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

Vigak, V. M.

doi:10.1007/bf02365249

Cited by 5 publications

(5 citation statements)

References 5 publications

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“…and the integral expressions ofσ are determined by (4.4). Finally, if E, G, ν = const, then (4.7) provides us with the same expressions for σ y and σ that have been found while solving the analogous problem for homogeneous material [14].…”

Section: Note That Ifmentioning

confidence: 72%

“…Following [14], we reduce the set of equations (2.1)-(2.4) to two governing equations for the normal stress σ y and total stress σ = σ x + σ y (we call them the key stresses).…”

Section: Reduction Of the Problem To Governing Equationsmentioning

confidence: 99%

“…The paper deals with construction of an analytical solution of the plane thermoelasticity problem in terms of stresses for a strip inhomogeneous in its cross-section. To solve the problem, we use a method of direct integration of equilibrium and compatibility equations proposed by Vihak [14]. Such an approach enables easy application of the method for solving the problems for inhomogeneous solids, since the equilibrium equations, which are integrated directly, are independent of the mathematical model of physical relations between stresses and strains.…”

Section: Introductionmentioning

confidence: 99%

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Reduction of Plane Thermoelasticity Problem in Inhomogeneous Strip to Integral Volterra Type Equation

Tokovyy

Rychahivskyy

2005

Mathematical Modelling and Analysis

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We have developed a method for analytical solving of the plane thermoelasticity problem in terms of stresses for a strip, which is infinite with respect to width. To derive the governing equations, we have used a method of direct integration of differential equilibrium and compatibility equations. Reducing the governing equations to the integral Volterra type equation of the second kind, we have solved it in Fourier transforms by applying a method of simple iteration. Straipsnyje vystomas analizinio sprendinio metodas nehomogeninio strypo termoelastiškumo uždaviniui strypo itempimams rasti, kai strypo ilgis yra begalinis pločio atžvilgiu. Pagrindines lygtys išvedamos naudojant diferencialines pusiausvyros ir suderinamumo lygtis ir tiesiogini integravima. Suvedus pagrindines lygtis i antrojo tipo Volterra integraline lygti, naudojant Furje transformacija, ji sprendžiama paprastosios iteracijos metodu.

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Section: Note That Ifmentioning

confidence: 72%

“…Following [14], we reduce the set of equations (2.1)-(2.4) to two governing equations for the normal stress σ y and total stress σ = σ x + σ y (we call them the key stresses).…”

Section: Reduction Of the Problem To Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reduction of Plane Thermoelasticity Problem in Inhomogeneous Strip to Integral Volterra Type Equation

Tokovyy

Rychahivskyy

2005

Mathematical Modelling and Analysis

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“…In paper [5], a semi-inverse method is used to deal with boundary value problem for a rectangular domain in second gradient elasticity. For rectangular domains, many authors try to solve the plane problem in elasticity and thermo-elasticity following various techniques (see [6][7][8][9][10][11][12]). In the present work, we introduce an analytical method to solve the uncoupled thermoelastostatic plane problems for rectangular regions.…”

Section: Introductionmentioning

confidence: 99%

Exact solution for certain regular thermoelasticity problems in a rectangular long rod by complex harmonics

Dhaba

Abou-Dina

2022

Preprint

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A simple expansion method is used to solve certain regular, uncoupled plane problems of thermoelastostatics for long rods in a rectangle. These solutions are exact when the involved boundary conditions are in polynomial form. A particular case involving heat source in the bulk and surface distributions of temperature and pressures is considered for an exact solution. The results are plotted and discussed.

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“…Using the method of the direct integration of the equations of equilibrium and continuity for stresses without additional potential functions proposed by Vihak in [2], we write the key equations for the stressed state of this layer in the form…”

mentioning

confidence: 99%

Determination of the triaxial distribution of residual stresses in welded joints of structural elements with rectilinear seams and estimation of their influence on joint strength in the presence of crack-type defects

Osadchuk¹,

Tsymbalyuk²,

Dzyubyk³

2012

J Math Sci

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On the basis of a three-dimensional theory, we consider a mathematical model of a welded joint of plane structural elements with rectilinear seams. These joints are simulated by a plane layer with rectilinear seam with residual stresses. Using the method of direct integration of key equations without additional potential functions, we obtain expressions for components of the stress tensor for a set of given inconsistent residual strains localized in the weld. We investigate the influence of residual stresses on the static strength of this welded joint with sharp defects simulated by external surface cracks. Using a twoparameter criterion for brittle-ductile fracture, we calculate the load factors for a welded joint with a crack in the weld.Welded joints with rectilinear seams are widely used in various elements of welded structures. Among them are plate-like structural elements, pipes, in particular, main pipelines, etc. The negative influence of nonrelaxed residual stresses on strength and load-carrying ability of welded structures, especially in the presence of sharp concentrators, is well known. Locality of residual plastic strains causes, in the general case, a triaxial stressed state of structural elements in the zone of the weld. To estimate this state, it is necessary to solve a three-dimensional problem. In the process of long-term operation and in the manufacture of structures, various sharp defects can appear in zones of welded joints, which are usually simulated by cracks. Analysis of the results of investigation presented in domestic works (see, e.g., [4]) shows that, in zones of axial (longitudinal) welds in cylindrical pipes (main pipelines), the distribution of welding stresses slightly differs from an analogous distribution in plates, i.e., in this case, the effect of curvature of a pipe is insignificant. In the general case, this distribution is triaxial. For its analysis in zones of welds, we use a mathematical model of a plane layer with rectilinear seam. For structures in long-term operation, a calculated-experimental method is used. The method is based on mathematical models of bodies with residual stresses and inverse problems with the use of experimental data. The essence of the method is presented, e.g., in [3,5,8,9]. In [7], based on this method, the strength of a part of the pipeline with a circumferential crack in a weld is considered. In the case where the circumferential crack lies outside the weld and a part of the pipeline is subjected to the action of torques, the process of propagation of the crack in linear and nonlinear cases is investigated in [14] on the basis of finite-element analysis. Calculated-Experimental Determination of Residual Stresses in Joints with WeldConsider an infinite plane layer of thickness 2h in the Cartesian coordinate system X 1 , X 2 , X 3 . We introduce the dimensional coordinates system x = X 1 /h, y = X 2 /h , z = X 3 /h , with axis y directed along the

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

Cited by 5 publications

References 5 publications

Reduction of Plane Thermoelasticity Problem in Inhomogeneous Strip to Integral Volterra Type Equation

Reduction of Plane Thermoelasticity Problem in Inhomogeneous Strip to Integral Volterra Type Equation

Exact solution for certain regular thermoelasticity problems in a rectangular long rod by complex harmonics

Determination of the triaxial distribution of residual stresses in welded joints of structural elements with rectilinear seams and estimation of their influence on joint strength in the presence of crack-type defects

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