The relationship between two- and three-dimensional elastic-wave propagation

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

Percussive drilling, that uses rotary torque and axial impact simultaneously, is an established method of making holes in concrete. This paper deals with the development of a finite element model, based on three-dimensional wave propagation, which represents the impact process of percussive drilling in concrete. The solid model of the percussive drilling system consists of five separate components, viz. shank, fluted body, socket tip, carbide blade and concrete target. After integration and meshing, the finite element model is analysed using three-dimensional wave propagation techniques. The bias weight and the impact impulse were set up as inputs to the drilling system. A "surface to surface eroding" interface was selected for the drill tip-concrete contact surface while a "rigid wall" interface was chosen for the concrete-ground support contact surface. Simulation results showed the usefulness of the model in understanding how the three-dimensional wave, due to impact, propagates through the drilling system resulting in penetration of the drill bit into concrete. The accuracy of the model was successfully verified with prior-known one-dimensional analytical model and experimental penetration results. Finally, to illustrate the potential use of this model to design and analyse percussive drilling system, the effect of variation of rake angle, helix angle, point angle, and bias weight on the extent of penetration is presented. The results clearly showed the importance of simulating the impact process for designing new percussive drill bits.

The present paper is concerned with two-dimensional wave propagation in a homogeneous isotropic elastic solid medium underlying a liquid layer, under the influence of a gravitational field, when a compressional wave source is present in the liquid. In the absence of gravity the result is in good agreement with the corresponding result of the classical problem.

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

Guided wave is a type of wave propagation in which the waves are guided in plates, rods, pipes or elongated structures such as rails and I-beams. In order to extract the guided wave velocities and wavestructures, for waveguides of arbitrary cross section, a theoretical framework is developed. Here, a semi-analytical finite element (SAFE) method is used for the calculation of the wave propagation characteristics in elastic waveguides immersed in vacuum. The method couples an approximate displacement field over the cross-section of the waveguide and assumes time harmonic representation of the propagating waves along the length of the guide. The Hamilton’s principle and the finite element discretization lead to a discrete weak form of the energy balance equation. The wave propagation problem reduces to a system of algebraic equations, from which the dispersive equation can be obtained. The solution, which depends on both time t and propagation coordinate z, i.e. e i(ξz − ωt) results in a two-parameter eigensystem. By specifying a real axial wavenumber ξ, the eigenproblem permits real frequencies of propagating modes to be determined. Giving instead real frequency ω, both real and complex axial wavenumbers can be extracted, where real values pertain to propagating modes and the complex ones to the evanescent modes. The method allows us to model a generic cross-section of solid waveguide and it is well suited for computing the phase velocity, the group velocity and the wavestructure or cross-sectional mode shape.

In order to better understand bone conduction sound propagation across the skull, three-dimensional (3D) wave propagation on the skull surface was studied, along with its dependence on stimulation direction and location of a bone conduction hearing aid (BCHA) actuator. Experiments were conducted on five Thiel embalmed whole head cadaver specimens. Stimulation, in the 0.1-10 kHz range, was sequentially applied at the forehead and mastoid via electromagnetic actuators from commercial BCHAs, supported by a 5-N steel band. The head response was quantified by sequentially measuring the 3D motion of ∼200 points (∼15-20 mm pitch) across the ipsilateral, top, and contralateral skull surface via a 3D laser Doppler vibrometer (LDV) system, guided by a robotic positioner. Low-frequency stimulation (<1 kHz) resulted in a spatially complex rigid-body-like motion of the skull that depended on both the stimulation condition and head support. The predominant motion direction was only 5-10 dB higher than other components below 1 kHz, with no predominance at higher frequencies. Sound propagation direction across the parietal plates did not coincide with stimulation location, potentially due to the head base and forehead remaining rigid-like at higher frequencies and acting as a large source for the deformation patterns across the parietal sections.

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

A multiple point source model is developed in this research for studying both shear and compressional spherical wave propagation in a non-viscous fluid-saturated elastic porous medium. Relative displacement between the fluid and solid of the medium is quantified by the spherical wave governing equations, such that the waves described are more representative to that in engineering practices. The shear wave has shown significant influences on the characteristics of superposed shear and compressional waves generated by multiple point sources. Utilization of multiple point sources shows higher efficiency and effectiveness in generating desired waves, in comparing with that of single source. Specifically, the multiple sources model is more energy effective in comparing with the single source model by always producing larger magnitudes of relative displacement than a single source with the same energy, which becomes more significant when the distance between the source and the considered geological particles increases. Multiple point sources also show advantages on duration, direction control and magnitude adjustment for the waves generated. Numerical analyses are performed for comparing different shear, compressional and the superposed wave responses under single and multiple sources.

During the seismic damage investigation of Wenchuan earthquake, it is found that some of the entrance lining structures, such as that of the Longxi Tunnel, suffered significant pavement blowup, which can be attributed to the effect of Rayleigh wave. Unlike the conventional analysis under shear wave propagating upwards, numerical dynamic analysis under the effect of Rayleigh wave in time domain is much more complicated in practice. In this paper, a practical wave motion input method for three-dimensional Rayleigh wave propagation problems in time domain is proposed and consequently applied in a numerical model to analysis the dynamic response characteristics of tunnel entrance. Desirable simulation result and sufficient accuracy is accomplished with the proposed approach. The particular seismic damage of tunnel entrance that was observed in Longxi tunnel is reproduced successfully in the simulation.

Elastic metamaterials possess band gaps, or frequency ranges that are forbidden to wave propagation. Existing solutions for impeding three-dimensional (3D) wave propagation largely rest on high-volume fractions of mass inclusions that induce and tailor negative effective density-based local resonances. This study introduces a class of elastic metamaterials that achieve low-frequency band gaps with a volume fraction as low as 3% (mass density as low as $0.034\phantom{\rule{0.1em}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$). The working of the proposed design hinges on a 3D trampolinelike mode behavior that gives rise to wide, omnidirectional, and low-frequency band gaps for elastic waves despite very low-mass densities. Such a 3D trampoline effect is derived from a network of overhanging nodal microarchitectures that act as locally resonating elements, which give rise to band gaps at low frequencies. The dynamic effective properties of the metamaterial are numerically examined, which reveal that the band gap associated with the trampoline effect is resulted from a negative effective modulus coupled with a near-zero yet positive effective density. The experimental characterization is then made possible by fabricating the metamaterial via a light-based printing system that is capable of realizing microarchitectures with overhanging microfeatures. This design strategy could be useful to applications where simultaneous light weight and vibration control is desired.

This study aims to systematically investigate the wave propagation behavior in curved sandwich nanoplates incorporating zirconia–Ti6Al4V cores and graphene-enhanced foam layers, subjected to combined thermal and magnetic fields. To achieve this, a high-order shear deformation plate theory coupled with nonlocal strain gradient elasticity theory is employed. Three distinct foam distribution patterns—U-foam, S1-foam, and S2-foam—are considered to assess their influence on wave characteristics. The results reveal that phase and group velocities decrease with increasing foam content, thermal softening effects, and reduced curvature radius, whereas the presence of a magnetic field contributes to increased wave velocities. Interestingly, the impact of foam distribution varies significantly among configurations, with U-foam showing the most pronounced effect (40%), followed by S1-foam (10%) and S2-foam (3%). These findings underscore the potential applicability of such nanostructures in advanced engineering systems, particularly in radar-absorbing materials; high-temperature nanoelectromechanical systems; and lightweight, thermally responsive sensors for wearable healthcare technologies.

A modification of the wave-propagation algorithm is used as a tool for determining contact quantities in a finite-volume scheme for the numerical simulation of two-dimensional thermoelastic wave propagation in inhomogeneous media. The modification is needed to provide the satisfaction of the thermodynamic consistency conditions between adjacent elements. It appears that the algorithm is thermodynamically consistent except for the limiter functions. Therefore, a composite scheme is used where the Godunov step is applied after each three second-order Lax–Wendroff steps. Elimination of source terms is made following the method of balancing source terms after independent solution of the heat conduction equation.

A multiscale model for the two-dimensional nonlinear wave propagation in a locally microdamaged medium is presented, and numerical simulations are analyzed in view of nondestructive testing applications. The multiscale model uses a statistical distribution of hysterons and upscales their microscopic stress-strain relations to a mesoscopic level. Macroscopic observations are then predicted by finite integration techniques. The influence of a small region with hysteretic nonlinearity on the generation of harmonics is investigated, and numerical results for different amplitudes of the input signal and different analysis techniques of the response signal are presented. Second, a study is conducted on the interaction of a Rayleigh wave with a microdamaged zone with hysteretic nonlinearity at the surface of an otherwise linear body, and the influence of the microdamaged zone on the surface wave velocity and on the generation of harmonics is examined. It is found that the effect of hysteresis on the Rayleigh wave propagation can be barely seen in the surface wave velocity measurement, but shows up nicely in the wave spectrum. The potential of a nonlinearity based depth profiling technique is explored by evaluating the nonlinear responses at different frequencies for a vertically stratified medium with spatially varying hysteresis properties.

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy–Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev–Petviashvili (KP), Zabolotskaya–Khokhlov (ZK) and Khokhlov–Zabolotskaya–Kuznetsov (KZK) equations of incompressible solid mechanics.

On the three-dimensional wave propagation through cascading screens having a periodic system of arbitrary openings