Hyperbolicity, Mach Lines, and Super-Shear Mode III Steady-State Fracture in Magneto-Flexoelectric Materials, Part I: Methodology

This work examines the sub-shear and super-shear steady-state growth of mode III fractures in flexoelectric materials, nonetheless, exhibiting Mach type shock wave patterns that resemble reported lattice dynamics results and three-dimensional calculations and experiments. Our mathematical models provide weak discontinuous solutions of the steady-state dynamic equations. In flexoelectric solids, super-shear rupture is possible with Mach lines appearing at sub-shear as well as super-shear crack rupture velocities. This is contrary to classical singular elastodynamics, where the notions of super-shear growth and hyperbolicity coincide. The results show that the deformation near the crack-tip agrees with studies based on lattice dynamics. In the first part of this work, a novel finite element approach has been developed where the problem is decomposed into two prestressed plates that are interconnected, resulting into the predicted radiation patterns and Mach lines. The polarization field is obtained from the calculated displacement field and is used in turn to calculate the magnetic and the electric fields. The analysis offers an analogy to the co-seismic magnetic fields encountered during mode III dominated earthquake rupture events.