Measurements of elastic constants in anisotropic media

Abstract

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.

A comparative study of the anisotropic dynamic and static elastic moduli of unconventional reservoir shales: Implication for geomechanical investigations

“…However, this is not the case at oblique angles. This has been overcome by use of transducer arrays to allow for unambiguous determination of phase speeds (e.g., Mah and Schmitt, 2003), by exploiting complementary Rayleigh wave modes (Abell and Pyrak-Nolte, 2013;Abell et al, 2014), by assuming that group speeds are what is being measured and developing appropriate relations usually with a minimization procedure (Every and Sachse, 1990;Cheadle et al, 1991;Jakobsen and Johansen, 2000;Dewhurst and Siggins, 2006;Sarout and Guéguen, 2008a), or, most commonly, by assuming that the sample and transducer geometries are appropriate for the direct determination of phase speed (Vernik and Liu, 1997;Hornby, 1998) with small errors estimated to be less than 1% judged as acceptable. We follow these last workers approach here, but note that proper analysis of this problem will require a full modeling of the beam propagation that must include the geometry of the transmitting and receiving piezoelectrics using beam propagation (Bouzidi and Schmitt, 2006) or Kirchoff-type summation of point sources over the transmitter aperture (e.g., Dellinger and Vernik, 1994).…”

Aggregate Elasticity, Crystal Structure, and Tableting Performance for p-Aminobenzoic Acid and a Series of Its Benzoate Esters

“…Multiple techniques are available for determining the elasticity tensor for single crystals: e.g., pulse-echo ultrasound, resonant ultrasound spectroscopy (RUS), and Brillouin light scattering (BLS). − Both pulse-echo ultrasound and RUS require large, single-crystal samples that are manufactured to a specific geometry. However, crystal growth at the millimeter scale in all dimensions is challenging and limits application of these techniques .…”

FEA-Based Ultrasonic Focusing Method in Anisotropic Media for Phased Array Systems

“…Thus, for practical applications of phased array systems in welds, information on grain orientation distribution, anisotropic material properties, and precise simulation of ultrasonic wave propagation behavior are required. As a result, many studies have been conducted to determine the distribution of grain orientation [1,3,5] and the elastic constants [6][7][8] in austenitic welds. In addition, several studies have applied ultrasonic array data to NDEs for austenitic welds [3,6,[9][10][11][12][13][14][15][16][17].…”