Journal of Elasticity

, Volume 114, Issue 2, pp 197–211 | Cite as

Internal Constraints in Finite Elasticity: Manifolds or not



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.


Internal constraints Finite elasticity Isotropic materials 

Mathematics Subject Classification (2010)




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


  1. 1.
    Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. Dover, New York (1980) Google Scholar
  2. 2.
    Boulos, P., Achcar, N.: Reduced constraint manifolds. Appl. Mech. Rev. 46(11), S105–S109 (1993) ADSCrossRefGoogle Scholar
  3. 3.
    Carroll, M.M.: On isotropic constraints. Int. J. Eng. Sci. 47, 1142–1148 (2009) CrossRefMATHGoogle Scholar
  4. 4.
    Cohen, H., Wang, C.C.: On the response and symmetry of elastic materials with internal constraints. Arch. Ration. Mech. Anal. 99, 1–36 (1987) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Destrade, M., Scott, N.H.: Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint. Wave Motion 40, 347–357 (2004) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1985) CrossRefMATHGoogle Scholar
  7. 7.
    Gurtin, M.E., Podio-Guidugli, P.: The thermodynamics of constrained materials. Arch. Ration. Mech. Anal. 51(3), 192–208 (1973) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Negahban, M.: Single and multiple material constraints in thermoelasticity. Math. Mech. Solids 12, 623–664 (2007) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Podio-Guidugli, P., Vianello, M.: Constraint manifolds for isotropic solids. Arch. Ration. Mech. Anal. 105(2), 105–121 (1989) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Saccomandi, G.: Universal results in finite elasticity. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications, pp. 97–134. Cambridge University Press, Cambridge (2001) CrossRefGoogle Scholar
  11. 11.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Scott, Foresman and Company, Glenview (1971) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations