# Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures

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## Abstract

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## 1. Introduction

“The term “attenuation” is used throughout to mean energy losses (as measured by amplitude decay) arising from all causes when ultrasonic waves are propagated through a solid medium. These “total” losses can be classed broadly as scattering and absorption arising from the intrinsic physical character of the solid under study, as well as diffraction, geometrical, and coupling losses.”

## 2. Methods

#### 2.1. Scattering Theory

#### 2.1.1. Geometric Correlation Function

#### 2.1.2. Spatial Scattering Function

#### 2.2. Geometric Modelling of Polycrystalline Microstructures

#### 2.2.1. Laguerre Tessellations

#### 2.2.2. Grain Size Distribution

#### 2.2.3. Fitting the Geometric Model Based on 2D Image Data

#### 2.3. Simulation of Wave Propagation

#### 2.3.1. Reciprocity Relations

#### 2.3.2. Modelling of the Bandwidth

#### 2.3.3. Evaluation Tools

- the low-frequency Rayleigh regime$\frac{\pi f}{{v}_{\alpha}}\phantom{\rule{0.166667em}{0ex}}{d}_{\mathrm{eff}}\ll 1$,
- the stochastic regime$\frac{\pi f}{{v}_{\alpha}}\phantom{\rule{0.166667em}{0ex}}{d}_{\mathrm{eff}}\le 1$ and
- the high-frequency geometric limit$\frac{\pi f}{{v}_{\alpha}}\phantom{\rule{0.166667em}{0ex}}{d}_{\mathrm{eff}}>1$.

## 3. Materials

#### 3.1. Computation Environment

#### 3.2. Inconel-617

#### 3.3. Titanium

## 4. Results

#### 4.1. Inconel-617

#### 4.1.1. Microstructure Model

#### 4.1.2. Model Based Scattering Investigation in Inconel-617

#### 4.2. Titanium

#### 4.2.1. Model Fit

#### 4.2.2. Model Based Scattering Investigation in Titanium

#### 4.3. Summary

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FFT | Fast Fourier Transformation |

DCT | Diffraction Contrast Tomography |

ESRF | European Synchrotron Radiation Facility |

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**Figure 1.**(

**a**) Experimental setup assumed throughout including the particular choice of the global Cartesian coordinate system with top-down z-axis. The transducer is both transmitter and receiver for longitudinally polarized waves. In (

**b**), dots mark the centers of the cells visualized in (

**a**). The entire microstructure contributes to the received backscattered signal.

**Figure 3.**Spatial scattering functions $\eta (\vartheta ,\phi ,2\pi \phantom{\rule{0.166667em}{0ex}}10\mathrm{MHz})$ for a single scatterer of size ${d}_{\mathrm{eff}}=54.5\phantom{\rule{4pt}{0ex}}$µm in nickel. For every unit vector $({x}_{1},{x}_{2},{x}_{3})=\left(sin\vartheta cos\phi ,\phantom{\rule{0.166667em}{0ex}}sin\vartheta sin\phi ,\phantom{\rule{0.166667em}{0ex}}cos\vartheta \right)$, the function value $\eta (\vartheta ,\phi ,2\pi \phantom{\rule{0.166667em}{0ex}}10\mathrm{MHz})$ is represented here as the length of the vector in the corresponding direction.

**Figure 4.**2D illustration for the construction of a Laguerre tessellation. The cyan circles are the generators with their weights, the red solid lines are the resulting faces of the Laguerre tessellation.

**Figure 5.**Volume renderings of force biased packings of 500 spheres in the unit window ${[0,1]}^{3}$. Both with volume fraction ${V}_{V}=52.1\%$.

**Left**: with constant sphere volume $V=0.011$.

**Right**: with log-normally distributed sphere volumes (${\mu}_{{v}_{\mathrm{s}}}=-7.67$ and ${\sigma}_{{v}_{\mathrm{s}}}=1.26$).

**Figure 6.**Simulated frequency bandwidth with left bound ${f}_{\mathrm{l}}=0.5f$ and right bound ${f}_{\mathrm{r}}=1.5f$.

**Figure 7.**Micrograph of the Inconel-617 under investigation, cell boundaries are emphasized by etching. Micrographs from [32] courtesy of Thomas Schwender (Fraunhofer IZFP).

**Figure 8.**Rendering of the DCT image data of the titanium sample. The imaged cylinder has a diameter of 518 µm and a height of 400 µm. In (

**a**), boundary grains on top and bottom are removed to emphasize the shape and packing of grains. Subfigure (

**b**) shows a cluster of grains from this data set. The grain colors are randomly chosen in order to visually separate neighboring grains and their shapes.

**Figure 9.**Visualization of the model fit for the Inconel-617.

**Left**: Rendering of a realization of the random Laguerre tessellation with overall 2000 cells in a cube of edge length 3.91 mm, and volume 60 mm${}^{3}$.

**Right**: 2D sections compared with the micrographs. The cells are visualized by false color mapping.

**Figure 10.**One of the original micrographs of the Inconel-617 and a rendering of a realization of the fitted Laguerre tessellation model with ${\mathrm{cv}}_{\mathrm{s}}=3.5$ and 2000 cells.

**Figure 11.**Histograms of maximal Feret diameters measured in the micrographs of the Inconel-617 and in 2D sections of realizations of the fitted Laguerre tessellation model with (${\mathrm{cv}}_{\mathrm{s}}=3.5$ and 2000 cells) slices.

**Figure 12.**Effective cell diameters in ten realizations of the Inconel-617 microstructure model. The solid and dashed horizontal lines correspond to the upper boundaries of the Rayleigh regime for the pressure and shear waves, respectively. Colors code the frequency, see legend. Clearly, the Rayleigh regime is violated in the 5 MHz case.

**Figure 13.**Spatial scattering function ${\eta}_{\mathrm{P}\to \mathrm{P}}\left(\vartheta ,\phi ,2\pi f\right)$ (in mm${}^{-3}$) for a single scatterer of size ${d}_{\mathrm{eff}}=329$ µm, testing frequency (

**a**) $f=0.5$ MHz, ${d}_{\mathrm{eff}}\pi f/{v}_{\alpha}=0.15$ and (

**b**) $f=2.25$ MHz, ${d}_{\mathrm{eff}}\pi f/{v}_{\alpha}=0.68$, respectively. Thus, according to Equation (16), for $f=2.25$, we are in the stochastic regime.

**Figure 14.**Backscattered signals (indicated by ten arbitrary colors) for a sequence of testing frequencies $f=$ 0.5, 1, 2.25, and 5 MHz. The corresponding transducer is modeled by a circle with radius 3 mm, which emits and receives longitudinal waves.

**Figure 15.**Histograms of the grain volumes of the titanium sample and realizations of the model fitted to it (${\mathrm{cv}}_{\mathrm{s}}=2.34$).

**Figure 17.**Effective cell diameters in ten realizations of the microstructure model for the titanium. Clearly, the Rayleigh regime is violated in the 50 MHz case.

**Figure 18.**Spatial scattering function ${\eta}_{\mathrm{P}\to \mathrm{P}}\left(\phi ,\vartheta ,2\pi f\right)$ (in mm${}^{-3}$) for a single scatterer size ${d}_{\mathrm{eff}}=36.4$ µm, testing frequency (

**a**) $f=5$ MHz, ${d}_{\mathrm{eff}}\pi f/{v}_{\alpha}=0.09$ and (

**b**) $f=30$ MHz, ${d}_{\mathrm{eff}}\pi f/{v}_{\alpha}=0.57$, respectively. Thus, according to Equation (16), for $f=30$ we are in the stochastic regime.

**Figure 19.**Received backscattered signals in titanium for a sequence of testing frequencies $f=$ 5, 10, 20, 30 and 50 MHz. The corresponding transducer is modeled as a circle with diameters 3 mm for frequency 50 MHz and 6 mm for all other frequencies, which emits and receives longitudinal waves.

Single Crystal | Polycrystal | Density | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{44}$ | ${\mathit{\lambda}}_{\mathbf{Lam\xe9}}$ | ${\mathit{\mu}}_{\mathbf{Lam\xe9}}$ | $\mathit{\varrho}$ | Featured | |

$\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\mathbf{GPa}\right]$ | $\left[\frac{\mathit{g}}{{\mathbf{cm}}^{3}}\right]$ | in Section | |

nickel | 250 | - | - | 160 | 118.5 | - | - | 8.905 | Section 2.1.2 |

Inconel-617 | 243.3 | - | - | 163.05 | 134.3 | 134.5 | 82.9 | 8.36 | Section 3.2 and Section 4.1 |

titanium | 160 | 66 | 181 | 90 | 46.5 | 76.17 | 43.39 | 4.51 | Section 3.3 and Section 4.2 |

**Table 2.**Ensemble averaged elastic constants (in GPa${}^{2}$) as used for the calculation of backscattering coefficients [40].

Abbreviation | Ensemble Averaged Elastic Constants | Nickel | Inconel-617 | Titanium |
---|---|---|---|---|

${A}_{1}$ | $\langle {\left({C}_{1133}^{\prime}\right)}^{2}\rangle $ | $370.44$ | $608.15$ | $39.86$ |

${A}_{2}$ | $\langle {\left({C}_{3333}^{\prime}\right)}^{2}\rangle $ | $658.55$ | $1\phantom{\rule{0.166667em}{0ex}}081.12$ | $30.96$ |

${A}_{4}$ | $\langle {\left({C}_{1233}^{\prime}\right)}^{2}\rangle $ | $205.80$ | $337.86$ | $29.56$ |

${A}_{5}$ | $\langle {C}_{1133}^{\prime}{C}_{3333}^{\prime}\rangle $ | $-329.27$ | $-540.58$ | $-17.84$ |

${A}_{6}$ | $\langle {\left({C}_{1333}^{\prime}\right)}^{2}\rangle $ | $411.59$ | $675.72$ | $7.14$ |

${A}_{7}$ | $\langle {C}_{1113}^{\prime}{C}_{2213}^{\prime}\rangle $ | $-102.90$ | $-168.93$ | $-12.71$ |

${A}_{8}$ | $\langle {\left({C}_{1213}^{\prime}\right)}^{2}\rangle $ | $205.80$ | $337.86$ | $15.53$ |

${A}_{9}$ | $\langle {C}_{1113}^{\prime}{C}_{1333}^{\prime}\rangle $ | $-308.69$ | $-506.79$ | $3.79$ |

${A}_{10}$ | $\langle {\left({C}_{1313}^{\prime}\right)}^{2}\rangle $ | $370.44$ | $608.15$ | $21.16$ |

**Table 3.**Summary of the Inconel-617 microstructure model (${\mathrm{cv}}_{\mathrm{s}}=3.5$ and 16,000 cells) realizations.

Realization | $\underset{\mathit{i}}{std}\left({\mathit{V}}_{\mathbf{eff}}\left({\mathit{C}}_{\mathit{i}}\right)\right)$ | $\underset{\mathit{i}}{min}\left({\mathit{d}}_{\mathbf{eff}}\left({\mathit{C}}_{\mathit{i}}\right)\right)$ | $\underset{\mathit{i}}{max}\left({\mathit{d}}_{\mathbf{eff}}\left({\mathit{C}}_{\mathit{i}}\right)\right)$ |
---|---|---|---|

Nr. | $\left({\mathbf{mm}}^{3}\right)$ | $\left(\mathsf{\mu}\mathit{m}\right)$ | $\left(\mathsf{\mu}\mathit{m}\right)$ |

1 | 0.0715 | 62 | 1039 |

2 | 0.0739 | 64 | 1131 |

3 | 0.0693 | 45 | 967 |

4 | 0.0693 | 57 | 1002 |

5 | 0.0723 | 60 | 1032 |

6 | 0.0701 | 62 | 896 |

7 | 0.0684 | 59 | 956 |

8 | 0.0729 | 59 | 1137 |

9 | 0.0669 | 61 | 860 |

10 | 0.0656 | 56 | 1099 |

mean | 0.0700 | 58.5 | 1011 |

**Table 4.**Results for the titanium model realizations (${\mathrm{cv}}_{\mathrm{s}}=2.34$ and 17,344 cells).

Nr. | $\underset{\mathit{i}}{std}\left({\mathit{C}}_{\mathit{i}}\right)$ | $\underset{\mathit{i}}{min}\left({\mathit{d}}_{\mathbf{eff}}\left({\mathit{C}}_{\mathit{i}}\right)\right)$ | $\underset{\mathit{i}}{max}\left({\mathit{d}}_{\mathbf{eff}}\left({\mathit{C}}_{\mathit{i}}\right)\right)$ |
---|---|---|---|

in ${10}^{4}\left(\mathsf{\mu}{\mathit{m}}^{3}\right)$ | $\left(\mathsf{\mu}\mathit{m}\right)$ | ($\mathsf{\mu}\mathit{m}$) | |

1 | 5.540 | 5 | 79 |

2 | 5.607 | 7 | 81 |

3 | 5.888 | 7 | 83 |

4 | 6.426 | 7 | 108 |

5 | 5.908 | 7 | 88 |

6 | 5.615 | 6 | 81 |

7 | 6.509 | 7 | 112 |

8 | 6.496 | 7 | 104 |

9 | 5.922 | 7 | 91 |

10 | 5.897 | 7 | 81 |

mean | 5.980 | 6.7 | 90.8 |

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**MDPI and ACS Style**

Dobrovolskij, D.; Schladitz, K. Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures. *Acoustics* **2022**, *4*, 139-167.
https://doi.org/10.3390/acoustics4010010

**AMA Style**

Dobrovolskij D, Schladitz K. Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures. *Acoustics*. 2022; 4(1):139-167.
https://doi.org/10.3390/acoustics4010010

**Chicago/Turabian Style**

Dobrovolskij, Dascha, and Katja Schladitz. 2022. "Simulation of Ultrasonic Backscattering in Polycrystalline Microstructures" *Acoustics* 4, no. 1: 139-167.
https://doi.org/10.3390/acoustics4010010