Ultrasonic wave propagation predictions for polycrystalline materials using three-dimensional synthetic microstructures: Attenuation

As an ultrasonic wave propagates in a polycrystalline material, it scatters energy in all directions because of the interactions with grain boundaries. Two adjacent grains typically have different orientations such that their elastic moduli differ relative to a fixed reference frame. Therefore, the wave propagates with different velocities through neighboring grains. The impedance mismatch at the grain boundary causes scattering to occur. Because of such scattering events, the amplitude of the incident wave loses energy as it propagates. Attenuation is the rate of energy loss due to these scattering events. The loss of energy due to the grain boundaries was first proposed by Mason and McSkimin¹ as the source of ultrasonic attenuation. Since then, several theoretical models of attenuation have been developed among which the models of Stanke and Kino² and Weaver³ are the most foundational ones. The unified theory of Stanke and Kino² uses the second-order Keller approximation to obtain the mean wave displacement. Their approach can be applied to frequencies ranging from the Rayleigh regime up to the geometric limit. Weaver's model was developed using a statistical physics approach. Although he applied a specific approximation in order to simplify the analysis, his overall approach is equivalent to that of Stanke and Kino.² Recently, Kube⁴ presented an iterative approach to provide a formal mathematical connection between the two techniques. Transcendental relationships associated with the real and imaginary parts of the wave number allow the phase velocity and attenuation to be determined. His zeroeth-order solution reduces to the approximation used by Weaver,³ but continued iteration shows convergence to the complex wave number from Stanke and Kino.² 

A commonality found in all of these theories is the way by which the statistics of the grains within the polycrystal are modeled. Going back to the work of Karal and Keller,⁵ Stanke and Kino,² and Weaver,³ a certain set of assumptions are usually considered in the modeling process. For a medium comprised of an infinite number of grains, the spatial and orientational properties are assumed uncorrelated. This assumption is applied by decoupling the spatial and tensorial components of the two-point statistics of the elastic modulus covariance function. In addition, the two-point statistics are considered to follow a specific exponential form. These assumptions are primarily made to simplify the form of the ultrasonic attenuation. With ongoing developments in computational tools for materials science, direct elastic wave simulations of three-dimensional (3D) numerical versions of polycrystals are now possible. These digital microstructures have proven to be helpful for assessment of the accuracy of the ultrasonic models. Using these simulations, the elastodynamic wave equation can be solved through numerical schemes such as the finite element (FE) or finite difference methods to obtain the ultrasonic attenuations. Comparisons of these solutions can then be made to the results of the analytical models. Such an approach can be found in the works of Van Pamel^6–8 and Ryzy.⁹ Such studies apply a Voronoi tessellation^10,11 to simulate the polycrystal geometry. Voronoi microstructures are created through random seeding by which perfectly planar grain boundaries are ultimately achieved. However, grain boundaries of real microstructures are more curved than planar. Other limitations of Voronoi tessellations are also described in Ref. 12. These limitations can be overcome by using a recently developed software called dream.3d.¹³ Microstructures simulated using dream.3d have grains that are morphologically more realistic than their Voronoi counterparts.¹² 

Moreover, polycrystals generated computationally are prescribed to have a specific grain size distribution and orientation distribution function (ODF). Because of the finite number of grains in these polycrystals, simulations performed with the same statistics will not result in identical volumes for each realization. Hence, it is important for studies that use such synthetic volumes to provide the uncertainty bounds for any predicted quantities. Unfortunately, this approach is not always practical when the full 3D elastodynamic wave equation is solved. In fact, the computational cost increases tremendously with the simulation volume. Because of this cost, a small number of simulations is used in studies which solve the full wave equation. In this article, an alternative approach is followed to examine ultrasonic attenuations. Here, spatial statistics for synthetic microstructures are calculated directly on the full 3D polycrystalline volumes. These statistics are then used within the analytical model developed by Weaver.³ With this approach, no assumptions are required because the microstructural information is obtained directly from the synthetic volumes. Due to the low cost of this approach, attenuation values can be studied from a statistical standpoint. Here, 10 sets of polycrystalline volumes, each with 50 realizations, are used for the calculations. The number of grains in these 500 volumes ranges from 2000 to more than 122 000. All synthetic volumes are created using dream.3d due to the advantages of the resulting microstructures over classical Voronoi tessellations.

In Sec. II of this article, the computational framework is presented in which Weaver's theoretical model³ is applied to the synthetic microstructures through discretization of the continuous forms. Section III covers the typical assumptions included in the current ultrasonic models. The formulation of the two-point statistics for the synthetic polycrystals are presented in Sec. IV. Section V provides the parameters used for generation of the volumes and the attenuation results of these synthetic volumes are given in Sec. VI. The validity of the assumptions are then examined in Sec. VII. The attenuation results provided in Sec. VI reveal specific aspects of the microstructures simulated with dream.3d. These aspects are discussed in depth in Sec. VIII.

The energy of an ultrasonic wave traveling through a heterogeneous microstructure scatters in all directions. Therefore, a scattering coefficient can be defined from which the attenuation can be obtained. This coefficient is controlled by the covariance function, $Λ$⁠, of the elastic modulus tensor which describes the spatial distribution of material heterogeneity. It is defined as

where δC is the modulus fluctuation about the average elastic modulus $〈C〉$⁠, and x and x′ are two points in the microstructure. Under the assumption of statistical homogeneity, the covariance function can be expressed as a function of the distance between the two points x and x′ rather than their absolute positions. Such an assumption is considered here to obtain the scattering coefficients. Consider an incident wave propagating in the $p̂$ direction with polarization direction $û$ that scatters into the $ŝ$ direction with polarization direction $v̂$ as shown schematically in Fig. 1. For this configuration, a scattering coefficient can be defined as

where ω is the frequency, ρ is material density and r is the separation vector between x and x′. $CIû$ and $CSv̂$ define the phase velocity of the incident wave polarized in the $û$ direction and the phase velocity of the scattered wave polarized in the $v̂$ direction, respectively.

The inner product between the covariance function, and the propagation and polarization vectors included in Eq. (2) is given in direct notation by

For each propagation and scattering direction considered in Eq. (1), nine scattering coefficients can be defined. For example, the total scattering coefficient for an incident quasi-longitudinal (qL) wave propagating in the $p̂$ direction which scatters into the $ŝ$ direction is composed of three scattering coefficients which account for the three outgoing wave types. It is given by

where qT1 and qT2 represent the two quasi-shear polarizations. The attenuation is calculated as the integral of the total scattering coefficient over all scattering directions. For example, the longitudinal attenuation is given by

The theory provided thus far treats the microstructure as a continuous medium. However, in the digital microstructures used here, material information is known at discrete locations (voxels). Therefore, a discrete version of each integral may be defined. This expression can be achieved using the Monte Carlo technique¹⁴ by which the integration $I=∫Vf(X)dX$ is discretized as $I=V/NΣi=1Nf(Xi)$⁠, where the integration domain Γ has a volume of ∫_Γdx = V. Hence, the scattering coefficient given in Eq. (2) can be rewritten as

where R is the largest distance used to calculate the two-point statistics, N is the number of discrete points, $μ=cos(θ)$⁠, and q is defined as $q=(ω/CIûj)p̂−(ω/CSv̂k)ŝ$⁠. Typically, in polycrystals with equiaxed randomly oriented grains, the covariance function $Λ(r)$ decreases with distance because the spatial correlation between grains is localized. Therefore, convergence is expected for the scattering coefficient as a function of R. However, R must be selected with care due to its direct impact on computational cost. In addition, integration using Monte Carlo requires careful sampling of the integration volume. Here r, μ and ϕ are chosen using a set of random numbers $λ∈[0,1]3$⁠. To ensure appropriate sampling of the integration volume, they are defined by

where λ₁, λ₂, and λ₃ are the elements belonging to the set $λ$⁠. The terms given in Eq. (7) are obtained by a change of variable in the spherical integration and assuming a random distribution for the new variables. For a function f(r, θ, ϕ), the triple integral in spherical coordinates is defined as $I=∫02π∫0π∫0Rf(r,θ,ϕ)r2 sin(θ)drdθdϕ$⁠. Defining new variables $r̃=(r/R)3$⁠, and $μ=cos(θ)$⁠, the integration can be rewritten as $I=(R3/3)∫02π∫−11∫01g(r̃,μ,ϕ)dr̃dμdϕ$⁠, where $g(r̃,μ,ϕ)=f(Rr̃1/3, cos−1(μ),ϕ)$⁠. This integral can be obtained directly using Monte Carlo for which random numbers are assigned to the integration domain. Equation (7) is the result of such a process and ensures uniform sampling. The attenuation discretization is obtained in an analogous manner.

The scattering coefficient expressed in the form of Eq. (6) includes only the statistical homogeneity assumption. However, many current ultrasonic models include several other assumptions related to the covariance function. Here, these assumptions are discussed for polycrystals comprised of randomly oriented equiaxed grains.

One common assumption is associated with the behavior of the two-point spatial correlation function. This function is directly controlled by the morphology of the microstructure. It defines the probability that two random points separated by r lie within the same crystallite. Stanke and Kino² and many others have assumed an exponential form for the spatial correlation function with a single correlation length L. For a microstructure with equiaxed grains, the two-point spatial correlation function η is defined as² 

Although the limitations of this assumption have been acknowledged by several authors,^15–19 a thorough examination of realistic three-dimensional microstructures has not been completed previously.

The second assumption most often used is that the covariance tensor can be decoupled into its tensorial and spatial components. This assumption implies that the orientations of crystallites are uncorrelated. Therefore, Eq. (1) can be simplified as

where the eighth-rank tensor $Ξ$ is the covariance of the elastic modulus tensor. With this assumption, the statistics of the grain orientations can be determined without the need to track their spatial position. For cubic and hexagonal symmetries, Weaver³ and Yang et al.,²⁰ respectively, defined $Ξ$ in a general form using combinations of Kronecker deltas and single-crystal elastic constants.

Finally, the last of these assumptions is that the covariance function is statistically isotropic. This assumption implies that the covariance function does not depend upon the orientation of the wave propagation. Therefore, the covariance can be expressed as

Subsequent calculations based on these assumptions are usually implemented for the limit of a polycrystal with an infinite number of randomly oriented grains that are assumed to be spherical. However, such assumptions are known to have limitations with regard to real microstructures. In this article, these assumptions are examined directly using synthetic three-dimensional polycrystals.

Equation (1) expresses the covariance function as the ensemble average of modulus fluctuations at two fixed positions. This expression implies the need for a large number of samples for accurate statistics. In most cases, it is not practical to create such ensembles. Hence, a different approach is needed. Torquato²¹ addresses this problem by using the ergodic hypothesis. According to ergodicity, an ensemble average is equivalent to a volume average for which an infinite volume is needed. Here, spatial averages on finite volumes are used to derive the quantities of interest, but the statistics of each quantity are examined across an ensemble of realizations of different sizes. Hence, Eq. (1) can be expressed as

The eighth-rank covariance tensor given by Eq. (11) is in its most general form, without any assumptions regarding the orientation or morphology of the microstructure. The accuracy of the calculation is dependent upon the number of pairs N_p.

The two-point spatial correlation function is calculated in an analogous way but using an indicator I. For two random points x and x' belonging to grains $G$ and $G′$⁠, respectively, the indicator I is defined as²¹ 

According to Eq. (12), if the two points are within the same grain, the indicator becomes unity and is zero otherwise. For N_p number of equally distant pairs of points selected in a statistically homogeneous polycrystal, the two-point spatial correlation function can be defined as

Equation (13) can be used to assess the validity of the decoupling assumption in which the spatial component of the covariance function decouples from its tensorial part. To that end, the tensorial component of the covariance function needs to be obtained for a digital microstructure. Using the covariance function given in Eq. (11), the covariance tensor $Ξ$ can be calculated simply by setting $r=0→$⁠. Therefore, the covariance tensor may be defined as

where V is the volume of the polycrystal, N_g is the total number of grains, and δC_n and V_n are the modulus fluctuation and volume of the nth grain. Finally, the decoupled covariance function is given by the product of Eqs. (13) and (14), or

Comparing the results obtained using Eqs. (11) and (15), the validity of decoupling can be assessed.

Attenuation and scattering coefficient values are calculated for polycrystals created with dream.3d.¹³ These calculations are carried out across 10 ensembles, each with a different volume and each containing 50 realizations of grains. Holland Computing Center (HCC)³⁰ supercomputer is used to create these 500 realizations in parallel. All microstructures are created with grains that follow a log-normal grain size distribution and contain grains of cubic single-crystal symmetry. The microstructures are assumed to be made of nickel with single crystal elastic properties of c₁₁ = 247, c₁₂ = 153, and c₄₄ = 122 GPa (Ref. 22) and density of ρ = 8900 kg/m³. It should be noted that the single-crystal elastic properties and the density are attributed to the microstructures after each realization is created as they are not inputs required by dream.3d. In order to study the influence of the number of grains on the attenuation uncertainty bounds, a narrow grain size distribution is considered with 30 µm average grain diameter and 3 µm standard deviation (SD). Table I provides the statistics of the synthetic volumes created. Examples of these synthetic realizations are shown in Fig. 2 for 300³, 600³, and 900³µm³ volumes. In addition, a sample pipeline with which the microstructures can be created is included as a supplementary file.³¹ The pipeline is in JSON format and generates cubes with 300³µm³ volumes. The dimensions of the blocks can be modified in the filter called Initialize Synthetic Volume. The microstructural information is exported into a text file using the Export ASCII Data filter. The output path in this filter should be modified accordingly.

The scattering coefficients, and hence attenuation, are quantities that are directly influenced by the model used for the microstructure. Typically, in ultrasonic scattering theories, a microstructure is described by its two-point statistics. Such a property is obtained by calculating the covariance function $Λ(r)$⁠. As discussed in Sec. III, the covariance function is conventionally calculated using several assumptions. To assess the validity of these assumptions, the covariance function should be calculated directly from the simulated microstructure using Eq. (11). This covariance function is then used in Eq. (6) in order to calculate the scattering coefficient from which the attenuation values are obtained. Here, the Voigt theory²³ is applied to obtain the covariance functions, although other averaging methods such as Reuss,²⁴ Hill,²⁵ and self-consistent²⁶ could be used as well.

In the process of calculating the scattering coefficient, the separation distance r should be truncated once convergence is observed. This maximum separation distance R, is used in Eq. (6). To see the influence of R on the convergence of the scattering coefficient, Eq. (6) is examined for a 15 MHz wave propagating in the [001] direction and scattering into the [100], [010], [001], and [111] directions. These directions are defined with respect to the laboratory coordinates and do not relate to the individual grains. A total of 5000 discrete points, N, and 50 000 number of pairs, N_p, are used for the results. The results, shown in Fig. 3, include ensembles of cubes with 300³, 600³, and 1200³µm³ volumes. For each scattering direction, the average and standard deviation of scattering coefficients are obtained across the ensemble of 50 realizations. In Fig. 3, the solid lines are the average values and the shaded regions show the corresponding standard deviations.

Upon inspection of Fig. 3, direct correlation between the standard deviations and the maximum separation distance, R, becomes apparent. Such a correlation is present in all four scattering directions and three volumes. The increase in the standard deviation with R indicates that at larger separation distances, the covariance function varies more across the ensemble. However, convergence is observed in the average values. These results suggest that at larger separation distances, the two-point statistics are obtained from an insufficient number of grains. Because periodicity is not assumed here, the number of combinations of two grains satisfying a specific separation distance r, decreases with r. Therefore, higher fluctuations in the covariance function are expected at larger separation distances. Moreover, the fluctuations are also not necessarily symmetric about zero. In fact, the asymmetry of the fluctuations is more apparent at larger separation distances. This behavior is more random in nature because across the ensemble of realizations, convergence is observed in the average properties. More specifically, Fig. 4 shows the inner product of the normalized covariance function which defines the longitudinal polarization of a wave propagating in the [0 0 1] direction and scattering backward. The normalization is made with respect to the square of the anisotropy coefficient, ν = c₁₁ – c₁₂ – 2c₄₄. This figure is obtained using a cube with a volume of 300³µm³. The asymmetry at larger separation distance is clear.

In addition, variations of the scattering coefficients are inversely related to the volume of the polycrystal. For example, the standard deviations observed for the 300³µm³ cubes at the maximum separation distance are about 2.8 and 7.6 times larger than those observed in the 600³ and 1200³µm³ volumes, respectively. This property is due to the larger number of grains for the larger volumes, hence, the larger number of combinations of grains satisfying a separation distance. Finally, considering that the average scattering coefficient across the ensembles converges at about one grain diameter but the standard deviation never converges, the truncation value appears to be somewhat arbitrary. In this work, 45 µm is selected for the maximum separation distance, R, that is used for the calculations. This value is about 1.5 times the length of the average grain diameter and is within the region for which convergence of the average is observed.

To calculate the attenuation, Eq. (5) is discretized analogously to that of the scattering coefficient. In all attenuation calculations here, a total of 1000 scattering directions are used for the integration. For each direction, the scattering coefficient is calculated using N_p = 5 × 10⁴ pairs and N = 5000 discrete points. For the simulated nickel microstructure, the quasi-longitudinal and quasi-shear attenuations are calculated for a 15 MHz wave propagating in the [001] direction. These attenuation results are demonstrated in Fig. 5 as a function of the number of grains. The quasi-shear attenuations are separated into the first and second quasi-shear modes which correspond with the fast-shear and slow-shear polarizations, respectively. For each number of grains, the mean attenuation is obtained from the 50 synthetic volumes. The error bars represent the attenuation standard deviation.

Figure 5 shows that volumes with a larger number of grains exhibit smaller variations across the ensemble. Such behavior implies the increase in the statistical homogeneity of the microstructures with more grains. This conclusion can be drawn due to the stochastic nature of dream.3d microstructure creation. It is important to note that the longitudinal and shear attenuation values based on an exponential form³ with a correlation length of L = 15 µm are α_L = 0.24 Np/cm and α_T = 0.93 Np/cm, respectively. Thus, the dream.3d microstructures result in attenuations 3–4 times smaller than values obtained from commonly used theories. The distinction between the attenuations of these different results is also shown in Fig. 6 for a polycrystalline nickel with 1200³µm³ volume as a function of the frequency. In this figure, α_qT1 and α_qT2 refer to the attenuations of the fast and slow quasi-shear polarizations, respectively. At higher frequencies, the differences between the theory with exponential statistics and those from dream.3d become more evident. Such results clearly demonstrate the limitations of the assumptions often used in scattering theories. In addition, the microstructures generated in dream.3d are more realistic from a morphological standpoint compared with Voronoi tessellations.^12,27 Classical Voronoi microstructures have planar grain boundaries, while more realistic curved grain boundaries can be created by dream.3d. Although generalized conclusions cannot be drawn based on limited data, these results suggest that dream.3d microstructures may lead to a new view of traditional scattering theories.

In Sec. III, three typical assumptions used in most ultrasonic models were introduced. By calculating the attenuation directly using the microstructural information, the validity of these assumptions can be tested.

The first assumption assessed is the two-point spatial correlation function. Conventionally, the spatial correlation function η(r) is expressed as an exponential function using a correlation length which corresponds with the average grain radius. This form is given in Eq. (8). To test the validity of this assumption, the spatial correlation function for a simulated microstructure is calculated directly using Eq. (13). Then, comparisons of the dream.3d statistics can be made with the exponential form. Such a comparison is given in Fig. 7 for a 300³ µm³ volume.

The results shown in Fig. 7 illustrate that the exponential form based on the average grain radius has longer range correlation in comparison with the spatial correlation function obtained from the synthetic volume. Ultimately, this longer correlation length will increase the scattering coefficients and the resulting attenuation values as shown in Fig. 6. Similar results are observed for other volumes, but are not shown here for brevity. Thus, for a microstructure with a narrow grain size distribution, an exponential form, such as Eq. (8) is not acceptable. Similar observations were made by Ryzy et al.⁹ in which digital microstructures were made with a Voronoi tessellation. Hence, a revised expression to describe the two-point statistics is clearly needed, but it likely will require inclusion of the grain size distribution and grain morphology and is beyond the scope of this article.

In order to assess the decoupling assumption, attenuation values are calculated by integrating the scattering coefficients given in Eq. (9). In this method, the tensorial component of the covariance function is decoupled from the spatial part. The validity of such an assumption implies that the orientations of grains are not correlated with each other. Comparisons are then necessary between the decoupled attenuation values and those calculated directly. For a 15 MHz wave propagating in the [001] direction, the attenuations are calculated using both approaches for polycrystalline nickel. These results are illustrated in Fig. 8.

It is clear from Fig. 8 that the decoupling assumption is valid for statistically isotropic microstructures with a narrow grain size distribution. However, attenuations calculated using the direct method show higher uncertainty bounds compared with their decoupled counterparts. Such behavior is attributed to the differences in the sufficiency of the statistics for the two methods. As mentioned earlier, fluctuations of the covariance function increase with the separation distance because a smaller number of individual pairs of grains exist that can satisfy larger separation distances. Because no two blocks with the same volume are identical, variations are expected across the ensemble. For the decoupled calculations, the covariance function is the product of the spatial correlation function with the covariance tensor evaluated at r = 0. The only source of fluctuation for the covariance tensor is the finite number of grains comprising the microstructures. For the spatial correlation function obtained directly from the geometry, small variations are present due to the large numbers of voxels. Hence, the decoupled attenuations are expected to show lower uncertainty bounds compared with the direct technique. Future studies of this assumption that are associated with texture or other microstructures are needed to determine the conditions for which the assumption still holds.

In Sec. VI, a methodology was explained with which the maximum separation distance R can be identified for a microstructure. The technique is general and considers none of the assumptions listed in Sec. III. With the validity of decoupling, the maximum separation distance can be viewed from a different perspective. In this case, the influence of R is directly applied upon the two-point spatial correlation function, η(r). This function gives the probability of two points residing inside a grain. Therefore, it can be argued that for distances larger than the largest grain diameter, such probability becomes infinitesimal. In this work, the average value of the largest grain diameter across all of the ensembles is 44.6 µm. This value is almost identical to the R = 45 µm obtained from the general technique. Hence, the maximum grain diameter can be used to identify R for microstructures with randomly oriented grains.

Typically, polycrystals with randomly oriented grains are considered statistically isotropic. The degree of isotropy, however, depends upon the total number grains comprising the polycrystal.²⁸ Absolute isotropy is achieved only when the polycrystal is considered to contain an infinite number of grains. Hence, statistical anisotropy always exists in finite volumes. This concept is studied in detail by Norouzian and Turner²⁸ using the directionality of phase velocity. dream.3d microstructures are finite in size and contain grains that do not fully follow random orientation in the Euler space. Therefore, a slightly lower degree of statistical isotropy is expected for these volumes. Table II provides the average longitudinal attenuation for all realizations for a 15 MHz wave propagating in the [100], [010], [001], and [111] directions.

The attenuations from the [100], [010], and [001] directions are very similar in all volumes. However, the attenuations in the [111] direction are up to 8% larger than other directions. For the shear attenuation, this value is about −7%. Therefore, slight texture is present in the dream.3d simulated blocks. Considering such texture in Fig. 8, it can be argued that the decoupling results discussed in Sec. VII B are valid for weakly textured polycrystals.

Attenuation results obtained from the dream.3d polycrystals showed an interesting behavior across the ten volumes. In Fig. 5, the two largest sets of grains show higher attenuation values compared with smaller volumes. However, the lower uncertainty bounds of the decoupled attenuation values demonstrated in Fig. 8 clarifies that the jump is not restricted only to the two largest volumes. For example, the quasi-longitudinal results based on the decoupling assumption show no overlap between the bounds of the 400³ and 700³µm³ volumes. Therefore, it is concluded that the two-point statistics are changing across the different volumes. This behavior is interesting because all microstructures were generated with the same grain statistics. To find the source of this jump, the results are studied from a tensorial and morphological stand point. This view to the problem is possible due to the validity of the decoupling assumption explained in Sec. VII. Hence, the influences of orientation and morphology on such differences can be investigated independently.

The role of the Euler angles can be studied by assigning random orientations to the grains of each volume. The attenuations are then calculated using the decoupling assumption and comparisons are made with the original dream.3d Euler angles. To ensure statistical isotropy, a set of random numbers X = {X₁, X₂, X₃} in the interval of [0,13] is used, such that the Euler angle triplets are defined as ϕ = 2πX₁, Θ = acos(2X₂ − 1), and ζ = 2πX₃.²⁹ Figure 9 shows the attenuation values for a 15 MHz wave propagating in the [001] direction of the nickel polycrystals for randomly oriented grains along with results based on the original dream.3d orientations.

Important information regarding the texture and morphology of the synthetic microstructures is contained in Fig. 9. First, microstructures with grain orientations originally generated by dream.3d show lower attenuations compared with those containing randomly oriented grains. Such a deviation from the statistically isotropic assumption implies a slight texture in the attenuation of the dream.3d microstructures. In addition, although the attenuation increases for truly random orientation, its trend relative to the number of grains remains unchanged. Hence, the choice of the Euler angles is not the source of the variation with number of grains.

To study the influence of the morphology, the ensembles of 50 realizations with volumes of 800³ and 1200³µm³ are selected from which to sample. Smaller volumes are sampled from the center of the larger cubes to create a new ensemble of 50 realizations. The attenuation values of the new ensembles are then calculated using the decoupling assumption for a 15 MHz wave propagating in the [001] direction in nickel. Comparisons of these attenuations with those obtained from the original microstructures are shown in Fig. 10.

The average attenuations obtained from the sectioned volumes, demonstrated in Fig. 10, exhibit very small variations as a function of the number of grains. Such a small variation agrees well with the initial assumption of statistical homogeneity. On the contrary, relatively larger variations are observed in the average attenuations of the original microstructures. This behavior is evidence that morphological differences are present within the dream.3d microstructures as a function of the number of grains. In addition, those microstructures that are sectioned from the 800³µm³ volumes have different attenuations than those obtained from the 1200³µm³ ones. In fact, the attenuation values of the sectioned cubes are closest to their parent microstructures. Such behavior agrees with the changes in the morphology of the dream.3d microstructures as a function of the number of grains. To determine the source of these changes, the mean and standard deviations of the grain diameter are calculated for all synthetic blocks. Here, these quantities are denoted as μ_d and σ_d, respectively. Such statistics are then averaged over the ensemble of 50 realizations for each volume. For the standard deviation, the averaging is performed using $〈σd2〉$⁠, where the angular brackets represent the ensemble average. Afterwards, the coefficient of variation (CV) is obtained as $〈σd2〉/〈μd〉$⁠. When the CV is compared with the ensemble average quasi-longitudinal attenuations, $〈αqL〉$⁠, of the original volumes shown in Fig. 10, a clear correlation is observed between the two quantities. This comparison is demonstrated in Fig. 11 in which both quantities are normalized to unity. Figure 11 exhibits an important aspect of the synthetic microstructures. Although, all microstructures are created using identical grain statistics, the normalized CV indicates that the grain size distribution of the microstructures are statistically dependent upon the simulation volume. Because similar dependency on the simulation volume is observed in $〈αqL〉$⁠, the variation in the ensemble average attenuation is attributed to the changes in the final grain size statistics.

Another observation made from Fig. 10 is with regard to the differences between the average number of grains obtained from the original and the sampled microstructures. The results show that on average, the sectioned volumes contain more grains. The relative difference between the average number of grains is about 30% for the smallest cube and decreases to about 7% for the 1100³µm³ volume. Such behavior signifies the role of the boundary grains. For smaller volumes, the ratio between the surface area to the volume is higher than the larger cubes. In fact, for cubes, this ratio is inversely proportional to the cube edge length. Figure 12 demonstrates that the surface to volume ratio decreases at almost the same rate as the relative difference between the number of grains, when both are normalized to unity. This result implies that microstructures simulated with dream.3d do not have similar grain statistics on the boundary of the volume as in the interior. Hence, dream.3d may prohibit the boundary grains to be small sections of larger grains, a condition that is not true for sectioned volumes.

Figure 13 compares the microstructures of one face of a realization with 300³µm³ volume with a cross-section of the same volume from the middle. In the microstructure on the left, Fig. 13(a), the image obtained from a face contains 156 grains. However, on the right image, Fig. 13(b), there are a total of 214 grains. This difference is apparent in the microstructure images. To capture this phenomenon more quantitatively, the number of grains on the 6 faces of the volumes is calculated as a percentage of the total number of grains. This operation is performed for the original and sectioned microstructures and the results are given in Table III. The relative difference between the original and sectioned results provided in Table III shows that the microstructures sectioned from the original 1200³µm³ volumes contain about 28% more boundary grains, on average, than those from the non-sectioned volumes. Thus, it is clear that care must be taken when grain statistics are determined from such synthetic microstructures.

In this article, a methodology has been described to calculate ultrasonic attenuation values in synthetic three-dimensional voxellated microstructures. To investigate the influence of the number of grains on attenuation, a total of 500 synthetic polycrystals were used, with all realizations created using dream.3d. The theoretical formulation used to calculate the attenuation values was based on the theory of Weaver.³ In addition, typical assumptions made in the theory have been investigated using the simulated volumes. It has been observed that the exponential form of the spatial correlation function used by many authors is clearly limited in its validity. In fact, the spatial correlation function obtained using the synthetic volumes consistently showed shorter correlation lengths. Such differences in the two-point grain statistics resulted in attenuations that were three to four times smaller than those based on the traditional exponential form of the correlation function.

Moreover, the results support the validity of the assumption regarding the decoupling of the spatial and tensorial components of the covariance function. It is emphasized that such a conclusion applies only to polycrystals with a narrow grain size distribution and a small degree of texture. For wider distributions and higher texture, further investigations are required. The attenuation results as a function of the number of grains showed an inverse relationship between the number of grains and the attenuation uncertainty bounds. Such information is useful with respect to experiments. Expected variations in attenuation based on statistical homogeneity can be used to assess differences in measured attenuation with respect to statistical differences between locations on a sample.

Furthermore, the variations in the average attenuation with respect to volumes of the dream.3d polycrystals have been demonstrated to be a result of morphological changes. Such a conclusion was drawn by studying the influence of the tensorial and spatial components of the covariance function on the attenuation. While assigning random orientations to the microstructures increased the overall attenuation, the variations across the volumes remained the same. This outcome was an indication of the slight morphological changes in different volumes generated by dream.3d. Similar changes were also noticed by comparing the attenuation results of the original and sectioned microstructures. The morphological changes were ultimately attributed to the dependencies of the resulting grain size statistics on the simulation volumes. The underlying causes of such dependencies should be investigated in the future by studying the process by which the microstructures are created in dream.3d.

Finally, a study on the sectioned and original volumes showed around 28% difference between the percentage of boundary grains. This difference indicates that the dream.3d simulated microstructures have boundary grains that do not satisfy the desired statistics, although the overall volume does. This result implies that the grains inside the volume must compensate for the deviation of the boundary grains. Such synthetic blocks slightly deviate from realistic samples because, in reality, many of the measurements are performed on samples that are sectioned from larger volumes. Therefore, on the boundary, there are grains that are only a small section of the larger ones. The implication of such behavior will be apparent when the software is used to simulate microstructures for comparison with grain size statistics obtained through microscopy techniques. The deviation from the desired statistics decreases with the total number of grains. Hence, studies that use dream.3d software for synthetic polycrystalline simulation should section the desired volume from a larger volume to ensure that the distributions of the sectioned blocks match with the desired statistics. The work presented in this article is a step towards modeling the statistical behavior of polycrystals more accurately and lays the groundwork for precise grain size characterization using ultrasonic methods.

This work was supported by the Air Force Research Laboratory under prime Contract No. FA8650-15-D-5231 and was completed utilizing the Holland Computing Center of the University of Nebraska, which receives support from the Nebraska Research Initiative.