The phase velocity dispersion of longitudinal waves in polycrystals with elongated grains of arbitrary crystallographic symmetry is studied in all frequency ranges by the theoretical second-order approximation (SOA) and numerical three-dimensional finite element (FE) models. The SOA and FE models are found to be in excellent agreement for three studied polycrystals: cubic Al, Inconel, and a triclinic material system. A simple Born approximation for the velocity, not containing the Cauchy integrals, and the explicit analytical quasi-static velocity limit (Rayleigh asymptote) are derived. As confirmed by the FE simulations, the velocity limit provides an accurate velocity estimate in the low-frequency regime where the phase velocity is nearly constant on frequency; however, it exhibits dependence on the propagation angle. As frequency increases, the phase velocity increases towards the stochastic regime and then, with further frequency increase, behaves differently depending on the propagation direction. It remains nearly constant for the wave propagation in the direction of the smaller ellipsoidal grain radius and decreases in the grain elongation direction. In the Rayleigh and stochastic frequency regimes, the directional velocity change shows proportionalities to the two elastic scattering factors even for the polycrystal with the triclinic grain symmetry.

## I. INTRODUCTION

Scattering occurs when elastic waves propagate through a random polycrystalline medium, exhibiting scattering-induced attenuation and velocity dispersion. Studying these wave phenomena is an important topic of interest in non-destructive evaluation and microstructure characterization.^{1,2} Both scattering-induced attenuation and velocity are important for practical applications. Elastic wave velocity is often used for elastic property determination and macroscopic texture assessment. When comparing different measurements made for different frequency ranges and microstructures, a clear understanding of the velocity dispersion is needed; however, the availability of theoretical models to make such assessments is limited.

Significant theoretical progress for elastic wave scattering in polycrystals was made by Stanke and Kino^{3} and Weaver,^{4} who have used different approaches that have resulted in equivalent approximations and final dispersion equations for the perturbed wave number of propagating elastic waves in equiaxed cubic polycrystals, with the results applicable in all frequency ranges. Following Stanke and Kino,^{3} we collectively call these model approximations the second-order approximation (SOA),^{5} since they are accurate to the second order of the inhomogeneity parameter, $\epsilon 2$ (represented by $Q$ factors in this study). Those models enable the consideration of some degree of multiple scattering. Most later studies, focusing on scattering-induced attenuation, have extended the Weaver approach (as more suitable for generalizations) to other polycrystalline materials with texture, duplex microstructures, and lower crystal symmetries.^{2,5–13} Most extensions^{2,6,7,9,10} invoked the Born approximation to obtain explicit expressions for attenuation and limited their applicability to frequencies below the geometric regime.^{3}

The scattering-induced velocity dispersion of elastic waves received much less attention. The earlier known general theoretical studies of scattering effects on both the phase velocity and attenuation were performed by Lifshits and Parkhomovski,^{14} Usov *et al.*,^{15} and Hirsekorn.^{16} Stanke and Kino^{3} have performed a detailed study of the phase velocity dispersion for equiaxed cubic polycrystals in all frequency ranges. They have also obtained explicit low frequency and stochastic asymptotes for the perturbed wave number (the phase velocity is determined from its real part); they noted that those asymptotes are the same as those of Lifshits and Parkhomovski.^{14} Calvet and Margerin^{8} have studied the velocity dispersion for equiaxed cubic polycrystals by the spectral function method based on the Weaver model.^{4} The velocity results^{8} were compared by Rokhlin *et al.*^{13} with those of Stanke and Kino,^{3} showing an excellent agreement in all frequency ranges. Maurel *et al.*^{17} have predicted the elastic wave velocities in textured polycrystalline ice and showed that neglecting scattering in their model leads to errors of 0.5%. The dispersion of elastic wave phase velocity was studied theoretically and by the three-dimensional (3D) finite element (FE) method in Ref. 5. They have investigated all frequency ranges for equiaxed polycrystals of general crystallite symmetry, where the explicit solutions for the quasi-static and stochastic asymptotes were obtained and compared with the 3D FE method.

The theoretical models are based on different levels of approximations in accounting for the multiple scattering and the assumption of small inhomogeneity perturbations (related to crystallite anisotropy). Alternatively, the recently deployed 3D FE method presents no such limitation since it incorporates all multiple scattering events as has been demonstrated for elastic waves in equiaxed polycrystals.^{5,18–21} In studies,^{5,19,20} the phase velocity dispersion was compared between the SOA (the type of Stanke and Kino^{3}) and the 3D FE results showing an excellent agreement. In Van Pamel *et al.*,^{20} those results were also compared with the measurement of longitudinal phase velocity^{22} for polycrystalline Cu in a wide frequency range. However, the availability of experimental studies suitable for the velocity dispersion comparison is limited due to the difficulty of accurate velocity measurements in a sufficiently large frequency range and lack of needed microstructure data in those studies.

Polycrystals with elongated grains are more commonly found in practice,^{7,12,23,24} in which case the geometric shapes of the grains are stretched (or more generally, deformed) in a preferred direction. The geometrical deformation of grain shape needs to be differentiated from the crystallographic texture in polycrystals with crystallographic symmetry axes of the grains having a preferred orientation, although grain elongation and texture usually occur simultaneously.^{2} The majority of the available elastic wave propagation studies in such polycrystals have been mostly focused on the scattering-induced attenuation^{1,2,9,10,12,13,23,25} by extending the Stanke and Kino^{3} and Weaver^{4} models to elongated polycrystals and invoking the Born approximation. Only recently, Calvet and Margerin^{26,27} have considered in detail the phase velocity dispersion in elongated cubic polycrystals by the spectral function method.

In this work, the SOA and the 3D FE methods are developed for the elongated polycrystals of different crystallite symmetries and anisotropies. Al, Inconel, and triclinic copper sulfate pentahydrate (CSP) are selected as the model material systems. Our goals are to better understand the scattering effects and to evaluate the applicability limits of the theoretical SOA model for the determination of the scattering-induced dispersion of the elastic wave phase velocity.

To the best of the authors' knowledge, the appropriate experimental studies are currently not available for the detailed comparison with the theoretical models of the phase velocity dispersion in elongated polycrystals. One of the experimental difficulties is the accurate measurement of the anisotropic two-point correlation function for nonequiaxed polycrystals.^{28,29} Also, comparing to attenuation, the overall change of phase velocity with frequency is small (for the studied material systems, the velocity change from the quasi-static to the geometric region transition is below two percent). Thus, measurements in a sufficiently large frequency range are needed. In addition, even small crystallographic macrotexture in the samples may significantly mask the scattering-induced anisotropy of velocity. The 3D FE method naturally averts those difficulties due to the full control and knowledge of microstructure statistics and the creation of polycrystals without crystallographic macrotexture; the 3D FE simulation methods have significant advantages over possible actual experiments for the assessment of the model predictions. While studying polycrystals with crystallographic macrotexture is important, it is useful to separate its effect from that of the grain shape. Such separation is practically important because there are application scenarios where preferred crystallographic orientation is partially eliminated by specific material processing while grain elongation remains to affect the scattering of waves. It also allows a better understanding of the effect of model approximations when applied to the elongated grains scattering. The investigation of the combined effect, as it was done for attenuation,^{2} is beyond the scope of this study and is expected to be addressed in future work.

This work implements a theoretical SOA model and a numerical 3D FE model for studying the frequency and angle behavior of elastic wave phase velocity in elongated polycrystals with different crystallographic symmetries and grain anisotropies. The SOA model incorporates the accurate two-point correlation (TPC) function as determined from the FE material model and thus allows for a direct comparison of the two models. A new representation for the Born approximation for the phase velocity is derived from the SOA model to obtain the explicit angle-dependent Rayleigh limit for quasi-static velocity. The solutions are compared with the FE model to reveal the dependences of velocity on grain shape and the elastic properties of the grains. The obtained results are fundamentally important for understanding the elastodynamic behaviors of polycrystals. Also, the explicit analytical quasi-static velocity limit obtained in this work may be of interest to the elastostatic community because it can be used to estimate the effective, static elastic constants of elongated polycrystals, as will be further discussed in Sec. IV E.

Section II describes the SOA model for elongated polycrystals of arbitrary crystal symmetry and presents the Born approximation and the Rayleigh velocity asymptote. The 3D FE model for simulating waves in elongated polycrystals is summarized in Sec. III. The velocity results of the SOA and FE models are compared in Sec. IV, with discussions of the phase velocity dependences on frequency, grain elongation, and elastic properties. Conclusions are provided in Sec. V.

## II. THEORETICAL MODEL: SOA

The theoretical SOA model is introduced in Sec. II A to calculate the phase velocity dispersion (and attenuation) of elastic waves in elongated polycrystals. A simplified model that invokes the Born approximation in the SOA model is presented in Sec. II B. Based on the Born approximation, Sec. II C derives the Rayleigh asymptote for velocity. The model descriptions are suitable for single-phase polycrystals of arbitrary crystal symmetries with statistically ellipsoidal shaped grains that are densely packed and fully bonded. For the Rayleigh asymptote, the model description is restricted to grains of ellipsoid-of-rotation shape.

### A. The SOA model

The polycrystals studied in this work are assumed to be statistically homogeneous, namely, their elastic tensors are position-independent on the macro scale. An inhomogeneous polycrystal can thus be approximated to a continuous random medium with small spatial variations of elastic tensor.^{3,4} In this case, a solution can be found for the ensemble-averaged wave response $k$, representing the perturbed wave number of a plane wave, from the dispersion equation^{4,20}

where the incident wave has a mode of $M$ and a propagation direction of $p$ as shown in Fig. 1, and therefore the wave vector is $k=kp$. $\omega =2\pi f$ denotes the angular frequency ($f$ is frequency). $V0M$ represents the phase velocity of the wave mode $M$ in the reference medium, which is a homogeneous, non-scattering medium with its elastic tensor $\u27e8cijkl\u27e9$ being obtained by ensemble averaging (Voigt average) that of the polycrystal $cijkl(X)$.^{20}

By numerically solving the dispersion equation, Eq. (1) of the SOA model, one obtains the perturbed wave number $k(f)$ for the propagating wave $M$. This further leads to the solutions for phase velocity $V(f)$ and attenuation coefficient $\alpha (f)$ that are calculated from the real and imaginary parts of the perturbed wave number by

This work focuses on the phase velocity part of the solution.

In the dispersion equation, Eq. (1), the mass operator $mM$ is the determining factor that accounts for the random scattering events that occur in the polycrystal. Considering the mode conversion on the grain boundaries, the mass operator of a given wave $M$ can be expressed as a sum of three terms, $mM=\u2211N=13mM\u2192N$, with each term representing the scattering of a wave $M$ into a wave of mode $N$.

For a general case, the incident wave mode $M$ and the scattered wave mode $N$ can be any of the three independent modes: longitudinal (L), fast transverse (T1), and slow transverse (T2). This work considers a specific case of the crystallite orientations being statistically uncorrelated. However, even for this specific case in the absence of crystallographic macrotexture, Calvet and Margerin^{26} have shown that the two transverse modes have slightly different velocities as a result of the macroscopic anisotropy caused by grain elongation. Since this difference is small, it can be neglected for the analysis of longitudinal-to-transverse wave scattering and we can assume that the two transverse scattered waves degenerate into two waves having the same velocities but different mutually orthogonal polarizations, allowing the two waves to be combined into one as $T=T1+T2$. We note that this crystallographically untextured case does not represent all practical elongated polycrystals because the material processing that produces elongated grains may also create preferred crystallographic orientations, showing elastic anisotropy on the macro scale. Nonetheless, as discussed in Sec. I, studying this case is an essential first step to see the effects of geometry without the influence of the crystallographic macrotexture.

As a result of the isotropic assumption, the mass operator can be written as $mM=\u2211N=L,TmM\u2192N$. By adopting the first-order smoothing approximation^{4,30} (or equivalently the Bourret approximation^{30–32}), the mass operator component $mM\u2192N$ is given by

where $\rho $ is the constant mass density for single-phase polycrystals as considered in this work. $k0N=\omega /V0N$ denotes the wave number for wave mode $N$ in the reference medium. P.V. represents the Cauchy principal value and $\xi $ is a dimensionless variable. The factor $\eta $ takes the values of 1 and 2 for longitudinal ($M=L$) and transverse ($M=T$) propagating waves, respectively.

In Eq. (3), the factor $fM\u2192N$ is the spectral representation of the spatial correlation function, $\u27e8\delta cijkl(x)\delta c\alpha \beta \gamma \delta (x\u2032)\u27e9$, between points $x$ and $x\u2032$^{5}; $\delta cijkl(x)$ is a small random spatial fluctuation of the elastic tensor $cijkl(x)$ around its mean $\u27e8cijkl\u27e9$. The correlation function can be factored into $\u27e8\delta cijkl(x)\delta c\alpha \beta \gamma \delta (x\u2032)\u27e9=\u27e8\delta cijkl\delta c\alpha \beta \gamma \delta \u27e9w(x\u2212x\u2032)$ for statistically homogeneous media, with the first part representing the elastic covariance and the second part denoting the geometric TPC function. The validity of this factorization has been demonstrated by the direct numerical comparison in Ref. 33 for statistically isotropic polycrystals with equiaxed grains. The good agreement between the analytical and 3D FE models for the phase velocity, Sec. IV, indirectly supports the applicability of the factorization to polycrystals with elongated grains that exhibit statistical anisotropy; a similar conclusion can be drawn from the agreement for wave attenuation in elongated polycrystals. In the spectral domain, the factor $fM\u2192N$ can be similarly factored into two parts,

where $IPM\u2192N$ is the inner product corresponding to the elastic covariance $\u27e8\delta cijkl\delta c\alpha \beta \gamma \delta \u27e9$ and it also accounts for the propagation and polarization directions of both incident and scattered waves. For the wave propagation setup shown in Fig. 1, the inner product is given by^{2,13}

where coefficients $AMN$, $BMN$, and $CMN$ ($M,N\u2208{L,T}$) are invariants^{2} and are represented by linear combinations of the seven quadratic invariants. The coefficients are the sums of the quadratic terms of elastic constants, their expressions are summarized in Appendix A of our prior work^{5} for arbitrary crystal symmetries. $cos\u2009\theta ps$ is the cosine of the angle between the incident and scattered waves: $cos\u2009\theta ps=p\u22c5s=cos(\phi \u2212\phi p)\u2009sin\u2009\theta \u2009sin\u2009\theta p+cos\u2009\theta \u2009cos\u2009\theta p$.

$WM\u2192N(k,\omega ,\xi ,\theta ,\phi )$ is the spatial Fourier representation of the geometric TPC function $w(x\u2212x\u2032)$. This latter function describes the probability of two points $x$ and $x\u2032$ separated by a vector $r=x\u2212x\u2032$ falling into the same grain. As aforementioned, we consider elongated grains with an average shape of ellipsoidal. When the axes of the average grain are perfectly aligned with the coordinate axes as shown in Fig. 1, the TPC function of the elongated grains can be described by

where the function is expressed as the sum of an exponential series. In direct contrast to the conventional single exponential form,^{3,4} this generalized TPC function has the advantage of accurately representing the actual TPC statistics of experimental^{28} and numerical^{20,21,33} samples. The coefficients $axi$, $ayi$, $azi$, and $Ai$ are determined by best fitting the actual TPC data of the polycrystal. Note that we do not attribute physical meaning to individual exponential terms, and instead, we use the sum of the terms to describe the physical TPC statistics of the polycrystal. The spectral representation $WM\u2192N(k,\omega ,\xi ,\theta ,\phi )$ can be obtained from Eq. (6) and is given by^{13}

where the vector $(qx,qy,qz)$ specifies the difference between the incident and scattered wave vectors, namely,

### B. Born approximation of the SOA model

The Born approximation of the SOA model is introduced here because: (1) it delivers an explicit expression for velocity (and attenuation) and thus considerably improves computation efficiency; further improvement is achieved below by obtaining the analytical formula of the Cauchy integral; (2) it enables the derivation of the Rayleigh velocity asymptote as provided in Sec. II C, which is a key element for interpreting the explicit dependence of velocity on grain elongation and elastic properties in the low-frequency range. We shall see below that the Born approximation is only valid for frequencies below the geometric regime, but it has a generally good agreement with the SOA model in its valid range.

The Born approximation is obtained by replacing the perturbed wave vector $k$ in the mass operator $mM(k;\omega )$ [Eqs. (1) and (3)] by the unperturbed reference wave vector $pk0M$ and by substituting the small wave number perturbation $(\omega /V0M)2\u2212k2$ in the dispersion equation [Eq. (1)] by $2k0M(k0M\u2212k)$. As a result, the perturbed wave number $k$ is explicitly expressed from Eq. (1) and its real and imaginary parts can be conveniently separated. The real and imaginary parts of the wave number $k$ determine, Eq. (2), the phase velocity and attenuation coefficient of the propagating wave. In this work, we focus on the phase velocity, and using Eq. (3) we obtain

where $\Delta M\u2192N$ is the scattering-induced velocity change given by

The triple integral in Eq. (10) can be written in the following form by substituting Eq. (4) into the equation and rearranging its terms

where the functions $C0i$, $C1i$, and $C2i$ are introduced to simplify notation

The innermost Cauchy integral in Eq. (11) is evaluated by using the contour integral and the residue theorem because its integrand approaches zero when $\xi \u2192\u221e$. This leads to the following solution $\chi (\theta ,\phi )$ for the Cauchy integral:

where $Di=4C0iC2i\u2212(C1i)2$. Note that the evaluation result is originally complex but only its real part is given in Eq. (13) because its imaginary part cancels out during the later double integration over $\theta $ and $\phi $ for the velocity change $\Delta M\u2192N$,

The Born approximation is valid only for frequencies below the geometric regime. Its range of validity is illustrated in Fig. 2 by comparing with the SOA model for longitudinal wave propagation in triclinic CSP with elongated grains of statistically ellipsoid-of-rotation shape ($ax=ay$; the elastic properties of the material are given in Ref. 5, and the coefficients of the generalized TPC function are specified in Sec. III). It is shown that the Born approximation has a very good agreement with the SOA model in the low-frequency range, but their difference becomes larger as frequency approaches the geometric regime. Due to its simplicity and its good agreement with the SOA model in its range of validity, the Born approximation is suitable for elucidating the relation of velocity to the properties of the polycrystal. It follows from Eq. (14) [and Eq. (9)] that the phase velocity of an elongated polycrystal depends on frequency, grain shape, and elastic properties.

### C. Quasi-static velocity limit

While the above Born approximation is given for elongated grains of generally ellipsoidal shape, here the Rayleigh velocity asymptote (or the quasi-static limit) is only provided for statistically ellipsoid-of-rotation grains, which is the case mainly studied in this work. Based on the Born approximation, the asymptote is given as

where $\Delta M\u2192N$ represents the relative change of phase velocity at the Rayleigh limit; it is derived in the Appendix; its final expression, which is independent of frequency, is given by

where $\theta p$ is the angle between the directions of wave propagation and grain elongation, see Fig. 1. $\u03c20$, $\u03c22$, and $\u03c24$ are the shape factors

and $f(Ri)$ denotes

where $Ri=azi/axi$ is the ratio of parameters $ax,zi$ in the fitted exponential series [Eq. (6)] between the elongated *z* and shortened *x* directions ($axi=ayi$). For the triclinic CSP polycrystal, the Rayleigh asymptotes (quasi-static limits) are shown in Fig. 2 by dashed-dotted lines.

While in this study we are not evaluating the far-field approximation (FFA) model,^{13} it is useful to note that the Rayleigh velocity asymptote $VMR$, Eq. (15), obtained in this work for polycrystals with elongated grains can be used to correct the phase velocity obtained in Ref. 13 (as explained in Rokhlin *et al.*,^{13} in the FFA, a constant phase velocity shift of $VMR\u2212V0M$ should be applied across the whole frequency range, see also Ref. 5).

For polycrystals with statistically equiaxed grains, the ratio $Ri=azi/axi$ is unity for all terms. The Rayleigh velocity limit for this equiaxed case is obtained by taking the limit of Eq. (16) at $Ri\u21921$ and accounting for $limRi\u21921f(Ri)=1$ and the denominators are canceled. This results in the same expression as in our previous study,^{5}

where $QMM*=(AMM+BMM/3+CMM/5)/(4\eta \rho 2V0M4)$ is an elastic factor introduced to simplify the above equation. $QM\u2192N=(AMN+BMN/3+CMN/5)/(4\eta \rho 2V0M2V0N2)$ is the mode-converted scattering factor.^{13}

For the extreme case of $Ri\u21920$ that corresponds to the infinitely thin (in the *z*-direction) pancake shape grains, the Rayleigh velocity limit can be obtained by substituting $Ri=0$ into Eq. (16), leading to

For the extreme case of $Ri\u2192\u221e$ that corresponds to ellipsoid-of-rotation grains with infinitely large axial dimension, its Rayleigh velocity asymptote resulting from Eq. (16) is

It is important to realize that the Rayleigh velocity limit as influenced by grain elongation is bounded by the extreme values at $Ri\u21920$ [Eq. (20)] and $Ri\u2192\u221e$ [Eq. (21)]. Also, it immediately follows from Eqs. (20) and (21) that the velocity limit in the larger radius direction is the same for $Ri\u21920$ and $Ri\u2192\u221e$: $VMR\u21920(\theta p=\pi /2)\u2261VMR\u2192\u221e(\theta p=0)$.

It is clear from Eq. (19) that the coefficients of the TPC function cancel out for equiaxed polycrystals, suggesting that the phase velocity limit is solely related to the elastic properties of the polycrystals and is independent of grain geometry. As follows from Eqs. (15) and (16), however, the phase velocity limit for elongated polycrystals depends on grain shape and is thus dependent on wave propagation direction $\theta p$ (this is discussed later in Sec. IV). The quasi-static limit is independent of frequency and the absolute values of the correlation radii $ax,zi$; it depends only on their ratio. The independence of the Rayleigh velocity (quasi-static) limit on frequency and the dependence on grain shape are in direct contrast to the Rayleigh attenuation that depends on frequency and is independent on the wave propagation direction.^{2,25} The analytical work provided here may be useful for investigating the influence of grain shape on static elastic moduli of polycrystals.

## III. NUMERICAL MODEL: 3D FINITE ELEMENT (FE) METHOD

The 3D FE method is described to numerically calculate effective wave parameters, such as scattering-induced phase velocity dispersion and attenuation, for plane longitudinal waves propagating through polycrystals with elongated grains. In comparison to the preceding SOA model, the 3D FE method captures the actual wave interactions with grains without approximations involved in the analytical models, and it thus delivers an accurate velocity dispersion in the FE model polycrystals. This method is generally suitable for polycrystals with any grain shape of any crystal symmetry, but the model description is given for grains of statistically ellipsoid-of-rotation shape.

The 3D FE method starts with the generation of a numerical material model with the dimensions of $dx\xd7dy\xd7dz$ along the three coordinate axes. The model is composed of random grains created by the Voronoi tessellation method.^{34} The grains are densely packed and fully bonded, and they have a statistically equiaxed shape; typical equiaxed grains are shown in Fig. 3(a). The equiaxed model is then scaled by using the scale factors of $Sx\xd7Sy\xd7Sz=(1\xd71\xd7R)/R3$, leading to a stretched model with the dimensions of $dxSx\xd7dySy\xd7dzSz$. The constituting grains of the stretched model are elongated in the $z$-direction with an elongation ratio of $R$ and the average shape of the grains is ellipsoid-of-rotation; typical elongated grains are shown in Fig. 3(b). The scale factors are chosen such that $Sx\u22c5Sy\u22c5Sz=1$, which are used to ensure that the volume of each grain remains unchanged during the scaling and to enable a direct comparison between equiaxed and elongated cases.

To simulate the wave propagation in the off-symmetry axis direction with an angle of $\theta p$ to the grain elongation direction, the stretched model is rotated about the *x*-axis with the desired angle $\theta p$. The outer surfaces of the rotated model are no longer normal to the coordinate axes, which complicates wave excitation and monitoring. This problem is averted by creating a sufficiently large equiaxed model and then by cropping its stretched and rotated counterpart to achieve the dimensions of $dxSx\xd7dySy\xd7dzSz$ along the coordinate axes. The models created for this work are listed in Table I. Note that the average grain diameter $D$ in the table denotes the cubic root of average grain volume $v\xaf$, i.e., $D=v\xaf3$. It differs from the mean grain radius, $ax$ for example, extensively used later in this work that corresponds to the slope at the origin of the generalized TPC function in a specific direction.

Model name . | $R$ . | $dx\xd7dy\xd7dz$ . | $D$ . | $N$ . | $h$ . | d.o.f. . | $fc$ . | ||
---|---|---|---|---|---|---|---|---|---|

Al . | Inconel . | CSP . | |||||||

N11520 | 1 | 12 × 12×10 | 0.5 | 11520 | 0.025 | 278 × 10^{6} | 5,10 | 2,5 | 2,5 |

5 | 7.02 × 7.02 × 29.24 | 0.29 × 0.29 × 1.46 | |||||||

N16000 | 1 | 20 × 20×5 | 0.5 | 16000 | 0.020 | 755 × 10^{6} | 15 | 10,20 | 10,20 |

5 | 11.70 × 11.70 × 14.62 | 0.29 × 0.29 × 1.46 | |||||||

N12250 | 1 | 17.5 × 17.5 × 5 | 0.5 | 12250 | 0.017 | 967 × 10^{6} | 28 | — | — |

5 | 10.23 × 10.23 × 14.62 | 0.29 × 0.29 × 1.46 |

Model name . | $R$ . | $dx\xd7dy\xd7dz$ . | $D$ . | $N$ . | $h$ . | d.o.f. . | $fc$ . | ||
---|---|---|---|---|---|---|---|---|---|

Al . | Inconel . | CSP . | |||||||

N11520 | 1 | 12 × 12×10 | 0.5 | 11520 | 0.025 | 278 × 10^{6} | 5,10 | 2,5 | 2,5 |

5 | 7.02 × 7.02 × 29.24 | 0.29 × 0.29 × 1.46 | |||||||

N16000 | 1 | 20 × 20×5 | 0.5 | 16000 | 0.020 | 755 × 10^{6} | 15 | 10,20 | 10,20 |

5 | 11.70 × 11.70 × 14.62 | 0.29 × 0.29 × 1.46 | |||||||

N12250 | 1 | 17.5 × 17.5 × 5 | 0.5 | 12250 | 0.017 | 967 × 10^{6} | 28 | — | — |

5 | 10.23 × 10.23 × 14.62 | 0.29 × 0.29 × 1.46 |

The generated model is then discretized with uniform linear eight-node “brick” elements; cross-sectional views of the discretized grains are given in Fig. 3. Although an unstructured mesh can better reproduce grain boundaries,^{18,19} we have chosen the structured mesh type because it performs equally well to the unstructured type for a sufficient number of elements per grain^{18} and it is easier for implementation and better for numerical error control. The edge size, $h$, of the uniform elements is chosen to be smaller than $D/20$ and $\lambda /10$, where $\lambda $ denotes the simulated wavelength. These sampling rates are selected to ensure a fairly good representation of the grains as well as to deliver satisfactory numerical convergence and accuracy.^{18} However, we note that some very sharp tips of the elongated grains may not be well represented by the structured mesh. In addition to the space discretization, the time is sampled with a time step of $\Delta t$ satisfying the Courant-Friedrichs-Levy condition: $\Delta t=Ch/Vmax$. $Vmax$ represents the fastest phase velocity in the studied polycrystal. $C$ denotes the Courant number, which is required to be no greater than unity, namely, $C\u22641$. This work utilizes a large Courant number of $C=0.98$ to suppress the numerical phase velocity error induced by FE approximations.^{35}

The grains of the created model are further assigned with material properties. This work employs, in separate analyses, three single-phase polycrystalline materials with properties being given in Table II. Al and Inconel possess the highest crystal symmetry of cubic and greatly differing anisotropy factors; CSP has the lowest symmetry of triclinic and a universal anisotropy factor^{36} between those of Al and Inconel. The grains of each model share the same material mass and elastic properties but face different directions in terms of crystallographic orientation. To enable a direct comparison with the theoretical SOA model, the FE model is made statistically homogeneous by uniformly randomly assigning crystallographic orientations to the grains. In contrast to equiaxed polycrystals, the randomness in crystallographic orientation does not make an elongated polycrystal elastically isotropic on the macro scale. Instead, an elongated polycrystal exhibits statistical anisotropy as a result of grain elongation, and it is specifically transverse isotropy for a polycrystal with ellipsoid-of-rotation grains as addressed in the present FE model.

. | $\rho $ . | $AU$ . | $V0L$ . | $V0T$ . | $QL\u2192L$ . | $QL\u2192T$ . |
---|---|---|---|---|---|---|

Al (Ref. 20) | 2700 | 0.054 | 6317.52 | 3128.13 | 7.80 × 10^{−5} | 3.34 × 10^{−4} |

Inconel (Ref. 20) | 8260 | 1.420 | 6025.37 | 3365.54 | 2.26 × 10^{−3} | 7.59 × 10^{−3} |

CSP (Ref. 5) | 2286 | 0.948 | 4874.33 | 2303.21 | 3.23 × 10^{−3} | 7.19 × 10^{−3} |

Once the model space and time are discretized and material properties are defined, boundary and loading conditions are specified to create a plane longitudinal wave propagating in the $z$-direction. Symmetry boundary conditions (SBCs) are defined for the four outer surfaces in parallel to the $z$-direction, which is necessary for accommodating the plane wave. A $z$-direction uniform force, in the form of a three-cycle Hann-windowed toneburst, is applied to all nodes of the transmitting surface $z=0$. This initiates a pure plane longitudinal wave for $\theta p=0\xb0/90\xb0$. For other propagation angles, because of macro-anisotropy, it may create small amplitude quasi-transverse waves in addition to the quasi-longitudinal wave. However, despite large time delays of the arrival of the possible quasi-transverse wave pulses, we did not observe any quasi-transverse signals. The excitation of the quasi-transverse waves is negligible because the elastic anisotropy caused by grain elongation is very small for the studied cases (for reference, the largest skew angle for the quasi-wave displacements in the studied materials is estimated to be smaller than 0.1°, indicating negligible effects on the transverse waves excitation).

The FE model is solved in the time domain by using the FE program Pogo,^{37} which is tailored for fast, large-scale elastodynamic simulations via parallel GPU computing. Over the course of simulation, the *z*-direction displacements are monitored across all nodes and at all time steps on the transmitting, $z=0$, and receiving, $z=dz$, surfaces. Spatial averaging these two sets of displacements over the respective nodes at each time step leads to the coherent transmitted and received signals, denoted as $U(0;t)$ and $U(dz;t)$. The frequency-domain signals, $U(0;f)$ and $U(dz;f)$, are then obtained by Fourier transforming the windowed time-domain signals. Frequency-dependent phase velocity is then calculated from the signals by $V(f)=2\pi fdz/[\varphi (0;f)\u2212\varphi (dz;f)]$, where $\varphi $ represents the unwrapped phase of $U$.

It is important to realize that different models are used in different frequency ranges to achieve a similar calculation accuracy across the entire frequency range; the center frequencies used for the models are provided in Table I. Also, a large number of grains are included in each FE model, ensuring that the phase velocity determined from a single FE material model incorporates a significant amount of scattering information. Despite this statistical significance of a single model, we use a combination of 15 realizations to get converged phase velocity (as well as attenuation) results for a given material in a given frequency range, see Table I. For each case, the multi-realizations are generated by re-randomizing the crystallographic orientations of the grains.

To compare FE simulation results with the predictions of the SOA model, accurate TPC data are numerically measured from the FE material models. For equiaxed polycrystals, the measured data points are shown as squares in Fig. 4, which are fitted into a generalized TPC function shown as the dashed line. The coefficients of the generalized TPC function, $w(r)=\u2211i=1nAi\u2009exp(\u2212r/ai)$, are provided in Table III. For elongated polycrystals, we simulate wave propagation at eight angles $\theta p$ of 0°, 15°, 18.75°, 30°, 45°, 60°, 75°, and 90°, and TPC statistics are thus measured in these directions. The data were collected for the elongated N11520 material model; the measurement procedure is similar to that of the polycrystal with equiaxed grains.^{5} Each measured TPC data point, $(r,w(r))$, in the figure is determined by dropping a significant number of random point pairs into the material model with each aligned in the desired propagation direction. Specifically, each dropped pair of points is separated by the variable distance of $r$, and then counted to determine the probability $w(r)$ of point pairs falling into the same grain in a given propagation direction. We note that the TPC statistics provided in the figure well represent the microstructure of other models in Table I. The data points for five angles are shown as circles in Fig. 4. The generalized TPC function of the elongated case is obtained by scaling that of the equiaxed case, given by

where the coefficients $Ai$ are the same as that of the equiaxed case, while $axi$, $ayi$, and $azi$ are scaled from that of the equiaxed case $ai$ by:$(axi,ayi,azi)=ai(1,1,R)/R3$. The resulting coefficients are given in Table III and the TPC function is plotted in Fig. 4 as solid lines. The scaled function shows a very good agreement with the measured data points, validating the effectiveness of the scaling approach. The TPC function, Eq. (22), is incorporated into the analytical models for the direct comparison of velocity in Sec. IV.

TPC terms . | Equiaxed case . | Elongated case . | ||||
---|---|---|---|---|---|---|

$Ai$ . | $ai$ . | $Ai$ . | $axi$ . | $ayi$ . | $azi$ . | |

1 | −2922.66 | 0.1157 | −2922.66 | 0.0677 | 0.0677 | 0.3384 |

2 | −10.62 | 0.1722 | −10.62 | 0.1007 | 0.1007 | 0.5036 |

3 | 3914.37 | 0.1108 | 3914.37 | 0.0648 | 0.0648 | 0.3240 |

4 | −3305.68 | 0.1032 | −3305.68 | 0.0604 | 0.0604 | 0.3018 |

5 | 696.92 | 0.0910 | 696.92 | 0.0532 | 0.0532 | 0.2662 |

6 | 54.36 | 0.1523 | 54.36 | 0.0890 | 0.0890 | 0.4452 |

7 | −42.47 | 0.0724 | −42.47 | 0.0423 | 0.0423 | 0.2116 |

8 | 1616.78 | 0.1108 | 1616.78 | 0.0648 | 0.0648 | 0.3240 |

TPC terms . | Equiaxed case . | Elongated case . | ||||
---|---|---|---|---|---|---|

$Ai$ . | $ai$ . | $Ai$ . | $axi$ . | $ayi$ . | $azi$ . | |

1 | −2922.66 | 0.1157 | −2922.66 | 0.0677 | 0.0677 | 0.3384 |

2 | −10.62 | 0.1722 | −10.62 | 0.1007 | 0.1007 | 0.5036 |

3 | 3914.37 | 0.1108 | 3914.37 | 0.0648 | 0.0648 | 0.3240 |

4 | −3305.68 | 0.1032 | −3305.68 | 0.0604 | 0.0604 | 0.3018 |

5 | 696.92 | 0.0910 | 696.92 | 0.0532 | 0.0532 | 0.2662 |

6 | 54.36 | 0.1523 | 54.36 | 0.0890 | 0.0890 | 0.4452 |

7 | −42.47 | 0.0724 | −42.47 | 0.0423 | 0.0423 | 0.2116 |

8 | 1616.78 | 0.1108 | 1616.78 | 0.0648 | 0.0648 | 0.3240 |

## IV. RESULTS AND DISCUSSIONS

Now we proceed to discuss the FE and the SOA phase velocity results for Al, Inconel, and CSP elongated polycrystals. We analyze separately the dependencies of velocity on frequency, elastic properties, and grain geometry and investigate in detail the quasi-static velocity limit.

### A. Comparison of the SOA and FE models

The SOA and finite element model (FEM) velocity results are compared in the three panels of Fig. 5 for Al, Inconel, and CSP, respectively. It qualitatively predicts the dependences of phase velocity on frequency, grain geometry, and elastic property. The dimensionless frequency, $2k0Lax$, is used for the *x*-axis of each panel; note that at $2k0Lax=1$ a wavelength is about ten times the mean grain radius $ax$ in the shortened direction (about two times the mean grain radius $az$ in the elongated direction). The *y*-axis of each panel represents the normalized variation in phase velocity $VL$ from the Voigt velocity, $VL/V0L\u22121$. The leftmost solid points in each panel represent quasi-static velocities obtained from FE simulations and details for these results will be provided in Sec. IV E. Although these points correspond to $2k0Lax\u21920$, they are plotted at $2k0Lax\u22480.1$ to fit the scales of the plots. We note that small discontinuities can be found as we look along sequential FEM points, which is clearer for those of Al shown in Fig. 5(a). These discontinuities occur at the interfaces of neighboring FE material models (Table I) that are used in this work to cover different frequency ranges, as was explained in Sec. III. The discontinuities arise because neighboring models have slightly different levels of numerical error that is related to frequency and mesh size. The numerical error has been minimized by using a Courant number $C$ as close as possible to unity ($C=0.98$ in this work) but has not been fully eliminated.^{35} The FEM points are provided in slightly different frequency ranges for the shortened *x* and elongated *z* directions to achieve the same level of numerical accuracy.

The quality of agreement between the SOA and FE models is excellent. The overall difference between the two models for the phase velocities across the entire frequency range is further quantified by the root-mean-square deviation (RMSD), normalized by the FEM velocity. The results are given in Table IV for the studied materials (the equiaxed Al polycrystal is also included for comparison). The table shows that: (1) the level of agreement for the elongated case is about the same as that of the equiaxed case; (2) for all materials the SOA model has a very good agreement with the FE model in the shortened direction, and the agreement decreases in the elongated direction due to the increased scattering; (3) the overall discrepancy between the two models is smallest for Al and largest for Inconel, suggesting that the discrepancy increases with the universal anisotropy factor, which was also observed for equiaxed polycrystals.^{5} In general, the observations for the RMSD between the SOA and the FE models are similar to those for the attenuation;^{5} however, the velocity RMSD is about two orders of magnitude smaller. The point-by-point difference between the SOA and FE models is annotated as $\delta max$ in the figure, showing the largest value in the elongated direction for all materials. The maximum discrepancy of −0.59% is found for CSP in the grain elongation direction at the highest normalized frequency of $2k0Laz=44.56$ (the high-frequency range will be discussed below). From our experience, the effect of SBCs may cause a discrepancy of −0.2% between the SOA and FE models for equiaxed Inconel.^{35} Although scattering is weak at low frequencies, a noticeable difference between the SOA and FE models is observed for Inconel in Fig. 5(b) for the quasi-static velocity. This difference may be attributed to the approximations involved in both the SOA and FE models, and further study is required to fully understand it.

Materials . | RMSD for equiaxed case . | RMSD for elongated case . | |
---|---|---|---|

Shortened . | Elongated . | ||

Al | 0.006% | 0.007% | 0.014% |

Inconel | – | 0.225% | 0.243% |

CSP | – | 0.178% | 0.183% |

Materials . | RMSD for equiaxed case . | RMSD for elongated case . | |
---|---|---|---|

Shortened . | Elongated . | ||

Al | 0.006% | 0.007% | 0.014% |

Inconel | – | 0.225% | 0.243% |

CSP | – | 0.178% | 0.183% |

The excellent agreement of phase velocity between the SOA and FE models indicates that the SOA model is suitable for accurate velocity estimation.

In the low-frequency Rayleigh regime, the phase velocity is independent of frequency, i.e., nondispersive, for all material cases and in all directions. While in the transition from the Rayleigh to the stochastic regimes, the phase velocity is dispersive in all cases. In the high-frequency stochastic regime, the weak dispersion is observable for the shortened direction until the transition to the geometric regime. In the elongated direction, after reaching maximum, the phase velocity starts to rapidly decrease even for Al, which has the lowest degree of inhomogeneity. These will be discussed in-depth below.

### B. Frequency dependence

Phase velocity is in general dispersive across the entire frequency range but also exhibits nondispersive behaviors at low and high frequencies.

In the low-frequency Rayleigh regime, the phase velocity dispersion is small and is well approximated by the quasi-static limit and a reasonable agreement between the theoretical SOA and numerical FE results is observed in Fig. 5. With frequency decrease, the velocity extends asymptotically to the quasi-static velocity limit, which is significantly below the Voigt average; see further discussions in Sec. IV E. The FE and the Born approximations of the phase velocity for the CSP polycrystal are shown in Fig. 6, which also presents the contributions of the $L\u2192T$ and $L\u2192L$ scattering components in addition to the total Born velocity. The figure demonstrates that the longitudinal phase velocity is dominated by the $L\u2192T$ scattering component in the Rayleigh region, which is consistent with that of attenuation.^{2,25}

The nondispersive high-frequency region is first discussed below. It is followed by the discussion of the overall variation of the scattering-induced phase velocity dispersion within a frequency range that is usually used in experimental studies.

#### 1. High-frequency region

We observe from both panels of Fig. 6 that in the high-frequency stochastic regime the $L\u2192T$ velocity components are weakly dispersive and behave similarly on frequency for both propagation directions; interestingly, they are slightly higher than the Voigt velocity $V0L$. This phenomenon is not well understood especially that the transverse wave is much slower than the longitudinal. Figures 5 and 6 show that the total velocity in the shortened grain direction is also weakly dispersive in the high-frequency stochastic regime. For Al, this region is much longer in frequency than those for Inconel and CSP, which have much higher material inhomogeneity (for equiaxed polycrystals the non-pronouncement of the stochastic regime was indicated in Ref. 3).

In contrast to the propagation in the shortened direction, the total phase velocity in the elongated direction dramatically decreases with frequency even for the weakly scattering Al. It is obvious from Fig. 6 that this directional difference is caused by the $L\u2192L$ scattering velocity component, rather than by the $L\u2192T$ component. The dominant $L\u2192L$ component in Fig. 6(b) decreases rapidly with frequency in the high-frequency range, leading to a similar decrease for the total phase velocity. The average grain radius in the elongated direction is significantly longer than the average wavelength of the waves (their ratio is represented by $2k0Laz$, where $az=5ax$). The velocity decrease may be associated with the bent path of refracted waves; due to longer propagation paths, the phase delays may be sufficient to cause a decrease of effective mean phase velocity. Importantly, the velocity decrease phenomenon is supported by the FE results, see Figs. 5 and 6. Comparing Figs. 5(c) and 6(b), we observe that the FE velocity deviates at the highest frequency points from the SOA velocity, Fig. 5(c); interestingly, the Born velocity agrees better with the FE velocity at those points, Fig. 6(b). The high-frequency behavior of the phase velocity merits further investigation by extending the FE simulations to a higher frequency; however, this is currently limited by our computational capacity and by very high scattering-induced attenuation.

#### 2. Comments on a relation of phase velocity dispersion with experiment

Here we use the predictions of the theoretical SOA model and the results of the numerical FE simulations, Fig. 5, to estimate the range of velocity dispersion. The accuracy of our FE method to measure frequency-dependent phase velocity is estimated as 0.03% (accurate to about three significant digits); the estimate is performed based on the maximum FE signal phase error obtained in Ref. 35 for the highest-frequency FE point of the CSP polycrystal in Fig. 5.

As we have mentioned in Sec. I, the authors are not aware of any experimental studies that exhibit sufficient detail and precision to allow focus on or account for the dispersion of the phase (or group) velocity in polycrystals, except for a recent^{20} comparison of the FE and the SOA phase velocity results with experimental data^{22} for equiaxed polycrystalline Cu. Based on our own experience and Ref. 22, three-to-four significant digits in absolute velocity can be measured in laboratory conditions on well-prepared polycrystalline samples (i.e., comparable to the above-mentioned FE accuracy). The effects of the finite transducer size and beam profile can be accounted for by a diffraction correction,^{38,39} and thus the measured apparent phase velocity may accurately represent the plane-wave phase velocity^{38,39} (for homogeneous nondispersive samples even higher accuracy was reported^{38}). This further enables the direct comparison of actual experiments with theoretical and numerical modeling. In addition, it is useful to note that both phase and group velocities can be measured in real experiments.^{38} In the dispersive frequency range, they are not equal but are related.^{38}

In most experimental studies, the velocity measurements are done in a relatively narrow frequency range to observe measurable dispersion that can be attributed to microstructure. However, often such measurements are compared with static measurements or low-frequency vibration measurements; see for example the intensive experimental study,^{40} where the elastic wave velocity was also investigated and compared for different material processing. In such cases of comparison, the scattering-induced dispersion needs to be considered for significantly different frequency ranges or different microstructures due to material processing. Therefore, it is practically important to have simple estimates of the range of the phase velocity dispersion.

Between static, vibration, and ultrasonic measurements, the range of the 2*ka* parameter is about 0–10. Due to grain scattering, it is very difficult to receive an ultrasonic wave at a meaningful distance above that frequency limit and to make ultrasonic velocity measurements in polycrystals with sufficient signal-to-noise ratio. For all material cases shown in Fig. 5, phase velocity in the normalized frequency range 0–10 is approximately between the Rayleigh asymptote (quasi-static velocity) and the Voigt velocity ($VL/V0L\u22121=0$). The phase velocity change for Al is about 0.1%, for Inconel and CSP is about 2% (for Inconel in the shortened direction is about 3%). The quasi-static velocity limit, Eqs. (15) and (16), is a good approximation for the lower velocity bound. The significant dispersion change starts at about $2k0Lax\u22481$ for the shortened and elongated propagation directions. As follows from Fig. 5, for both the SOA and FE models this dispersion frequency range continues until the stochastic range emerges at $2k0Lax\u22485$ in the shortened grain propagation direction and $2k0Laz\u224815$ in the elongated direction, where the phase velocity reaches a peak that is close to the Voigt velocity. In the elongation propagation direction, it is already on the high end of our assumed practical frequency range.

In the Rayleigh regime, the phase velocity is smaller in the shortened direction than in the elongated direction, while at the end of the transition regime, this reverses and the velocity in the shortened direction is slightly above that in the elongated direction. As a result, in the normalized frequency range $2k0Lax=1\u22125$, the change of the phase velocity with frequency is larger in the shortened propagation direction than that in the elongated direction.

Finally, based on the good agreement between the velocity in the infinite space obtained from the SOA model and the FE calculations, one can use the size of our FE material systems in the direction of wave propagation to estimate the number of grains in the experimental samples needed to obtain representative velocity (a material property) for a polycrystal. The elongated FE material models, Table I, contain at least 11 520 grains, translating to 7 × 7 × 29 mm in dimensions and 24 × 24 × 20 in grain numbers in the three coordinate directions. Thus, at least 10–20 grains in the propagation direction is needed to emulate our FE experiments and maximize measurement accuracy.^{35,41} This will assure a satisfactory statistical significance for measurement.^{19}

### C. Normalization by elastic scattering factors

In this section, we consider the scaling of the relative directional phase velocity difference $\xi V=(VLz\u2212VLx)/VLx$ by the elastic scattering factors. In the low-frequency range ($2k0Lax\u22642$), we divide this relative directional difference by $QL\u2192TV0L3/V0T3$ and in the high-frequency range ($2k0Lax\u22656$) by $QL\u2192L$. $QL\u2192T$ is the longitudinal-to-transverse mode-converted elastic scattering factor used in Eq. (19) and $QL\u2192L$ is the longitudinal-to-longitudinal wave elastic scattering factor.^{5,13} As was shown in Ref. 13, the attenuation and phase velocity of the longitudinal wave depend mainly on a combination of the elastic constants through parameters $QL\u2192L$ and $QL\u2192T$ even for crystallites of general anisotropy. Thus, when those parameters are nearly identical for different materials systems, those systems are approximately equivalent in terms of scattering induced attenuation and velocity dispersion. For this reason, they are used as normalization factors to evaluate this universality; more specifically, parameters $QL\u2192L$ and $QL\u2192T$ show explicit proportionalities to directional phase velocity changes in the respective low-frequency Rayleigh, see Eq. (19), and high-frequency stochastic, see Ref. 5, ranges for equiaxed polycrystals. The scaling results are plotted in Fig. 7. After the elastic property scaling, both the FE and the SOA directional velocity differences are almost overlapped for the different materials studied. We note that the elastic scattering factors are approximately proportional to the relative directional velocity difference $\xi V$ rather than the directional ratio $VLz/VLx$.

The longitudinal-to-transverse velocity ratio $V0L3/V0T3$ of the Voigt reference medium depends on the Poisson's ratio only and is weakly varying for most structural materials. For this reason, it is sufficient in the Rayleigh regime to use only the $QL\u2192T$ factor for material property scaling. Thus, we conclude that the anisotropy of wave velocity in polycrystals with elongated grains is also governed by the degrees of inhomogeneity $QL\u2192T$ and $QL\u2192L$ in the Rayleigh and stochastic regimes, respectively. These are identical to the elastic dependencies of attenuation for equiaxed polycrystals^{5} and elongated polycrystals as discussed in Ref. 13. As indicated in Ref. 13, if different material systems have nearly the same elastic scattering factors $QL\u2192T$ and $QL\u2192L$, they have very similar scattering behavior in all frequency ranges including the Rayleigh-to-stochastic transition.

### D. Geometric dependence: Effect of grain elongation

#### 1. Dependence on propagation direction

We have mentioned in Sec. IV A that phase velocity varies with propagation direction due to grain elongation. Conversely, this directional dependence may help infer average grain shape from phase velocity measurement. Thus, we further investigate this dependence by evaluating the change of phase velocity with propagation direction $\theta p$, which denotes the angle between the propagation and grain elongation directions as illustrated in Fig. 3. We use CSP as the evaluation material and simulate wave propagation in the directions of 0° (elongated direction), 15°, 18.75°, 30°, 45°, 60°, 75°, and 90° (shortened direction). The FE results and the predictions of the SOA model are plotted in Fig. 8 for the directions of 0°, 15°, 18.75°, 30°, 45°, and 90°; the results for 60° and 75° will be used later in Fig. 9. The results in the figure are limited to the high-frequency range; readers are referred to Sec. IV E for the discussion of directional dependence for the low-frequency range.

The figure shows a generally good agreement between the FE and SOA models for all directions, especially for the three directions closer to the shortened direction (90°) where $2k0La\theta p$ and therefore scattering is relatively smaller than in the other three directions. The two models start to deviate when the transition into the geometric regime begins at the normalized frequency of about 6–8. The largest deviation is about 0.8%.

The mean grain radius in the wave propagation direction decreases as $\theta p$ deviates from the elongated direction (0°). This leads to a decrease of scattering level, which is manifested in two aspects in the figure as $\theta p$ deviates from 0°: phase velocity becomes faster, and the transition point into the geometric regime occurs later in frequency, which is observable from the SOA curves.

The directional dependence is further evaluated in Fig. 9 by polar plotting the normalized phase velocity against propagation direction $\theta p$ for the normalized frequencies of 5.56 and 8.34. The *r*-axis of each panel represents the normalized phase velocity difference $VL/V0L\u22121$ and the $\theta $-axis denotes the propagation direction $\theta p$. The FEM points are provided for all simulated directions. The SOA and Born approximation results are provided as solid and dash lines in the figure.

The agreement between the FE and theoretical models is generally very good in all directions. In directions nearer the shortened direction (90°/270°), the two theoretical models both provide slightly larger velocities than the FE model. Closer to the elongated direction (0°/180°), the FE model matches better with the Born approximation than with the SOA model. This is because the SOA model predicts the transition to the geometric regime; however, at the highest FE frequencies achieved in this study, the FE velocity does not follow this transition and coincides with the Born approximation.

The absolute value of normalized phase velocity difference in Fig. 9 increases as the propagation direction changes from the shortened direction (90°/270°) to the elongated direction (0°/180°), which is observable from both the numerical FE and theoretical SOA models. This means that as a function of the propagation angle the scattering-induced variation of phase velocity (characterized by the normalized phase velocity $VL/V0L\u22121$) increases with the mean grain radius in the direction of propagation. This indicates that the phase velocity dispersion in the high-frequency range carries information about the average geometry of the grains. The absolute difference of FEM phase velocity between the shortened and elongation directions is 19.19 and 48.73 m/s at the normalized frequencies of 5.56 and 8.34, respectively.

#### 2. Dependence on grain aspect ratio

The above evaluation of directional dependence has used a representative aspect ratio of $R=5$. Since in practical materials grain aspect ratio may vary in a wide range, an important point would be how directional dependence changes with grain aspect ratio $R$. To consider pancake-shaped grains, aspect ratio rather than elongation ratio is used here to refer to the ratio of average grain radius between the axial *z* and transverse *x* directions.

In Fig. 10(a), we plot the relative directional ratio of phase velocity, $VLz/VLx\u22121$, against the normalized frequency for CSP with four aspect ratios of 0.1, 0.2, 5, and 10. The predictions of the SOA and the Born approximation models are provided for all cases, while the FEM points are only provided for $R=5$. In the low-frequency Rayleigh range, the directional ratio has a flat part observable for all cases. This is because phase velocity is nondispersive as explained in Sec. IV B. The flat parts appear on the opposite sides of $VLz/VLx\u22121=0$ for $R<1$ and $R>1$. They collectively indicate that phase velocity is larger for a longer mean grain radius in the direction of propagation. Beyond the nondispersive Rayleigh regime, the directional ratio exhibits a complicated behavior. A non-monotonic variation is visible in the Rayleigh-stochastic transition range, and it occurs relatively early in frequency for the two cases with $R>1$. This variation ends at the frequency where the ratio starts to decrease (for $R>1$) or increase (for $R<1$) continuously. The continuous decrease or increase at high frequencies may be caused by the dominant $L\u2192L$ scattering component as has been discussed in Sec. IV B 1.

In Fig. 10(b), we further evaluate how the directional velocity ratio varies with the grain aspect ratio. We choose the normalized frequencies of 3, 5, 7, 10, 15, and 20 for the evaluation. The lowest frequency corresponds to the peak ratio in Fig. 10(a) for $R=5$, so the chosen frequencies are mostly in the high-frequency range. A similar discussion will be given for the low-frequency Rayleigh range (the quasi-static limit) in Sec. IV E. We have used the simpler Born approximation, instead of the SOA model, to obtain the curves in Fig. 10(b) because it is computationally efficient and its prediction is very close to that of the SOA model as can be seen in Fig. 10(a). As the aspect ratio approaches zero, the directional velocity ratio tends to be independent of the aspect ratio and it converges to the same value for all frequencies. This converged value is below zero, which is consistent with the above finding that the shorter axial direction has a slower phase velocity than the longer transverse direction. On the right side of the plot, the directional velocity ratio exhibits a continuous decrease with the aspect ratio in the shown range. This may be due to the dominant $L\u2192L$ scattering component that tends to have larger phase delays in the elongated *z* direction as a wave propagates through longer grains. We note that a nonpropagative case ($VLz/VLx\u22121=\u22121$) will not happen because each curve in the figure will eventually reach a flat part (with the directional ratio being greater than −1) as the aspect ratio further increases.

### E. Quasi-static velocity limit

In this final section, we study the effect of grain elongation on the velocity at the quasi-static limit. It is important for understanding the wave behavior of polycrystals at low *ka* and also because it is directly related to the study of effective, static elastic constants. Our prior studies^{5,20} have evaluated the quasi-static velocity limit by using both the SOA and FE models, but they have been limited to polycrystals with equiaxed grains. To our knowledge, limited attention has been received for studying the quasi-static limit with the consideration of grain shape. While Kocks *et al.*^{24} have included tables showing simulated results for the combined effect of macro-texture and grain shape on elastic moduli, it is difficult to make a conclusion on the grain shape effect. This section aims to evaluate the effect of grain elongation on the quasi-static velocity limit.

In addition to the analytical quasi-static velocity limit (the low-frequency Rayleigh velocity limit) obtained above in Eq. (15), we also employ the FE method to calculate the quasi-static limit; readers are referred to our previous studies for this quasi-static FE method.^{5,20} We note that the continuity of stress and strain at grain boundaries is naturally satisfied in the chosen FE method, which thus delivers a very accurate calculation of the quasi-static velocity.^{5}

The quasi-static velocities obtained from the FE and SOA models have already been provided in Fig. 5, but they are more elaborately summarized in Table V for all three polycrystalline materials. As indicated by the relative difference $\delta $, the SOA results are mostly indistinguishable from the FEM results. The SOA model gives a generally higher quasi-static velocity but the maximum difference between the SOA and FE models for the studied cases is as small as 0.36%. Furthermore, quasi-static velocity shows an identical trend to attenuation and dynamic phase velocity in that the increase of anisotropy (characterized by the universal anisotropy factor as shown in Table II) directly causes the increase of difference between the SOA and FE models.

Material . | Shortened $x$ direction . | Elongated $z$ direction . | $\xi V$ (%) . | $\xi V/[QL\u2192TV0L3/V0T3]$ . | ||||
---|---|---|---|---|---|---|---|---|

FEM . | SOA . | $\delta $ (%) . | FEM . | SOA . | $\delta $ (%) . | |||

Al | 6312.2 | 6312.4 | 0.003 | 6313.4 | 6313.1 | −0.005 | 0.019 | 0.069 |

Inconel | 5890.8 | 5911.6 | 0.353 | 5911.1 | 5927.6 | 0.280 | 0.344 | 0.079 |

CSP | 4780.2 | 4784.7 | 0.094 | 4799.5 | 4802.3 | 0.060 | 0.403 | 0.059 |

Material . | Shortened $x$ direction . | Elongated $z$ direction . | $\xi V$ (%) . | $\xi V/[QL\u2192TV0L3/V0T3]$ . | ||||
---|---|---|---|---|---|---|---|---|

FEM . | SOA . | $\delta $ (%) . | FEM . | SOA . | $\delta $ (%) . | |||

Al | 6312.2 | 6312.4 | 0.003 | 6313.4 | 6313.1 | −0.005 | 0.019 | 0.069 |

Inconel | 5890.8 | 5911.6 | 0.353 | 5911.1 | 5927.6 | 0.280 | 0.344 | 0.079 |

CSP | 4780.2 | 4784.7 | 0.094 | 4799.5 | 4802.3 | 0.060 | 0.403 | 0.059 |

The grain shape effect in the table is manifested as the directional difference between the elongated and shortened directions. To further evaluate this effect, we plot normalized quasi-static velocity versus propagation direction in Fig. 11(a). In the figure, FE results are only provided for the elongated and shortened directions, but the excellent agreement between the FE and SOA models means that FE results should be close to the SOA curves in all directions. In the figure, the SOA curves are all close to elliptical and their major axes overlap with the direction of grain elongation, revealing a positive correlation between quasi-static velocity and grain geometry.

The directional difference is further quantified in the table by the ratio $\xi V$ as used earlier in the above sections. It is clear from the table that this ratio can differ significantly from one material to another, but it tends to deliver the same value when it is divided by the elastic factor $QL\u2192TV0L3/V0T3$. This is an important finding that grain shape effect is also positively correlated with the elastic inhomogeneity perturbation of polycrystals.

In addition, the analytical expression in Eq. (15) is used to evaluate the directional ratio of quasi-static velocity between the axial *z* and transverse *x* directions versus the grain aspect ratio, Fig. 11(b). Both the cases for the pancake shape ($R<1$) and the needle shape (elongated, $R>1$) are shown in the figure. For this reason, we have used in Fig. 11(b) the notation $VLz/VLx\u22121$ for the velocity ratio, where the ellipsoid axes *z* and *x* are defined in Fig. 1. The FE results are provided as points in the figure, corresponding to the elongation ratio of $R=5$. This work is not intended to extend the quasi-static FE simulations to other elongation ratios because it has a very large demand on computational costs, this may be pursued in future work.

The figure shows a general increase in the directional velocity ratio as the grain aspect ratio gets larger. Obviously, the directional difference vanishes for the equiaxed case of $R=1$, given by Eq. (19). The velocity ratio reaches its lower bound as $R$ approaches zero ($R\u21920$), Eq. (20), which is shown in dotted lines. Similarly, the quasi-static velocity ratio reaches its upper bound as grain elongation increases ($R\u2192\u221e$), Eq. (21). These bounds are further listed in Table VI, showing a generally larger directional difference for the case of $R\u21920$ than for $R\u2192\u221e$. For the latter case, the absolute quasi-static velocity differences between the elongated axial and shortened transverse directions are, respectively, 0.9, 19.3, and 21.3 m/s for Al, Inconel, and CSP. The velocity differences of Inconel and CSP may be measured in practical experiments.

Materials . | Lower bound at $R\u21920$, Eq. (20) . | Upper bound at $R\u2192\u221e$, Eq. (21) . | ||||
---|---|---|---|---|---|---|

$VLz$ (m/s) . | $VLx$ (m/s) . | $VLz/VLx\u22121$ . | $VLz$ (m/s) . | $VLx$ (m/s) . | $VLz/VLx\u22121$ . | |

Al | 6312 | 6313 | −0.0003 | 6313 | 6312 | 0.0001 |

Inconel | 5892 | 5930 | −0.0064 | 5930 | 5911 | 0.0033 |

CSP | 4764 | 4805 | −0.0085 | 4805 | 4784 | 0.0044 |

Materials . | Lower bound at $R\u21920$, Eq. (20) . | Upper bound at $R\u2192\u221e$, Eq. (21) . | ||||
---|---|---|---|---|---|---|

$VLz$ (m/s) . | $VLx$ (m/s) . | $VLz/VLx\u22121$ . | $VLz$ (m/s) . | $VLx$ (m/s) . | $VLz/VLx\u22121$ . | |

Al | 6312 | 6313 | −0.0003 | 6313 | 6312 | 0.0001 |

Inconel | 5892 | 5930 | −0.0064 | 5930 | 5911 | 0.0033 |

CSP | 4764 | 4805 | −0.0085 | 4805 | 4784 | 0.0044 |

We have revealed from Eqs. (20) and (21) that the phase velocities for $R\u21920$ and $R\u2192\u221e$ are the same along the long axes of the grains (*x* direction for the former and *z* direction for the latter). This equality is shown in the third and fifth columns of Table VI. We note that the equality is only valid for this extreme case, meaning that phase velocity differs between a $R<1$ case and a $R>1$ case even though they have the same grain radius in the long axes of the grains.

## V. SUMMARY AND CONCLUSIONS

The SOA and 3D FE models are utilized to study the phase velocity dispersion of elastic waves in single-phase polycrystals with elongated grains of arbitrary crystal symmetry. The SOA model is suitable for the scattering of longitudinal and shear waves by statistically ellipsoidal shaped grains. It accounts for some possibilities of multiple scattering and is valid for all frequencies. The Born approximation for the phase velocity is obtained in a new form for the elongated polycrystals with the statistical grain shape of a general ellipsoid; this form of the Born approximation results from the explicit evaluation of the Cauchy integral. From the Born approximation, the explicit expression is derived for the Rayleigh quasi-static velocity asymptote. The 3D FE model is implemented for longitudinal waves in polycrystals with statistically ellipsoid-of-rotation grains. It accurately represents wave scattering in realistic sample volumes with a large number of grains that are densely packed and fully bonded. The TPC function of the FE material model is accurately determined and incorporated into the SOA model to achieve a comparative study of the two models. The study employs three polycrystalline materials, Al, Inconel, and CSP, that cover different crystal symmetries and anisotropy factors to identify the relation of velocity to frequency, grain geometry, and elastic properties.

An excellent agreement is found between the SOA and FE models for the studied materials across the investigated frequency ranges. The overall difference between the two models, as characterized by the RMSD, is smaller than 0.3% for all cases. The elongated polycrystals show a similar RMSD to the equiaxed case.^{5} The RMSD is slightly larger in the elongated direction of the grains than in the shortened direction, showing a direct relation to the level of scattering. The RMSD is the smallest for Al and largest for Inconel, exhibiting a positive correlation with the universal anisotropy factor. The largest point-by-point difference between the SOA and FE models is around −0.6%, which is found for CSP in the elongated direction.

The comparative study of the SOA and FE models shows that the relative directional difference of phase velocity exhibits anisotropy as a result of grain elongation. This grain elongation effect varies from one scattering regime to another. In the Rayleigh regime, the phase velocity is larger in the direction of grain elongation. This relation is reversed in the stochastic regime, where the velocity is slower in the elongated direction than in the shortened direction.

The comparative study also shows a direct influence of elastic scattering factors on phase velocity. This influence is manifested as the proportionalities of directional velocity difference to the two elastic scattering factors, $QL\u2192T$ and $QL\u2192L$, in the Rayleigh and stochastic regimes, respectively. These proportionalities suggest that the elastic property affects the phase velocity via the two combinations of the elastic constants of the crystallites.

The analytical and numerical analyses show that the phase velocity at the quasi-static limit follows a similar pattern as found for phase velocity in the low-frequency Rayleigh regime. Namely, the quasi-static velocity is angle-dependent in elongated polycrystals, and its directional change is related to the grain aspect ratio and the elastic scattering factor $QL\u2192T$. The analytical analysis shows that the quasi-static velocity is lower and upper bounded as the grain aspect ratio decreases and increases, respectively.

## ACKNOWLEDGMENTS

M.H. was supported by the Chinese Scholarship Council and the Beijing Institute of Aeronautical Materials. P.H. was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) fellowship EP/M020207/1. M.J.S.L. was partially sponsored by the EPSRC. G.S. and S.I.R. were partially supported by the AFRL (USA) under the prime contract FA8650-10-D-5210.

### APPENDIX: DERIVATION OF THE RAYLEIGH VELOCITY ASYMPTOTE

The derivation in this section is limited to elongated grains that are axially symmetric about the *z*-axis of the global coordinate system. In this case, the coefficients of the generalized TPC function in Eq. (7) satisfy $axi=ayi$, and an elongation ratio can be defined as $Ri=azi/axi$. In the Born approximation, the functions $C0i$, $C1i$, and $C2i$ in Eq. (11) can thus be written in a simpler form as

For the function $C0i$, its second term is bounded in the interval [$k0M2(axi)2$, $k0M2(azi)2$] when $Ri\u22651$ and in the inverted interval when $Ri<1$. This function can thus be approximated as $C0i=1$, because in the low-frequency Rayleigh regime, the dimensionless wave number $k0Maxi$ and $k0Mazi$ are small. Similarly, the function $C2i$ is bounded by $k0N2(axi)2$ and $k0N2(azi)2$, and thus $C2i\u226a1$ at the low-frequency limit. The function $C1i$ is not always positive and its bounds can be given as [$\u22122k0Mk0N(azi)2$, $2k0Mk0N(azi)2$] and [$\u22122k0Mk0N(2(axi)2\u2212(azi)2)$, $\u22122k0Mk0N(azi)2$] for $Ri\u22651$ and $Ri<1$, respectively. The absolute value of $C1i$ is always much smaller than unity in the Rayleigh limit.

Equation (11) can now be written as

where the factor $[1\u2212C1i\xi /(1+C2i\xi 2)]2$ can be approximated as unity. This conclusion is reached differently for different signs of the function $C1i$. When $C1i>0$, the factor $C1i\xi /(1+C2i\xi 2)$ is a positive function of $\xi $ in the range of $[0,\u221e]$. Its maximum occurs at $\xi =1/C2i$ and the maximum value is $C1i/(2C2i)$, which is smaller than unity according to the above-mentioned bounds of $C1i$ and $C2i$. Thus, the factor $1/(1\u2212C1i\xi /(1+C2i\xi 2))2$ can be expressed in the Taylor series as $\u2211j=1\u221ej[C1i\xi /(1+C2i\xi 2)]j\u22121$. By neglecting higher-order terms, the factor becomes unity. Similarly, for $C1i\u22640$, we arrive at the same conclusion. As a result of the Rayleigh limit, the double integral in Eq (A2) is independent of $\xi $ and can be pull-out from the Cauchy integral, and it can be evaluated.

The limit of the integrand $\xi 4/[(1\u2212\xi 2)(1+C2i\xi 2)2]$ is zero as $\xi $ approaches infinity. This allows the Cauchy integral to be solved by the contour integral and residue theorem,

The result can be further simplified by taking the first term of its Taylor series (with respect to $C2i$), leading to the integral being given as $\u2212\pi /4/(C2i)3/2$. Substituting the result for the triple integral in Eq. (A2) into Eq. (10), we obtain the relative change of velocity as

Evaluating the inner integration of Eq. (A4) over $\phi $ and denoting $x=cos\u2009\theta $ leads to

where

and