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