# Internal Constraints in Finite Elasticity: Manifolds or not

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## Abstract

An internal constraint for an elastic material is described either by a submanifold Open image in new window of the space of deformation gradients or by a submanifold Open image in new window of the space of symmetric strain tensors. It was proved in Podio-Guidugli and Vianello (Arch. Ration. Mech. Anal. 105(2):105–121, 2001) that the dimension of an *isotropic* constraint Open image in new window is always 8, or, equivalently, that the dimension of an *isotropic* constraint Open image in new window is always 5, rigidity and conformality being the only exceptions. Recently, this statement has been questioned in Carroll (Int. J. Eng. Sci. 47:1142–1148, 2009), where it is suggested that isotropic constraints might exist which do not conform to the above prescriptions. It is shown here that this is not the case, because the proposed counterxamples lack a proper manifold structure.

## Keywords

Internal constraints Finite elasticity Isotropic materials## Mathematics Subject Classification (2010)

74A05## Notes

### Acknowledgements

The Author is grateful to the anonymous Referee who took great care to give useful suggestions for improving this work.

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