Finite Amplitude Wave Propagation in Anisotropic Materials

Often, ultrasound used in nondestructive evaluation is applied to materials that are elastically anisotropic. A few example materials include composites, welds, and rolled metal plates. The influence of elastic anisotropy on the propagation of ultrasound in materials that are linearly elastic is well understood. For example, elastic constants of a composite can be determined reasonably well by measuring phase velocities for propagation in certain directions. However, the influence of elastic anisotropy on nonlinear ultrasonic techniques has received much less attention. In this work, finite amplitude bulk wave propagation is considered for materials with general elastic anisotropy of the second-, third-, and fourth-order elastic constants (anisotropy associated with triclinic symmetry). Three displacement solutions are obtained for arbitrary propagation directions of the three possible bulk wave modes (one quasi-longitudinal and two quasi-transverse). The solution corresponding to each wave mode is a harmonic series having contributions from the fundamental, second-, and third-harmonic waves. The second-harmonic wave amplitude is a function of the quadratic (β) nonlinearity parameter while the thirdharmonic amplitude is a function of both the quadratic and cubic () nonlinearity parameters. β is given in terms of displacement and propagation directions along with elastic tensors that define the second- and third-order elastic constants of the material. An additional contribution from the elastic tensor defining the fourth-order elastic constants is needed to define . Closed-form evaluation of β and for the three different wave modes has been conducted for a variety of materials having different crystallographic point group symmetries. Surfaces will be presented for selected materials, which illustrate the three-dimensional spatial distribution of β and for any propagation direction of the fundamental wave. The vanishing of β for shear waves propagating within material planes of symmetry causes the surface to display the symmetry of the material elegantly. Lastly, straightforward expressions for β and are given for some pure mode directions

Acoustoelasticity of Polycrystalline Materials; a Formalism based on the Self-Consistent Elastic Constants

Elastic constants of polycrystalline materials can be obtained through methods of ensemble averages of the elastic constants belonging to individual grains. Assumptions are often made to relate the local strains (stresses) within individual grains as a result of a macroscopic strain (stress) on the polycrystal. The different assumptions lead to different estimates for the elastic constants of polycrystals. However, an exact formulation is possible, which enforces continuity (at the grain boundaries) between the macroscopic strain (stress) and the strain (stress) in the grain. The resulting estimates of the polycrystal’s elastic constants are known as selfconsistent because either a stress or strain formalism leads to the same estimates. This presentation extends the idea of macroscopic and local continuity of stress and strain and applies it to the theory of acoustoelasticity. Acoustoelasticity describes the dependence of the properties of an elastic wave on the stress state in the material supporting the wave. The selfconsistent formalism enters the elastic constitutive relation developed by C.-S. Man and coworkers. Such a constitutive relation is a function of initial stress, which can be either residual stress resulting from a series of inhomogeneous plastic deformations or generated from external mechanisms. The constitutive relation is used to derive the stress-dependent Christoffel equations for the polycrystal. Solutions to the Christoffel equation yield expressions for the phase velocities and displacement directions of elastic waves in a stressed polycrystal. A comparison is made between phase velocity values based on ensemble averaging originating from the self-consistent formalism and the phase velocities arriving from previous models. The cases in which the present model shows considerable differences from the previous models are presented. This overall goal of this work is to provide a better understanding of the influence of polycrystalline microstructure on acoustoelasticit

Mode-converted ultrasonic scattering in polycrystals with elongated grains

Elastic wave scattering is used to study polycrystalline media for a wide range of applications. Received signals, which include scattering from the randomly oriented grains comprising the polycrystal, contain information from which useful microstructural parameters may often be inferred. Recently, a mode-converted diffuse ultrasonic scattering model was developed for evaluating the scattered response of a transverse wave from an incident longitudinal wave in a polycrystalline medium containing equiaxed single-phase grains with cubic elastic symmetry. In this article, that theoretical mode-converted scattering model is modified to account for grain elongation within the sample. The model shows the dependence on scattering angle relative to the grain axis orientation. Experimental measurements were performed on a sample of 7475-T7351 aluminum using a pitch-catch transducer configuration. The results show that the mode-converted scattering can be used to determine the dimensions of the elongated grains. The average grain shape determined from the experimental measurements is compared with dimensions extracted from electron backscatter diffraction, an electron imaging technique. The results suggest that mode-converted diffuse ultrasonic scattering has the potential to quantify detailed information about grain microstructure

Mode-converted ultrasonic scattering in polycrystals with elongated grains

Elastic wave scattering is used to study polycrystalline media for a wide range of applications. Received signals, which include scattering from the randomly oriented grains comprising the polycrystal, contain information from which useful microstructural parameters may often be inferred. Recently, a mode-converted diffuse ultrasonic scattering model was developed for evaluating the scattered response of a transverse wave from an incident longitudinal wave in a polycrystalline medium containing equiaxed single-phase grains with cubic elastic symmetry. In this article, that theoretical mode-converted scattering model is modified to account for grain elongation within the sample. The model shows the dependence on scattering angle relative to the grain axis orientation. Experimental measurements were performed on a sample of 7475-T7351 aluminum using a pitch-catch transducer configuration. The results show that the mode-converted scattering can be used to determine the dimensions of the elongated grains. The average grain shape determined from the experimental measurements is compared with dimensions extracted from electron backscatter diffraction, an electron imaging technique. The results suggest that mode-converted diffuse ultrasonic scattering has the potential to quantify detailed information about grain microstructure

Generalized ultrasonic scattering model for arbitrary transducer configurations

Ultrasonic scattering in polycrystalline media is directly tied to microstructural features. As a result, modeling efforts of scattering from microstructure have been abundant. The inclusion of beam modeling for the ultrasonic transducers greatly simplified the ability to perform quantitative, fully calibrated experiments. In this article, a theoretical scattering model is generalized to allow for arbitrary source and receiver configurations, while accounting for beam behavior through the total propagation path. This extension elucidates the importance and potential of out-of-plane scattering modes in the context of microstructure characterization. The scattering coefficient is explicitly written for the case of statistical isotropy and ellipsoidal grain elongation, with a direct path toward expansion for increased microstructural complexity. Materials with crystallites of any symmetry can be studied with the present model; the numerical results focus on aluminum, titanium, and iron. The amplitude of the scattering response is seen to vary across materials, and to have varying sensitivity to grain elongation and orientation depending on the transducer configuration selected. The model provides a pathway to experimental characterization of microstructure with optimized sensitivity to parameters of interest

Measuring the Neutrino Cross Section Using 8 years of Upgoing Muon Neutrinos Observed with IceCube

The IceCube Neutrino Observatory detects neutrinos at energies orders of magnitude higher than those available to current accelerators. Above 40 TeV, neutrinos traveling through the Earth will be absorbed as they interact via charged current interactions with nuclei, creating a deficit of Earth-crossing neutrinos detected at IceCube. The previous published results showed the cross section to be consistent with Standard Model predictions for 1 year of IceCube data. We present a new analysis that uses 8 years of IceCube data to fit the ν$_{μ}$ absorption in the Earth, with statistics an order of magnitude better than previous analyses, and with an improved treatment of systematic uncertainties. It will measure the cross section in three energy bins that span the range 1 TeV to 100 PeV. We will present Monte Carlo studies that demonstrate its sensitivity