Convexity conditions and existence theorems in nonlinear elasticity

John M. Ball

doi:10.1007/bf00279992

journal article Dec 01, 1976

Convexity conditions and existence theorems in nonlinear elasticity

John M. Ball

Archive for Rational Mechanics and Analysis Vol. 63 No. 4 pp. 337-403 · Springer Science and Business Media LLC

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References

97

[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. 10.1215/s0012-7094-56-02343-2

[2]

A.R. Amir-Moéz & A. Horn [1] Singular values of a matrix, Amer. Math. Monthly (1958), 742–748. 10.1080/00029890.1958.11991995

[3]

S.S. Antman [1] Equilibrium states of nonlinearly elastic rods, J. Math. Anal. Appl. 23 (1968), 459–470. 10.1016/0022-247x(68)90083-8

[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. 10.1512/iumj.1971.20.20025

[5]

S.S. Antman [3] Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rational Mech. Anal. 40 (1971), 329–372. 10.1007/bf00251796

[6]

S.S. Antman [4] “The theory of rods”, in Handbuch der Physik, Vol VIa/2, ed. C. Truesdell, Springer, Berlin, 1972.

[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.

[8]

S.S. Antman [6] Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl. 44 (1973), 333–349. 10.1016/0022-247x(73)90063-2

[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. 10.1007/bf00250722

[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. 10.1007/bf00250723

[11]

J.M. Ball [1] Weak continuity properties of mappings and semigroups, Proc. Roy. Soc. Edin (A) 72 (1973/4), 275–280. 10.1017/s008045410000964x

[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. 10.1007/bfb0087322

[13]

M.F. Beatty [1] Stability of hyperelastic bodies subject to hydrostatic loading, Non-linear Mech. 5 (1970), 367–383. 10.1016/0020-7462(70)90001-6

[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. 10.1007/bf00282314

[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.

[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. 10.1007/bf01343323

[17]

H. Busemann & G.C. Shephard [1] Convexity on nonconvex sets, Proc. Coll. on Convexity, Copenhagen, Univ. Math. Inst., Copenhagen, (1965), 20–33.

[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.

[19]

L. Cesari [1] Closure theorems for orientor fields and weak convergence, Arch. Rational Mech. Anal. 55 (1974), 332–356. 10.1007/bf00250438

[20]

L. Cesari [2] Lower semicontinuity and lower closure theorems without seminormality conditions, Annali Mat. Pura. Appl. 98 (1974), 381–397. 10.1007/bf02414036

[21]

L. Cesari [3] A necessary and sufficient condition for lower semicontinuity, Bull. Amer. Math. Soc. 80 (1974), 467–472. 10.1090/s0002-9904-1974-13452-x

[22]

A. Clebsch [1] Über die zweite Variation vielfacher Integrale, J. Reine Angew. Math., 56 (1859), 122–149. 10.1515/crll.1859.56.122

[23]

B.D. Coleman & W. Noll [1] On the thermostatics of continuous media, Arch. Rational Mech. Anal., 4 (1959), 97–128. 10.1007/bf00281381

[24]

T.K. Donaldson & N.S. Trudinger [1] Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal., 8 (1971), 52–75. 10.1016/0022-1236(71)90018-8

[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.

[26]

N. Dunford & J.T. Schwartz [1] “Linear operators”, Pt. 1., Interscience, New York 1958.

[27]

D.G.B. Edelen [1] The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal., 11 (1962), 117–121. 10.1007/bf00253934

[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.

[29]

I. Ekeland & R. Témam [1] “Analyse convexe et problèmes variationnels”, Dunod, Gauthier-Villars, Paris, 1974.

[30]

J.L. Ericksen [1] Nilpotent energies in liquid crystal theory, Arch. Rational Mech. Anal., 10 (1962), 189–196. 10.1007/bf00281186

[31]

J.L. Ericksen [2] Loading devices and stability of equilibrium, in “Nonlinear Elasticity”, ed. R.W. Dickey, Academic Press, New York 1973.

[32]

J.L. Ericksen [3] Equilibrium of bars, J. of Elasticity, 5 (1975), 191–201. 10.1007/bf00126984

[33]

J.L. Ericksen [4] Special topics in elastostatics, to appear.

[34]

G. Fichera [1] Existence theorems in elasticity, in Handbuch der Physik, ed. C. Truesdell, Vol. VIa/2, Springer, Berlin, 1972.

[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. 10.1007/bf00281227

[36]

A. Fougères [1] Thesis, Besançon, 1972.

[37]

A. Friedman [1] “Partial differential equations”, Holt Rinehart and Winston, New York, 1969.

[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. 10.1090/s0002-9947-1974-0342854-2

[39]

L.M. Graves [1] The Weierstrass condition for multiple integral variation problems, Duke Math. J., 5 (1939), 656–660. 10.1215/s0012-7094-39-00554-5

[40]

A.E. Green & W. Zerna [1] “Theoretical elasticity”, 2nd edition, Oxford Univ. Press, 1968.

[41]

J. Hadamard [1] Sur une question de calcul des variations, Bull. Soc. Math. France, 30 (1902), 253–256.

[42]

J. Hadamard [2] “Leçons sur la propagation des ondes”, Paris, Hermann, 1903.

[43]

J. Hadamard [3] Sur quelques questions de calcul des variations, Bull. Soc. Math de France, 33 (1905), 73–80. 10.24033/bsmf.741

[44]

R. Hill [1] On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5 (1957), 229–241. 10.1016/0022-5096(57)90016-9

[45]

R. Hill [2] On constitutive inequalities for simple materials, I, J. Mech. Phys. Solids, 16 (1968), 229–242. 10.1016/0022-5096(68)90031-8

[46]

R. Hill [3] Constitutive inequalities for isotropic elastic solids under finite strain. Proc. Roy. Soc. London A314 (1970), 457–472 10.1098/rspa.1970.0018

[47]

J.T. Holden [1] Estimation of critical loads in elastic stability theory, Arch. Rational Mech. Anal., 17 (1964), 171–183. 10.1007/bf00282436

[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.

[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. 10.1007/bf00126996

[50]

M.A. Krasnosel'skii & Ya.B. Rutickii [1] “Convex functions and Orlicz spaces”, trans. L.F. Boron, Nordhoff, Groningen, 1961.

Showing 50 of 97 references

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Published: Dec 01, 1976
Vol/Issue: 63(4)
Pages: 337-403
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John M. Ball

Cite This Article

John M. Ball (1976). Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis, 63(4), 337-403. https://doi.org/10.1007/bf00279992

Convexity conditions and existence theorems in nonlinear elasticity

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