On the Rayleigh Wave Speed in Orthotropic Elastic Solids

Vinh, Pham Chi; Ogden, Ray W.

doi:10.1007/s11012-005-1603-6

Cited by 43 publications

(41 citation statements)

References 10 publications

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“…The benefits of using a Rayleigh wave are as follows: only one sample is required, the same S-wave transducer used on the bulk can be used to measure the surface wave, and the propagation direction is aligned with one of the symmetry axes (and thus the phase and group velocities are equal) removing any uncertainty as to which velocity is measured. The relationship between the elastic constants of an orthotropic medium and the Rayleigh-wave velocity has been recently obtained by Vinh and Ogden (2004a). The Rayleigh velocity, V ray , in an orthotropic medium is expressed as…”

Section: Rayleigh-wave Methodsmentioning

confidence: 99%

“…This solution was later analyzed by several authors to develop an even simpler approximate expression for the velocity in isotropic and anisotropic media using various mathematical techniques (Vinh and Ogden, 2004b;Li, 2006;Rahman and Michelitsch, 2006;Nkemzi, 2008). Among the explicit solutions derived for the Rayleigh-wave velocity, Vinh and Ogden (2004a) obtained a simple expression for the Rayleigh velocity in orthotropic media as a function of the elastic constants (see equation 3.28 in Vinh and Ogden, 2004a). The expression is easily extended from orthotropic to higher symmetries following the formulation of Chadwick (1976).…”

Section: Introductionmentioning

confidence: 99%

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Measurements of elastic constants in anisotropic media

2014

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Simulation of elastic-wave propagation in rock requires knowledge of the elastic constants of the medium. The number of elastic constants required to describe a rock depends on the symmetry class. For example, isotropic symmetry requires only two elastic constants, whereas transversely isotropic symmetry requires five unique elastic constants. The off-diagonal elastic constant depends on a wave velocity measured along a nonsymmetry axis. The most difficult barrier when measuring these elastic constants is the ambiguity between the phase and group velocity in experimental measurements. Several methods to eliminate this difficulty have been previously proposed, but they typically require several samples, difficult machining, or complicated computational analysis. Another approach is to use the surface (Rayleigh) wave velocity to obtain the off-diagonal elastic constant. Rayleigh waves propagated along symmetry axes have phase and group velocities that are equal for materials with no frequency dispersion, thereby eliminating the ambiguity. Using a theoretical secular equation that relates the Rayleigh velocity to the elastic constants enable determination of the offdiagonal elastic constant. Laboratory measurements of the elastic constants in isotropic and anisotropic materials were made using ultrasonic transducers (central frequency of 1 MHz) for the Rayleigh-wave method and a wavefront-imaging method. The two methods indicated agreement within 1% and 3% for isotropic and transversely isotropic samples, respectively, demonstrating the ability of the Rayleigh-wave method to measure the off-diagonal elastic constant. The surface-wave approach eliminates the need for multiple samples, expensive computational calculations, and most importantly, it removes the ambiguity between the phase and group velocity in the measured data for materials with no frequency dispersion because all measurements are made along symmetry axes.

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Section: Rayleigh-wave Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Measurements of elastic constants in anisotropic media

2014

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“…By using Equations (17) and (18) in which f() ¼ x(), x() defined by Equations (7)-(9) and a ¼ 0, b ¼ 0.5, we obtain:…”

Section: Improved Nesvijski's Approximationsmentioning

confidence: 99%

“…By using Equations (17) and (18) in which f() ¼ s(), a ¼ 0, b ¼ 0.5 we have: c 0 ¼1:09158536649509, c 1 ¼ À0:04865048166307, c 2 ¼ 0:00389293077594,…”

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confidence: 99%

Improved Approximations of the Rayleigh Wave Velocity

Vinh

Malischewsky

2008

Journal of Thermoplastic Composite Materials

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In this article we have derived some approximations for the Rayleigh wave velocity in isotropic elastic solids which are much more accurate than the ones of the same form, previously proposed. In particular: (1) A second (third)-order polynomial approximation has been found whose maximum percentage error is 29 (19) times smaller than that of the approximate polynomial of the second (third) order proposed recently by Nesvijski [Nesvijski, E. G., J. Thermoplas. Compos. Mat. 14 (2001), 356-364].(2) Especially, a fourth-order polynomial approximation has been obtained, the maximum percentage error of which is 8461 (1134) times smaller than that of Nesvijski's second (third)-order polynomial approximation. (3) For Brekhovskikh-Godin's approximation [Brekhovskikh, L. M., Godin, O. A. 1990, Acoustics of Layered Media: Plane and Quasi-Plane Waves. Springer-Verlag, Berlin], we have created an improved approximation whose maximum percentage error decreases 313 times. (4) For Sinclair's approximation [Malischewsky, P. G., Nanotechnology 16 (2005), 995-996], we have established improved approximations which are 4 times, 6.9 times and 88 times better than it in the sense of maximum percentage error. In order to find these approximations the method of least squares is employed and the obtained approximations are the best ones in the space L 2 [0, 0.5] with respect to its corresponding subsets.KEY WORDS: Rayleigh wave velocity, the best approximation, method of least squares.

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“…While a large number of formulas for the Rayleigh wave velocity have been derived, see for examples, [6][7][8][9][10][11][12][13][14][15][16], only few formulas for the Rayleigh wave H/V ratio have been obtained. They are, for example, the exact Rayleigh wave H/V ratio formula for a compressible layered half-space with traction-free surface [2], the exact and an approximate formula for that model of incompressible media [17].…”

Section: Introductionmentioning

confidence: 99%

Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

2017

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This paper is concerned with the propagation of Rayleigh waves in an incompressible orthotropic elastic half-space coated with a thin incompressible orthotropic elastic layer. The main purpose of the paper is to establish an approximate formula for the Rayleigh wave H/V ratio (the ratio between the amplitudes of the horizontal and vertical displacements of Rayleigh waves at the traction-free surface of the layer). First, the relations between the traction amplitude vector and the displacement amplitude vector of Rayleigh waves at two sides of the interface between the layer and the half-space are created using the Stroh formalism and the effective boundary condition method. Then, an approximate formula for the Rayleigh wave H/V ratio of third-order in terms of dimensionless thickness of the layer has been derived by using these relations along with the Taylor expansion of the displacement amplitude vector of the thin layer at its traction-free surface. It is shown numerically that the obtained formula is a good approximate one. It can be used for extracting mechanical properties of thin films from measured values of the Rayleigh wave H/V ratio.Keywords: Rayleigh waves, the Rayleigh wave H/V ratio, incompressible orthotropic elastic half-space, thin incompressible orthotropic elastic layer, approximate formula for the Rayleigh wave H/V ratio.

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On the Rayleigh Wave Speed in Orthotropic Elastic Solids

Cited by 43 publications

References 10 publications

Measurements of elastic constants in anisotropic media

Measurements of elastic constants in anisotropic media

Improved Approximations of the Rayleigh Wave Velocity

Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

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