Rayleigh Wave Speed Research Articles

What Happens When Two Ruptures Collide?

AbstractWe investigate the interaction between two rupture fronts as they propagate toward each other and ultimately collide. This phenomenon was observed during laboratory experiments conducted on poly methyl methacrylate. Subsequently, we used numerical simulations to elucidate key aspects of these observations and draw broader conclusions. Our findings indicate that the collision of the rupture fronts generates interface waves that propagate along the sliding interface at the Rayleigh wave speed. Additionally, the rupture fronts interact with the starting and stopping S‐wave phases radiated by the opposite rupture fronts, which can locally change their velocity and generate additional interface waves. We discuss the implications of these results for understanding earthquake source phenomena.

Estimation of ground vibration emission increase with respect to velocity in high-speed railways

During the design of a new line, it is often necessary to estimate railway vibration emissions at a given future speed from experimental data in similar circumstances, but at a different speed. Future levels are usually estimated according to a logarithmic evolution law with respect to speed, a typical one being that in USA FTA Report nº0123 "Transit Noise and Vibration Impact Assessment Manual", a de facto standard in the sector. Also, it is known that for high-speed lines a drastic increase in vibrations is observed when railway speed is higher than the propagation speed of surface Rayleigh waves. This phenomenon, known as trans-Rayleigh boom, produces a sharp discontinuity in the evolution of vibration emissions. However, vibration levels observed in the vicinity of a high-speed line have shown values well above the prediction rule suggested in FTA Manual, which could not be explained by a trans-Rayleigh boom. A more accurate extrapolation method based on local rail roughness measurements is capable of producing improved predictions compared to FTA rule of thumb. This contribution shows the details of the method suggested, and the improvement in the prediction (with respect to FTA extrapolation law) obtained in the specific case of a high-speed railway.

Transonic and Supershear Crack Propagation Driven by Geometric Nonlinearities.

Linear elastic fracture mechanics theory predicts that the speed of crack growth is limited by the Rayleigh wave speed. Although many experimental observations and numerical simulations have supported this prediction, some exceptions have raised questions about its validity. The underlying reasons for these discrepancies and the precise limiting speed of dynamic cracks remain unknown. Here, we demonstrate that tensile (mode I) cracks can exceed the Rayleigh wave speed and propagate at supershear speeds. We show that taking into account geometric nonlinearities, inherent in most materials, is sufficient to enable such propagation modes. These geometric nonlinearities modify the crack-tip singularity, resulting in different crack-tip opening displacements, cohesive zone behavior, and energy flows towards the crack tip.

A Gaussian process approach for rapid evaluation of skin tension

Skin tension plays a pivotal role in clinical settings, it affects scarring, wound healing and skin necrosis. Despite its importance, there is no widely accepted method for assessing in vivo skin tension or its natural pre-stretch. This study aims to utilise modern machine learning (ML) methods to develop a model that uses non-invasive measurements of surface wave speed to predict clinically useful skin properties such as stress and natural pre-stretch. A large dataset consisting of simulated wave propagation experiments was created using a simplified two-dimensional finite element (FE) model. Using this dataset, a sensitivity analysis was performed, highlighting the effect of the material parameters and material model on the Rayleigh and supersonic shear wave speeds. Then, a Gaussian process regression model was trained to solve the ill-posed inverse problem of predicting stress and pre-stretch of skin using measurements of surface wave speed. This model had good predictive performance (R2 = 0.9570) and it was possible to interpolate simplified parametric equations to calculate the stress and pre-stretch. To demonstrate that wave speed measurements could be obtained cheaply and easily, a simple experiment was devised to obtain wave speed measurements from synthetic skin at different values of pre-stretch. These experimental wave speeds agree well with the FE simulations, and a model trained solely on the FE data provided accurate predictions of synthetic skin stiffness. Both the simulated and experimental results provide further evidence that elastic wave measurements coupled with ML models are a viable non-invasive method to determine in vivo skin tension. Statement of significanceTo prevent unfavourable patient outcomes from reconstructive surgery, it is necessary to determine relevant subject-specific skin properties. For example, during a skin graft, it is necessary to estimate the pre-stretch of the skin to account for shrinkage upon excision. Existing methods are invasive or rely on the experience of the clinician. Our work aims to present an innovative framework to non-invasively determine in vivo material properties using the speed of a surface wave travelling through the skin. Our findings have implications for the planning of surgical procedures and provides further motivation for the use of elastic wave measurements to determine in vivo material properties.

Investigation on ground vibration displacements induced by high-speed trains moving on elastic-plastic multi-layered ground by 2.5D FEM

Considering the soil as nonlinear material, the ground vibration displacements caused by high-speed train (HST) moving on the elastic-plastic multi-layered ground are investigated using a 2.5-dimensional finite element model (2.5D FEM), in which the improved Mohr-Coulomb model and the post-Eulerian integration algorithm are also introduced. The accuracy of the established model is validated with published data. Results show that, under different train speeds, the ground vibrations at the track center for the multi-layered soil media are predominantly affected by the properties of the topsoil layer. The ground vibration displacements within 2 m from the track center are controlled by resonance conditions. The displacement attenuation curves fluctuate more in the soft layer than the hard one when the train speed reaches or exceeds the Rayleigh wave speed of all soil layers. Additionally, increasing the elastic modulus of topsoil can effectively reduce the ground vibration displacements in the range of 0.0 m–2.5 m from the track center, and the effects of buried depth for the soft soil layer being slight and thus ignorable.

Propagation of extended fractures by local nucleation and rapid transverse expansion of crack-front distortion.

Fractures are ubiquitous and can lead to the catastrophic material failure of materials. Although fracturing in a two-dimensional plane is well understood, all fractures are extended in and propagate through three-dimensional space. Moreover, their behaviour is complex. Here we show that the forward propagation of a fracture front occurs through an initial rupture, nucleated at some localized position, followed by a very rapid transverse expansion at velocities as high as the Rayleigh-wave speed. We study fracturing in a circular geometry that achieves an uninterrupted extended fracture front and use a fluid to control the loading conditions that determine the amplitude of the forward jump. We find that this amplitude correlates with the transverse velocity. Dynamic rupture simulations capture the observations for only a high transverse velocity. These results highlight the importance of transverse dynamics in the forward propagation of an extended fracture.

Influence of rupture velocity on risk assessment of concrete moment frames: Supershear vs. subshear ruptures

The possibility of ground motions propagating at speeds greater than the Rayleigh wave speed was accepted not long ago, and thus studies on them are limited. These supershear motions are believed to damage specific kinds of infrastructure, and thus studying the risk associated is important. As a part of this investigation, we use state-of-the-art tools from the Natural Hazards Engineering Research Infrastructure to estimate normalized economic losses and injuries, by considering one particular building type and occupancy category listed in HAZUS-MH. Faults of similar dimensions are shown to replicate subshear and supershear earthquake instances with four groups of station points parallel to the fault and four more perpendicular to the fault. Both fault parallel and fault normal components of ground motion are stimulated. The economic loss % and injury is standardized in metrics of repair cost and population respectively and is estimated along with the associated uncertainty for 3, 6 and 9-story commercial concrete moment frame structures. Inclusion of complex phenomena such as supershear earthquake events in evaluating economic losses and risk will contribute to the United Nation Office for Disaster Risk Reduction (UNDRR) vision of disaster-resilient development goals.

Волны Рэлея на гладкой кривой поверхности

A model has been developed that describes the propagation of surface waves on smooth curved surfaces. This model is developed using the hyperbolic wave equation and is applicable to studying the propagation of surface waves on any curved surface described by a related orthogonal curvilinear coordinate system and for any values of the product of the wave vector and the radius of curvature. The behavior of Rayleigh waves on canonical curved surfaces (cylinder, sphere) is simulated, and the results are used to quantify the effect of curvature on the speed of Rayleigh waves. The condition for the existence of a Rayleigh wave is the validity of the values of the wave vectors K obtained from the dispersion equations. The actual values of the wave vectors K determine the waves propagating along the surface without carrying the energy deep into the body. The speed of the surface wave increases with increasing curvature of the concave surface, and decreases on the concave surface. The Rayleigh wave has limited growth as the positive radius of curvature decreases. At positive values of the radius of curvature (outer surface), as the radius decreases, a maximum is observed, and then the speed of waves decreases and breaks of curves are observed at negative curvature (inner surface), which is associated with resonance along the circumference of the envelope of a cylinder or great circle of a sphere.

Fast atomic crack kinking and branching in WS2

Abstract Dynamic nanocrack propagation in 1T- and 2H-WS2 strips is studied by molecular dynamics, and the T-stress and circumferential stress in linear elastic fracture mechanics are considered. As the crack propagates, the crack-tip speed (v) experiences a rapid acceleration, and then oscillates at ∼55% (1T) and ∼65% (2H) of the Rayleigh-wave speed followed by crack kinking/branching. The critical energy release rates of crack instability are estimated to be ∼1.5 J/m2 (1T) and ∼4.0 J/m2 (2H). The crack path in 1T-WS2 exhibits higher sensitivity of strain rates for atomic asymmetry around the crack tip. Examination of the dynamic crack-tip field shows that the T-stress obtained by the over-deterministic method always fluctuates in negative, and the theoretical circumferential stress curve does not accurately capture the v-dependent atomic stress distribution. Consequently, both T-stress and circumferential stress are limited in predicting the crack kinking/branching directions, which can be attributed to the discrete crystal lattice and local anisotropy of WS2, where a preferred crack path along the zigzag surface is observed. The fracture properties of WS2 might provide useful physics for its applications in nano-devices.

Influence of rate-dependent damage phase-field on the limiting crack-tip velocity in dynamic fracture

This contribution explores three thermodynamically-consistent damage phase-field formulations for rate-dependent dynamic fracture in viscoelastic materials. By means of a numerical study on a uniform displacement strip benchmark, the formulations and modelling assumptions are compared, and the corresponding limiting crack-tip speeds are discussed. In essence, in addition to recalling the existing phase-field model for viscoelastic materials, a damage phase-field formulation for rate-dependent toughness is introduced as a function of the damage-rate at the crack-tip and is contrasted to existing strain-rate dependent toughness models. Dynamic fracture simulations have demonstrated the significant role of rate dependency in suppressing crack branching and accelerating the crack propagation. The rate dependency is analysed through two main mechanisms: (i) the viscous dissipation, which can promote fracture, and (ii) an increase in the energy required to evolve a crack, through rate-dependent toughness models. Depending on the specific choice of parameters, the numerical simulations show crack propagation at speeds that exceed the theoretical limit depicted by the Rayleigh wave speed. Indeed, these high speeds are attributed to the viscoelastic and viscoelastic-like (observed in the case of strain-rate dependent fracture toughness) stiffening at the crack-tip, which translates to faster running surface waves and enables supersonic crack-speeds; a never-seen-before result in damage phase-field simulations.

Resonance with surface waves induces forbidden velocity bands in dislocation glide

We use molecular dynamics simulations to demonstrate a pronounced influence of free surfaces on the mobility of dislocations at near-sonic velocities. By correlating instantaneous velocities v and resolved stresses τ, we show that the mobility relations of both edge and screw characters are composed of multiple ranges of stable uniform motion separated by bands of forbidden velocity. The lower limiting velocity for uniform motion of both edge and screw dislocations is the Rayleigh wave speed, cR. When forced to propagate at an average velocity greater than cR, dislocations exhibit cyclic, intermittent jumps in instantaneous velocity between cR and higher velocity branches of the mobility relation. Bands of forbidden velocity may be calculated directly from the velocity dependence of dislocation drag functions, which contain peaks near cR and—in the edge case—one other characteristic surface wave speed, which we associate with shear-horizontal (SH) waves. We propose that peaks in the drag function are due to resonance of free surface vibrations forced by nearby moving dislocations.

Asymptotic formula of Rayleigh wave speed in an orthotropic half-space coated by a thin piezoelectric film

ABSTRACT Piezoelectric thin films have been widely used in electro-acoustic devices including ones measuring Rayleigh wave speed. The main objective of this work is to find an explicit formula of Rayleigh speed taking into account the effect of the thin piezoelectric film. First the problem of Rayleigh surface waves propagating along an orthotropic substrate coated by a thin piezoelectric film is considered using Stroh’s formalism approach. Then, the exact secular equation of Rayleigh wave is obtained in implicit form. In case when the thickness of the thin layer is much smaller than the considering wavelength of Rayleigh waves, an asymptotic formula of Rayleigh speed is derived from the asymptotic form of the exact secular equation. The obtained asymptotic formula is simple enough and could be used in practice when Rayleigh speed is measured by using piezoelectric device. Numerical illustrations are carried out to show the validity of the obtained approximate formulas and its potential application in inverse problem.

Tensile cracks can shatter classical speed limits.

Brittle materials fail by means of rapid cracks. Classical fracture mechanics describes the motion of tensile cracks that dissipate released elastic energy within a point-like zone at their tips. Within this framework, a "classical" tensile crack cannot exceed the Rayleigh wave speed, [Formula: see text]. Using brittle neo-hookean materials, we experimentally demonstrate the existence of "supershear" tensile cracks that exceed shear wave speeds, [Formula: see text]. Supershear cracks smoothly accelerate beyond [Formula: see text], to speeds that could approach dilatation wave speeds. Supershear dynamics are governed by different principles than those guiding "classical" cracks; this fracture mode is excited at critical (material dependent) applied strains. This nonclassical mode of tensile fracture represents a fundamental shift in our understanding of the fracture process.

Laboratory earthquakes decipher control and stability of rupture speeds

Earthquakes are destructive natural hazards with damage capacity dictated by rupture speeds. Traditional dynamic rupture models predict that earthquake ruptures gradually accelerate to the Rayleigh wave speed with some of them further jumping to stable supershear speeds above the Eshelby speed (~sqrt{2} times S wave speed). However, the 2018 Mw 7.5 Palu earthquake, among several others, significantly challenges such a viewpoint. Here we generate spontaneous shear ruptures on laboratory faults to confirm that ruptures can indeed attain steady subRayleigh or supershear propagation speeds immediately following nucleation. A self-similar analysis of dynamic rupture confirms our observation, leading to a simple model where the rupture speed is uniquely dependent on a driving load. Our results reproduce and explain a number of enigmatic field observations on earthquake speeds, including the existence of stable subEshelby supershear ruptures, early onset of supershear ruptures, and the correlation between the rupture speed and the driving load.

An Important Historical Milestone: The Classification of the Cubic Equations

This article investigates the use of the history of mathematics as a pedagogical tool for the teaching and learning of mathematics, using the history of the cubic equation as a specific example.Cubic equations arise intrinsically in many applications in natural sciences and mathematics. For example, in physics, the solutions of the equations of state in thermodynamics, or the computation of the speed of seismic Rayleigh waves require the solutions of cubic equations. In mathematics, they are instrumental in solving the quartic equations, for in the process, these are reduced to cubic equations. The impossibility of trisecting an angle or doubling a cube using only a straightedge and compass is equivalent to solving some cubic equations. As the name implies, the cubic spline approximation, an important tool in numerical analysis, also entails working with cubic functions. Although cubic equations were explored by the ancient Babylonian, Greek, Chinese, Indian, and Egyptian scholars, it took the collective work of many well-known mathematicians such as Diophantus, Archimedes, Fibonacci, del Ferro, Khayyam, Tartaglia, Cardano, Viète, Descartes, and Lagrange to finally obtain a full solution. Our goal in this paper is to investigate one of the most formidable steps in this extensive and prolific history, namely the complete classification of the cubic equations by Omar Khayyam in eleventh century, who was the first scholar to classify cubic equation and hence facilitate a methodical and logical approach to obtaining a general solution.

Dynamic acoustoelastic testing (DAET) with a thermal strain pump for characterization of closed fatigue cracks

Dynamic acoustoelastic testing (DAET) is a nonlinear acoustic technique used to characterize the nonlinear elastic behavior of materials. To enable the in-situ implementation of DAET, we propose a modification to the conventional DAET setup by replacing the mechanical strain pump with a thermal strain pump. This study presents the feasibility of the proposed thermal-pump DAET setup for in-situ detection of closed fatigue cracks in an aluminum test specimen. Our setup employs a hot air blower to induce a small (∼4 °C) thermal gradient near the surface of the specimen. The surface temperature evolution is captured by an infrared camera allowing thermal strain evaluation on the surface. A pair of high frequency angle-beam Rayleigh surface wave transducers are used to probe the thermal strain-induced changes in near-surface elastic properties. The slope of the relative change in Rayleigh wave speed with the applied thermal strain is taken as a measure of nonlinearity (γ) evaluated at several locations across the specimen along the intact and cracked regions. Our results show that γ measures higher near the crack and decreases as the probe moves towards the crack tip, demonstrating the potential of thermal-pump DAET for in-situ evaluation of closed fatigue cracks.

Osmotic instability in soft materials under well-controlled triaxial stress

Living tissues and some engineering materials contain water. When a wet material loses water, high triaxial tensile stress may build up and cause instability. The mechanism of instability under triaxial tension has attracted great attention, but quantitative study remains an ongoing challenge. Here we develop an experimental method to apply well-controlled triaxial tensile stress and observe osmotic instability in situ. We synthesize a hydrogel in an elastomer tube with strong adhesion between them. The elastomer dissolves minute amount of water, but allows water to diffuse out and places the hydrogel under homogeneous, equal-triaxial, tensile stress. We develop a method to determine the stress as a function of time. The transparent setup enables observation of various types of osmotic instabilities, including cavity nucleation, crack propagation, and surface undulation. Notably, our method enables the measurement of crack speed from ∼10−5 m/s to a limit comparable to the Rayleigh wave speed ∼1 m/s. We observe a large jump in crack speed at a critical energy release rate. This work opens opportunities to study the physics of soft materials under high triaxial tension.

Simultaneous determination of Young's modulus and Poisson's ratio in metals from a single surface using laser-generated Rayleigh and leaky surface acoustic waves

Material elastic moduli are used to assess stiffness, elastic response, strength, and residual life. Ultrasound (US) measurements of propagation wave speeds (for longitudinal and shear waves) are now primary tools for non-destructive evaluation (NDE) of elastic moduli. Most US techniques measure the time-of-flight of through-transmission signals or reflected signals from the back wall. In both cases, an independently determined sample thickness is required. However, US methods are difficult for complex (non-flat) samples. When the local thickness is unknown, the propagation speed cannot be determined. On the other hand, the propagation speed of Rayleigh waves can be calculated without knowledge of sample thickness, but another independent measurement is still required to compute both Young's modulus and Poisson's ratio. We present a comprehensive theoretical background, numerical simulations, and experimental results that clearly show that when the material density is assumed known, both elastic constants of an isotropic metal sample can be determined with laser-ultrasound by tracking two types of surface propagating waves without any sample contact (both signal excitation and detection are performed optically). In addition to a conventional surface, or Rayleigh, acoustic wave, a leaky surface wave can also be launched with nanosecond laser pulses in the thermoelastic regime of excitation (i.e., without material ablation) close to the source that propagates along the sample surface with speed close to that of bulk longitudinal waves. Samples can be of arbitrary shape and their thickness need not be measured.

Nanoscale Damage Production by Dynamic Tensile Rupture in α‐Quartz

AbstractThe creation of new fractures during an earthquake produces rock damage and contributes to the dissipation of strain energy. During dynamic rupture propagation, tensile microfractures can form in the earthquake process zone and in the domains around a fault that host large transient tensile stress. These microfractures can produce rock fragments with a wide range of sizes. Using molecular dynamics simulations, we model tensile rupture propagation in α‐quartz under conditions of stress that occur during earthquake propagation. Our results show that for rupture speeds below 15% of the Rayleigh wave speed, fractures propagate linearly. At higher speeds, fracture propagation undergoes path instabilities with crack oscillations and microbranching leading to the formation of nanoscale roughness and fragments. This nanoscale damage can form in and around the earthquake process zone before any significant slip has occurred on the fault. The produced nanoparticles may control further energy dissipation during frictional slip.

Propagation of interfacial (Stoneley-type) waves at the joint boundary of two initially-stressed incompressible half-spaces

The presence of initial stress in solids changes the material’s response to finite and infinitesimal deformations. This paper presents a study on the effects of initial stress, finite deformation, and material properties on the speed of interfacial waves which are assumed to travel at the joint boundary of two incompressible half-spaces. The theory of infinitesimal deformations superimposed on finite deformations is used along with the theory of invariants. Boundary conditions at the interface result in an explicit secular equation, which is analyzed generally. For each half-space, a prototype strain–energy function is assumed, which is a function of the deformation gradient and initial stress tensors. The effect of different governing parameters is observed on the wave speed and is analyzed theoretically. Graphical illustrations are also presented to elaborate on the theoretical results. It is observed that for admissible values of parameters, a real wave speed exists and is bounded by values depending on the governing parameters. A comparison between Rayleigh and interfacial wave speeds is also made and supported with graphs of dimensionless wave speeds with respect to initial stresses.