Nonlinear Problems of Elasticity (Applied Mathematical Sciences Book 107)

This second edition is an enlarged, completely updated, and extensively revised version of the authoritative first edition. It is devoted to the detailed study of illuminating specific problems of nonlinear elasticity, directed toward the scientist, engineer, and mathematician who wish to see careful treatments of precisely formulated problems. Special emphasis is placed on the role of nonlinear material response. The mathematical tools from nonlinear analysis are given self-contained presentations where they are needed. This book begins with chapters on (geometrically exact theories of) strings, rods, and shells, and on the applications of bifurcation theory and the calculus of variations to problems for these bodies. The book continues with chapters on tensors, three-dimensional continuum mechanics, three-dimensional elasticity, large-strain plasticity, general theories of rods and shells, and dynamical problems. Each chapter contains a wealth of interesting, challenging, and tractable exercises. Reviews of the first edition: ``A scholarly work, it is uncompromising in its approach to model formulation, while achieving striking generality in the analysis of particular problems. It will undoubtedly become a standard research reference in elasticity but will be appreciated also by teachers of both solid mechanics and applied analysis for its clear derivation of equations and wealth of examples.'' --- J. M. Ball, (Bulletin of the American Mathematical Society), 1996. ``It is destined to become a standard reference in the field which belongs on the bookshelf of anyone working on the application of mathematics to continuum mechanics. For graduate students, it provides a fascinating introduction to an active field of mathematical research.'' --- M. Renardy, (SIAM Review), 1995. ``The monograph is a masterpiece for writing a modern theoretical treatise on a field of natural sciences. It is highly recommended to all scientists, engineers and mathematicians interested in a careful treatment of uncompromised nonlinear problems of elasticity, and it is a `must' for applied mathematicians working on such problems.'' --- L. v Wolfersdorf, (Zeitschrift fur Angewandte Mathematik und Mechanik), 1995. The scientists of the seventeenth and eighteenth centuries, led by Jas. Bernoulli and Euler, created a coherent theory of the mechanics of strings and rods undergoing planar deformations. They introduced the basic con­ cepts of strain, both extensional and flexural, of contact force with its com­ ponents of tension and shear force, and of contact couple. They extended Newton's Law of Motion for a mass point to a law valid for any deformable body. Euler formulated its independent and much subtler complement, the Angular Momentum Principle. (Euler also gave effective variational characterizations of the governing equations.) These scientists breathed life into the theory by proposing, formulating, and solving the problems of the suspension bridge, the catenary, the velaria, the elastica, and the small transverse vibrations of an elastic string. (The level of difficulty of some of these problems is such that even today their descriptions are sel­ dom vouchsafed to undergraduates. The realization that such profound and beautiful results could be deduced by mathematical reasoning from fundamental physical principles furnished a significant contribution to the intellectual climate of the Age of Reason.) At first, those who solved these problems did not distinguish between linear and nonlinear equations, and so were not intimidated by the latter. By the middle of the nineteenth century, Cauchy had constructed the basic framework of three-dimensional continuum mechanics on the founda­ tions built by his eighteenth-century predecessors. This Second Edition Is An Enlarged, Completely Updated, And Extensively Revised Version Of The Authoritative First Edition. It Is Devoted To The Detailed Study Of Illuminating Specific Problems Of Nonlinear Elasticity, Directed Toward The Scientist, Engineer, And Mathematician Who Wish To See Careful Treatments Of Precisely Formulated Problems. Special Emphasis Is Placed On The Role Of Nonlinear Material Response. The Mathematical Tools From Nonlinear Analysis Are Given Self-contained Presentations Where They Are Needed.--book Jacket. Ch. 1. Background -- Ch. 2. The Equations Of Motion For Extensible Strings -- Ch. 3. Elementary Problems For Elastic Strings -- Ch. 4. Planar Steady-state Problems For Elastic Rods -- Ch. 5. Introduction To Bifurcation Theory And Its Applications To Elasticity -- Ch. 6. Global Bifurcation Problems For Strings And Rods -- Ch. 7. Variational Methods -- Ch. 8. Theory Of Rods Deforming In Space -- Ch. 9. Spatial Problems For Rods -- Ch. 10. Axisymmetric Equilibria Of Shells -- Ch. 11. Tensors -- Ch. 12. 3-dimensional Continuum Mechanics -- Ch. 13. 3-dimensional Theory Of Nonlinear Elasticity -- Ch. 14. Problems In Nonlinear Elasticity -- Ch. 15. Large-strain Plasticity -- Ch. 16. General Theories Of Rods -- Ch. 17. General Theories Of Shells -- Ch. 18. Dynamical Problems -- Ch. 19. Appendix : Topics In Linear Analysis -- Ch. 20. Appendix : Local Nonlinear Analysis -- Ch. 21. Appendix : Degree Theory And Its Applications. Stuart S. Antman. Includes Bibliographical References (pages 781-818) And Index. Front Matter....Pages i-xviii Background....Pages 1-10 The Equations of Motion for Extensible Strings....Pages 11-48 Elementary Problems for Elastic Strings....Pages 49-84 Planar Equilibrium Problems for Elastic Rods....Pages 85-123 Introduction to Bifurcation Theory and its Applications to Elasticity....Pages 125-172 Global Bifurcation Problems for Strings and Rods....Pages 173-226 Variational Methods....Pages 227-257 The Special Cosserat Theory of Rods....Pages 259-324 Spatial Problems for Cosserat Rods....Pages 325-342 Axisymmetric Equilibria of Cosserat Shells....Pages 343-370 Tensors....Pages 371-383 Three-Dimensional Continuum Mechanics....Pages 385-455 Elasticity....Pages 457-530 General Theories of Rods and Shells....Pages 531-601 Nonlinear Plasticity....Pages 603-628 Dynamical Problems....Pages 629-664 Appendix. Topics in Linear Analysis....Pages 665-673 Appendix. Local Nonlinear Analysis....Pages 675-681 Appendix. Degree Theory....Pages 683-698 Back Matter....Pages 699-752 Within the past few decades, there has been an accelerating development of methods for studying nonlinear equations. Nonlinear analysis offers exciting prospects for certain specific areas of nonlinear problems with continuum mechanics. The objective of this book is to carry out such studies for problems in nonlinear elasticity. This book is directed toward scientists, engineers, and mathematicians who wish to see careful treatments of uncompromised problems. The author's aim is to retain the orientation toward fascinating problems that characterizes the best engineering texts on structural stability while retaining the precision of modern continuum mechanics and employing powerful methods of nonlinear analysis. The author's approach is to lay down a general theory for each kind of elastic body, carefully formulate specific problems, introduce the pertinent mathematical methods, and then conduct rigorous analyses of the problems. Ch. I. Background -- Ch. Ii. The Equations Of Motion For Extensible Strings -- Ch. Iii. Elementary Problems For Elastic Strings -- Ch. Iv. Planar Equilibrium Problems For Elastic Rods -- Ch. V. Introduction To Bifurcation Theory And Its Applications To Elasticity -- Ch. Vi. Global Bifurcation Problems For Strings And Rods -- Ch. Vii. Variational Methods -- Ch. Viii. The Special Cosserat Theory Of Rods -- Ch. Ix. Spatial Problems For Cosserat Rods -- Ch. X. Axisymmetric Equilibria Of Cosserat Shells -- Ch. Xi. Tensors -- Ch. Xii. Three-dimensional Continuum Mechanics -- Ch. Xiii. Elasticity -- Ch. Xiv. General Theories Of Rods And Shells -- Ch. Xv. Nonlinear Plasticity -- Ch. Xvi. Dynamical Problems -- Appendix Xvii. Topics In Linear Analysis -- Appendix Xviii. Local Nonlinear Analysis -- Appendix Xix. Degree Theory. Stuart S. Antman. Includes Bibliographical References (p. 699-735) And Index. Enlarged, updated, and extensively revised, this second edition illuminates specific problems of nonlinear elasticity, emphasizing the role of nonlinear material response. Opening chapters discuss strings, rods, and shells, and applications of bifurcation theory and the calculus of variations to problems for these bodies. Subsequent chapters cover tensors, three-dimensional continuum mechanics, three-dimensional elasticity , general theories of rods and shells, and dynamical problems. Each chapter includes interesting, challenging, and tractable exercises. This monograph provides a general theory of each type of elastic body, formulates specific problems, introduces the pertinent mathematical methods and conducts rigorous analyses of the problems, with solutions.