Statistical Moments of Coverage Maps of Ultrasonic Waves in Layered Polycrystalline Solids

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• Tony Mulholland (University of Bristol)
• Thursday 23 March 2023, 11:30-12:00
• Seminar Room 1, Newton Institute.

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MWSW02 - Theory of wave scattering in complex and random media

This talks considers the propagation of high frequency elastic waves (ultrasound waves) in a polycrystalline material (titanium alloys used in high value manufacturing such as the aerospace industry).  In this high frequency regime, we assume that the wave ’sees’ the complex media as a series of locally anisotropic layers with varying thicknesses, where the distribution of layer thicknesses and orientations follow a stochastic (Markovian) process [1].  The reflected wave subsequently has a complex structure and therefore a homogenisation or effective medium approach is not appropriate.  Indeed this complex wave has encoded within it the journey that the wave has undertaken and hence opens up tomography opportunities via full wave inversion.  In ultrasonic non-destructive testing it is of value to be able to model the attenuation within such media and comment on the uncertainty in doing so.  This motivates this study where a probabilistic framework is adopted and wherein a probability density function of the reflected/transmitted wave is determined [2].  At its heart is a set of stochastic differential equations which describe the statistics of the energy in the system.  The material properties are captured within this system by a correlation integral which encapsulates the coupling of length-scales between the random media and the probing wave. Using experimentally obtained EBSD (electron backscatter diffraction) data for an austenitic steel weld, and subsequent processing of the data via a ray based probing technique, we show to calculate this correlation integral [3].  A diffusion approximation was then used to obtain the statistical moments of the transmission/reflection coefficients.  A Ricatti equation for the reflection coefficient was derived, allowing a study into the frequency autocorrelation function for the reflected wave.  Using this solution to the transport equations an expression was obtained for the reflected intensity of the wave at each element of an ultrasonic transducer array on the surface of the material.  This was then used to create coverage maps via the Total Focusing Method. [1] J. P. Fouque, J. Garnier, G. Papanicolaou, and K. Sølna. Wave Propagation and Time Reversal in Randomly Layered Media. New York, Springer, (2007). [2] J. Garnier and K. Sølna. “Apparent Attenuation of ShearWaves Propagating through a Randomly Anisotropic Medium”. In: Stochastics and Dynamics, No. 4. Vol. 16 (2015). [3] A. S. Ferguson, K. M. M. Tant and A. J. Mulholland, “Modelling of Ultrasonic Waves in Layered Elastic Heterogeneous Materials,” 2021 IEEE International Ultrasonics Symposium (IUS), 2021, pp. 1-4, doi: 10.1109/IUS52206.2021.9593907.

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