Asymptotics of an Ultrasonic Sounding Field in Anisotropic Materials

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Abstract

To model the wave field of an ultrasonic transducer in materials with strong anisotropy (monocrystalline alloys of turbine blades, composite materials, welded joints, etc.), a physically descriptive asymptotic representation is obtained for quasi-spherical body waves excited by a surface source in an arbitrarily anisotropic elastic half-space. The asymptotics is derived by the stationary phase method from the integral representation of the solution in terms of contour integrals of the inverse Fourier transform. The peculiarities of their derivation and numerical implementation are discussed on the examples of a transversely isotropic composite material and a monocrystalline nickel alloy with cubic anisotropy. The dependence of the stationary points on the direction is more complicated here than in the isotropic case, up to the appearance of multiple stationary points and folds, giving rise to additional wave fronts and caustics. A comparison is made with the plane waves described by eigensolutions of the classical Christoffel equation. It is shown that, despite the phenomenon of multiple wave fronts, varying the plane-wave orientation allows us to obtain the same group velocity vectors as for any of the waves described by the asymptotics.

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About the authors

Evgeny V. Glushkov

Kuban State University

Author for correspondence.
Email: evg@math.kubsu.ru
Russian Federation, 149, Stavropolskaya St., Krasnodar, 350040

Natalia V. Glushkova

Kuban State University

Email: nvg@math.kubsu.ru
Russian Federation, 149, Stavropolskaya St., Krasnodar, 350040

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2. Fig. 1. Geometry of the problem.

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3. Fig. 2. Schematic illustration of the dependence of the position of stationary points on the angle θ in the isotropic (a) and anisotropic (b) cases.

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4. Fig. 3. Surfaces σ^n (top) and their projections on the (β1, β2) plane (bottom) for the GE transversal-isotropic composite material.

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5. Fig. 4. The same as in Fig. 3, but for nickel alloy NS with cubic anisotropy.

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6. Fig. 5. Motion of stationary points in regions D2 and D3 when the angle θ changes from 0 (large markers) to 0.45π; φ = 0, NS material.

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7. Fig. 6. Dependence of the group velocity vn of quasi-spherical waves (10) on θ at φ = 0, NS material; marker-circles show the values plotted below in Fig. 7d.

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8. Fig. 7. Dependence of the phase velocity cn (a), group velocity vn (b), and guiding angle θV of the group velocity vector of plane waves on the front orientation angle θ at φ = 0 (c); group velocities vn as a function of θV (d); marker-circles show those taken from Fig. 7. 6 group velocities of quasi-spherical waves (10) for three directions θ / π = 0, 0.2 and 0.4, φ = 0; material NS.

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9. Fig. 8. Directional diagrams (dependences of |anm| on θ at φ = 0) for quasi-spherical waves excited by vertical (top, a-b) and tangential (bottom, d-e) concentrated loads.

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