Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T08:26:59.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Linearization stability and Signorini Series for the traction problem in elastostatics

Published online by Cambridge University Press:  14 November 2011

J. E. Marsden  and
Y. H. Wan
J. E. Marsden
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, U.S.A
Y. H. Wan
Affiliation:
Department of Mathematics, State University of New York, Buffalo, New York 14214, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper uses previous results of Chillingworth, Marsden and Wan on symmetry and bifurcation for the traction problem in three dimensional elastostatics to establish new results on the Signorini expansion. We show that the Signorini compatibility conditions are necessary and sufficient for linearization stability and analogies with results known for other field theories are pointed out. Under an explicit non-degeneracy condition, a new series expansion is given in which successive terms are inductively determined in pairs rather than singly. Our results include as special cases, classical results of Signorini, Tolotti and Stoppelli.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 1-2 , 1983 , pp. 171 - 180
Copyright
Copyright © Royal Society of Edinburgh 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arms, J., Marsden, J. and Moncrief, V.. Bifurcations of momentum mappings. Comm. Math. Phys. 78 (1981), 455–478.Google Scholar
2Arms, J., Marsden, J. and Moncrief, V.. The structure of the space solutions of Einstein's equations; II Several killing fields and the Einstein-Yang-Mills equations. Ann. Physics 144 (1982), 81–106.CrossRefGoogle Scholar
3Capriz, G. and Podio-Guidugli, P.. On Signorini's perturbation method in nonlinear elasticity. Arch. Rational Mech. Anal. 57 (1974), 1–30.CrossRefGoogle Scholar
4Chillingworth, D. R. J., Marsden, J. E. and Wan, Y. H.. Symmetry and bifurcation in three dimensional elasticity, Part I. Arch. Rational Mech. Anal. 80 (1982), 295–331.CrossRefGoogle Scholar
5Chillingworth, D. R. J., Marsden, J. E. and Wan, Y. H.. Symmetry and bifurcation in three dimensional elasticity, Part II. Arch. Rational Mech. Anal, (to appear).Google Scholar
6Fichera, G.. Existence theorems in elasticity. Handbuch der Physik, Bd VIa/2, 347–389 (Berlin: Springer, 1972).Google Scholar
7Fischer, A. and Marsden, J.. Linearization stability of the Einstein equations. Bull. Amer. Math. Soc. 79 (1973), 995–1001.Google Scholar
8Fischer, A. and Marsden, J.. Linearization stability of nonlinear partial differential equations. Proc. Sympos. Pure Math. Stanford 1973, vol. 27; Part 2, 219–263 (Providence, R.I.: Amer. Math. Soc, 1975).Google Scholar
9Fischer, A., Marsden, J. and Moncrief, V.. The structure of the space solutions of Einstein's equations. I. One killing field. Ann. Inst. H. Poincaré Sect. B 33 (1980), 147–194.Google Scholar
10Grioli, G.. Mathematical Theory of Elastic Equilibrium. Ergebnisse der Ang. Mat. 67 (Berlin: Springer, 1962).CrossRefGoogle Scholar
11Marsden, J. E. and Hughes, T. J. R.. Mathematical Foundations of Elasticity (Englewood Cliffs: Prentice-Hall, 1983).Google Scholar
12Signorini, S.. Sulle deformazioni termoelastiche finite. Proc. 3rd Int. Cong,. Appl. Mech. 2 (1930), 80–89.Google Scholar
13Stoppelli, F.. Sulla svilluppabilità in serie di potenze di un parametro delle soluzioni delle equazioni dell'elastostatica isoterma. Ricerche Mat. 4 (1955), 58–73.Google Scholar
14Stoppelli, F.. Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio. Ricerche Mat. 6 (1957), 241–287; 7 (1958), 71–101, 138–152.Google Scholar
15Tolotti, C.. Orientamenti principali di un corpo elastico rispetto alia sua sollecitazione totale. Atti Accad. Italia Mem. Cl. Sci. Fis. Mat. Nat. 13 (1943), 1139–1162.Google Scholar
16Truesdell, C. and Noll, W.. The nonlinear field theories of mechanics. Handbuch der Physik Bd III/3, S. Flügge, ed. (Berlin: Springer, 1965).Google Scholar