×

zbMATH — the first resource for mathematics

Ordinary differential equations of non-linear elasticity. I: Foundations of the theories of non-linearly elastic rods and shells. (English) Zbl 0354.73046

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K25 Shells
47E05 General theory of ordinary differential operators
74K15 Membranes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. S. Antman (1970), The shape of buckled nonlinearly elastic rings. Z.A.M.P. 21, 422–438. · Zbl 0214.52005
[2] S. S. Antman (1971), Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells. Arch. Rational Mech. Anal. 40, 329–372. · Zbl 0254.73072
[3] S. S. Antman (1972), The Theory of Rods. Handbuch der Physik, Vol. VI a/2. Springer-Verlag, Berlin, Heidelberg, New York.
[4] S. S. Antman (1973a), Nonuniqueness of equilibrium states for bars in tension. J. Math. Anal. Appl. 44, 333–349. · Zbl 0267.73031
[5] S. S. Antman (1973b), Monotonicity and invertibility conditions in one-dimensional nonlinear elasticity. Symposium on Nonlinear Elasticity, Mathematics Research Center, Univ. Wisconsin, Academic Press, New York, 57–92. · Zbl 0294.73042
[6] S. S. Antman (1974a), Qualitative Theory of the Ordinary Differential Equations of Nonlinear Elasticity. Mechanics Today, 1972, Pergamon, New York, 58–101.
[7] S. S. Antman (1974b), Kirchhoff’s problem for nonlinearly elastic rods. Q. Appl. Math. 32, 221–240. · Zbl 0302.73031
[8] S. S. Antman & E. Carbone (1976), Shear and necking instabilities in nonlinear elasticity. J. Elasticity, to appear. · Zbl 0356.73048
[9] S. S. Antman & K. B. Jordan (1975), Qualitative aspects of the spatial deformation of nonlinearly elastic rods. Proc. Roy. Soc. Edinburgh 73A, 85–105. · Zbl 0351.73076
[10] J. M. Ball (1974), personal communication.
[11] R. C. Batra (1972), On non-classical boundary conditions. Arch. Rational Mech. Anal. 48, 163–191. · Zbl 0252.73010
[12] I. Beju (1971), 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, 1–23. · Zbl 0224.73015
[13] J. F. Bell (1973), The Experimental Foundations of Solid Mechanics. Handbuch der Physik, Vol. VI a/1. Springer-Verlag, Berlin, Heidelberg, New York.
[14] H. Brezis (1968), Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18, 115–175. · Zbl 0169.18602
[15] F. E. Browder (1970), Existence theorems for nonlinear partial differential equations. Proc. Symp. Pure Math., Vol. 16, Amer. Math. Soc., Providence, 1–60.
[16] M. M. Carroll & P. M. Naghdi (1972), The influence of the reference geometry, on the response of elastic shells. Arch. Rational Mech. Anal. 48, 302–318. · Zbl 0283.73034
[17] B. D. Coleman & W. Noll (1959), On the thermostatics of continuous media. Arch. Rational Mech. Anal. 4, 97–128. · Zbl 0231.73003
[18] G. Duvaut & J. L. Lions (1972), Les inéquations en mécanique et en physique. Dunod, Paris. · Zbl 0298.73001
[19] I. Ekeland & R. Temam (1974), Analyse convexe et problèmes variationnels. Dunod, Gauthier-Villars. Paris.
[20] J. L. Ericksen (1972), Symmetry transformations for thin elastic shells. Arch. Rational Mech. Anal. 47, 1–14. · Zbl 0242.73045
[21] J. L. Ericksen (1974): Plane waves and stability of elastic plates. Q. Appl. Math. 32, 343–345. · Zbl 0327.73041
[22] J. L. Ericksen & R. S. Rivlin (1954), Large elastic deformations of homogeneous anisotropic materials. J. Rational Mech. Anal. 3, 281–301. · Zbl 0055.18103
[23] G. Fichera (1972a), Existence Theorems in Elasticity. Handbuch der Physik, Vol.IV a/2. Springer-Verlag, Berlin, Heidelberg, New York.
[24] G. Fichera (1972b), Boundary Value Problems of Elasticity with Unilateral Constraints. Handbuch der Physik, Vol.VI a/2. Springer-Verlag, Berlin, Heidelberg, New York.
[25] G. E. Hay (1942), The finite displacement of thin rods. Trans. Am. Math. Soc. 51, 65–102. · Zbl 0061.42206
[26] M. Hayes (1969), Static implications of the strong ellipticity condition. Arch. Rational Mech. Anal. 33, 181–191. · Zbl 0201.26502
[27] R. Hill (1970), Constitutive inequalities for isotropic elastic solids under finite strain. Proc. Roy. Soc. (Ser. A) 314, 457–472. · Zbl 0201.26601
[28] F. John (1975), A priori estimates, geometric effects, and asymptotic behavior. Bull. Am. Math. Soc. 81, 1013–1023. · Zbl 0322.35004
[29] R. J. Knops & L. E. Payne (1971), Uniqueness Theorems in Linear Elasticity. Springer-Verlag, New York, Heidelberg, Berlin. · Zbl 0224.73016
[30] M. A. Krasnosel’skii & Ya. B. Rutitskii (1958), Convex functions and Onlicz spaces (in Russian), Fizmatgiz, Moscow. (English translation by L. F. Boron (1961), Noordhoff, Groningen).
[31] J. E. Lavery (1976), Conjugate quasilinear Dirichlet and Neumann problems and a posteriori error bounds (to appear).
[32] S. H. Leventhal (1975), The methods of moments and its optimization. Int. J. Num. Methods in Engg. 9, 337–351. · Zbl 0306.65071
[33] J. L. Lions (1969), Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris.
[34] C. B. Morrey (1966), Multiple Integrals in the Calculus of Variations. Springer-Verlag, Berlin, Heidelberg, New York. · Zbl 0142.38701
[35] P. M. Naghdi (1972), The Theory of Shells. Handbuch der Physik Vol. VI a/2. Springer-Verlag, Berlin, Heidelberg, New York.
[36] W. E. Olmstead & D. J. Mescheloff (1974), Buckling of a nonlinear elastic rod. J. Math. Anal. Appl. 46, 609–634. · Zbl 0308.73023
[37] J. F. Pierce (1973), Dissertation, Univ. of Houston.
[38] E. L. Reiss & S. Locke (1961), On the theory of plane stress. Q. Appl. Mech. 19, 195–203. · Zbl 0107.18303
[39] A. Rigolot (1972), Sur une théorie asymptotique des poutres. J. de Méc. 11, 674–703. · Zbl 0257.73013
[40] R. T. Rockafellar (1970), Convex Analysis. Princeton Univ. Press, Princeton. · Zbl 0193.18401
[41] M. J. Sewell (1967), On configuration-dependent loading. Arch. Rational Mech. Anal. 23, 327–351. · Zbl 0166.43406
[42] M. Shahinpoor (1974), Plane waves and stability in thin elastic circular cylindrical shells. Arch. Rational Mech. Anal. 54, 267–280. · Zbl 0285.73095
[43] T. W. Ting (1974), St. Venant’s compatibility conditions. Tensor, N. S. 28, 5–12. · Zbl 0291.46022
[44] C. Truesdell & W. Noll (1965), The Non-Linear Field Theories of Mechanics. Handbuch der Physik, Vol.III/3. Springer-Verlag, Berlin, Heidelberg, New York.
[45] C. Truesdell & R. Toupin (1960), The Classical Field Theories. Handbuch der Physik Vol. III/1. Springer-Verlag, Berlin, Heidelberg, New York.
[46] C. Truesdell & R. Toupin (1963), Static grounds for inequalities in finite elastic strain. Arch. Rational Mech. Anal. 12, 1–33. · Zbl 0119.19201
[47] C.-C. Wang (1972). Material uniformity and homogeneity in shells. Arch. Rational Mech. Anal. 47, 343–368. · Zbl 0266.73054
[48] C.-C. Wang (1973), On the response functions of isotropic elastic shells. Arch. Rational Mech. Anal. 50, 81–98. · Zbl 0276.73045
[49] C.-C. Wang & C. Truesdell (1973), Introduction to Rational Elasticity. Noordhoff, Leyden.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.