Mach Cone Research Articles

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.

Backward Mach cone of shear waves induced by a moving force in soft anisotropic materials

When a point force travels in a solid with a speed greater than the velocity of the elastic wave induced, the interfering elastic wave fronts will form a Mach cone. This phenomenon is called the elastic Cherenkov effect (ECE). In this study, the ECE in soft matter was investigated with emphasis on backward Mach cone formation. Phase diagrams were proposed based on the theoretical analysis to elucidate key features of the ECE in a wide material parameter space, including the critical conditions for the onset of backward Mach cones and the cone angles. Subsequently, finite element models were developed to validate the theoretical solutions. Our results show that backward Mach cones can be formed in some typical soft tissues under the described conditions, which is important for the use of the ECE in characterizing the mechanical properties of these soft tissues in vivo. Our method and results also illustrate that backward Mach cones can be generated in soft phononic crystals with periodic microstructures, indicating that they are promising material systems for studying the ECE in soft matter.

Application of PDS–FEM to simulate dynamic crack propagation and supershear rupture

Dynamic crack propagation problems, including intersonic rupture, are simulated with two Hamiltonian based formulations of particle discretization scheme (PDS) FEM: the traditional displacement momentum form and the strain momentum form, for which consistent momentum conserving and symplectic integration schemes are derived. Numerical results are verified, and validated by comparing with photoelastic observations of a dynamic mode-I crack captured with a 1 Mfps camera. Both methods are successful in accurately reproducing the crack patterns observed in classical 2D and 3D dynamic rupture scenarios, as well as the near crack tip stress field during the propagation. The two methods are found to be numerically indifferentiable, although the displacement based method offers a significantly better computational performance. As a demonstrative application, we simulate the super-shear rupture in earthquakes, modeling the contact at the fault surface by a linear slip weakening friction law. The Burridge–Andrews mechanism naturally appears in the simulations, making the crack front jump from the sub-Rayleigh regime to the intersonic regime and propagate while producing shear Mach cones.

How fast can Evangelion run? Application of aerodynamics and scaling laws to the Super Robot

Super robots are huge, powerful robots that protect mankind from various invaders, and thus these superheroes are the main figures in many science fiction movies and Japanese animations. Among them, Evangelions have been a very popular type of super robot since the 1990s given that the animation series Neon Genesis Evangelion has been globally influential in various pop cultures. Evangelions (also called Evas) are cyborgs comprised of huge human body and robotic systems, and in the animation series, they often run at seemingly high speeds, which is quite different from traditional super robots. In this paper, we attempt to estimate the running speed of Evangelions based on known scientific facts. First, we measured the running speed of Eva Unit 01 (Eva-01) to be between 910 and 980 m/s based on its step length measured in movie scenes, and the Mach cone formed behind Eva-01. Second, we employed scaling laws known for animals and find that the maximum running speed of Eva-01 is 0.9 m/s. This striking difference between the anime-based speed and the physics-based speed raises a question as to how Eva-01 can run at such a high speed, and we conjecture that the cyborg can do so due to internally stored electrical power.

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 work18had 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.

Experimental study on a Mach cone and trailing Rayleigh waves in a stress wave chasing running crack problem

This paper focuses on a blast-induced Mach cone followed by trailing Rayleigh waves during blast stress waves chasing a running crack in an epoxy plate, using the photoelasticity method and high-speed photography. A Mach cone and trailing Rayleigh waves were visualized and interpreted based on isochromatic fringes, agreeing well with theoretical wave paths. The Mach cone was the coalescence of shear wavelets generated by P waves on crack surfaces, which was verified by the half-cone angle, equal to arcsin (CS/CP). Effects of the Mach cone and trailing Rayleigh waves on the running crack were evaluated by measuring crack velocity and dynamic stress intensity factors (SIFs) of the crack tip. It was found that the Mach cone decreased both crack velocity and SIFs, while trailing Rayleigh waves had the opposite effect. Finally, because of similarities, the findings in our experiments were discussed with those in the supershear rupture. In particular, effects of the Mach cone and trailing Rayleigh waves on the running crack were consistent with particle motion near the fault in the supershear rupture.

Uniformly moving screw dislocation in flexoelectric materials

The anti-plane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls within the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic anti-plane problems of flexoelectric or couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy is materialized in a finite element code. Screw dislocations play prominent role in crystal growth, plasticity, development of thin epitaxial films, micro-components and opto-mechanical devices and are important for our modern understanding of seismology. In general, screw dislocations are treated in the context of anti-plane elasticity. In this work we solve the steady-state problem of a screw dislocation in flexoelectric materials, moving with constant velocity. We investigate the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are attributed to the interplay of the elastic and the flexoelectric parameters. We also investigate the connection of the electric boundary conditions with the boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigate the subsonic and the supersonic steady state dislocation motion and find that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The presented results are important for all crystalline dielectrics such as ceramics, ice and perovskites that exhibit strong flexoelectric effect, often uncoupled from piezoelectricity.

Analysis on the influence mechanism of conical shock wave on pulsed laser forward detection

In order to study the influence mechanism of conical shock waves on pulsed laser forward detection, an analytical method is proposed for arbitrary Mach number and cone angle conditions. Based on traditional gas dynamics and physical optics, a simplified model of Gaussian beam transmission across the conical shock wave flow field is established. On this basis, combined with the process of pulsed laser detection, the analytical expression of pulsed laser echo waveform under the influence of shock wave is derived. Simulation results show that the conical shock wave under different conditions can lead to beam deflection, focus, and reflection to different degrees. Furthermore, the variation of echo waveform caused by aero-optical effect is also analyzed, and it is shown that the peak power, pulse width and arrival time of echo all fluctuate greatly with rolling motion at different shock angle and cone angle, among which the shift of the arrival time and the reflection effect of shock wave have the largest impact on laser detection.

Wake Potential and Wake Effects on the Ionospheric Plasma Density Measurements With Sounding Rockets

AbstractA new method is developed to systematically study variations in the electrostatic potential around a spinning spacecraft in flowing plasma. We argue that the main modulation of this potential is due to wake leading to different depletions in electron and ion densities. This wake potential can be calculated with the help of electric currents measured by Langmuir probes mounted on a boom system and the assumption of Boltzmann‐distributed electrons. The wake position is determined with the electric field measurements. We apply this method to data from the Investigation of Cusp Irregularities 3 ionospheric sounding rocket and find the observed wake to be larger than expected from the flow velocity and the Mach cone formation, which can be explained by the presence of booms. By using the wake potential to filter the rocket data, we show that this method is robust during highly dynamic plasma conditions and that it can be used to improve the quality of plasma density measurements and reduce the wake effects.

Numerical Investigation of Influence of Mesh Property in Nearfield Sonic Boom Prediction

To investigate the influence of mesh property in nearfield sonic boom prediction, the over-pressures of several typical configurations are commutated and analyzed with an in-house Reynolds Average Navier-Stokes solver. The property of mesh alignment, mesh type and mesh distribution are analyzed based on the double cone configuration and the 69-degree Delta Wing Body configuration. The experimental data from wind tunnel are used as comparisons. The results show that the Mach cone aligned method has significant improvement to the accuracy for the prediction of sonic boom in nearfield. Both structured and un-structed mesh with a cylinder zone near the model can help to capture the distribution of the over-pressure, however, it is more convenient to use unstructured mesh when deal with complex configurations. The aspect ratio and circumferential spacing affect the captured shock wave peak value and the shapes. The accuracy of the prediction can be satisfied by carefully selecting these two factors. In the last, the summarized strategies for mesh generation are applied in the sonic boom prediction of a complex supersonic transport configuration, the Lockheed Martin 1021 to verify its practicability.

Singularities in vorticity-induced velocity field eliminated from linearized supersonic flow

The origin of singularities along the Mach cones in a horseshoe line-vortex distribution have been identified and eliminated.

Effects of multiscale phase-mixing and interior conductance in the lunar-like pickup ion plasma wake. First results from 3-D hybrid kinetic modeling

The study of multiscale pickup ion phase-mixing in the lunar plasma wake with a hybrid model is the main subject of our investigation in this paper. Photoionization and charge exchange of protons with the lunar exosphere are the ionization processes included in our model. The computational model includes the self-consistent dynamics of the light (H+ or H2+ and He+), and heavy (Na+) pickup ions. The electrons are considered as a fluid. The lunar interior is considered as a weakly conducting body. In this paper we considered for the first time the cumulative effect of heavy neutrals in the lunar exosphere (e.g., Al, Ar), an effect which was simulated with one species of Na+ but with a tenfold increase in total production rates. We find that various species produce various types of plasma tail in the lunar plasma wake. Specifically, Na+ and He+ pickup ions form a cycloid-like tail, whereas the H+ or H2+ pickup ions form a tail with a high density core and saw-like periodic structures in the flank region. The length of these structures varies from 1.5RM to 3.3RM depending on the value of gyroradius for H+ or H2+ pickup ions. The light pickup ions produce more symmetrical jump in the density and magnetic field at the Mach cone which is mainly controlled by the conductivity of the interior, an effect previously unappreciated. Although other pickup ion species had little effect on the nature of the interaction of the Moon with the solar wind, the global structure of the lunar tail in these simulations appeared quite different when the H2+ production rate was high.

Interaction of a supersonic particle with a three-dimensional complex plasma

The influence of a supersonic projectile on a three-dimensional complex plasma is studied. Micron sized particles in a low-temperature plasma formed a large undisturbed system in the new “Zyflex” chamber during microgravity conditions. A supersonic probe particle excited a Mach cone with Mach number M ≈ 1.5–2 and double Mach cone structure in the large weakly damped particle cloud. The speed of sound is measured with different methods and particle charge estimations are compared to the calculations from standard theories. The high image resolution enables the study of Mach cones in microgravity on the single particle level of a three-dimensional complex plasma and gives insight to the dynamics. A heating of the microparticles is discovered behind the supersonic projectile but not in the flanks of the Mach cone.

Videos of ultrasonic wave propagation through transparent acrylic objects in water for introductory physics courses produced using refracto-vibrometry

This project used refracto-vibrometry to produce full-field videos of ultrasonic wave propagation in water; these videos could be incorporated into high school and college physics courses to illustrate wave behaviors. Refracto-vibrometry (RV) is an interferometric method for optically measuring sound waves. The measurement beam from a Polytec PSV-400 scanning laser Doppler vibrometer was directed through a water tank towards a stationary retroreflective surface. Acoustic wave fronts (density variations) which pass through the measurement laser cause variations in the integrated optical path length. By superimposing tens of thousands of scan points, videos enable visualization of the ultrasound propagation. In one experiment, ultrasonic waves in water (1484 m/s) are incident on a submerged acrylic block at a non-zero angle. The waves are refracted as they travel at about 2700 m/s through the block. Students can determine this speed of sound by measuring the angle of refraction and lateral deflection of the wave fronts in the video. In another experiment, ultrasonic waves passed through an acrylic cylinder, producing a Mach cone in water. Measurement of the Mach cone angle enables determination of the acrylic speed of sound. Links for videos and sample worksheets will be provided.

Stagnation temperature effect on the conical shock with application for air

The aim of this work is to realize a new numerical program based on the development of a mathematical model allowing determining the parameters of the supersonic flow through a conical shock under hypothesis at high temperature, in the context of correcting the perfect gas model. In this case, the specific heat at constant pressure does not remain constant and varies with the increase of temperature. The stagnation temperature becomes an important parameter in the calculation. The mathematical model is presented by the numerical resolution of a system of first-order nonlinear differential equations with three coupled unknowns for initial conditions. The numerical resolution is made by adapting the higher order Runge Kutta method. The parameters through the conical shock can be determined by considering a new model of an oblique shock at high temperature. All isentropic parameters of after the shock flow depend on the deviation of the flow from the transverse direction. The comparison of the results is done with the perfect gas model for low stagnation temperatures, upstream Mach number and cone deviation angle. A calculation of the error is made between our high temperature model and the perfect gas model. The application is made for air.

INVESTIGATION OF MACH CONE DEVELOPMENT IN BALLASTLESS RAILWAY TRACK STRUCTURES OVER SOFT CLAYS

High speed trains (HST) have many challenges from a geodynamic perspective; such as the case of HST operating in areas with soft soils like soft clays in Delta areas. The speeding trains can easily reach critical velocity due to the resonance of Rayleigh surface waves developed by the train and the low shear wave velocity of the soft soil. When this resonance happens it can develop a phenomenon called “Mach cones” which is similar to the ones the air jets produce at the speed of sound. This phenomenon can be observed in theoretical analytical measurements as well as in practical measurements. This paper studies the occurrence of mach cones in the soft clay soil under ballastless track structure systems and compares this phenomenon with the traditional ballasted track. Results of the displacement contours in the clay layer of the ballasted track show concentric pattern with train velocity of 250 km/h which indicated mach cone development around this velocity. Other ballastless track types modeled show that the displacement contours are distributed with no concentric pattern and indicating no mach cones even with velocities reaching 250 km/h.

Analytical expressions for the borehole seismic signatures excited by an explosive source buried in an elastic half‐space

ABSTRACTBorehole guided waves that are excited by explosive sources outside of the borehole are important for interpreting borehole seismic surveys and for rock property inversion workflows. Borehole seismograms are typically modelled using numerical methods of wave propagation. In order to benchmark such numerical algorithms and partially to interpret the results of modelling, an analytical methodology is presented here to compute synthetic seismograms. The specific setup is a wavefield emanating from a monopole point source embedded within a homogeneous elastic medium that interacts with a fluid‐filled borehole and a free surface. The methodology assumes that the wavelength of the seismic signal is much larger than the borehole radius. In this paper, it is supposed that there is no poroelastic coupling between the formation and the borehole.The total wavefield solution consists of P, PP, and PS body waves; the surface Rayleigh wave; and the low‐frequency guided Stoneley wave (often referred as the tube wave) within the borehole. In its turn, the tube wave consists of the partial responses generated by the incident P‐wave and the reflected PP and PS body waves at the borehole mouth and by the Rayleigh wave, as well as the Stoneley wave eigenmode. The Mach tube wave, which is a conic tube wave, additionally appears in the Mach cone in a slow formation with the tube‐wave velocity greater than the shear one. The conditions of appearance of the Mach wave in a slow formation are formulated. It is shown that the amplitude of the Mach tube wave strongly depends on Poisson's ratio of the slow surrounding formation. The amplitude of the Mach tube wave exponentially decreases when the source depth grows for weakly compressible elastic media with Poisson's ratio close to 0.5 (i.e., saturated clays and saturated clay soils). Asymptotic expressions are also provided to compute the wavefield amplitudes for different combinations of source depth and source‐well offset. These expressions allow an approximate solution of the wavefield to be computed much faster (within several seconds) than directly computing the implicit integrals arising from the analytical formulation.

Underwater noise from pile driving of conductor casing at a deep-water oil platform.

Underwater noise from impact pile driving of 512 -m-long conductor casings was measured at a deep-water offshore oil platform in the Santa Barbara Channel. Beamforming measurements, obtained with a vertical array, confirmed that the primary wave front generated by hammering the conductor casing was a Mach cone propagating at an angle of 17.6° below the horizontal. Analysis of the processed array data also revealed the presence of high-frequency secondary waves at angles steeper than 45° below the horizontal. These secondary waves, which appeared to be generated near the sea-surface, dominated the acoustic spectrum of the pulses at frequencies above 1 kHz. Shallow hydrophone measurements outside the Mach cone showed clear evidence of a surface shadow zone, which was caused by the strong downward directivity of the source. Although reflected waves, diffraction, and secondary waves still produced sound inside the surface shadow zone, sound levels were 10-15 dB lower in this region. Long-term hydrophone measurements showed that there was little difference (±1 dB) in mean sound levels from impact hammering of different conductors installed at the same platform over three months.

Visualization of Mach cones in a viscoelastic fluid by the sensitive-tint method

Visualization of Mach cones in a viscoelastic fluid by the sensitive-tint method

Simulations of galaxy cluster collisions with a dark plasma component

Dark plasma is an intriguing form of self-interacting dark matter with an effective fluid-like behavior, which is well motivated by various theoretical particle physics models. We aim to find an explanation for an isolated mass clump in the Abell 520 system, which cannot be explained by traditional models of dark matter, but has been detected in weak lensing observations. We performed N-body smoothed particle hydrodynamics simulations of galaxy cluster collisions with a two component model of dark matter, which is assumed to consist of a predominant non-interacting dark matter component and a 10-40 percent mass fraction of dark plasma. The mass of a possible dark clump was calculated for each simulation in a parameter scan over the underlying model parameters. In two higher resolution simulations shock-waves and Mach cones were observed to form in the dark plasma halos. By choosing suitable simulation parameters, the observed distributions of dark matter in both the Bullet Cluster (1E 0657-558) and Abell 520 (MS 0451.5+0250) can be qualitatively reproduced.