Kirchhoff’s problem for nonlinearly elastic rods
Author:
Stuart S. Antman
Journal:
Quart. Appl. Math. 32 (1974), 221-240
MSC:
Primary 73.53
DOI:
https://doi.org/10.1090/qam/667026
MathSciNet review:
667026
Full-text PDF Free Access
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Additional Information
- Stuart Antman, Existence of solutions of the equilibrium equations for non-linearly elastic rings and arches, Indiana Univ. Math. J. 20 (1970/71), 281–302. MR 266478, DOI https://doi.org/10.1512/iumj.1970.20.20025
---, Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rat. Mech. Anal. 40, 329–372 (1971)
---, The theory of rods, Handbuch der Physik Vla/2, Springer-Verlag, Berlin, 1972
---, Qualitative theory of the ordinary differential equations of nonlinear elasticity, Mechanics Today 1 (1972), Pergamon, Oxford
---, One-dimensional boundary value problems of nonlinear elasticity, in preparation
- Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60. MR 0269962
- G. F. Carrier, On the buckling of elastic rings, J. Math. Phys. Mass. Inst. Tech. 26 (1947), 94–103. MR 22525, DOI https://doi.org/10.1002/sapm194726194
A Clebsch, Theorie der Elasticität fester Körper, Leipzig, 1862
H. Cohen, A non-linear theory of elastic directed curves, Int. J. Eng. Sci. 4, 5ll–524 (1966)
E. and F. Cosserat, Théorie des corps déformables, Hermann, Paris, 1909
C. N. DeSilva and A. B. Whitman, A thermodynamic theory of directed curves, J. Math Phys. 12, 1603–1609 (1971)
J. L. Ericksen, Simpler static problems in nonlinear theories of rods, Int. J. Solids Structures 6, 371–377 (1970)
L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne, 1744
A. E. Green and N. Laws, A general theory of rods, Proc. Roy. Soc. (London) A293, 145–155 (1966)
- G. Kirchhoff, Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. Reine Angew. Math. 56 (1859), 285–313 (German). MR 1579104, DOI https://doi.org/10.1515/crll.1859.56.285
- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
A. Nadai, Theory of flow and fracture of solids, Vol. I, 2nd ed., McGraw-Hill, New York, 1950
B. de St. Venant, Mémoire sur la torsion des prismes, etc., Mém. de Savants étrangers 14, 233–560 (1855)
J. T. Schwartz, Lectures on nonlinear functional analysis, New York Univ., 1964
- C. Truesdell, A new chapter in the theory of the elastica, Proceedings of the First Midwestern Conference on Solid Mechanics, April, 1953, The Engineering Experiment Station, University of Illinois, Urbana, Ill., 1954, pp. 52–55. MR 0061546
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
A. B. Whitman and C. N. DeSilva, An exact solution in a nonlinear theory of rods, to appear
S. S. Antman, Existence of solutions of the equilibrium equations for nonlinearly elastic elastic rings and arches, Indiana Univ. Math. J. 20, 281–302 (1970)
---, Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rat. Mech. Anal. 40, 329–372 (1971)
---, The theory of rods, Handbuch der Physik Vla/2, Springer-Verlag, Berlin, 1972
---, Qualitative theory of the ordinary differential equations of nonlinear elasticity, Mechanics Today 1 (1972), Pergamon, Oxford
---, One-dimensional boundary value problems of nonlinear elasticity, in preparation
F. E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis III, American Math. Soc., Providence, 1970
G. F. Carrier, On the buckling of elastic rings, J. Math. Phys. 26, 94–103 (1947)
A Clebsch, Theorie der Elasticität fester Körper, Leipzig, 1862
H. Cohen, A non-linear theory of elastic directed curves, Int. J. Eng. Sci. 4, 5ll–524 (1966)
E. and F. Cosserat, Théorie des corps déformables, Hermann, Paris, 1909
C. N. DeSilva and A. B. Whitman, A thermodynamic theory of directed curves, J. Math Phys. 12, 1603–1609 (1971)
J. L. Ericksen, Simpler static problems in nonlinear theories of rods, Int. J. Solids Structures 6, 371–377 (1970)
L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne, 1744
A. E. Green and N. Laws, A general theory of rods, Proc. Roy. Soc. (London) A293, 145–155 (1966)
G. Kirchhoff, Über des Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine angew. Math. (Crelle) 56, 285–313 (1859)
A. E. H. Love, A treatise on the mathematical theory of elasticity, 4th ed., Dover, New York, 1944
A. Nadai, Theory of flow and fracture of solids, Vol. I, 2nd ed., McGraw-Hill, New York, 1950
B. de St. Venant, Mémoire sur la torsion des prismes, etc., Mém. de Savants étrangers 14, 233–560 (1855)
J. T. Schwartz, Lectures on nonlinear functional analysis, New York Univ., 1964
C. Truesdell, A new chapter in the theory of the elastica, Proc. First Midwestern Conf. on Solid Mechanics 1953, 52–55, 1954
C. Truesdell and W. Noll, The nonlinear field theories of mechanics, Handbuch der Physik III/3, Springer-Verlag, Berlin, 1965
A. B. Whitman and C. N. DeSilva, An exact solution in a nonlinear theory of rods, to appear
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© Copyright 1974
American Mathematical Society
