Solution of one-dimensional problems of elasticity and thermoelasticity in stresses for a cylinder

Vigak, V. M.

doi:10.1007/bf02169001

Cited by 8 publications

(6 citation statements)

References 2 publications

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“…the equation of compatibility in terms of stresses obtained with regard for the equation of equilibrium (1) and relationships (2) d dρ…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Among numerous works, semianalytic methods that enable one to write approximate formulas for the determination of the stress-strain state in these bodies occupy an especially important place. This strongly increases the efficiency of solution of inverse problems of thermoelasticity obtained as a result of the solution of problems of control over the heating of bodies optimal in terms of speed under the restrictions imposed on stresses or temperature and the problems of identification of temperature, stresses, and strains according to the data obtained on a part of the surface [2,7].…”

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confidence: 98%

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Fredholm Equations of the Second Kind for Radial Stresses Aimed at the Determination of the Thermoelastic State of an Inhomogeneous Hollow Long Cylinder

Kalynyak

2015

J Math Sci

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The problem of thermoelasticity in terms of stresses for an inhomogeneous hollow long cylinder with an arbitrary dependence of the physicomechanical characteristics of the material on the radial coordinate is reduced to the solution of a Fredholm equation of the second kind for radial stress. We obtain this equation by direct integration of the equations of equilibrium and continuity and solve it by reducing to a system of algebraic equations. The results of calculations are compared with the known exact solutions of the problem of thermoelasticity for individual dependences of the characteristics of the material on the radial coordinate. We determine the characteristics of materials, the temperature field, and loading guaranteeing the equality of the radial stress in the cylinder to zero.In recent years, the number of investigations of the stress-strain state of bodies of simple shape made of functionally gradient materials has increased, which is explained by their intensification of their practical application [8][9][10]. These problems are reduced to the solution of differential equations with variable coefficients [3]. Among numerous works, semianalytic methods that enable one to write approximate formulas for the determination of the stress-strain state in these bodies occupy an especially important place. This strongly increases the efficiency of solution of inverse problems of thermoelasticity obtained as a result of the solution of problems of control over the heating of bodies optimal in terms of speed under the restrictions imposed on stresses or temperature and the problems of identification of temperature, stresses, and strains according to the data obtained on a part of the surface [2, 7].As one of the methods revealing its high efficiency in the investigation of the thermal stressed state of thermosensitive bodies and bodies made of functionally gradient materials, we can mention the method of direct integration of the equations of motion and integrity in stresses with subsequent reduction of the corresponding problem of thermoelasticity to a set of integral equations and integral conditions [3][4][5]10]. In this case, we arrive at the problem of choosing determining stresses for which we construct the corresponding integral equations [6].For the determination of the thermal stressed state of a long hollow cylinder whose characteristics depend on the radial coordinate and/or a thermosensitive cylinder under the action of constant pressures upon its bounding surfaces and known mass forces and temperature distributions depending on the radial coordinate and, possibly, on time (as a parameter), we construct a Fredholm integral equation of the second kind in which the boundary conditions are also taken into account. Unlike the previous works [3,4,10], where the sum of the radial and circumferential stresses was chosen as determining stresses, in the present work, we choose the radial stresses as determining.

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“…the equation of compatibility in terms of stresses obtained with regard for the equation of equilibrium (1) and relationships (2) d dρ…”

Section: Formulation Of the Problemmentioning

confidence: 99%

mentioning

confidence: 98%

Fredholm Equations of the Second Kind for Radial Stresses Aimed at the Determination of the Thermoelastic State of an Inhomogeneous Hollow Long Cylinder

Kalynyak

2015

J Math Sci

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“…Since the equations of equilibrium (1) must also hold on the boundary of the region, taking account of the second equilibrium equation (1), we see that the boundary conditions (2) and (3) for the tangential stresses can be replaced by conditions on the first derivative with respect to the y-coordinate of the normal stresses o/ (4) Oy y=o for a half-plane and…”

Section: As Ix] -~ Oo Conditionsmentioning

confidence: 99%

“…Such a method would make it possible to construct exact closed-form solutions of the direct problems of elasticity and thermoelasticity, not only for unbounded regions, but also for bounded regions with corners. The studies that have been carried out [3][4][5] make it possible to develop a very general direct method of integrating the equations of equilibrium and continuity in the stresses without applying supplementary potential functions. The direct method is most effective, in that, besides the solutions of the problems just posed, it makes it possible to find a number of relations among the components of the stress tensor and the integral conditions of equilibrium for the individual components of the stress state that are valid for arbitrary physical models of the mechanics of a deformable solid body and are needed to construct solutions of the corresponding inverse problems of thermomechanics [2].…”

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confidence: 99%

A direct method of integrating the equations of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials

Vigak¹

1998

J Math Sci

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For a plane, a half-plane, and a strip, we propose a direct method of integrating the differential equations of equilibrium and continuity with respect to the stresses in the case of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials. We find the relations between the components of the stress tensor, the key integro-differerttial equation and the equation of continuity equivalent to it for determining one of the comvonents of the normal stresses.Methods of constructing exact closed-form solhtions of problems of elasticity and thermoelasticity for homogeneous isotropic and orthotropic bodies are in the main directed toward the application of auxiliary potential functions of stress or displacement types [1,6,7,9], leading to the determination of quantities that have no physical meaning when solving mechanical problems.In solving inverse problems of thermoelasticity in application to control of thermal stresses or displacements [2] the need in mechanics for a fundamental method of separation of variables in the equations of the theory of elasticity and thermoelasticity, similar to the Fourier method in heat conduction, becomes manifest. Such a method would make it possible to construct exact closed-form solutions of the direct problems of elasticity and thermoelasticity, not only for unbounded regions, but also for bounded regions with corners. The studies that have been carried out [3][4][5] make it possible to develop a very general direct method of integrating the equations of equilibrium and continuity in the stresses without applying supplementary potential functions. The direct method is most effective, in that, besides the solutions of the problems just posed, it makes it possible to find a number of relations among the components of the stress tensor and the integral conditions of equilibrium for the individual components of the stress state that are valid for arbitrary physical models of the mechanics of a deformable solid body and are needed to construct solutions of the corresponding inverse problems of thermomechanics [2].For the two-dimensional problem of the mechanics of a deformable solid body the differential equations of equilibrium can be written in the following form [7,9]: c~cr +aC~y+x=o, c3(ry cqCry+y=O.ax Oy 8y + Ox

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“…Therefore, when the stress-tensor components are found, then the displacement-vector components are also found automatically. Such relations have been derived for the case of one-dimensional problem for a thermoelastic hollow cylinder (Vigak, 1999a) and plane problems for elastic and thermoelastic semi-plane Vigak, 2004;Rychahivskyy & Tokovyy, 2008). Since application of this method rests upon the direct integration of the equilibrium equations, the proposed solution scheme offers ample opportunities for efficient analysis of inhomogeneous solids.…”

Section: Introductionmentioning

confidence: 99%

Steady-State Heat Transfer and Thermo-Elastic Analysis of Inhomogeneous Semi-Infinite Solids

Tokovyy¹,

Ma²

2011

Heat Conduction - Basic Research

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Solution of one-dimensional problems of elasticity and thermoelasticity in stresses for a cylinder

Cited by 8 publications

References 2 publications

Fredholm Equations of the Second Kind for Radial Stresses Aimed at the Determination of the Thermoelastic State of an Inhomogeneous Hollow Long Cylinder

Fredholm Equations of the Second Kind for Radial Stresses Aimed at the Determination of the Thermoelastic State of an Inhomogeneous Hollow Long Cylinder

A direct method of integrating the equations of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials

Steady-State Heat Transfer and Thermo-Elastic Analysis of Inhomogeneous Semi-Infinite Solids

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