Three-dimensional problem of the theory of elasticity in strains

Borodachev, N. M.

doi:10.1007/bf02208501

Cited by 10 publications

(9 citation statements)

References 3 publications

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“…Similar reasoning is valid for the «Beltrami-Michell type» strain equations obtained from the strain compatibility conditions using Hooke's law and the equilibrium equation expressed with respect to deformations [7].…”

Section: Introductionmentioning

confidence: 64%

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

2023

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The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation. Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written. The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

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Section: Introductionmentioning

confidence: 64%

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

2023

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“…Three different approaches may be used to solve problems of the theory of elasticity (Borodachev 1995;2001). In the first approach, the displacement vector is determined first, and this vector is then used to determine the stress and strain tensors (known as problem in displacements).…”

Section: Theoretical Developmentmentioning

confidence: 99%

Numerical modelling of flexible pavement incorporating cross-anisotropic material properties. Part II: Surface rectangular loading

Maina

Kawana

Matsui

2017

J. S. Afr. Inst. Civ. Eng.

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In order to better understand the impact of increased loading on roads, studies on tyre-road interaction have gained prominence in recent years. Tyres form an essential interface between vehicles and road pavement surfaces. These are the only parts of the vehicle that are in contact with the road and transmit the vehicle loading to the road surface. The use of the Cartesian coordinate system is convenient in dealing with a uniform/non-uniform tyre load acting over a rectangular area, but few research reports are available that provide any form of theoretical solutions for pavement responses. This paper presents analytical solutions of responses due to rectangular loading acting on the surface of a multi-layered pavement system. The solutions developed incorporate both isotropic and cross-anisotropic material properties. The method followed is based on classical trigonometric integral and Fourier transformation of Navier's equations. Accuracy and validity of the solutions are verified through comparisons with a proprietary finite element method (FEM) package. For this purpose, a pavement structure composed of five main layers constituted by isotropic and cross-anisotropic (also known as transversely isotropic) material properties is analysed. In order to vary some of the layer properties with depth, the main layers were sub-layered, resulting in a 17-layer pavement system.

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“…The works of B.E. Pobedri [3,4] and N.M.Borodachev [7,8] are devoted to the study and development of model equations for deformations. It is known that boundary value problems in stresses are based on the compatibility condition, and usually reduced to the Beltrami-Mitchell equations [6,18].…”

Section: Introductionmentioning

confidence: 99%

Model Equations of the Theory of Elasticity in Strains: Classical and New Formulations

Khaldjigitov,

Djumayozov

2024

E3S Web Conf.

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The article is devoted to the construction of model equations of the theory of elasticity with respect to deformations. Classical and new versions of boundary value problems of the theory of elasticity in strains are considered. In the classical version, model equations in strains are constructed within the framework of the Beltrami-Michell equations. A new version of model equations in strains is based on a new formulation of boundary value problems of the theory of elasticity in stresses. Discrete equations are constructed using the finite-difference method for two-dimensional problems. The well-known problem of tension a rectangular plate with a parabolic load applied on opposite sides has been solved. By comparing the numerical results of boundary value problems in classical and new formulations, as well as the Timoshenko-Goodier solution, the validity of the formulated model equations in strains and the reliability of the obtained numerical results are ensured.

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Three-dimensional problem of the theory of elasticity in strains

Cited by 10 publications

References 3 publications

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

Numerical modelling of flexible pavement incorporating cross-anisotropic material properties. Part II: Surface rectangular loading

Model Equations of the Theory of Elasticity in Strains: Classical and New Formulations

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