Hamiltonian Duality Equation on Three-Dimensional Problems of Magnetoelectroelastic Solids

摘要

Hamiltonian system used in dynamics is introduced to formulate the three-dimensional problems of the transversely isotropic magnetoelectroelastic solids. The Hamiltonian dual equations in magnetoelectroelastic solids are developed directly from the modified Hellinger-Reissner variational principle derived from generalized Hellinger-Ressner variational principle with two classes of variables. These variables not only include such origin variables as displaces, electric potential and magnetic potential, but also include such their dual variables as lengthways stress, electric displacement and magnetic induction in the symplectic space. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate so that the method of separation of variables can be used. The governing equations are a set of first order differential equations in z, and the coefficient matrix of the differential equations is Hamiltonian in (x,y).