Theory Of Elastic Wave Propagation Research Articles

Deep learning-based inversion of soil and rock compaction density utilizing Falling Weight Deflectometer (FWD) and Surface Waves

Methods relying on a single dynamic parameter to estimate the material density for the compaction quality testing of earth-rock dams and roadbed projects, present certain limitations in terms of applicability, testing accuracy, and work efficiency. The theory of elastic wave propagation reveals that the compaction density of the medium results from the combined interaction of multiple material properties. This study proposes a method that utilizes the transient surface wave method and a vehicle-mounted Falling Weight Deflectometer (FWD) to jointly enhance the accuracy of soil and rock material compaction detection. The transient surface wave method is employed to extract surface wave velocity and compressional wave velocity. The analysis of the FWD signal involves extracting dynamic parameters closely related to material density, including stiffness coefficients, central frequencies, and stiffness impedances, from time-domain, frequency-domain, and mechanical impedance analyses. A fully connected deep neural network is introduced to intelligently estimate the compaction density. The deep learning model is optimized by selecting the optimal activation function and optimization algorithm, as well as additional tuning. Extensive testing shows that deep learning model produce results closely approximating the true compaction density values. The average relative errors for the wet density and the dry density are 2.9% and 2.85%, respectively, meeting the quality control requirements for density testing. The proposed approach enables rapid detection and control of the compaction quality of soil and rock materials, providing a new method for the in-situ rapid detection of compaction quality.

Theory of elastic wave propagation in a fluid-saturated multi-porous medium with multi-permeability

This paper establishes the concept of elastic wave propagation in a multi-porous medium with different permeabilities by assuming there are n distinct pore fluid phases. The dynamic equation of motion of elastic wave propagation through this multi-porous medium is derived based on Lagrangian mechanics. In this regard, the generalized form of mass coefficients and then the energy loss due to the fluid phases in terms of dissipation coefficients are presented for low-frequency limits with the help of Darcy’s Law of multi-phases system. The elastic coefficients of the constitutive equation in terms of compliance matrix are identified using a series of Gedanken experiments. Some significant results regarding the compressional and rotational waves in a multi-porosity medium are derived. The validation of the theory has been shown by comparing it with the existing theory of single and double porosity. It is observed that there are ( n + 1 ) compressional waves corresponding to solid and fluid phases, whereas only one rotational wave is associated with the solid phase. The concept of multi-porosity theory can contribute to a deeper understanding of wave behaviour in a porous medium.

Dynamics of stress waves in graded density impactors during the internal ballistic process

Quasi-isentropic loading and unloading, employing graded density impactors (GDIs) as flyers in gas gun-driven plate impact experiments, can provide novel and valuable insights into the equation of state and strength properties of the loaded material. However, the internal ballistic process may lead to spalling or debonding of the GDI due to the intricate interactions between stress waves and interfaces. In this study, the wave propagation in the GDI was analyzed using the multimaterial Lagrangian elastic-plastic model and elastic wave propagation theory. The impact of gradient direction, power-law constant p, and thickness of the first and last layers on the tensile stress was investigated. The outcomes reveal that the mechanism of generating tensile stress varies for two gradient directions. Moreover, adjusting the constant p and the layer thickness may decrease the maximum tensile stress by 74.1% (forward graded) and 95.8% (reverse graded), respectively. The outcomes of this research provide a theoretical and simulation basis for designing and fabricating GDIs to be utilized in quasi-isentropic experiments.

Instrumentation of Field-Testing Sites for Dynamic Characterization of the Temperature-Dependent Stiffness of Pavements and Their Layers

Falling weight deflectometer (FWD) tests are performed worldwide for assessing the health of pavement structures. Interpretation of FWD-measured surface deflections turns out to be challenging because the behavior of pavement structures is temperature-dependent. In order to investigate the influence of temperature on the overall pavement performance and on the stiffness of individual layers, temperature sensors, asphalt strain gauges, and accelerometers were installed into one rigid (concrete) and two flexible (asphalt) pavement structures, mostly at layer interfaces. Three different methods for installation of the strain gauges are compared. From correspondingly gained experience, it is recommended to install a steel dummy as a place-holder into the surface of hot asphalt layers, immediately after their construction and right before their compaction, and to replace the dummy with the actual sensor right before the installation of the next layer. Concerning the first data obtained from dynamic testing at the field-testing sites, FWD tests performed at different temperatures deliver, as expected, different surface deflections. As for the rigid pavement, sledgehammer strokes onto a metal plate, transmitted to the pavement via a rubber pad, yield accelerometer readings that allow for detection of curling (=temperature-gradient-induced partial loss of contact of the concrete slab from lower layers). In the absence of curling, the here-proposed sledgehammer tests yield accelerometer readings that allow for quantification of the runtime of longitudinal waves through asphalt, cement-stabilized, and unbound layers, such that their stiffness can be quantified using the theory of elastic wave propagation through isotropic media.

Vibrational loss analysis of a new type of phononic crystal with a tungsten block embedded inside a rubber matrix

Vibration and noise pollution is one of the main pollution in modern society. In order to obtain good vibration damping effect, more effective phononic crystals should be designed. Based on the theory of elastic wave propagation in solids, three types of phononic crystals are designed in this paper using tungsten blocks embedded in a rubber matrix, which are cup-shaped phononic crystal, solid cylindrical phononic crystal, and hollow cylindrical phononic crystal. Firstly, the band gap characteristics and vibration losses of the three phononic crystals are analyzed by using the finite element method. Secondly, the physical mechanism of band gap formation is explored by vibration modes. Finally, the cup-shaped phononic crystal was introduced into the core layer of the sandwich plate to form the cup-shaped phononic crystal sandwich plate, and its vibration damping performance was analyzed. The results show that the three phononic crystals can form three band gaps in the range of 0–800 Hz, and the first low-frequency band gap starts at about 140 Hz and is all wider than 200 Hz. It is noteworthy that the average loss of vibration transmission of the three types of phononic crystals is more than 69 dB, which possesses stronger damping capability and wider low- and middle-frequency band gaps than the flat plate type phononic crystals. The vibration direction of the phononic crystal is at an angle of 90° to the wave vector, which prevents the propagation of elastic waves. The phononic crystal embedded in the sandwich panel plate is more in line with the actual demand of vibration damping and obtains good vibration damping effect. The research in this paper can provide more feasibility for phononic crystal damping.

Predictive model and physical interpretation of asymmetric features in pyroshock signals

In aerospace engineering, electronic devices can be easily damaged under pyroshock environments, and therefore, shock testing is required. The shock response spectrum (SRS) is the widely adopted specification tool to compare shock severities between field pyroshocks and testing shocks. However, two shocks with almost the same SRS but different temporal features could still lead to large differences in the failure of a device. The test specification should consider temporal features of shocks in order to be more representative of the field shocks. During shock propagation, shock signal evolves from an exponentially decaying shape to an asymmetric shape with an increased initial rise stage, which is characterized by the temporal moment or asymmetric basis functions. This study explains the evolution mechanism of shock asymmetry based on elastic wave propagation theory and proposed a predictive model. The theoretical foundation for the application of signal asymmetry in pyroshock characterization, device failure analysis and SRS reconstruction is provided.

Stress wave propagation and incompatible deformation mechanisms in rock discontinuity interfaces in deep‐buried tunnels

AbstractComplex weak structural planes and fault zones induce significant heterogeneity, discontinuity, and nonlinear characteristics of a rock mass. When an earthquake occurs, these characteristics lead to extremely complex seismic wave propagation and vibrational behaviors and thus pose a huge threat to the safety and stability of deep buried tunnels. To investigate the wave propagation in a rock mass with different structural planes and fault zones, this study first introduced the theory of elastic wave propagation and elastodynamic principles and used the Zoeppritz equation to describe wave field decomposition and develop a seismic wave response model accordingly. Then, a physical wave propagation model was constructed to investigate seismic waves passing through a fault, and dynamic damage was analyzed by using shaking table tests. Finally, stress wave attenuation and dynamic incompatible deformation mechanisms in a rock mass with fault zones were explored. The results indicate that under the action of weak structural planes, stress waves appear as a complex wave field decomposition phenomenon. When a stress wave spreads to a weak structural plane, its scattering may transform into a tensile wave, generating tensile stress and destabilizing the rock mass; wave dynamic energy is absorbed by a low‐strength rock through wave scattering, which significantly weakens the seismic load. Wave propagation accelerates the initiation and expansion of internal defects in the rock mass and leads to a dynamic incompatible deformation. This is one of the main causes for large deformation and even instability within rock masses. These findings provide an important reference and guide with respect to stability analysis of rock mass with weak structural planes and fault zones.

Ray theory for elastic wave propagation in graded metamaterials

We present a ray theory for modeling elastic wave propagation in spatially graded mechanical metamaterials. Wave propagation in periodic metamaterials has been well studied, motivated by their beneficial wave steering and bandgap properties. By contrast, comparably little work has explored wave propagation in spatially graded metamaterials despite the increased design opportunities, largely due to the lack of efficient modeling techniques. We develop a ray theory to model waves in graded metamaterials based on high-frequency asymptotics and the assumption of local periodicity. This work builds upon the well-developed ray theories that are fundamental in a wide range of fields, from optics to seismology. Our derivations produce a practical framework for computing approximate wave fields in graded metamaterials. Ray trajectories are computed by independently solving a system of ordinary differential equations for each ray, requiring only knowledge of local dispersion relations throughout the metamaterial, which vary smoothly in space due to grading. Equations for the wave amplitude along rays are also derived in the two-dimensional setting. We show that the form of the ray tracing equations are nearly identical to those for smooth solids in seismic ray theory, with the primary difference being the dispersion relations. A numerical framework for computing ray solutions is demonstrated on a mass–spring network with analytical dispersion relations as well as a truss metamaterial that requires the numerical evaluation of dispersion relations. Through these examples, we demonstrate that ray theory provides an efficient means of studying the fascinating behavior of waves in graded metamaterials such as wave guiding along curved trajectories.

Особенности оценки сплошности свай методом сейсмоакустической дефектоскопии

Introduction. The authors present the results of pile continuity studies conducted using seismoacoustic defectoscopy. Tests were conducted in the Rostov region and Moscow. Reflectograms of tested piles, as well as the subsequent interpretation and confirmation of the design length and continuity of the pile shaft are provided. Materials and methods. The research results were obtained using Spektr 4.3, whose operating principle is based on the theory of elastic wave propagation. This device is used for the recording of the pile response to external impacts with the preset pulse parameters. The signal is recorded by means of an accelerometer sensor, connected to a special computer with the software for the digitization and interpretation of incoming information. Data processing results are presented in the form of computer graphic images (reflectograms), which show the pile length, continuity, damages (if any) and their location. Results. The dependence between soil homogeneity and signal purity has been identified. At the interface between two types of soils that are different in density, the sound wave was reflected and partially passed further, creating characteristic peaks of false defects. To eliminate false defects, studies of layered soils are necessary. Conclusions. The results confirm the effect of the soil homogeneity on signal purity. The higher the heterogeneity of the soil structure, the bigger the number of false peaks that the reflectogram demonstrates. A prerequisite for the correct interpretation of the results of pile continuity testing is the study of the geological section and preliminary instrumental identification of the ultrasound propagation velocity for each individual pile.

A helical elastic wave metasurface based on GSL

Based on the Generalized Snell’s Law (GSL), combined with the elastic wave propagation theory in the thin plate and beam structure, a new type of elastic wave metasurface is constructed. The metasurface structure realizes the design of equal gradient of transmitted wave phase according to the difference of the length of different helical curved beams. Through this structure, the transmission wavefront of the Lamb wave in the thin plate is controlled, and acoustic phenomena such as negative refraction and asymmetric transmission are realized, which proves the accuracy of the GSL theory and the feasibility of the structure. And on this basis, the use of the design of symmetrical attachment to the metasurface solves the problem of destroying the plate structure in the traditional elastic wave metasurface, and provides an effective new idea for the elastic wave metasurface.

A Methodology for Reconstructing Source Properties of a Conical Piezoelectric Actuator Using Array-Based Methods

We investigated the force produced by a conical piezoelectric (PZT, lead zirconate titanate) transducer actuated by high voltage pulses (HVP) in contact with a steel transfer plate. Using elastic wave propagation theory in a semi-infinite plate, we aimed to quantify the magnitude and estimate the shape of the force–time function via the body waves produced in the transfer plate using the displacement field recorded on an array of 20 absolutely calibrated PZT receivers. We first calibrated the receiver array using glass capillary fracture. We proceeded to use a conical PZT transducer to actively produce a source at the origin, allowing us to study the displacement field produced on the now calibrated PZT receiver array. We studied two types of HVP: An impulsive and step source. The calibrated receiver array was used to estimate the general shape of the force–time functions for each type of HVP. From our hypothesized force–time functions we were able to estimate the peak force produced by the PZT actuator: The impulsive source generated a force of f_{peak} = 2.90 pm 0.42 N and the step source generated f_{peak} = 1.79 pm 0.30 N, respectively, for a peak applied voltage of 273 V. This translates to an applied force of sim 0.011 N/V and 0.007 N/V for the impulse and step force–time functions, respectively, which is similar to estimates found in the literature for other conical transducers in contact with metallic transfer media. This measurement was verified directly by independent measurements of the peak force f_{peak} using a dynamic force transducer. We found that our methodology correctly estimated the magnitude of the force but is limited to transducers with incident angles theta< 53^{circ }. Beyond this angle, overestimates of the force were observed due to the lack of body wave energy produced by the source. These results allow us to quantitatively determine the forces produced by active PZT techniques using only the measurement of the displacement field captured on a calibrated conical PZT array. Quantitative understanding of active PZT sources additionally constrains the transfer functions approach, which is commonly used in the non-destructive testing of materials and in other fields, such as rock physics and laboratory seismology.

What kind of waves are measured in trabecular bone?

The paper discusses the fundamental mechanisms underlying the interaction between ultrasound and trabecular bone, which is considered a two-phase material. When fluid-saturated cancellous bone is interrogated by ultrasound, in some cases, one or two wave modes are observed. Many authors claim that these waves correspond to the fast and slow waves predicted by Biot’s theory of elastic wave propagation in fluid-saturated porous media. Within our analysis of the physical conditions, predictions of the existing two-phase models of the propagation of ultrasonic waves in the material as well as numerical simulations for fluid-saturated trabecular bone were performed. On the basis of the theoretical results (from numerical studies) and arguments presented in this paper, we aimed to answer the question of whether two waves observed in ultrasonic wave transmission studies can be interpreted as the fast and slow waves predicted by Biot’s theory.

Elastic immersive wavefield modelling

Many modelling studies of wave scattering require repeated numerical simulations through models with properties that differ only in a small sub-domain. Hence, it is of interest to recompute the wavefields that account for wave propagation through the whole domain, using simulations that are performed only in the sub-domain. Immersive boundary conditions (IBCs) can be used to establish such a local wavefield modelling scheme which enables accurate wavefield recomputation, including all interactions between the locally-perturbed medium and the full domain. We develop IBC theory for elastic wave propagation, in which the boundary conditions are updated dynamically at each time step of a simulation in the local domain. These updates are calculated by wavefield extrapolation based on the Kirchhoff-Helmholtz integrals using Green's functions in the background medium. Wavefield recording and injection in IBCs can be implemented either using finite-difference (FD) injection methods, or using the method of multiple point sources (MPS). The latter method is significantly less computationally demanding in terms of both memory and number of calculations. We therefore both extend acoustic FD injection methods to elastic media, and propose a new second-order accurate MPS method to implement elastic IBCs, which is numerically exact. In higher-order FD modelling, the MPS method is not numerically exact but still produces highly accurate IBC wavefields when compared to global-domain simulations.

On a bidirectional bending Hopkinson tension test method

In order to achieve bidirectional high strain rate dynamic tension of materials or structures, based on the elastic stress wave propagation theory in bending bars and the principle of the Hopkinson bar, a symmetrical herringbone bending bar was designed. The designed structure can generate and transmit two compression waves at the same time, and convert them into two-way tension waves propagating along the tension bar through the contact adapters. In order to understand the influence of the herringbone bending bar geometric configuration on the propagation of elastic compression waves, the dynamic analysis and ABAQUS finite element analysis (FEA) were carried out for the device. The study shows that after the square compression elastic wave propagates through the bending bar, the platform section of the square wave will incline in high front and low back, and as the bending angle increases, the slope is larger, and the waveform distortion caused by the large curvature rod is more serious. In order to realize the platform segment of the square wave or trapezoidal wave, the tapered impact bar is optimized so that it can be used to generate load waves with low front and high back to offset the tilt distortion in the transmission. In the end, to verify the feasibility and effect of the bidirectional dynamic tensile loading device based on the bidirectional bending Hopkinson bar, a small verification device was built. The results show that the device realized bidirectional tension loading for the pulse width of about 54 μs with good synchronization, the time difference between the starting point of the two waves was less than 2.5 μs, and the amplitude difference was less than 6×10−6. The bidirectional tensile test was carried out on the 2024 aluminum alloy samples, and the good test results were obtained. This confirms that the proposed method can be used for bidirectional dynamic tension and lays the foundation for the expansion of the device to biaxial tensile loading.

Research on attenuation law and application of elastic wave propagation in multi-coal seam fracture

The micro-seismic technology based on elastic wave propagation has been applied to the monitoring of hydraulic fracturing in coal mine and the early warning of dynamic disasters. As a typical stratified medium, the propagation of elastic waves in the stratified coal rocks is affected by the heterogeneity of the media and the reflection of the layers. Based on the elastic wave propagation theory, the phased attenuation characteristics of elastic wave propagation in layered coal rock are obtained through COMSOL numerical simulation. The results show that in layered coal rock, the attenuation amplitude of elastic wave propagation decays fast at first and then slow, the attenuation gradient in coal seam is larger than that in rock layer, showing a phased attenuation law. The propagation attenuation coefficients of different medium layer densities are different: the greater the medium layer density, the slower the amplitude attenuation, and the smaller the absorption attenuation coefficient. The attenuation of elastic wave amplitude in the coal-rock matched medium layer is more obvious in vertical direction than that in horizontal, which is characterized by the amplitude attenuation gradient. The research results can provide a theoretical basis for the optimization of the arrangement method of the micro-seismic pickup.

Fluids Alter Elasticity Measurements: Porous Wave Propagation Accounts for Shear Wave Dispersion in Elastography

In shear wave elastography, rotational wave speeds are converted to elasticity measures using elastodynamic theory. The method has a wide range of applications and is the gold standard for non-invasive liver fibrosis detection. However, the observed shear wave dispersion of in vivo human liver shows a mismatch with purely elastic and visco-elastic wave propagation theory. In a laboratory phantom experiment we demonstrate that porosity and fluid viscosity need to be considered to properly convert shear wave speeds to elasticity in soft porous materials. We extend this conclusion to the clinical application of liver stiffness characterization by revisiting in vivo studies of liver elastography. To that end we compare Biot’s theory of poro-visco-elastic wave propagation to Voigt’s visco-elastic model. Our results suggest that accounting for dispersion due to fluid viscosity could improve shear wave imaging in the liver and other highly vascularized organs.

Testing Method for the Range of Fracture Zone of Rock Slope under Blasting Load

In engineering blasting, the determination of the range of rock blasting fracture zone has important guiding significance for blasting construction. This paper proposes a method that can accurately and directly obtain the range of rock blasting fracture zone. Based on the theory of elastic wave propagation, test rods which are made of appropriate material are selected and prepared. A certain number of boreholes are drilled for subsequent insertion of the test rods along the direction perpendicular to the free surface of the excavation at a certain distance from the blast hole. Based on the field blasting test results, the deepest fracture position of the test rod is used as the boundary of the blasting fracture zone, and the range of the rock blasting fracture zone is obtained. A numerical analysis model is established according to the Mohr–Coulomb constitutive relationship and the Von Mises yield criterion. Then, the range of the fracture zone and the maximum horizontal radius of the fracture zone are analyzed and obtained. The numerical analysis results are compared with the field measured data. It is demonstrated that the range of the fracture zone obtained by the numerical simulation is in good agreement with the blasting test results of the pre‐embedded test rods. The research results can provide references for the safety control of blasting and excavation of rock slopes.

Editorial: Biographical Memoirs, Volume 69

Editorial: Biographical Memoirs, Volume 69

Experimental Analysis of the Effects of a Polyurethane Foam on Geotechnical Seismic Isolation

ABSTRACT A new experimental apparatus for performing impact tests is presented, to analyse the wave propagation phenomenon in pure sand and in composite sand-polyurethane specimens, with the final aim to explore the seismic mitigation given by a polyurethane inserted into a real soil under existing structures. Polyurethane layers at three different thicknesses are considered. Experimental results show that composite specimens provide acceleration reductions increasing with the increase of polyurethane thickness. Result interpretation in the light of the elastic wave propagation theory is presented, pointing out the experimental set-up limitations, mainly due to the lack of sufficient specimen confinement.

Peter Chadwick. 23 March 1931—12 August 2018

Peter Chadwick studied mathematics as an undergraduate at the University of Manchester, graduating with first-class honours in 1952, from where he moved to Cambridge and completed a PhD on the thermal history of the Earth in the Department of Geodesy and Geophysics under the supervision of Dr Robert Stoneley. His research then developed to focus primarily on the propagation of waves, and he made a major contribution to the mathematical theory of elastic wave propagation and became a world-leading authority in this area. He also made fundamental advances in the modelling of the thermo-elastic properties of rubberlike materials. At the University of East Anglia, where he was a professor for 26 years, he was the driving force behind the development of a research group in theoretical mechanics in the School of Mathematics and Physics, leading by example and supporting and encouraging fellow faculty members, especially the younger staff, academic visitors and students. He gave considerable service to the University of East Anglia in a number of capacities, including a period as Dean of the School, and to the scientific community, through substantial journal editorial activities and as a member of several national and international committees.