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# The Inverse Problem in Materials Characterization through Ultrasonic Attenuation and Velocity Measurements

• Emmanuel P. Papadakis
Chapter

## Abstract

The inverse problem in materials characterization is most often skipped over in favor of a correlation because the inverse problem is so difficult and the correlation is only tedious, not hard. In this talk the basic difficulty in the inverse problem field will be illustrated by two examples: finding grain size in metals when ultrasonic attenuation is measured, and finding graphite shape in cast iron when ultrasonic velocity is measured.

The basic difficulty arises because the measured quantity, attenuation or velocity, is a function of several variables. Thus, any single variable among the latter cannot be written as a single-valued function of the measured quantity.

For instance, attenuation α, caused by grain scattering, is a function of frequency f, grain diameter D, grain substructure μ, grain size distribution “GSD”, and the ratio of the grain diameter to the ultrasonic wavelength D/λ. The total attenuation “ATT” is also a function of geometrical beam spreading “BS” (which depends on sample anisotropy) as well as on physical absorption mechanisms “ABS” which in turn are functions of frequency and other parameters.

We have
$$ATT = \alpha + ABS + BS$$
(1)
The portion BS can be calculated and subtracted out immediately as long as the macroscopic sample anisotropy is known. The portion ABS may then be separable by multi-frequency measurements or by changing other environmental parameters (temperature, magnetic field) which may influence it, or by calculating it in the simplest cases like thermoelastic losses. The remainder, a, is frequently numerically the largest of the three terms. The term a is written symbolically as
$$\alpha = \alpha (D,{\kern 1pt} GSD,{\kern 1pt} \mu ,{\kern 1pt} D/\lambda ,{\kern 1pt} f)$$
(2)
The variables combine in such a way that there is a function F of D, GSD, and D/λ; another function n of D/λ; a coefficient A which is a function of n; and the term μ2. The result is
$$\alpha = AF{\kern 1pt} {\mu ^{2}}{\kern 1pt} {f^{n}}$$
(3)
The multiple factors discussed above make it obvious that the simplistic hope of finding
$$\bar{D} = \bar{D}{\kern 1pt} (\alpha )$$
(4)
will not be fulfilled in engineering materials, and that many factors must be taken into account.

The talk will present further clarifications of these challenging research opportunities using ultrasonic attenuation and velocity.

## Keywords

Inverse Problem Grain Size Distribution Physical Acoustics Ultrasonic Velocity Ductile Iron
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Plenum Press, New York 1984

## Authors and Affiliations

• Emmanuel P. Papadakis
• 1
1. 1.Manufacturing Processes LaboratoryFord Motor CompanyRedfordUSA