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

Abstract

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].A.E. Green je 1973. godine ukazao na nekoliko interesantnih for?mula, koje vaze u elasticnosti konacnih deformacija. Jedna od njih (for?mula (2.10), p.75 [1]) se cesto u literaturi citira kao Green-ova identicnost. Ova veoma znacajna identicnost je uspesno primenjivana u razlicitim kontekstima. Ovde pominjemo samo njenu centralnu ulogu u teoremama jedinosti u elastostatici [2,3]. Dublje sagledavanje Green-ove identicnosti pokazuje sustinsku vezu ove formule sa Eshelby-jevim tenzorom kao i sa zakonom materijalnog balansa ravnoteze [4,5]. U homogenim hiperelasticnim materijalima , ovaj zakon prelazi u Eshelby-jevu identicnost, te se moze dokazati da Green-ova identicnost direktno sledi iz Eshelby-jeve identicnosti. Moguce je da se ove identicnosti protezu na generalisane kontinuume, kao sto su kontinuumi sa mikrostructurom i elasticni dielektrici. U ovom radu se diskutuje mogucnost vazenja Eshelby-jeve identicnosti za takve materijale. Na osnovu nove prosirene Eshelby-jeve identicnosti moze se, pri specificnim pretpostavkama, takodje uspostaviti odgovarajuca Green-ova identicnost. U slucaju dielektrica, dva ekvivalentna oblika Eshelby-jevog tenzora slede iz predlozenog pristupa pri cemu oba zadovoljavaju Eshelby-jevu identicnost. Jedan od njih je podesniji za izvodjenje zeljene identicnosti Green-ovskog tipa. Drugi tenzor, koji je redukovani oblik prvoga, reprezentuje fizicki Eshelby-jev tenzor u dielektricima [4-5].