Singularity Theory, Rod Theory, and Symmetry-Breaking Loads
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1377
John F. Pierce
Singularity Theory, Rod Theory, and Symmetry-Breaking Loads
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
John F. Pierce Department of Mathematics United States Naval Academy Annapolis, MD 21402, USA
Mathematics Subject Classification (1980): 58F 14, 73C50, 58C27, 58F05 ISBN 3-540-51304-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51304-3 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations. recitation, broadcasting. reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
OF CONTENTS
I.
Introduction
1
II.
The 1. 2. 3.
7
III.
IV.
V.
VI.
Spaces of Configurations The Spaces of Classical Configurations for a Rod The Spaces of Infinitesimal Displacements The Manifolds of Generalized and constrained Configurations
The Spaces of Loads Loads in the special Cosserat Theory 12. The Space of Loads for the Kirchhoff Theory 3. The Co-Adjoint Group Action on the Space of Loads 4. The Generalization of the Load Spaces
The Rod The 12. The 3. The 4. The
Equilibrium Variational Euler Field Constrained Bifurcation
Problem Functions for the Kirchhoff Problem Equilibrium Problem Problem for the Kirchhoff Rod
The Reduction of the Bifurcation Problem 1. The Decomposition of the Spaces 2. The Liapunov-Schmidt Reduction
The Analysis of the Reduced Problem The Symmetrically Perturbed Problem 12. The Critical Manifolds for the Symmetry-Breaking Loads 3. The Classification of the critical Manifolds
7
13 24
32 32 52 58 62
66 66 72 76 82
90 90
101
117 117 120 125
IV
VII.
The Results of the Bifurcation Problem 1The Reduction of e and its Analysis 2. Perturbations of Class a 3. Nondegenerate Perturbations of Classes
VIII. Conclusions and Additional Problems
140 140 144
and 0
149
163
References
167
Index
172
I.
INTRODUCTION
Take an initially straight rod of circular cross section which is composed of an isotropic material. Apply an axially sYmmetric compressive load to it. In general, the rod will assume an equilibrating configuration. However, this equilibrium is not isolated. Because of the symmetry of the rod and the load, the configuration will determine an entire family or "orbit" of other equilibrating configurations which is gained by rotating the image of the original configuration about the axis of sYmmetry through any angle. Now perturb the compressive load by additional loads which break the axial sYmmetry. How does this perturba
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