Scattering of the fundamental anti-symmetric Lamb wave at delaminations in composite laminates

An analysis of the scattering characteristics of the fundamental anti-symmetric (A(0)) Lamb wave at a delamination in a quasi-isotropic composite laminate is presented. Analytical solutions for this problem do not exist due to the anisotropic nature and multilayer characteristics of composite laminates. This study uses a three-dimensional finite element (FE) method and experimental measurements to provide physical insight into the scattering phenomena. Good agreement is found between simulations and experimental measurements. The results show that the A(0) Lamb wave scattering at a delamination in composite laminates is much more complicated than the scattering at a defect in isotropic plates. Scatter amplitudes and scatter directivity distributions depend on the delamination size to wavelength ratio and the through-thickness location of the delamination damage. The study also investigates the feasibility of the common experimental practice of simulating delamination damage by bonding masses to the surface of composite laminates for guided wave damage detection and characterization methodologies verifications. The results suggest that care is required to use bonded masses to simulate delamination damage for verifying and optimizing damage characterization techniques. In summary, the results of the investigation help to further advance the use of the A(0) Lamb wave for damage detection and characterization.     

Scattering characteristics of Lamb waves from debondings at structural features in composite laminates

This article investigates the scattering characteristics of Lamb waves from a debonding at a structural feature in a composite laminate. This study specifically focuses on the use of the low frequency fundamental antisymmetric (A(0)) Lamb wave as the incident wave for debonding detection. Three-dimensional finite element (FE) simulations and experimental measurements are used to investigate the scattering phenomena. Good agreement is obtained between the FE simulations and experimental results. Detailed parameter studies are carried out to further investigate the relationship between the scattering amplitudes and debonding sizes. The results show that the amplitude of the scattered A(0) Lamb wave is sensitive to the debonding size, which indicates the potential of using the low frequency A(0) Lamb wave as the interrogating wave for debonding detection and monitoring. The findings of the study provide improved physical insights into the scattering phenomena, which are important to further advance damage detection techniques and optimize transducer networks.     

Generalized thermoelastic Lamb waves in a plate bordered with layers of inviscid liquid

The propagation of thermoelastic waves in a homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid or half-space of inviscid liquid on both sides is investigated in the context of generalized theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases. The different regions of secular equations are obtained and special cases, such as Lame modes, thin plate waves and short wavelength waves of the secular equations are also discussed. The secular equations for thermoelastic leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for an aluminum-epoxy composite and aluminum materials plate bordered with water. The dispersion curves for symmetric and antisymmetric thermoelastic wave modes and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement. The results have been deduced and compared with the relevant publications available in the literature at the relevant stages of the work.     

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.     

Influence of stacking sequence on scattering characteristics of the fundamental anti-symmetric Lamb wave at through holes in composite laminates

This paper investigates the scattering characteristics of the fundamental anti-symmetric (A(0)) Lamb wave at through holes in composite laminates. Three-dimensional (3D) finite element (FE) simulations and experimental measurements are used to study the physical phenomenon. Unidirectional, bidirectional, and quasi-isotropic composite laminates are considered in the study. The influence of different hole diameter to wavelength aspect ratios and different stacking sequences on wave scattering characteristics are investigated. The results show that amplitudes and directivity distribution of the scattered Lamb wave depend on these parameters. In the case of quasi-isotropic composite laminates, the scattering directivity patterns are dominated by the fiber orientation of the outer layers and are quite different for composite laminates with the same number of laminae but different stacking sequence. The study provides improved physical insight into the scattering phenomena at through holes in composite laminates, which is essential to develop, validate, and optimize guided wave damage detection and characterization techniques.     

Effect of misalignment of air-coupled probes on Ao Lamb mode propagating in a metal plate

Proper alignment of air-coupled ultrasonic transducers for generation and reception of Lamb waves is vital in order to acquire a high amplitude wave group. Any misalignment with either the transmitter or the receiver or both adversely influences the amplitude of a Lamb mode. This paper reports a systematic attempt to quantify the reduction in the amplitude of the fundamental anti-symmetric Lamb mode (A_o) in a metal plate caused by misalignments in air-coupled probes. Three different types of misalignments – linear, orientation and synchronised orientation were deliberately introduced in the transducers, and experiments were performed on a 6 mm thick aluminium plate. Amplitudes of A_o mode measured at various configurations were normalised with that of A_o mode, captured in a reference configuration. Suitable curves fitted over the experimental data points revealed that Gaussian curves represent appropriately the variations in normalised amplitudes of A_o mode. Moreover, analytical expressions were derived to predict the difference in arrival times of Lamb mode(s) due to orientation and synchronised orientation misalignments.     

Pion-nucleon backward scattering: The location of the amplitude zeros

The complex zero trajectories for the πN helicity amplitudes with definite u-channel isospin are determined from phase shifts. Some of these trajectories display systematic behaviour, having fixed values of (u?u₀), where u₀ corresponds to the backward direction, and small values of Im u. On the assumption that this systematic behaviour will persist to higher energies one can make predictions about the locations of zeros in these helicity amplitudes at high energies. These zero locations are very relevant to an understanding of the mechanisms involved in πN backward scattering. The results, which show quite a striking pattern for the zero positions in the various amplitudes, are discussed from the point of view (a) of the experimental data for πN backward scattering and (b) of a theoretical interpretation of the interaction mechanism.     

On the behavior of the πN partial wave amplitudes ats=0

The behavior of theπN partial wave amplitudes in the limitss→+0 ands→? 0 is related to backward scattering in thes- andt-channel, respectively. Assuming Mandelstam analyticity we prove with the aid of the Phragmen-Lindelöf theorem that the amplitudes for high energy backward scattering inπN→πN andππ→N¯N are equal and therefore dominated by the same exchange mechanism, namely Reggeized Fermion exchange. The dominating Regge trajectory is the Δ_δ-trajectory, and it is shown that theπN partial wave amplitudes diverge fors→±0 as \(s^{ - \alpha \Delta _\delta (0) - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) . — Reduced amplitudes are defined which are regular ats=0. — Using recent results ofπN backward scattering real and imaginary part of thes-wave amplitudef ₀₊ ^(?) are calculated in the interval 0≦s≦7.     

Reflection amplitudes of boundary Toda theories and thermodynamic Bethe ansatz

We study the ultraviolet asymptotics in A_n affine Toda theories with integrable boundary actions. The reflection amplitudes of non-affine Toda theories in the presence of conformal boundary actions have been obtained from the quantum mechanical reflections of the wave functional in the Weyl chamber and used for the quantization conditions and ground-state energies. We compare these results with the thermodynamic Bethe ansatz derived from both the bulk and (conjectured) boundary scattering amplitudes. The two independent approaches match very well and provide the non-perturbative checks of the boundary scattering amplitudes for Neumann and (+) boundary conditions.     

Asymmetric propagation of the first order antisymmetric lamb wave in a tapered plate based on time domain analysis

The asymmetric propagation of the first order antisymmetric (A₁) Lamb wave in a tapered plate respectively carved with sharp bottom corner and round bottom corner is theoretically investigated. Through numerical simulation of A₁ Lamb wave in time domain, we find that when the thickness of the waveguide abruptly decreases to below the cut-off thickness, about half of the A₁ mode is converted into the fundamental symmetrical S₀ and antisymmetrical A₀ modes to pass through the defected region. Furthermore, the transmitted modes A₀ and S₀ are completely apart from each other and can be quantitatively evaluated. Conversely, when the thickness change is very smooth, most of the energy of A₁ Lamb wave is reflected back. It is the unique mode conversion behavior that leads to great transmission difference value of A₁ Lamb wave along the opposite directions. Finally, the influence of geometrical parameters on the transmission coefficient is also studied. The higher efficiency and proper working frequency range can be realized by adjusting the slope angle θ, height h ₁ and h ₂. The simple asymmetric systems will be potentially significant in applications of ultrasound diagnosis and therapy.     

Crossing characteristics of lamb wave modes in [001]c and [011]c polarized Pb(Mg1/3Nb2/3)O3-PbTiO3 crystal plates

Multiple crossings of Lamb modes found in [001]_c and [011]_c polarized (1−x)Pb(Mg_1/3Nb_2/3)O₃-xPbTiO₃ (PMN-xPT, x=0.33 and 0.29, respectively) plates have been studied by comparing their slowness patterns with their dispersion patterns. Multiple bandgaps and crossings of Lamb wave dispersion curves are directly related to the multivalued curves in region II, and a pair of complex conjugate b² values in region I of their slowness curves. An expression composed of elastic and piezoelectric constants has been derived to judge whether S₀ and A₀ modes cross with each other. This expression can be greatly simplified for non-piezoelectric materials, and there are no crossings of Lamb modes for isotropic materials.     

Experimental study of the A0 and S0 Lamb waves interaction with symmetrical notches

The aim of this work is to study the fundamental Lamb modes interaction with defects in isotropic plates. For these experimental investigations, symmetrical notches with various depths milled in aluminum plates are considered. Moreover, the incident Lamb wave of a specific mode is generated by means of two identical thin piezoceramic transducers placed at the opposite sides of the plate. The waves scattered by the notch are recorded with conventional transducers located on the plate surface in front and behind the defect. The selection of the A₀ or the S₀ modes is obtained by exciting the transducers with anti-phased or in-phased signals, respectively. Furthermore, a calibration process is investigated to correct errors caused by the presence of the receiver between the emitters and the defects. The power reflection and transmission coefficients are then obtained and the power balance is verified. Finally, these measurements are compared successfully with those obtained by a numerical method using the finite-element modeling described in a previous work.     

Wave vector conservation in the first order resonant Raman scattering

The dependence of the shape and the intensity of the first order resonant Raman line 1LO on the direction and magnitude of phonon wave vector has been investigated in CdS crystals. Comparison of RRS in different orientations shows that in the most pure samples the phonon wave vector is determined, to a great extent, by the momentum conservation law. In Ni doped crystals one can observe violation of momentum conservation, resulting in a sharp increase of 1 LO intensity. The shape of the 1 LO line in such sample does not depend on the experimental orientation. For the first time the dependence of the Raman intensity on the scattering angle has been observed by the comparison of forward and backward scattering spectra. Observation of this dependence shows that the free excitons are the dominating intermediate states in the resonant Raman scattering in A₂B₆ compounds.     

c向切割LiNbO3∶Fe晶体中光折变光散射

提出了一个新的理论机理:c向切割LiNbO₃∶Fe晶体中光折变背向光散射是由光生伏特电流的不均匀性引起的,并经过理论计算,比较成功的解释了实验观察到的各种现象 关键词： 光折变 散射 光生伏特     

Theoretical investigation of parity violation in pα scattering

Calculations are presented of the parity-non-conserving (PNC) analyzing power A_z for longitudinally polarized incoming protons in elastic $\text{p}\text{α}$ scattering below ～50 MeV. The results are given in terms of the PNC as well as the regular, parity-conserving (PC) meson-nucleon coupling constants for single π-, ρ- and ω-exchange. The reliability and parameter dependence of the calculations are discussed in detail. The PC on-shell scattering parameters play an important role in particular for the angular dependence of A_z, which must be known for the actual analysis of measurements and are taken from experiments directly (phase analysis) rather than model calculations. The PNC scattering amplitudes are calculated in DWBA with optical-model wave functions properly antisymmetrized. Short-range correlations between nucleon pairs are taken into account by a Jastrow factor whose appropriate choice is discussed in detail. While absorption from the elastic channel is taken into account by the optical-model wave function, intermediate breakup channels are not included explicitly in the matrix elements.     

Exact solution of a model for πd induced reactions and its application to pion-nucleus reactions

Reduction techniques are applied to πd elastic scattering and π-absorption in a theory without anti-nucleons. In a one-pion approximation we derive two sets of exact coupled-channel equations for respectively the amplitudes T_πd,NN, T_{Nδ, NN} and T_{πd, πd}, T_{d, NΔ}- Alternatively we express in terms of the absorption amplitude T_{πd, NN} available solutions for a three-body problem restricted to the πd and NΔ channels. We explicitly demonstrate that our model (which comes close to the one of Thomas, Mizutani and Koltun) strictly respects the Pauli principle and avoids double-counting. Using the same technique we determine amplitudes for the (π, 2N) reaction and for π (in)elastic scattering on general nuclei in terms of amplitudes amongst the NN, NNπ channels. Both the elastic amplitude and the πA optical potential are shown to decompose into a multiple scattering part based on an input πN amplitude without the P₁₁ partial wave and calculable absorption corrections.     

Analysis of the methods proposed for computing the scattering of atoms from a hard corrugated surface model of He/LiF (001)

It appears that the scattering of atoms on crystal surfaces is accounted for by a corrugated infinite wall model of the surface. To determine the “corrugation function” which represents best the experimental scattering intensities, it is necessary to choose among many possible versions, the simplest and most effective computational method of the theoretical scattering amplitudes. It is the purpose of this paper to analyze the relative merits of three particular versions which we call HDM, NHM and GR, and which are described in section 3. This comparison is carried out on the He/LiF(001) system for variable incident wave vector (k) and orientation (θ_i) as well as a number of Fourier components (ζ_G) used to describe the “corrugation function”. We conclude that the GR method presents advantages over the other two: (i) it reduces the computer time necessary for the HDM and NHM methods, (ii) converges and computes to within 10^?4 in unitarity the scattering amplitudes in a larger range of energies as well as angular and corrugation parameters, and (iii) it presents a great simplicity and the corrugation is very handy. The size of that range depends on the computer facilities used. In our case, the GR method works for any calculation in which kC₀ < 9 and $\text{C}{}_0\text{< 0.35}\text{A}\text{?}$ (where k is the incident wave vector and C₀ twice the amplitude of the corrugation). These characteristics prove to make the GR method a good tool for the study of surface crystallography by parametrization of the surface corrugation.     

Fredholm approximants for potential scattering

An approximate method based on Fredholm approximants is used to solve Lippmann-Schwinger equation fors-andp-wave Yukawa ands-wave exponential potential scattering problems. The method is then applied to a problem ofπ-π scattering i nvolving the use of a well known equivalent potential. The equation for partial wave amplitudes is solved to generate the first threeπ-π resonances,ρ, f ₀, andg, in a self-consistent manner.     

An experimental study on the behavior of Ao mode transduced using misaligned air-coupled transducers in composite laminates

Air-coupled transducers can be employed for generation and reception of Lamb waves in composite laminates. To generate and receive a particular Lamb mode, the transducers are oriented in a particular angle. The orientations of transducers also determine amplitude of a particular Lamb mode of interest. Any deviation (misalignment) from the right orientation results in a reduction in the amplitude. Therefore, an attempt has been made to establish variation in amplitude of the fundamental anti-symmetric Lamb mode (A_o) when misalignments were purposefully introduced in the air-coupled probes. In the present work, three different misalignments – linear, orientation and synchronized orientation were envisaged. Experiments were performed on two laminates – unidirectional ([0₃]_s) and cross-ply ([0/90₂/0]_s) of GFRP (Glass Fiber Reinforced Plastic) material. Curves were fitted on experimental data to characterize variation in amplitude of A_o mode in each misalignment. Moreover, wave splitting and merging was observed when the probe(s) was (were) rotated in a particular direction in orientation and synchronized orientation misalignments.     

Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms

The two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms is described in detail. This method provides a simple procedure for deriving formulas for the energy shift of a single level and for the energies and wave functions of degenerate and quasi-degenerate states. It also allows one to derive formulas for the transition and scattering amplitudes. Application of the method to resonance scattering processes yields a systematic theory for the spectral line shape. The practical ability of the method is demonstrated by deriving formulas for the QED and interelectronic-interaction corrections to energy levels and transition and scattering amplitudes in one-, two-, and three-electron atoms. Numerical calculations of the Lamb shift, the hyperfine splitting, the bound-electron g factor, and the radiative recombination cross section in heavy ions are also reviewed.