Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry

Panicaud, Benoît; Rouhaud, Emmanuelle; Altmeyer, Guillaume; Wang, Mingchuan; Kerner, Richard; Roos, Arjen; Ameline, Olivier

doi:10.1007/s00707-015-1470-8

Cited by 12 publications

(4 citation statements)

References 37 publications

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“…Attempts to formulate Relativistic Elasticity in the General Relativity framework go back to 1916 with the pioneering work of Nordström [68], in Dutch. Since then, several authors have first aimed at proposing constitutive equations for Relativistic fluids [87,54,14,60] and, then, at modeling Relativistic continuous media, most often at the astrophysics scale [80,86,19,74,82,5,70,50,43,44,4,27,90,36,10], for instance for the modeling of the solid crust of neutron stars, but also at a local scale [57,58,59,72,72,73,63], for mechanical engineering applications.…”

Section: Introductionmentioning

confidence: 99%

Souriau's Relativistic general covariant formulation of hyperelasticity revisited

Kolev¹,

Desmorat²

2023

Preprint

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We present and modernize Souriau's 1958 geometric framework for Relativistic continuous media, and enlighten the necessary and the ad hoc modeling choices made since, focusing as much as possible on the Continuum Mechanics point of view. We describe the general covariant formulation of Hyperelasticity in (Variational) General Relativity, and then in the particular case of a static spacetime. Different relativistic strain and stress tensors are formulated and discussed. Finally, we apply Souriau's formalism to Schwarzschild's metric, and recover the Classical Galilean Hyperelasticity with gravity, as the Newton-Cartan infinite light speed limit of this formulation.

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Section: Introductionmentioning

confidence: 99%

Souriau's Relativistic general covariant formulation of hyperelasticity revisited

Kolev¹,

Desmorat²

2023

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Souriau’s general covariant formulation of Relativistic Hyperelasticity revisited

Kolev,

Desmorat

2023

Journal of the Mechanics and Physics of Solids

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“…So called objective time-derivatives (rates) of objective mechanical quantities have been proposed in the literature [86,35,61,79,27], in order to introduce some kind of elasticity for viscous fluids or to derive computationally efficient formulations of finite strain elasto-(visco-)plasticity [42,43,47]. Some extensions to four-dimensional formalism can be found in [68,62]. Since a lot of objective derivatives are available in the literature, the natural question arose of how to unify them all [32], as well as of better understanding their intrinsic nature [48].…”

Section: Introductionmentioning

confidence: 99%

“…Acknowledgements. It is a pleasure to thank Emmanuelle Rouhaud for stimulating discussions concerning the concept of objective derivative and related to her own work [68,62].…”

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confidence: 99%

Objective rates as covariant derivatives on the manifold of Riemannian metrics

Kolev,

Desmorat

2021

Preprint

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The subject of so-called objective derivatives (in Continuum Mechanics) has a long history and is somehow controversial. Several works concern the formulation of the correct mathematical definition of what they are really and try to unify them all into one definition (or one family). In this paper, we show, finally, that all of them correspond in fact to covariant derivatives on the infinite dimensional manifold Met(B) of all Riemannian metrics on the body. Moreover, a natural Leibniz rule, which allows to define an objective derivative on contravariant tensor fields given one on covariant tensor fields and vice versa makes the distinction between those of them who are of "Lie type" or of "co-rotational type" useless. We calculate furthermore, for an exhaustive list of objective derivatives found in the literature, the corresponding covariant derivative on Met(B).

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Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry

Cited by 12 publications

References 37 publications

Souriau's Relativistic general covariant formulation of hyperelasticity revisited

Souriau's Relativistic general covariant formulation of hyperelasticity revisited

Souriau’s general covariant formulation of Relativistic Hyperelasticity revisited

Objective rates as covariant derivatives on the manifold of Riemannian metrics

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