Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The traction problem for incompressible materials


Author: Y. H. Wan
Journal: Trans. Amer. Math. Soc. 291 (1985), 103-119
MSC: Primary 73C50; Secondary 58E99, 73H99
MathSciNet review: 797048
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The traction problem for incompressible materials is treated as a bifurcation problem, where the applied loads are served as parameters. We take both the variational approach and the classical power series approach. The variational approach provides a natural, unified way of looking at this problem. We obtain a count of the number of equilibria together with the determination of their stability. In addition, it also lays down the foundation for the Signorini-Stoppelli type computations. We find second order sufficient conditions for the existence of power series solutions. As a consequence, the linearization stability follows, and it clarifies in some sense the role played by the linear elasticity in the context of the nonlinear elasticity theory. A systematic way of calculating the power series solution is also presented.


References [Enhancements On Off] (What's this?)

  • [1] J. M. Ball and D. G. Schaeffer, Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 315–339. MR 715037, 10.1017/S030500410006117X
  • [2] D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. I, Arch. Rational Mech. Anal. 80 (1982), no. 4, 295–331. MR 677564, 10.1007/BF00253119
  • [3] D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. II, Arch. Rational Mech. Anal. 83 (1983), no. 4, 363–395. MR 714980, 10.1007/BF00963840
  • [4] David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2) 92 (1970), 102–163. MR 0271984
  • [5] G. Fichera, Existence theorems in elasticity, Handbuch der Physik, Bd. VIa/2, Springer-Verlag, Berlin and New York, 1972, pp. 347-389.
  • [6] Arthur E. Fischer and Jerrold E. Marsden, Linearization stability of nonlinear partial differential equations, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 219–263. MR 0383456
  • [7] A. E. Green and E. B. Spratt, Second-order effects in the deformation of elastic bodies, Proc. Roy. Soc. London. Ser. A. 224 (1954), 347–361. MR 0063234
  • [8] G. Grioli, Mathematical theory of elastic equilibrium, Ergeb. Math. Grenzgeb., Band 7, Springer-Verlag, Berlin and New York, 1962.
  • [9] M. Gurtin, The linear theory of elasticity, Handbuch der Physik, vol. VIa/2, Springer-Verlag, Berlin, 1972, pp. 1-295.
  • [10] J. Marsden and T. Hughes, The mathematical foundations of elasticity, Prentice-Hall, Englewood Clifs, N.J., 1982.
  • [11] J. E. Marsden and Y. H. Wan, Linearization stability and Signorini series for the traction problem in elastostatics, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 1-2, 171–180. MR 723105, 10.1017/S0308210500015870
  • [12] A. Signorini, Sulle deformazioni termoelastiche finite, Proc. 3rd Internat. Congr. Appl. Mech., vol. 2, 1930, pp. 80-89.
  • [13] F. Stoppelli, Sulle esistenza di soluzioni delli equazioni delĺ elastostatica isoterma rel case di sollectizioni detate di assi di equilibria, Richerche Mat. 6 (1957), 241-287; 7 (1958), 71-101, 138-152.
  • [14] Carlo Tolotti, Orientamenti principali di un corpo elastico rispetto alla sua sollecitazione totale, Atti Accad. Italia. Mem. Cl. Sci. Fis. Mat. Nat. (7) 13 (1942), 1139–1162=Ist. Naz. Appl. Calcolo (2) no. 145 (1942) (Italian). MR 0013342
  • [15] C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
  • [16] C. C. Wang and C. Truesdell, Introduction to rational elasticity, Noordhoff International Publishing, Leyden, 1973. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Continua. MR 0468442

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 73C50, 58E99, 73H99

Retrieve articles in all journals with MSC: 73C50, 58E99, 73H99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0797048-0
Keywords: Traction problem, incompressible materials, variational principle, linearization stability, Signorini-Stoppelli Schemes
Article copyright: © Copyright 1985 American Mathematical Society