Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres

Akhmedov, N.K.; Sofiyev, A.H.

doi:10.1016/j.tws.2019.03.022

Cited by 25 publications

(29 citation statements)

References 28 publications

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“…Substituting (64) into (81) on the basis of (79) and taking into account (80), we find that for the third group of zeros the main term of the asymptotic solution of Equation (15) for ε → 0 takes the following form [22]…”

Section: Construction Of Asymptotic Formulae For Displacements Andmentioning

confidence: 99%

“…The system of infinite linear algebraic equations (98) is always solvable under the physically correctly imposed conditions on the right-hand side of (98) [6–8,10,22,26].…”

Section: Satisfying the Boundary Conditions At The Ends Of The Spherementioning

confidence: 99%

“…Solutions (65), (66), (75), (76) have the character of a boundary layer and are absent in the Kirchhoff-Love theory. The first terms of its asymptotic expansion are equivalent to the Saint-Venant edge effect of an inhomogeneous isotropic plate [8,22].…”

Section: Analysis Of Stress-strain States Corresponding To Various Tymentioning

confidence: 99%

“…The results obtained by the spline collocation method and the finite element method are compared. In [22], using the asymptotic integration of equations of the elasticity theory and three iterative processes, the axisymmetric problem of the theory of elasticity was studied for the radially inhomogeneous transversely isotropic sphere of small thickness. It is assumed that the elasticity modules are arbitrary positive continuous functions along the radius, the values of which can vary within the same order.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

Akhmedov

Gasanova

2020

Mathematics and Mechanics of Solids

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In the paper the axisymmetric problem of elasticity theory is studied for the radially inhomogeneous sphere of small thickness that does not contain any of the poles 0 and [Formula: see text]. Here the case is considered when the elasticity modules vary linearly with respect to the radius. It is assumed that the lateral surface of the sphere is free of stresses, and at the ends of the sphere (at the conical sections) the stresses are set, leaving it in equilibrium. A characteristic equation is obtained and, based on its asymptotic analysis, the existence of three groups of roots is established with respect to the small parameter characterizing the thickness of the sphere. The corresponding homogeneous solutions are constructed, depending on the roots of the characteristic equation. It is shown that the penetrating solution corresponds to the first group of roots. The second group of roots corresponds to the solution of the edge effect type, similar to the edge effect in the applied theory of shells. The third group of roots corresponds to the boundary layer type solution localized in the conical sections. The solution corresponding to the first and second groups of roots determines the internal stress–strain state of the sphere. In the first term of the asymptotic, they can be considered as a solution in the applied theory of shells. The question of satisfying the boundary conditions at the ends (on the conical sections) of the sphere is considered using the variational Lagrange principle.

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Section: Construction Of Asymptotic Formulae For Displacements Andmentioning

confidence: 99%

“…The system of infinite linear algebraic equations (98) is always solvable under the physically correctly imposed conditions on the right-hand side of (98) [6–8,10,22,26].…”

Section: Satisfying the Boundary Conditions At The Ends Of The Spherementioning

confidence: 99%

Section: Analysis Of Stress-strain States Corresponding To Various Tymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

Akhmedov

Gasanova

2020

Mathematics and Mechanics of Solids

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“…Sarkar and Mondal (2019) investigated transient responses in a two-temperature thermoelastic infinite medium having cylindrical cavity due to moving heat source with memory-dependent derivative. Akhmedov and Sofiyev (2019) discussed asymptotic analysis of the three-dimensional problem of elasticity theory for radially inhomogeneous transversely isotropic thin hollow spheres. Sur et al (2019) reported finite element analysis in a fiber-reinforced cylinder due to memory-dependent heat transfer.…”

Section: Introductionmentioning

confidence: 99%

Vibration analysis of transversely isotropic hollow cylinder considering three different theories using the matrix Frobenius method

Biswas

2019

MMMS

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Purpose The purpose of this paper is to deal with the three-dimensional analysis of free vibrations in a stress-free and rigidly fixed homogeneous transversely isotropic hollow cylinder in the context of three-phase-lag (TPL) model of hyperbolic thermoelasticity. Design/methodology/approach The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. Findings The natural frequency, dissipation factor and inverse quality factor in the stress-free and rigidly fixed hollow cylinder get significantly affected due to thermal vibrations and thermo-mechanical coupling. Originality/value The modified Bessel functions and matrix Frobenius method have been directly used to study the vibration model of a homogeneous, transversely isotropic hollow cylinder in the context of TPL model based on three-dimensional thermoelasticity.

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Investigation of elasticity problem for the radially inhomogeneous transversely isotropic sphere

Akhmedov

Yusubova

2022

Math Methods in App Sciences

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The axisymmetric problem of the theory of elasticity for the radially heterogeneous transverse‐isotopic nonclosed spheres is studied, which does not contain any of the poles 0 and π. The elasticity modules are taken as the linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and in the conical sections, the arbitrary stresses are set that provide equilibrium for the sphere. After consideration of the homogeneous boundary conditions, set on the lateral surfaces of the sphere, the characteristic equation for the spectral parameter is obtained. On the basis of the asymptotic analysis, a classification of the roots of the characteristic equation is made relatively small parameter that characterizes the thickness of the sphere. Corresponding asymptotic solutions are constructed depending on the roots of the characteristic equation. The behavior of the constructed solutions is studied in the internal parts of the sphere, as well as in the neighborhood of the conical sections.

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Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres

Cited by 25 publications

References 28 publications

Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

Vibration analysis of transversely isotropic hollow cylinder considering three different theories using the matrix Frobenius method

Investigation of elasticity problem for the radially inhomogeneous transversely isotropic sphere

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