Anti-plane Yoffe-type crack in flexoelectric material

Abstract

This work investigates the deformation and polarization fields around a finite anti-plane shear (mode III) crack growing dynamically under steady-state conditions. The leading tip of this finite crack breaks the material while the trailing tip heals it. This fast moving finite crack (referred to as a rupture “pulse” in the geophysics literature) propagates with a constant velocity and with the mechanical and the electrical fields that remain invariant with respect to an observer moving with the crack-tips. This problem belongs to the first type of steady state crack growth problems according to the classification of Freund. The “prototype” problem which refers to an isotropic, body subjected to fracture under tensile loading was first proposed and solved by Yoffe, while finite cracks (or shear pulses) were also analyzed by Freund and by Rice. In the above cases the material was assumed to be linear elastic. Our analysis extends these studies to flexoelectric materials, and it is both theoretical and numerical. It discusses the asymptotic structure of the crack-tip displacement and the polarization fields; it calculates the dynamic energy release rate and presents their dependence on crack-tip velocity. Comparisons are made to the available, classical, elasto-dynamic solutions and to the static case. The influence of the electrical properties of the material on strengthening is also analyzed. Dynamic fracture of flexoelectric materials is of relevance to both the study of earthquake source mechanics and to the analysis of the reliability of micro-electronic devices. This is because both rocks and ceramics are flexoelectric. Indeed, during earthquake rupture processes, dynamic, in-plane shear (Mode-II) and out of plane shear (Mode-III), cracks propagate along faults and exhibit both mechanical and electrical polarization signatures. At an entirely different length scale, flexoelectric ceramics are currently used as sensors and transducers and can experience dynamic shear failure along interfaces when subjected to dynamic loading (e.g. impact.). Failure by dynamic fracture can be detrimental to both their mechanical reliability and electrical functionality.