Analysis of the structure of the boundary layer in the problem of the torsion of a laminated spherical shell

Akhmedov, N.K.; Ustinov, Yu. A.

doi:10.1016/j.jappmathmech.2009.07.010

Cited by 5 publications

(5 citation statements)

References 3 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 e ! 0 takes the following form [22]…”

Section: ð þ : ð64þmentioning

confidence: 99%

“…A quantitative analysis of the stress state of the three-layer spherical shell of an asymmetric structure with soft aggregate is given. In [15], based on the perturbation theory of the operators, the torsion problem of the radially layered spherical shell with an arbitrary number of alternating hard and soft layers was studied. It is shown that for the radially layered sphere with alternating hard and soft layers, weakly damped boundary layer solutions exist and they can significantly affect the internal stress–strain state of the sphere.…”

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: ð þ : ð64þmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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 Boyev and Ustinov, 20 the analysis of the three‐dimensional stress–strain state of the three‐layer sphere with a soft filler was presented. In Akhmedov and Ustinov, 21 the torsion problem for the radially layered sphere with an arbitrary number of alternating hard and soft layers was studied. The existence of weakly damped boundary layer solutions and a possible violation of the Saint‐Venant principle in its classical formulation were shown.…”

Section: Introductionmentioning

confidence: 99%

“…Asymptotic methods have made a significant contribution to the solution of three-dimensional problems of the theory of elasticity. [10][11][12][13]17,18,20,21,36 In previous studies, 37,38 on the basis of three-dimensional equations of elastodynamics, using a discrete-analytical method, the natural oscillations of a three-layer sphere with an average radially inhomogeneous layer were studied. It was assumed that the moduli of elasticity and the density of the material of the middle layer are a power law along the radius.…”

Section: Introductionmentioning

confidence: 99%

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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“…In [8], an analysis of the three-dimensional stress-strain state of a three-layer sphere with soft filler is given. In [9], the problem of torsion was studied for a radially layered sphere with an arbitrary number of alternating hard and soft layers. The existence of weakly damped boundary layer solutions and the possible violation of the Saint-Venant principle in its classical formulation are shown.…”

Section: Introductionmentioning

confidence: 99%

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

Yusubova¹

2022

TAPR

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The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on three-dimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ=const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state.

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Analysis of the structure of the boundary layer in the problem of the torsion of a laminated spherical shell

Cited by 5 publications

References 3 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

Investigation of elasticity problem for the radially inhomogeneous transversely isotropic sphere

Construction of homogeneous solutions in the torsion problem for a transversally isotropic sphere with variable elastic moduli

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