Introduction to mathematical elasticity

This book provides the general reader with an introduction to mathematical elasticity, by means of general concepts in classic mechanics, and models for elastic springs, strings, rods, beams and membranes. Functional analysis is also used to explore more general boundary value problems for three-dimensional elastic bodies, where the reader is provided, for each problem considered, a description of the deformation; the equilibrium in terms of stresses; the constitutive equation; the equilibrium equation in terms of displacements; formulation of boundary value problems; and variational principles, generalized solutions and conditions for solvability. Introduction to Mathematical Elasticity will also be of essential reference to engineers specializing in elasticity, and to mathematicians working on abstract formulations of the related boundary value problems.

1. Models and ideas of classical mechanics. 1.1. Orientation. 1.2. Some words on the fundamentals of our subject. 1.3. Metric spaces and spaces of particles. 1.4. Vectors and vector spaces. 1.5. Normed spaces and inner product spaces. 1.6. Forces. 1.7. Equilibrium and motion of a rigid body. 1.8. D'Alembert's principle. 1.9. The motion of a system of particles. 1.10. The rigid body. 1.11. Motion of a system of particles ; comparison of trajectories ; notion of operator. 1.12. Matrix operators and matrix equations. 1.13. Complete spaces. 1.14. Completion theorem. 1.15. Lebesgue integration and the L[symbol] spaces. 1.16. Orthogonal decomposition of Hilbert Space. 1.17. Work and energy. 1.18. Virtual work principle. 1.19. Lagrange's equations of the second kind. 1.20. Problem of minimum of a functional. 1.21. Hamilton's principle. 1.22. Energy conservation revisited -- 2. Simple elastic models. 2.1. Introduction. 2.2. Two main principles of equilibrium and motion for bodies in continuum mechanics. 2.3. Equilibrium of a spring. 2.4. Equilibrium of a string. 2.5. Equilibrium boundary value problems for a string. 2.6. Generalized formulation of the equilibrium problem for a string. 2.7…