Table of Contents ECNDT '98 Session: Materials Characterization | Ultrasonic Backscattering Formulation Applied to Cast Iron CharacterizationSilvio Elton Kruger, Joao Marcos Alcoforado Rebello Jacques Charlier |
TABLE OF CONTENTS |
A model for simulation of the spectrum of the backscattered ultrasonic signal in nodular cast iron with direct contact by a broadband transducer is proposed. Spherical discrete graphite particles are assumed, with dimensions much smaller than the incident ultrasonic wave length. The simulated signal shows to be very sensitive to the graphite size. A preliminary comparison with experimental results obtained in two cast iron samples with a slight difference in their averaged graphite size showed a good correlation.
An ultrasonic wave that propagates through an engineering material will be partially scattered by the present inhomogeneities, that could be grains, porosity, inclusions, precipitates, microcracks, etc or any other microstructrural components that have different acoustic impedance of the matrix. The scattering by microstructural components of a material causes frequently difficulties to discontinuity detection, as it reduces the signal to noise ratio, but by other side, it could be used also to characterize these scatterers. The application of the backscattering technique to characterize materials would have some advantages in relation to the attenuation and velocity techniques as it does not need the backwall echo. So evaluations of materials with nonparallel surfaces, rough back surface, unknown thickness, etc. could be readily possible.
Many authors have proposed the use of backscattering technique to characterization of the propagation medium mainly in biological tissues and solid [1-7]. Most of these works propose data reduction methods for the evaluation of the backscatterer coefficient [1-6], that is defined as the differential cross section for 180° scattering from a volume of the sample normalized to that volume. The backscatterer coefficient is a frequency dependent function that is a system independent characteristic of an ensemble of scatterers. Although this parameter showed to be in some cases very effective and consistent with theory for scatterers evaluation, in this paper the formulation will be rather the proposition of a simulation of the received signal by a known experimental system. This formulation has in mind the characterization of cast irons problem with direct contact technique by comparison of the received backscattering amplitude spectrums.
The general scattering formulation is a complex physic/mathematical problem. Although many research have been made in this subject, solutions are achieved only for particular geometries and/or wavelength to particle size ratio. Fortunately, the analytical solution obtained by Ying and Truel [8] for spherical scatterers in a solid isotropic matrix, could be a reasonably approximation for the nodular cast iron case. This solution is considerably simplified for low frequencies, were the ultrasonic wavelength is much longer than the scatterer dimension ( >> a) and the total scattering cross section can be stated as,
[m^{2}] | (1) |
where is the angular frequency, v is the longitudinal velocity, a is the radius of the scatterer and g is a parameter that contains the matrix and scatterer elastic properties [8]. The scattering cross section relates the energy scattered and normalized by the incident intensity, so the scattered pressure P_{s} at a point distanced r of the scatterer can be written by,
[N/m^{2}] | (2) |
where P_{inc} is the incident pressure of a longitudinal wave and assuming the low frequency approximation were the scattered wave is spherical. The scattering by the inhomogeneities present in a material can cause also a significant attenuation. The attenuation coefficient due to the scattering is also readily deduced by the cross section and can be written by,
[Nepers/m] | (3) |
where n is the volume density of particles [m^{-3}].
The problem to be treated here is the expected spectrum of the scattered signal that will be received by a broad band transducer in direct contact with an inhomogeneous material, as illustrated in figure 1.a. The scatterers are spherical and randomly spatially distributed in the material. The region of interest is selected by the time window in the received signal (figure 1.b), so only the scatterers with a distance sufficient long to the far field assumption be valid will be treated. The scatterers are so numerous in typical problems for engineering materials that the simulations of the signal received by all scatterers would be to much time consuming although it have already been done [7]. Statistical treatment of the problem is necessary. Fortunately, very simple formulation can be adopted for this case on considering that in the frequency domain, the (mathematically) expected spectrum of the backscattered signal for many scatterers is the same as for a single scattering of same dimension, as stated by
(4) |
Where is the expected spectrum for the backscattered signal of many particles,E_{s}( ) is the spectrum of the backscatterd signal for a single particle and N is the number of particles that contributes to the bacscattered signal. This is valid when the position of the particles are independently distributed, that is there is no correlation between the particle positions [5]. To evaluate this, a simulation was performed by the summation of a finite number of broadband echoes in random positions in the time domain and then transformed to the frequency domain. Figure 2 shows the single and the summed echoes in the time domain. In the frequency domain, both are represented in figure 3, where the spectrum for the summed echoes are processed to become more smoothed, and divided by . As expected, the curves are very similar.
Fig 1: Backscattering signal parameters for direct contact measurements. (a) Geometry and main parameters of the problem and (b) display of the ultrasonic signal in the time domain. |
Fig 2: Time domain single echo (left) and summed in random positions (right). | Fig 3: Spectrums for the single echo of figure 2-left (dotted line) and for the random position summed echoes (solid line). |
The spectrum of the scattered echo of a single particle can be replaced by the voltage at the transducer V_{s}( ). This voltage is related to the pressure P_{s}( ) at the transducer by the system response. So we consider a broadband transducer excited by a impulse voltage V_{i}( ) that is transformed in a mechanical pulse P_{i}( ) by the transmitting factor X_{T}( )
(5.a) |
this pulse propagates through the material and reaches the particle at a distance x away from the transducer with a pressure P_{inc} given by
(5.b) |
Where is the attenuation coefficient and D_{T} is the transmission coefficient for diffraction. The scattered pressure by a scatterer with a cross section at a point r away from the particle is given by the equation 2 corrected by the attenuation, that is
(5.c) |
For the backscattering configuration where the transmission transducer is also the receiving transducer we have r = x and the received voltage at the transducer will be given by the pressure at the transducer (P_{s}) corrected by the receiving diffraction factor (D_{R}) and the efficiency factor of the transducer at the reception (X_{R}), that is
(5.d) |
combining the equations 5.a-d we have,
(6) |
for a perfectly reversible transducer (X_{T} = X_{R} = X) and (D_{T} = D_{R} = D). The expected voltage (V_{sx}) received due the scattering of many particles inside a cylinder with volume Adx at a distance x (figure 1.a) can by given by the equation 4, or
(7) |
Where A is the area illuminated by the transducer at a distance x and n is the density of particles [particles/m^{3}].
Substituting (6) in (7) we have,
(7.a) |
The total backscattered signal from a volume determined by the gating of the signal (figure 1.b) will be given by the integration of 7.a, or
We can solve this equation by assuming that there is a linear variation of in the range of variation of x (small (x = vt_{w}). We assume also that the material is homogeneous (n(x) = constant) and the area of the ultrasonic beam is independent of x. Then we obtain,
(8) |
- mean distance to the region of scattering = t_{i} + t_{w}/2 ;
N = (x.n.A number of particles that contribute to the signal.
Substituting the scattering cross section (equation 1) and taking the square root of both sides we have
,
(8.a) |
where k_{s} is and _{g} contains other attenuation effects than the scattering by the particles. The coupling factor X² and the impulse voltage (V_{i}) could be evaluated by a simple experiment in using a plane reflector at on a material with very low attenuation. Using equation 6.a for a plane reflector, instead of a scatterer, the coupling and the impulse voltage could be estimated to be
where V_{ref} is the spectrum of the echo by the plane reflector, R_{ref} is the reflection coefficient of the reflector and D_{ref} is the expected diffraction correction for the reference reflector and as D it can be evaluated theoretically [3]. The equation 8.a can then be written as
(8.b) |
The influence of the scattering in the signal, i. e., considering unitary response of the system in the frequency domain, we will have,
(9) |
Simulations for Nodular Cast Irons
Nodular cast irons have a distribution of quasi spherical particles of graphite in a matrix of iron as shown in figure 4.a by a typical micrography of this material.
Fig 4: (a-left) Micrography of a nodular cast iron without etching and (b-right) simulated backscattered signal for two particle sizes and assuming unitary system response (equation 9). |
The elastic properties of the particles and the matrix used for a simulation was found in the literature [9]. The expected curve for the backscattered signal applying equation 9 with the longitudinal and transversal velocity of the matrix of 5830 m/s and 3090m/s and for the graphite 4210 m/s and 2030 m/s respectively is depicted in figure 4.b with two particle diameters. Other assumed parameters are the density of particles of 10^{12}, the average distance from the transducer =20mm, the number of particles that contribute to the signal of N = 10^{8}, and the density of the iron matrix and the graphite particle of 7920 kg/m^{3} and 2170 kg/m^{3} respectively. As can be readily seem, the simulated curves are very sensitive to nodule diameters.
An experimental verification of the model was done on two cast iron samples with a direct contact ultrasonic technique. A pulser/receiver equipment send a spike pulse to a broadband transducer with a nominal frequency of 5 MHz that is also used to receive the signals and is amplified by the pulser/receiver. The amplified signal is digitized by an A/D card in a PC-microcomputer and processed to have a smoothed spectrum. Figure 6 shows the averaged spectrums taken in 8 random positions in the sample surface with the simulated spectrums. Image processing of the micrographies was used to estimate the nodules diameters.
Fig 6: Experimental (solid lines) and simulated (dotted lines) spectrums for the backscattered signal for direct contact in two nodular cast iron samples with different nodules sizes. |
A it can be readily seem, the model seems to fit the experimental curve near the central frequency, but have some discrepancies for low and high frequencies. The nodules diameters for the two samples are not so different (29 and 36µm), but both experimental and theoretical backscattered spectrums can differentiate them.
This paper develop a simple model to simulate the backscattered signal for direct contact in a material with discrete inhomogeneities that are much smaller than the wavelength. The simulated spectrums curves indicate that the technique should be very sensible to the nodule size of the graphite precipitated in cast irons (figure 4.b). A preliminary experimental verification shows that the model, whereas very simple, could indicate the general behavior of the spectrum curves and their center frequency. Possible discrepancies between the model and the experimental curves could be the fact that the model assumes an unique particle size when the cast iron nodules presents a distribution of the sizes. In the comparison of the spectrums it was taken the average diameter of the nodules. The spherical form of the scatterers is not always true for real cast iron nodules, but when the incident ultrasonic wavelength is much larger than the particle size, the form of the scatterer is not important. So the model could be applied to evaluate other than nodular graphite shape, like vermicular and lamelar.
The authors wish to thank CNPq and CAPES from Brazil and FNRS from Belgium for financial support. We would like to thank the Foundry Center/Univ. of Ghent for providing the cast iron samples.
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