Acoustic Scattering
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,
| [m2]
| (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 Ps at a point distanced r of the scatterer can be written by,
| [N/m2]
| (2)
|
where Pinc 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].
Backscattering Formulation
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,Es( ) 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).
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Fig 3: Spectrums for the single echo of figure 2-left (dotted line) and for the random position summed echoes (solid line).
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The spectrum of the scattered echo of a single particle can be replaced by the voltage at the transducer Vs( ). This voltage is related to the pressure Ps( ) at the transducer by the system response. So we consider a broadband transducer excited by a impulse voltage Vi( ) that is transformed in a mechanical pulse Pi( ) by the transmitting factor XT( )
| (5.a)
|
this pulse propagates through the material and reaches the particle at a distance x away from the transducer with a pressure Pinc given by
| (5.b)
|
Where is the attenuation coefficient and DT 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 (Ps) corrected by the receiving diffraction factor (DR) and the efficiency factor of the transducer at the reception (XR), that is
| (5.d)
|
combining the equations 5.a-d we have,
| (6)
|
for a perfectly reversible transducer (XT = XR = X) and (DT = DR = D). The expected voltage (Vsx) 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/m3].
Substituting (6) in (7) we have,
| (7.a)
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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 = vtw). 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 = ti + tw/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 ks is 
and g contains other attenuation effects than the scattering by the particles. The coupling factor X² and the impulse voltage (Vi) 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 Vref is the spectrum of the echo by the plane reflector, Rref is the reflection coefficient of the reflector and Dref 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)
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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.
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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).
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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 1012, the average distance from the transducer =20mm, the number of particles that contribute to the signal of N = 108, and the density of the iron matrix and the graphite particle of 7920 kg/m3 and 2170 kg/m3 respectively. As can be readily seem, the simulated curves are very sensitive to nodule 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.