Imperfect bifurcation in structures and materials. Engineering use of group-theoretic bifucation theory
type: Item
instance of: scholarly article
title: Imperfect bifurcation in structures and materials. Engineering use of group-theoretic bifucation theory
zbMATH Open document ID: 1005.74001
author: Kiyohiro Ikeda
author: Kazuo Murota
published in: Applied Mathematical Sciences
publication date: 2002-08-08T00:00:00Z
review text: Many physical and mechanical systems lose or get stability, and exhibit pattern formation through bifurcation behaviour. The group-theoretic bifurcation theory is an established tool to deal with formation and selection of patterns due to symmetry-breaking bifurcations. The present book gives a wide and deep description of imperfect bifurcation behaviour in engineering problems. Although the authors' approach is rather pragmatic than mathematically rigorous, the book offers a number of systematic methods based on contemporary mathematics. Chapter 1 gives an introduction to bifurcational behaviour with examples which clarify the mechanism of bifurcation and the influence of initial imperfections. It also contains an overview of the whole book divided into three parts. Part I (chapters 2-6) is devoted to imperfect behaviour in a neighborhood of simple critical point. Such important aspects as Lyapunov-Schmidt bifurcation equation with classification of simple critical points, imperfection sensitivity laws, critical initial imperfections, and random initial imperfections are introduced accompanied by numerous examples from mechanics of structures and materials. Here the authors also discuss experimentally obtained bifurcation diagrams. Part II ``Imperfect bifurcation of symmetric systems'' extends the results of part I to multiple critical points of nonlinear equations admitting group symmetry. A brief account of the main ideas of the group-theoretic bifurcation analysis is presented in chapter 7 as an informal introduction for engineers. Chapter 8 is devoted to the simplest symmetry related to dihedral and cyclic groups for both perfect and imperfect bifurcations. This theory is applied to spherical truss dome structures. Critical and random initial imperfections are studied in chapters 9-11, again with mechanical applications, especially in chapter 11 (truss dome structures and soil specimens). Part III studies the bifurcational behaviour of various physical and stru…
Mathematics Subject Classification ID: 74-02
Mathematics Subject Classification ID: 74G60
zbMATH DE Number: 1777543
zbMATH Keywords: group-theoretic bifurcation theory
zbMATH Keywords: structural analysis
zbMATH Keywords: symmetry-breaking bifucations
zbMATH Keywords: Lyapunov-Schmidt bifurcation equation
zbMATH Keywords: simple critical points
zbMATH Keywords: imperfection sensitivity laws
zbMATH Keywords: critical initial imperfections
zbMATH Keywords: random initial imperfections
zbMATH Keywords: multiple critical points
zbMATH Keywords: group symmetry
zbMATH Keywords: spherical truss dome structures
zbMATH Keywords: soil specimens
zbMATH Keywords: echelon-mode formation
zbMATH Keywords: recursive bifurcation
zbMATH Keywords: clustered bifurcation point
zbMATH Keywords: parametric double critical point
zbMATH Keywords: explosive bifurcation
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1777543