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Journal of Elasticity

, 82:1 | Cite as

Equivalence between Rank-One Convexity and Polyconvexity for Some Classes of Elastic Materials

  • Márcio José Horta Dantas
Article

Abstract

Let W(F) = φ(λ 1 s + λ 2 s + λ 3 s ) + ψ(λ 1 r λ 2 r + λ 1 r λ 3 r + λ 2 r λ 3 r ) + f(λ 1 λ 2 λ 3) be a stored energy function. We prove that, for this function, rank-one convexity is equivalent to polyconvexity.under suitable assumptions on φ, ψ and f.

Key words

convexity polyconvexity rank-one convexity finite elasticity 

Mathematics Subject Classifications (2000)

74A20 74B20 

References

  1. 1.
    J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337–403.MATHGoogle Scholar
  2. 2.
    J.M. Ball, Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51 (1984) 699–728.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    J.M. Ball and F. Murat, W 1,p-Quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    P.G. Ciarlet, Mathematical Elasticity, Vol. I. North-Holland, Amsterdam, New York (1988).Google Scholar
  5. 5.
    J.F. Dunn and R. Fosdick, The Weierstrass condition for a special class of elastic materials. J. Elasticity 34 (1994) 167–184.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Podio-Guidugli, G. Vergara-Cafarelli and E.G. Virga, Cavitation and phase transition of hyperelastic fluids. Arch. Ration. Mech. Anal. 92 (1986) 121–136.MATHCrossRefGoogle Scholar
  7. 7.
    P. Rosakis, Characterization of convex isotropic functions. J. Elast. 49 (1998) 257.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    D.J. Steigmann and A.C. Pipkin, Stability of harmonic materials in plane strain. Quart. Appl. Math. 56 (1988) 559–568.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil

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