Voigt, Reuss, Hill, and Self-Consistent Techniques for Modeling Ultrasonic Scattering

An elastic wave propagating in a metal loses a portion of its energy from scattering caused by acoustic impedance differences existing at the boundaries of anisotropic grains. Theoretical scattering models capture this phenomena by assuming the incoming wave is described by an average elastic moduli tensorC0 i jkl (x) that is perturbed by a grain with elasticityCi jkl (x ′) where the scattering event occurs when x= x′. Previous models have assumed thatC0 i jkl (x) is the Voigt average of the singlecrystal elastic moduli tensor. However, this assumption may be incorrect because the Voigt average overestimates the wave’s phase velocity. Thus, the use of alternate definitions ofC0 i jkl (x) to describe the incoming wave is posed. Voigt, Reuss, Hill, and self-consistent definitions of C0 i jkl (x) are derived in the context of ultrasonic scattering models. The scattering-based models describing ultrasonic backscatter, attenuation, and diffusion are shown to be highly dependent on the definition ofC0 i jkl (x).