Imperfect bifurcation in structures and materials. Engineering use of group-theoretic bifucation theory
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 structural systems by appropriately modeling their symmetries: the recursive change of shapes of cylindrical sand specimens undergoing bifurcations (chapter 12); the mechanism of echelon-mode formation on sand, kaolin and steel specimens as the bifurcations of \(O(2)\times O(2)\)-equivariant system (chapter 13); recursive bifurcation of rectangular parallelepiped steel speciments (chapter 14). In the last chapter 15 the authors describe miscellaneous aspects of bifurcation phenomena: a clustered bifurcation point which is a set of closely located simple bifurcation points; size effect observed in experiments on rocks; hilltop bifurcation point which is a parametric double critical point occuring as a coincidence of a limit point and a simple pitchfork point; and the so-called explosive bifurcation. On balance, the reviewed book is very useful as it develops a modern static imperfect bifurcation theory and fills gap between mathematical theory and engineering practice.
scientific article; zbMATH DE number 1777543