On the figure of elastic planets I: gravitational collapse and infinitely many equilibria

A classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity the initial radius is given and the density is constant. With gravity and for s...

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Asıl Yazarlar: Jai, F, Kodio, O, Chapman, S, Goriely, A
Materyal Türü: Journal article
Baskı/Yayın Bilgisi: Royal Society 2019
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author Jai, F
Kodio, O
Chapman, S
Goriely, A
author_facet Jai, F
Kodio, O
Chapman, S
Goriely, A
author_sort Jai, F
collection OXFORD
description A classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity the initial radius is given and the density is constant. With gravity and for small planets, the elastic deformations are small enough so that the spherical equilibria can be readily obtained by using the theory of linear elasticity. For larger or denser planets, large deformations occur and the general theory of nonlinear elasticity is required to obtain the solution. Depending on the elastic model, we show that there may be parameter regimes where there exist no equilibrium or arbitrarily many equilibria. Yet, at most two of them are dynamically stable with respect to radial disturbances. In some of these models, there is a critical initial radius at which spherical solutions cease to exist. For planets with larger initial radii, there is no spherical solution as the elastic forces are not sufficient to balance the gravitational force. Therefore, the system undergoes gravitational collapse, an unexpected phenomenon within the framework of classical continuum mechanics.
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spelling oxford-uuid:cf62e881-f9ec-4b09-a57a-4075130f9d9c2022-03-27T07:42:04ZOn the figure of elastic planets I: gravitational collapse and infinitely many equilibriaJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:cf62e881-f9ec-4b09-a57a-4075130f9d9cSymplectic Elements at OxfordRoyal Society2019Jai, FKodio, OChapman, SGoriely, AA classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity the initial radius is given and the density is constant. With gravity and for small planets, the elastic deformations are small enough so that the spherical equilibria can be readily obtained by using the theory of linear elasticity. For larger or denser planets, large deformations occur and the general theory of nonlinear elasticity is required to obtain the solution. Depending on the elastic model, we show that there may be parameter regimes where there exist no equilibrium or arbitrarily many equilibria. Yet, at most two of them are dynamically stable with respect to radial disturbances. In some of these models, there is a critical initial radius at which spherical solutions cease to exist. For planets with larger initial radii, there is no spherical solution as the elastic forces are not sufficient to balance the gravitational force. Therefore, the system undergoes gravitational collapse, an unexpected phenomenon within the framework of classical continuum mechanics.
spellingShingle Jai, F
Kodio, O
Chapman, S
Goriely, A
On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title_full On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title_fullStr On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title_full_unstemmed On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title_short On the figure of elastic planets I: gravitational collapse and infinitely many equilibria
title_sort on the figure of elastic planets i gravitational collapse and infinitely many equilibria
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AT kodioo onthefigureofelasticplanetsigravitationalcollapseandinfinitelymanyequilibria
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AT gorielya onthefigureofelasticplanetsigravitationalcollapseandinfinitelymanyequilibria