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This book presents the foundational issues of linear elasticity in a compact, unabridged manner; it is directed to mathematicians and physical scientists who care for approaching this classical subject with rigor and depth.
There are four chapters: the first two illustrate, respectively, the concepts of deformation and strain and of force and stress; the third is devoted to a study of constitutive relations; the last discusses the posing of equilibrium problems. The emphasis is in the description of elasticity as a model whose construction calls for a delicate interplay between physics and mathematics. The conceptual links with general continuum mechanics are carefully indicated. It would not be easy to find in one other book a treatment of such issues as exact and linearized equilibria, the constitutive problems of classification and representation, internal constraints and material symmetries, elastic equilibrium with the Cauchy relations, and elastic equilibrium in the presence of internal constraints.
The book can be be used to teach one-semester advanced undergraduate and graduate courses in elasticity theory to students in applied mathematics and engineering; for this purpose, it contains one hundred exercises of variable difficulty.
Preface. I: Strain. 1. Deformation Displacement. 2. Rigid Deformations. Pure Strains. 3. Strain Measures. 4. Small Strain. 5. Simple Deformations. 6. Divergence Identities. II: Stress. 7. Forces. Balances. 8. Stress. Dynamical Processes. 9. Simple Equilibrium Solutions. Normal and Shear Forces. 10. Alternative Forms of the Basic Balance Laws. 11. Power. Stress Power. 12. Exact and Linearized Equilibrium Theories. III: Constitutive Assumptions. 13. Linearly Elastic Materials. 14. Material Symmetry. 15. Fourth-Order Tensors. 16. Problems of Classification and Representation. 17. Internal Constraints. 18. Constraints and Material Symmetries. 19. Interpretation of Material Moduli. IV: Equilibrium. 20. Classical, Strong, and Weak Formulations. 21. Variational Formulation. The Principle of Minimum Potential Energy. 22. Minimum Complementary Energy. Variational Principles. 23. Compatible Field and Boundary Operators. 24. Generalized Boundary Conditions. 25. Elastic Equilibrium with the Cauchy Relations. 26. Elastic Equilibrium in the Presence of Internal Constraints. References. Subject Index.