A Broad Field
TABLE 2.2 Spreading Laws
2.10.2.6 Scattering in Practice
In practice, an ultrasonic wave propagating in a scattering medium encounters many scatterers. Typical examples are grain boundaries in metals, porosity in any media, and bubbles in liquids. The total effect on the attenuation is a summation of indi-vidual effects. The mechanisms that occur in the interaction between an ultrasonic wave and a polycrystalline material are a function of the scale, which again can be defined in terms of ka or D/λ, where D is a grain diameter.
In measurements, as the ultrasonic frequency is scanned, the attenuation can vary over many orders of magnitude. This is illustrated with the case of a polycrys-talline aluminum, mean grain diameter 0.6 mm for frequencies between 100 kHz TABLE 2.7
Field region Near or far Near or far Near or far** Near or far Near or far.
Dimension 2-D and 3-D 2-D 2-D 2-D 2-D
Shape of scatterer Limited (square)*
*** Limited ** **
Included material Most Most Most Most Most
Mode conversion *** *** *** *** ***
Incident wave Arbitrary 2-D Arbitrary 2-D Arbitrary 2-D Arbitrary 2-D Arbitrary 2-D
Short pulse *** *** *** *** ***
Multiple scattering (2 body)
*** *** *** *** ***
Source: Reprinted from Proc. IUTAM Symposium on Elastic Wave Propagation and Nondestructive Evaluation (invited paper), L. J. Bond, ed. S. K. Datta, J. D. Achenbach, and Y. D. S. Rajapakse, Numerical techniques and their use to study wave propagation and scattering, 17–28, Copyright 1990, with permission from Elsevier.
Rating system: *** Very good; ** Good; * Copes; + Poor; × Very poor.
and 1 GHz. Data are shown in Figure 1.13 [46], and this includes data from Merkulov [47,48]. It is seen that as the D/λ ratio changes from long wavelength, with Rayleigh scattering, and where attenuation is 10−5 Np/cm to 1 GHz, where hysteresis and ther-moelastic effects dominate and attenuation is up to 10 Np/cm.
With the need to understand damage/crack precursors, early crack growth, and cumulative damage, as well as to provide more quantitative material structure characterization, there has been growth in the investigation of ultrasound-grain response. Such signals were previously considered to be “just” noise in traditional NDT [49], are giving material signatures, and are being investigated to give what is now called back-scatter and diffuse field ultrasound response, and these signatures are found as different parts of the pulse-excitation response time window that is investigated [50].
According to Mason and McSkimin [51], the attenuation in a metal is
α β= 1f +β2f4 λ≥3D (2.111) where D is the average diameter of the grains, β1 is a constant corresponding to hys-teresis, and β2 is a constant corresponding to scattering.
Scattering loss from grain boundaries in metals results from the random ori-entation of the grains with respect to the direction of wave propagation. Since, in an anisotropic material, the elastic constant varies with orientation, scattering occurs at intersections between grain boundaries. Materials at the grain boundaries, which differ in density and elasticity from the body of the grains, also contribute to scattering.
According to Rayleigh [38], scattering of energy from a single particle is given by the formula
where SA/IA is the ratio of the amplitude of the scattered wave to that of the incident wave, V is the volume of the scatterer, R is the radius of the scatterer, κ is the elas-ticity of the medium, Δκ is the difference in elasticity between the particle and the medium, ρ is the density of the medium, and Δρ is the difference in density between the particle and the medium.
According to Mason and McSkimin [51], the Rayleigh scattering law leads to the following scattering term for longitudinal waves in metals:
α π
Elastic Wave Propagation and Associated Phenomena 85 where cL is the longitudinal-wave velocity in the metal, cS is the shear-wave velocity in the metal, and a is attenuation (Np/m).
= + + + +
c11 c11( 12 m12 2) c m33 14 (2c13 4c n44) (12 122+ m )12
(for hexagonal crystals, any orientation)
= + −
(for hexagonal crystals, any orientation)
= + + + + (for cubic crystals, any orientation)
= + + +
(for cubic crystals, any orientation)
where ℓ1, ℓ2, ℓ3, m1, …, n3 are the direction cosines between the new set of axes and the crystallographic axes x, y, and z according to the relation
x y z
x′ ℓ1 m1 nl
y′ ℓ2 m2 n2
z′ ℓ3 m3 n3
and c11, c12, …, c44 are the elastic constants corresponding to the various crystallo-graphic orientations. Mason and McSkimin [51] give the scattering factors for cubic metals as
and for hexagonal metals
Later studies have shown Equation 2.111 to be applicable to a limited range of ratios D/λ. Discrepancies between scattering factors α given by Equations 2.133 through 2.122 and later studies are common. Papadakis (1981 and 1968) [52,53] gives a more thorough discussion of such scattering. These discrepancies show the extreme dif-ficulty of accounting for all practical conditions in a simple theory. However, the ultrasonic specialist should be aware of the factors that influence attenuation, and Equations 2.111 through 2.122 serve this purpose for hysteresis and scattering from grain boundaries in metals. Scattering is also caused by inclusions, microcracks, and other forms of discontinuities.
The functional form for the attenuation due the grain scattering in a polycrys-talline media, in the different scattering regimes, is shown in Table 2.8. From a practical perspective in determining response, a simple approach is to measure the attenuation as a function of frequency. However, before using attenuation data, it is necessary to determine if there is a need to correct for transducer diffraction or beam spread effects. When the attenuation is at a low level, when compared with diffraction loss, correction is essential. For media with very high attenuation, the
Elastic Wave Propagation and Associated Phenomena 87
diffraction loss effects can, in many cases, be negligible. It is however advisable to evaluate the relative magnitudes for both attenuation and beam effects when data are collected. An example of the effect of correction for beam spread on a measured attenuation is illustrated with the data given in Figure 2.30 [28]. In this case, the raw data and the magnitude of the apparent attenuation are significantly impacted when the correction for beam spread is applied.
TABLE 2.8
Functional Dependence of Attenuation
Range Dependencea
λ > 2π D B1f + A4D3f4
λ > 2π D A2Df4
λ << Dmin B1f +B2f2 +Ao/D After [51,52]
a B1f is elastic hysteresis loss and B2f2 is thermoelastic loss D is mean grain diameter.
Raw data
Corrected
d, dB/μsec
3.0
1.0
0.5 0.3
0.1
.05 .03
.015.0 10 30
f, Mc/sec 50 Correction
100
FIGURE 2.30 Example of the correction of attenuation data for diffraction loss. The diffraction corrections change the apparent frequency dependence of the attenuation.
(Reprinted from Papadakis, E. P., J Acoust Soc Am, 40(4):863–67, 1966. With permission.)
2.10.3 attenuation due to hySteReSiS
Hysteresis in an ultrasonic wave refers to a lag between the imposed stress and the resulting strain. Consequently, a plot of strain with increasing stress does not coincide with that for decreasing stress. As the stress is varied over a complete cycle, the stress–
strain curve forms a characteristic hysteresis loop. The area inside this loop corre-sponds to energy lost by hysteresis during the cycle. Therefore, this type of absorption is proportional to the frequency (corresponding to the term β1f in Equation 2.111).
The significance of hysteresis effects is shown in the data for aluminum given in Figure 1.13. In metals, this effect becomes significant at higher frequencies. In some materials, a variety of viscoelastic phenomena are encountered. In polymers, this typically becomes significant at low megahertz frequencies [2].
2.10.4 attenuation due to otheR mechaniSmS
Attenuation is caused by many factors other than scattering. Any mechanism that results in the removal of energy or conversion of energy from the original state to a new form is a cause of attenuation. Some of these causes are (1) frictional losses due to relative motion between adjacent surfaces, for instance, adjacent surfaces in laminated structures and powder metal compactions of less than 100% theoretical density; (2) conduction of heat from high-stress regions to low-stress regions and to regions adjacent to the ultrasonic beam; (3) micro eddy currents; (4) motions of atoms in a lattice caused by stresses, which effect a divergence from a preferential distribu-tion of the atoms; (5) viscosity (gases and liquids); and (6) dislocadistribu-tions in solids.
Mason [54] and others (e.g., [2,35,55]) discuss many of these mechanisms in some detail.
2.10.5 meaSuRement SyStem modelS
Recent advances in computer hardware and digitization capabilities are enabling new generations of measurement tools to be developed. Much of the fundamental physics and many measurement methodologies are long established. They have been employed to meet needs in solid state physics [22] and to solve a diverse range of process measurement problems (e.g., [56]). The full range of ultrasonic methods used in process monitoring and chemical analysis has been extensively reviewed by Asher [35] and Hauptmann [57].
To facilitate the evaluation of feasibility and design of systems for many of the new applications of ultrasonics, both a metrology and tools are needed. The needed framework can best be understood by starting by considering the example of a gen-eral pulse-echo and two-transducer system, as shown in Figure 1.15. This gengen-eral two-transducer model can be considered a linear system, and its response has been modeled in terms of a series of convolutions [38]. Such a one-dimensional model was presented by Newhouse and Furgason [58] as an extension of work by McGillen and Cooper [59] and Seydel [60]. In this work, the various elements in the system, namely the basic impulse generator, transducers, and the flaw response, are all represented as linear filters with effects that are combined by a series of convolutions.
Elastic Wave Propagation and Associated Phenomena 89
The elements in such a model are shown in Figure 2.31 and can be described with an equation using notation similar to that of Newhouse and Furgason [58] that can be given as
e t( )=x t( )* ( )* ( )* ( )*h t m t g t m t( )* ( )h t +n t( ) (2.123) Where * denotes convolution and the various terms are identified with the compo-nents in Figure 1.15. For the case of a single transducer in pulse-echo operation, reciprocal action for the transmitter and receiver can be assumed. In such a system, at least in principle, the desired, and usually unknown, defect or flaw response can be extracted from the complex echo signal by a series of deconvolutions.
The application of a Fourier transformation and deconvolution procedure to Equation 2.123 can be shown for a system with sufficient bandwidth to give improved resolution [58].
The overall response of a two-dimensional pulse-echo system has been considered by Staphanishen [61], who combined the transducer and acoustic media response in a “transfer function” model. The main aim of this model is to combine both the transducer’s dynamic response and the acoustic diffraction, which cause the near- and far-field phenomena. A Fourier transform formulation is applied, and the effects of the system elements are combined as convolutions so as to calculate the receiver voltage that is given for a particular form of input voltage.
A new model [40] has recently been developed by Berkhout et al. [62] for the description of acoustic wave phenomena in inhomogeneous media and demonstrated for the case of scatters in a water tank. The model considers a forward-traveling pressure or acoustic wave problem in a layered media, and it is formulated in terms of a matrix equation representation. The system is described by a matrix equation of the form
Generated signal
x(t) h(t) m(t)
Input
transducer Receiver
transducer Defect
Amplifier noise
g(t) m′(t) h(t)
n(t) e(t) Propagation
medium Propagation
medium Echo
signal
FIGURE 2.31 Schematic for a linear system model for basic ultrasonic NDT. (After Rayleigh, L., The Theory of Sound, 1st Am. ed., Vol. 2., Dover, New York, 1945, and Newhouse, V. L., and E. S. Furgason, Ultrasonic correlation techniques, In Research Techniques in Non-Destructive Testing, ed. R. S. Sharpe, Vol. 3, 101–34, Academic Press, London, 1984.)
P z( )0 =D z( )0
∑
W z z R z W z z( ,0 m) ( ) ( , )m m 0 S z(m
00) (2.124) where
S(z0) is the source vector representing the source configuration at the surface (z0) W(zm, z0) is the downward propagation matrix representing the propagation
properties from the surface (z0) to depth level zm
R(zm) is the scattering matrix representing the scattering properties of the inhomogeneities at depth level zm
W(z0, zm) is the upward propagation matrix representing the propagation properties from depth level zm to the surface (z0). It can be shown that for a time-invariant medium W(zm, z0) = WT (z0, zm), where T denotes a matrix transposition.
D(z0) is the detector matrix representing the detector configuration at the surface (z0)
P(z0) is the data vector representing the single-scattered echo data at the sur-face (z0) due to source S(z0)
In Equation 2.124, W is defined by the wave equation for a pressure wave in inho-mogenous absorptive fluids, R is defined by the elastic boundary conditions, and S and D are defined by the geometrical and the acoustical properties of the transducer in emission and in reception, respectively.
This model has been used by Berkhout et al. [62] to form inversion techniques based on focusing and deconvolution [62].
This general approach is analogous to the so called SONAR equations [34] and models that have been used in NDT [40]. Such linear models use data in dB and are the easiest ways to estimate signal-to-noise, range and detection sensitivity.
2.10.5.1 Resolution
The basic equations and phenomena of the pulse-echo NDT systems are well known, and the concept of resolution has been treated by various workers (e.g., Newhouse and Furgason, [58]). In outline, for a pulse-echo system in normal operation, the time delay (τ) between the transmitted and returned pulse from a target at a given range (R) is
τ = 2R
V (2.125)
where V is the wave velocity in the test piece. The pulse-repetition frequency for the system must be such that there is no overlap in wave trains, therefore,
τmax max
>2R
V (2.126)
Assuming no frequency dependence in the reflection coefficients of the waves, for two features with separation (ΔR), the time spacing in the wave train for resolution must be the length of the pulse used (Δτ), therefore,
Elastic Wave Propagation and Associated Phenomena 91
τ =2 R
V (2.127)
It can also shown that the half-power bandwidth and the length of the pulse are related as
B = 1/ τ (2.128)
and the resolution and bandwidth are related by
R V B= /2 (2.129)
Therefore, mode-conversion to a wave-type with a lower velocity (e.g., shear wave) can give better resolution. The advantages of using the compression wave as the input pulse are to give higher power inputs for the same initial electrical impulse, or in those cases where media encountered in the system have complex properties.