Symplectic elasticity

Exact analytical solutions in some areas of solid mechanics, in particular problems in the theory of plates, have long been regarded as bottlenecks in the development of elasticity. In contrast to the traditional solution methodologies, such as Timoshenko's approach in the theory of elasticity for which the main technique is the semi-inverse method, this book presents a new approach based on the Hamiltonian principle and the symplectic duality system where solutions are derived in a rational manner in the symplectic space. Dissimilar to the conventional Euclidean space with one kind of variables, the symplectic space with dual variables thus provides a fundamental breakthrough. A unique feature of this symplectic approach is the classical bending problems in solid mechanics now become eigenvalue problems and the symplectic bending deflection solutions are constituted by expansion of eigenvectors. The classical solutions are subsets of the more general symplectic solutions. This book explains the new solution methodology by discussing plane isotropic elasticity, multiple layered plate, anisotropic elasticity, sectorial plate and thin plate bending problems in detail. A number of existing problems without analytical solutions within the framework of classical approaches are solved analytically using this symplectic approach. Symplectic methodologies can be applied not only to problems in elasticity, but also to other solid mechanics problems. In addition, it can also be extended to various engineering mechanics and mathematical physics fields, such as vibration, wave propagation, control theory, electromagnetism and quantum mechanics.

1. Mathematical preliminaries. 1.1. Linear space. 1.2. Euclidean space. 1.3. Symplectic space. 1.4. Legengre's transformation. 1.5. The Hamiltonian principle and the Hamiltonian canonical equations. 1.6. The Reciprocal theorems -- 2. Fundamental equations of elasticity and variational principle. 2.1. Stress analysis. 2.2. Strain analysis. 2.3. Stress-strain relations. 2.4. The fundamental equations of elasticity. 2.5. The principle of virtual work. 2.6. The principle of minimum total potential energy. 2.7. The principle of minimum total complementary energy. 2.8. The Hellinger-Reissner variational principle with two kinds of variables. 2.9. The Hu-Washizu variational principle with three kinds of variables. 2.10. The principle of superposition and the uniqueness theorem. 2.11. Saint-Venant principle -- 3. The Timoshenko beam theory and its extension. 3.1. The Timoshenko beam theory. 3.2. Derivation of Hamiltonian system. 3.3. The method of separation of variables. 3.4. Reciprocal theorem for work and adjoint symplectic orthogonality. 3.5. Solution for non-homogeneous equations. 3.6. Two-point boundary conditions. 3.7. Static analysis of Timoshenko beam. 3.8. Wave propagation analy…