Iterative solution to bulk wave propagation in ...

Iterative solution to bulk wave propagation in polycrystalline materials Christopher M. Kubea) Army Research Laboratory, Weapons & Materials Research Directorate, Building 4600, Aberdeen Proving Ground, Maryland 21005-5069, USA

(Received 12 May 2016; revised 25 January 2017; accepted 15 February 2017; published online 16 March 2017) This article reevaluates two foundational models for bulk ultrasonic wave propagation in polycrystals. A decoupling of real and imaginary parts of the effective wave number permits a simple iterative method to obtain longitudinal and shear wave attenuation constants and phase velocity relations. The zeroth-order solution is that of Weaver [J. Mech. Phys. Solids 38, 55–86 (1990)]. Continued iteration converges to the unified theory solution of Stanke and Kino [J. Acoust. Soc. Am. 75, 665–681 (1984)]. The converged solution is valid for all frequencies. The iterative method mitigates the need to solve a nonlinear, complex-valued system of equations, which makes the models more robust and accessible to researchers. An analysis of the variation between the solutions is conducted and is shown to be proportional to the degree of inhomogeneity in the polycrystal. C 2017 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4978008] V [JDM]

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I. INTRODUCTION

Ultrasound is abundantly used in a variety of physics and engineering applications such as materials characterization, biomedical diagnosis, nondestructive evaluation, and structural health monitoring.1 Many of the specific methods in these applications rely on understanding the material dependent dissipation mechanisms acting on the ultrasonic wave. As an example, consider an ultrasonic system that generates a coherent ultrasonic wave or primary wave propagating in a given direction in a host solid medium. This system can be used to quantify the dimensionality of internal defects by careful modeling of all transfer processes and inverting (via deconvolution) for the scattering amplitude(s) from the defect.2 In this case, dissipation of the primary wave can occur prior to reaching the defect if the host medium contains scattering or absorption mechanisms. The interaction of the primary wave with the defect results in a secondary or scattered field, the energy transfer to the secondary field causes the primary wave to dissipate. Ultrasonic methods applied in situations with polycrystalline host mediums present unique challenges and opportunities for the researcher or practitioner. In strongly scattering polycrystals, the contribution of the secondary field from defect scattering can be partially or completely obscured by the scattering from the grain boundaries. This is especially true when the characteristic length scales of the defects are near the length scales of the grainy microstructure. In these cases, models of the grain noise, which can be treated as a transfer function in the ultrasonic system model, are of vital importance when inverting the model to solve for defect scattering amplitudes.2,3 On the other hand, vast opportunities exist for material and, specifically, microstructure characterization by appropriately analyzing the secondary field from a)

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grain boundary scattering. Historically, there was an implicit analysis of scattering conducted by measuring the attenuation of a primary coherent wave, in which energy is directed out of its propagation path via grain scattering. An overview of some of the historic theoretical and experimental work was given by Papadakis.4 Attenuation based techniques have seen a resurgence in recent years partly because of advancements in robust modeling of complex microstructures.5–18 The advances in microstructural modeling has also led to an emergence in explicit measurements of ultrasonic grain noise or scattering for microstructural characterization.19–39 In these instances, it is typically the scattered incoherent field that is of interest. For example, backscattered grain noise can be observed when using pulse/echo transducers with the help of amplification. Inversion measurements of the incoherent scattered field with ultrasonic system models have been used to measure or quantify a variety material properties such as average grain dimensions,24–28 singlecrystal elastic constants,29 elastic anisotropy,30–33 and internal stresses.34,35 The resurgence of attenuation methods (implicit scattering measurements) and the emergence of explicit scattering measurements can be heavily attributed to two seminal theories. The first of which was Stanke and Kino’s Unified Theory, which was the first model that could be used to define attenuations (and dispersive phase velocities) at all frequencies.5 Prior theories considered the Rayleigh, stochastic or intermediate, and geometric scattering regimes separately. The second, and more general theory, was Weaver’s analysis of the diffusivity of ultrasound in polycrystals.6 Weaver derived the elastodynamic Dyson equation, which governs the (mean or average) Green’s function for the attenuating coherent field. Additionally, he derived the elastodynamic Bethe-Salpeter equation, which governs the covariance of the Green’s function. It is the BetheSalpeter equation that forms the starting point to descriptions

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of the incoherent scattered field.24 Descriptions of both fields are required in order to arrive at an equation of ultrasonic radiative transfer, which provides the quantity of ultrasonic diffusivity in the limit of long-length scales (multiple scattering regime). A meticulous examination of both theories will suggest that it is probable that the attenuations obtained from the theories are equivalent [compare Eq. (3.15) of Ref. 6 to the volume integral found in Eq. (93) of Ref. 5]. However, at least in the author’s experience, it is common to find researchers who treat the models as distinct or who choose to apply one model without mention of the other. Thus, it is the partial intent of this article to give a formal and explicit link between the two attenuation models in an effort to further unify the theories. In doing so, it is demonstrated in Sec. II that the attenuations from the Unified Theory (denoted throughout as UT) can be solved through a simple iteration procedure, where the zeroth-order solution is shown to be equal to Weaver’s result (denoted as W). For solving the UT, the iterative solution procedure circumvents the need for a nonlinear solver capable of handling simultaneous equations with complex roots. Last, in Sec. III, the iterative solution is used to compare the attenuations obtained from the two models. The comparison is made using input single-crystal elastic constant data of 748 materials obtained from density functional theory.40 The breadth of materials examined allows the effect of degree of inhomogeneity to be analyzed. The degree of inhomogeneity proves to be an important parameter that drives the variation between UT and W attenuation estimates.

The theory is organized as follows. First, a descriptive background is given describing a high-level overview of the UT and W models. The original references should be consulted for additional detail and further model assumptions not noted here.5,6,41,42 The background is used to highlight the model’s inherent similarities and equivalence under certain approximations. The equivalence motivates the iterative solution procedure that is provided in Sec. II B.

wave, and includes effects from the average of all possible single-scattering events. Truncation at 2 is known as the second-order Keller approximation of hui, which is a transcendental equation with hui appearing on both sides [see Eq. (8) of Ref. 41]. The first-order Born approximation places hui  u0 into the right-hand side, where u0 is the displacement vector of the non-random, homogenized medium. Stanke and Kino5 applied the second-order Keller approximation and analytically evaluated all of the involved integrations, thus avoiding the use of the Born approximation. They obtained transcendental expressions for the effective propagation constants for both longitudinal and shear waves [see Eqs. (101) and (102) of Ref. 5]. Unless further assumptions are made, the effective propagation constants typically require solution by numerical methods. It is these expressions that will be solved by iteration. Weaver6 derived elastodynamic forms of the Dyson and Bethe-Salpeter equation, which follow from the diagrammatic methods outlined by Frisch.42 The elastodynamic Dyson equation, governs the coherent propagation of the mean Green’s function hGi. The Dyson equation for hGi is exact and contains contributions of all multiple scattering events, which is represented by the so-called mass operator m. It is also a transcendental equation with hGi appearing in both sides. Weaver solves for hGi by introducing the first-order smoothing approximation of m (and its ~ which, similar to the Keller approxiFourier transform m), mation, neglects contributions of multiple scattering. A dispersion equation equivalent to Eqs. (101) and (102) of Ref. 5 was then given. Numerically solving the dispersion relation would have resulted in the same attenuation estimates as those from the UT. Alternatively, Weaver derived elegant closed-form expressions for the attenuations by using a “Born-like” approximation in the dispersion equation. This approximation replaces the effective wave num~ with the wave number of the non-random ber in m homogenized medium. The remaining integrals were fully evaluated, which resulted in closed-form expressions of the longitudinal and shear wave attenuations.

A. Background

B. Iterative solution

The derivations used to arrive at the attenuations in the UT (Ref. 5) and W (Ref. 6) models were partially based on the historical work of Karal and Keller41 and Frisch,42 respectively. Karal and Keller studied wave propagation in a random medium that differs only slightly from a nonrandom homogeneous medium.41 Their approach, based on formal perturbation theory, sought an effective propagation constant41 (or effective wave number) of an assumed plane wave. They denoted  as a small perturbation parameter that depends on the mean square deviation between the effective propagation constant and the wave number of a non-random homogeneous medium. They derived an expression for the mean displacement hui up to order 2. Higher-order terms that represent multiple scattering contributions were neglected. At this level of approximation, the mean displacement represents the coherent or primary

Consider the mean wave displacement of an initially ^ at unit amplitude plane wave propagating in the direction n angular frequency x,   ^  x  xtÞ ; ^ p exp iðKp ‘1 n (1) hup ðx; tÞi ¼ u   ^  x  xtÞ ; ^ s exp iðKs ‘1 n hus ðx; tÞi ¼ u (2)

II. THEORY

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where p and s are used to indicate longitudinal and shear waves, respectively. The complex-valued effective wave numbers are denoted as Kp ¼ p – iap and Ks ¼ s – ias where the real parts define the phase velocity (vp ¼ x‘/p, vs ¼ x‘/s) while the imaginary parts are attenuation constants (ap, as). These mean displacements are those that would be found from solving the UT.5 Like Stanke and Kino5 and Weaver,6 the presented theory considers only statistically isotropic Christopher M. Kube

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polycrystals such that a second shear wave solution is degenerate. For convenience, the effective wave numbers are made dimensionless by absorbing a factor of the correlation length ‘, which is a statistical length scale of the microstructure related to the average grain radius. Thus, the true attenuation constants and wave numbers are ap/‘, as/‘ and p/‘, s/‘, respectively. Note that a factor of ‘–1 is included into Eqs. (1) and (2) to return the correct dimensionality to the wave displacements. Using the UT, the squared effective wave numbers ðK2p ; K2s Þ can be written as p2  2iap p  a2p ¼ p20 þ f ðp; ap Þ;

(3)

s2  2ias s  a2s ¼ s20 þ gðs; as Þ;

(4)

where the right-hand sides contain the wave numbers of the homogeneous medium in addition to the factors f and g, which contain the contributions of scattering. The functions f(p, ap) and g(s, as) are, in part, spatial convolutions of the Green’s function of the medium and the two-point grain statistics of the polycrystal’s elastic properties. These convolutions have been fully evaluated, but are too lengthy to reproduce here. p0 and s0 are the wave numbers of the homogenized polycrystal. They are obtained by applying an (Voigt) average of the elastic constants of a single crystallite over all possible orientations.5,6 The bare wave numbers for a statistically isotropic polycrystal having crystallites of cubic crystallographic symmetry are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5q ; (5) p0 ¼ x‘ 3c11 þ 2c12 þ 4c44 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3c11 þ 2c12 þ 4c44 s0 ¼ p0 ; (6) c11  c12 þ 3c44 where c11, c12, and c44 are single-crystal elastic constants and the factor of ‘ is included to make the bare wave numbers dimensionless. Equations (3) and (4) can be solved for the phase velocities and attenuations by equating the real and imaginary parts, which give qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   p ¼ p20 þ a2p þ Re f ðp; ap Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ s20 þ a2s þ Re½gðs; as Þ; (7)   1 ap ¼  Im f ðp; ap Þ ; 2p

a0p

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1 as ¼  Im½gðs; as Þ: 2s

(8)

Equations (7) and (8) are transcendental, which makes closed-form solutions difficult to obtain. f and g contain the trigonometric functions atan(p, ap) and atan(s, as), respectively [see Eqs. (101) and (102) of Ref. 5]. It is convenient to decouple the real and imaginary parts of the atan() functions using the identities, h i 2 1 Im½atanð x þ iyÞ ¼  log 1  x2  y2 þ 4x2 4 h i 1 þ log ð1 þ yÞ2 þ x2 ; (9) 2   p 1 1  x 2  y2 ; (10) Re½atanð x þ iyÞ ¼  atan 2x 4 2 where log is the base-e logarithm. Equations (9) and (10) are valid for x > 0, i.e., positive values of the wave numbers and attenuations. The resulting expressions for the wave numbers and attenuation constants remain transcendental. However, the decoupling of the real and imaginary parts allows an iterative solution to be constructed. The transcendental, closed-form expressions for the wave numbers in Eq. (7) and attenuations in Eq. (8) can be located in the supplemental file.43 The iterative solutions are constructed by applying the approximations p ¼ p0, s ¼ s0, and ap ¼ as ¼ 0 in the right-hand sides of Eqs. (7) and (8). This approximation is sometimes known as the Born approximation.5 It is the same approximation employed by Weaver.6 This terminology is not fully adopted here to avoid confusion. Instead, this will be referred to as the zeroth-order approximation because it is the zeroth-order iterative solutions to the wave numbers and attenuation constants, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi p20 þ Re f ðp ¼ p0 ; ap ¼ 0Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s0 ¼ s20 þ Re½gðs ¼ s0 ; as ¼ 0Þ;

p0 ¼

1 Im½ f ðp ¼ p0 ; ap ¼ 0Þ; 2p0 1 Im½gðs ¼ s0 ; as ¼ 0Þ: a0s ¼  2s0

a0p ¼ 

(12)

The zeroth-order attenuations can be written in closedform as

(

 2   3s0 s20 þ 1 3 12 12 26s30  6s0 3  3s ¼ þ þ 48p þ p 40 þ  þ 0 0 0 48p0 p30 p0 4p20 þ 1 p40 p20    240s50 3   5 32p60 þ 24p40 þ 6p20 þ 1 log 4p20 þ 1 þ     2 4p0 p40  2p20 s20  1 þ s20 þ 1 " #)   4  1 þ ðp0  s0 Þ2 3  2 2 4 2 2 2 2 ;  5 p0 þ s0 þ 1  p0 þ s0 þ 2p0 þ 2s0  10p0 s0 þ 1 log 4p0 1 þ ðp0 þ s0 Þ2 n2p

(11)

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(13)

Christopher M. Kube

(

 2   3p0 s20 þ 1 2p30 13s20  3 3 12 3p50 336s50  þ þ 34s0  4 þ 2 þ s40 s0 4s0 þ 1 s40 s30 s0    240p50 3   5 20s60 þ 18s40 þ 6s20 þ 1 log 4s20 þ 1 þ     2 4s0 p40  2p20 s20  1 þ s20 þ 1 " #)   4  1 þ ðp 0  s0 Þ2 3  2 2 4 2 2 2 2 ;  5 p0 þ s0 þ 1  p0 þ s0 þ 2p0 þ 2s0  10p0 s0 þ 1 log 4s0 1 þ ðp 0 þ s0 Þ2

n2 a0s ¼ s 54s0

where np and ns are factors related to the degree of inhomogeneity present in the polycrystal. The closed-form expressions for the wave numbers are omitted for brevity. a0p and a0s are equivalent to the attenuations defined by Weaver,6 which are constructed from Eqs. (7.11)–(7.14) in Ref. 6. The first-order iterative solution is obtained by substituting p0 ; s0 ; a0p , and a0s into the right-hand sides of Eqs. (7) and (8) and evaluating the resulting expressions. Continuing this iterative process quickly produces the values of p, s, ap, and as from the unified theory of Stanke and Kino.5 Thus, providing the link between the two models. The provided supplemental file contains a MATLAB function that generates the solutions of various order.43 III. ANALYSIS

In this section, the converged iterative solution (equivalent to the unified theory5) is compared to the zeroth-order solution (equivalent to Weaver’s expressions6). Specific examples amongst 748 materials are highlighted to show instances when the zeroth-order solution is suitable, which is less computationally demanding. Parameters that define the degree of inhomogeneity in the polycrystal () and the related anisotropy index (AL) are used to facilitate this comparison. The degree of inhomogeneity is an important parameter because it is the necessarily small expansion parameter that is fundamental to employing perturbation techniques to model scattering in polycrystals. The anisotropy index is a scalar parameter that quantifies the degree of elastic anisotropy present in the crystallites. No scattering will occur at grain boundaries if the elastic anisotropy of the grains approaches isotropy. For polycrystals containing crystallites of cubic symmetry, the degree of inhomogeneity is approximated by the expressions

(14)

The degrees of inhomogeneity defined in Eqs. (20) and (21) in Stanke and Kino5 are the squared factors np, ns defined here in Eqs. (17) and (18), respectively. n2p and n2s are amplitude factors located in the quantities f and g and they are observed explicitly in Eqs. (13) and (14). The exponential correlation function exp ðr=‘Þ is missing in Eqs. (20) and (21) of Stanke and Kino,5 which describes the twopoint Poisson statistics of the polycrystal. Spatial homogeneity is implied, which allows the decoupling of the spatial and tensorial parts of the elastic covariance. Thus, the correct degrees of inhomogeneity will be less than those predicted by Stanke and Kino5 because 0  exp ðr=‘Þ  1. 2p and 2s are important parameters because they are the expansion parameters in the perturbation theory. Application of these models5,6 should be restricted to cases where 2p  1 and 2s  1. The degree of inhomogeneity is closely related to crystallite anisotropy. For cubic crystals, a commonly used anisotropy factor is  ¼ c11 – c12 – 2c44, which can be seen in Eqs. (17) and (18). Here, the log-Euclidean anisotropy index AL will be used to investigate the effect of crystallite anisotropy on the variation between the zeroth-order and converged solutions. AL is a universal index that can be applied to crystallites of any symmetry.44 It was derived by evaluating the log-Euclidean distance45 between the calculated Voigt and Reuss bounds on the elastic properties of polycrystals. Here, AL will be used to estimate the anisotropy of crystallites with cubic symmetry. Many other anisotropy indexes could be used for this case.46–49 However, AL is chosen because it displays less sparsity for cases of highly anisotropic crystals.44 For cubic symmetry, AL can be written as44 " # p ffiffi ffi ð Þ ð Þ c  c þ 3c  3c þ 4c 3c 11 12 44 11 12 44 AL ¼ 5 log : 25c44 ðc11  c12 Þ (19)

2p  n2p exp ðr=‘Þ; 2s



n2s

exp ðr=‘Þ;

(15) (16)

where n2p ¼

4 ðc11  c12  2c44 Þ2 ; 21 ð3c11 þ 2c12 þ 4c44 Þ2

(17)

n2s ¼

3 ðc11  c12  2c44 Þ2 : 28 ðc11  c12 þ 3c44 Þ2

(18)

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As an example, first consider a polycrystalline material composed of lithium single crystals. These single crystals have elastic constants of c11 ¼ 13.4 GPa, c12 ¼ 11.3 GPa, c44 ¼ 9.6 GPa, and density q ¼ 534 kg/m3 (obtained from density functional theory40). This material has a large anisotropy index of AL ¼ 2.254. The amplitude factors related to the degree of inhomogeneity are np ¼ 0.074 and ns ¼ 0.18, which are relatively large compared to many other materials. However, note that 2p  0:0054  exp ðr=‘Þ and 2s  0:033  exp ðr=‘Þ, which satisfies the condition of squared perturbations much less than unity. Figures 1(a) Christopher M. Kube

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FIG. 1. Longitudinal and shear wave attenuations of polycrystalline lithium versus bare wave number for (a) longitudinal and (b) shear waves. (c) and (d) show the respective variation between the zeroth-order and converged solutions.

and 1(b) displays the dimensionless attenuation constants for longitudinal and shear waves versus the respective Voigt averaged bare wave numbers. The bare wave numbers are defined in Eqs. (5) and (6) and are proportional to the wave’s angular frequency and the correlation length. The attenuations were calculated from the iterative solution technique, which was truncated at five iterations where the variation in attenuation values between the previous and successive iterations were observed to be less than 0.01%. p0 and s0 values outside of the range 102 and 101 are not shown because they displayed numerical instabilities; a result of the nearly singular points in the solutions for small or large values of the bare wave numbers. The zeroth-order solutions underestimate the attenuations by a considerable amount for most values of the bare wave numbers. However, the variation in the solutions approaches zero for longitudinal waves in the stochastic scattering regime (near 100 and 100.5). The variations are more severe for shear wave attenuation. Similarly, the phase velocities illustrated in Fig. 2 show some variation between the solutions. The phase velocities are obtained using the relations v0p ¼ x‘=p0 ; vp ¼ x‘=p; v0s ¼ x‘=s0 ; vs ¼ x‘=s. v0p and v0s are the Voigt averaged phase velocities of the bare medium and vp and vs are the phase velocities influenced by the polycrystal’s heterogeneity. Figure 2 is sometimes known as the dispersion relations5,7 for the polycrystal because it gives the frequency dependence of the phase velocity (p0 and s0 are functions of the frequency x). The most notable variation in the phase velocities is observed for large values of p0 and s0 (high 1808

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frequencies or large correlation lengths). The converged solutions of the phase velocities at low values of p0 and s0 are closely related to the phase velocities one would calculate when using the Hashin-Shtrikman upper bound.5 This is a result of the Voigt-type averaging procedure that is employed; an alternative formalism based on self-consistency of the polycrystal’s elastic properties was recently given by the author.15 In order to further investigate the variation between the solutions, the elastic constants of 748 materials of cubic crystallite symmetry were obtained from density functional theory40 and used as inputs into the model. A list of these materials and the corresponding calculations presented in this article can be located in a supplemental resource.43 For the attenuation and phase velocity calculations, the correlation length and center frequency were assumed to be ‘ ¼ 10 lm and 10 MHz, respectively. These parameters are commonly encountered when preforming ultrasonic testing of metals. The percent change between the longitudinal and shear wave attenuations constants are compared against the anisotropy index AL in Figs. 3(a) and 3(b), respectively. In general, the variation between the attenuation constants increases with anisotropy. The zeroth-order longitudinal attenuation is within 10% of the unified theory (converged solution) values for 91% of the materials considered. However, only 69% of the materials considered have shear wave attenuation constants that deviate by less than 10%. The longitudinal and shear wave attenuations of all the materials deviated by less than 10% for an anisotropy index Christopher M. Kube

FIG. 2. Longitudinal and shear wave phase velocities of polycrystalline lithium versus bare wave number for (a) longitudinal and (b) shear waves. (c) and (d) show the respective variation between the zeroth-order and converged solutions.

AL < 0.4. As some examples; AL ¼ 0.023 for aluminum, AL ¼ 0.69 for copper, AL ¼ 0.42 for iron, and AL ¼ 0.47 for nickel. For the case of polycrystalline copper, ap =a0p ¼ 1:057 and as =a0s ¼ 1:17. Interestingly, two branches appear for increasing levels of anisotropy. This feature is more prominent for shear waves. The cause of the two branches in Fig. 3 is related to the degrees of inhomogeneity. This is evident by analyzing the variation in the attenuation constants as a function of the respective degree of inhomogeneity as seen in Fig. 4. The variations have a distinct quadratic dependence to the degree of inhomogeneity for both longitudinal and shear waves. The upper and lower branches observed in

Fig. 3 correspond to cases where n > 0 and n < 0, respectively. Quadratic fits were generated for this dataset, which produced the equations of fit ðap =a0p  1Þ  100  2763  n2p ;

(20)

ðas =a0s  1Þ  100  1355  n2s ;

(21)

with R2 values of 0.96 and 0.928, respectively. All materials in this dataset displayed n2  1 where MbF3 had the largest squared degrees of inhomogeneity, which were n2p ¼ 0:013 and n2s ¼ 0:068. MbF3 also had the largest variation between

FIG. 3. The percent variation between the attenuation constants from the zeroth-order and converged solutions is given in (a) for longitudinal waves and (b) for shear waves plotted against the anisotropy index AL. Each data point is one of 748 materials with cubic crystallite symmetry (see Ref. 43 for a list of materials).

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FIG. 4. (Color online) The percent variation between the attenuation constants from the zeroth-order and converged solutions is given in (a) for longitudinal waves and (b) for shear waves plotted against the parameters np and ns. Each data point is one of 748 materials with cubic crystallite symmetry (see Ref. 43 for a list of materials). The solid red lines are quadratic fits given in Eqs. (20) and (21).

the attenuation constants of 41.59% and 151.3% for longitudinal and shear waves, respectively. This highlights that the perturbation model remains accurate for a wide range of materials. However, care should be taken when applying the zeroth-order solutions to obtain attenuation constants. IV. SUMMARY

The two foundational models of Stanke and Kino5 and Weaver6 describing the propagation of ultrasound in polycrystals were reevaluated. An iterative solution to the unified theory was developed based on decoupling the real and imaginary parts of the effective wave number. The zerothorder iterative solution recovered the solution from Weaver,6 while the converged iterative solution recovered the solution from the unified theory.5 This led to a formal mathematical connection between the solutions to the models. A dataset of 748 materials was used to survey the variation between the zeroth-order solution and the converged solution as a function of the degree of crystallite anisotropy and inhomogeneity. These materials and the corresponding calculations are included in a supplemental resource table.43 All 748 materials satisfied the condition of small inhomogeneity. The variation between the solutions had a quadratic dependence on the degree of inhomogeneity. Equations (20) and (21) provide estimates to the variation between the attenuation constants calculated from the two solutions. 1

T. Kundu, Ultrasonic and Electromagnetic NDE for Structure and Material Characterization (CRC Press, Boca Raton, FL, 2012). 2 L. W. Schmerr, Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach (Springer International Publishing, Switzerland, 2016). 3 F. J. Margetan, L. Yu, and R. B. Thompson, “Computation of grain noise scattering coefficients for ultrasonic pitch/catch inspections of metals,” in Review of Progress in Quantitative NDE, edited by D. O. Thompson and D. E. Chimenti (AIP, Melville, NY, 2005), Vol. 24, pp. 1300–1307. 4 E. P. Papadakis, “Ultrasonic attenuation caused by scattering in polycrysatlline media,” in Physical Acoustics: Principles and Methods, edited by W. P. Mason (Academic Press, New York, 1968), Vol. 4B, pp. 269–328. 5 F. E. Stanke and G. S. Kino, “A unified theory for elastic wave propagation in polycrystalline materials,” J. Acoust. Soc. Am. 75, 665–681 (1984). 6 R. L. Weaver, “Diffusivity of ultrasound in polycrystals,” J. Mech. Phys. Solids 38, 55–86 (1990). 1810

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