Linearized forward and inverse problems of the resonant ultrasound spectroscopy for the evaluation of thin surface layers

Guided acoustic waves in thin epitaxial films: Experiment and inverse problem solution for NiTi

Despite the fundamental and technological importance of the elastic constants, a suitable method for their full characterization in epitaxial films is missing. Here we show that transient grating spectroscopy (TGS) with highly k-vector-selective generation and detection of acoustic waves is capable of determination of all independent elastic coefficients of an epitaxial thin film grown on a single-crystalline substrate. This experimental setup enables detection of various types of guided acoustic waves and evaluation of the directional dependence of their speeds of propagation. For the studied model system, which is a 3μm thin epitaxial film of the NiTi shape memory alloy on an MgO substrate, the TGS angular maps include Rayleigh-type surface acoustic waves as well as Sezawa-type and Love-type modes, delivering rich information on the elastic response of the film under different straining modes. The resulting inverse problem, which means the calculation of the elastic constants from the TGS maps, is subsequently solved using the Ritz-Rayleigh numerical method. Using this approach, tetragonal elastic constants of the NiTi film and their changes with the austenite→martensite phase transition are analyzed.

A new elastic characterization method for anisotropic bilayer specimens via Bayesian resonant ultrasound spectroscopy

A novel nondestructive method for complete elastic characterization of substrate-coating bilayer specimens with distinct anisotropic layers via resonant ultrasound spectroscopy (RUS) and Bayesian inversion is developed here. Bayesian formulations of the RUS inversion problem-of quantifying elastic properties given a measured list of resonance frequencies recorded from a single, typically small, precisely fabricated, macroscopically homogeneous, linear-elastic specimen-are a recent development. Here we report the first Bayesian formulation of the bilayer problem, and through a series of practical examples, demonstrate novel parameter estimation capabilities of our open-source CmdStan-RUS code. Finding specimen geometry and the number of resonance modes used for inversion strongly govern the ability to retrieve individual elastic moduli. The concept of "invertability" is explored for a range of relevant geometries using virtual specimens that resemble experimental bilayers of plasma sprayed ceramic coatings on single crystal metallic substrates. A range of Bayesian posterior evaluation methods are addressed, particularly considering the large computational cost of the bilayer forward model. Laplace approximation methods are thus developed and implemented for bilayer geometry design space modeling and expedient estimates of parameter uncertainties. Ideal specimen design, different noise models, the influence of prior distributions, dual-likelihood fits incorporating measurements of the bare substrate, and how Bayesian RUS methods differ from traditional RUS optimization are discussed.

Forward and inverse problems for surface acoustic waves in anisotropic media: a Ritz-Rayleigh method based approach

We show that the Ritz-Rayleigh method can be used for calculation of velocity of surface acoustic waves (SAWs) propagating in a general direction of an anisotropic medium of arbitrary symmetry class. The main advantage of this method is that expanding the displacement field of SAW into a fixed functional basis transforms the calculation of SAW velocities into a simple linear eigenvalue problem. The correctness and reliability of the proposed approach are verified on experimental SAW data obtained for generally oriented planes of an indium phosphide single crystal. The same experimental datasets are then used to discuss the invertibility of the method, i.e. the possibility of determination of elastic coefficients from SAW measurements in general directions. It is shown that the SAW data obtained on a single generally oriented plane are sufficient for such an inverse calculation for a cubic material only if they are complemented by measurements of velocities of bulk quasi-longitudinal (qL) waves propagating along the same free surface. Moreover, when the SAW and qL data are available from three almost perpendicular faces of a single specimen, the complete elastic tensor (21 independent constants) can be inversely determined, without considering a priori any symmetry constraints to the material.

Sensitivity of the resonant ultrasound spectroscopy to weak gradients of elastic properties

The applicability of resonant ultrasound spectroscopy on materials with weak spatial gradients in elastic coefficients and density is analyzed. It is shown that such gradients do not affect measurably the resonant spectrum but have a significant impact on the modal shapes. A numerical inverse procedure is proposed to explore the possibility of reconstructing the gradients from experimentally obtained modal shapes. This procedure is tested on synthetic data and applied to determine the gradient of the shear modulus in a continuously graded silicon nitride ceramic material. The results are in a good agreement with the gradient calculated for the examined material theoretically as well as with the results of other experimental methods.