State space formulation of magneto-electro-elasticity in Hamiltonian system and applications

A systematical method is presented to derive the state space equations of anisotropic magneto-electro-elastic materials of orthogonal curvilinear coordinates in Hamiltonian system. Based on the three-dimensional theory of magneto-electro-elasticity, the constitutive relations and the Lagrangian function of the total potential are rewritten in terms of the generalized displacements, i.e., displacements, electric potential and magnetic potential, and their derivatives. The Legendre transform is then applied to release the Hamiltonian function and the canonical equations are obtained through variational operation. The canonical equations in curvilinear coordinates, i.e., the desired state equations, can be degenerated readily to the ones of spherical coordinates, cylindrical coordinates and rectangular coordinates. These equations are also deduced conveniently for piezoelectric and elastic materials since these derivation are rendered in terms of matrix. As an example of application, a magneto-electro-elastic cylindrical shell with four simply supported edges is studied when it is bended by sinusoidal load.