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Symmetry and bifurcation in three-dimensional elasticity. Part II

Chillingworth, D. R. J. and Marsden, J. E. and Wan, Y. H. (1983) Symmetry and bifurcation in three-dimensional elasticity. Part II. Archive for Rational Mechanics and Analysis, 83 (4). pp. 363-395. ISSN 0003-9527. doi:10.1007/BF00963840.

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In part I of this paper (Chillingworth, Marsden and Wan [1982]--hereafter referred to as [1]), we reformulated the traction problem in elastostatics in various forms, gave a classification of loads and gave complete analysis of solutions of the traction problem that are nearly stress-free for loads near loads of type 0 and type 1. This part develops the basic theory as well as giving an analysis of solutions for loads of types 2, 3 and 4. It includes a count of the numbers of solutions and an analysis of their stability and the structural stability of the bifurcation diagrams. We begin in Section 2 with a derivation of a potential formulation of the problem on SO (3). The "second order potential" used in [1] can be recovered as a special case. It follows from this that the traction problem always has at least four solutions, at least one of which is neutrally stable. For loads of type 0, we showed in [1] that there are exactly four solutions near SO (3); for the other types there can be many more ... up to 40. Section 3, 4 and 5 examine types 2, 3 and 4 respectively, in a manner analogous to our treatment of types 0 and 1 in [1]. Loads of type 3 and 4 have some special features already studied in the literature in connection with parallel loads. These special features will be discussed and other connections with the existing literature will be made at appropriate points throughout the paper.

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Additional Information:© 1983 Springer-Verlag. Received December 15, 1982. Communicated by S. Antman. J. E. Marsden's research was supported in part by the U.S. National Science Foundation under Grant MCS-81-07086, by the Miller Institute, and by a contract from the Department of Energy, DE-AT03-82ER12097. Y. H. Wan's research was partially supported by the U.S. National Science Foundation under Grant MCS-81-02463 and the Department of Energy, Contract DE-AT03-82ER12097. D. R. J. Chillingworth's research was partially supported by the U. K. Science Research Council through the University of Warwick, 1980.
Funding AgencyGrant Number
NSFMCS 81-07086
Miller InstituteUNSPECIFIED
Department of Energy (DOE)DE-AT03-82ER-12097
U. K. Science Research Council, University of WarwickUNSPECIFIED
Issue or Number:4
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20411
Deposited By: Ruth Sustaita
Deposited On:27 Oct 2010 21:02
Last Modified:08 Nov 2021 23:59

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