Wave propagation in a fluid wedge over a solid half-space – Mesh-free analysis with experimental verification

Interaction between a bounded ultrasonic beam and a liquid wedge over a solid half-space is studied. A semi-analytical technique called distributed point source method (DPSM) is adopted for modeling the ultrasonic field in a wedge shaped fluid structure over a solid half space. This study is important for analyzing the ultrasonic waves used for the non-destructive inspections of partially immersed structures. It is also useful for studying the effect of underwater ultrasonic or acoustic wave experiments on marine lives near the shore. The problem geometry considers a bounded acoustic beam striking a fluid–solid interface between a fluid wedge and a solid half-space at steady-state. Solution of this problem is beyond the scope of the currently available analytical methods when the beam is bounded. Only numerical method (boundary element method (BEM) or finite element method (FEM)) based packages (e.g. PZFlex) are in principle capable of modeling ultrasonic fields in such structures. At high frequencies FEM and BEM based packages require huge amount of computation memory and time for their executions that DPSM technique can avoid. Effect of the angle of strike and the fluid wedge angle variation on the wave propagation characteristics is studied. Theoretical predictions are compared with some experimental results.     

Elastic wave scattering in a solid half-space with a circular cylindrical hole using the Distributed Point Source Method

Ultrasonic wave scattering in a solid half-space containing a circular cylindrical hole is studied. The solid is struck by a bounded ultrasonic beam. Distributed Point Source Method (DPSM) which is a semi-analytical technique is adopted to model the ultrasonic field. A finite size transducer is used to generate the ultrasonic field. The circular hole in the solid is modeled by passive point sources. These point sources along with the point sources placed near the fluid–solid interface contribute to the scattered wave field in the solid. Even though the wave scattering by a circular hole in a solid is a classical problem this is the first time the complete variation of the scattered wave field generated by a bounded ultrasonic beam in a solid half-space with a circular hole is shown. The solution of this problem will help us to understand the distortion of the ultrasonic field in a solid and in the neighboring fluid due to the presence of a cylindrical anomaly which plays an important role in sensitivity analysis and calibration of different non-destructive testing techniques. Numerical examples for 1 MHz and 2.25 MHz transducers are given and scattered wave fields generated by holes of different diameters in an aluminum half-space are compared with the wave fields generated in the defect-free environment.     

Surface displacements due to elastic wave scattering by buried planar and non-planar cracks

The scattering of time-harmonic plane longitudinal, shear, and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the far-field displacements as well as the stress-intensity factors for the crack tips. Results for vertical displacement on the free surface of the half-space are presented in this paper.     

A Mathematical Model for Short-Time Filtration in Poroelastic Media with Thermal Relaxation and Two Temperatures

In this work, we derive a set of governing equations for a mathematical model of generalized thermoelasticity in poroelastic materials. This model predicts finite speeds of propagation of waves contrary to the model of coupled thermoelasticity where an infinite speed of propagation is inherent. Next, we prove the uniqueness of solution of these equations under suitable conditions. We also obtain a reciprocity theorem for these equations. A thermal shock problem for a half-space composed of a poroelastic material saturated with a liquid is then considered. The surface of the half-space is assumed to be traction free, permeable, and subjected to heating. The Laplace transform technique is used to solve the problem. Numerical results for the temperature in the elastic body and fluid, displacement of the elastic body, velocity of the fluid, and stresses for both components are obtained and represented graphically.     

Orthogonality of 3D guided waves in viscoelastic laminates and far field evaluation to a local acoustic source

The fundamental properties of guided waves in a laminate with any homogeneous boundary conditions on its faces are considered. As shown, the waves satisfy orthogonality relations whose physical meaning is related to the additivity of the average power flow. The applications of this orthogonality for solving some particular boundary value problems are discussed. A method for exact calculation of the far field caused by an acoustic source of a finite size is suggested. The only restriction is that the distance required must exceed the longitudinal radius of the source. The obtained results can be used for evaluating the fields radiated by ultrasonic transducers of arbitrary aperture and by other realistic sources.     

Longitudinal waves at a micropolar fluid/solid interface

The possibility of plane wave propagation in a micropolar fluid of infinite extent has been explored. The reflection and transmission of longitudinal elastic wave at a plane interface between a homogeneous micropolar fluid half-space and a micropolar solid half-space has also been investigated. It is found that there can exist four plane waves propagating with distinct phase speeds in an infinite micropolar fluid. All the four waves are found to be dispersive and attenuated. The reflection and transmission coefficients are found to be the functions of the angle of incidence, the elastic properties of the half-spaces and the frequency of the incident wave. The expressions of energy ratios have also been obtained in explicit form. Frequency equation for the Stoneley wave at micropolar solid/fluid interface has also been derived in the form of sixth-order determinantal expression, which is found in full agreement with the corresponding result of inviscid liquid/elastic solid interface. Numerical computations have been performed for a specific model. The dispersion curves and attenuation of the existed waves in micropolar fluid have been computed and depicted graphically. The variations of various amplitudes and energy ratios are also shown against the angle of incidence. Results of some earlier workers have been deduced from the present formulation.     

Unsteady gravity-elastic and gravity-capillary ship waves

The development of three-dimensional waves generated by a region of pressures moving uniformly and rectilinearly over the surface of a thin elastic isotropic plate covering an ideal fluid layer of finite depth is investigated. The pressures act starting at a certain instant. A qualitative similarity between the waves occurring and gravity-capillary waves is noted. The calculations are made for an ice cover. This model problem permits examining a number of properties of the oscillations of the ice cover occurring when hauling freight over ice roads, landing and takeoff of aircraft from ice fields, etc. [1]. The development of ship waves in a fluid of finite depth in the absence of a floating plate was investigated in [2, 3] and gravity-capillary waves were studied in [4–6]. Certain properties of steady three-dimensional waves occurring during movement of a load over the surface of a floating elastic plate were established in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–32, September–October, 1978.     

Finite difference analysis of acoustic reflection and radiation from fluid-loaded plates

A finite difference/boundary integral procedure to determine the acoustic reflected pressure from a fluid-loaded bi-laminate plate is described. The bi-laminate is composed of a piezo-electric layer and an elastic layer in contact with the fluid. The plate is either of finite length and held at its two ends in an acoustically hard baffle or of infinite length with periodically etched electrodes. In the numerical model, the fluid pressure at fluid/solid interface is replaced by a continuum of point sources weighted by the normal acceleration of the elastic plate, and the governing equation system is solved in the solid domain. It is demonstrated that an appropriate applied voltage potential across the baffled piezoelectric layer has the effect of cancelling the reflected pressure at any chosen field points, and a piecewise constant voltage potential with properly chosen amplitude and phase in the periodic structure has the effect of cancelling the fundamental propagating mode of the reflected waves. The project supported by the National Natural Science Foundation of China (10172039)     

Shear-wave resonances in a fluid–solid–solid layered structure

An inhomogeneous solid layer is bounded on one side by a fluid half-space and on the other by a homogeneous solid half-space. An acoustic wave in the fluid is incident on the layer. Experiments suggest that some kind of shear-wave resonance of the layer exists. Here, the layer is modeled with exponential variations of the material properties (Epstein model). Solutions in terms of hypergeometric functions are found. Genuine resonances are found but only when the layer is not bonded to the solid half-space; these are analogous to Jones frequencies in fluid–solid interaction problems. When the solid half-space is present, the resonances become complex: they are scattering frequencies. Simple but accurate asymptotic approximations are found using known estimates for hypergeometric functions with large parameters.     

钢轨轨底垂直振动模式导波检测技术的实验研究

现有铁路钢轨超声探伤车技术无法检测线路钢轨轨底缺陷,给铁路运输安全带来很大隐患.本文开展了钢轨轨底超声导波传播特性和垂直振动模式导波检测技术研究,采用半解析有限元方法分析了我国60型钢轨轨底的各振动模态导波频散曲线和波结构.应用模态锤技术对自由状态钢轨轨底垂直振动模态导波传播特性进行了实验测量,结果表明,在0～100kHz频率范围内,钢轨轨底垂直振动模态优势模式与厚度为14 mm板中的A0模式兰姆波具有等效性.进一步研究了激励频率、激励脉冲周数、传播距离对轨底垂直振动模态导波传播的影响,设计了导波斜探头,选择合适的参数在钢轨轨底激励出垂直振动模态导波并检测出了轨底的人工缺陷.本文的研究结果为线路钢轨轨底的导波检测技术奠定了一定的基础.     

Frequency domain fundamental solutions for a poroelastic half-space

In frequency domain, the fundamental solutions for a poroelastic half-space are re-derived in the context of Biot＇s theory. Based on Biot＇s theory, the governing field equations for the dynamic poroelasicity are established in terms of solid displacement and pore pressure. A method of potentials in cylindrical coordinate system is proposed to decouple the homogeneous Biot＇s wave equations into four scalar Helmholtz equations, and the general solutions to these scalar wave equations are obtained. After that, spectral Green＇s functions for a poroelastic full-space are found through a decomposition of solid displacement, pore pressure, and body force fields. Mirror-image technique is then applied to construct the half-space fundamental solutions.Finally, transient responses of the half-space to buried point forces are examined.     

Evolution of three-dimensional gravitationally warped waves during the movement of a pressure zone of variable intensity

Three-dimensional, unestablished, gravitationally warped waves arising due to the motion of a harmonically time-varying pressure zone over a solid, thin plate floating on the surface of a homogeneous liquid of finite depth have been studied in the linear formulation. In the absence of a plate, three-dimensional waves are generated by the movement of a region of periodic perturbations, where established waves have been studied in [1, 2], and unestablished waves have been investigated in [3–5]. The evolution of three-dimensional, gravitationally warped waves formed during the motion of a constant load over a plate has been considered in [6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 54–60, September–October, 1986.     

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.     

Scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid

A scattering or T-matrix approach is presented for studying the scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid. A Kelvin-Voigt model is used to obtain the complex elastic moduli of the viscoelastic solid. The T-matris formulation is somewhat complicated because the wave equations and fields are quite different in the solid and fluid regions and are coupled by continuity conditions at the interface. We have obtained fairly extensive numerical results for prolate and oblate spheroids for a variety of scattering geometries. The backscattering, bistatic, absorption and extinction cross-section are presented as a function of the frequency of the incident wave.     

Stability of a rubber half-space

The paper treats the stability of surface waves generated when a rubber half-space is subjected to compression. At a load level below the critical one an asymptotic expansion is given for the difference between the potential energy of an adjacent state and that of the fundamental state. The displacement field is expressed approximately by a linear combination of two different buckling modes and a residual displacement field which is orthogonal to the former fields. This remainder permits us to take into account the effect of all other modes which have been neglected. The wavelength of the two modes are governed by the dominant imperfections of the half-space. Terms up to the fourth order in the amplitudes of the buckling modes are included.Results are presented to show the most severe post-buckling behaviour as presented by the line of steepest descent in the load displacement diagram. Furthermore curves showing the reduction of the critical load on account of the imperfections in the two dominant modes are also presented. The analysis is kept sufficiently general to include the effect of pre-straining the rubber half-space upon the post-buckling behaviour.     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.     

Numerical study on wave propagation in a low-rigidity elastic medium considering the effects of gravity

We discuss the effects of vertical gravity force on wave propagation when a material is intermediate between solid and fluid, especially we focus on what kinds of phase are generated and how it propagates on the surface. We introduce gravity terms into the 2D linear finite element method in order to account for the contribution from the gravity. Numerical simulations are conducted for a half-space model and a two-layered, single horizontal layer overlain on a half-space, model. Both models are compared between the results including and excluding the viscosity. The fastest phase propagating from a surface point source, a leaking Rayleigh wave for usual elastic material, is transformed into an interesting phase including some common features to the gravity wave when the gravity effect becomes significant. The viscosity does not affect the fastest phases, whereas it affects other latter phases appearing only for the two-layered model.     

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.     

Attenuated Wave Field in Fluid-Saturated Porous Medium with Excitations of Multiple Sources

This research addresses the investigation of an elastic wave field in a homogeneous and isotropic porous medium which is fully saturated by a Newtonian viscous fluid. A new methodology is developed for describing the wave field in the medium excited by multiple energy sources. To quantify the relative displacements between the fluid and solid of the medium, the governing equations of the elastic wave propagation are derived in the form of displacements specially. The velocities and attenuation of the waves are considered as functions of viscosity and frequency. Making use of the Hankel function and the moving-coordinate method, a model of the wave motion with multiple cylindrical wave sources is built. Making use of the model established in this research, the relative displacement between the fluid and the solid can be quantified, and the wave field in the porous media can then be determined with the given energy sources. Numerical simulations of cylindrical waves from multiple energy sources propagating in the porous medium saturated by viscous fluid are performed for demonstrating the practicability of the model developed.