Elastodynamic Green Research Articles

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Analytical solutions of peristatics and peridynamics for 3D isotropic materials

In a recent paper, Mikata (2019) has established two explicit analytical governing equations of peridynamics for 3D isotropic and orthotropic materials. In the present paper, general analytical solutions of peristatics and peridynamics for 3D isotropic peridynamic materials are obtained in infinite media for an arbitrary body force density, and arbitrary initial conditions in the case of peridynamics. In addition, peristatic and peridynamic Green's functions corresponding to a point load are also obtained. Classical limit of the peristatic and peridynamic Green's functions in the limit of the generalized material horizon δ going to 0 (δ→0) is analytically discussed in detail for one of the most commonly used nonlocality functions with a finite material horizon with a sharp cut-off. It is analytically and explicitly shown that the peristatic and peridynamic Green's functions converge to the elastostatic and elastodynamic Green's functions, respectively, as the generalized material horizon δ goes to 0 (δ→0). Numerical results are obtained and discussed for the peristatic Green's function and a 3D peristatic displacement field under a 1D-like loading. The governing equation investigated in the present paper is the first explicit analytical governing equation of peridynamics for 3D isotropic materials, which is comparable to Navier-Cauchy's equation. Thus, this equation opens up various ways to investigate peridynamics in 3D isotropic materials in an exact analytical fashion, which was not possible before.

Numerical evaluation of 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids

The paper presents a fast and accurate numerical technique for evaluation of the 3D time-harmonic elastodynamic Green's function and its derivatives for anisotropic solids. Following Wang & Achenbach the Green's function is presented in the form of the sum of singular (static) and regular parts, which are both reduced to integrals over a unit sphere. Singular part and its derivatives are then reduced to the integrals over a unit circle, which can be efficiently (fast and accurately) evaluated using the trapezoid rule, which is exponentially convergent for integrals over the period of a periodic integrand. The regular time-harmonic parts are presented through the double integrals. The outer integral is also efficiently evaluated using the trapezoid rule. A special quadrature is developed for evaluation of the inner integral, which accounts for the highly-oscillating behavior of the integrand. Numerical examples are presented, which shows the advantage of the proposed technique over others.

Elastodynamic Green’s functions of transversely isotropic n-layer half- and full-spaces subjected to a surface or buried time-harmonic annular loading and associated material degeneracy

Elastodynamic Green’s functions of transversely isotropic n-layer half- and full-spaces subjected to a surface or buried time-harmonic annular loading and associated material degeneracy

Elastic and magnetic properties of Tb-MnO based thin films

We report on the elastic and magnetic properties of multiferroic Tb0.3Mn20.9O78.8 thin films. The films were deposited on (001) Si using radio frequency magnetron (RF) sputtering at 50 W and ambient temperature. The elastic properties have been established by surface Brillouin scattering (SBS) after fitting the discrete phonon dispersion in the k||d range of 0 – 5 by surface Elastodynamic Green’s function. The uncertainties in elastic constants values were obtained from Taylor series expansion of the phonon phase velocity as a function of C11 and C44 values and subsequently fitted by the least-squares method. The optimum primary elastic constants and their uncertainties were determined as C11 = 180 ± 4.9 GPa and C44 = 43.0 ± 0.89 GPa. The derived elastic constants show that the films are ductile. Using a vibrating sample magnetometer (VSM), the films have been noted to behave like soft ferromagnetic materials at T⩽150 K. Incommensurate magnetic ordering and spin-glass behaviors have also been observed.

Surface Brillouin scattering study of tantalum nitride (TaN) thin films.

Elastic properties of TaN films on (001)Si are investigated by surface Brillouin scattering (SBS). The velocity dispersion is obtained from Brillouin frequency shifts and surface acoustic wave phase velocities derived from measured SBS spectra. The observed Rayleigh surface acoustic and Sezawa waves indicate a soft on hard configuration. The elastic stiffnesses from the best fit of the calculated and measured spectra were C11=288 and C44=80GPa. Using these elastic constants in the surface elastodynamic Green's function yielded theoretical Brillouin spectra in agreement with measured spectra. Employing a least squares fitting procedure to the elastic constants gives uncertainties in C11 and C44 as ±3.7 and ±2.5GPa, respectively.

Super-shear evanescent waves for non-contact elastography of soft tissues

We describe surface wave propagation in soft elastic media at speeds exceeding the bulk shear wave speed. By linking these waves to the elastodynamic Green's function, we derive a simple relationship to quantify the elasticity of a soft medium from the speed of this supershear evanescent wave (SEW). We experimentally probe SEW propagation in tissue-mimicking phantoms, human cornea ex vivo, and skin in vivo using a high-speed optical coherence elastography system. Measurements confirm the predicted relationship between SEW and bulk shear wave speeds, agreeing well with both theoretical and numerical models. These results suggest that SEW measurements may be a robust method to quantify elasticity in soft media, particularly in complex, bounded materials where dispersive Rayleigh-Lamb modes complicate measurements.

An Improved Technique for Elastodynamic Green's Function Computation for Transversely Isotropic Solids

Elastodynamic Green's function for anisotropic solids is required for wave propagation modeling in composites. Such modeling is needed for the interpretation of experimental results generated by ultrasonic excitation or mechanical vibration-based nondestructive evaluation tests of composite structures. For isotropic materials, the elastodynamic Green’s function can be obtained analytically. However, for anisotropic solids, numerical integration is required for the elastodynamic Green's function computation. It can be expressed as a summation of two integrals—a singular integral and a nonsingular (or regular) integral. The regular integral over the surface of a unit hemisphere needs to be evaluated numerically and is responsible for the majority of the computational time for the elastodynamic Green's function calculation. In this paper, it is shown that for transversely isotropic solids, which form a major portion of anisotropic materials, the integration domain of the regular part of the elastodynamic time-harmonic Green's function can be reduced from a hemisphere to a quarter-sphere. The analysis is performed in the frequency domain by considering time-harmonic Green's function. This improvement is then applied to a numerical example where it is shown that it nearly halves the computational time. This reduction in computational effort is important for a boundary element method and a distributed point source method whose computational efficiencies heavily depend on Green's function computational time.

Analytical and numerical modeling of rock blasting operations using a two-dimensional elasto-dynamic Green's function

Rock media may be exposed to dynamic forces and time-dependent stresses. Dynamic phenomena in rock such as blasting and earthquakes need to propagate waves and dynamic stresses through the body. Therefore, stresses, strains, and displacements in rocks should be defined as time-dependent due to the dynamic phenomena. In this study, a general Green's function solution of an elastic wave propagation due to rock blasting has been analytically derived. The Navier's equations of motion have been assumed as governing equations and a general two-dimensional elastodynamic Green's function has been obtained in terms of displacement. The strain and stress fields related to the displacement Green's function have also been obtained using the theory of elasticity. Then, a generic two-dimensional problem containing an explosion in a blast hole has been solved using the resulting analytical solution. Moreover, a similar numerical modeling has been conducted using Finite Difference Method (FDM). Eventually, results from the analytical scheme have been compared with the finite difference numerical modeling and the valuable results from a previous experimental and numerical study.

A method based on 3D stiffness matrices in Cartesian coordinates for computation of 2.5D elastodynamic Green's functions of layered half-spaces

This article elaborates on an extension to the classical stiffness matrix method to obtain the Green's functions for two-and-a-half dimensional (2.5D) elastodynamic problems in homogeneous and horizontally layered half-spaces. Exact expressions for the three-dimensional (3D) stiffness matrix method for isotropic layered media in Cartesian coordinates are used to determine the stiffness matrices for a system of horizontal layers underlain by an elastic half–space. In the absence of interfaces, virtual interfaces are considered at the positions of external loads. The analytic continuation is used to find the displacements at any receiver point placed within a layer. The responses of a horizontally layered half-space subjected to a unit harmonic load obtained using the present method are compared with those calculated using a well-established methodology, achieving good agreement.

Theory and implementation of [formula omitted]-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

In this work, we study the accuracy and efficiency of hierarchical matrix (H-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard H-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. H2-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard H-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the H-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.

Influence of flood-induced scour on dynamic impedances of pile groups considering the stress history of undrained soft clay

Flood-induced scour causes loss of lateral support at bridge foundations and thus modifies their dynamic impedances, which will in turn influence the dynamic behavior and the seismic response of bridges located in seismically active flood-prone regions. When evaluating the seismic performance of scoured bridges, however, few efforts have been made to analyze the dynamic impedances of their foundations, and possible changes in stress history of the remaining soils are also ignored. In reality, the remaining soils around pile foundations undergo an unloading process due to scour and their over-consolidation ratios are increased, which can be expected to change the properties of the remaining soils. Therefore, a three-dimensional numerical model, in which the pile-soil-pile interaction and the radiation problem are accounted for by means of the elastodynamic Green's functions that permit the mutual interactions between all the piles of the group consistently and simultaneously, is employed in this paper to investigate the influence of flood-induced scour on the dynamic impedances of pile groups in soft clay. Accordingly, the key soil parameter of the elastodynamic Green's functions, i.e., Young's modulus of soft clay, is modified in this model to take into account the stress history effect due to scour. A case study to evaluate the dynamic impedances of 3×3 and 4×4 pile groups with various scour depths and pile spacing, considering or ignoring stress history effect, is discussed in this paper. The results show that ignoring the effect of stress history of the remaining soft clay will overestimate the scour affected dynamic impedances of pile groups, which are found to be reduced significantly as the scour depth increases.

Soil-structure interaction effects in single bridge piers founded on inclined pile groups

This paper investigates the seismic response of bridge piers founded on inclined pile groups in different soil deposits, evaluating effects of soil-structure interaction induced by different pile group geometries and piles inclinations. Analyses are performed in the frequency domain by means of the direct approach taking advantage of a numerical model developed by the authors for the analysis of inclined pile groups. Both the superstructure and piles are modelled with beam elements and the soil is schematized as a visco-elastic medium constituted by independent infinite horizontal layers. The soil-pile and the pile-soil-pile interaction are captured in the frequency domain by means of elastodynamic Green's functions that also allow including the hysteretic and radiation damping. The significance of kinematic stress resultants in piles, the foundation filtering effect and the rotational component of the input motion due to the coupled roto-translational behaviour of the soil-foundation system are also investigated; to this purpose kinematic interaction analyses are performed. These analyses revealed essential for the understanding of the general phenomena governing the dynamic response of the whole soil-foundation-superstructure systems. Results of numerical investigations highlight that conventional design approaches suggested by codes do not provide reliable predictions of the superstructure displacements and stress resultants.

Green's function calculation from equipartition theorem.

A method is presented to calculate the elastodynamic Green's functions by using the equipartition principle. The imaginary parts are calculated as the average cross correlations of the displacement fields generated by the incidence of body and surface waves with amplitudes weighted by partition factors. The real part is retrieved using the Hilbert transform. The calculation of the partition factors is discussed for several geometrical configurations in two dimensional space: the full-space, a basin in a half-space and for layered media. For the last case, it results in a fast computation of the full Green's functions. Additionally, if the contribution of only selected states is desired, as for instance the surface wave part, the computation is even faster. Its use for full waveform inversion may then be advantageous.

A computer code for forward calculation and inversion of the H/V spectral ratio under the diffuse field assumption

During a quarter of a century, the main characteristics of the horizontal-to-vertical spectral ratio of ambient noise HVSRN have been extensively used for site effect assessment. In spite of the uncertainties about the optimum theoretical model to describe these observations, over the last decade several schemes for inversion of the full HVSRN curve for near surface surveying have been developed.In this work, a computer code for forward calculation of H/V spectra based on the diffuse field assumption (DFA) is presented and tested. It takes advantage of the recently stated connection between the HVSRN and the elastodynamic Green's function which arises from the ambient noise interferometry theory.The algorithm allows for (1) a natural calculation of the Green's functions imaginary parts by using suitable contour integrals in the complex wavenumber plane, and (2) separate calculation of the contributions of Rayleigh, Love, P-SV and SH waves as well. The stability of the algorithm at high frequencies is preserved by means of an adaptation of the Wang's orthonormalization method to the calculation of dispersion curves, surface-waves medium responses and contributions of body waves.This code has been combined with a variety of inversion methods to make up a powerful tool for passive seismic surveying.

A numerical model for the dynamic analysis of inclined pile groups

SummaryThe paper presents a numerical model for the dynamic analysis of pile groups with inclined piles in horizontally layered soil deposits. Piles are modelled with Euler–Bernoulli beams, while the soil is supposed to be constituted by independent infinite viscoelastic horizontal layers. The pile–soil–pile interaction as well as the hysteretic and geometric damping is taken into account by means of two‐dimensional elastodynamic Green's functions. Piles cap is considered by introducing a rigid constraint; the condensation of the problem permits a consistent derivation of both the dynamic impedance matrix of the soil–foundation system and the foundation input motion. These quantities are those used to perform inertial soil–structure interaction analyses in the framework of the substructure approach. Furthermore, the model allows evaluating the kinematic stress resultants in piles resulting from waves propagating in the soil deposit, taking into account the pile–soil–pile interactions. The model validation is carried out by performing accuracy analyses and comparing results in terms of dynamic impedance functions, kinematic response parameters and pile stress resultants, with those furnished by 3D refined finite element models. To this purpose, classical elastodynamic solutions are adopted to define the soil–pile interaction problem. The model results in low computational demands without significant loss of precision, compared with more rigorous approaches or refined finite element models. Copyright © 2015 John Wiley & Sons, Ltd.

Longitudinal shear wave and transverse dilatational wave in solids.

Dilatation wave involves compression and extension and is known as the curl-free solution of the elastodynamic equation. Shear wave on the contrary does not involve any change in volume and is the divergence-free solution. This letter seeks to examine the elastodynamic Green's function through this definition. By separating the Green's function in divergence-free and curl-free terms, it appears first that, strictly speaking, the longitudinal wave is not a pure dilatation wave and the transverse wave is neither a pure shear wave. Second, not only a longitudinal shear wave but also a transverse dilatational wave exists. These waves are shown to be a part of the solution known as coupling terms. Their special motion is carefully described and illustrated.

Elastodynamic Green's function retrieval through single-sided Marchenko inverse scattering.

The solution of the inverse scattering problem for the one-dimensional Schrödinger equation is given by the Marchenko equation. Recently, a Marchenko-type equation has been derived for three-dimensional (3D) acoustic wave fields, whose solution has been shown to recover the Green's functions from points within the medium to its exterior, using only single-sided scattered data. Here we extend this approach to 3D vectorial wave fields that satisfy the elastodynamic wave equation and recover Green's functions from points interior to an elastic, solid-state medium from purely external and one-sided measurements. The method is demonstrated in a solid-earth-like model to construct Green's functions using only subsurface sources, from earth-surface force and deformation sources and particle velocity and stress measurements.

Induced water pressure profiles due to seismic motions

About 90% of all natural earthquakes have epicenters in offshore areas and may cause damage to subsea and floating structures. These excitations can have effects on the performance of facilities installed on the seabed, like foundations, pipelines and subsea equipment for the oil industry. Several studies on this subject have been carried out to show the importance of seaquake analyses and their effects have been highlighted. In this paper the boundary element method is used to analyze the influence that some parameters, involved in this kind of problems, have on the dynamic response of marine waters under the incidence of theoretical seismic events. This method takes advantage of the elastodynamic Green's functions and, after the application of boundary conditions, a Fredholm system of integral equations is achieved and solved in frequency domain. Results in time domain are also obtained by applying a DFT algorithm. Flat and non-flat seabed configurations excited for normal and oblique incidence of P and S waves are considered. Pressure profiles through water depth are provided and synthetic seismograms of pressure are also shown. Amplifications of water pressures are emphasized.

Laser thermoelastic generation in metals above the melt threshold

An approach is presented for calculating thermoelastic generation of ultrasound in a metal plate exposed to nanosecond pulsed laser heating, sufficient to cause melting but not ablation. Detailed consideration is given to the spatial and temporal profiles of the laser pulse, penetration of the laser beam into the sample, the appearance and subsequent growth and then contraction of the melt pool, and the time dependent thermal conduction in the melt and surrounding solid throughout. The excitation of the ultrasound takes place during and shortly after the laser pulse and occurs predominantly within the thermal diffusion length of a micron or so beneath the surface. It is shown how, because of this, the output of the thermal simulations can be expressed as axially symmetric transient radial and normal surface force distributions. The epicentral displacement response to these force distributions is obtained by two methods, the one based on the elastodynamic Green's functions for plate geometry determined by the Cagniard generalized ray method and the other using a finite element numerical method. The two approaches are in very close agreement. Numerical simulations are reported on the epicentral displacement response of a 3.12 mm thick tungsten plate irradiated with a 4 ns pulsed laser beam with Gaussian spatial profile, at intensities below and above the melt threshold.