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

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.

Applicability of 3D Spectral Element Method for Computing Close-Range Underwater Piling Noises

Pile driving is used for constructing foundation supports for offshore structures. Underwater noise, induced by in-water pile driving, could adversely impact marine life near the piling location. Many studies have computed this noise in close ranges by using semi-analytical models and Finite Element Method (FEM) models. This work presents a Spectral Element Method (SEM) wave simulator as an alternative simulation tool to obtain close-range underwater piling noise in complex, fully three-dimensional, axially-asymmetric settings in the time domain for impacting force signals with high-frequency contents (e.g., frequencies greater than 1000[Formula: see text]Hz). The presented numerical results show that the flexibility of SEM can accommodate the axially-asymmetric geometry of a model, its heterogeneity, and fluid-solid coupling. We showed that there are multiple Mach Cones of different angles in fluid and sediment caused by the difference in wave speeds in fluid, a pile, and sediment. The angles of Mach Cones in our numerical results match those that are theoretically evaluated. A previous work18 had shown that Mach Cone waves lead to intense amplitudes of underwater piling noise via a FEM simulation in an axis-symmetric setting. Since it modeled sediment as fluid with a larger wave speed than that of water, we examined if our SEM simulation, using solid sediment–fluid coupling, leads to additional Mach Cones. Because this work computes the shear wave in sediment and the downward-propagating shear wave in a pile, we present six Mach Cones in fluid and sediment induced by downward-propagating P- and S-waves in a pile in lieu of two previously-reported Mach Cones in fluid and sediment (modeled as fluid) induced by a downward-propagating P-wave in a pile. We also showed that the amplitudes of the close-range underwater noise are dependent on the cross-sectional geometry of a pile. In addition, when a pile is surrounded by a solid of an axially-asymmetric geometry, waves are reflected from the surface of the surrounding solid back to the fluid so that constructive and destructive interferences of waves take place in the fluid and affect the amplitude of the underwater piling noise.

The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space

This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.