A Hamiltonian State Space Approach to Elasticity and Piezoelasticity
The conventional Lagrangian approach aims to eliminate the unknowns from the basic equations, resulting in coupled partial differential equations which are difficult to solve without further considerations. In this presentation we describe a state space formalism for anisotropic elasticity and piezoelasticity. On the basis of Hamiltonian mechanics by letting one of the spatial coordinates play the role of the time variable in the dynamic system, the basic equations of elasticity and piezoelasticity are formulated into a state equation and an output equation in which the generalized displacement vector and the conjugate generalized traction vector are dual variables. The symplectic characteristics of the Hamiltonian system provide an essential basis for the solution by means of separation of variables in conjunction with eigenfunction expansion. Many problems which would be intractable analytically become solvable in the state space setting.
In this presentation, the background of the state space approach is provided, followed by a comparison of the Lagrangian and Hamiltonian approaches of applied mechanics. On describing the Hamiltonian state space formalism, some unsolved problems of elasticity and piezoelasticity, which are solvable by the Hamiltonian state space approach are discussed.
譚建國教授 美國杜克大學博士(1976年), 是台灣成功大學土木工程系榮譽講座教授, 現為浙江大學土木工程系與工程力學系兼任教授, 講授彈性力學課程。