in an infinite body with circular cavities and cracks. One figure. Six tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25. 1995, pp. 56–62.     

On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions

We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h ² ln h ⁻¹), where h is the step of a cubic grid.     

Steady planar vibrations of an isotropic body with cavities

We solve the planar-deformation problem for a multiconnected isotropic body deformed by a pulsing load. For a circular cylinder with a circular central cavity we give the results of numerical studies of the stress distribution and the frequencies of the natural axisymmetric vibrations. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 69–74.     

Radially symmetric solutions of the Poisson–Boltzmann equation

The electrostatic potential u outside of a charged cylinder contained in an infinitely large container of electrolyte satisfies the Poisson–Boltzmann equation Δu = shu and boundary condition ?u/?n = ? σ on the cylinder and u(∞) = 0. We show that there exists a unique radially symmetric solution of the problem and consider its properties.     

Antiplane strain of an elliptical cylinder with a crack

The antiplane strain of an anisotropic elliptical cylinder with a crack is examined. An infinite system of linear algebraic equations is obtained from the boundary conditions on the cylinder surface to determine the constants in the complex potential of the problem. Detailed numerical investigations are performed of the influence of the geometric and elastic characteristics of the cylinder on the magnitude of the stress intensity coefficients near the outer edge.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 28–34, 1987.     

Spectral asymptotics for an elliptic operator in a locally periodic perforated domain

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.     

Steady-state motions of a massif with a circular cavity under an antiplane deformation

We give a solution of the problem of antiplane deformation of an isotropic massif with a cavity of circular cross-section, reinforced by a multilayer cylinder under steady-state motions. We give the results of numerical studies that characterize the influence of reinforcement on the stress distribution in a neighborhood of the cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 78–81.     

The quasisteady contact problem of thermoelasticity for a two-layered cylinder under frictional heating and nonideal thermal contact

In this article we give an analytic solution of the polar-symmetric quasisteady thermoelastic contact problem for a two-layer hollow circular cylinder. The problem is solved taking account of frictional heat production and thermal resistance on the mutually tangent surfaces of the components of the cylinder. On the exterior boundary of the two-layer system we study the condition of Winkler elastic fixing. In the solution we apply the Laplace transform with respect to time. We carry out a numerical analysis whose results are shown as graphs. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 104–110.     

The coupled problem of thermoelastic vibrations of a cylinder with cavities

Using a method based on the machinery of analytic functions of complex variables, we construct the solution of the problem of steady planar strain of an isotropic cylinder with curvilinear cavities caused by uneven heating of the surfaces of the cylinder and the cavities. It is an exact solution of the equations and an approximate solution of the boundary conditions. The results of numerical studies are given for a hollow cylinder undergoing thermoelastic radially symmetric vibrations. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 82–86.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

Regularization of a Nonstandard Cauchy Problem for a Dynamic Lame System

We consider the Cauchy problem for dynamic Lame systems in the cylinder G_T = D × (0,T) constructed over a domain D in a three-dimensional space, where the initial data are given in some strip in the lateral surface of the cylinder. The strip has the form S × (0,T), where S is an open subset of the boundary surface of the domain D. This problem is ill-posed. Under certain requirements to the configuration of S, we derive an explicit formula for solutions to this problem.

     

The uncoupled problem of thermoelastic vibrations of a cylinder

Using the theory of functions of a complex variable, we construct an approximate solution of the problem of steady thermoelastic vibrations of a cylinder with curvilinear cavities, not taking account of the interaction of the strain and temperature fields. The results of numerical computations are given in the case of radially symmetric vibrations of a cylinder with a cavity. Two figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 107–112.     

Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids

The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of a compressible non-Newtonian fluid. In the case of the potential stress tensor, we develop a technique for deriving energy identities that do not contain derivatives of density. On the basis of these identities, in the case of sufficiently rapidly increasing potentials, we obtain an extended system ofa priori estimates for the equations mentioned above. We also study the related problem of estimating solutions to the nonlinear elliptic system generated by the stress tensor. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 360–376, September, 2000.     

End effects in three-phase-lag heat conduction

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Solution of Hilbert’s problem for an annulus in a specific case and its application to an explosion problem

A method for solving Hilbert's boundary value problem with discontinuous coefficients is studied for a function single-valued and analytic in an annulus in the case in which the solution may have power-law singularities at finitely many points on the boundary of the annulus. To illustrate the results obtained, we consider an explosion problem for two pinching charges in a homogenous medium with a circular cylinder lying in the flow caused by the explosion. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 135–144, July, 1999.     

A Magneto-Thermo-Elastic Identification Problem with a Moving Boundary in a Micro-Device

In this paper we model a thermally actuated device in the microdomain: a hollow metallic cylinder, coaxially located with respect to a current-carrying cable inside the cylinder is considered. Owing to induction heating, we have to deal with a (nonlinear) magneto-thermo-elastic problem with a moving boundary that we consider in the quasi static case. Assuming that the intensity of the current is unknown, we have to solve an identification problem consisting of identifying the waveform of the electric current, when an additional information on the magnitude of the magnetic field is supplied on the inner face of the cylinder. After setting our direct problem in the general form, we change it to a simpler, but handful, formulation. Working in the class of classical Hölder spaces, we can prove the existence and uniqueness of the solution to our identification problem.