A Hamiltonian state space approach to anisotropic elasticity and piezoelasticity

Abstract

We present a Hamiltonian state space approach for problems of anisotropic elasticity and piezoelasticity. By means of Legendre’s transformation, the basic equations of piezoelasticity are formulated into a state equation and an output equation in terms of the state vector that comprises the generalized displacement vector and the conjugate generalized traction vector as the dual variables. The Hamiltonian features and symplectic orthogonality of the system, which are essential for the solution approach using eigenfunction expansion, are delineated at length. We show that the solution to 3D problems of a prismatic body hinges upon a 2D Hamiltonian eigensystem and the eigensolution associated with the zero eigenvalue leads to the solution to the generalized plane problem naturally. Based on the formalism, the solution to a problem of piezoelasticity is no more difficult than its elastic counterpart.