The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals

We extend the layer multiple-scattering theory (LMST) to elastic waves propagating in two-dimensional (2D) periodical composites. The formalism to calculate the reflection and transmission coefficients for elastic waves through finite slabs is presented. In this spirit, the crystal is viewed as a sequence of identical monolayer which has one-dimensional (1D) periodicity along a given direction. The reflection and transmission coefficients for a multilayer slab can be obtained by a double-layer scheme through the calculation of the scattering matrix of a monolayer. To demonstrate the application of this formalism, we calculate transmission coefficients for systems consisting of pure solid components or mixing (solid and fluid) components. The validity of this method is checked by both band structure calculations and transmission measurement of ultrasonic experiment.     

The use of an orthogonality relation for reducing the size of finite element models for 3D guided waves scattering problems

The scattering of guided waves by complex shaped defects in three-dimensional (3D) waveguides is considered. For such problems, analytical solutions do not exist, and modal decomposition techniques based on the establishment of the displacement and stress fields in the vicinity of the scatterer are quite heavy and complicated to perform. On the other hand, finite elements (FE)-based methods constitute a powerful way to obtain solutions, but they are known to be very memory consuming. This paper proposes a post-processing technique, based on a 3D orthogonality relation, to decompose a complex acoustic field produced by a scatterer and predicted by a 3D FE model, into plane waves, the amplitudes of which are quantified in the far field. This technique allows important reductions in the size of the FE models to be made. Two applications are presented to demonstrate the potential of this method. The first one concerns the scattering of the S₀ Lamb wave incident on a flat bottom circular hole. In this example, the amplitude of each mode is calculated via the orthogonality relation-based method, and compared to that obtained by simply monitoring the displacements at appropriate through-thickness positions. In the second application, the incident S₀ Lamb mode is converted into five modes scattered by a defect of complex geometry.     

Attenuation coefficient estimation using experimental diffraction corrections with multiple interface reflections

The ultrasonic attenuation coefficient of a fluid or solid is an acoustic parameter routinely estimated for the purpose of materials characterization and defect/disease detection. This paper describes a broadband attenuation coefficient estimation technique that combines two established estimation approaches. The key elements of these two approaches are: (1) the use of magnitude spectrum ratios of front surface, first back surface, and second back surface reflections from interfaces of materials with plate-like geometries, and (2) the use of an experimental diffraction correction approach to avoid diffraction losses. The combined estimation approach simplifies the attenuation coefficient estimation process by eliminating the need to explicitly make diffraction corrections or calculate reflection/transmission coefficients. The approach yields estimates of the attenuation coefficient, reflection coefficient, and material density. Models, experimental procedures, and signal analysis procedures, which support implementation of the approach, are presented. Attenuation coefficient and reflection coefficient estimates are presented for water and solid samples with estimates based on measurements made with multiple transducers.     

Impact of localized inhomogeneity on the surface-wave velocity and bulk-wave reflection in solids

The effect of a weak surface, near-surface and interfacial inhomogeneity on the frequency dependence of the surface wave velocity and of the SH (shear horizontal) wave reflectivity in isotropic elastic media is studied analytically and numerically. The inhomogeneity is modeled as an infinite planar layer with continuously varying properties. Weak inhomogeneity may markedly affect the dispersion of the Rayleigh velocity and especially of the reflectivity. It is demonstrated how this effect, particularly pronounced at high frequency, depends on the extent of inhomogeneity. The material data for damaged and ideal concrete and several simple examples of inhomogeneity profiles are utilized for the numerical calculations based on the Peano expansion. The use of explicit low- and high-frequency approximations is also exemplified. Among these, simple WKB asymptotics are shown to be particularly helpful for the Rayleigh velocity in the case of a prominent inhomogeneity attached to the surface and for the reflection on weak interfaces.     

Generalized eigenproblem of hybrid matrix for Floquet wave propagation in one-dimensional phononic crystals with solids and fluids

A method based on the solution to a generalized eigenproblem of hybrid matrix is presented for stable analysis of Floquet wave propagation in one-dimensional phononic crystals with solids and fluids. The method overcomes the numerical instability in the standard eigenproblem of transfer matrix, thus enabling Floquet waves to be determined reliably. The recursion relations of hybrid matrix for periodic multilayered structure of various solid and/or fluid phases are formulated. Dispersion relation and omnidirectional reflection for one-dimensional phononic crystals with solids and fluids are discussed. The frequency-thickness range of phononic bandgap is determined conveniently based on the Floquet wavenumbers.     

Reflection of structural waves at a solid/liquid interface

This paper investigates the reflection characteristics of structural or guided waves in rods at a solid/liquid interface. Structural waves, whose wavelengths are much larger than the diameter of the rod, are described in a first approximation by classical one-dimensional wave theory. The reflection characteristics of such waves at a solid/liquid (melting) interface has been reported by two different ultrasonic measurement techniques: first, measuring the fast regression rate of a melting interface during the burning of metal rod samples in an oxygen-enriched environment, and second, monitoring the propagation of the solid/liquid interface during the slow melting and solidification of a rod sample in a furnace. The second work clearly shows that the major reflection occurs from the solid/liquid interface and not the liquid/gas interface as predicted by plane longitudinal wave reflectivity theory. The present work confirms this observation by reporting on the results of some specially designed experiments to identify the main interface of reflection for structural waves in rods. Hence, it helps in explaining the fundamental discrepancy between the reflection characteristics at a solid/liquid interface between low frequency structural waves and high frequency bulk waves, and confirms that the detected echo within a burning metallic rod clearly represents a reflection from the solid/liquid interface.     

Ultrasonic wave propagation in heterogeneous solid media: theoretical analysis and experimental validation

This research deals with the ultrasonic characterization of thermal damage in concrete. This damage leads to the appearance of microcracks which then evolve in terms of volume rate and size in the material. The scattering of ultrasonic waves from the inclusions is present in this type of medium. The propagation of the longitudinal wave in the heterogeneous media is studied via a homogenization model that integrates the multiple scattering of waves. The model allows us to determine the phase velocity and the attenuation according to the elements which make the medium. Simulations adapted to the concrete are developed in order to test the responses of the model. These behaviors are validated by an experimental study: the measurements of phase velocity and attenuation are performed in immersion, with a comparison method, on a frequency domain which ranges from 160 kHz to 1.3 MHz. The analysis of different theoretical and experimental results obtained on cement-based media leads to the model validation, on the phase velocity behavior, in the case of a damage simulated by expanded polystyrene spheres in granular media. The application to the case of a thermally damaged concrete shows a good qualitative agreement for the changes in velocity and attenuation.     

The detectability of kissing bonds in adhesive joints using ultrasonic techniques

This paper concerns a study of the detectability of dry contact kissing bonds in adhesive joints using three ultrasonic inspection techniques. Conventional normal incidence longitudinal and shear wave inspection were conducted on dry contact kissing bonds using a standard damped ultrasonic transducer and an electro-magnetic acoustic transducer (EMAT) respectively. The detectability of the dry contact kissing bonds was assessed by calculating the reflection coefficient of the imperfect interface at varying loads for a number of surface roughnesses. A high power ultrasonic method was also employed to determine the non-linear behavior of the adhesive interface. The non-linearity of the interface was determined by the ratio of the amplitudes of the first harmonic and fundamental frequencies of the transmitted waveform. It was found that the high power technique showed the greatest sensitivity to these kissing bonds at low contact pressures, however at high loads conventional longitudinal wave testing was more sensitive. It was also noted that a combination of two or more techniques could provide enhanced information about the kissing bond compared to a single technique alone.     

Normal modes of a poroelastic plate and their relation to the reflection and transmission coefficients

A poroelastic plate that obeys the Biot theory is considered. Compact new forms of its reflection and transmission coefficients, similar to those of the resonance scattering theory for an elastic plate, are derived. A numerical comparison of the reflection coefficient modulus with the plate normal modes, at low frequency, shows that a study of the reflection or transmission coefficient does not provide the same kind of information on the poroelastic plate than an investigation of guided leaky waves propagation.     

Minimization of finite beam effects in the determination of reflection and transmission coefficients of an elastic layer

A special data acquisition technique was applied to determine the acoustic plane-wave reflection and transmission properties of a plane-parallel aluminum plate. In this technique, the reflected and transmitted wavefield along a plane or line normal to the reflected or transmitted wave vector is recorded at equidistant receiver positions. The obtained traces are subsequently added up in the temporal domain to satisfy plane-wave conditions, thus effectively removing the effect of the limited beam of commonly used transducers. The agreement between plane-wave theory and experiment was found to be excellent, both in the temporal and in the frequency domain.     

A semi-analytical model for predicting multiple propagating axially symmetric modes in cylindrical waveguides

A semi-analytical model for multiple mode axially symmetric wave propagation in finite solid cylindrical waveguides is presented. The model is designed as a tool for predicting and interpreting experimental signals. The model is based on a common experimental configuration and considers the excitation, propagation and reception of the ultrasonic signal in the waveguide. The Pochhammer-Chree solution for an infinite cylinder is the basis for the model. Extensions are made to enable comparison to experimental results. Comparisons with experiment are performed in the time, frequency and joint-time frequency domain for both narrow band and broad band excitation of the piezo-electric transducer.     

Excitations of thermoelastic waves in plates by a pulsed laser

The method of the eigenfunction expansion, also known as the expansion in normal modes, is employed to study numerically the axisymmetric excitation of the thermoelastic waves in plates by a pulsed laser. This method gives a systematic treatment and allows one to investigate not only the quasistatic and dynamic thermoelastic responses of pulsed photothermal deformation on the time scale of 1 s, but also the thermoelastic generation of longitudinal, transverse, and surface acoustic waves in thick materials, as well as the excitations of the Rayleigh-Lamb wave modes in thin plates. The formalism is particularly suitable for waveform analyses of the excitations of transient Lamb waves in thin plates because one needs only to calculate the contributions of several lower eigenmodes. The numerical technique provides a quantitative tool for the experimental determination of material properties, especially the mechanical and elastic properties of free-standing films and thicker sheet materials by thermoelastic detection.     

Optimal Acoustic Attenuation of Weakly Compressible Media Permeated with Air Bubbles

Based on fuzzy logic （FL） and genetic algorithm （GA）, we present an optimization method to obtain the optimal acoustic attenuation of a longitudinal acoustic wave propagating in a weakly compressible medium permeated with air bubbles. In the optimization, the parameters of the size distribution of bubbles in the medium are optimized for providing uniformly high acoustic attenuation in the frequency band of interest. Compared with other traditional optimization methods, the unique advantage of the present method is that it can locate the global optimum quickly and effectively in need of knowing the mathematical model precisely. As illustrated by a numerical simulation, the method is effective and essential in enhancing the acoustic attenuation of such a medium in an optimal manner. The bubbly medium with optimized structural parameters can effectively attenuate longitudinal waves at intermediate frequencies with an acoustic attenuation approximating a constant value of lO（dB/cm）. Such bubbly media with optimal acoustic attenuations may be applied to design acoustic absorbent by controlling broader attenuation band and higher efficiency.     

Ultrasonic sizing of short surface cracks

This paper presents a method for ultrasonic sizing of surface cracks based on time domain and frequency domain Rayleigh wave near-field analysis. The procedure allows for the entire range of ratio of crack depth to Rayleigh wavelength a/λ to be covered with one single measurement. In the time domain the time-of-flight method was extended to cracks smaller than the wavelength by correlation of the time delay of the transmitted Rayleigh wave with the crack depth. In the frequency domain, the inverse scattering problem was solved by comparison of the measured scattering coefficients and central frequencies of the reflected and transmitted Rayleigh waves with theoretical curves. The sizing procedure was demonstrated experimentally with narrow slots and real fatigue cracks. The out-of-plane displacement component was measured pointwise in the scattered near field by means of laser interferometry. The determination of the scattering parameters in the near field was enabled by a procedure that allows for the Rayleigh wave to be separated from the other modes scattered at the defect. The experimental results showed good accuracy and repeatability down to the smallest available ratio of crack depth to Rayleigh wavelength a/λ = 0.15.     

Extraordinary Resonant Scattering in Imperfect Acoustic Cloak

We present a systematic study on the extraordinary resonant scattering in imperfect acoustic cloak by means of acoustic scattering theory. Analysis results demonstrate that the resonances are inevitable due to the perturbation to the ideal clo~k, and specific resonance modes are excited by specific order waves. The strength of resonance is determined by the magnitude of perturbation and each order wave＇s sensitivity to the perturbation. Further studies reveal the unique scattering characters of different resonance modes.     

Tri-component phononic crystals for underwater anechoic coatings

Localized resonance in phononic crystal, composed of three-dimensional arrays of composite units, has been discovered recently. The composite unit is a high-density sphere coated by soft silicon rubber. In this Letter, the absorptive properties induced by the localized resonance are systemically investigated. The mode conversions during the Mie scattering of a single coated lead sphere in unbounded epoxy are analyzed by referring the elements of the scattering matrix. Then the anechoic properties of a slab containing a plane of such composite scatterers are investigated with the multiple-scattering method by accounting the effects of the multiple scattering and the viscous dissipation. The results show that the longitudinal to transverse mode conversion nearby the locally resonant region is an effective way to enhance the anechoic performance of the finite slab of phononic crystal. Then, the influences of the viscoelasticity of the silicon rubber and the coating thickness on the acoustic properties of the finite slab are investigated for anechoic optimization. Finally, we synthetically consider the destructive scattering in the finite slab of phononic crystal and the backing, and design an anechoic slab composed of bi-layer coated spheres. The results show that the most of the incident energy is absorbed at the desired frequency band.     

On the separability of relativistic electron propagators

In this contribution we examine the separability of relativistic electron propagators. Both, magnetic and non-magnetic systems are studied on the basis of the Kohn-Sham-Dirac equation. We find a Dirac-Green's function in excellent agreement with recent calculations utilizing the left and right-handed solutions to the Dirac equation. Starting from these Dirac-Green's functions we re-derive a rotation matrix formalism that was shown to result in separable scattering matrices in the non-relativistic case. It turns out, that spin-dependent scattering matrices can be formulated which are closely related to their non-relativistic counterparts. These matrices incorporate spin-flip and non spin-flip processes on an equal footing, but are irreducible to sums over composite rotation matrices. The latter result is a major drawback for numerical applications since electron scattering in terms of composite rotations had drawn a lot of attention recently. Received 1st July 1997     

A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic,multilayered structures

A three-dimensional finite element method (FEM) for the analysis of plane wave diffraction by a bi-periodic slab is described and implemented. A scattering matrix formalism based on the FEM allows the efficient treatment of light reflection and transmission by multilayer bi-periodic structures, and the computation of Bloch modes of three-dimensional arrays. Numerical simulations, which show the accuracy and flexibility of the FEM, are presented.     

A New Method for Calculation of Single Seismic Phase of Cylindrically Multilayered Media Including Liquid Interlayer

A new method based on generalized reflection and transmission （R/T） coefficients method is proposed to calculate the single seismic phase （SSP） of cylindrically multilayered media including liquid interlayer. The use of normalization factors and normalized Lamé coefficients makes the algorithm stable numerically. Using the modified R/T matrices, we derive the iterative expressions of generalized R/T matrices, and by using the iterative relation we determine the SSP of each interface and the full waveforms. To show the superiority of this new approach for investigating of reflection and transmission properties of cylindrically multilayered media, we simulate the full waveforms and SSPs of cased hole model with annulus I （casing-cement interface） channelling （or, cross-flow）. The generalized reflection coefficient spectra and SSPs of interfaces obtained show the propagation mechanism of each component of full waveform clearly.     

Modelling the attenuation in the ATHENA finite elements code for the ultrasonic testing of austenitic stainless steel welds

Multipass welds made in austenitic stainless steel, in the primary circuit of nuclear power plants with pressurized water reactors, are characterized by an anisotropic and heterogeneous structure that disturbs the ultrasonic propagation and makes ultrasonic non-destructive testing difficult. The ATHENA 2D finite element simulation code was developed to help understand the various physical phenomena at play. In this paper, we shall describe the attenuation model implemented in this code to give an account of wave scattering phenomenon through polycrystalline materials. This model is in particular based on the optimization of two tensors that characterize this material on the basis of experimental values of ultrasonic velocities attenuation coefficients. Three experimental configurations, two of which are representative of the industrial welds assessment case, are studied in view of validating the model through comparison with the simulation results. We shall thus provide a quantitative proof that taking into account the attenuation in the ATHENA code dramatically improves the results in terms of the amplitude of the echoes. The association of the code and detailed characterization of a weld’s structure constitutes a remarkable breakthrough in the interpretation of the ultrasonic testing on this type of component.