Rayleigh Wave Speed Research Articles

Stability of Penetrative Natural Convection in a Non-Newtonian Fluid-Saturated Vertical Porous Layer

The stability of natural penetrative convection arising due to a uniform internal heat source in a vertical porous layer saturated with an Oldroyd-B fluid is investigated. The vertical walls of the porous layer are impermeable and maintained at different uniform temperatures. The energy stability analysis performed reveals that the system is unconditionally stable even in the presence of internal heating in the case of Newtonian fluids, while for viscoelastic fluids the base flow is found to be unstable. As the energy stability analysis of Gill type is unable to decide the stability of the system, the Galerkin method is used to solve the complex eigenvalue problem. The internal heating introduces asymmetry in the basic flow and amounts to the existence of different set of onset modes. The internal heating and stress relaxation parameter facilitates instability of the system while increasing strain retardation parameter discloses stabilizing effect on the system. Moreover, the critical Darcy–Rayleigh number, wave number and wave speed become invariant as Ns becomes large. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state.

On Rayleigh wave in a generalized porothermoelastic solid half-space

The governing equations of generalized porothermoelasticity are formulated in context of Green and Naghdi theory of thermoelasticity without energy dissipation. A problem on Rayleigh type surface in a generalized porothermoelastic solid half-space is considered. The governing equations are solved for particular surface wave solutions satisfying required radiation conditions in the half-space. A secular equation for Rayleigh wave speed is derived after applying these solutions to relevant boundary conditions at stress free thermally insulated surface of half-space. The Rayleigh wave speed is computed for relevant physical constants of material and plotted against various material parameters to observe the effects of porosity, coefficients of thermal expansion, coefficients of thermoelastic coupling and characteristics of solid and fluid phases.

Study of the dynamic fracture of hollow spheres under compression using the Discrete Element Method

Hollow sphere structure (HSS) belongs to cellular solids that have been studied recently for its multiples properties. In our case, HSS aims to absorb soft impacts energy on an airliner cockpit. This structure is investigated because of its promises in term of specific energy dissipated (J.kg1) during impact. First of all, quasi- static and dynamic (v = 5mmmin_1 to v = 2ms4) uniaxial compression tests are conducted at room temperature on a single sphere (D = 30 mm). Rapid crack propagation (RCP) is observed to be predominant at macroscopic scale. The formalism of Linear Elastic Fracture Mechanics (L.E.F.M.) is therefore used to estimate the dynamic energy release rate GIdc. The crack tip location is measured during the crack propagation using a high speed camera. The Discrete Element Method (DEM) is used to simulate the dynamic fracture by implementing a node release technique to perform a generation phase simulation. The dynamic energy release rate can be determined using the experimentally measured crack history. In hollowed spherical structures the numerical results reveal a high proportion of energy dissipated through inertial effects as well as a dependence of the thickness of the skin over the range of 0.04 mm to 1.2 mm. At a crack tip velocity of 0.6 times the Rayleigh wave speed of the material, the dynamic correction factor is less than 0.05. Similar results have been shown for the longitudinal dynamic fracture of polymer pipes. The quantitative results of GIdc are in good agreement with the literature and the present model offers an alternative to the finite element method to simulate the rapid crack propagation.Its use reveals to be an interesting way to model the mechanical behavior of brittle materials.

Disturbance and recovery in high speed (110) cleavage in single crystalline silicon

Stress perturbations and material defects can significantly affect the fracture initiation and propagation behaviors in brittle materials. In this work, we show that (110)[110] cleavage in silicon deflects onto (111) plane in the presence of contact stresses. The deflection is however not permanent as the crack returns to the (110) plane after a certain length of propagation, even in the case where the crack velocity is up to 78% of the Rayleigh wave speed. The recovery behavior indicates that the (110)[110] cleavage is invariably prevailing when perpendicular to the maximum stress. Following this indication, it can be concluded that the observed (110)[110]–(111) deflection in previous literature is more likely driven by the external disturbance rather than the crack velocity induced toughness evolution. We also highlight that the extra energy for the (110) recovery is minimized at the expense of a large propagation distance upon the plane switch.

Brittle crack propagation resistance inside grain and at high angle grain boundary in 3% Si-Fe alloy

Brittle fracture in carbon steel has a strong impact on the safety of the structures. Especially, the arresting technology of the running crack is the measure of last resort for ensuring structural integrity. Due to such high importance, many experimental and theoretical studies of brittle crack propagation have been conducted from both mechanical and microstructural viewpoints.It is thought that the elementary step of the brittle fracture of polycrystalline steel is the cleavage in each crystal grain and their connection process. However, the detailed mechanisms of brittle fracture have not been fully understood; for example, it is still unclear why the propagation rate under a large driving force is not increased up to the Rayleigh wave speed. Several difficulties hinder the achievement of a detailed understanding so far: 1) crack propagation is quite rapid, 2) the crystal grain is usually too small (10–100 μm) to collect sufficient information, 3) the microstructure is very complicated in most case, i.e., containing different phases, microstructures, precipitates and inclusions.In this study, to eliminate such difficulties, 3% silicon steel in which microstructure is the ferrite single phase and the grain size is increased to 4–5 mm is used. This steel can be fractured in a brittle manner, even under ambient temperature. For this steel, by using a high speed camera and strain gauge data with a high sampling rate, the elementary process of brittle crack propagation is elucidated.As a result, it is revealed that the brittle crack propagation rate even in a single-crystal grain is much slower than the Rayleigh wave speed. This seems to be due to the presence of twin deformations and twin boundary cracks in crystal grain as observed on the fracture surface. Using the analysis of electron back scatter diffraction (EBSD) data, the mechanism of twin deformation and twin boundary crack is revealed. Additionally, it is shown that the brittle crack propagation rate where the path includes crystal grain boundaries is much slower. This delay seems to be related to the misorientation angle at the GB. By applying our simplified model, the delay effect at the grain boundary can be successfully explained.

Connection between the properties of 40kh-steel bar stock and the speed of bulk and rayleigh waves

Results of the research of dependence between the structural states of 40Kh-steel bar samples obtained with different modes of thermal processing and the velocity of longitudinal, transverse, and Rayleigh waves are presented. The mirror-shadow multiple reflection method with the use of dedicated pass-through and attachable electromagnetic–acoustic transducers was employed. A connection between the wave velocity and the ultimate strength, yield point, plasticity, and rigidity of the examined samples is demonstrated. Dependences of the acoustic wave velocity on mechanical tensile stresses are presented. The acoustoelastic coefficients are determined.

Rayleigh surface wave in a porothermoelastic solid half-space

In the present paper, the Rayleigh wave at a stress free thermally insulated surface of a generalized porothermoelastic solid half-space is considered. The governing equations of generalized porothermoelasticity are solved for general surface wave solution. The particular solutions satisfying the required radiation conditions are obtained. These solutions are applied to boundary conditions at stress free thermally insulated surface. In order to satisfy the relevant boundary conditions, a secular equation for wave speed of Rayleigh wave is obtained. Numerical simulations are done using an experimental data given by Yew and Jogi (1976) [J. Acoust. Soc. Am. 60, 2-8]. The wave speed is computed to observe the effects of frequency, porosity, coefficients of thermal expansion, and thermoelastic coupling coefficients.

The influence of soil nonlinear properties on the track/ground vibration induced by trains running on soft ground

The deflections of the track under a moving train depend on the stiffness of the underlying soil as well as the properties of the track and the train. In many situations, small-strain linear properties can be assumed for the soil. However, particularly for soft soil, as the load speed approaches the speed of Rayleigh waves in the ground, the deflections increase considerably. In such situations the use of the small-strain soil stiffness may lead to inaccuracies in the estimates of track deflections or of the critical speed. A finite element model of the track and ground has been developed to study the deflections induced by trains running on soft ground. Soil nonlinearity is introduced through a user-defined subroutine. The nonlinearity is specified in terms of the shear modulus reduction as a function of octahedral shear strain, which can be based on data obtained from laboratory tests on soil samples. The model is applied to the soft soil site at Ledsgard in Sweden, from which extensive measurements are available from the late 1990s. It is shown that the use of a linear model based on the small-strain soil parameters leads to an underestimation of the track displacements when the train speed approaches the critical speed, whereas the nonlinear model gives improved agreement with measurements. In addition, an equivalent linear model is considered, in which the equivalent soil modulus is derived from the laboratory curve of shear modulus reduction using an ‘effective’ shear strain. For this approach it is shown that the predictions in this specific case are improved by using a value of 20% of the maximum strain as the effective strain rather than the value of 65% commonly used in earthquake studies.

Dynamic Shakedown Analysis of Flexible Pavements under Rolling and Sliding Contact Considering Moving Speed

Pavement subjected to the traffic load above shakedown limit is predicted to exhibit accumulated plastic strains that eventually lead to failure within the pavement system. Shakedown theory is a valid approach to estimating the critical shakedown load level. In this study, a dynamic shakedown method for computing the shakedown limit of pavement is presented based on the lower-bound theorem of shakedown. To this end, a combined Finite element-Infinite element (FE-IE) method is utilized to calculate the dynamic elastic stress field due to rolling and sliding contact between wheel and pavement surface considering the effect of moving speed. The rolling and sliding contact is approximated by a moving Hertz distribution load, which is performed through moving vertical and horizontal body forces simultaneously modeled by thin-layered element above the pavement surface. Shakedown limits for both uniform and two-layered pavement system at various moving speeds have been investigated, respectively. It is found that the shakedown limit decreases with the increasing moving speed and then turns to increase when the moving speed exceeds the Rayleigh wave speed. This dynamic effect is more profound when the horizontal force is small. The influence of horizontal force on shakedown limit at various moving speed is also discussed. Eventually, the impact of frictional coefficient and material properties on shakedown limit for two-layered pavement is also investigated as can then be applied to the optimal pavement design.

Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms

We address the simulation of dynamic crack propagation in brittle materials using a regularized phase-field description, which can also be interpreted as a damage-gradient model. Benefiting from a variational framework, the dynamic evolution of the mechanical fields are obtained as a succession of energy minimizations. We investigate the capacity of such a simple model to reproduce specific experimental features of dynamic in-plane fracture. These include the crack branching phenomenon as well as the existence of a limiting crack velocity below the Rayleigh wave speed for mode I propagation. Numerical results show that, when a crack accelerates , the damaged band tends to widen in a direction perpendicular to the propagation direction, before forming two distinct macroscopic branches. This transition from a single crack propagation to a branched configuration is described by a well-defined master-curve of the apparent fracture energy Γ as an increasing function of the crack velocity. This Γ(v) relationship can be associated, from a macroscopic point of view, with the well-known velocity-toughening mechanism. These results also support the existence of a critical value of the energy release rate associated with branching: a critical value of approximately 2Gc is observed i.e. the fracture energy contribution of two crack tips. Finally, our work demonstrates the efficiency of the phase-field approach to simulate crack propagation dynamics interacting with heterogeneities, revealing the complex interplay between heterogeneity patterns and branching mechanisms.

On the coseismic ionospheric disturbances after the Nepal Mw7.8 earthquake on April 25, 2015 using GNSS observations

Nepal Mw7.8 earthquake occurred at 06:11:26 UTC on April 25, 2015. The epicenter was located at 28.147°N, 84.708°E, with the focal depth of 15km. In this paper, the coseismic ionospheric disturbances were analyzed using Vertical Total Electron Content from Global Navigation Satellite System (GNSS) observations obtained from 309 continuous tracking stations around the epicenter. The results show that significant ionospheric disturbances occurred within 10min after the earthquake. The maximum anomaly reached 1.34 TECU; the maximum duration was 10min; the farthest anomaly was found beyond 3000km from the epicenter. The ionospheric response was attributed to two modes: shock acoustic waves and Rayleigh wave. Within 500km from the epicenter, ionosphere anomaly was induced by both waves, with average propagating speed of 0.61 and 1.62km/s, respectively. With increasing epicentral distance, the ionospheric disturbances caused by acoustic waves gradually weakened, whereas Rayleigh waves induced ionosphere disturbances reached up to 3000km from the epicenter. The propagation speed of Rayleigh waves gradually increased to 2.74km/s while the distance from the epicenter to sub-ionospheric point was over 1500km. The occurrence of the disturbances appeared strong directivity. The amplitudes of the disturbances were large to the south of the epicenter. In contrast, to the northwest of the epicenter, few disturbances were observed, and the amplitudes of these disturbances were small. In general, the amplitude of the anomaly decreased with increasing epicentral distance. However, the amplitude of the anomaly increased at 1500–2500km to the northeast of the epicenter.

A note on formulas for the Rayleigh wave speed in elastic solids

In the present paper, new analytical, numerical and approximate methods have been presented for the determination of Rayleigh wave speed in isotropic and anisotropic media. The Lagrange’s method is used to provide exact expression for the roots of the secular equation for Rayleigh waves in isotropic media. Then, a simple non-iterative type quadrature method is used to numerically determine the Rayleigh wave speed in isotropic and anisotropic media. Further, an approximate method is presented to determine the velocity of Rayleigh waves. The discrete least square approximation on Chebyshev – Gauss - Lobatto nodes is suggested to transform secular equations to quadratic equations, thereby, providing improved approximations to the Rayleigh wave speed. The analysis is complemented with numerical examples.

Accurate Young's modulus measurement based on Rayleigh wave velocity and empirical Poisson's ratio.

This paper presents a method for Young's modulus measurement based on Rayleigh wave speed. The error in Poisson's ratio has weak influence on the measurement of Young's modulus based on Rayleigh wave speed, and Poisson's ratio minimally varies in a certain material; thus, we can accurately estimate Young's modulus with surface wave speed and a rough Poisson's ratio. We numerically analysed three methods using Rayleigh, longitudinal, and transversal wave speed, respectively, and the error in Poisson's ratio shows the least influence on the result in the method involving Rayleigh wave speed. An experiment was performed and has proved the feasibility of this method. Device for speed measuring could be small, and no sample pretreatment is needed. Hence, developing a portable instrument based on this method is possible. This method makes a good compromise between usability and precision.

The secular equation for non-principal Rayleigh waves in deformed incompressible doubly fiber-reinforced nonlinearly elastic solids

The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiber-reinforced incompressible nonlinearly elastic half-space are obtained. Hence, the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. One of the principal planes of the primary pure homogeneous strain coincides with the free surface while the surface wave is not restricted to propagate in a principal direction. Results are illustrated with numerical examples. In particular, an isotropic material reinforced with two families of fibers is considered. Each family of fibers is characterized by defining a privileged direction. Furthermore, the fibers of each family are located throughout the half space and run parallel to each other and perpendicular to the depth direction, i.e. the free surface is a plane of symmetry of the anisotropy. The wave speed depends strongly on the anisotropic character of the material model as well as the direction of propagation.

The effect of initial stress on the propagation of surface waves in a layered half-space

In this paper the propagation of small amplitude surface waves guided by a layer with a finite thickness on an incompressible half-space is studied. The layer and half-space are both assumed to be initially stressed. The combined effect of initial stress and finite deformation on the speed of Rayleigh waves is analyzed and illustrated graphically. With a suitable simple choice of constitutive law that includes initial stress, it is shown that in many cases, as is to be expected, the effect of a finite deformation (with an associated pre-stress) is very similar to that of an initial stress (without an accompanying finite deformation). However, by contrast, when the finite deformation and initial stress are considered together independently with a judicious choice of material parameters different features are found that do not appear in the separate finite deformation or initial stress situations on their own.

Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects.

The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the 'local' problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.

An Observation of Brittle Crack Propagation in Coarse Grained 3% Silicon Steel

Brittle fracture in carbon steel affects serious impact for the safety of fracture of in steel. Especially, the arresting technology of crack propagation is the last stand for the structure. It is so important issue that the conditions which can be stopped reliably crack propagation should be clarified thoroughly. On the background of such social importance, lots of experimental and theoretical researches have been conducted from both mechanical and microstructural viewpoints.Though it has been reported that the limit speed of brittle crack propagation is the Rayleigh wave speed and the speed is about 2,900 m/s in steels, the real speed of brittle crack propagation in steels is about 1,000 m/s and lower. The reason for that is considered braking effects with crack propagation, for example unevenness in the facet, tear ridge, microcrack, twin deformation and side ligament, namely the elements that dominate the arresting toughness.In this study, the 3% silicon steel where microstructure is ferrite single phase and its grain size is 4-5mm of very large is used. It is assumed that the brittle crack propagation speed in single crystal grain is very fast because there is no need to consider tear ridge between crystal grains. In this experiment, it is succeeded in shooting of brittle fracture of steel by high speed camera. As a result of that, it was revealed that the brittle crack propagation rate of 3% silicon steel is much lower than ordinary carbon steel. Also, twin deformation and unevenness in the facet are observed by scanning electron microscope (SEM).

On Propagation of Rayleigh Type Surface Wave in a Micropolar Piezoelectric Medium

In the present paper, the governing equations of a linear transversely isotropic micropolar piezoelectric medium are specialized for x-z plane after using symmetry relations in constitutive coefficients. These equations are solved for the general surface wave solutions in the medium. Following radiation conditions in the half-space, the particular solutions are obtained, which satisfy the appropriate boundary conditions at the stress-free surface of the half-space. A secular equation for Rayleigh type surface wave is obtained. An iteration method is applied to compute the non-dimensional wave speed of the Rayleigh surface wave for specific material parameters. The effects of piezoelectricity, non-dimensional frequency and non-dimensional material constant, charge free surface and electrically shorted surface are shown graphically on the wave speed of Rayleigh wave.

Seismic response of the outer shell of a liquefied natural gas storage tank using a semi-analytical dynamic substructuring technique

This paper deals with the seismic response of the outer shell of a liquefied natural gas storage tank. The concrete shell is considered as three inhomogeneous subsystems, namely; a circular plate of varying thickness on an elastic foundation, a cylindrical wall and a spherical dome. The linear dynamic response of the complete system, subjected to a strong seismic ground motion, is derived in the frequency domain using a semi-analytical approach. The concept of frequency- and wavenumber-dependent dynamic spring stiffness is introduced for the exact satisfaction of the displacement continuity and force equilibrium at the interfaces between the three subsystems. The ground motion is described kinematically in the form of a Rayleigh wave propagating along the free surface of the soil. The influence of the Rayleigh wave speed and of the interference of the structural responses to the horizontal and the vertical seismic ground motion are examined for a particular case.

Dynamic Shakedown Analysis of Flexible Pavement under Traffic Moving Loading

Shakedown theory is often used to analyze the plastic behaviors of structures subjected to variable complex loads. It is an effective way to predict the maximum load under which the failure due to plastic collapse or excessive permanent deformation of pavement will not occur. Based on Melan's lower-bound shakedown theory, this work has proposed a numerical method for estimating the shakedown limit involving the effect of traffic moving speed. The dynamic response of elastic stress to traffic moving speed is computed using combined Finite Element-Infinite Element (FE-IE) method. The shakedown limits for a two-layered pavement system have been investigated at various traffic moving speeds. It shows that the shakedown limit early reduces and subsequently turns to grow as the moving speed increases. The shakedown limit decreases to the minimum when the traffic speed reaches the Rayleigh wave speed of subsoil. In order to provide an optimized design of the pavement system, the dependence of shakedown limits on the material properties of pavement system has also been estimated. Eventually, the characteristic distribution of critical residual stress is discussed.