Theory Of Elastic Waves Research Articles

Modeling the longitudinal wave in a nanorod based on a novel theory of elastic waves with surface effects

This paper investigates the impact of surface effects on the propagation behavior of longitudinal waves in a nanorod. A theoretical model has been established on the basis of a newly proposed theory of elastic waves with surface effects. The surface effects comprise two components: the effect of surface energy and the effect of surface inertia. An analytical formula for the longitudinal wave velocity of a nanorod has been derived. Two inherent lengths at nanoscale have been deduced to characterize these two types of surface effects. The results indicate that the longitudinal wave in a nanorod is still nondispersive. However, an attractive phenomenon uncovered is that when the size of a rod reduces to the inherent lengths at nanoscale, the longitudinal wave velocity becomes size-dependent due to the effects of surface energy and surface inertia. The former increases the longitudinal wave velocity, whereas the latter decreases it. This can be understood as the former equivalently increasing the stiffness of the nanorod, whereas the latter enhancing its effective density. On the other hand, when the rod is at the macroscale, the longitudinal wave velocity degenerates to the classical velocity for a macroscopic rod without any surface effects. The current findings not only enhance our understanding of the size-dependent wave velocity of longitudinal waves in nanorods but also facilitate precisely designing the elastic wave nanodevices.

Rock-like material under large diameter SHPB dynamic splitting tension: meso-damage mechanical behavior and stress wave propagation model

The mechanical behavior of splitting tensile damage and the law of stress wave propagation of rock-like materials (RLM) are of great significance to further reveal the dynamic disaster mechanism of the deep rock mass. The meso-damage mechanical behavior and stress wave propagation characteristics of RLM disks under impact splitting were studied by using a large diameter split Hopkinson pressure bar (SHPB). In terms of dynamic damage, the splitting tensile stress-compression strain curves of RLM disks obviously showed three stages of mechanical behavior evolution: initial elastic-plastic deformation, pre-peak plastic damage, and post-peak brittle fracture failure. The macro-damage of RLM disks increased with the increase of strain rate. The meso-tensile fracture was the result of both the initial meso-damage and the impact splitting meso-damage. The dynamic splitting damage variable defined based on the damage fracture energy can accurately describe the damage evolution characteristics of impact splitting on RLM disks. In the aspect of stress wave propagation, the peak value of transmission stress showed an advanced effect with the increase of incident stress wave. In the early stage (0–50 μs), the transmission stress wave ratio (σT/σI) increased with the increase of strain rate, while in the later stage (82–200 μs), the transmission stress wave ratio (σT/σI) decreased with the increase of strain rate. The stress wave propagation law in the process of impact splitting on RLM disks was clearly revealed based on the stress wave propagation model established by the one-dimensional elastic stress wave theory. Finally, the dynamic mechanical mechanism of splitting damage and fracture of RLM disks under different strain rates was discussed deeply.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

Exploring Dynamic Spalling Behavior in Rock–Shotcrete Combinations: A Theoretical and Numerical Investigation

Spalling is a widespread dynamic disaster during blasting excavation in underground engineering. To clarify the coupled dynamic response and spalling behavior of an underground tunnel with a spray anchor, an investigation based on the rock–shotcrete combination was conducted using theoretical and numerical methods. The mathematical representation of stress wave propagation between rock and shotcrete was deduced based on the elastic stress wave theory. A novel method for predicting the location and time of initial spalling in a rock–shotcrete combination was proposed. A numerical simulation was conducted to verify the validity of the proposed theoretical method. In addition, the effect of the material’s tensile strength, the loading amplitude, and the thickness of shotcrete on the stress evolution and spalling characteristics was studied. The results demonstrate that the initial spalling locations are sensitive to the relationship between the normalized tensile strength of the rock, shotcrete, and interface. A high incident amplitude can cause the initial spalling in rock, and the shotcrete or rock–shotcrete interface can cause initial spalling due to a low incident amplitude. The stress evolution and spalling characteristics are sensitive to the thickness of shotcrete. The location of the initial spalling failure changes with the thickness of the shotcrete. An appropriate increment in thickness and normalized strength of the shotcrete is beneficial to the dynamic stability of underground engineering.

How Fast are Elastic Domino Waves?

The paper is concerned with the problem of toppling propagation velocity in elastic, domino-like mechanical systems. We build on the work of Efthimiou and Johnson, who developed the theory of perfectly elastic collisions of thin rigid dominoes on a frictional foundation. This theory has been criticised for the lack of correspondence with the experimental observations, in particular, prediction of infinite propagation velocity for zero spacing between dominoes, as well as the inability to represent the collective nature of collisions in real domino systems. In our work we consider a more realistic scenario of dominoes of finite stiffness and obtain a theory of fast elastic domino waves, taking into account a limit velocity of the perturbation propagation in the system of dominoes. Moreover, finite collision time allows to extract dynamic quantities of collisions and establish upper and lower borders for domino separations where the theory could still be applied. Our discrete element simulations support our theoretical findings and shed light on the nature of collective interactions in the nearly-elastic domino chains.Graphical abstract

Analytical solution for the free transverse vibration of an elastically connected annular plate system with discontinuities

Due to the performance requirements, structural discontinuities are inevitable in engineering structures. Free vibration of an elastically connected annular plate system with discontinuities has not yet been reported in the literature. In this study, an analytical method is developed for the free vibration of an annular plate system with discontinuities along its radius. The annular plates are connected continuously using an elastic layer. The coupled vibration equations of the annular plate system were decoupled into a series of uncoupled general “vibration” equations which are then transferred into a symplectic dual system. The general “vibration” state can then be analytically described in terms of waves utilizing the elastic wave theory. Using the analytical wave modes and satisfying the compatibility and boundary conditions at the discontinuities, a frequency equation can be obtained analytically. In numerical examples, the free vibrations of systems consist of two and three annular plates were investigated. The accuracy of the proposed method is verified by comparison with literature and finite element method (FEM).

An Analytical Seismic Response for Rigid Underground Rectangular Tunnel Subjected to Incident P Waves

A diffraction model based on the diffraction theory of elastic waves for the dynamic response of underground rectangular tunnel subjected to incident P waves is established. The rectangular structure is assumed to be a movable rigid body embedded in a homogenous, isotropic, and linear elastic medium. One particular advantage of the model is that the effects of vital factors such as the rigid scattering, radiation scattering, as well as the two‐dimensional scattering effect, and the wedge diffraction during movement are fully considered. The numerical method is verified by comparing with experimental data and other solutions in the literature. In addition, a parametric analysis of the effects of the incident angle, surrounding rock medium, and aspect ratio on the tunnel is also presented. The results show that with the increase of the incident angle, different parts of the tunnel exhibit obvious nonuniform distribution characteristics, and the bending moment of the sidewall is greater than that of the top slab. When the impedance ratio exceeds 0.26, the influence of surrounding rock characteristics on the dynamics of structure is more obvious; with the increase of the aspect ratio, the peak values of axial force at the top slab increases, whereas it decreases at the sidewall.

Study on transmission and reflection characteristics of plane S1 wave at the interface between saturated frozen soil and elastic solid bedrock

Based on the propagation theory of elastic waves in frozen poroelastic media and elastic media, a theoretical model of saturated frozen soil and elastic solid bedrock is established. The saturated frozen soil is lying on an elastic solid bedrock and is modeled as a porous material containing a three-phase medium that contains two solids (soil skeleton and ice particles) and a compressible viscous fluid. Based on the Helmholtz vector decomposition and the boundary conditions on the interface, the theoretical expressions of the amplitude ratio and displacement of the plane S1 wave incident from the saturated frozen soil to the elastic solid bedrock interface are obtained. The effects of incidence angle, frequency, cementation parameters, saturation, and contact parameters on amplitude ratio and displacement were investigated. The results indicate that only S wave and horizontal displacement exist when the incident angle [Formula: see text] = 0°. As cementation parameters, saturation, porosity, and contact parameters increase, the critical angle appears earlier. Each wave has different degrees of pulses when reaching the critical angle, among which the transmitted P wave and the reflected S2 wave are the most significant. At higher saturation, it is observed that the amplitude ratio of the reflected S2 wave decreases significantly. The horizontal and vertical displacements increase significantly with increasing frequency.

1256 Fabricating Artificial Kidney Stones of Different Physical Properties for Ex Vivo Experimentation

Abstract Introduction There have been several new technologies introduced recently for lithotripsy. The optimal settings of these devices are unknown for different kidney stone densities. This study aims to develop artificial kidney stones for ex vivo studies. Method BegoStone was prepared with a powder to water ratio (by weight) ranging from 15:3 to 15:6. The phantoms' acoustic properties were characterised by using an ultrasound transmission technique, from which the corresponding mechanical properties can calculated based on elastic wave theory. Results The measured parameters for BegoStone phantoms of different water contents were assessed with regard to longitudinal wave speed & density. If transverse wave speeds were measured then longitudinal acoustic impedance, transverse acoustic impedance, Young's modulus, bulk modulus, and shear modulus can be derived. Longitudinal (CL (m/s)) and density (ρ (Kg/m3)) of BegoStone phantoms with different powder to water ratios corresponded to the known properties reported in natural kidney stones (Calcium Oxalate Monohydrate, Brushite, Uric acid and Struvite). A BegoStone water ratio of 15:3 had similar properties to Calcium Oxalate Monohydrate, 15:4 to be similar to Brushite, 15:5 was similar to Uric acid, and 15:6 was similar to Struvite. Conclusions This BegoStone preparation method can be used to fabricate artificial stones with physical properties matched with those of natural kidney stones of various chemical compositions.

Numerical investigation into hydrodynamic behavior of graded aggregates of high-speed ballastless track considering the degree of saturation

The ballastless track inevitably suffers from some functional defects, such as expansion joint cracking, due to the long term combined effect of dynamic train loads and environmental factors. As a result, water can be somehow retained in the graded aggregates of the roadbed surface layer under ballastless track, which is a key structural component subject to high-speed train loads. Accordingly, it is of significance to study the hydrodynamic behavior of roadbed surface layer that can be prone to depend on variation of the degree of saturation (Sr) in graded aggregates. A coupled hydrodynamic model of roadbed surface layer under ballastless track subject to high speed train loads was established based on homogenization theory and elastic wave theory involving Sr of graded aggregates. The proposed model is validated by degrading to Biot's dynamic model in saturated porous medium and analytical method of unsaturated soil. The effect of different Sr on the pore fluid pressure, spatial seepage velocity of graded aggregates is investigated based on this model with the practical seepage boundaries in ballastless track considered. On this basis, the influence of different spatial locations on the dynamic seepage index is discussed, and the differences in the stress paths in the roadbed surface layer under different Sr at different depths are analysed. The numerical results show that the increase of Sr has a spatial and critical effect on the increase of pore water pressure and seepage velocity in the roadbed surface layer. When the expansion joint of ballastless track cracks, the hydrodynamic erosion in the surface layer of ballastless track roadbed has been accumulating continuously. The erosion is more serious within the depth of 15 cm under the surface of roadbed. For Sr above 0.7, extra attention should be given to prevent internal erosion diseases in the roadbed surface layer during maintenance of ballastless track.

Steeply dipping structural target oriented viscoacoustic prismatic reverse time migration in frequency domain and its application

AbstractSteeply dipping structural imaging is a great challenge due to its poor illumination. Conventional migration methods are unable to produce an accurate image of complex steeply dipping structures. The prismatic wave can improve the illumination of steeply dipping structures and is often used to improve the imaging results of such structures. Traditional elastic wave theory assumes that seismic waves do not attenuate when propagating through subsurface media. However, during seismic wave propagation, the wave energy decays exponentially due to the absorption and attenuation of the ground layer. Subsurface attenuation leads to amplitude loss and phase distortion of seismic waves, resulting in blurring of migration amplitudes when this attenuation is not taken into account during imaging. To address this issue, a frequency‐domain Q‐compensated prismatic reverse time migration method is proposed, which derives Q‐compensated prismatic wavefield propagation operators. In the proposed frequency‐domain Q‐compensated prismatic reverse time migration, Q attenuation is fully compensated along three propagation paths and two propagation types of prismatic waves. The optimized four‐order mixed 25‐point difference format and LU decomposition method are used to solve the Q‐compensated prismatic wavefield propagation equations with high computational efficiency. Numerical and field data examples demonstrate that the proposed frequency‐domain Q‐compensated prismatic reverse time migration method can compensate for deep attenuation energy and improve the imaging resolution of steeply dipping structures.

Analytical solution for circular tunnel under obliquely incident shear waves

The incident direction plays an important role in the seismic response of long lined tunnels. A simplified analytical solution is developed for estimating the seismic response of the circular tunnel under obliquely incident shear waves, including the principle stress and the internal forces and deformations on the cross section and longitudinal of the tunnel. Based on the elastic wave theory, the oblique incident shear wave can be decomposed into an axially propagating shear wave and a shear wave propagating transversely to the tunnel axis. A modified elastic foundation beam model is used to calculate the tunnel longitudinal response, which is based on Timoshenko beams resting on Winkler foundations with considering different tangential contact conditions. Furthermore, by adopting wave theory and Fourier transform, longitudinal response of tunnels under arbitrary seismic load can be obtained. Moreover, the flattening effect and out-plane response of tunnel cross-section caused by longitudinal bending are considered. The proposed analytical solution is verified by comparing with the dynamic numerical results. The effect of the incident angle, ground condition and contact condition on the seismic response of long lined tunnel is investigated by the simplified analytical solution. It is demonstrated that the proposed solution can be adopted to predict internal forces and deformations of circular tunnels under obliquely incident shear wave in preliminary design.

Reflection and transmission of plane harmonicPwave at planar interface between elastic medium and frozen poroelastic medium

SUMMARYBased on the propagation theory of elastic waves in three-phase frozen poroelastic media and single-phase elastic media, the energy transmission problem of a plane harmonic P-wave incident at the planar interface between elastic medium and frozen poroelastic medium is studied. According to the boundary conditions at the planar interface, the analytical expressions of the transmission and reflection coefficients and energy ratios of the transmitted and reflected waves are derived. The relationship between the energy of plane harmonic P-wave incident on the planar interface between elastic medium and frozen poroelastic medium and the incident angle, frequency of incident wave, cementation parameters, porosity, saturation and contact parameters are investigated. The results show that the incident angle significantly affects the reflection and transmission energy ratios; the incidence frequency only has a large effect on the transmission energy ratios of the transmitted P2, P3 and S2 waves; the contact parameters have a relatively large effect on the reflected S wave and the transmitted P2, P3, S1 and S2 waves, but have a smaller effect on the reflected P wave and the transmitted P1 wave; cementing parameters, porosity, saturation has a great influence on the energy ratios.

Design of shock wave attenuation effects on multi-impedance-matched laminated composites

To improve the protective performance of composite armor materials, three types of laminated composites with different arrangements were designed by the principle of impedance matching. The impedance combination schemes used in this work were high impedance→low impedance (type I), low impedance→high impedance (type II), and high and low impedance (type III). The microstructures and properties of the test samples prepared in a vacuum hot-pressing furnace were investigated by micro-indentation and dynamic impact experiments. According to the propagation theory of one-dimensional elastic waves in multilayered media, the propagation law of shock waves in the as-prepared laminated composites was examined. Micro-indentation test results revealed that the hardness of the Fe2Al5 intermetallic compound formed by the reaction of Al and 304 SS was the highest and ranged from 917.5 HV to 1026.3 HV. The type II layered composite had the highest compressive strength (1139 MPa) followed by the type I composite and the type III composite. The type III composite manifested the highest energy absorption efficiency (73%) followed by the type II layered composite and the type I composite.

Dynamic characteristics and safety control of mortar bolts under tunnel blasting vibration loads

Mortar bolts are an important part of the tunnel support structure, and tunnel blasting can cause vibration hazards to the existing bolt support structure. According to the theory of blasting elastic wave theory, a dynamic mathematical model of bolts mortar bolts considering initial surrounding rock stress was established. Based on the blasting excavation example of the Longnan tunnel of a high-speed railway from Ganzhou to Shenzhen in China, the blasting dynamic response characteristics of mortar bolts in 0 ∼ 3 days, 3 ∼ 7 days, and 7 ∼ 28 days curing time under the blasting excavation of Classification-II, Classification-III, Classification-IV, and Classification-V surrounding rock were calculated and studied. The dynamic failure mechanism of mortar bolts with different curing time under blasting vibration was clarified. According to the limit axial force of the mortar bolts, the blasting safety distance and vibration velocity of the tunnel mortar bolts support structure was proposed.

Isolation effects of S-wave by composite multilayer wave impeding block in unsaturated soil

Based on the propagation theory of elastic waves in unsaturated porous medium and single-phase elastic medium, considering that a composite multilayer wave impeding block (WIB) with a certain thickness is set in unsaturated soil, and using Helmholtz vector decomposition theorem, the analytical solutions of transmitted/reflected amplitude ratio of S-wave passing through the composite multilayer wave impeding block in the unsaturated foundation are derived. The influences of physical and mechanical parameters such as wave impedance ratio, saturation, shear modulus and density of the wave impeding block on the propagation characteristics of S-wave passing through the composite multilayer wave impeding block in unsaturated foundation are analyzed by numerical examples. The results show that the saturation has little influence on the transmission/reflection coefficient of P1-wave and S-wave, but has a significant influence on the transmission/reflection coefficient of P2-wave and P3-wave. The wave impedance ratio and shear modulus of interlayer wave impeding block material have a significant influence on the transmission/reflection coefficient. The best vibration isolation effect can be obtained by selecting the optimal wave impedance ratio and shear modulus of interlayer wave impeding block material, which may be provide a design criteria for the application of composite multilayer wave impeding block in foundation vibration control.

Modeling the Effect of Multiscale Heterogeneities on Wave Attenuation and Velocity Dispersion

Wave attenuation and velocity dispersion are of great significance to fluid evaluation in fluid-saturated rocks. Inspired by the double-porosity and cracked porous elastic wave theory, we develop a multi-scale heterogeneities elastic wave theory to better describe the attenuation characteristics caused by wave-induced fluid flow at various scales in saturated rocks. First, we introduce penny-shaped and narrow-shaped coupled cracks into double-porosity model. The potential energy function is given by the stress-strain relationship. The kinetic energy function and dissipation function are derived based on the generalized Biot theory. Next, we derive the multi-scale wave equations through the Lagrange equation and obtain three compressional waves and one shear wave. According to the novel wave equations, we work out the phase velocity and inverse quality factor in double-porosity and cracked media based on plane-wave analysis. The numerical simulation results show that there are four attenuation peaks in the whole frequency band, corresponding to mesoscopic fluid flow, two kinds of squirt flow and Biot flow respectively. Then, we investigate the effect of poroelastic parameters on wave propagation. It is found that porosity, inclusion size, crack aspect ratio and other parameters significantly affect the dispersion and attenuation. Finally, we compare experimental data and theoretical prediction. Moreover, the new elastic wave theory can degenerate into classical Biot’s theory, double porosity theory and pore-crack theory under certain conditions, which demonstrates the consistency of this method. The proposed theory can better predict and simulate the wave propagation by incorporating the effects of microscopic, mesoscopic, and macroscopic heterogeneities.

The dispersion of the axisymmetric waves propagating in the pre-strained highly elastic plate under bi-lateral contact with a fluid

The paper is concerned with the study of axisymmetric waves in an initially axisymmetric, finitely strained plate of highly compressible elastic material immersed in a fluid whose flow is confined by rigid walls. The fluid is modeled as compressible and inviscid and its flow is described by the linearized Euler equations. The motion of the plate is described by the so-called three-dimensional linearized equations and relations of elastic wave theory in initially stressed bodies. The dispersion equation in question is obtained and solved numerically. Concrete numerical results are obtained for the cases where the material of the plate is chosen as Soft-rubber and as Lucite, and water is chosen as the fluid. Based on these results, the influence of the initial axisymmetric finite strains in the plate on the dispersion curves at different fluid depths is analyzed. In particular, it is found that bilateral contact of the plate with the fluid leads to the appearance of asymmetric and symmetric quasi-Scholte waves, and that the propagation velocity of these waves increases (decreases) with the initial stretching (compression) of the plate.

Theoretical and experimental study on interface stiffness measurement of rough surface using improved acoustic model

Structural health monitoring (SHM) of interface stiffness is of great significance in many fields, such as accurate rotor dynamics modeling and interface friction performance analysis. Acoustic wave is an important interface detection method. However, the interface mechanics and elastic wave theory are relatively independent, and the interaction mechanism between the contact asperity parameters and the energy transfer of elastic waves has not been thoroughly explored. The existing interface stiffness detection indexes do not consider the uncertainty of boundary parameters, which easily leads to the degradation of detection performance and limits the promotion of interface stiffness detection. In this paper, we proposed an acoustic model considering contact interface and boundary effects integrating microwave transmission line theory. A new interface stiffness detection index is constructed to ensure the robustness of the detection results. The relationship between contact parameters and elastic wave energy transfer is analyzed, and the robustness of different detection indexes and the uncertainty of different detection methods are compared. The experimental results of the pressure-bearing connection structure are obtained to verify the correctness of the acoustic model and the feasibility of the detection method. The proposed method is further applied to multi-bolted structures, and the robustness of the proposed method is discussed. The results show that the proposed method has the ability of in-situ interface stiffness detection and potential engineering utility.

Dynamic Sustainable Processes Simulation to Study Transport Object Efficiency

The development of reliability theory has led to the setting of tasks requiring consideration of the efficiency and functional safety of technical objects of transport over the life cycle. The paper demonstrates the possibility of using the universal laws of elastic wave theory to describe natural phenomena occurring in complex dynamic systems, on the examples of solving issues arising in the interaction of rolling stock and the railway track. The accounting of the time component and the ability of elastic waves to propagate energy in time and space allowed considering any interaction process as a chain of processes, incidence-reflection-refraction of force impulses of interaction. Understanding the physics of dynamic processes that occur in objects while performing their intended functions allows developers to improve objects in such a way as to minimize their life cycle cost and maximize their ability to perform their intended functions under different operating conditions. In addition, it allows the expansion of existing methods and approaches to diagnostics of dynamic transport systems. All this is a base for making it possible to develop an innovative and effective tool for engineers and scientists to assess the impact of technosphere transport objects on human habitats.