Eigenstresses in a Nonlinearly Elastic Sphere with Distributed Dislocations

Goloveshkina, Evgeniya V.; Зубов, Л. М.

doi:10.1007/978-3-030-13307-8_11

Cited by 1 publication

(4 citation statements)

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“…6. With the help of Formulas (1), (22), and ( 27), we can derive the expressions for the gradient, curl, and divergence of a second-rank spherically symmetric tensor field P…”

Section: Spherically Symmetric Tensor Fieldsmentioning

confidence: 99%

“…Some problems for elastic sphere with dislocations, the distribution of which is of spherical symmetry, are considered in the works [20][21][22][23]49]. The paper [20] describes the stress state of a hollow sphere made of a compressible semi-linear (harmonic) material containing edge dislocations and subject to external or internal hydrostatic pressures.…”

Section: Spherically Symmetric State Of Elastic Hollow Sphere With Distributed Dislocations In Radial Directionmentioning

confidence: 99%

“…In [21], a universal solution for the class of isotropic incompressible elastic bodies was found, and eigenstresses in a solid sphere (in a space with a spherical cavity) with a special distribution of edge and screw dislocations, as well as stresses in the presence of hydrostatic pressure applied to the surface of the sphere (cavity), were determined. In [22], for compressible and incompressible materials, the stresses in a hollow sphere, caused by edge and screw dislocations, are numerically determined. The paper [49] contains an exact formulation and solution of the stability problem for a hollow sphere made of a semi-linear material with edge dislocations in the framework of the three-dimensional theory.…”

Section: Spherically Symmetric State Of Elastic Hollow Sphere With Distributed Dislocations In Radial Directionmentioning

confidence: 99%

“…This paper solves the problem for an elastic sphere containing continuously distributed dislocations. Problems about a sphere with dislocations were previously considered in the papers [20][21][22][23]. The main result of this work is a new exact solution to the nonlinear theory of dislocations, which is of spherical symmetry.…”

Section: Introductionmentioning

confidence: 98%

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Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Goloveshkina

Зубов

2021

Symmetry

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The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

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“…6. With the help of Formulas (1), (22), and ( 27), we can derive the expressions for the gradient, curl, and divergence of a second-rank spherically symmetric tensor field P…”

Section: Spherically Symmetric Tensor Fieldsmentioning

confidence: 99%