Skip to main content
Log in

Quasiconvexity at the boundary, positivity of the second variation and elastic stability

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • V. I. Arnold [1969]. On an a priori estimate in the theory of hydrodynamic stability, Am. Math. Soc. Transl. 79, 267–269.

    Google Scholar 

  • J. M. Ball [1977a]. Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403.

    Google Scholar 

  • J. M. Ball [1977b]. Constitutive inequalities and existence theorems in elasticity, in Nonlinear Analysis and Mechanics, Vol. I, R. J. Knops (ed), Pitman.

  • J. M. Ball [1981]. Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh 88A, 315–328.

    Google Scholar 

  • J. M. Ball [1982]. Discontinuous equilibrium solutions and cavitation in non-linear elasticity, Phil. Trans. R. Soc. London A 306, 557–611.

    Google Scholar 

  • J. M. Ball [1984]. Differentiability properties of symmetric and isotropic functions, Duke Math. J., to appear.

  • J. M. Ball, J. C. Currie & P. J. Olver [1981]. Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Functional Anal. 41, 135–174.

    Google Scholar 

  • J. M. Ball & F. Murat [1984]. W1,p-quasiconvexity and variational problems for multiple integrals J. Functional Anal., to appear.

  • J. M. Ball, R. J. Knops & J. E. Marsden [1978]. Two examples in nonlinear elasticity, Springer Lecture Notes in Mathematics, No. 466, 41–49.

  • O. Bolza [1904]. Lectures on the Calculus of Variations, Reprinted by Chelsea, N.Y., [1973].

  • R. C. Browne [1978]. Dynamic stability of one dimensional nonlinearly viscoelastic bodies, Arch. Rational Mech. Anal. 68, 257–282.

    Google Scholar 

  • H. J. Buchner, J. Marsden & S. Schecter [1983]. Examples for the infinite dimensional Morse Lemma, SIAM J. Math. An. 14, 1045–1055.

    Google Scholar 

  • H. Busemann & G. C. Shephard [1965]. Convexity on nonconvex sets, Proc. Coll. on Convexity, Copenhagen, Univ. Math. Inst., Copenhagen, 20–33.

  • P. G. Ciarlet & G. Geymonat [1982]. Sur les lois de comportement en élasticité nonlinéaire compressible, C. R. Acad. Sc. Paris, 295, 423–426.

    Google Scholar 

  • R. J. DiPerna [1983]. Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82, 27–70.

    Google Scholar 

  • J. L. Ericksen [1966a]. Thermoelastic stability, Proc. Fifth U.S. Cong. on Appl. Mech. 187–193.

  • J. L. Ericksen [1966b]. A thermokinetic view of elastic stability theory, Int. Journal Solids Structures, 2, 573–580.

    Google Scholar 

  • J. L. Ericksen [1975]. Equilibrium of bars, J. of Elasticity 5, 191–201.

    Google Scholar 

  • J. L. Ericksen [1980]. Some phase transitions in crystals, Arch. Rational Mech. Anal. 73, 99–124.

    Google Scholar 

  • M. Golubitsky & J. Marsden [1982]. The Morse Lemma in infinite dimensions via singularity theory, SIAM J. Math. An. 14, 1037–1044.

    Google Scholar 

  • L. M. Graves [1939]. The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5, 656–660.

    Google Scholar 

  • M. E. Gurtin [1975]. Thermodynamics and stability, Arch. Rational Mech. Anal. 59, 63–96.

    Google Scholar 

  • M. E. Gurtin [1983]. Two-phase deformations of elastic solids, (preprint).

  • J. Hadamard [1902]. Sur une question de calcul des variations, Bull. Soc. Math. France 30, 253–256.

    Google Scholar 

  • P. Hartman [1964]. Ordinary Differential Equations. New York: John Wiley & Sons, Inc., reprinted by Birkhauser, Boston, 1982.

    Google Scholar 

  • M. R. Hestenes [1966]. Calculus of variations and optimal control theory, Wiley.

  • D. D. Holm, J. E. Marsden, T. Ratiu & A. Weinstein [1983]. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Physics Letters 98A, 15–21.

    Google Scholar 

  • T. Hughes, T. Kato & J. Marsden [1977]. Well-posed quasi-linear hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63, 273–294.

    Google Scholar 

  • R. D. James [1979]. Co-existent phases in the one-dimensional static theory of elastic bars, Arch. Rational Mech. Anal. 72, 99–140.

    Google Scholar 

  • R. D. James [1980]. The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158.

    Google Scholar 

  • R. D. James [1981]. Finite deformation by mechanical twinning, Arch. Rational Mech. Anal. 77, 143–176.

    Google Scholar 

  • W. T. Koiter [1945]. On the stability of elastic equilibrium, Dissertation. Delft, Holland (English translation: NASA Tech. Trans. F10, 833 (1967)).

  • W. T. Koiter [1976]. A basic open problem in the theory of elastic stability, Springer Lecture Notes in Math. 503, 366–373.

    Google Scholar 

  • W. T. Koiter [1981]. Elastic stability, buckling and post-buckling behaviour, in Proc. IUTAM Symp. on Finite Elasticity, pp. 13–24, D. E. Carlson and R. T. Shield, (eds.), Martinus Nijhoff Publishers.

  • R. J. Knops & L. E. Payne [1978]. On potential wells and stability in nonlinear elasticity, Math. Proc. Camb. Phil. Soc. 84, 177–190.

    Google Scholar 

  • R. J. Knops & E. W. Wilkes [1973]. Theory of elastic stability, in Handbuch der Physik VIa/3, C. Truesdell, ed., Springer.

  • J. E. Marsden & T. J. R. Hughes [1983]. Mathematical Foundations of Elasticity, Prentice-Hall.

  • R. Martini [1979]. On the Fréchet differentiability of certain energy functionals, Proc. Kon Ned. Akad. Wet. B 82: 42–45.

    Google Scholar 

  • N. G. Meyers [1965]. Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc. 119, 225–249.

    Google Scholar 

  • C. B. Morrey [1952]. Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 25–53.

    Google Scholar 

  • C. B. Morrey [1966]. Multiple Integrals in the Calculus of Variations, Springer.

  • M. Potier-Ferry [1982]. On the mathematical foundations of elastic stability theory. I., Arch. Rational Mech. Anal. 78, 55–72.

    Google Scholar 

  • H. Rund [1963]. On the Weierstrass excess function of parameter-invariant multiple integrals in the calculus of variations. Tydskr. Natuurwetensk 3, 168–179.

    Google Scholar 

  • H. Rund [1974]. Integral formulae associated with the Euler-Lagrange operators of multiple integral problems in the Calculus of Variations, Aeq. Math. 11, 212–229.

    Google Scholar 

  • D. H. Sattinger [1969]. On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30, 148–172.

    Google Scholar 

  • R. T. Shield & A. E. Green [1963]. On certain methods in the stability theory of continuous systems, Arch. Rational Mech. Anal. 12, 354–360.

    Google Scholar 

  • T. Valent [1981]. Local theorems of existence and uniqueness in finite elastostatics, in Proc. IUTAM Symp. on Finite Elasticity, pp. 401–421, D. T. Carlson & R. T. Shield (eds.), Martinus Nijhoff Publishers.

  • L. van Hove [1949]. Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues, Koninkl. Belg. Acad., Klasse der Wetenschappen, Verhandelingen, Vol. 24.

  • Y. H. Wan, J. E. Marsden, T. Ratiu & A. Weinstein [1983]. Nonlinear stability of circular vortex patches (to appear).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to Jerry Ericksen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ball, J.M., Marsden, J.E. Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal. 86, 251–277 (1984). https://doi.org/10.1007/BF00281558

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00281558

Keywords

Navigation