Andrei L. Smirnov - Böcker

The construction of solutions of singularly perturbed systems of equations and boundary value problems that are characteristic for the mechanics of thin-walled structures are the main focus of the book. The theoretical results are supplemented by the analysis of problems and exercises. Some of the topics are rarely discussed in the textbooks, for example, the Newton polyhedron, which is a generalization of the Newton polygon for equations with two or more parameters. After introducing the important concept of the index of variation for functions special attention is devoted to eigenvalue problems containing a small parameter. The main part of the book deals with methods of asymptotic solutions of linear singularly perturbed boundary and boundary value problems without or with turning points, respectively. As examples, one-dimensional equilibrium, dynamics and stability problems for rigid bodies and solids are presented in detail. Numerous exercises and examples as well as vast references to the relevant Russian literature not well known for an English speaking reader makes this a indispensable textbook on the topic.

The construction of solutions of singularly perturbed systems of equations and boundary value problems that are characteristic for the mechanics of thin-walled structures are the main focus of the book.

This book contains solutions to the most typical problems of thin elastic shells buckling under conservative loads. The linear problems of bifurcation of shell equilibrium are considered using a two-dimensional theory of the Kirchhoff-Love type. The explicit approximate formulas obtained by means of the asymptotic method permit one to estimate the critical loads and find the buckling modes.The solutions to some of the buckling problems are obtained for the first time in the form of explicit formulas. Special attention is devoted to the study of the shells of negative Gaussian curvature, the buckling of which has some specific features. The buckling modes localized near the weakest lines or points on the neutral surface are constructed, including the buckling modes localized near the weakly supported shell edge. The relations between the buckling modes and bending of the neutral surface are analyzed. Some of the applied asymptotic methods are standard; the others are new and are used for the first time in this book to study thin shell buckling. The solutions obtained in the form of simple approximate formulas complement the numerical results, and permit one to clarify the physics of buckling.