Dynamics of Flexoelectric Materials: Subsonic, Intersonic, and Supersonic Ruptures and Mach Cone Formation

Abstract

Motivated by recent, unexpected, experimental observations of “intersonic” rupture growth in which both shear and dilatational Mach fronts were observed at the tips of dynamic frictional ruptures propagating at rupture speeds below the dilatational wave speed of the surrounding solid, and we formulate the general dynamic flexoelectric problem and we investigate its plane strain/plane polarization specialization. The coupling of the mechanical problem is analogous to a problem of Toupin–Mindlin gradient elasticity, where two micromechanical characteristic lengths and two microinertial lengths emerge as a combination of the mechanical, dielectric, and flexoelectric constants. The solution of the rupture growth problem allows us to provide an explanation of the experimental results. This becomes possible since flexoelectricity predicts a new aspect that was not observed in the classical analysis: subsonic super shear and supersonic crack tip (or rupture) motions are not related exclusively with the problem being elliptic or hyperbolic, respectively. This is due to the influence of the microinertial lengths, which, in addition to the ratios of the rupture to the wave speeds, also affect the slopes of the Mach cones. Moreover, we are able to explain the experimental paradox of the observation of double Mach cone pairs at the tips of supershear, but subsonic, frictional, ruptures in poly-methyl-methacrtylate (PMMA) by demonstrating that both dilatational and shear Mach cones could appear in flexoelectric solids at rupture speeds below the material dilatation wave speed, something that is impossible from the classical elasticity analysis and is due to the dispersive nature of the present problem. Our analysis is of relevance to the dynamic deformation and fracture of both synthetic and naturally occurring flexoelectric materials and systems, with implications to both engineering and earthquake source mechanics.

Deriving the slip front propagation velocity with the slip- and slip-velocity-dependent friction laws via the use of the linear marginal stability hypothesis

“…These cases correspond to the slip-and/or slip-velocity-weakening behaviors with the critical condition just before the slip. Though the treatment in this study is not the same as that of the crack-tip propagation, the front propagation obtained here may be consistent with the supersonic crack-tip propagations addressed in previous studies [23][24][25][26].…”

“…Actualy, from Eqs. ( 19), (26), and ( 27), we can obtain the relationship C 2 = C 1 /3 for C 1 > 0 [Eq. ( 27)] and for C 1 < 0 [Eq.…”

“…( 27)] and for C 1 < 0 [Eq. (26)]. This forms a boundary between regions in which solutions for f in (ω i ) = 0 fall into cases 5 and 6, and cases 9 and 10.…”

Nucleation of frictional slip: A yielding or a fracture process?

“…The post-nucleation processes have been shown to be well described by Linear Elastic Fracture Mechanics (LEFM). For instance, LEFM has proven very successful in determining the conditions for propagation and arrest of rupture fronts mediating the onset of dynamic frictional motion [4][5][6], including the stress fields near the front-tip [7][8][9][10] or the propagation velocity of such fronts [11,12]. However, it remains unclear whether or not LEFM can describe the nucleation of frictional sliding.…”