Book contents
- Frontmatter
- Contents
- Preface
- 1 Elements of the Theory of Finite Elasticity
- 2 Hyperelastic Bell Materials: Retrospection, Experiment, Theory
- 3 Universal Results in Finite Elasticity
- 4 Equilibrium Solutions for Compressible Nonlinearly Elastic Materials
- 5 Exact Integrals and Solutions for Finite Deformations of the Incompressible Varga Elastic Materials
- 6 Shear
- 7 Elastic Membranes
- 8 Elements of the Theory of Elastic Surfaces
- 9 Singularity Theory and Nonlinear Bifurcation Analysis
- 10 Perturbation Methods and Nonlinear Stability Analysis
- 11 Nonlinear Dispersive Waves in a Circular Rod Composed of a Mooney-Rivlin Material
- 12 Strain-energy Functions with Multiple Local Minima: Modeling Phase Transformations Using Finite Thermo-elasticity
- 13 Pseudo-elasticity and Stress Softening
- Subject Index
9 - Singularity Theory and Nonlinear Bifurcation Analysis
Published online by Cambridge University Press: 09 October 2009
- Frontmatter
- Contents
- Preface
- 1 Elements of the Theory of Finite Elasticity
- 2 Hyperelastic Bell Materials: Retrospection, Experiment, Theory
- 3 Universal Results in Finite Elasticity
- 4 Equilibrium Solutions for Compressible Nonlinearly Elastic Materials
- 5 Exact Integrals and Solutions for Finite Deformations of the Incompressible Varga Elastic Materials
- 6 Shear
- 7 Elastic Membranes
- 8 Elements of the Theory of Elastic Surfaces
- 9 Singularity Theory and Nonlinear Bifurcation Analysis
- 10 Perturbation Methods and Nonlinear Stability Analysis
- 11 Nonlinear Dispersive Waves in a Circular Rod Composed of a Mooney-Rivlin Material
- 12 Strain-energy Functions with Multiple Local Minima: Modeling Phase Transformations Using Finite Thermo-elasticity
- 13 Pseudo-elasticity and Stress Softening
- Subject Index
Summary
In this chapter we provide an introductory exposition of singularity theory and its application to nonlinear bifurcation analysis in elasticity. Basic concepts and methods are discussed with simple mathematics. Several examples of bifurcation analysis in nonlinear elasticity are presented in order to demonstrate the solution procedures.
Introduction
Singularity theory is a useful mathematical tool for studying bifurcation solutions. By reducing a singular function to a simple normal form, the properties of multiple solutions of a bifurcation equation can be determined from a finite number of derivatives of the singular function. Some basic ideas of singularity theory were first conjectured by R. Thorn, and were then formally developed and rigorously justified by J. Mather (1968, 1969a, b). The subject was extended further by V. I. Arnold (1976, 1981). In two volumes of monographs, M. Golubitsky and D. G. Schaeffer (1985), and M. Golubitsky, I. Stewart and D. G. Schaeffer (1988) systematized the development of singularity theory, and combined it with group theory in treating bifurcation problems with symmetry. Their work establishes singularity theory as a comprehensive mathematical theory for nonlinear bifurcation analysis.
The purpose of this chapter is to give a brief exposition of singularity theory for researchers in elasticity. The emphasis is on providing a working knowledge of the theory to the reader with minimal mathematical prerequisites. It can also serve as a handy reference source of basic techniques and useful formulae in bifurcation analysis.
- Type
- Chapter
- Information
- Nonlinear ElasticityTheory and Applications, pp. 305 - 344Publisher: Cambridge University PressPrint publication year: 2001
- 2
- Cited by