Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods
Highlights
► Full-wave numerical solution at the surface of arbitrary anisotropic media. ► Focus on media of cubic and hexagonal symmetries. ► Simulations in heterogeneous media with large impedance variations at the interface.
Introduction
The problem of surface acoustic wave (SAW) propagation in anisotropic media has been studied for many decades. Nevertheless, anisotropy induces great difficulties in analytically and explicitly studying wave propagation because the anisotropic behavior of the medium considerably modifies the existence and the structure of the SAW that propagates at the free surface of the medium (see a companion paper [1] for a detailed review). Few problems in elastodynamics have a closed-form analytical solution and some can be investigated with semi-analytical methods, but often one cannot be sure if these methods give reliable solutions. Being able to accurately simulate wave propagation numerically is therefore essential in a wide range of fields, including ultrasonics, earthquake seismology and seismic prospecting. The emergence of ultrasonic techniques for nondestructive evaluation has provided a strong impulse to the study of wave propagation and its numerical simulation [2], [3], [4], [5], [6], [7], [8]. Ultrasonic theory and numerical modeling is applied to the detection of flaws and micro-cracks, inhomogeneous stress field evaluation, and the characterization of effective mechanical properties of fibers and composites with imperfect interface bonding. These systems generally possess anisotropic properties, described, in their most general form, by 21 elastic coefficients and by the mass density of the material. Numerical simulations therefore become an attractive method to describe the propagation of SAWs generated by a point source at a free surface that can be different from a symmetry plane of a given anisotropic medium and for which no analytical solution can be derived.
In the following sections we use two full-wave numerical methods to solve the problem without any approximation regarding the type of symmetry nor the orientation of the free surface. The methods are highly accurate because they are based on spectral representations of the wave field. We present some examples in hexagonal and cubic media, validation benchmarks against the analytical solution in known cases, and snapshots of propagation in more complex heterogeneous media.
Section snippets
Equation of motion
In a heterogeneous elastic, anisotropic medium, the linear wave equation may be written aswhere u denotes the displacement vector, σ the symmetric, second-order stress tensor, ε the symmetric, second-order strain tensor, C the fourth-order stiffness tensor, ρ the density, and f an external source force. A dot over a symbol denotes time differentiation, a colon denotes the tensor product, and a superscript ⊤ denotes the transpose.
In the case of a fully anisotropic
Time-domain modeling methods
We propose algorithms to simulate surface waves in a material with arbitrary symmetry. The computations are based on two different numerical techniques, namely, the Fourier-Chebyshev pseudospectral method (PSM) [13], [14], [12] and the spectral finite-element method (SEM) [15], [16], [17], [18], [19], [20]. The first is based on global differential operators in which the field is expanded in terms of Fourier and Chebyshev polynomials, while the second is an extension of the finite-element
Numerical simulations
We consider the materials whose properties are given in Table 1, and which are dissimilar: apatite, beryllium and zinc have hexagonal symmetry and copper has cubic symmetry, with c22 = c11, while epoxy is isotropic.
The pseudospectral method uses a mesh composed of 81 grid points along the three Cartesian directions, with a constant grid spacing of 2.5 mm along the x- and y-directions and a total mesh size of 20 cm in the z-direction with varying grid spacing. The surface of the sample is the (x, y, z =
Conclusions
The two numerical modeling methods compute the full wave field and have spectral accuracy. At each grid point these methods allow us to model an anisotropic medium of arbitrary crystal symmetry, i.,e., a triclinic medium or a medium of lower symmetry whose symmetry axes can be rotated by any angle. We have shown numerical examples for media of hexagonal or cubic symmetry, for which we obtained time histories and snapshots at the surface and at vertical sections. The wavefronts have been
Acknowledgements
The authors thank Arthur G. Every for fruitful discussion.
Some of the calculations were performed on an SGI cluster at Centre Informatique National de l’Enseignement Suprieur (CINES) in Montpellier, France. This material is based in part upon research supported by European FP6 Marie Curie International Reintegration Grant MIRG-CT-2005-017461.
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Also at: Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France. Formerly at: Université de Pau et des Pays de l’Adour, CNRS and INRIA, Laboratoire de Modélisation et d’Imagerie en Géosciences (UMR 5212) and IPRA, Avenue de l’Université, 64013 Pau Cedex, France.