Hamiltonian analysis of a magnetoelectroelastic notch in a mode III singularity

Abstract

The stress intensity factor (SIF) of a multi-material magnetoelectroelastic wedge in anti-plane deformation is analytically determined by the symplectic method. The Lagrangian equations in configuration variables alone are transformed to Hamiltonian equations in dual variables (configuration and momentum) which allow the use of the method of separation of variables. The solutions of the Hamiltonian equations can be expanded analytically in terms of the symplectic eigenfunctions with coefficients to be determined by the boundary conditions. For the wedge problem, the pairs of anti-plane displacements and shear stresses, electric fields and electric displacements, and magnetic fields and magnetic inductions are proved to be the dual (momentum) variables of the configuration variables. The singularity orders depend directly on the first few eigenvalues whose real parts are less than one but greater than zero. Numerical results for various conditions show the variations of the singularity orders. In particular, special behaviors of the order of the singularity for some special wedge angles are noted.