Archive for Rational Mechanics and Analysis

, Volume 63, Issue 4, pp 337–403 | Cite as

Convexity conditions and existence theorems in nonlinear elasticity

  • John M. Ball


Neural Network Complex System Nonlinear Dynamics Electromagnetism Existence Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. A.R. Amir-Moéz [1] Extreme properties of eigenvalues of a Hermitian transformation and singular values of sum and product of linear transformations, Duke Math. J., 23 (1956), 463–476.Google Scholar
  2. A.R. Amir-Moéz & A. Horn [1] Singular values of a matrix, Amer. Math. Monthly (1958), 742–748.Google Scholar
  3. S.S. Antman [1] Equilibrium states of nonlinearly elastic rods, J. Math. Anal. Appl. 23 (1968), 459–470.Google Scholar
  4. S.S. Antman [2]S.S. Antman Existence of solutions of the equilibrium equations for nonlinearly elastic rings and arches, Indiana Univ. Math. J. 20 (1970) 281–302.Google Scholar
  5. S.S. Antman [3] Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rational Mech. Anal. 40 (1971), 329–372.Google Scholar
  6. S.S. Antman [4] “The theory of rods”, in Handbuch der Physik, Vol VIa/2, ed. C. Truesdell, Springer, Berlin, 1972.Google Scholar
  7. S.S. Antman [5] Monotonicity and invertibility conditions in one-dimensional nonlinear elasticity, in “Nonlinear Elasticity”, ed. R.W. Dickey, Academic Press, New York, 1973.Google Scholar
  8. S.S. Antman [6] Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl. 44 (1973), 333–349.Google Scholar
  9. S.S. Antman [7] Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells, Arch. Rational Mech. Anal. 61 (1976), 307–351.Google Scholar
  10. S.S. Antman [8] Ordinary differential equations of nonlinear elasticity II: Existence and regularity theory for conservative boundary value problems, Arch. Rational Mech. Anal. 61 (1976), 353–393.Google Scholar
  11. J.M. Ball [1] Weak continuity properties of mappings and semigroups, Proc. Roy. Soc. Edin (A) 72 (1973/4), 275–280.Google Scholar
  12. J.M. Ball [2] On the calculus of variations and sequentially weakly continuous maps, Proc. Dundee Conference on Ordinary and Partial Differential Equations 1976, Springer Lecture Notes in Mathematics, to appear.Google Scholar
  13. M.F. Beatty [1] Stability of hyperelastic bodies subject to hydrostatic loading, Non-linear Mech. 5 (1970), 367–383.Google Scholar
  14. I. Beju [1] Theorems on existence, uniqueness and stability of the solution of the place boundary-value problem, in statics, for hyperelastic materials, Arch. Rational Mech. Anal., 42 (1971), 1–23.Google Scholar
  15. I. Beju [2] The place boundary-value problem in hyperelastostatics, I. Differential properties of the operator of finite elastostatics, Bull. Math. Soc. Sci. Math. R.S. Roumanie 16 (1972), 132–149, II. Existence, uniqueness and stability of the solution, ibid. 283–313.Google Scholar
  16. H. Busemann, G. Ewald & G.C. Shephard [1] Convex bodies and convexity on Grassman cones, Parts I–IV, Math. Ann., 151 (1963), 1–41.Google Scholar
  17. H. Busemann & G.C. Shephard [1] Convexity on nonconvex sets, Proc. Coll. on Convexity, Copenhagen, Univ. Math. Inst., Copenhagen, (1965), 20–33.Google Scholar
  18. C. Cattaneo [1] Su un teorema fondamentale nella teoria delle onde di discontinuità, Atti. Accad. Sci. Lincei Rend., Cl. Sci. Fis. Mat. Nat. Ser 8, 1 (1946), 66–72.Google Scholar
  19. L. Cesari [1] Closure theorems for orientor fields and weak convergence, Arch. Rational Mech. Anal. 55 (1974), 332–356.Google Scholar
  20. L. Cesari [2] Lower semicontinuity and lower closure theorems without seminormality conditions, Annali Mat. Pura. Appl. 98 (1974), 381–397.Google Scholar
  21. L. Cesari [3] A necessary and sufficient condition for lower semicontinuity, Bull. Amer. Math. Soc. 80 (1974), 467–472.Google Scholar
  22. A. Clebsch [1] Über die zweite Variation vielfacher Integrale, J. Reine Angew. Math., 56 (1859), 122–149.Google Scholar
  23. B.D. Coleman & W. Noll [1] On the thermostatics of continuous media, Arch. Rational Mech. Anal., 4 (1959), 97–128.Google Scholar
  24. T.K. Donaldson & N.S. Trudinger [1] Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal., 8 (1971), 52–75.Google Scholar
  25. P. Duhem [1] Recherches sur l'élasticité, troisième partie. La stabilité des milieux élastiques, Ann. Ecole Norm., 22 (1905), 143–217. Reprinted Paris, Gauthier-Villars 1906.Google Scholar
  26. N. Dunford & J.T. Schwartz [1] “Linear operators”, Pt. 1., Interscience, New York 1958.Google Scholar
  27. D.G.B. Edelen [1] The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal., 11 (1962), 117–121.Google Scholar
  28. D.G.B. Edelen [2] “Non local variations and local invariance of fields”, Modern analytic and computational methods in science and engineering No. 19, Elsevier, New York, 1969.Google Scholar
  29. I. Ekeland & R. Témam [1] “Analyse convexe et problèmes variationnels”, Dunod, Gauthier-Villars, Paris, 1974.Google Scholar
  30. J.L. Ericksen [1] Nilpotent energies in liquid crystal theory, Arch. Rational Mech. Anal., 10 (1962), 189–196.Google Scholar
  31. J.L. Ericksen [2] Loading devices and stability of equilibrium, in “Nonlinear Elasticity”, ed. R.W. Dickey, Academic Press, New York 1973.Google Scholar
  32. J.L. Ericksen [3] Equilibrium of bars, J. of Elasticity, 5 (1975), 191–201.Google Scholar
  33. J.L. Ericksen [4] Special topics in elastostatics, to appear.Google Scholar
  34. G. Fichera [1] Existence theorems in elasticity, in Handbuch der Physik, ed. C. Truesdell, Vol. VIa/2, Springer, Berlin, 1972.Google Scholar
  35. R.L. Fosdick & R.T. Shield [1] Small bending of a circular bar superposed on finite extension or compression, Arch. Rational Mech. Anal., 12 (1963), 223–248.Google Scholar
  36. A. Fougères [1] Thesis, Besançon, 1972.Google Scholar
  37. A. Friedman [1] “Partial differential equations”, Holt Rinehart and Winston, New York, 1969.Google Scholar
  38. J-P. Gossez [1] Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163–204.Google Scholar
  39. L.M. Graves [1] The Weierstrass condition for multiple integral variation problems, Duke Math. J., 5 (1939), 656–660.Google Scholar
  40. A.E. Green & W. Zerna [1] “Theoretical elasticity”, 2nd edition, Oxford Univ. Press, 1968.Google Scholar
  41. J. Hadamard [1] Sur une question de calcul des variations, Bull. Soc. Math. France, 30 (1902), 253–256.Google Scholar
  42. J. Hadamard [2] “Leçons sur la propagation des ondes”, Paris, Hermann, 1903.Google Scholar
  43. J. Hadamard [3] Sur quelques questions de calcul des variations, Bull. Soc. Math de France, 33 (1905), 73–80.Google Scholar
  44. R. Hill [1] On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5 (1957), 229–241.Google Scholar
  45. R. Hill [2] On constitutive inequalities for simple materials, I, J. Mech. Phys. Solids, 16 (1968), 229–242.Google Scholar
  46. R. Hill [3] Constitutive inequalities for isotropic elastic solids under finite strain. Proc. Roy. Soc. London A314 (1970), 457–472Google Scholar
  47. J.T. Holden [1] Estimation of critical loads in elastic stability theory, Arch. Rational Mech. Anal., 17 (1964), 171–183.Google Scholar
  48. R.J. Knops & E.W. Wilkes [1] “Theory of elastic stability”, in Handbuch der Physik, Vol. VIa/3, ed. C. Truesdell, Springer, Berlin, 1973.Google Scholar
  49. J.K. Knowles & E. Sternberg [1] On the ellipticity of the equations of nonlinear elastostatics for a special material, J. of Elasticity, 5 (1975), 341–361.Google Scholar
  50. M.A. Krasnosel'skii & Ya.B. Rutickii [1] “Convex functions and Orlicz spaces”, trans. L.F. Boron, Nordhoff, Groningen, 1961.Google Scholar
  51. M-T. Lacroix [1] Espaces de traces des espaces de Sobolev-Orlicz, J. de Math. Pures Appl., 53 (1974), 439–458.Google Scholar
  52. E.J. McShane [1] On the necessary condition of Weierstrass in the multiple integral problem of the calculus of variations, Annals of Math. Series 2, 32 (1931), 578–590.Google Scholar
  53. N.G. Meyers [1] Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125–149.Google Scholar
  54. L. Mirsky [1] On the trace of matrix products, Math. Nach. 20 (1959), 171–174.Google Scholar
  55. L. Mirsky [2] A trace inequality of John von Neumann, Monat. für Math., 79 (1975), 303–306.Google Scholar
  56. J.J. Moreau [1] Fonctionnelles convexes, Séminaire sur les équations aux dérivées partielles, Collège de France, 1966–1967.Google Scholar
  57. C.B. Morrey, Jr. [1] Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53.Google Scholar
  58. C.B. Morrey, Jr. [2] “Multiple Integrals in the Calculus of Variations”, Springer, Berlin, 1966.Google Scholar
  59. J. Nečas [1] “Les méthodes directes en théorie des équations elliptiques”, Masson, Paris, 1967.Google Scholar
  60. P. Niederer [1] A molecular study of the mechanical properties of arterial wall vessels, Z.A.M.P., 25 (1974), 565–578.Google Scholar
  61. J.T. Oden [1] Approximations and numerical analysis of finite deformations of elastic solids, in “Nonlinear Elasticity” ed. R. W. Dickey, Academic Press, New York, 1973.Google Scholar
  62. R.W. Ogden [1] Compressible isotropic elastic solids under finite strain — constitutive inequalities, Quart, J. Mech. Appl. Math., 23 (1970), 457–468.Google Scholar
  63. R.W. Ogden [2] Large deformation isotropic elasticity — on the correlation of theory and experiment for incompressible rubberlike solids, Proc. Roy. Soc. London A326 (1972), 565–584.Google Scholar
  64. R.W. Ogden [3] Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London A328 (1972), 567–583.Google Scholar
  65. A. Phillips [1] Turning a surface inside out, Scientific American, May 1966.Google Scholar
  66. Y. G. Reshetnyak [1] On the stability of conformal mappings in multidimensional spaces, Sibirskii Math. 8 (1967), 91–114.Google Scholar
  67. Y. G. Reshetnyak [2] Stability theorems for mappings with bounded excursion, Sibirskii Math. 9 (1968), 667–684.Google Scholar
  68. R. S. Rivlin [1] Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformation, Phil. Trans. Roy. Soc. London 240 (1948), 491–508.Google Scholar
  69. R.S. Rivlin [2] Some restrictions on constitutive equations, Proc. Int. Symp. on the Foundations of Continuum Thermodynamics, Bussaco, 1973.Google Scholar
  70. R.S. Rivlin [3] Stability of pure homogeneous deformations of an elastic cube under dead loading, Quart. Appl. Math. 32 (1974), 265–272.Google Scholar
  71. R.T. Rockafellar [1] “Convex analysis”, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  72. H. Rund [1] “The Hamilton-Jacobi theory in the calculus of variations”, Van Nostrand, London, 1966.Google Scholar
  73. H. Rund [2] Integral formulae associated with the Euler-Lagrange operators of multiple integral problems in the calculus of variations, Aequationes Math., 11 (1974), 212–229.Google Scholar
  74. L. Schwartz [1] “Théorie des distributions”, Hermann, Paris, 1966.Google Scholar
  75. C.B. Sensenig [1] Instability of thick elastic solids, Comm. Pure Appl. Math., 17 (1964), 451–491.Google Scholar
  76. M.J. Sewell [1] On configuration-dependent loading, Arch. Rational Mech. Anal., 23 (1967), 327–351.Google Scholar
  77. F. Sidoroff [1] Sur les restrictions à imposer à l'énergie de déformation d'un matériau hyperélastique, C.R.Acad. Sc. Paris A, 279 (1974), 379–382.Google Scholar
  78. E. Silverman [1] Strong quasi-convexity, Pacific J. Math., 46 (1973), 549–554.Google Scholar
  79. S. Smale [1] A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90, (1959), 281–290.Google Scholar
  80. F. Stoppelli [1] Un teorema di esistenza e di unicità relativo allé equazioni dell'elastostatica isoterma per deformazioni finite, Ricerche Matematica, 3 (1954), 247–267.Google Scholar
  81. R. Temam [1] On the theory and numerical analysis of the Navier-Stokes equations, Lecture notes in Mathematics No. 9, University of Maryland.Google Scholar
  82. F.J. Terpstra [1] Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166–180.Google Scholar
  83. C.M. Theobald [1] An inequality for the trace of the product of two symmetric matrices, Math. Proc. Camb. Phil. Soc., 77 (1975), 265–268.Google Scholar
  84. R.C. Thompson [1] Singular value inequalities for matrix sums and minors, Linear Algebra and Appl., 11 (1975), 251–269.Google Scholar
  85. R.C. Thompson & L.J. Freede [1] On the eigenvalues of sums of Hermitian matrices, Linear Algebra and Appl., 4 (1971), 369–376.Google Scholar
  86. R.C. Thompson & L.J. Freede [2] On the eigenvalues of sums of Hermitian matrices II, Aequationes Math., 5 (1970), 103–115.Google Scholar
  87. R.C. Thompson & L.J. Freede [3] Eigenvalues of sums of Hermitian matrices III, J. Research Nat. Bur. Standards B, 75B (1971), 115–120.Google Scholar
  88. L.R.G. Treloar [1] “The physics of rubber elasticity”, 3rd edition, Oxford Univ. Press, Oxford, 1975.Google Scholar
  89. C. Truesdell [1] The main open problem in the finite theory of elasticity (1955), reprinted in “Foundations of Elasticity Theory”, Intl. Sci. Rev. Ser. New York: Gordon and Breach 1965.Google Scholar
  90. C. Truesdell & W. Noll [1] “The non-linear field theories of mechanics”, in Handbuch der Physik Vol. III/3, ed. S. Flügge, Springer, Berlin, 1965.Google Scholar
  91. W. van Buren [1] “On the existence and uniqueness of solutions to boundary value problems in finite elasticity”, Thesis, Department of Mathematics, Carnegie-Mellon University, 1968. Research Report 68-ID7-MEKMA-RI, Westinghouse Research Laboratories, Pittsburgh, Pa. 1968.Google Scholar
  92. L. van Hove [1] Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues, Proc. Koninkl. Ned. Akad. Wetenschap 50 (1947), 18–23.Google Scholar
  93. L. van Hove [2] Sur le signe de la variation seconde des intégrales multiples à plusieurs fonctions inconnues, Koninkl. Belg. Acad., Klasse der Wetenschappen, Verhandelingen, 24 (1949).Google Scholar
  94. J. von Neumann [1] Some matrix-inequalities and metrization of matric-space, Tomsk Univ. Rev. 1 (1937), 286–300. Reprinted in Collected Works Vol. IV Pergamon, Oxford, 1962.Google Scholar
  95. C.-C. Wang & C. Truesdell [1] “Introduction to rational elasticity,” Noordhoff, Groningen, 1973.Google Scholar
  96. Z. Wesołowski [1] “Zagadnienia dynamiczne nieliniowej teorii sprezystosci”, Polska Akad. Nauk. IPPT, Warsaw, 1974.Google Scholar
  97. E.W. Wilkes [1] On the stability of a circular tube under end thrust, Quart. J. Mech. Appl. Math. 8 (1955), 88–100.Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • John M. Ball
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburgh

Personalised recommendations