# Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background about Attenuation Modelling in Textured Polycrystals

#### 2.1. Orientation Distribution Function (ODF)

#### 2.1.1. Generalized Spherical Harmonics ODF

_{1}, Φ, φ

_{2}, which are shown in Figure 1. To rotate the global texture coordinate to the crystallite coordinate, one has to follow such a procedure [43]: first, rotate about the T

_{Z}axis through the angle φ

_{1}; then rotate about x’ axis by the angle Φ and finally rotate about z-axis by the angel φ

_{2}. Therefore, the rotation matrix in Bunge’s Euler rotation angles is:

#### 2.1.2. Gaussian form ODF

_{φ1}, σ

_{Φ}, σ

_{φ2}are texture parameters and the factor F

_{0}=S

_{1}csch(S

_{1})/[I(0,S

_{2})I[0,S

_{3}]] satisfies this equality $\frac{1}{8{\pi}^{2}}{\int}_{0}^{\pi}{\int}_{0}^{2\pi}{\int}_{0}^{2\pi}F\left(\Phi ,{\phi}_{1},{\phi}_{2}\right)sin\Phi d\Phi d{\phi}_{1}d{\phi}_{2}=1$. Here, S

_{1}=1/(2σ

_{Φ}), S

_{2}= 1/(2σ

_{φ1}), S

_{3}=1/(2σ

_{φ2}) and I(0,S

_{2}), I(0,S

_{3}) are modified Bessel functions of the first kind [50], and φ

_{1}, Φ, φ

_{2}are three Euler angles defined by Bunge notation as shown in Figure 1. In this representation, the texture parameters σ

_{φ1}, σ

_{Φ}, σ

_{φ2}in the Gaussian ODF can be directly related to the full width at half maxima (FWHMs) of texture angles φ

_{1}, Φ, φ

_{2}from experimental orientation imaging microscopy [39,47]. When all three parameters σ

_{φ1}, σ

_{Φ}, σ

_{φ2}approach infinity the ODF degenerates to the case of statistically isotropic polycrystals. When two of them go to infinity, the ODF would be equivalent to fiber (axisymmetric) texture like Refs. [39,53]. Note that ODFs in different notations are related to each other, for example, the relation between Bunge’s ODF and Roe’s ODF has been investigated in Ref. [54] and the conversion from 1D Gaussian ODF to Roe’s ODF has been proposed in Refs. [49,53].

#### 2.2. Theoretical Background of Attenuation Modelling

_{jα}means Kronecker delta (a second-order tensor) and ${C}_{ijkl}^{0}$ stands for the homogenized elastic constants. ω is the angular frequency. The elastic constant at a given location

**X**is equal to the elastic constants of its reference medium plus a spatial fluctuation term, namely:

**Y**and location

**Z**. The mass operator may be expressed in a diagrammatic form including infinite series [56]. However, the exact equation for the mass operator is difficult to obtain but finite order approximation is employed instead in references [35,56]. Under the weak scattering assumption $\left|\delta {c}_{ijkl}\right|/{C}_{ijkl}^{0}<<1$, the mass operator after first order smoothing approximation (FOSA) is [6]:

_{ijql}is the spatial variation of elastic constants. The mass operator in Equation (9) includes some second-order scattering events according to Refs. [35,56]. From Equation (9), the mass operator is relevant to the covariance of elastic constants fluctuation at two different points, namely the two-point correlation [5].

**p**is the unit wave normal vector, and $\otimes $ denotes dyadic product. The phase velocity V

_{M}and polarization vector ${\mathit{u}}^{M}$ at a certain wave propagation direction are determined through Christoffel’s Equation [40]:

_{j}is a component of wave normal vector

**p**and ρ denotes the mass density of the medium. The phase velocity V

_{M}is the square root of the eigenvalue of the matrix $\frac{{C}_{ijml}^{0}{n}_{j}{n}_{l}}{\rho}$ for wave mode M and polarization vector

**u**is the corresponding eigenvector. In a textured polycrystal, there are three wave modes: quasi-longitudinal (L) mode, quasi-transverse fast (T1) mode and quasi-transverse slow (T2) mode. Unlike the isotropic media, the polarization direction of a quasi-longitudinal wave generally deviates from its wave propagation direction while the polarization directions of a quasi-transverse wave are not perpendicular to its wave propagation direction. However, three polarization vectors are still mutually perpendicular to each other.

#### 2.3. Two-point Statistics for Textured Polycrystals

_{ijkl}(

**X**)δc

_{αβγδ}(

**X**’), a covariance of elastic constants fluctuation from two points

**X**and

**X**’ (see the schematic of a polycrystal in Figure 2). Since we are solely interested in the mean wave response in heterogeneous polycrystals, the ensemble average of this covariance is critical. From statistics of numerous grains, the ensemble average of elastic covariance can be further decomposed into two parts: the volumetric average of elastic constants variation covariance and geometric two-point correlation (GTPC) function [6,40]. The mathematical expression is:

_{ijk}δc

_{αβγδ}> is the ensemble average of elastic constants variation covariance for textured polycrystals by Voigt averaging [58]. It can be numerically calculated through single-crystal elastic constants and ODF by the following equation [40,50,59]:

_{ijkl}(φ

_{1}, Φ, φ

_{2}) and f(φ

_{1}, Φ, φ

_{2}) are the rotated single-crystal elastic tensor and ODF (either Gaussian ODF or generalized spherical harmonics ODF), respectively. The elastic constants rotation matrix and details about elastic constants covariance calculation can be found in Refs. [39,40,50,53,59]. It is worthy to mention that Voigt effective elastic constants (or called Voigt reference media) are used in this whole paper because it is more suitable for wave scattering modeling [23,60], although different homogenization methods like self-consistent reference medium are available for polycrystalline media [58,61]. Ref. [23] has compared the SOA models (for texture-free polycrystals) with Voigt reference medium and self-consistent reference medium to finite element modeling and found that the SOA model with Voigt reference medium agrees much better with FEM results.

**q**) is:

**k**and the scattered wavenumber ${\mathit{k}}^{S,N}$. Here N stands for a scattering wave mode in an anisotropic medium, namely a quasi-longitudinal mode, a quasi-transverse fast mode or a quasi-transverse slow mode.

## 3. Second-order Scattering Model for Textured Polycrystals with Ellipsoidal Triclinic Grains

#### 3.1. Dispersion Equation

**u**and

**v**are the polarization vectors corresponding to the incident wave and scattered wave, respectively.

#### 3.2. Analytical Equations for Attenuation Coefficient and Phase Velocity

#### 3.2.1. Attenuation Coefficient and Phase Velocity by the Born Approximation

#### 3.2.2. Attenuation at Rayleigh Limit

_{M}approaches zero and $W\left(\mathit{k}-{\mathit{k}}_{}^{S}\right)=\frac{{R}_{0}{R}_{1}{a}_{X}^{3}}{{\pi}^{2}}$, thus the attenuation coefficient at Rayleigh limit ${\alpha}_{M}^{R}\left(\mathit{p}\right)$ by Equation (35) becomes:

#### 3.2.3. Quasi-Static Phase Velocity

_{M}reaches zero, the Cauchy principal value can be analytically obtained by contour integral. After much simplification, the real part of the perturbed wavenumber at Rayleigh limit is obtained as:

**p**also can be obtained as:

_{0}and R

_{1}), texture (affecting wave velocity and inner product) and wave scattering. For a polycrystal with known texture coefficient, single-crystal elastic constants, and grain sizes, the quasi-static velocity of an incident wave (QL, QT1 or QT2) at a certain direction can be predicted by Equation (40) and the elastic constants of the homogenized polycrystal also can be obtained inversely if sufficient phase velocities at different angles are available. For textured polycrystals with equiaxed grains, the expression for the quasi-static velocity can be further simplified as:

## 4. Computational Results and Discussion

#### 4.1. Comparison of the SOA Model to Available FEM and Experimental Results

#### 4.2. Applicability and Limitation of the Born Approximation on Textured Polycrystals

_{X}so that the horizontal and vertical axes are dimensionless. It can be seen from Figure 5 that the Born approximation is in reasonable agreement with our SOA model below the geometric region (also refer to the mean deviation in Table 2) and the transition frequency to geometric region varies in different directions. The deviation is defined as $\left({\alpha}_{Born}-{\alpha}_{SOA}\right)/{\alpha}_{SOA}$. The signs for all mean deviations at different wave propagation directions in Table 2 are negative, which indicates that the Born approximation underestimates the wave attenuation because it accounts for fewer scattering events than the SOA model. In the geometric region, the Born approximation [40] breaks due to its intrinsic limitation, and the break frequency ${k}_{L}{a}_{X}$ of the Born approximation for each direction is also listed in Table 2. Since the departure frequency ${k}_{L}{a}_{X}$ of the Born approximation from the SOA model is comparable to the stochastic-to-geometric transition frequency in the SOA model (sharp corner shown in Figure 5 from our SOA model), this departure frequency is also governed by the grain elongation and longitudinal wave scattering strength at high frequency. The attenuation at the geometric region from this work for textured polycrystals is also inversely proportional to grain size in the wave propagation direction as texture-free polycrystal case in Ref. [64,80], thus attenuation coefficients in geometric region predicated by the SOA model obey this relation: ${\alpha}_{X}>{\alpha}_{Y}>{\alpha}_{Z}.$ In the geometric region, the attenuation is so high that the penetration depth of an elastic wave is limited to a couple of grains [81].

#### 4.3. Comparison of Quasi-Static Velocity from the SOA Model with Other Velocity Bounds

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Mason, W.P.; McSkimin, H.J. Attenuation and Scattering of High Frequency Sound Waves in Metals and Glasses. J. Acoust. Soc. Am.
**1947**, 19, 725. [Google Scholar] [CrossRef][Green Version] - Papadakis, E.P. Rayleigh and Stochastic Scattering of Ultrasonic Waves in Steel. J. Appl. Phys.
**1963**, 34, 265–269. [Google Scholar] [CrossRef] - Stanke, F.E.; Kino, G.S. A Unified Theory for Elastic Wave Propagation in Polycrystalline Materials. J. Acoust. Soc. Am.
**1984**, 75, 665. [Google Scholar] [CrossRef] - Hirsekorn, S. The Scattering of Ultrasonic Waves by Polycrystals. J. Acoust. Soc. Am.
**1982**, 72, 1021–1031. [Google Scholar] [CrossRef] - Weaver, R.L. Diffusivity of Ultrasound in Polycrystals. J. Mech. Phys. Solids
**1990**, 38, 55–86. [Google Scholar] [CrossRef] - Turner, J.A. Elastic Wave Propagation and Scattering in Heterogeneous, Anisotropic Media: Textured Polycrystalline Materials. J. Acoust. Soc. Am.
**1999**, 106, 541. [Google Scholar] [CrossRef][Green Version] - Yang, L.; Turner, J.A. Attenuation of Ultrasonic Waves in Rolled Metals. J. Acoust. Soc. Am.
**2004**, 116, 3319–3327. [Google Scholar] [CrossRef][Green Version] - Yang, L.; Lobkis, O.I.; Rokhlin, S.I. Explicit Model for Ultrasonic Attenuation in Equiaxial Hexagonal Polycrystalline Materials. Ultrasonics
**2011**, 51, 303–309. [Google Scholar] [CrossRef] - Lobkis, O.I.; Yang, L.; Li, J.; Rokhlin, S.I. Ultrasonic Backscattering in Polycrystals with Elongated Single Phase and Duplex Microstructures. Ultrasonics
**2012**, 52, 694–705. [Google Scholar] [CrossRef] - Du, H.; Turner, J.A. Ultrasonic Attenuation in Pearlitic Steel. Ultrasonics
**2014**, 54, 882–887. [Google Scholar] [CrossRef] [PubMed] - Papadakis, E.P. Physical Acoustics and Microstructure of Iron Alloys. Int. Met. Rev.
**1984**, 29, 1–24. [Google Scholar] [CrossRef] - Guo, C.B.; Höller, P.; Goebbels, K. Scattering of Ultrasonic Waves in Anisotropic Polycrystalline Metals. Acta Acust. united with Acust.
**1985**, 59, 112–120. [Google Scholar] - Gobran, N.K.; Youssef, H. Viscous Ultrasonic Attenuation in Metals. J. Appl. Phys.
**1967**, 38, 3291–3293. [Google Scholar] [CrossRef] - Nowick, A.S.; Berry, B.S.; Katz, J.L. Anelastic Relaxation in Crystalline Solids; Academic Press: New York, NY, USA; London, UK, 1972; p. 667. [Google Scholar]
- Ryzy, M.; Grabec, T.; Österreicher, J.A.; Hettich, M.; Veres, I.A. Measurement of Coherent Surface Acoustic Wave Attenuation in Polycrystalline Aluminum. AIP Adv.
**2018**, 8, 125019. [Google Scholar] [CrossRef][Green Version] - Papadakis, E.P. Grain-Size Distribution in Metals and Its Influence on Ultrasonic Attenuation Measurements. J. Acoust. Soc. Am.
**1961**, 33, 1616–1621. [Google Scholar] [CrossRef] - Bai, X.; Zhao, Y.; Ma, J.; Liu, Y.; Wang, Q. Grain-Size Distribution Effects on the Attenuation of Laser-Generated Ultrasound in $α$-Titanium Alloy. Materials (Basel).
**2019**, 12, 102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sha, G. A Simultaneous Non-Destructive Characterisation Method for Grain Size and Single-Crystal Elastic Constants of Cubic Polycrystals from Ultrasonic Measurements. Insight - Non-Destructive Test. Cond. Monit.
**2018**, 60, 190–193. [Google Scholar] [CrossRef] - Li, X.; Song, Y.; Liu, F.; Hu, H.; Ni, P. Evaluation of Mean Grain Size Using the Multi-Scale Ultrasonic Attenuation Coefficient. NDT E Int.
**2015**, 72, 25–32. [Google Scholar] [CrossRef] - Van Pamel, A.; Brett, C.R.; Huthwaite, P.; Lowe, M.J.S. 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] - Bai, X.; Tie, B.; Schmitt, J.-H.; Aubry, D. Finite Element Modeling of Grain Size Effects on the Ultrasonic Microstructural Noise Backscattering in Polycrystalline Materials. Ultrasonics
**2018**, 87, 182–202. [Google Scholar] [CrossRef] - Lhuillier, P.E.; Chassignole, B.; Oudaa, M.; Kerhervé, S.O.; Rupin, F.; Fouquet, T. Investigation of the Ultrasonic Attenuation in Anisotropic Weld Materials with Finite Element Modeling and Grain-Scale Material Description. Ultrasonics
**2017**, 78, 40–50. [Google Scholar] [CrossRef] [PubMed] - Van Pamel, A.; Sha, G.; Lowe, M.J.S.; Rokhlin, S.I. Numerical and Analytic Modelling of Elastodynamic Scattering within Polycrystalline Materials. J. Acoust. Soc. Am.
**2018**, 14, 2394–2408. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, L.; Lobkis, O.I.; Rokhlin, S.I. Shape Effect of Elongated Grains on Ultrasonic Attenuation in Polycrystalline Materials. Ultrasonics
**2011**, 51, 697–708. [Google Scholar] [CrossRef] [PubMed] - Kube, C.M.; Turner, J.A. Stress-Dependent Second-Order Grain Statistics of Polycrystals. J Acoust Soc Am
**2015**, 138, 2613–2625. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, L.; Turner, J.A. Wave Attenuations in Solids with Perfectly Aligned Cracks. Acoust. Res. Lett. Online
**2005**, 6, 99–105. [Google Scholar] [CrossRef] - Kube, C.M.; Turner, J.A. Stress-Dependent Ultrasonic Scattering in Polycrystalline Materials. J. Acoust. Soc. Am.
**2016**, 139, 811–824. [Google Scholar] [CrossRef][Green Version] - Ahmed, S.; Thompson, R.B.; Panetta, P.D. Ultrasonic Attenuation as Influenced by Elongated Grains. Rev. Quant. Nondestruct. Eval. ed. by D. O. Thompson D. E. Chimenti
**2003**, 22, 109–116. [Google Scholar] - Karal, F.C.; Keller, J.B. Elastic, Electromagnetic, and Other Waves in a Random Medium. J. Math. Phys.
**1964**, 5, 537. [Google Scholar] [CrossRef] - Hirsekorn, S. The Scattering of Ultrasonic Waves by Polycrystals. II. Shear Waves. J Acoust Soc Am
**1983**, 73, 1160–1163. [Google Scholar] [CrossRef] - Page, J.H.; Sheng, P.; Schriemer, H.P.; Jones, I.; Jing, X.; Weitz, D.A. Group Velocity in Strongly Scattering Media. Science
**1996**, 271, 634–637. [Google Scholar] [CrossRef][Green Version] - Calvet, M.; Margerin, L. Velocity and Attenuation of Scalar and Elastic Waves in Random Media: A Spectral Function Approach. J. Acoust. Soc. Am.
**2012**, 131, 1843. [Google Scholar] [CrossRef] [PubMed] - Kube, C.M. Iterative Solution to Bulk Wave Propagation in Polycrystalline Materials. J. Acoust. Soc. Am.
**2017**, 141, 1804–1811. [Google Scholar] [CrossRef] [PubMed] - Calvet, M.; Margerin, L. Impact of Grain Shape on Seismic Attenuation and Phase Velocity in Cubic Polycrystalline Materials. Wave Motion
**2016**, 65, 29–43. [Google Scholar] [CrossRef] - Frisch, U. Probabilistic Methods in Applied Mathematics; Vols. I II, Ed.; Bharucha-Reid Academic Press: New York, NY, USA, 1968. [Google Scholar]
- Rokhlin, S.I.; Li, J.; Sha, G. Far-Field Scattering Model for Wave Propagation in Random Media. J. Acoust. Soc. Am.
**2015**, 137, 2655–2669. [Google Scholar] [CrossRef] [PubMed] - Kocks, U.F.; Tomé, C.N.; Wenk, H.-R. Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Ahmed, S.; Thompson, R.B. Propagation of Elastic Waves in Equiaxed Stainless-Steel Polycrystals with Aligned [001] Axes. J. Acoust. Soc. Am.
**1996**, 99, 2086. [Google Scholar] [CrossRef] - Yang, L.; Rokhlin, S.I. Ultrasonic Backscattering in Cubic Polycrystals with Ellipsoidal Grains and Texture. J. Nondestruct. Eval.
**2013**, 32, 142–155. [Google Scholar] [CrossRef] - Li, J.; Rokhlin, S.I. Propagation and Scattering of Ultrasonic Waves in Polycrystals with Arbitrary Crystallite and Macroscopic Texture Symmetries. Wave Motion
**2015**, 58, 145–164. [Google Scholar] [CrossRef] - Hirsekorn, S. The Scattering of Ultrasonic Waves in Polycrystalline Materials with Texture. J. Acoust. Soc. Am.
**1985**, 77, 832–843. [Google Scholar] [CrossRef] - Norouzian, M.; Turner, J.A. Ultrasonic Wave Propagation Predictions for Polycrystalline Materials Using Three-Dimensional Synthetic Microstructures: Attenuation. J. Acoust. Soc. Am.
**2019**, 145, 2181–2191. [Google Scholar] [CrossRef] [PubMed] - Bunge, H.-J. Texture Analysis in Materials Science: Mathematical Methods, 1st ed.; Armco Inc.: Middeletown, OH, USA, 1982. [Google Scholar]
- Bache, M.R.; Evans, W.J.; Suddell, B.; Herrouin, F.R.M. The Effects of Texture in Titanium Alloys for Engineering Components under Fatigue. Int. J. Fatigue
**2001**, 23, 153–159. [Google Scholar] [CrossRef] - Roe, R.J.; Krigbaum, W.R. Description of Crystallite Orientation in Polycrystalline Materials Having Fiber Texture. J. Chem. Phys.
**1964**, 40, 2608. [Google Scholar] [CrossRef] - Roe, R.J. Description of Crystallite Orientation in Polycrystalline Materials. III. General Solution to Pole Figure Inversion. J. Appl. Phys.
**1965**, 36, 2024–2031. [Google Scholar] [CrossRef] - Cho, J.H.; Rollett, A.D.; Oh, K.H. Determination of Volume Fractions of Texture Components with Standard Distributions in Euler Space. Metall. Mater. Trans. A Phys. Metall. Mater. Sci.
**2004**, 35, 1075–1086. [Google Scholar] [CrossRef] - Yang, L.; Turner, J.A.; Li, Z. Ultrasonic Characterization of Microstructure Evolution during Processing. J. Acoust. Soc. Am.
**2007**, 121, 50. [Google Scholar] [CrossRef][Green Version] - Li, J.Y. Effective Electroelastic Moduli of Textured Piezoelectric Polycrystalline Aggregates. J. Mech. Phys. Solids
**2000**, 48, 529–552. [Google Scholar] [CrossRef] - Li, J.; Rokhlin, S.I. Elastic Wave Scattering in Random Anisotropic Solids. Int. J. Solids Struct.
**2016**, 78–79, 110–124. [Google Scholar] [CrossRef] - Mason, J.K.; Schuh, C.A. Hyperspherical Harmonics for the Representation of Crystallographic Texture. Acta Mater.
**2008**, 56, 6141–6155. [Google Scholar] [CrossRef] - Huang, M. Perturbation Approach to Elastic Constitutive Relations of Polycrystals. J. Mech. Phys. Solids
**2004**, 52, 1827–1853. [Google Scholar] [CrossRef] - Sha, G. Explicit Backscattering Coefficient for Ultrasonic Wave Propagating in Hexagonal Polycrystals with Fiber Texture. J. Nondestruct. Eval.
**2018**, 37, 37–51. [Google Scholar] [CrossRef] - Bunge, H.J.; Esling, C.; Bechler-Ferry, E. Three- Dimensional Texture Analysis after Bunge and Roe: Correspondence between the Respective Mathematical Techniques. Textures Microstruct.
**1982**, 5, 95–125. [Google Scholar] - Frisch, U. Wave Propagation in Random Media. In Probabilistic Methods in Applied Mathematics, Volume 1; Academic Press: New York, NY, USA, 1968; pp. 75–198. [Google Scholar]
- Bourret, R.C. Stochastically Perturbed Fields, with Applications to Wave Propagation in Random Media. Nuovo Cim. Ser. 10
**1962**, 26, 1–31. [Google Scholar] [CrossRef] - Budreck, B.D.E. An Eigenfunction Expansion of the Elastic Wave Green ’ S Function for Anisotropic Media. Q. J. Mech. Appl. Math.
**1993**, 46, 1–26. [Google Scholar] [CrossRef] - Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc. Sect. A
**2002**, 65, 349–354. [Google Scholar] [CrossRef] - Li, J.; Yang, L.; Rokhlin, S.I. Effect of Texture and Grain Shape on Ultrasonic Backscattering in Polycrystals. Ultrasonics
**2014**, 54, 1789–1803. [Google Scholar] [CrossRef] - Van Pamel, A.; Sha, G.; Rokhlin, S.I.; Lowe, M.J.S. 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] - Berryman, J.G. Bounds and Self-Consistent Estimates for Elastic Constants of Random Polycrystals with Hexagonal, Trigonal, and Tetragonal Symmetries. J. Mech. Phys. Solids
**2005**, 53, 2141–2173. [Google Scholar] [CrossRef] - Stanke, F.E. Spatial Autocorrelation Functions for Calculations of Effective Propagation Constants in Polycrystalline Materials. J. Acoust. Soc. Am.
**1986**, 80, 1479–1485. [Google Scholar] [CrossRef] - Hu, P.; Kube, C.M.; Koester, L.W.; Turner, J.A. Mode-Converted Diffuse Ultrasonic Backscatter. J Acoust Soc Am
**2015**, 134, 982–990. [Google Scholar] [CrossRef] - Yang, L.; Lobkis, O.I.; Rokhlin, S.I. An Integrated Model for Ultrasonic Wave Propagation and Scattering in a Polycrystalline Medium with Elongated Hexagonal Grains. Wave Motion
**2012**, 49, 544–560. [Google Scholar] [CrossRef] - Ahmed, S.; Thompson, R.B. Attenuation of Ultrasonic Waves in Cubic Metals Having Elongated, Oriented Grains. Nondestruct. Test. Eval.
**1992**, 8–9, 525–531. [Google Scholar] [CrossRef] - Wu, R.-S. Attenuation of Short Period Seismic Waves Due to Scattering. Attenuation short period Seism. waves due to Scatt.
**1982**, 9, 9–12. [Google Scholar] - Hashin, Z.; Shtrikman, S. A Variational Approach to the Theory of the Elastic Behaviour of Polycrystals. J. Mech. Phys. Solids
**1962**, 10, 343–352. [Google Scholar] [CrossRef] - Brown, J.M. Determination of Hashin-Shtrikman Bounds on the Isotropic Effective Elastic Moduli of Polycrystals of Any Symmetry. Comput. Geosci.
**2015**, 80, 95–99. [Google Scholar] [CrossRef] - Kube, C.M.; Arguelles, A.P. Bounds and Self-Consistent Estimates of the Elastic Constants of Polycrystals. Comput. Geosci.
**2016**, 95, 118–122. [Google Scholar] [CrossRef] - Li, Y.; Thompson, R.B. Relations between Elastic Constants Cij and Texture Parameters for Hexagonal Materials. J. Appl. Phys.
**1990**, 67, 2663–2665. [Google Scholar] [CrossRef][Green Version] - Man, C.S.; Huang, M. A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries. J. Elast.
**2011**, 105, 29–48. [Google Scholar] [CrossRef] - Böhlke, T.; Lobos, M. Representation of Hashin-Shtrikman Bounds of Cubic Crystal Aggregates in Terms of Texture Coefficients with Application in Materials Design. Acta Mater.
**2014**, 67, 324–334. [Google Scholar] [CrossRef] - Huang, M.; Man, C.S. Explicit Bounds of Effective Stiffness Tensors for Textured Aggregates of Cubic Crystallites. Math. Mech. Solids
**2008**, 13, 408–430. [Google Scholar] [CrossRef] - Lobos Fernández, M.; Böhlke, T. Representation of Hashin--Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials. J. Elast.
**2019**, 134, 1–38. [Google Scholar] [CrossRef] - Tane, M.; Yamori, K.; Sekino, T.; Mayama, T. Impact of Grain Shape on the Micromechanics-Based Extraction of Single-Crystalline Elastic Constants from Polycrystalline Samples with Crystallographic Texture. Acta Mater.
**2017**, 122, 236–251. [Google Scholar] [CrossRef] - Rokhlin, S.I.; Chimenti, D.E.; Nagy, P.B. Physical Ultrasonics of Composites; Oxford University Press: Oxford, UK, 2011. [Google Scholar]
- Gubernatis, J.E.; Krumhansl, J.A. Macroscopic Engineering Properties of Polycrystalline Materials: Elastic Properties. J. Appl. Phys.
**1975**, 46, 1875–1883. [Google Scholar] [CrossRef] - Ploix, M.A.; Guy, P.; Chassignole, B.; Moysan, J.; Corneloup, G.; Guerjouma, R.E. Measurement of Ultrasonic Scattering Attenuation in Austenitic Stainless Steel Welds: Realistic Input Data for NDT Numerical Modeling. Ultrasonics
**2014**, 54, 1729–1736. [Google Scholar] [CrossRef] [PubMed] - Pal, B. Pulse-Echo Method Cannot Measure Wave Attenuation Accurately. Ultrasonics
**2015**, 61, 6–9. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, L.; Rokhlin, S.I. On Comparison of Experiment and Theory for Ultrasonic Attenuation in Polycrystalline Niobium. J. Nondestruct. Eval.
**2012**, 31, 77–79. [Google Scholar] [CrossRef] - Sha, G.; Rokhlin, S.I. Universal Scaling of Transverse Wave Attenuation in Polycrystals. Ultrasonics
**2018**, 88, 84–96. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Hurley, D.H.; Hua, Z.; Sha, G.; Raetz, S.; Gusev, V.E.; Khafizov, M. Nondestructive Characterization of Polycrystalline 3D Microstructure with Time-Domain Brillouin Scattering. Scr. Mater.
**2019**, 166, 34–38. [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]

**Figure 1.**The orientation of the crystallite coordinate (x, y, z) in the texture coordinate system (Tx, Ty, Tz).

**Figure 3.**The geometry of an ellipsoidal grain with three main axes ${a}_{X},{a}_{Y}$ and ${a}_{Z}$, wave propagation $\mathit{p}\left(\tau ,{\varphi}_{\tau}\right)$ and scattering direction $\mathit{s}\left(\theta ,\varphi \right)$. Angle ${\theta}_{ps}$ is the angle between $\mathit{p}\left(\tau ,{\varphi}_{\tau}\right)$ and $\mathit{s}\left(\theta ,\varphi \right)$.

**Figure 4.**Attenuation comparison of the second-order attenuation (SOA) model (this work) to available experimental results (from Ref. [77]) and finite element modeling (FEM) results (from Ref. [22]) on a textured 316L stainless steel polycrystal with mean grain sizes ${a}_{x}={a}_{y}=0.125\text{mm}$, ${a}_{z}=2.5\text{mm}$. The error bars of the FEM results are also from Ref. [22].

**Figure 5.**Comparison of the longitudinal attenuation results from SOA model and the Born approximation [40] for a textured polycrystal of triclinic Ti crystallites with main grain radii ${a}_{X}=0.1\text{mm},{a}_{Y}=0.3\text{mm}$ and ${a}_{Z}=0.5\text{mm}$. The macrotexture parameters are ${\sigma}_{\theta}=0.03,{\sigma}_{\varphi}={\sigma}_{\theta}$ and ${\sigma}_{\zeta}=3{\sigma}_{\theta}$. The Rayleigh region, ${k}_{L}{a}_{X}<0.2$; geometric region, ${k}_{L}{a}_{X}>\mathrm{departure}\mathrm{frequency}$ (given in Table 2); estimated stochastic region, $7<{k}_{L}{a}_{X}<\mathrm{departure}\mathrm{frequency}$.

**Figure 6.**Comparison of the longitudinal attenuation results from the SOA model and the Born approximation [40] for a textured Ti polycrystal of triclinic crystallites with mean grain radii ${a}_{X}={a}_{Y}={a}_{Z}=0.1mm$. The macrotexture parameters are ${\sigma}_{\theta}=0.03,{\sigma}_{\varphi}={\sigma}_{\theta}$ and ${\sigma}_{\zeta}=3{\sigma}_{\theta}$. Rayleigh region, ${k}_{L}{a}_{X}<0.2$; geometric region, ${k}_{L}{a}_{X}>\mathrm{departure}\mathrm{frequency}$ (given in Table 3); estimated stochastic region, $10<{k}_{L}{a}_{X}<\mathrm{departure}\mathrm{frequency}$.

**Figure 7.**Phase velocity results from the SOA model for two Ti polycrystals with triclinic grains: (

**a**) the ellipsoidal grain case and (

**b**) the equiaxed grain case. ${V}_{l}$ is the phase velocity predicted by the SOA model while V

_{L}is the Voigt phase velocity in the corresponding wave propagation direction.

${c}_{11}$ | ${c}_{12}$ | ${c}_{13}$ | ${c}_{14}$ | ${c}_{15}$ | ${c}_{16}$ | ${c}_{22}$ | ${c}_{23}$ | ${c}_{24}$ | |

Triclinic Ti | 161.6 | 67.8 | 88.2 | 1.8 | 1.1 | −0.64 | 184.4 | 68.0 | −2.5 |

${c}_{25}$ | ${c}_{26}$ | ${c}_{33}$ | ${c}_{34}$ | ${c}_{35}$ | ${c}_{36}$ | ${c}_{44}$ | ${c}_{45}$ | ${c}_{46}$ | |

Triclinic Ti | −1.9 | −2.6 | 161.3 | −0.18 | 0.83 | 2.4 | 50.9 | −5.1 | 0.20 |

${c}_{55}$ | ${c}_{56}$ | ${c}_{66}$ | ρ | ||||||

Triclinic Ti | 39.0 | −5.2 | 50.9 | 4.54 |

**Table 2.**The mean deviation between the SOA model and the Born approximation below break frequency (SOA model is treated as the reference) and the break frequency ${k}_{L}{a}_{X}$ of the Born approximation for a textured polycrystal with ellipsoidal Ti grains.

Direction | X | Y | Z |
---|---|---|---|

Mean deviation | −13% | −9.7% | −14% |

Departure frequency | 31.6 | 12.7 | 25.1 |

**Table 3.**The mean deviation between SOA and the Born approximation below break frequency (SOA model is treated as reference) and the break frequency ${k}_{L}{a}_{X}$ of the Born approximation for a textured polycrystal with equiaxed Ti grains.

Direction | X | Y | Z |
---|---|---|---|

Mean deviation | −10.9% | −7.5% | −14.4% |

Departure frequency | 31.0 | 34.4 | 102.7 |

**Table 4.**Comparison of the longitudinal velocities (in km/s) at different propagation directions for a macroscopically orthotropic copper with equiaxed grains [73,74]. Hashin–Shtrikman (H-S) bounds and self-consistent (SC) velocity are directly from Ref. [73,74]. Assume copper density is 8.935g/cm

^{3}.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sha, G. Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry. *Acoustics* **2020**, *2*, 51-72.
https://doi.org/10.3390/acoustics2010005

**AMA Style**

Sha G. Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry. *Acoustics*. 2020; 2(1):51-72.
https://doi.org/10.3390/acoustics2010005

**Chicago/Turabian Style**

Sha, Gaofeng. 2020. "Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry" *Acoustics* 2, no. 1: 51-72.
https://doi.org/10.3390/acoustics2010005