Universal spherically symmetric solution of nonlinear dislocation theory for incompressible isotropic elastic medium

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

doi:10.1007/s00419-018-1403-9

Cited by 13 publications

(5 citation statements)

References 26 publications

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“…The two integrations (equations (65) and (66)) can be shown to be identical using equation (57). The agreement of the compatibility conditions for Skyrme's field and Nye's tensor is by no means accidental.…”

Section: Skyrme's Theory With Compatibility Conditionmentioning

confidence: 86%

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Compatibility conditions of continua using Riemann–Cartan geometry

Böhmer

Lee

2020

Mathematics and Mechanics of Solids

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The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann–Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

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Section: Skyrme's Theory With Compatibility Conditionmentioning

confidence: 86%

“…The three-dimensional Einstein tensor hence plays an important role in continuum mechanics. Further applications of the compatibility condition in solving non-linear systems with non-trivial dislocations and disclinations in both classical and micropolar theories can be found in [62][63][64][65][66].…”

Section: An Application To Axisymmetric Problemsmentioning

confidence: 99%

Compatibility conditions of continua using Riemann–Cartan geometry

Böhmer

Lee

2020

Mathematics and Mechanics of Solids

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“…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%

“…Thanks to the selected material model and dislocation distribution, an exact solution was built. 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.…”

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

confidence: 99%

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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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Eigenstresses in a Nonlinearly Elastic Sphere with Distributed Dislocations

Goloveshkina

Зубов

2019

Advanced Structured Materials

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Universal spherically symmetric solution of nonlinear dislocation theory for incompressible isotropic elastic medium

Cited by 13 publications

References 26 publications

Compatibility conditions of continua using Riemann–Cartan geometry

Compatibility conditions of continua using Riemann–Cartan geometry

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

Eigenstresses in a Nonlinearly Elastic Sphere with Distributed Dislocations

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