An analogical study of wave equations,physical quantities,conservation and reciprocity equations between electromagnetic and elastic waves

This paper presents an analogical study between electromagnetic and elastic wave fields,with a one-to-one correspondence principle established regarding the basic wave equations,the physical quantities and the differential operations.Using the electromagnetic-to-elastic substitution,the analogous relations of the conservation laws of energy and momentum are investigated between these two physical fields.Moreover,the energy-based and momentum-based reciprocity theorems for an elastic wave are also derived in the time-harmonic state,which describe the interaction between two elastic wave systems from the perspectives of energy and momentum,respectively.The theoretical results obtained in this analysis can not only improve our understanding of the similarities of these two linear systems,but also find potential applications in relevant fields such as medical imaging,non-destructive evaluation,acoustic microscopy,seismology and exploratory geophysics.

On Klein tunneling of low-frequency elastic waves in hexagonal topological plates

Incident particles in the Klein tunnel phenomenon in quantum mechanics can pass a very high potential barrier.Introducing the concept of tunneling into the analysis of phononic crystals can broaden the application prospects.In this study,the structure of the unit cell is designed,and the low frequency(<1 k Hz)valley locked waveguide is realized through the creation of a phononic crystal plate with a topological phase transition interface.The defect immunity of the topological waveguide is verified,that is,the wave can propagate along the original path in the cases of impurities and disorder.Then,the tunneling phenomenon is introduced into the topological valley-locked waveguide to analyze the wave propagation,and its potential applications(such as signal separators and logic gates)are further explored by designing phononic crystal plates.This research has broad application prospects in information processing and vibration control,and potential applications in other directions are also worth exploring.

Mathematical framework of nonlinear elastic waves propagating in pre-stressed media

Acoustic nonlinearity holds potential as a method for assessing material stress.Analogous to the acoustoelastic effect,where the velocity of elastic waves is influenced by third-order elastic constants,the propagation of nonlinear acoustic waves in pre-stressed materials would be influenced by higher-order elastic constants.Despite this,there has been a notable absence of research exploring this phenomenon.Consequently,this paper aims to establish a theoretical framework for governing the propagation of nonlinear acoustic waves in pre-stressed materials.It delves into the impact of pre-stress on higher-order material parameters,and specifically examines the propagation of one-dimensional acoustic waves within the contexts of the uniaxial stress and the biaxial stress.This paper establishes a theoretical foundation for exploring the application of nonlinear ultrasonic techniques to measure pre-stress in materials.

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids. The propagation of three longitudinal waves is represented through three scalar potential functions. The lone transverse wave is presented by a vector potential function. The displacements of particles in different phases of the aggregate are defined in terms of these potential functions. It is shown that there exist three longitudinal waves and one transverse wave. The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated. For the presence of viscosity in pore-fluids, the waves refracted to the porous medium attenuate in the direction normal to the interface. The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a non- singular system of linear algebraic equations. These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave. The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model. The conservation of the energy across the interface is verified. The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed,

Harten-Lax-van Leer-contact(HLLC)approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact （HLLC） approximate Riemann solver is built with elastic waves （HLLCE） for one-dimensional elastic-plastic flows with a hypo- elastic constitutive model and the von Mises＇ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the pre- sented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves （TRRSE）

Propagation behavior of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with line defects

Using a stiffness matrix method, we in- vestigate the propagation behaviors of elastic waves in one-dimensional （1D） piezoelectric/piezomagnetic （PE/PM） phononic crystals （PCs） with line defects by calculating energy reflection/transmittion coefficients of quasi-pressure and quasi-shear waves. Line defects are created by the re- placement of PE or PM constituent layer. The defect modes existing in the first gap are considered and the influences on defect modes of the material properties and volume fraction of the defect layers, the type of incident waves, the location of defect layer and the number of structural layers are discussed in detail. Numerical results indicate that defect modes are the most obvious when the defect layers are inserted in the middle of the perfect PCs; the types of incidence wave and material properties of the defect layers have important effects on the numbers, the location of frequencies and the peaks of defect modes, and the defect modes are strongly de- pendent on volume fraction of the defect layers. We hope this paper will be found useful for the design of PE/PM acoustic filters or acoustic transducer with PCs structures.

WATER WAVES IN AN ELASTIC VESSEL

Linear and nonlinear analyses of water waves in an elastic vessel are carried out to study the dramatic phenomena of Dragon Wash as well as related controllable experiments. It is proposed that the capillary edge waves are generated by parametric resonance, which is shown to be a possible mechanism for both rectangular an circular vessels. For circular vessel, the normal geometric resonance is also operating, thus greatly. enhance the dramatic effect. The mechanism of nonlinear mode-mode interaction is proposed far the generation of axisymmetric low-frequency gravity waves by the high-frequency external excitation. A simple model system is studied numerically to demonstrate explicitly this interaction mechanism.

Diffraction of plane SV waves by a cavity in poroelastic half-space

This paper presents an indirect boundary integration equation method for diffraction of plane SV waves by a 2-D cavity in a poroelastic half-space.The Green＇s functions of compressive and shear wave sources are derived based on Biot＇s theory. The scattered waves are constructed using fictitious wave sources close to the boundary of the cavity, and their magnitudes are determined by the boundary conditions. Verification of the accuracy is performed by： （1） checking the satisfaction extent of the boundary conditions, （2） comparing the degenerated solutions of a single-phased case with well- known solutions, and （3） examining the numerical stability of the solutions. The nature of diffraction of plane SV waves around a cavity in a poroelastic half-space is investigated by numerical examples.

Hydroelastic Response of A Circular Plate in Waves on A Two-Layer Fluid of Finite Depth

The hydroelastic response of a circular, very large floating structure(VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the platecovered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young’s modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.

THEORY OF WATER WAVES IN AN ELASTIC VESSEL

Recent experiments related to the Dragon Wash phenomena showed that axisymmetric capillary waves appear first from excitation, and circumferential capillary waves appear after increase of the excitation strength. Based on this new finding, a theory of parametric resonance is developed in detail to explain the on- set of the prominent circumferential capillary waves. Numerical computation is also carried out and the results agree generally with the experiments. Analysis and nu- merical computation are also presented to explain the generation of axisymmetric low-frequency gravity waves by tile high-frequency external excitation.

On the Interaction of Surface Waves with An Elastic Plate of Finite Length in Head Seas

An eigen-function expansion method based on a new orthogonal inner product is proposed by Sahoo et al. (2000) for the study of the hydroelastic response of mat-type VLFS in head seas. However, their main emphasis is on the effect of edge conditions and they assume that the plate is of a semi-infinite length. In reality, the plate is of finite length. For consideration of the finite length effect, the reflection and transmission from the other end must be considered. The effect of this reflection and transmission on the hydroelastic response of VLFS is of interest for practical application. Furthermore, the physical meaning of the new inner product was not given in their paper. In this paper, it is shown that the new inner product can he derived from the governing equation and the bottom boundary conditions. Then the same eigen-function expansion method is adopted for the study of the hydroelastic response of an elastic plate of finite length in surface waves. Detailed comparisons are made between the present finite length model and the semi-infinite model and between the present model predictions and the experimental results. It is found that that the finite length effect is significant and the accuracy of present model is higher than the semi-infinite model. Furthermore, a new phenomenon, which is not mentioned in Sahoo et al. (2000), is found. Taht is, for larger L/h ratios, the reflection and transmission coefficients will oscillate with the non-dimensional parameter k(0) h. Further study is needed for full understanding of this phenomenon.

Research on propagation properties of elastic waves in two-phase anisotropic media

With the development of seismic engineering and seismic exploration of energy, the underground media that westudy are more and more complicated. Conventional anisotropy theory or two-phase isotropy theory is difficult todescribe anisotropic media containing fluid, such as fractures containing gas, shales containing water Based onBlot theory about two-phase anisotropy, with the use of elastic plane wave equations, we get Christoffel equations.We calculate and analyze the effects of frequency on phase velocity, attenuation, amplitude ratio and polarizationdirection of elastic waves of two-phase, transversely isotropic media. Results show that frequency affects slow Pwave the greatest among the four kinds of waves, i.e., fast P wave, slow P wave, fast S wave and slow S wave.Fluid phase amplitude to solid phase amplitude ratio of fast P wave, fast S wave and slow S wave approaches unitfor large dissipation coefficients. Polarization analysis shows that polarization direction of fluid phase displacement is different from, not parallel to or reverse to, that of solid phase displacement in two-phase anisotropic media.

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.

Evaluation of soil fabric using elastic waves during load-unload

It is essential to assess the evolution of soil fabric as it has an important role in the mechanical responses of soils during complex loading conditions.This contribution carries out the physical experiments using three granular materials in the laboratory.The variations of compression and shear wave velocities(Vp and Vs)are investigated during load-unload cycles under dry and drained conditions.Supplementary discrete element method(DEM)simulations are performed to understand the evolution of soil fabric during the equivalent load-unload cycles using spherical particles.Vp and Vs are not always reversible even though the stress state regains its isotropic condition after unload,indicating that Vp and Vs are governed by not only the stress state but also the fabric change.The variations of Vp/Vs are density-and stress-dependent;a higher level of stress ratio(s01/s03)threshold is observed for denser packings to trigger a significant change in wave velocity ratio(Vp/Vs)for experimental results using spherical glass beads and simulation data using spherical particles.Considering the particle shape,a higher s01/s03 threshold is found for more angular particles than rounded particles.The DEM result reveals that Vp/Vs of spherical particles can be correlated linearly with the evolution of fabric ratio(Fver/Fhor)during loadunload in a pre-peak range under dry and drained conditions.

SCATTERING OF HARMONIC ANTI-PLANE SHEAR WAVES BY AN INTERFACE CRACK IN MAGNETO-ELECTRO-ELASTIC COMPOSITES

The dynamic behavior of an interface crack in magneto-electro-elastic composites under harmonic elastic anti-plane shear waves is investigated for the permeable electric boundary conditions. By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable was the jump of the displacements across the crack surfaces. To solve the dual integral equations, the jump of the displacements across the crack surface was expanded in a series of Jacobi polynomials. Numerical examples were provided to show the effect of the length of the crack, the wave velocity and the circular frequency of the incident wave on the stress, the electric displacement and the magnetic flux intensity factors of the crack. From the results, it can be obtained that the singular stresses in piezoelectric/piezomagnetic materials carry the same forms as those in a general elastic material for anti-plane shear problem.

Diffraction of plane P waves by a canyon of arbitrary shape in poroelastic half-space (Ⅰ): Formulation

This paper presents an indirect boundary integration equation method for diffraction of plane P waves by a two-dimensional canyon of arbitrary shape in poroelastic half-space. The Green＇s functions of compressional and shear wave sources in poroelastic half-space are derived based on Biot＇s theory. The scattered waves are constructed using the fictitious wave sources close to the boundary of the canyon, and magnitude of the fictitious wave sources are determined by the boundary conditions. The precision of the method is verified by the satisfaction extent of boundary conditions, the comparison between the degenerated solutions of single-phased half-space and the well-known solutions, and the numerical stability of the method.

SCATTERING OF ELASTIC WAVES IN AN ELASTIC MATRIX CONTAINING AN INCLUSION WITH INTERFACES

Based on the theory of elastic dynamics, the scattering of elasticwaves and dynamic stress concentration of fiber-reinforced compositewith interfaces are studied. Analytical expressions of elastic wavesin different medium areas are presented and an analytic method ofsolving this problem is established. The mode coefficients aredetermined by means of the continuous conditions of displacement andstress on the boundary of the interfaces. The influence of materialproperties and structural size on the dynamic stress con- centrationfactors near the interfaces is analyzed.

Reflection and refraction of plane waves at the boundary of an elastic solid and double-porosity dual-permeability materials

Phenomena of reflection and refraction of plane harmonic waves at a plane interface between an elastic solid and doubleporosity dual-permeability material are investigated. The elastic solid behaves non-dissipatively, while double-porosity dual-permeability materials behave dissipatively to wave propagation due to the presence of viscosity in pore fluids. All the waves(i.e., incident and reflected) in an elastic medium are considered as homogeneous(i.e., having the same directions of propagation and attenuation), while all the refracted waves in double-porosity dual-permeability materials are inhomogeneous(i.e., having different directions of propagation and attenuation). The coefficients of reflection and refraction for a given incident wave are obtained as a non-singular system of linear equations. The energy shares of reflected and refracted waves are obtained in the form of an energy matrix. A numerical example is considered to calculate the partition of incident energy among various reflected and refracted waves. The effect of incident direction on the partition of the incident energy is analyzed with a change in wave frequency, wave-induced fluid-flow, pore-fluid viscosity and double-porosity structure.It has been confirmed from numerical interpretation that during the reflection/refraction process, conservation of incident energy is obtained at each angle of incidence.

ANTI-PLANE SHEAR WAVES SCATTERING FROM A PARTIALLY DEBONDED MAGNETO-ELECTRO-ELASTIC CIRCULAR CYLINDRICAL INHOMOGENEITY

This article presents an analysis of the scattering of anti-plane shear waves from a single cylindrical inhomogeneity partially bonded to an unbounded magneto-electro-elastic matrix. The magneto-electric permeable boundary conditions are adopted. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients. Some examples are calculated and the results are illustrated. The results show that the COD increases when the piezomagnetic coefficient of the inhomogeneity bonded to the piezoelectric matrix becomes larger, and that the COD decreases when the piezomagnetic coefficient of the matrix with the piezoelectric inhomogeneity increases.

GLOBAL ATTRACTOR OF NONLINEAR STRAIN WAVES IN ELASTIC WAVEGUIDES

The initial-boundary value problem of the propagation of nonlinear longitudinal elastic waves in an initially strained rod is considered. The rod is assumed to interact with the surrouding elastic and viscous external medium. The long time behavior of solutions are derived and global attractors in E-1 space is obtained.