Finite element evaluation of a simple model for elastic waves in strongly scattering elongated polycrystals

Polycrystalline materials tend to have elongated grains when produced by traditional forging, rolling, and extrusion technologies^1,2 as well as by the fast-growing additive manufacturing techniques.³ Prior experimental studies showed that the elongated grains exert an anisotropic effect on the scattering-induced attenuation and velocity dispersion of elastic waves propagating in such materials.^4–6 Therefore, an analytical model accurately describing this effect is essential to non-destructively characterize the elongated grains from elastic wave measurements. Early analytical studies were performed by directly extending the Stanke and Kino⁷ and Weaver⁸ models suitable for elastic waves in cubic polycrystals with statistically equiaxed grains to the case of elongated grains,^9–12 but these studies are limited to grains of cubic symmetry⁹ or frequencies below the geometric regime due to the invoking of the Born approximation.^10–12 The subsequently developed far-field approximation (FFA)¹³ and spectral function^14,15 models are valid at all frequency ranges for statistically ellipsoidal grains of essentially any symmetry.¹³ Recently, we advanced a second-order approximation (SOA) model for wave propagation in polycrystals with statistically ellipsoidal grains of arbitrary symmetry described by the generalized two-point correlation statistics.^16,17 This SOA model maintains the same second-order accuracy of material inhomogeneity perturbation as the Stanke and Kino model.⁷ The approximations of the SOA model were quantitatively evaluated by comparison to accurate three-dimensional (3D) grain-scale finite element (FE) simulations for polycrystals with statistically ellipsoidal grains.^16,17 The comparative study demonstrates good agreement between the SOA model and finite element model (FEM) for weakly scattering aluminum (Zener anisotropy factor $A=1.24$⁠) across a wide frequency range for both attenuation and phase velocity. However, the SOA model exhibits an apparent deviation from the FE results for relatively highly scattering Inconel (⁠$A=2.83$⁠), and unexpectedly, the deviation is more pronounced in the low-frequency Rayleigh regime than in the high-frequency stochastic regime as one would expect. In this work, we will demonstrate in Sec. 2 that the low-frequency validity of the SOA model is greatly challenged by a more strongly scattering material (lithium, $A=9.14$⁠). We attribute this pronounced low-frequency difference between the SOA model and FEM to the occurrence at low frequencies of multiple scattering from grains in all directions in polycrystals with strongly anisotropic grains, which is not accounted for by the SOA model. To overcome this analytical difficulty, we propose a simple semi-analytical model in Sec. 3 that is valid for strongly scattering cubic polycrystals, and we evaluate the applicability of this model in Sec. 4 by comparing with 3D FE simulation results. Summary and conclusions are provided in Sec. 5.

Here, we use our prior FE and SOA results for aluminum and Inconel^16,17 and a set of new results for the more highly anisotropic lithium to demonstrate how the existing SOA model is challenged by strongly scattering elongated polycrystals in the low-frequency Rayleigh range.

As detailed in our prior work,^16,17 the 3D FE method uses Voronoi tessellation¹⁸ and subsequent scaling^16,17 to generate grain-scale spatial polycrystal models [see Fig. 1(a)]. The grains have a statistically ellipsoid-of-rotation shape and an elongation ratio of $R=5$⁠, meaning that the grain radius in the elongated z-direction is 5 times that in the shortened x-direction. The grains within each model are assigned with the same mass density and elastic constants, but their crystallographic axes are uniformly randomly oriented; therefore, the elastic property of each model is macroscopically isotropic (untextured) in an average sense, and the anisotropy in the scattering-induced attenuation and velocity dispersion is solely attributed to the grain elongation. Each model is spatially discretized with structured “brick” elements and temporally sampled to discrete time steps. To simulate a plane longitudinal wave propagating in the elongated z-direction (similarly in the shortened x-direction), symmetry boundary conditions are defined for the four lateral surfaces, as illustrated in Fig. 1(a), and a uniform force in the form of a three-cycle Hann-windowed toneburst is applied in the surface normal direction to all nodes on the $z=0$ surface. The axes of the grains are not necessarily aligned with the coordinate axes as shown in Fig. 1(a) for simplicity.^16,17 The wave propagation problem is solved with the graphics processing unit (GPU)-accelerated Pogo program.¹⁹ The propagating wave scatters on the grain boundaries as a result of impedance mismatch between adjacent grains, owing to the anisotropic grains and the different crystallographic orientations of the grains. Spatially averaged coherent displacement signals are recorded on the transmitting $z=0$ surface and its opposite receiving $z=dz$ surface, and the Fourier transforms of the two signals are used to calculate the frequency-dependent attenuation and phase velocity of the mean coherent wave.

In Fig. 2, the FEM points for aluminum and Inconel are reproduced from our prior work,^16,17 whereas the results for lithium are newly produced in this work. Figure 2(a) shows attenuation results, while Fig. 2(b) displays the phase velocity variation versus normalized frequency. Since the phase and group velocities are related in the dispersive frequency range,²⁰ this work focuses just on the phase velocity. The three materials have grains of cubic crystal symmetry with greatly different Zener anisotropy indices A given in Table 1. The material properties for aluminum and Inconel are summarized in Table 1 of Ref. 16, while lithium has the properties²¹ of $c11=13.40$ GPa, $c12=11.30$ GPa, $c44=9.60$ GPa, and $ρ=534$ kg/m³, and the Voigt velocities of lithium are $V0L=6157$ m/s and $V0T=3402$ m/s for longitudinal and transverse waves, respectively. For each material in Fig. 2, multiple spatial FEMs were used, each using different center frequencies for the applied toneburst force, to cover a wide frequency range; the spatial models and center frequencies are outlined in Table 2 of Ref. 16. At each center frequency, 15 realizations of the same spatial model but randomly reshuffled crystallographic orientations are utilized to obtain the averaged results shown in Fig. 2; the corresponding standard deviations are not plotted because they are too small to be visible. A high degree of numerical accuracy has been achieved for the shown FEM results that accurately account for all possible multiple scattering events occurring in the polycrystalline media.^16,17,22

The analytical SOA model^16,17 is applied to the same material systems for plane longitudinal wave scattering in polycrystals with statistically ellipsoidal grains; in contrast to the FEM, the SOA model only partially accounts for multiple scattering effects. The model treats a polycrystal as a statistically continuous medium with random elastic fluctuations and describes the spatial property of the medium by the two-point correlation (TPC) function. The TPC function $w(r)$ represents the probability of two points $x$ and $x′$ separated by $r=x−x′$ falling into the same grain, and it statistically describes the size and shape of the grains. For direct comparison, the direction-dependent TPC function is numerically measured from the FE spatial models and fitted into a generalized function, $w(r)=∑iAi exp[−rx2/(axi)2+ry2/(ayi)2+rz2/(azi)2]$⁠, which is valid^16,17 in an arbitrary direction of wave propagation $θp$ [Fig. 1(b)]. The coefficients $Ai$ and $ax,y,zi$ are provided in Table 3 of Ref. 16, and $ax,y,zi$ have a scaling relation of $(axi,ayi,azi)=ai(1,1,R)/R3$ to the coefficients of the equiaxed case $ai$⁠.^16,17 The SOA model provides a dispersion equation for the effective wave number $k$ of the mean coherent wave perturbed by the elastic fluctuations in the medium along the wave propagation^13,16,17

where the perturbed wave has a wave vector $k=kp$⁠, propagating in the direction $p$ as shown in Fig. 1(b). $ω=2πf$ (where $f$ is the frequency) denotes the angular frequency. The mass operator $mM$ accounts for the random scattering events that occur in the polycrystal, and its expression can be found in Refs. 16 and 17 (for the FFA model utilized below, its expression can be found in Ref. 13). The imaginary and real parts of the wave number solution determine the attenuation and phase velocity of the propagating wave by $VL=ω/Rek$ and $αL=Imk$⁠, respectively. The resulting SOA predictions are displayed as solid lines in Fig. 2.

Figure 2 shows that the SOA attenuation and phase velocity predictions have an excellent agreement with the FE results for aluminum in both the shortened and elongated grain directions. However, for the more strongly scattering Inconel and especially lithium, a distinctive difference can be observed between the SOA predictions and the FE results. Surprisingly, the difference is more pronounced in the low-frequency Rayleigh range than in the high-frequency stochastic range, where the degree of scattering would usually be expected to be stronger. At the normalized frequency of $2k0Lax=1$⁠, for example, the relative difference in attenuation is –33% for Inconel and −58% for lithium in the grain elongated direction; the respective phase velocity differences are 0.29% and 1.66%. These results suggest that in strongly scattering polycrystals, a substantial multiscattering occurs at low frequencies, which is not accounted for by the SOA model. Thereby, it is desirable to have an analytical model that is applicable to highly scattering materials.

It was shown in Ref. 13 that the terms of the dispersion equation for the effective wave number are factorized to the elastic and geometric factors with the two elastic scattering factors $QL→L$ and $QL→T$ describing, respectively, longitudinal-to-longitudinal and longitudinal-to-transverse scattering in macroscopically isotropic polycrystals (three factors for polycrystals with macrotexture). It was shown¹³ that the factorization is also approximately satisfied for the SOA model. By comparison with the FEM results, it was shown^16,17 that the factorization holds in the low frequency with the elastic scattering factor $QL→T$ (at low frequency, the propagating longitudinal wave mainly scatters into the transverse wave). Due to the factorization and the independence of low-frequency attenuation on grain geometry,^16,17 we can relate the attenuation coefficient to $QL→T$ using the FE results for the polycrystals with equiaxed grains for six cubic materials studied in our prior work.²³ Based on these FE results, quadratic relations between the elastic scattering factor $QL→T$ and the attenuation and phase velocity variation at $2k0La=1$ are obtained by fitting

and for the real and imaginary parts of the effective wave number $kL$⁠, we have

We note that the seemingly meaningless numbers 1.39, 0.09, and 0.78 are nearly equal to those obtained at this frequency parameter from the Born approximation of the FFA model.¹³ $a=1/∑i(Ai/ai)$ is the mean line intercept of the equiaxed grains.

The respective relation for the SOA predictions is found to be linear at the same frequency for both attenuation and phase velocity variation. This suggests that an iterative approach²⁴ may be applied to the SOA model to incorporate the quadratic term on the elastic scattering factor $QL→T$⁠. However, even though a quadratic term can be successfully introduced by one step of the iteration, the coefficient for this term is too small; further analysis reveals that the model with infinite steps of iteration still underestimates the level of attenuation and velocity variation, meaning that only a subset of the infinite scattering events is considered.

Instead, we propose a semi-analytical model based on the simpler Born approximation of the FFA model, which has closed-form expressions for the attenuation and phase velocity of the mean coherent wave that explicitly depend on the elastic scattering factors. We first perform one iteration of the model to introduce a corrective quadratic term on the scattering factor and then increase the coefficient of this corrective term for the resulting model prediction to match with the FEM results. For a longitudinal wave propagating in a polycrystal with statistically ellipsoid-of-rotation grains, the Born approximation of the FFA model is given by^13,16

where the first three terms are obtained from the original model,¹³ while the last two terms are included to obtain the proper Rayleigh velocity limit of the SOA model.^13,16 The integral $IM→Ni$ and the relative velocity change $ΔM→N$ are provided in the  Appendix. By performing one iteration for the two $L→T$ terms in Eq. (4), we obtain

where the corrective terms $2piReQL→T$ and $4piImQL→T$ for the real and imaginary parts of the perturbed wave number are generated by the iteration, whereas the respective corrective coefficients $π3/2$ and $π3$ are found by fitting the model to Eq. (3). Since in the FE material model, the elongated grains have a scaling relation to the equiaxed grains^16,17 and the real and imaginary parts of the perturbed wave number are interrelated in the dispersion equation, we have utilized the imaginary part of the perturbed wave number of the equiaxed case to define the corrective factors $piRe$ and $piIm$⁠, both having the form of $1/[(1+k0L2ai2−k0T2ai2)2+4k0T2ai2]$⁠.¹³ Additionally, because our intention is to retain the same frequency behavior in the stochastic regime as that of the original FFA model in Eq. (4), we have modified the expressions of $piRe$ and $piIm$ to improve the transition of the semi-analytical model into the stochastic regime, leading to the final expressions

At the Rayleigh limit of $k0L→0$⁠, the real part of the integrand in $IM→Ni$ is symmetric about $x=0$⁠, and the real part of the integral is thus zero. Therefore, it follows from Eq. (5) that $RekR=k0L+k0LΔL→L+k0LΔL→T∑iAi(1+π3QL→T)$⁠, and the Rayleigh (quasi-static) velocity asymptote is frequency-independent and given by

The Rayleigh attenuation asymptote is easier to obtain from the Born approximation of the FFA model in the spatial frequency domain, and thus we replace the imaginary part of $IM→Ni$ with¹³ 

where $(qx,qy,qz)=k0Mp−k0Ns$ is the difference between the incident and scattered wave vectors [see Fig. 1(b)]. Since $qx2(axi)2+qy2(ayi)2+qz2(azi)2$ vanishes at $k0L→0$⁠, Eq. (8) can be analytically evaluated, leading to $ImIM→Ni=4axiayiazik0N$⁠. Consequently, the Rayleigh attenuation asymptote can be obtained from Eq. (5) as

where $Veffg=8π∑iAiaxiayiazi$ is the effective grain volume.^7,8,23

The attenuation and phase velocity predictions of the semi-analytical model [Eq. (5)] are compared with the FEM and SOA results in Fig. 2. The figure shows that the semi-analytical model mostly overlaps with the FE points in the low-frequency range for both attenuation and phase velocity in both the shortened and elongated directions of all three materials. The normalized root-mean-square deviation (RMSD) of the models from the FEM values, shown in Table 1, further demonstrates that for lithium, the semi-analytical model performs almost 1 order of magnitude better than the SOA model in the low-frequency range. The table also shows that the difference between the semi-analytical model and FEM barely depends on material anisotropy. In most cases, the semi-analytical model has a better agreement with the FEM results than the SOA model in the transition region, as can be seen from both Fig. 2 and Table 1. The semi-analytical model approaches the SOA model in the stochastic region.

The Rayleigh asymptotes of the semi-analytical model [Eqs. (7) and (9)] are also provided in Fig. 2, and it can be hypothesized that the asymptotes would have a very high degree of accuracy in predicting low-frequency attenuation. For phase velocity, we have provided the quasi-static FEM limits for all three materials in Fig. 2. These points are replotted in Fig. 3 against the elastic scattering factor and compared with the Rayleigh asymptote of the semi-analytical model [Eq. (7)] and the Rayleigh asymptote of the SOA model, whose expression is obtained by removing the correction term in Eq. (7). Whereas the SOA results show an apparent deviation from the FEM points and a linear relationship to the elastic scattering factor, the semi-analytical results are indistinguishable from the FEM points and demonstrate quadratic dependence on the elastic scattering factor. The relative differences of the semi-analytical and SOA models to the FEM values for the quasi-static velocity are provided in the insets of Fig. 3 and are additionally listed in Table 2. As shown in the table, the semi-analytical model has a less than 0.05% difference from the FEM results; the figure for the SOA model is an order of magnitude larger.

Whereas the FE evaluation is performed only for the grain shortened and elongated angles (⁠$θp=90°$ and $θp=0°$⁠, Fig. 1), we note that the semi-analytical model predicts the same angular dependences on $θp$ as the SOA model for both attenuation and phase velocity. Therefore, the angular validity of the SOA model established in our prior work^16,17 can be extended to the semi-analytical model, meaning that the semi-analytical model is applicable to any propagation angle $θp$⁠.

The semi-analytical model performs extremely well in the low-frequency range for strongly scattering materials. Interestingly, the corrective empirical coefficients appearing in the quadratic terms for the scattering factor $QL→T$ are the natural constants, $π3/2$ and $π3$⁠, and we hope that future theoretical studies will find the underlying physical bases for this finding. Also, we would like to emphasize that the simple, but accurate, quasi-static velocity expression, Eq. (7), might be of interest to the elastostatic community because it can be used to estimate the effective elastic constants of elongated polycrystals.

The validity of the existing SOA model is greatly challenged by strongly scattering polycrystals with statistically elongated grains, in particular, in the low-frequency Rayleigh regime. It was shown that in this frequency regime, the deviation between the SOA model and the FEM results increases with the anisotropy of grains, reaching 63% for lithium. This work further revealed that the SOA predictions have a linear relationship to the elastic scattering factor $QL→T$ (dependent on the grain anisotropy), whereas the FEM results show a distinctive quadratic dependence. Based on this analysis, this work proposed a simple semi-analytical model for strongly scattering materials with elongated inhomogeneities by iterating the Born approximation of the FFA model and empirically increasing the corrective coefficients of the quadratic terms on the elastic scattering factor. The FE evaluation demonstrates that the proposed model performs extremely well in the low-frequency Rayleigh region for both attenuation and phase velocity and is comparable to the SOA model in the transition and the stochastic regions. The agreement of the proposed model with the FE results for strongly scattering cubic polycrystals is mostly an order of magnitude better than that of the SOA model; such agreement improvement is also evident for the quasi-static velocity limit. The corrective empirical coefficients in the model are the natural constants, $π3/2$ and $π3$⁠, that are obtained for cubic polycrystals covering a large span of Zener anisotropy indices A. We hope future theoretical studies of elastic waves in strongly scattering materials will be stimulated by the findings of this work.

The integral $IM→Ni$ in Eq. (4) is given by¹³ 

where $x=cos θ$⁠, $aeli(θ)−1=(1−x2)/(axi)2+x2/(azi)2=(1−x2)R2/3+x2R−4/3/ai$⁠. $θp$ is the angle between the wave propagation and grain elongation directions; $θp=π/2$ and $θp=0$ correspond to the shortened x-direction and the elongated z-direction, respectively [see Fig. 1(b)].

The relative velocity change $ΔM→N$ in Eq. (4) is given by¹⁶ 

where the shape factors $ς0$⁠, $ς2$⁠, and $ς4$ are given by

where $f(R)=R/R2−1arccoshR$⁠. For longitudinal waves in cubic polycrystals addressed in this work, the coefficients $AMN$⁠, $BMN$⁠, and $CMN$ are given by $ALL=3c2/175$⁠, $ALT=c2/35$⁠, $BLL=BLT=2c2/175$⁠, and $CLL=−CLT=c2/525$⁠, where $c=c11−c12−2c44$ is the invariant anisotropy coefficient.²⁵