*Corr. Author’s Address: DEEC, Pólo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal, marioj@deec.uc.pt 245
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255 Received for review: 2020-12-22
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-04-02
DOI:10.5545/sv-jme.2020.7078 Original Scientific Paper Accepted for publication: 2021-05-03
Ultrasonic Scattering Attenuation in Nodular Cast Iron: 
Experimental and Simulation Studies
Santos, M. – Santos, J.
Mário Santos* – Jaime Santos
University of Coimbra, CEMMPRE, Department of Electrical and Computer Engineering, Portugal
This work evaluates the ultrasonic scattering attenuation of structures with complex scatterer distributions via experimental and simulation 
studies. The proposed approach uses experimental attenuation knowledge to infer the scatterer size and its concentration in the studied 
structures, which are important for the effective construction of simulated models. The MATLAB k-Wave toolbox has been used to implement 
the simulator. Several cast-iron samples have been used to demonstrate the importance of simulation in the characterization of such 
structures. First, the scattering attenuation was evaluated using the Truell and Papadakis models, and then the results were compared 
with experimental ones. Emphasis was given to the Papadakis approach because it takes into account the scatterer size distribution. It is 
demonstrated that both analytical models provide results that are far from the experimental ones. The developed simulator for the studied 
samples led to a predictive model, in which the attenuation was proportional to the fifth power of the scatterer size, and the corresponding 
formulation is close to the one proposed by the analytical models.
Keywords: modelling, anisotropy, pulse-echo, simulation, ultrasonic attenuation
Highlights
• 	 Experimental attenuation in cast iron samples was carried out. 
• 	 Scattering attenuation theoretical models do not apply to complex nodular cast-iron structures.
• 	 Simulation models as a strategy to predict the experimental performance of cast iron.
• 	 A k-Wave simplified simulation model is used to characterize complex structures.
0  INTRODUCTION
There are many applications for nodular cast iron due 
to its castability, high thermal conductivity, and good 
mechanical properties, specifically tensile strength 
and ductility. The mechanical properties of a metal 
greatly depend on the microstructure; in the case of 
nodular cast iron, which is produced by adding, shortly 
before solidification, a small amount (lower than 0.04 
%) of substances such as magnesium or cerium are 
present. These substances give rise to the growth of 
nodular graphite, whose shape and distribution are of 
fundamental importance in the behaviour of the metal 
[1] to [3]. Thus, the non-destructive evaluation of such 
structures is very important for the identification of 
the nodularity and matrix phases.
Ultrasonic characterization offers great 
advantages when compared with destructive 
metallographic methods. The interaction of 
ultrasound waves with the material microstructure 
can be evaluated, measuring the acoustic parameters, 
including velocity and attenuation. Two ultrasound 
attenuation mechanisms are generally identified: 
absorption and scattering. Absorption is related to 
thermal conduction loss, hysteresis, and a viscous loss 
mechanism [4]. Scattering is due to heterogeneities 
such as grain boundaries, voids, inclusions, second-
phase particles or porosity [5] to [9]. This attenuation 
mechanism is commonly accepted as the most 
important in heterogeneous materials, such as the 
case of cast iron [10] to [13]. It is important to take 
into account the fact that the scattering effects of the 
matrix grains are too small when compared to the 
nodular scattering effects and can be ignored [6], [7], 
[14] to [16].
Several authors have extensively studied 
scattering attenuation. At the beginning of the last 
century, Rayleigh presented a scattering formula [17], 
later adapted by Mason and McSkimin [18] and [19]; 
the case of polycrystalline aluminium. Huntington 
[20] used a stochastic theory to explain the scattering 
effects in polycrystalline structures. Lifshitz and 
Parkhomovskii [21] proposed a theory that considers 
the mode conversion at the grain boundaries. 
Moreover, a great contribution was made by 
Papadakis [5], [6], [22] to [25] with several published 
works related to that topic. The author classified 
the scattering in three classical regimes depending 
on the relation between the grain size (D) and the 
wavelength ( λ): (1) Rayleigh regime (for λ>>D), 
where the attenuation is proportional to the fourth 
power of frequency; (2) stochastic regime (for λ≅D), 
where the attenuation is proportional to the square of 
frequency; (3) and geometrical regime (for λ<<D), 
where the attenuation is frequency independent. For 
each regime, the microstructure is assumed to be 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
246 Santos, M. – Santos, J.
composed of spherical grains of the same size filling 
the medium. As the Mason and McSkimin models 
[18] are very complex, due to multiple scattering, 
their theoretical treatment is a laborious task. That 
led other authors to restrict the scattering analysis 
to a limited frequency range [5], [24] to [27]. Later, 
Hirsekorn [28] and [29], Stanke and Kino [30], and 
Weaver [31] developed general solutions valid for 
all grain size to wavelength ratios and for cubic 
symmetry polycrystalline materials. However, other 
researchers demonstrated that those general solutions 
failed in more complex microstructures, such as the 
commercial aluminium alloys [32] and two-phase 
sintered powders [33]. More recent works deal with 
advances related to the interaction of ultrasound waves 
with polycrystalline materials. Yang et al. [34] studied 
the shape effect of elongated grains on attenuation. 
Arguelles and Turner [35] evaluated the errors 
resulting from neglecting the grain size distribution 
and using an average grain size. Ryzy et al. [36] used 
a semi-analytical attenuation model that considers the 
grain morphology and incorporates an exact spatial 
two-point correlation function. The authors concluded 
that the grain shape has a strong effect on attenuation 
in the Rayleigh-stochastic transition region. Rokhlin et 
al. [37] presented a model that includes second-order 
multiple scattering, applicable to all frequency ranges, 
providing small relative errors on both longitudinal 
and transverse attenuations for low anisotropy. Sha 
[38] extended the second-order attenuation (SOA) 
model for elastic waves in texture-free materials 
to textured polycrystals with ellipsoidal grains of 
arbitrary crystal symmetry. The predicted attenuation 
results of this work agree well with the literature on a 
textured stainless steel polycrystal.
Numerical or grid-based methods are also 
powerful tools in the analysis of the propagation in 
scattering media. The improvement of computational 
resources makes possible the implementation of those 
methods to study increasingly complex interactions 
[39]. Recent works used finite element methods 
(FEM) to accurately mimic wave propagation through 
a rather complex material [40] to [44]. Norouzian, et 
al. [45] use DREAM.3D tool to construct material 
volumes with lognormal grain-size distributions. The 
results show that the correlation between attenuation 
and distribution width can be modelled with a power 
law, and the frequency dependence of attenuation has 
been shown to be strongly depend on the distribution 
width.
The main advantage of these methods, when 
compared to analytical approaches, is related to the 
fact that they allow microscopic scale analysis instead 
of an averaged effective medium. No assumptions 
have to be made concerning grain statistics, anisotropy 
degree, or multiple scattering events [36]. Limitations 
related to frequency range, model dimensions and 
grid discretization tend to be strongly reduced, in the 
future, with the increase of computational resources.
One technique widely established for the 
generation of polycrystalline material morphologies 
is the Voronoi tessellation [40] to [50] for 2-D or 
3-D dimensions. Details about that approach can be 
found in [50]. The major drawback of elastic wave 
models based on low-order finite difference or finite 
element schemes is the large number of grid points per 
wavelength required to avoid numerical dispersion. 
The k-Wave toolbox for Matlab can be used as an 
alternative to those approaches. It uses a Fourier 
domain pseudospectral method for the simulation and 
reconstruction of photoacoustic wave fields in a faster 
way. Also, it uses less memory and is user friendly [51] 
and [52]. Fewer spatial and temporal grid points are 
needed for accurate simulations [51]. Recently many 
authors have used the k-Wave toolbox in applications 
related to attenuation in ultrasonic computed 
tomography [53], guided waves in layered structures 
[54] and [55] time domain, power law attenuation in 
breast and liver tissues [56], one-sided ultrasonic non-
destructive evaluation [57], high intensity focused 
ultrasound [58], nonlinear ultrasound propagation 
in absorbing media [59], ultrasonic transducers field 
modelling [60], and 3-D ultrasound imaging [61].
In the present work, the authors propose a 
simulation model to evaluate the scattering attenuation 
in structures with complex scatterer distributions. 
It uses the experimental attenuation knowledge in 
nodular cast iron to infer the scatterer size and its 
power dependence, as well as the concentration 
that best mimics the structure behaviour. The 
k-Wave toolbox is used to construct the simulator. 
Additionally, the results provided by the theoretical 
and simulation models are presented and a discussion 
is also made about the feasibility of the Truell et 
al. [62] and Papadakis [22] models for attenuation 
evaluation in nodular cast iron.
1  ULTRASONIC SCATTERING ATTENUATION
1.1  Theory
For the Rayleigh domain, characterized by a 
wavelength much higher than the scatterer size, 
Ying and Truell [27] presented a model for the total 
scattering cross-section, assuming the scatterers are 
spherical, solid, elastic, isotropic, have uniform size 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
247 Ultrasonic Scattering Attenuation in Nodular Cast Iron: Experimental and Simulation Studies
and are embedded in a solid elastic matrix. Based 
on those assumptions, Truell et al. [62] presented an 
expression for the ultrasonic scattering attenuation 
evaluation of longitudinal waves:
 

sT
l
Ng
v
r 
2
9
4
4
6
, (1)
where N is the number of scatterers per unit volume, ω 
is the angular frequency, v
l
 represents the longitudinal 
propagation velocity, r is the scatterer radius, and 
g denotes a factor related to the elastic properties of 
the medium. The original g expression in [27] can 
be changed to encompass the longitudinal (v
l
) and 
transversal (v
t
) velocities:
 
g
v
v
v
v
l
t
l
t


































3
34 4
1
1
1
2
2
2
2
2
1


 






























2
1
1
3
12
21
2
1
3
12 1
4
v
v
v
v
l
t
t
t


0 02 3
1
1
5
















v
v
l
t
 

































2
1
1
1
2
2
1
1
1
2
1
23 29 4
v
v
v
v
l
t
l
t
 


















2
, (2)
where µ is the shear modulus given by v
t
2
ρ ; ρ is 
the density, and the indexes 1 and 2 are related to the 
surrounding medium and scatterer, respectively.
Based on a similar work developed by Bathia 
and Moore [63], Papadakis [6] and [22] introduced 
a correction factor for the attenuation considering 
the effect of the grain size distribution. The author 
also showed a way of obtaining the correction factor 
in solids using two-dimension micrographs. The 
scatterer volume correction factor (T ) is:
 T
r
r
n
n

4
3
6
3

, (3)
where r
n
6
 and r
n
3
 are the average scatterer radius 
values of the sixth and third power, respectively, 
obtained from micrograph samples. From Eq. (3), 
it is clear that the scattering effects of a single large 
scatterer are much greater than the effect summation 
of many small scatterers having the same volume. 
Taking the correction factor into account, the 
scattering attenuation is given as [6]:
   





sP
l
NT
f
v
L
L
L
LL

 





2
34
4
2
1
2
2
1
2
1
2
2
1
2
4
1
3
8
3
32
15
  
 





























LL
L
1
1
3
2 2
1
2
1
1
2
1
2
2
3
8
15 

 
 
 

, (4)
where f is the frequency, L
1
 and μ
1
 are the longitudinal 
and the shear modulus of surrounding medium, 
respectively, and Δρ, ΔL, Δµ are the differences in 
density, longitudinal modulus, and shear modulus 
between scatterers and the surrounding medium, 
respectively. Although Eq. (4) uses a different 
approach for the attenuation calculation, the results 
are similar to those provided by the Truell model for 
uniform scatterer sizes.
1.2  Experimental Attenuation
The attenuation is the result of acoustic wave 
interactions with the propagation medium. 
Experimentally, the attenuation can be calculated 
by collecting the front surface reflection and two 
back wall ultrasound pulses from samples with 
parallel faces, as illustrated in Fig. 1. Following that 
approach, the authors studied different methodologies 
to calculate the ultrasonic attenuation in nodular 
cast iron samples [64], and concluded that Eq. (5) 
provided more reliable results, because it does not 
take into account the reflection coefficient between 
water and sample, which is frequency-dependent, as 
demonstrated experimentally [64]:
  







1
2
1
1
2
2
L
AAD
AA A
sc
s
ln . (5)
In Eq. (5), L is the sample thickness, A
1
 and A
2
 are 
the first and second back wall signals from samples, 
A
s
 is the reflected signal on the sample front face, 
and D
c
 is the diffraction correction coefficient for the 
sample path. The attenuation (α) unit is [Np/m], where 
Np is the symbol of neper, which is a logarithmic unit 
for ratios of measurements of physical quantities. The 
neper and dB are related by the following relationship: 
1 Np = 8.686 dB

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
248 Santos, M. – Santos, J.
Fig. 1.  Setup for attenuation measurement
1.3  k-Wave Simulation
The simulation of elastic wave propagation has many 
applications in ultrasonic non-destructive testing. 
The open-source k-Wave toolbox is an easy-to-use 
time-domain forward model based on a k-space 
pseudospectral time-domain solution to couple 
first-order acoustic equations for homogeneous 
and heterogeneous media in one, two, and three 
dimensions [51]. That tool makes possible the 
modelling of arbitrary sources, detecting surfaces 
with directional elements, and wave propagation that 
can account for nonlinearity, acoustic heterogeneities 
and power-law absorption. It is also possible to use 
optional input parameters to adjust the visualization 
and performance, including making wave propagation 
movies for presentations and running simulations 
on graphics processing units (GPU). The available 
geometry creation functions allow both Cartesian and 
grid-based geometries, such as circles, arcs, disks, 
spheres, shells or balls. As an example, Fig. 2 shows 
a three-dimensional model for a circular transducer 
(top) radiating into a sample with randomly distributed 
scatterers. The main drawback of k-Wave in ultrasonic 
non-destructive testing is related to the existence of 
large contrasts between media. Fourier-based methods 
such as k-Wave are effective when everything is 
smooth, and the contrasts are not too big (the code 
was originally designed for modelling ultrasound in 
biological tissue, where the contrasts are low). For 
large contrasts, oscillatory errors can be accumulated, 
because the steep edges cannot be represented very 
well by a small number of Fourier components. Some 
caution must be taken in such situations. Smoothing 
the density and propagation velocity or keeping the 
time step small will help to reduce that effect.
Fig. 2.  Three-dimensional simulation model
2  RESULTS AND DISCUSSION
2.1  Samples Granulometry Analysis
Six machined cast iron samples with parallel surfaces 
within 15 µm and 13.1 ± 0.05 mm in thickness were 
used in this study (Fig. 3). The samples were prepared 
using standard metallographic methods. First, they 
were mechanically polished using metallographic 
carbon silicate sandpaper with decreasing 
granulometry (P180, P1000, and P2500) and a final 
polishing by a 3-μm diamond suspension. The 
microstructure of the processed zones was examined 
using optical microscopy, with a 200× magnification.
The typical microstructure of such samples is 
shown in Fig. 4, where the precipitated graphite 
nodules (in black) embedded in a ferrite/perlite matrix 
are easily identified, as are the grain boundaries. It 
was observed in all samples that the nodules present 
non-uniform shapes and have a large wide range of 
sizes that can vary from less than one micrometre 
to several tens of micrometres (the larger measured 
radius was 35 µm).
Fig. 3.  Cast iron samples

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
249 Ultrasonic Scattering Attenuation in Nodular Cast Iron: Experimental and Simulation Studies
Fig. 4.  Optical micrograph of a nodular cast iron sample
Ten micrograph images were taken from each 
sample, corresponding to an analysis section of 
10.32 mm². The ImageJ free package software 
was used to determine the nodules’ size [65], which 
were considered to be spherical. The nodule size 
distribution is presented in Fig. 5. Most of the nodules 
have a radius less than 10 µm. The spatial variability 
of the nodules’ size in different regions of each sample 
is low. The standard deviation of the nodule size 
divided by its average value obtained for all images of 
each sample was about 8.9 %. Also, for each sample, 
the total number of nodules was used to obtain the 
lognormal distribution parameters. The mean and 
standard deviation are presented in Table 1.
2.2  Experimental Attenuation Evaluation Setup and 
Results
The experimental immersion setup for the attenuation 
evaluation is presented in Fig. 6. A pulser/receiver is 
used to excite a broadband 8 MHz central frequency 
transducer and to collect, amplify, and filter the 
reflected signals. Then, the acquired signals are 
displayed in an oscilloscope and transferred to a 
computer for further processing. The transducer is 
moved using a computer-controlled micro-positioning 
system.
Fig. 5.  Samples’ nodule size distribution
Fig. 6.  Experimental setup for attenuation evaluation
All samples characterized in section 2.1 were used 
in the study. Four acquisitions were accomplished in 
different regions of each sample to take into account 
their structural variation. Then, using Eq. (5) an 
average attenuation of 11.96 Np/m was obtained, 
Table 1.  Lognormal fitting parameters for each sample
Sample 1 2 3 4 5 6
Mean [µm] 0.2335 0.1178 0.4310 0.1007 0.0671 0.2420
Standard deviation [µm] 0.9492 0.8026 0.7862 0.8751 0.8545 0.9161
Table 2.  Acoustic properties of cast iron components
Longitudinal velocity  
[m/s]
Transversal velocity  
[m/s]
Longitudinal modulus  
[GPa]
Shear modulus  
[GPa]
Density  
[kg/m³]
Pearlite/Ferrite 5830 3090 269 75.6 7920
Graphite 4210 2030 38 8.9 2170

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
250 Santos, M. – Santos, J.
with a standard deviation (SD) of 1.22 Np/m and root 
mean square error (RMSE) of 1.10 Np/m for the 24 
measurements.
2.3  Attenuation Evaluation Using Analytical Models
In order to evaluate the scattering attenuation in the 
cast iron samples, important parameters as longitudinal 
and shear velocities, longitudinal and shear moduli, 
and density related to the matrix (pearlite/ferrite) 
and scatterers (graphite), must be known. They are 
represented in Table 2 [24].
The scatterer concentration knowledge is also 
required to estimate the attenuation. Thus, the two-
dimensional scatterer concentration is obtained 
relating the total amount of nodules (n
T
) with the area 
of micrograph (A
T
), as follows:
 N
n
A
A
T
T
= . (6)
Then, the scatterer concentration (N) can be 
calculated using the following equation [66]:
 N
f
N
g
NA












6
1
2 3
2
, (7)
where f
g
 is the graphite nodule fraction that is about 
3.33 % for the analysed images, and α
N
 = 1.25, which 
is a parameter related to the width of the nodule size 
distribution.
To infer the real scatterer effect in the attenuation, 
the scatterers that contribute to the attenuation should 
be determined. To do so both the Truell and Papadakis 
models will be used and their performance analysed.
Due to the wide nodular distribution observed 
in the samples, the Truell model given by Eq. (1) 
cannot be used to calculate the scattering attenuation, 
because it uses the same size scatterers. Thus, in 
order to encompass all scatterer sizes found in the 
samples, the authors present a new equation for the 
attenuation, based on Eq. (1), which takes into account 
the summation of the different classes of scattering 
nodules:
 

sTm
i
n
l
ii
g
v
Nr 


1
4
4
6
2
9
. (8)
In Eq. (8), n is the total number of scatterer 
classes, r
i
 and N
i
 are the average radius and scatterer 
concentration of each class, respectively. Thirty-five 
scatterer classes with 1 µm step were considered for 
the attenuation calculation. The scatterer concentration 
of each class is a fraction of the one obtained by Eq. 
(7). From Eq. (8), resulted an attenuation of 2.67 
Np/m.
The different attenuation values provided by the 
Truell model and experimental approach could be 
due to the fact that the Truell model makes use of 
spherical, solid, elastic, and isotropic scatterers. In 
addition, the referred model assumes that there is a 
sharp variation of the properties between the scatterers 
and the surrounding media. However, the nodular 
cast iron structure is rather more complex, especially 
its matrix, which is usually formed by ferrite around 
the nodules, with a homogeneous structure, and by 
pearlite in other regions with a lath-type structure 
with some degree of heterogeneity [24]. Therefore, 
the boundary effect between scatterers and matrix is 
certainly more complex than the one presented in the 
model.
The Papadakis model takes into account the 
scatterer size distribution, which is an improved 
version of the Truell theory. Generally, in practical 
materials, such as cast iron, the scatterers are not all 
the same size and follow a distribution that can be 
evaluated. The correction factor (see Eq. (3)), requires 
the knowledge of 
r
n
6
 and r
n
3
:
 r
r
n
n
i
n
i 6
1
6



, (9)
 r
r
n
n
i
n
i 3
1
3



. (10)
Using the experimental scatterer distribution 
(see Fig. 5), resulted r
n
6
= 3.5 × 10
5
 μm
6
 and  
r
n
3
= 54.73 μm
3
 for the whole sample area analysed, 
and T = 2.68 × 10
–14
 m
3
.
Then, using Eq. (4), where the right side is 
essentially composed by the physical parameters of 
matrix and scatterers presented in Table 1, resulted 
α
sP
 = 370 Np/m, which is very large compared to 
the experimental attenuation. In addition to the 
explanation made for the Truell model deviation in 
relation to the experimental attenuation, which also 
applies to the Papadakis model, other important factors 
can contribute to the observed discrepancy between 
theory and experiment. Perhaps, the most important 
one deals with the scatterer distribution obtained from 
the micrograph images. In the Papadakis model, all 
scatterer sizes and shapes were taken into account 
for the attenuation calculation that originates a very 
high scatterer concentration value. However, the 
authors concluded the attenuation decreases sharply if 
the lower size scatterers are discarded, as illustrated 
in Fig. 7. That tendency is expected because the 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
251 Ultrasonic Scattering Attenuation in Nodular Cast Iron: Experimental and Simulation Studies
scatterer concentration (N) falls more quickly than 
the correction factor (T). For instance, if all scatterers 
lower than 7 µm are discarded, the attenuation is 15 
Np/m, which is very close to the experimental one.
Finally, the Papadakis model also presents 
limitations in predicting the scattering attenuation in 
the cast iron samples.
Fig. 7.  Attenuation behaviour versus scatterer size
2.4  Simulation Model
The simulation of the cast iron nodular structure 
shown in Fig. 4 is difficult due to the non-uniformity of 
scatterers in size and shape. Here, the authors propose 
a simplified model assuming all scatterers with the 
same size and known concentration that leads to the 
experimental attenuation value. For that purpose, it is 
assumed that the attenuation is directly proportional to 
the scatterer concentration (see Eq. (11)), which is in 
accordance with the theoretical models [21] and [60] 
and is a common-sense assumption. The equivalent 
scatterer size (r
eq
) and its exponent dependence (n) will 
be determined based on the inverse problem, using the 
experimental attenuation. The simplified attenuation 
expression based on the previous assumptions is as 
follows,
  

CN r
eq
n
, (11)
where C and n are constants to be determined. Eq. 
(11) follows the Rayleigh scattering model, and only 
the geometric parameters are of interest, such as the 
scatterer concentration, equivalent scatterer size, and 
its exponent dependence. The frequency dependence 
and the matrix and scatterer properties are included 
in the constant C, because the study related to the 
attenuation variation with these parameters is outside 
the scope of this work.
Using k-Wave, a simulation model with 
dimensions x = 6 mm, y = 6 mm and z = 40 mm, 
similar to the one illustrated in Fig. 2, was defined 
by a computational grid for simulation purposes. For 
the scattering attenuation evaluation, an ultrasonic 
probe is located at the bottom of the model (z = 0), 
working as a transmitter and another at z = 30 mm 
working as a receiver. Two signals are collected at 
the receiver: a reference signal a
0
 (model without 
scatterers) and a signal a
s
 considering the model filled 
up with randomly distributed scatterers. The scattering 
attenuation is then given by:
 
sim
s
d
a
a







1
0
ln , (12)
where d = 30 mm is the distance between the probes. 
The two probes are 6 mm in diameter, like the one 
used in the experimental measurements (see section 
2.2). The probe excitation was made by a tone burst, as 
shown in Fig. 8, to mimic the experimental transducer 
response illustrated in Fig. 9.
The first set of simulations were carried out 
for scatterers of r
eq 
= 60 µm, corresponding to a 
grid point spacing of 120 µm (cuboid edge). The 
scatterer concentration was varied from 0 mm
–3
 to 
100 mm
–3
 with a step of 10 mm
–3
, and ten signals 
a
s
 were collected for each concentration value. The 
corresponding attenuation was then calculated using 
Eq. (12), and the results are shown in Fig. 10. An 
average attenuation value for each concentration 
was considered. As expected, the attenuation 
increases as N also increases. That behaviour is more 
pronounced for lower concentrations. In the boxplot 
of the simulated data (Fig. 10) the central mark is 
the median, the edges of the box are the 25
th
 and 75
th
 
percentiles, the whiskers extend to the most extreme 
data points (not considering the outliers, which are 
plotted individually as crosses).
Fig. 8.  Simulated tone burst

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
252 Santos, M. – Santos, J.
Fig. 9.  Experimental transducer response
For the simulated scatterer size, the concentration 
value that leads to the experimental attenuation 
calculated in section 2.2 (α
exp 
= 11.96 Np/m), is about 
3.2 mm
–3
. This result is more clearly observed in Fig. 
11, which corresponds to an expansion of Fig. 10.
Fig. 10.  Simulated attenuation for r
eq
 = 60 µm  
versus scatterer concentration
Fig. 11.  Zoomed version of Fig. 10 for lower concentrations
In order to establish a correlation between the 
scatterer concentration (N) and size (r
eq
), additional 
attenuation simulations were carried out using 
different scatterer sizes, resulting in the concentrations 
(presented in Table 2) that lead to the experimental 
attenuation. A power fitting applied to the values in 
Table 2 gave rise to Eq. (13), whose behaviour is 
illustrated in Fig. 12, where a remarkable goodness of 
fit with R
2
 
= 0.99 is observed:
 Nr
eq




4 527 10
13
5 173
..
.
 (13)
Table 2.  Scatterers dimensions and concentration for  
α
exp
 = 11.96 Np/m
r
eq
 [µm] 40 50 60 70 80
N [mm
–3
] 25.5 8.0 3.2 1.4 0.68
Fig. 12.  Power fitting of scatterer concentration N  
as function of r
eq
Solving the proposed simplified model given by 
Eq. (11), in order to extract the scatterer concentration:
 N
C
r
eq
n



11 96 .
, (14)
and comparing the Eqs. (13) and (14), results 
C = 2.65 × 10
13
 and n = 5.173, which allows writing Eq. 
(11) as:
  

26 51 0
13
5 173
.,
.
Nr
eq
 (15)
where the attenuation [Np/m] is a function of the 
fifth power of the scatterer size, which is not far 
from the trend of Eq. (1), as expected. It should be 
pointed out that Eq. (15) resulted from the studied cast 
iron samples illustrated in Fig. 4, whose attenuation 
was experimentally calculated by Eq. (5). Thus, 
based on the previous knowledge of experimental 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)5, 245-255
253 Ultrasonic Scattering Attenuation in Nodular Cast Iron: Experimental and Simulation Studies
attenuation, simulation models can be created 
varying the parameters N and r
eq
, according to Eq. 
(14). This allows mimicking structures with complex 
scatterer distributions in terms of uniform scatterer 
concentration and size.
The presented simulation model considers 
the geometrical factors (size and concentration of 
scatterers) and can be used to mimic the microstructure 
of cast iron for a specific excitation frequency. The 
scatterer equivalent size can be defined a priori and 
the concentration calculated by Eq. (14) to establish 
the simulation model.
3  CONCLUSIONS
This work aimed to evaluate the ultrasonic scattering 
attenuation of structures with complex scatterer 
distributions. The idea was to predict the attenuation 
behaviour for such complex structures by developing 
a simulation model based on the experimental 
attenuation results. That was motivated by the 
limitations presented by the Truell and Papadakis 
theoretical models that provide ultrasonic scattering 
attenuation results quite inconsistently when 
compared with the experimental one. That conclusion 
was observed for nodular cast-iron structures, and 
the authors believe that the same results are expected 
for materials having inhomogeneous scatterer sizes 
and concentrations. The k-Wave simulation model 
developed in this work allows an easy and customized 
implementation, where the scatterer size and 
concentration can be varied to mimic the experimental 
ultrasonic scattering attenuation measured in 
inhomogeneous structures.
4  ACKNOWLEDGEMENTS
This research is sponsored by FEDER funds through 
the program COMPETE – Programa Operacional 
Factores de Competitividade – and by national funds 
through FCT – Fundação para a Ciência e a Tecnologia 
– under the project UIDB/00285/2020.
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