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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.
A.R. Amir-Moéz & A. Horn [1] Singular values of a matrix, Amer. Math. Monthly (1958), 742–748.
S.S. Antman [1] Equilibrium states of nonlinearly elastic rods, J. Math. Anal. Appl. 23 (1968), 459–470.
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.
S.S. Antman [3] Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rational Mech. Anal. 40 (1971), 329–372.
S.S. Antman [4] “The theory of rods”, in Handbuch der Physik, Vol VIa/2, ed. C. Truesdell, Springer, Berlin, 1972.
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.
S.S. Antman [6] Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl. 44 (1973), 333–349.
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.
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.
J.M. Ball [1] Weak continuity properties of mappings and semigroups, Proc. Roy. Soc. Edin (A) 72 (1973/4), 275–280.
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.
M.F. Beatty [1] Stability of hyperelastic bodies subject to hydrostatic loading, Non-linear Mech. 5 (1970), 367–383.
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.
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.
H. Busemann, G. Ewald & G.C. Shephard [1] Convex bodies and convexity on Grassman cones, Parts I–IV, Math. Ann., 151 (1963), 1–41.
H. Busemann & G.C. Shephard [1] Convexity on nonconvex sets, Proc. Coll. on Convexity, Copenhagen, Univ. Math. Inst., Copenhagen, (1965), 20–33.
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.
L. Cesari [1] Closure theorems for orientor fields and weak convergence, Arch. Rational Mech. Anal. 55 (1974), 332–356.
L. Cesari [2] Lower semicontinuity and lower closure theorems without seminormality conditions, Annali Mat. Pura. Appl. 98 (1974), 381–397.
L. Cesari [3] A necessary and sufficient condition for lower semicontinuity, Bull. Amer. Math. Soc. 80 (1974), 467–472.
A. Clebsch [1] Über die zweite Variation vielfacher Integrale, J. Reine Angew. Math., 56 (1859), 122–149.
B.D. Coleman & W. Noll [1] On the thermostatics of continuous media, Arch. Rational Mech. Anal., 4 (1959), 97–128.
T.K. Donaldson & N.S. Trudinger [1] Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal., 8 (1971), 52–75.
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.
N. Dunford & J.T. Schwartz [1] “Linear operators”, Pt. 1., Interscience, New York 1958.
D.G.B. Edelen [1] The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal., 11 (1962), 117–121.
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.
I. Ekeland & R. Témam [1] “Analyse convexe et problèmes variationnels”, Dunod, Gauthier-Villars, Paris, 1974.
J.L. Ericksen [1] Nilpotent energies in liquid crystal theory, Arch. Rational Mech. Anal., 10 (1962), 189–196.
J.L. Ericksen [2] Loading devices and stability of equilibrium, in “Nonlinear Elasticity”, ed. R.W. Dickey, Academic Press, New York 1973.
J.L. Ericksen [3] Equilibrium of bars, J. of Elasticity, 5 (1975), 191–201.
J.L. Ericksen [4] Special topics in elastostatics, to appear.
G. Fichera [1] Existence theorems in elasticity, in Handbuch der Physik, ed. C. Truesdell, Vol. VIa/2, Springer, Berlin, 1972.
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.
A. Fougères [1] Thesis, Besançon, 1972.
A. Friedman [1] “Partial differential equations”, Holt Rinehart and Winston, New York, 1969.
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.
L.M. Graves [1] The Weierstrass condition for multiple integral variation problems, Duke Math. J., 5 (1939), 656–660.
A.E. Green & W. Zerna [1] “Theoretical elasticity”, 2nd edition, Oxford Univ. Press, 1968.
J. Hadamard [1] Sur une question de calcul des variations, Bull. Soc. Math. France, 30 (1902), 253–256.
J. Hadamard [2] “Leçons sur la propagation des ondes”, Paris, Hermann, 1903.
J. Hadamard [3] Sur quelques questions de calcul des variations, Bull. Soc. Math de France, 33 (1905), 73–80.
R. Hill [1] On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5 (1957), 229–241.
R. Hill [2] On constitutive inequalities for simple materials, I, J. Mech. Phys. Solids, 16 (1968), 229–242.
R. Hill [3] Constitutive inequalities for isotropic elastic solids under finite strain. Proc. Roy. Soc. London A314 (1970), 457–472
J.T. Holden [1] Estimation of critical loads in elastic stability theory, Arch. Rational Mech. Anal., 17 (1964), 171–183.
R.J. Knops & E.W. Wilkes [1] “Theory of elastic stability”, in Handbuch der Physik, Vol. VIa/3, ed. C. Truesdell, Springer, Berlin, 1973.
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.
M.A. Krasnosel'skii & Ya.B. Rutickii [1] “Convex functions and Orlicz spaces”, trans. L.F. Boron, Nordhoff, Groningen, 1961.
M-T. Lacroix [1] Espaces de traces des espaces de Sobolev-Orlicz, J. de Math. Pures Appl., 53 (1974), 439–458.
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.
N.G. Meyers [1] Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125–149.
L. Mirsky [1] On the trace of matrix products, Math. Nach. 20 (1959), 171–174.
L. Mirsky [2] A trace inequality of John von Neumann, Monat. für Math., 79 (1975), 303–306.
J.J. Moreau [1] Fonctionnelles convexes, Séminaire sur les équations aux dérivées partielles, Collège de France, 1966–1967.
C.B. Morrey, Jr. [1] Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25–53.
C.B. Morrey, Jr. [2] “Multiple Integrals in the Calculus of Variations”, Springer, Berlin, 1966.
J. Nečas [1] “Les méthodes directes en théorie des équations elliptiques”, Masson, Paris, 1967.
P. Niederer [1] A molecular study of the mechanical properties of arterial wall vessels, Z.A.M.P., 25 (1974), 565–578.
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.
R.W. Ogden [1] Compressible isotropic elastic solids under finite strain — constitutive inequalities, Quart, J. Mech. Appl. Math., 23 (1970), 457–468.
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.
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.
A. Phillips [1] Turning a surface inside out, Scientific American, May 1966.
Y. G. Reshetnyak [1] On the stability of conformal mappings in multidimensional spaces, Sibirskii Math. 8 (1967), 91–114.
Y. G. Reshetnyak [2] Stability theorems for mappings with bounded excursion, Sibirskii Math. 9 (1968), 667–684.
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.
R.S. Rivlin [2] Some restrictions on constitutive equations, Proc. Int. Symp. on the Foundations of Continuum Thermodynamics, Bussaco, 1973.
R.S. Rivlin [3] Stability of pure homogeneous deformations of an elastic cube under dead loading, Quart. Appl. Math. 32 (1974), 265–272.
R.T. Rockafellar [1] “Convex analysis”, Princeton University Press, Princeton, New Jersey, 1970.
H. Rund [1] “The Hamilton-Jacobi theory in the calculus of variations”, Van Nostrand, London, 1966.
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.
L. Schwartz [1] “Théorie des distributions”, Hermann, Paris, 1966.
C.B. Sensenig [1] Instability of thick elastic solids, Comm. Pure Appl. Math., 17 (1964), 451–491.
M.J. Sewell [1] On configuration-dependent loading, Arch. Rational Mech. Anal., 23 (1967), 327–351.
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.
E. Silverman [1] Strong quasi-convexity, Pacific J. Math., 46 (1973), 549–554.
S. Smale [1] A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90, (1959), 281–290.
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.
R. Temam [1] On the theory and numerical analysis of the Navier-Stokes equations, Lecture notes in Mathematics No. 9, University of Maryland.
F.J. Terpstra [1] Die Darstellung biquadratischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166–180.
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.
R.C. Thompson [1] Singular value inequalities for matrix sums and minors, Linear Algebra and Appl., 11 (1975), 251–269.
R.C. Thompson & L.J. Freede [1] On the eigenvalues of sums of Hermitian matrices, Linear Algebra and Appl., 4 (1971), 369–376.
R.C. Thompson & L.J. Freede [2] On the eigenvalues of sums of Hermitian matrices II, Aequationes Math., 5 (1970), 103–115.
R.C. Thompson & L.J. Freede [3] Eigenvalues of sums of Hermitian matrices III, J. Research Nat. Bur. Standards B, 75B (1971), 115–120.
L.R.G. Treloar [1] “The physics of rubber elasticity”, 3rd edition, Oxford Univ. Press, Oxford, 1975.
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.
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.
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.
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.
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).
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.
C.-C. Wang & C. Truesdell [1] “Introduction to rational elasticity,” Noordhoff, Groningen, 1973.
Z. Wesołowski [1] “Zagadnienia dynamiczne nieliniowej teorii sprezystosci”, Polska Akad. Nauk. IPPT, Warsaw, 1974.
E.W. Wilkes [1] On the stability of a circular tube under end thrust, Quart. J. Mech. Appl. Math. 8 (1955), 88–100.
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Communicated by S.S. Antman & C.M. Dafermos
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Ball, J.M. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337–403 (1976). https://doi.org/10.1007/BF00279992
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DOI: https://doi.org/10.1007/BF00279992