Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension

Discrete and Continuous Dynamical Systems - Series B

Volumn 1, Issue 2, 2001, Pages 257-263

  Dacorogna, Bernard ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4944267168     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2001.1.257     Document Type: Article

References (21)

1. G. Aubert : Necessary and sufficient conditions for isotropic rank one functions in dimension 2; Journal of Elasticity, 39 (1995), 31-46.
2. G. Aubert and R. Tahraoui : Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3; Rendiconti Cire. Mat. Palermo, 34 (1985), 460-488
3. G, Aubert and R. Tahraoui : Sur la faible fermeture de certains ensembles de contraintes en élasticité non linéaire plane; Archive for Rational Mechanics and Analysis, 97 (1987), 33-58.
4. M. Baker and J.L. Ericksen : Inequalities restricting the form of the stress deformation relations for isotropic elastic solids and Reiner-Rivlin fluids; J. Wash. Acad. Sci. 44 (1954), 33-45.
5. J.M. Ball : Convexity conditions and existence theorems in nonlinear elasticity; Archive for Rational Mechanics and Analysis, 63 (1977), 337-403
6. J.M. Ball : Differentiability properties of symmetric and isotropic functions; Duke Mathematical J. 51 (1984), 699-728.
7. P. Chadwick and R.W. Ogden : On the definition of elastic moduli; Archive for Rational Mechanics and Analysis, 44 (1971), 41-53.
8. P. Chadwick and R.W. Ogden : A theorem of tensor calculus and its application to isotropic elasticity; Archive for Rational Mechanics and Analysis, 44 (1971), 54-68.
9. B. Dacorogna : Direct methods in the calculus of variations; Springer Verlag, Berlin (1989).
10. B. Dacorogna and P. Marcellini : Implicit partial differential equations; Birkhauser, Boston (1999).
11. P.J. Davies : A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain; Journal of Elasticity, 26 (1991), 291-296.
12. K.P. Hadeler : On copositive matrices; Linear Algebra and its Applications, 49 (1983), 79-89.
13. R. Hill : Constitutive inequalities for isotropic elastic solids under finite strain; Proc. Roy. Soc. London A 314 (1970), 457-472.
14. J.K. Knowles and E. Sternberg : On the failure of ellipticity of the equation of the finite elastostatic plane strain; Archive for Rational Mechanics and Analysis, 63 (1977), 321-336.
15. H. Le Dret : Sur les fonctions de matrices convexes et isotropes; G. R. Acad. Sci. Paris Ser. I Math. 310 (1990), 617-620.
16. T.S. Motzkin : Copositive quadratic forms; National Bureau of Standards report 1818 (1952), 11-22.
17. M. Silhavy : T/ie mechanics and thermodynamics of continuous media; Springer Verlag, Berlin (1997).
18. M. Silhavy : On isotropic rank one convex functions; Proc. Royal Soc. Edinburgh, 129A (1999), 1081-1105.
19. H.C. Simpson and S.J. Spector : On copositive matrices and strong ellipticity for isotropic materials; Archive for Rational Mechanics and Analysis, 84 (1983), 55-68.
20. R.C. Thompson and L.J. Fteede : Eigenvalues of sums of Hermitian matrices; J. Research Nat. Bur. Standards B 75B (1971), 115-120.
21. L. Zee and E. Sternberg : Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids; Archive for Rational Mechanics and Analysis, 83 (1983), 53-90.