Nonlinear Problems of Elasticity

1. Front Matter

   Pages i-xviii

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2. Background

   • Stuart S. Antman

   Pages 1-10

3. The Equations of Motion for Extensible Strings

   • Stuart S. Antman

   Pages 11-48

4. Elementary Problems for Elastic Strings

   • Stuart S. Antman

   Pages 49-84

5. Planar Equilibrium Problems for Elastic Rods

   • Stuart S. Antman

   Pages 85-123

6. Introduction to Bifurcation Theory and its Applications to Elasticity

   • Stuart S. Antman

   Pages 125-172

7. Global Bifurcation Problems for Strings and Rods

   • Stuart S. Antman

   Pages 173-226

8. Variational Methods

   • Stuart S. Antman

   Pages 227-257

9. The Special Cosserat Theory of Rods

   • Stuart S. Antman

   Pages 259-324

10. Spatial Problems for Cosserat Rods

    • Stuart S. Antman

    Pages 325-342

11. Axisymmetric Equilibria of Cosserat Shells

    • Stuart S. Antman

    Pages 343-370

12. Tensors

    • Stuart S. Antman

    Pages 371-383

13. Three-Dimensional Continuum Mechanics

    • Stuart S. Antman

    Pages 385-455

14. Elasticity

    • Stuart S. Antman

    Pages 457-530

15. General Theories of Rods and Shells

    • Stuart S. Antman

    Pages 531-601

16. Nonlinear Plasticity

    • Stuart S. Antman

    Pages 603-628

17. Dynamical Problems

    • Stuart S. Antman

    Pages 629-664

18. Appendix. Topics in Linear Analysis

    • Stuart S. Antman

    Pages 665-673

19. Appendix. Local Nonlinear Analysis

    • Stuart S. Antman

    Pages 675-681

20. Appendix. Degree Theory

    • Stuart S. Antman

    Pages 683-698

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.

• Extension
• angular momentum
• bifurcation
• elasticity
• mechanics
• momentum
• tensor

From the reviews of the second edition:

"This second edition accounts for the developments since the first edition was published, and differs from the first edition in many points. The book has been reorganized and several parts have been added. … The already impressive body of references has been further expanded. The reviewer highly recommends this book both to graduate students and to scholars interested in the theory of elasticity." (Massimo Lanza de Cristoforis, Mathematical Reviews, Issue 2006 e)

"The second extended edition of the reviewed monograph gives a fundamental presentation of problems of nonlinear elasticity. Every chapter is equipped by instructive exercises, unsolved problems and exhaustive historical comments. The book could be very useful to applied mathematicians and engineers using in their works the elasticity theory and … to specialists dealing with applications of differential equations and bifurcation theory." (Boris V. Loginov, Zentralblatt MATH, Vol. 1098 (24), 2006)

"Antman’s impressive work is … a comprehensive treatise on nonlinear elasticity and a quintessential example of applied nonlinear analysis. … The text has been revised and updated, Several new sections have been added … This book is a ‘must’ for researchers and graduate students interested in nonlinear continuum mechanics and applied analysis. The work is scholarly and well written. … ‘This book is directed toward scientists, engineers, and mathematicians who wish to see careful treatments of uncompromised problems.’" (Timothy J. Healey, SIAM Review, Vol. 49 (2), 2007)

• Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, USA

  Stuart S. Antman

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