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The Signorini perturbation scheme in an abstract setting

Published online by Cambridge University Press:  14 November 2011

D. R. J. Chillingworth
D. R. J. Chillingworth
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH, UK
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Synopsis

The Signorini perturbation scheme is a series expansion algorithm that locates solution branches for a class of symmetry-breaking bifurcation problems in nonlinear elastostatics. The relationship of the formal steps in the algorithm to geometric aspects of the problem is brought out in work of J. E. Marsden and Y.-H. Wan, where an abstract formulation is also considered. In this paper, the abstract algorithm and its geometry are explored further: the logical structure is clarified, and it is shown how the scheme adapts to the presence of additional symmetry constraints.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 373 - 395
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Ball, J. M. and Schaeffer, D. G.. Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. Math. Proc. Cambridge Philos. Soc. 94 (1983), 315–339.CrossRefGoogle Scholar
2Bharatha, S. and Levinson, M.. Signorini's perturbation scheme for a general reference configuration in finite elastostatics. Arch. Rational Mech. Anal. 67 (1978), 365–394.CrossRefGoogle Scholar
3Buzano, E., Geymonat, G. and Poston, T.. Post-buckling behaviour of a nonlinearly hyperelastic thin rod with cross-section invariant under the dihedral group D n. Arch. Rational Mech. Anal. 89 (1985), 307–388.CrossRefGoogle Scholar
4Capriz, G. and Podio-Guidugli, P.. On Signorini's perturbation method in nonlinear elasticity. Arch. Rational Mech. Anal. 57 (1974), 1–30.CrossRefGoogle Scholar
5Capriz, G. and Podio-Guidugli, P.. The role of Fredholm conditions in Signorini's perturbation method. Arch. Rational Mech. Anal. 70 (1979), 261–288.CrossRefGoogle Scholar
6Capriz, G. and Podio-Guidugli, P.. A generalization of Signorini's perturbation method suggested by two problems of Grioli. Rend. Sem. Mat. Univ. Padova 68 (1982), 149–162.Google Scholar
7Chillingworth, D. R. J.. Bifurcation from an orbit of symmetry. In Singularities and Dynamical Systems, ed. Pnevmatikos, S. N., pp. 285–294 (Amsterdam: North-Holland, 1985).Google Scholar
8Chillingworth, D. R. J.. Bifurcation from a manifold. In Singularity Theory and its Applications, Warwick 1989 vol. 2, eds Roberts, R. M. and Stewart, I. N., pp. 22–37, Lecture Notes in Mathematics 1463 (Heidelberg, Springer 1991).CrossRefGoogle Scholar
9Chillingworth, D. R. J., Marsden, J. E. and Wan, Y.-H.. Symmetry and bifurcation in three-dimensional elasticity. Parts I, II. Arch. Rational Mech. Anal. 80 (1982), 295–331; 83 (1983), 363–395.CrossRefGoogle Scholar
10Chow, S.-N. and Hale, J. K.. Methods of Bifurcation Theory (New York: Springer 1982).CrossRefGoogle Scholar
11Dancer, E. N.. On the existence of bifurcating solutions in the presence of symmetries. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 321–336.CrossRefGoogle Scholar
12Dancer, E. N.. The G-invariant implicit function theorem in infinite dimensions. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 13–30.CrossRefGoogle Scholar
13Dancer, E. N.. Perturbation of zeros in the presence of symmetries. J. Austral. Math. Soc. (Ser. A) 36 (1984), 106–125.CrossRefGoogle Scholar
14Golubitsky, M. and Schaeffer, D.. Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67 (1979), 205–232.CrossRefGoogle Scholar
15Golubitsky, M., Stewart, I. and Schaeffer, D. G.. Singularities and Groups in Bifurcation Theory, vols. I, II (New York: Springer 1985, 1988).CrossRefGoogle Scholar
16Grioli, G.. Mathematical Theory of Elastic Equilibrium (Berlin: Springer 1962).CrossRefGoogle Scholar
17Hale, J. K.. Generic bifurcation with applications. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium Vol. I, ed. Knops, R. J., Research Notes in Mathematics 17 (London: Pitman, 1977).Google Scholar
18Hale, J. K. and Taboas, P. Z.. Interaction of damping and forcing in a second order equation. Nonlinear Anal. 2 (1978), 77–84.CrossRefGoogle Scholar
19Hale, J. K. and Taboas, P. Z.. Bifurcation near degenerate families. J. Appl. Anal. 11 (1980), 21–37.CrossRefGoogle Scholar
20Loginov, B. V. and Trenogin, V. A.. The application of continuous groups in branching theory. Dokl. Akad. Nauk SSSR 197 (1971), 36–39 (in russian); Soviet Math. Dokl. 12 (1971), 404–408.Google Scholar
21Magnus, R. J.. On perturbations of a translationally invariant differential equation. Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 1–25.CrossRefGoogle Scholar
22Marsden, J. E. and Hughes, T. J. R.. Mathematical Foundations of Elasticity (New Jersey: Prentice-Hall 1983).Google Scholar
23Marsden, J. E. and Wan, Y.-H.. Linearization stability and Signorini series for the traction problem in elastostatics. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 171–180.CrossRefGoogle Scholar
24Michel, L.. Points critiques des fonctions invariantes sur un G-variété. C.R. Acad. Sci. Paris Ser. 1 Math. 272 (1971), 433–436.Google Scholar
25Palais, R. S.. The classification of G-spaces. Mem. Amer. Math. Soc. 36 (1960).Google Scholar
26Pierce, J. F.. Singularity theory, Rod theory, and Symmetry-breaking Loads, Lecture Notes in Mathematics 1377 (Berlin: Springer, 1989).CrossRefGoogle Scholar
27Reeken, M.. Stability of critical points under small perturbations. Part I: Topological theory. Manuscripta Math. 7 (1972), 387–411.CrossRefGoogle Scholar
28Ruelle, D.. Bifurcations in the presence of a symmetry group. Arch. Rational Mech. Anal. 51 (1973), 136–152.CrossRefGoogle Scholar
29Sattinger, D. H.. Transformation groups and bifurcation at multiple eigenvalues. Bull. Amer. Math. Soc. 79 (1973), 709–711.CrossRefGoogle Scholar
30Sattinger, D. H.. Branching in the Presence of Symmetry, CBMS Reg. Conf. Ser. in App. Math. 40 (Philadelphia: SI AM, 1983).CrossRefGoogle Scholar
31Signorini, S.. Sulle deformazioni termoelastiche finite. Proc. 3rd Int. Cong. Appl. Mech. 2 (1930), 80–89.Google Scholar
32Thom, R.. Structural Stability and Morphogenesis, trans. Fowler, D. H. (Reading, Mass: Benjamin, 1975). (Original French edition 1972.)Google Scholar
33Truesdell, C. and Noll, W.. The nonlinear field theories of mechanics. In Handbuch der Physik, Bd. III/3, ed. Fliigge, S. (Berlin: Springer, 1965).Google Scholar
34Vanderbauwhede, A.. Symmetry and bifurcations near families of solutions. J. Differential Equations 36 (1980), 173–187.CrossRefGoogle Scholar
35Vanderbauwhede, A.. Local Bifurcation and Symmetry, Research Notes in Mathematics 75 (London: Pitman, 1982).Google Scholar
36Wan, Y.-H.. The traction problem for incompressible materials. Trans. Amer. Math. Soc. 291 (1985), 103–119.CrossRefGoogle Scholar
37Wan, Y.-H. and Marsden, J. E.. Symmetry and bifurcation in three-dimensional elasticity. Part III. Arch. Rational Mech. Anal. 84 (1983), 203–233.CrossRefGoogle Scholar
38Waterhouse, W. C.. Do symmetric problems have symmetric solutions? Amer. Math. Monthly 90 (1983), 378–387.CrossRefGoogle Scholar
39Weinstein, A.. Perturbation of periodic manifolds of Hamiltonian systems. Bull. Amer. Math. Soc. 77 (1971), 814–818.CrossRefGoogle Scholar