The Green's and the Eshelby's identities in generalised continua and in dielectrics

In 1973, A. E. Green pointed out several interesting formulae, which hold true in finite elasticity [1]. One of them (formula (2.10), p.75) is repeatedly quoted in the literature as the Green identity. This remarkable identity has been successfully employed in several contexte. We only mention here...

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Main Author: Trimarco Carmine
Format: Article
Language:English
Published: Serbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade 2003-01-01
Series:Theoretical and Applied Mechanics
Subjects:
Online Access:http://www.doiserbia.nb.rs/img/doi/1450-5584/2003/1450-55840301041T.pdf
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spelling doaj-c0f12c180ade44a7b74291973279147c2020-11-24T21:02:00ZengSerbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, BelgradeTheoretical and Applied Mechanics1450-55842003-01-01200330415210.2298/TAM0301041TThe Green's and the Eshelby's identities in generalised continua and in dielectricsTrimarco CarmineIn 1973, A. E. Green pointed out several interesting formulae, which hold true in finite elasticity [1]. One of them (formula (2.10), p.75) is repeatedly quoted in the literature as the Green identity. This remarkable identity has been successfully employed in several contexte. We only mention here its central role in theorems on uniqueness in elastostatics [2,3]. A deeper insight to the Green identity shows in evidence an intimate link of this formula with the Eshelby tensor and with the material balance law of equilibrium [4,5]. In homogeneous hyperelastic materials, this law turns out to an identity, the Eshelby identity, and one can easily prove that the Green identity stems straightforwardly from the Eshelby identity. These identities possibly extend to generalized continua, such as continua with microstructure and elastic dielectrics. Hereafter, the validity of the Eshelby identity is discussed for these materials. Basing on the novel extended Eshelby-like identity, the corresponding extended Green-like identity can be also established, under specifie assumptions. In the case of dielectrics, two equivalent forms for the Eshelby tensor emerge from the treatment, both satisfying the Eshelby identity. One of them is more appropriate for deriving the desired Green-like identity. The second one, which is a reduced form of the first one, represents the physical Eshelby tensor in dielectrics [4-5]. http://www.doiserbia.nb.rs/img/doi/1450-5584/2003/1450-55840301041T.pdfelasticitydielectricsmicrostructure
collection DOAJ
language English
format Article
sources DOAJ
author Trimarco Carmine
spellingShingle Trimarco Carmine
The Green's and the Eshelby's identities in generalised continua and in dielectrics
Theoretical and Applied Mechanics
elasticity
dielectrics
microstructure
author_facet Trimarco Carmine
author_sort Trimarco Carmine
title The Green's and the Eshelby's identities in generalised continua and in dielectrics
title_short The Green's and the Eshelby's identities in generalised continua and in dielectrics
title_full The Green's and the Eshelby's identities in generalised continua and in dielectrics
title_fullStr The Green's and the Eshelby's identities in generalised continua and in dielectrics
title_full_unstemmed The Green's and the Eshelby's identities in generalised continua and in dielectrics
title_sort green's and the eshelby's identities in generalised continua and in dielectrics
publisher Serbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade
series Theoretical and Applied Mechanics
issn 1450-5584
publishDate 2003-01-01
description In 1973, A. E. Green pointed out several interesting formulae, which hold true in finite elasticity [1]. One of them (formula (2.10), p.75) is repeatedly quoted in the literature as the Green identity. This remarkable identity has been successfully employed in several contexte. We only mention here its central role in theorems on uniqueness in elastostatics [2,3]. A deeper insight to the Green identity shows in evidence an intimate link of this formula with the Eshelby tensor and with the material balance law of equilibrium [4,5]. In homogeneous hyperelastic materials, this law turns out to an identity, the Eshelby identity, and one can easily prove that the Green identity stems straightforwardly from the Eshelby identity. These identities possibly extend to generalized continua, such as continua with microstructure and elastic dielectrics. Hereafter, the validity of the Eshelby identity is discussed for these materials. Basing on the novel extended Eshelby-like identity, the corresponding extended Green-like identity can be also established, under specifie assumptions. In the case of dielectrics, two equivalent forms for the Eshelby tensor emerge from the treatment, both satisfying the Eshelby identity. One of them is more appropriate for deriving the desired Green-like identity. The second one, which is a reduced form of the first one, represents the physical Eshelby tensor in dielectrics [4-5].
topic elasticity
dielectrics
microstructure
url http://www.doiserbia.nb.rs/img/doi/1450-5584/2003/1450-55840301041T.pdf
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