This is a preview of subscription content, log in via an institution to check access.
References
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403.
J. M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, “Nonlinear Analysis and Mechanics”: Heriot-Watt Symposium Vol. 1 (R. J. Knops Ed.), Pitman (London), 1977, 187–241.
J. M. Ball & J. E. Marsden, Quasiconvexity at the boundary, positivity of the second variation and elastic stability, Arch. Rational Mech. Anal. 86 (1984), 251–277.
O. Bolza, “Lectures on the Calculus of Variations”, Dover Publications Inc., 1961.
G. A. Bliss, “Lectures on the Calculus of Variations”, University of Chicago Press, 1946.
L. Cesari, “Optimisation—theory and applications”, Springer-Verlag, 1983.
B. Dacorogna, “Weak continuity and weak lower semi-continuity of nonlinear functionals”, Springer-Verlag, Lecture notes in Mathematics, vol. 992, 1982.
L. C. Evans, Quasiconvexity and partial regularity in the Calculus of Variations, Arch. Rational Mech. Anal. 95 (1986), 227–252.
P. J. Olver, Conservation laws and null divergences, Math. Proc. Camb. Phil. Soc. 94 (1983), 529–540.
P. J. Olver & J. Sivaloganathan, The structure of null Lagrangians, Nonlinearity 1 (1988), 389–398.
H. Rund, “The Hamilton-Jacobi theory in the Calculus of Variations”, Van Nostrand (London), 1966.
J. Sivaloganathan, A field theory approach to stability of radial equilibria in nonlinear elasticity, Math. Proc. Camb. Phil. Soc. 99 (1986), 586–604.
J. Sivaloganathan, Implications of rank one convexity, Analyse Non Linéaire 5 no. 2 (1988), 99–118.
H. Weyl, Geodesic Fields, Ann. Math. 36 (1935), 607–629.
P. Ponte Casteneda, On the overall properties of nonlinearly elastic composites, (Preprint) Dept. of Mech. Eng., The Johns Hopkins University, Baltimore, U.S.A. (1988).
D. R. S. Talbot & J. R. Willis, Variational principles for inhomogeneous nonlinear media, I.M.A. J. Appl. Math. 35 (1985), 39–54.
R. B. Vinter & R. M. Lewis, A verification theorem which provides a necessary and sufficient condition for optimality, I.E.E.E. Trans. on Automatic control AC-25 no. 1 (1980), 84–89.
P. L. Lions, “Generalised solutions of Hamilton-Jacobi equations”, Research notes in Mathematics, no. 69, Pitman, Boston, 1982.
D. G. B. Edelen, The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal. 11 (1984), 117–121.
J. M. Ball & V. J. Mizel, One-dimensional variational problems whose minimisers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), 325–388.
J. M. Ball & J. C. Currie & P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), 135–174.
L. Tonelli, “Fondamenti di Calcolo delle Variazioni”, Zanichelli 2 vols. I, II, 1921–1923 (in L. Tonelli, Opere Scelte vol. 13, Edizioni Cremonesi, Roma, 1961).
T. Valent, “Boundary value problems of finite elasticity”, Springer Tracts in Natural Philosophy vol. 31, Springer-Verlag, 1988.
Author information
Authors and Affiliations
Additional information
Communicated by J. M. Ball
Rights and permissions
About this article
Cite this article
Sivaloganathan, J. The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal. 107, 347–369 (1989). https://doi.org/10.1007/BF00251554
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00251554