# 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 |

## References

- Gros, X. NDT Data Fusion; Knovel Library: London, UK, 1997. [Google Scholar]
- Spies, M.; Rieder, H.; Rauhut, M.; Kreier, P.; Innotest, A. Surface, Near-Surface and Volume Inspection of Cast Components Using Complementary NDT Approaches. In Proceedings of the 19th World Conference on Non-Destructive Testing 2016, Munich, Germany, 13–17 June 2016. [Google Scholar]
- Ashcroft, N.W.; Mermin, N.D. Festkörperphysik; De Gruyter: Berlin, Germany, 2001. [Google Scholar]
- Okazaki, K.; Conrad, H. Grain size distribution in recrystallized alpha-titanium. Trans. Jpn. Inst. Met.
**1972**, 13, 198–204. [Google Scholar] [CrossRef] [Green Version] - Fan, Z.; Wu, Y.; Zhao, X.; Lu, Y. Simulation of polycrystalline structure with Voronoi diagram in Laguerre geometry based on random closed packing of spheres. Comput. Mater. Sci.
**2004**, 29, 301–308. [Google Scholar] [CrossRef] - Mason, W.P.; McSkimin, H. Attenuation and scattering of high frequency sound waves in metals and glasses. J. Acoust. Soc. Am.
**1947**, 19, 464–473. [Google Scholar] [CrossRef] - Truell, R.; Elbaum, C.; Chick, B.B. Ultrasonic Methods in Solid State Physics; Academic Press: New York, NY, USA, 1969. [Google Scholar]
- Stanke, F.E.; Kino, G. A unified theory for elastic wave propagation in polycrystalline materials. J. Acoust. Soc. Am.
**1984**, 75, 665–681. [Google Scholar] [CrossRef] - Ishimaru, A. Wave Propagation and Scattering in Random Media; Academic Press: New York, NY, USA, 1978; Volume 1–2. [Google Scholar]
- Rose, J.H. Ultrasonic backscatter from microstructure. Rev. Prog. Quant. Nondestruct. Eval. Vol. 11B
**1992**, 11, 1677–1684. [Google Scholar] - Dorval, V.; Jenson, F.; Corneloup, G.; Moysan, J. Accounting for structural noise and attenuation in the modeling of the ultrasonic testing of polycrystalline materials. In Proceedings of the AIP Conference Proceedings, Penang, Malaysia, 21–23 December 2010; Volume 1211, pp. 1309–1316. [Google Scholar]
- Ganjehi, L.; Dorval, V.; Jenson, F. Modelling of the ultrasonic propagation in polycrystalline materials. Acoustics
**2012**, 2012, 2621–2626. [Google Scholar] - Hirsekorn, S. Theoretical description of ultrasonic propagation and scattering phenomena in polycrystalline structures aiming for simulations on nondestructive materials characterization and defect detection. In Proceedings of the 11th ECNDT Conference, Prague, Czech Republic, 6–11 October 2014. [Google Scholar]
- Bachmann, F.; Hielscher, R.; Schaeben, H. Grain detection from 2d and 3d EBSD data—Specification of the MTEX algorithm. Ultramicroscopy
**2011**, 111, 1720–1733. [Google Scholar] [CrossRef] - Alpers, A.; Brieden, A.; Gritzmann, P.; Lyckegaard, A.; Poulsen, H.F. Generalized balanced power diagrams for 3D representations of polycrystals. Philos. Mag.
**2015**, 95, 1016–1028. [Google Scholar] [CrossRef] [Green Version] - Šedivỳ, O.; Brereton, T.; Westhoff, D.; Polívka, L.; Beneš, V.; Schmidt, V.; Jäger, A. 3D reconstruction of grains in polycrystalline materials using a tessellation model with curved grain boundaries. Philos. Mag.
**2016**, 96, 1926–1949. [Google Scholar] [CrossRef] - Šedivỳ, O.; Dake, J.M.; Krill III, C.E.; Schmidt, V.; Jäger, A. Description of the 3D morphology of grain boundaries in aluminum alloys using tessellation models generated by ellipsoids. Image Anal. Stereol.
**2017**, 36, 5–13. [Google Scholar] [CrossRef] [Green Version] - Šedivỳ, O.; Westhoff, D.; Kopeček, J.; Krill III, C.E.; Schmidt, V. Data-driven selection of tessellation models describing polycrystalline microstructures. J. Stat. Phys.
**2018**, 172, 1223–1246. [Google Scholar] [CrossRef] - Teferra, K.; Rowenhorst, D.J. Direct parameter estimation for generalised balanced power diagrams. Philos. Mag. Lett.
**2018**, 98, 79–87. [Google Scholar] [CrossRef] - Petrich, L.; Staněk, J.; Wang, M.; Westhoff, D.; Heller, L.; Šittner, P.; Krill, C.E.; Beneš, V.; Schmidt, V. Reconstruction of Grains in Polycrystalline Materials From Incomplete Data Using Laguerre Tessellations. Microsc. Microanal.
**2019**, 25, 743–752. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lyckegaard, A.; Lauridsen, E.M.; Ludwig, W.; Fonda, R.W.; Poulsen, H.F. On the Use of Laguerre Tessellations for Representations of 3D Grain Structures. Adv. Eng. Mater.
**2011**, 13, 165–170. [Google Scholar] [CrossRef] - Ghoshal, G.; Turner, J.A. Numerical model of longitudinal wave scattering in polycrystals. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2009**, 56, 1419–1428. [Google Scholar] [CrossRef] [PubMed] - Shivaprasad, S.; Krishnamurthy, C.; Balasubramaniam, K. Modeling and simulation of ultrasonic beam skewing in polycrystalline materials. Int. J. Adv. Eng. Sci. Appl. Math.
**2018**, 10, 70–78. [Google Scholar] [CrossRef] - Ryzy, M.; Grabec, T.; Sedlák, P.; Veres, I.A. Influence of grain morphology on ultrasonic wave attenuation in polycrystalline media with statistically equiaxed grains. J. Acoust. Soc. Am.
**2018**, 143, 219–229. [Google Scholar] [CrossRef] [Green Version] - Van Pamel, A.; Brett, C.R.; Huthwaite, P.; Lowe, M.J. Finite element modelling of elastic wave scattering within a polycrystalline material in two and three dimensions. J. Acoust. Soc. Am.
**2015**, 138, 2326–2336. [Google Scholar] [CrossRef] [Green Version] - Van Pamel, A.; Sha, G.; Rokhlin, S.I.; Lowe, M.J. Finite-element modelling of elastic wave propagation and scattering within heterogeneous media. Proc. R. Soc. A: Math. Phys. Eng. Sci.
**2017**, 473, 20160738. [Google Scholar] [CrossRef] - Van Pamel, A.; Sha, G.; Lowe, M.J.; Rokhlin, S.I. Numerical and analytic modelling of elastodynamic scattering within polycrystalline materials. J. Acoust. Soc. Am.
**2018**, 143, 2394–2408. [Google Scholar] [CrossRef] [Green Version] - Quey, R.; Dawson, P.; Barbe, F. Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing. Comput. Methods Appl. Mech. Eng.
**2011**, 200, 1729–1745. [Google Scholar] [CrossRef] [Green Version] - Huthwaite, P. Accelerated finite element elastodynamic simulations using the GPU. J. Comput. Phys.
**2014**, 257, 687–707. [Google Scholar] [CrossRef] [Green Version] - Standard ASTM E112; Standard Test Methods for Determining Average Grain Size. ASTM International: West Conshohocken, PA, USA, 2003.
- Margetan, F.J.; Nieters, E.; Haldipur, P.; Brasche, L.; Chiou, T.; Keller, M.; Degtyar, A.; Umbach, J.; Hassan, W.; Patton, T.; et al. Fundamental Studies of Nickel Billet Materials-Engine Titanium Consortium Phase II; National Technical Information Service (NTIS): Springfield, MA, USA, 2005.
- Walte, F.; Schwender, T.; Hirsekorn, S.; Schubert, F.; Spies, M. Reaktorsicherheitsforschung—Vorhaben-Nr.: 1501442 “Berechnung der Ultraschallstreuung für einen Verbesserten Nachweis von Rissartigen Fehlern in Austenitischen Schweissnähten. Phase 1: Berechnung der Ultraschallstreuung für 2DSchweissnahtmodelle”; Technical Report; Fraunhofer-Institut für Zerstörungsfreie Prüfverfahren IZFP: Saarbrücken, Germany, 2015. [Google Scholar]
- Tromans, D. Elastic anisotropy of HCP metal crystals and polycrystals. Int. J. Res. Rev. Appl. Sci
**2011**, 6, 462–483. [Google Scholar] - Shannon, C.E. Communication in the presence of noise. Proc. IRE
**1949**, 37, 10–21. [Google Scholar] [CrossRef] - Fließbach, T. Statistische Physik; Springer: Berlin/Heidelberg, Germany, 1993. [Google Scholar]
- Gaspard, P. Chaos, Scattering and Statistical Mechanics; Cambridge University Press: Cambridge, UK, 2005; Volume 9. [Google Scholar]
- Gubernatis, J.; Krumhansl, J. Macroscopic engineering properties of polycrystalline materials: Elastic properties. J. Appl. Phys.
**1975**, 46, 1875–1883. [Google Scholar] [CrossRef] - Weaver, R.L. Diffusivity of ultrasound in polycrystals. J. Mech. Phys. Solids
**1990**, 38, 55–86. [Google Scholar] [CrossRef] - Hirsekorn, S. Elastic properties of polycrystals: A review. Texture, Stress. Microstruct.
**1990**, 12, 1–14. [Google Scholar] [CrossRef] [Green Version] - Hirsekorn, S. The scattering of ultrasonic waves by multiphase polycrystals. J. Acoust. Soc. Am.
**1988**, 83, 1231–1242. [Google Scholar] [CrossRef] - Born, M. Quantenmechanik der stoßvorgänge. Z. Phys.
**1926**, 38, 803–827. [Google Scholar] [CrossRef] - Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 16. [Google Scholar]
- Illian, J.; Penttinen, A.; Stoyan, H.; Stoyan, D. Statistical Analysis and Modelling of Spatial Point Patterns; John Wiley & Sons: Hoboken, NJ, USA, 2008; Volume 70. [Google Scholar]
- Man, C.S.; Paroni, R.; Xiang, Y.; Kenik, E.A. On the geometric autocorrelation function of polycrystalline materials. J. Comput. Appl. Math.
**2006**, 190, 200–210. [Google Scholar] [CrossRef] - Arguelles, A.P.; Turner, J.A. Ultrasonic attenuation of polycrystalline materials with a distribution of grain sizes. J. Acoust. Soc. Am.
**2017**, 141, 4347–4353. [Google Scholar] [CrossRef] [PubMed] - Dobrovolskij, D.; Hirsekorn, S.; Spies, M. Simulation of Ultrasonic Materials Evaluation Experiments Including Scattering Phenomena due to Polycrystalline Microstructure. Phys. Procedia
**2015**, 70, 644–647. [Google Scholar] [CrossRef] - Chiu, S.N.; Stoyan, D.; Kendall, W.S.; Mecke, J. Stochastic Geometry and Its Applications; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Ohser, J.; Schladitz, K. 3D Images of Materials Structures: Processing and Analysis; John Wiley & Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- Xue, X.; Righetti, F.; Telley, H.; Liebling, T.M.; Mocellin, A. The laguerre model for grain growth in three dimensions. Philos. Mag. B
**1997**, 75, 567–585. [Google Scholar] [CrossRef] - Kühn, M.; Steinhauser, M.O. Modeling and simulation of microstructures using power diagrams: Proof of the concept. Appl. Phys. Lett.
**2008**, 93, 034102. [Google Scholar] [CrossRef] - Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S.N. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 501. [Google Scholar]
- Redenbach, C. Microstructure models for cellular materials. Comput. Mater. Sci.
**2009**, 44, 1397–1407. [Google Scholar] [CrossRef] - Geißendörfer, M.; Liebscher, A.; Proppe, C.; Redenbach, C.; Schwarzer, D. Stochastic multiscale modeling of metal foams. Probabilistic Eng. Mech.
**2014**, 37, 132–137. [Google Scholar] [CrossRef] - Kampf, J.; Schlachter, A.L.; Redenbach, C.; Liebscher, A. Segmentation, statistical analysis, and modelling of the wall system in ceramic foams. Mater. Charact.
**2015**, 99, 38–46. [Google Scholar] [CrossRef] - Abdullahi, H.; Liang, Y.; Gao, S. Predicting the elastic properties of closed-cell aluminum foams: A mesoscopic geometric modeling approach. SN Appl. Sci.
**2019**, 1, 380. [Google Scholar] [CrossRef] [Green Version] - Liebscher, A. Laguerre approximation of random foams. Philos. Mag.
**2015**, 95, 2777–2792. [Google Scholar] [CrossRef] - Mościński, J.; Bargieł, M.; Rycerz, Z.; Jacobs, P. The force-biased algorithm for the irregular close packing of equal hard spheres. Mol. Simul.
**1989**, 3, 201–212. [Google Scholar] [CrossRef] - Jodrey, W.S.; Tory, E.M. Simulation of random packing of spheres. Simulation
**1979**, 32, 1–12. [Google Scholar] [CrossRef] - Bezrukov, A.; Stoyan, D.; Bargieł, M. Spatial statistics for simulated packings of spheres. Image Anal. Stereol.
**2001**, 20, 203–206. [Google Scholar] [CrossRef] - Bezrukov, A.; Bargieł, M.; Stoyan, D. Statistical analysis of simulated random packings of spheres. Part. Part. Syst. Charact. Meas. Descr. Part. Prop. Behav. Powders Other Disperse Syst.
**2002**, 19, 111–118. [Google Scholar] [CrossRef] - He, D.; Ekere, N. Computer Simulation of Powder Compaction of Spherical Particles. J. Mater. Sci. Lett.
**1998**, 17, 1723–1725. [Google Scholar] [CrossRef] - Wu, Y.; Fan, Z.; Lu, Y. Bulk and interior packing densities of random close packing of hard spheres. J. Mater. Sci.
**2003**, 38, 2019–2025. [Google Scholar] [CrossRef] - Rhines, F.; Patterson, B. Effect of the degree of prior cold work on the grain volume distribution and the rate of grain growth of recrystallized aluminum. Metall. Trans. A
**1982**, 13, 985–993. [Google Scholar] [CrossRef] - Spies, M. Kirchhoff evaluation of scattered elastic wavefields in anisotropic media. J. Acoust. Soc. Am.
**2000**, 107, 2755–2759. [Google Scholar] [CrossRef] - Červenỳ, V. Reflection/transmission laws for slowness vectors in viscoelastic anisotropic media. Stud. Geophys. Geod.
**2007**, 51, 391–410. [Google Scholar] [CrossRef] - Auld, B. General electromechanical reciprocity relations applied to the calculation of elastic wave scattering coefficients. Wave Motion
**1979**, 1, 3–10. [Google Scholar] [CrossRef] - Scheben, R.; Rieder, H.; Spies, M.; Götz, S. Kopplung von EFIT und GPSS zur Schnellen Ultraschallsimulation; NDT.net. German Society of NDT: Berlin, Germany, 2010; Volume 2. [Google Scholar]
- Spies, M. Prediction of Ultrasonic Flaw Signals and Model-to-Experiment Comparison. In Proceedings of the AIP Conference Proceedings, Salt Lake, UT, USA, 10–11 August 2005; Volume 760, pp. 1851–1858. [Google Scholar]
- Frigo, M.; Johnson, S.G. FFTW: An adaptive software architecture for the FFT. In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP’98 (Cat. No. 98CH36181). Seattle, WA, USA, 12–15 May 1998; Volume 3, pp. 1381–1384. [Google Scholar]
- Frigo, M.; Johnson, S.G. FFTW: Fastest Fourier Transform in the West; Astrophysics Source Code Library, 2012; Available online: https://ui.adsabs.harvard.edu/abs/2012ascl.soft01015F/abstract (accessed on 21 July 2021).
- Dagum, L.; Menon, R. OpenMP: An industry standard API for shared-memory programming. IEEE Comput. Sci. Eng.
**1998**, 5, 46–55. [Google Scholar] [CrossRef] [Green Version] - Chandra, R.; Dagum, L.; Kohr, D.; Menon, R.; Maydan, D.; McDonald, J. Parallel Programming in OpenMP; Morgan Kaufmann: Burlington, MA, USA, 2001. [Google Scholar]
- Fraunhofer Institute for Industrial Mathematics ITWM. Cluster Homepage. 2021. Available online: https://www.itwm.fraunhofer.de/en/about-itwm/profile/central-it-infrastructure.html (accessed on 21 July 2021).
- Guo, Y.; Wang, B.; Hou, S. Aging precipitation behavior and mechanical properties of Inconel 617 superalloy. Acta Metall. Sin. (English Lett.)
**2013**, 26, 307–312. [Google Scholar] [CrossRef] [Green Version] - Poulsen, H.F. An introduction to three-dimensional X-ray diffraction microscopy. J. Appl. Crystallogr.
**2012**, 45, 1084–1097. [Google Scholar] [CrossRef] - Ludwig, W.; Reischig, P.; King, A.; Herbig, M.; Lauridsen, E.; Johnson, G.; Marrow, T.; Buffiere, J.Y. Three-dimensional grain mapping by x-ray diffraction contrast tomography and the use of Friedel pairs in diffraction data analysis. Rev. Sci. Instruments
**2009**, 80, 033905. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ducousso-Ganjehi, L.; Châtillon, S.; Dorval, V.; Gilles-Pascaud, C.; Jenson, F. Modelling of the Ultrasonic Propagation in Titanium Alloy Materials. In Proceedings of the AeroNDT 2012 4th International Symposium on NDT in Aerospace, Augsburg, Germany, 13–14 November 2012; NDT.net. German Society of NDT: Berlin, Germany, 2012. [Google Scholar]
- Pilchak, A.L.; Li, J.; Rokhlin, S.I. Quantitative comparison of microtexture in near-alpha titanium measured by ultrasonic scattering and electron backscatter diffraction. Metall. Mater. Trans. A
**2014**, 45, 4679–4697. [Google Scholar] [CrossRef]

**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