Abstract

Isotropic energy densities in finite-strain elasticity can be written in terms of the singular values of the deformation gradient, which is assumed to have nonnegative determinant. It is of importance to characterize convexity conditions like rank-one convexity, polyconvexity and full convexity in terms of the singular-value representation. Necessary and sufficient conditions for rank-one convexity (strict ellipticity) were found in 1977 and 1983. Polyconvexity is a more difficult notion since it is a nonlocal condition. The case of space dimension 2 was solved by Rosakis 1998 and Šilhavý 1999. This work is based on a different approach and leads to new necessary conditions in every space dimension. For dimensions 2 and 3 the conditions are also shown to be sufficient. The result in 3D relies on the fact that the extremal points of the set of all Schur products of two rotation matrices is given by 24 signed permutations.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexander Mielke

Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Present address: Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany

mielke@wias-berlin.de

Keywords

  • Polyconvexity
  • isotropic finite-strain elasticity
  • singular values
  • Schur product of matrices

References

  1. G. Aubert, R. Tahraoui: Sur la faible fermeture de certains ensembles de contrainte en élasticité non linéaire plane, C. R. Acad. Sci., Paris, Sér. A 290(12) (1980) 537–540.Google Scholar
  2. G. Aubert, R. Tahraoui: Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3, Rend. Circ. Mat. Palermo, II. Ser. 34 (1985) 460–488.https://doi.org/10.1007/BF02844538Google Scholar
  3. G. Aubert, R. Tahraoui: Sur la faible fermeture de certains ensembles de contraintes en élasticité non-linéaire plane, Arch. Rational Mech. Anal. 97 (1987) 33–58.https://doi.org/10.1007/BF00279845Google Scholar
  4. G. Aubert: On a counterexample of a rank 1 convex function which is not polyconvex in the case N = 2, Proc. R. Soc. Edinb., Sect. A, Math. 106(3-4) (1987) 237–240.https://doi.org/10.1017/S0308210500018382Google Scholar
  5. G. Aubert: Quelques remarques sur les notions de 1-rang convexité et de polyconvexité en dimensions 2 et 3, RAIRO Modél. Math. Anal. Numér. 22 (1988) 5–28.https://doi.org/10.1051/m2an/1988220100051Google Scholar
  6. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63(4) (1976) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  7. J. Ball: Constitutive inequalities and existence theorems in nonlinear elastostatics, in: Nonlin. Anal. Mech., Heriot-Watt Symp., Edinburgh 1976, Vol. I, Res. Notes in Math. 17, Pitman, London (1977) 187–241.Google Scholar
  8. J. Ball: Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51(3) (1984) 699–728.https://doi.org/10.1215/S0012-7094-84-05134-2Google Scholar
  9. J. M. Ball: Some open problems in elasticity, in: Geometry, Mechanics, and Dynamics, P. Newton (ed.) et al., Springer, New York (2002) 3–59.https://doi.org/10.1007/0-387-21791-6_1Google Scholar
  10. M. Buliga: Majorisation with applications to the calculus of variations, preprint, available at http://arxiv.org/pdf/math/0105044 (2001).Google Scholar
  11. M. Buliga: Lower semi-continuity of integrals with G-quasiconvex potential, Z. Angew. Math. Phys. 53 (2002) 949–961.https://doi.org/10.1007/PL00012621Google Scholar
  12. C. Carstensen, K. Hackl, A. Mielke: Non-convex potentials and microstructures in finite-strain plasticity, Proc. R. Soc. London, Ser. A, Math. Phys. Eng. Sci. 458 (2002) 299–317.https://doi.org/10.1098/rspa.2001.0864Google Scholar
  13. P. Cardaliaguet, R. Tahraoui: Sur l’équivalence de la 1-rang convexité et de la polyconvexité des ensembles isotropiques de R 2×2, C. R. Acad. Sci., Paris, Sér. I, Math. 331 (2000) 851–856.https://doi.org/10.1016/S0764-4442(00)01716-XGoogle Scholar
  14. P. Cardaliaguet, R. Tahraoui: Equivalence between rank-one convexity and polyconvexity for isotropic sets of R2×2, I & II, Nonlinear Anal. 50 (2002) 1179–1239.https://doi.org/10.1016/S0362-546X(01)00808-2Google Scholar
  15. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  16. B. Dacorogna: Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension, Discrete Contin. Dyn. Syst., Ser. B 1 (2001) 257–263.https://doi.org/10.3934/dcdsb.2001.1.257Google Scholar
  17. P. Davies: A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain, J. Elasticity 26(3) (1991) 291–296.https://doi.org/10.1007/BF00041893Google Scholar
  18. A. DeSimone, G. Dolzmann: Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Ration. Mech. Anal. 161(3) (2002) 181–204.https://doi.org/10.1007/s002050100174Google Scholar
  19. B. Dacorogna, J.-P. Haeberly: Some numerical methods for the study of the convexity notions arising in the calculus of variations, RAIRO Modél. Math. Anal. Numér. 32(2) (1998) 153–175.https://doi.org/10.1051/m2an/1998320201531Google Scholar
  20. J. Knowles, E. Sternberg: On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal. 63 (1977) 321–336.https://doi.org/10.1007/BF00279991Google Scholar
  21. A. Mielke: Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Cont. Mech. Thermodynamics 15 (2003) 351–382.https://doi.org/10.1007/s00161-003-0120-xGoogle Scholar
  22. A. Mielke: Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal. 36 (2004) 384–404.https://doi.org/10.1137/S0036141003429906Google Scholar
  23. A. Mielke, P. Sprenger: Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition, J. Elasticity 51 (1998) 23–41.https://doi.org/10.1023/A:1007542228169Google Scholar
  24. P. Rosakis: Characterization of convex isotropic functions, J. Elasticity 49 (1998) 257–267.https://doi.org/10.1023/A:1007468902439Google Scholar
  25. M. Šilhavý: The Mechanics and Thermodynamics of Continuous Media, Texts and Monographs in Physics, Springer, Berlin (1997).https://doi.org/10.1007/978-3-662-03389-0Google Scholar
  26. M. Šilhavý: Convexity conditions for rotationally invariant functions in two dimensions, in: Applied Nonlinear Analysis, A. Sequeira (ed.) et al., Kluwer / Plenum, New York (1999) 513–530.https://doi.org/10.1007/0-306-47096-9_35Google Scholar
  27. M. Šilhavý: On isotropic rank 1 convex functions, Proc. R. Soc. Edinb., Sect. A, Math. 129(5) (1999) 1081–1105.https://doi.org/10.1017/S0308210500031085Google Scholar
  28. M. Šilhavý: Differentiability properties of rotationally invariant functions, J. Elasticity 58 (2000) 225–232.https://doi.org/10.1023/A:1007663001108Google Scholar
  29. M. Šilhavý: Relaxation in a class of SO(n)–invariant energies related to nematic elastomers, preprint (2001).Google Scholar
  30. M. Šilhavý: Monotonicity of rotationally invariant convex and rank 1 convex functions, Proc. R. Soc. Edinb., Sect. A, Math. 132 (2002) 419–435.https://doi.org/10.1017/S0308210500001712Google Scholar
  31. H. C. Simpson, S. J. Spector: On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Rational Mech. Anal. 84 (1983) 55–68.https://doi.org/10.1007/BF00251549Google Scholar
  32. L. Zee, E. Sternberg: Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids, Arch. Rational Mech. Anal. 83 (1983) 53–90.https://doi.org/10.1007/BF00281087Google Scholar

Suggested citation

A. Mielke. “Necessary and Sufficient Conditions for Polyconvexity of Isotropic Functions.” Journal of Convex Analysis 12 (2005), No. 2, 291–314. https://doi.org/10.68381/jca12021

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2005. Rights now held by Banach Press.