Solutions of axially symmetric and two-dimensional dynamic problems of thermoelasticity for cylinders in terms of stresses

We present the mathematical statement of a dynamic problem of thermoelasticity in cylindrical coordinates in terms of the components of the stress tensor. We write the systems of basic equations for axially symmetric and two-dimensional problems satisfying the consistency conditions for strains. By identical transformations, these systems are reduced to hierarchies of wave equations, which can be solved by using standard methods of mathematical physics. In the absence of the influence of temperature and bulk forces, the indicated hierarchies turn into the well-known equations. “L'vivs'ka Politeknika” State University, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 3, pp. 39–42, May–June, 1997.     

Solution of plane dynamic boundary-value problems of electromagnetothermoelasticity for cylindrical bodies

We give the mathematical statement of a two-dimensional dynamic problem of electromagnetothermoelasticity for cylindrical bodies and deduce the primary equations of electrodynamics and thermoelasticity for the complex problem. The primary equations of the two-dimensional dynamic problem of thermoelasticity in stresses are reduced to a system of hierarchically connected wave equations. The boundary-value problems are formulated for a long hollow cylinder and a cylindrical beam whose cross section has the shape of a circular sector. These bodies are in the state of plane deformation and subjected to the action of a nonstationary electromagnetic field on their outer surfaces. We propose a method for the solution of these boundary-value problems. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36. No. 3, pp. 35–41. May-June. 2000.     

The solution of steady-state axtsymmetric thermoelastic problems by means of complex variables

In this paper is introduced a method of solution of steady-state axisymmetrie thermoelastic problems by means of functions of complex variable. There the equations of the problem are obtained by a rotation of the plane state about an axis of symmetry or by a linear translation of the axisymmetrie state. The equations are utilized for the solution of steady-state thermoelastic problem in a sphere. The stresses in a sphere are given in cylindrical coordinates and coincide with the known solution.     

A variational principle for nonlinear water waves

Summary This paper is concerned with a variational formulation of non-axisymmetric water waves and of two-dimensional surface waves in a running stream of finite depth. The full set of equations of motions for the non-axisymmetric water wave problem in cylindrical polar coordinates and for the two-dimensional surface waves in the running stream in Cartesian coordinates is obtained from a Lagrangian function which is equal to the pressure.With 1 Figure     

System of Equations of the Dynamic Problem of Thermoelasticity in Stresses for an Elliptic Cylinder

On the basis of the equations of motion, Cauchy formulas, generalized Hooke’s law, and compatibility conditions for the Saint-Venant strains, a system of determining equations of the dynamic problem of thermoelasticity in stresses is deduced for a homogeneous isotropic cylinder in an elliptic cylindrical coordinate system. This system is reduced to a system of consecutively correlated wave equations in which the equation for the first invariant of the stress tensor is independent. The initial conditions for the resolving functions are presented.__________Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 5, pp. 57–62, September–October, 2004.     

Field equations, equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness from a three-dimensional theory

Summary This work uses tensor calculus to derive a complete set of three-dimensional field equations well-suited for determining the behavior of thick shells of revolution having arbitrary curvature and variable thickness. The material is assumed to be homogeneous, isotropic and linearly elastic. The equations are expressed in terms of coordinates tangent and normal to the shell middle surface. The relationships are combined to yield equations of motion in terms of orthogonal displacement components taken in the meridional, normal and circumferential directions. Strain energy and kinetic energy functionals are also presented. The equations of motion and energy functionals may be used to determine the static or dynamic displacements and stresses in shells of revolution, including free and forced vibration and wave propagation.     

Optimization of plasmon excitation at structured apertures

Surface plasmon excitation that is due to a single or a structured circular aperture in a flat metallic screen is investigated theoretically and numerically with a view to enhancing the electric field close to the metallic surface. A systematic study of the homogeneous solution of the electromagnetic scattering problem is made with cylindrical coordinates, expanding Maxwell equations on a Fourier-Bessel basis. A perturbation analysis devoted to simple physical analyses of different types of cylindrical nanostructure is developed for the optimization of plasmon excitation by a normally incident linearly polarized monochromatic plane wave. The conclusions drawn from this analysis agree well with the results of rigorous electromagnetic calculations obtained with the differential theory of diffraction in cylindrical coordinates.     

Equations in Stresses for Two- and Three-Dimensional Dynamic Problems of Thermoelasticity in Spherical Coordinates

The basic system of differential equations for the six components of the dynamic stress tensor is deduced in a spherical coordinate system by using the equations of motion, Hooke's law, Cauchy relations, and Saint-Venant conditions of compatibility of strains. The obtained system of differential equations is reduced (without introducing additional potential functions) to a system of wave equations used for the successive evaluation of the first invariant and unknown components of the stress tensor. We also present the corresponding systems of wave equations for the axisymmetric, polar, and centrally symmetric problems of thermoelasticity.     

On the Theory of Acoustic Tomography of Stresses in Solids

We consider a mathematical model of propagation of weak elastic perturbations in acoustically inhomogeneous solids. The acoustic inhomogeneity and anisotropy of the body are induced by a field of initial strains. A linearized system of dynamic equations obtained for the Murnagan cubic potential of elasticity includes coefficients depending on the components of initial strains. An iterative approach proposed for the solution of this system of equations whose coefficients depend on space coordinates enables us to reduce the problem to the solution of a sequence of inhomogeneous wave equations with constant coefficients. Even in the zero-order approximation, this approach enables us to establish relations connecting the mean phase velocities of plane monochromatic waves of various types with linear integrals of the components of strains for arbitrary directions of propagation of waves. The corresponding relations are presented for the case of plane deformation.     

Redistribution of residual stresses in a cylindrical piecewise homogeneous shell with crack

We propose a nondestructive theoretical-experimental method for the evaluation of residual welding stresses in piecewise-homogenous cylindrical elements of shell structures based on the solution of inverse problems with the use of the available experimental data. For a structure weakened by a longitudinal crack, the problem of redistribution of stresses near the crack is reduced to a system of singular integral equations whose solution is constructed by the method of mechanical quadratures. As an example, we solve the problem for a piecewise homogeneous cylindrical shell welded from two semiinfinite shells. The dependences of the force and moment intensity factors on the distribution of residual stresses are investigated. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 5, pp. 39–45, September–October, 1999.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Thermal stress state of a bimetallic plate subjected to pulsed electromagnetic action

We formulate a problem of thermomechanics for an infinite bimetallic plate of constant thickness subjected to nonstationary electromagnetic action and present a relationship for the evaluation of the load-carrying capacity of the plate. A procedure of approximate determination of the parameters of the electromagnetic fields, temperature, and stresses by using the quadratic approximation of the distribution of key functions over the thickness of layers of the plate is proposed. The solution of this problem is obtained for the electromagnetic action in the mode with pulsed modulating signal. We present the results of numerical investigation of the components of the stress tensor and stress intensities in the plane of contact of the constituent layers for a frequency of the carrier signal lying outside the neighborhood of the resonance frequencies and for the first resonance frequency. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 6, pp. 30–40, November–December, 2008.     

Diffraction theory: application of the fast Fourier factorization to cylindrical devices with arbitrary cross section lighted in conical mounting

The differential theory of diffraction by arbitrary cross-section cylindrical objects is developed for the most general case of an incident field with a wave vector outside the cross-section plane of the object. The fast Fourier factorization technique recently developed for studying gratings is generalized to anisotropic and/or inhomogeneous media described in cylindrical coordinates; thus the Maxwell equations are reduced to a first-order differential set well suited for numerical computation. The resolution of the boundary-value problem, including an adapted S-matrix propagation algorithm, is explained in detail for the case of an isotropic medium. Numerical applications show the capabilities of the method for resolving complex diffraction problems. The method and its numerical implementation are validated through comparisons with the well-established multipole method.     

Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section

The diffraction of an electromagnetic wave by a cylindrical object with arbitrary cross section is studied by taking advantage of recent progress in grating theories. The fast Fourier factorization method previously developed in Cartesian coordinates is extended to cylindrical coordinates thanks to the periodicity of both the diffracting object and the incident wave with respect to the polar angle theta. Thus Maxwell equations in a truncated Fourier space are derived and separated in TE and TM polarization cases. The new set of equations for TM polarization is resolved numerically with the S-matrix propagation algorithm. Examples of elliptic cross sections and cross sections including couples of nonconcentric circles show fast convergence of the results, for both dielectric and metallic materials, as well as good agreement with previous published results. Thus the method is suitable for an extension to conical (out-of-plane) diffraction, which will allow studying mode propagation along microstructured fibers.     

Application of an analog of the δc-model to the analysis of strength of elastoplastic cracked shellsto the analysis of strength of elastoplastic cracked shells

We propose a method for the investigation of the limiting equilibrium of elastoplastic shells with systems of interacting cracks. It can be described as follows: By using an analog of the δ_c-model, the elastoplastic problem is reduced to an elastic problem of the limiting equilibrium of a shell with cracks of unknown length whose lips are subjected to the action of unknown forces and moments satisfying the conditions of plasticity for thin shells. By using equations of the general moment theory of shells and the theory of generalized functions, we reduce the problem to the solution of a system of singular integral equations with unknown limits of integration and singular right-hand sides. We construct an algorithm for the numerical solution of systems of this sort supplemented by the conditions of boundedness of stresses and conditions of plasticity. We investigate crack tip opening displacements in a closed cylindrical shell with a regular system of longitudinal cracks or two transverse cracks. For a cylindrical shell with a single crack, we present an approximate relation for the determination of the critical load or crack length. Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 5–15, May–June, 1996.     

Equations of the Dynamic Problem of Thermoelasticity in Stresses in a Three-Orthogonal Coordinate System

By using the system of source equations including the equations of motion, Cauchy relations, generalized Hooke’s law, and Saint-Venant compatibility equations for strains, we deduce the system of defining equations for the dynamic problem of thermoelasticity in stresses in an arbitrary three-orthogonal curvilinear coordinate system. This system is reduced to a system of successively coupled wave equations in which the equation for the first invariant of the stress tensor is independent. The initial conditions are presented for the resolving functions.__________Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 1, pp. 69–75, January–February, 2005.     

Plane problem for a piecewise-homogeneous anisotropic body with elastic inclusion in the form of a strip

A plane problem of the theory of elasticity for a piecewise-homogeneous anisotropic plane with elastic inclusions in the form of strips is solved by the method of jump functions. Inclusions are simulated by the jumps of the vector of stresses and the derivative of the vector of displacements on the median surfaces. By using complex potentials, we obtain the dependences of the components of the stress tensor and the vector of displacements on the load and unknown jump functions. In view of the conditions of interaction of a thin inclusion with an anisotropic medium, the problem is reduced to a system of singular integral equations for the jump functions. In the general case, this system is solved by the collocation method. For the cases of a slot and a perfectly rigid inclusion, we deduce the dependences of the generalized stress intensity factors on the concentrated forces and edge dislocations. Franko L'viv University, L'viv; Pidstryhach Institute of Applied Problems of Mechanics and Mathematics. Ukranian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 6, pp. 7–16. November–December, 1999.     

Plane stress problem of a piezothermoelastic plate

Summary The equations governing plane-stress behavior of a piezothermoelastic body, expressed in cylindrical coordinates, are presented. A solution technique based upon displacement and electric potential functions is developed, and application is made to the problem of a circular plate exposed to axisymmetric surface heating Numerical results obtained for the elastic displacement and stresses, and the electric displacements and electric potential are shown to be in good agreement with those found using a previously derived exact three-dimensional solution.