Singularity theory, rod theory, and symmetry-breaking loads / John F. Pierce

Auteur: Pierce, John Festus (1943-) - AuteurType de document: Livre numériqueCollection: Lecture notes in mathematics, (Online) ; 1377Langue: anglaisÉditeur: Berlin : Springer-Verlag, 1989 ISBN: 9783540513049 ISSN: 1617-9692Note: The author studies from a mathematically advanced point of view three- dimensional buckling of rods where both the rod geometry and the loading are rotationally symmetric. It is well-known that for such problems after loss of stability not only one equilibrium but a whole orbit of equilibria will be obtained. The main question the author asks is: How is the orbit changed under a perturbation of the loading? For this purpose a nice presentation of the Kirchhoff and Cosserat rod theories in modern mathematical notation is given. The analysis is performed first by reducing the infinite-dimensional problem defined on a function space by means of the Lyapunov-Schmidt-method to one which is defined on a finite-dimensional space and second by application of the concepts of singularity theory. The main result is a classification theorem for certain classes of perturbation of loads for the orbit breaking problem. Finally some comments are made concerning the general imperfection sensitivity problem making use of the concept of unfoldings. The presentation is clear and good to understand. The book should certainly find an interested readership both in the applied mathematics and in the theoretical mechanics community. (Zentralblatt) Sujets MSC: 58C25 Global analysis, analysis on manifolds -- Calculus on manifolds; nonlinear operators -- Differentiable maps
58Kxx Global analysis, analysis on manifolds -- Theory of singularities and catastrophe theory
57R70 Manifolds and cell complexes -- Differential topology -- Critical points and critical submanifolds
74K10 Mechanics of deformable solids -- Thin bodies, structures -- Rods (beams, columns, shafts, arches, rings, etc.)
58D05 Global analysis, analysis on manifolds -- Spaces and manifolds of mappings -- Groups of diffeomorphisms and homeomorphisms as manifolds
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The author studies from a mathematically advanced point of view three- dimensional buckling of rods where both the rod geometry and the loading are rotationally symmetric. It is well-known that for such problems after loss of stability not only one equilibrium but a whole orbit of equilibria will be obtained. The main question the author asks is: How is the orbit changed under a perturbation of the loading?

For this purpose a nice presentation of the Kirchhoff and Cosserat rod theories in modern mathematical notation is given. The analysis is performed first by reducing the infinite-dimensional problem defined on a function space by means of the Lyapunov-Schmidt-method to one which is defined on a finite-dimensional space and second by application of the concepts of singularity theory. The main result is a classification theorem for certain classes of perturbation of loads for the orbit breaking problem. Finally some comments are made concerning the general imperfection sensitivity problem making use of the concept of unfoldings.

The presentation is clear and good to understand. The book should certainly find an interested readership both in the applied mathematics and in the theoretical mechanics community. (Zentralblatt)

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