Damage-based fracture with electro-magnetic coupling

A coupled elastic and electro-magnetic analysis is proposed including finite displacements and damage-based fracture. Piezo-electric terms are considered and resulting partial differential equations include a non-classical wave equation due to the specific constitutive law. The resulting wave equation is constrained and, in contrast with the traditional solutions of the decoupled classical electro-magnetic wave equations, the constraint is directly included in the analysis. The absence of free current density allows the expression of the magnetic field rate as a function of the electric field and therefore, under specific circumstances, removal of the corresponding magnetic degrees-of-freedom. A Lagrange multiplier field is introduced to exactly enforce the divergence constraint, forming a three-field variational formulation (required to include the wave constraint). No vector-potential is required or mentioned, eliminating the need for gauges. The classical boundary conditions of electromagnetism are specialized and a boundary condition involving the electric field is obtained. The spatial discretization makes use of mixed bubble-based (of the MINI type) finite elements with displacement, electric field and Lagrange multiplier degrees-of-freedom. Three verification examples are presented with very good qualitative conclusions and mesh-independence.