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Correlation of elastic wave attenuation and scattering with volumetric grain size distribution for polycrystals of statistically equiaxed grains
Accepted Manuscript Correlation of elastic wave attenuation and scattering with three-dimensional grain size distribution for polycrystals of statistically equiaxed grains Gaofeng Sha
PII: DOI: Reference:
S0165-2125(18)30064-7 https://doi.org/10.1016/j.wavemoti.2018.08.012 WAMOT 2274
To appear in:
Wave Motion
Received date : 23 February 2018 Revised date : 7 July 2018 Accepted date : 23 August 2018 Please cite this article as: G. Sha, Correlation of elastic wave attenuation and scattering with three-dimensional grain size distribution for polycrystals of statistically equiaxed grains, Wave Motion (2018), https://doi.org/10.1016/j.wavemoti.2018.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Correlation of elastic wave attenuation and scattering with three-dimensional grain size distribution for polycrystals of statistically equiaxed grains Gaofeng Sha* Department of Mechanical and Aerospace Engineering 201 W 19th Ave, Columbus, Ohio 43210 [email protected] Abstract: This study establishes an explicit relation between spatial two-point correlation function (TPCF) and 3D grain size distribution for aggregates of statistically equiaxed grains. This relation is further validated by applying it to available TPCF and 3D grain size distribution in the literature. Based on this relation, analytical attenuation coefficients for longitudinal and transverse waves, accounting for grain size distribution, are derived from Born approximation for macroscopically isotropic polycrystals of equiaxed triclinic grains. These attenuation models are applicable for whole frequency range except geometric region. Moreover, scattering coefficients for a polycrystal of equiaxed triclinic grains with a 3D grain size distribution are obtained. Finally, the analytical attenuation model for the longitudinal wave is verified by comparison with existing 3D finite element simulation results in the literature. This theoretic study has practical applications to the inverse determination of 3D grain size distribution from ultrasonic measurements. Keywords: Grain size distribution, two-point correlation function, scattering, equiaxed grains
1.Introduction Modeling of elastic wave attenuation and scattering in polycrystalline materials has practical applications in nondestructive evaluation and seismology, thus extensive studies have been reported for wave attenuation and scattering in various polycrystalline media[1]–[20]. A series of
1
analytical attenuation and scattering models[3]–[20] have adopted a spatial two-point correlation function (TPCF) in exponential form, which is governed by one parameter, mean intercept length. The exponential form TPCF assumes that the line intercepts or cord length obeys a Poisson distortion[21], and the mean intercept length can be experimentally determined from many 2D cross-sections of a polycrystal[21]–[23]. However, the exponential form spatial TPCF does not always agree well with the realistic TPCF of a polycrystalline metal (see Ref.[23]) or the spatial TPCF of a Voronoi polycrystal (see Refs.[24], [25]), and it may result in a large discrepancy between predicted attenuation with 3D finite element method (FEM)[25]. Studies[24], [25] developed alternative approaches to incorporate the realistic spatial TPCF into the analytical attenuation models, and both found that analytical models with realistic TPCFs are in better agreement with 3D FEM than those with exponential TPCFs. Physically, the spatial TPCF addresses the probability of two points falling into the same grains[26], hence it is the grains microstructure that controls the spatial TPCF of a polycrystal. For a single phase polycrystal of equiaxed grains, a key factor is the 3D grain size distribution, which is of great importance to the grain boundary engineering[27]. Unlike the line intercept distribution, sophisticated techniques are necessary for the reconstruction of 3D grain size distribution because a direct measurement is difficult [28]–[30]. Note that chord length distribution from 2D cross-sections and 3D grain size distribution are correlated[31]–[33], but relevant theories are complicated. Although Ref.[34] proposed a general relation between spatial TPCF and line intercept distribution (2D grain size distribution), the correlation between 3D grain size distribution and TPCF is still unknown. Virtual Voronoi tessellation grains can represent natural polycrystalline materials and they are widely used in stereology, materials science and mechanics. Moreover, recent progress in 3D FEM modeling of wave scattering in
2
Voronoi grains makes it possible to numerically determine both 3D grain size distribution and wave attenuation coefficient simultaneously[24], [25], [35]. Although some studies[36]–[38] correlated asymptotic attenuation coefficients with 2D grain size distribution, no study attempted to develop analytical attenuation and scattering models with a 3D grain size distribution. Based on the observations in Refs.[24], [25], this study proposes a general relation between 3D grain size distribution and TPCF for a polycrystal of statistically equiaxed grains. Different grain size distribution functions including Gaussian, gamma and lognormal distribution functions are applicable to this relation. This relation is validated by applying it to available 3D grain size distribution and TPCF data in Ref.[25]. Furthermore, analytical attenuation coefficients applicable from Rayleigh region to stochastic region are derived to correlate attenuation with 3D grain size distribution. Besides, the attenuation model for longitudinal wave is verified by comparisons with available attenuation results in Ref.[25]. The scattering coefficients for different mode are also obtained to account for the 3D grain size distribution. These attenuation and scattering models can be used to inversely determine 3D grain size distribution, if experimental attenuation coefficient or scattering amplitude are given. This paper is structured as follows. The proposed relation between TPCF and 3D grain size distribution (probability density function) is addressed in Section 2, and this relation also gets verified in Section 2. The analytical attenuation and scattering models with 3D grain size distribution are derived in Section 3. It is followed by Section 4, validation of the attenuation model. Finally, conclusions are provided.
2. Relation between TPCF and 3D grain size distribution For a polycrystal of uniform equiaxed grains with a grain diameter d , the 3D grain size distribution function or the probability density function of grain diameter would be a Delta
3
function, r d . For such a polycrystal, if treating equiaxed grains as spheres, the TPCF has a simple expression as in Ref.[21]
3r r3 1 w(r , d ) 2d 2d 3 0
0rd
(1)
rd
where w(r , d ) denotes the probability of two points with a distance r falling into the same grain. When equiaxed grains in an aggregate obeys an arbitrary probability density function p(r ) , due
to this relation p(r ) r x p x dx the TPCF for a polycrystal of equiaxed grains with 0
a distribution can be written as:
W (r ) w r , x p( x)dx ,
(2)
0
where the 3D grain size distribution function p ( x ) is obtained from the grain volume distribution assuming spherical grains with equivalent volume as the actual grains. In the literature different 3D grains distribution functions have been reported, including gamma distribution[31], [32], [39]–[41], lognormal distribution[39]–[41] and Gaussian distribution[25]. Thus, in this study the probability density function p ( x) is not confined to a specific function and any form is applicable. Here three common grain size distribution functions are provided. The Gaussian form grain size distribution function is: p (d )
exp[(d ) 2 / 2 / 2 ]
(3)
2
where is the mean grain diameter and stands for distribution width. The lognormal grain diameter distribution function is [39]–[41]:
4
p ( x) where
ln 2 ( x / ) exp 2 d2 d x 2 1
(4)
is the median of the distribution and d stands for the distribution width. The mean 2
grain diameter of this distribution is e d 2 . The two-parameter gamma distribution function is:
xa 1 x p ( x) a exp , b (a) b
(5)
where a, b are two positive variables and ( a ) is a gamma function. The mean grain diameter from Eq. (5) is ab . More details about these three distributions can be found in the textbooks. There are also other distribution functions like Weibull distribution[41] and one parameter gamma distribution[31]; however, it is not necessary to list all of them. Again, any form probability density function can be substitute into Eq. (2), if this distribution function well presents the actual 3D grain size distribution of a polycrystal with statistically equiaxed grains. Following the definition of a spatial Fourier transform in Ref.[4], the spatial Fourier transform of the TPCF in Eq.(1) is:
w (q ,d )
1
2
3
3 dq cos dq 2 2sin dq 2 q r r ( , ) exp w r d d 2 d 3 q6
2
(6)
where bold letters stand for vectors, q = q and r = r . The spatial Fourier transform of the TPCF of a polycrystal with a grain size distribution p ( x ) can be written as: W (q) w q, x p( x)dx ,
(7)
0
where p ( x ) is the same grain size distribution function in Eq. (2). When 3D grain size distribution and TPCF are given, one can obtain some information about 2D grain size distribution from 3D grain size distribution. One study [23] has rigorously derived the 5
relation between mean cord length LC and spatial TPCF, and for a polycrystal with equiaxed grains this relation can be written as[23]:
LC 1 W ' r 0 ,
(8)
where W ' r 0 means the slope of a TPCF near the origin. After substituting Eqs. (1) and (2) into Eq. (8), one can obtain this expression: 1
2 1 LC p ( x)dx . 0 3 x
(9)
For a polycrystal with uniform equiaxed grains with a dimeter d, the mean cord length is
LC
2 d , which is consistent with the result in Ref.[21]. 3
Two examples are taken to validate the relation between TPCF and 3D grain size distribution. One is a polycrystal with 8k equiaxed Voronoi grains created by centroidal Voronoi tessellation (CVT) technique in Ref.[25], and the other is a Voronoi polycrystal with 2k CVT grains as introduced in Ref.[25]. The data of 3D grain size distribution and TPCF were numerically determined for both cases in Ref.[25]. As stated in Ref.[25], the numerical result for 3D grain diameter distribution calculated using equivalent spheres was well fitted by a Gaussian distribution. The fitted Gaussian distribution functions for 8k CVT grains and 2k CVT grains (see regular 8k grains and regular 2k grains in Table II of Ref.[25]) were [25]: p (d )
p (d )
exp[(d 97.76) 2 / 2 / 12.82 ] 12.8 2
exp[(d 154.7) 2 / 2 / 18.512 ] 18.51 2
(8k, regular)
(2k, regular)
(10a)
(10b)
where the unit for grain size is μm. After calculating the TPCFs for both examples through Eq. (10) and Eq. (2), one can compare them with available numerical TPCF data determined in 6
Ref.[25]]. The compparison of prredicted TP PCFs (lines) and measurred TPCFs iin Ref.[25] ((symbols) for thesee two exampples is show wn in Figuree 1. From Fiigure 1, in eeach examplle the prediccted TPCF by b Eq. (2) reeasonably aggrees with numerically n measured T TPCF in Reff.[25] exceppt for a small diiscrepancy ((relative diff fference 15% %) near the ttail. Such a discrepancyy may be caaused by the fittinng of 3D graain size disttribution. Ovverall, the aagreement between theooretical preddiction and 3D FEM is reasonable, thuus the propoosed relationn in Eq. (2) bbetween TP PCF and 3D grain size disttribution is jjustified.
Figuree 1 Compariison betweeen predictedd TPCFs throough 3D graain size disttribution (thhis work) and meeasured TPC CFs in 3D FEM by Ref. f.[25] for twoo Voronoi polycrystals p with 8k CV VT grains and 2k CV VT grains, respectively r y.
Therefoore, this secction establishes a reaasonable coorrelation between b TP PCF and 3D D grain size disstribution. T The TPCF with a graiin size distrribution wiill be used for attenuaation and scatteriing modelinng in the flowing secttion.
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3. Analytical models with 3D grain size distribution As mentioned above, this study is focused on macroscopically isotropic polycrystals of equiaxed grains but with arbitrary grain symmetry. Recent studies[14], [42] have developed attenuation models for texture free polycrystals of triclinic grains using an exponential form TPCF, and those models only account for scattering induced attenuation. Here following the Born approximation as Ref.[14], the spectral TPCF with a 3D grain size distribution, Eq.(7), will be incorporated into the model for scattering induced attenuation. Based on the derivation in Refs. [14], [42], the general expressions of attenuation coefficients under Born approximation for longitudinal and transverse waves in a polycrystal of triclinic grains are:
2 k04L L 2 2V04L 2 k04T T 4 2V04T
0
0
2 k0 L k03T S IPLL W ( k L k L ) sin d 2 2V02LV02T 2 k03L k0T S IPTT W ( kT kT ) sin d 4 2V02LV02T
0
0
IPLT W ( k L kTS ) sin d ;
(11a)
IPLT W ( kT k LS ) sin d ;
(11b)
where , V0 L , V0T , k0 L , k0T are grain or material density, longitudinal velocity, transverse velocity, longitudinal wavenumber and transverse wavenumber. Here IPLL , IPLT , IPTT means inner products, and a subscript stands for a specific scattering mode, for example, L-L scattering. k L ,kT are wavenumber vectors of incidence wave, k LS ,kTS are wavenumber vectors of scattered
wave, and is the angle between incidence wave vector and scatter wave vector. Two terms exist in each attenuation equation, Eq.(11a) and Eq.(11b), due to scattering mode conversion[4], [14], [42]. The wave velocities in statistically isotropic polycrystal of triclinic grains can be calculated through single crystal elastic constants[14], [42]:
V02L
1 2 4 c11 c22 c33 c12 c23 c31 c44 c55 c66 5 15 15 8
(12a)
V02T
1 1 1 c11 c22 c33 c12 c23 c31 c44 c55 c66 15 15 5
(12b)
In terms of the notations in Ref.[42], expressions for inner products in Eq. (11) are: 1 L0 L1 cos 2 L2 cos 4 1575 1 , IPLT M 0 L0 ( M 1 L1 ) cos 2 L2 cos 4 1575 1 IPTT N 0 2 M 0 L0 ( N1 2 M 1 L1 ) cos 2 L2 cos 4 1575
IPLL
(13)
where inner product coefficients L0 , L1 , L2 , M 0 , M 1 , N 0 , N1 for arbitrary symmetry grains are given through single crystal elastic constants in Ref.[14]. Hence theses inner product coefficients are not repeated here. Considering wave attenuation in a polycrystal of uniform equiaxed grains (diameter d), the spectral TPCF in Eq. (6) should be substituted into the attenuation coefficients, Eq. (11), and finally we obtain: L (d )
k04L 1050 2V04L
3 0 L 0T 2 2 2 0 L 0T
k k 1050 V V
T (d)
0
0
0
L1 cos 2 L2 cos 4
[dqLL cos(dqLL / 2) 2sin(dqLL / 2)]2 sin d 6 d 3 qLL
M 0 L0 ( M 1 L1 ) cos 2 L2 cos 4
0
k04T 21002V04T
k03L k0T 21002V02LV02T
L
0
(14a)
[dqLT cos(dqLT / 2) sin(dqLT / 2)] sin d 6 d 3 qLT
N0 2M0 L0 (N1 2M1 L1 )cos2 L2 cos4
2
[dqTT cos(dqTT / 2) sin(RqTT / 2)]2 sin d 6 d 3qTT
(14b)
[dqLT cos(dqLT / 2) sin(dqLT / 2)]2 M0 L0 (M1 L1 )cos2 L2 cos4 sin d 6 d3qLT
2 where qLL k0L 2 2cos , q LT k 0 L 1 V02L V 0T 2V0 L / V0T cos , qTT k0T 2 2cos .
When the 3D grain size obeys a distribution p ( x ) , the spectral TPCF in Eq. (7) should be substituted into general Eq. (11) to derive the attenuation coefficients for longitudinal wave and transverse wave. There would be a double integral in the attenuation coefficient. Since the order
9
of integration for r and are interchangeable, the longitudinal and transverse attenuation coefficients for an aggregate of triclinic equiaxed grains with a distribution p ( x ) are:
L L r p r dr ,
(15a)
0
T T r p r dr ,
(15b)
0
where L r , T r are given in Eq. (14). Like the 3D grain size distribution function in Eq. (2),
p ( x) in Eq. (15) can be any form, such as Gaussian distribution, gamma distribution and lognormal distribution. Hence the total attenuation coefficient in a polycrystal with a grain size distribution is a weighted average of these for polycrystals with different grain sizes. Since the scattering induced attenuation physically results from wave scattering over all angles, scattering coefficients for different scattering modes can be obtained from analytical attenuation models derived by the Born approximation[10], [15]. The respective scattering coefficients (differential cross-section from a unit volume) for longitudinal and transverse waves in a polycrystal of triclinic equiaxed grains with a grain size distribution are:
LL
0
LT
0
k 04L L0 L1 cos 2 L2 cos 4 [ rq LL cos( rq LL / 2) 2 sin( rq LL / 2)]2 p r dr 6 1050 2V04L r 3 q LL
k0L k03T M0 L0 (M1 L1 )cos2 L2 cos4 [rqLT cos(rqLT / 2) 2sin(rqLT / 2)]2 p r dr 6 1050 2V02LV02T r3 qLT
V02T LT 2V02L
TL TT
0
(16a)
(16b)
(16c)
k04T N0 2M0 L0 ( N1 2M1 L1 )cos2 L2 cos4 [rqTT cos(rqTT / 2) 2sin(rqTT / 2)]2 p r dr (16d) 6 r3 qTT 2100 2V04T
where all the symbols have the same definition as those in Eqs. (14) and (15). Similarly, the total scattering coefficient in a polycrystal with a grain size distribution is a weighted average of scattering coefficients for polycrystals with different grain sizes. For a special case, a polycrystal 10
of uniform equiaxed grains, the probability density function p ( x) would become a Delta function ( x d ) and the scattering coefficients is related to grain diameter d.
4. Verification of the analytical model The attenuation and scattering models in Eq. (15) and Eq. (16) are applicable for polycrystals of triclinic grains with a grain size distribution; however, these models need to be verified by other independent methods. Experimental measurements of 3D grain size distribution, attenuation and scattering for a polycrystal of triclinic equiaxed grains are laborious and inaccurate (experimental attenuation may deviate from theory by a factor of 2 [43]), so experimental validation of analytical models in Eq. (15) and Eq. (16) is not feasible. 3D FEM simulation is an alternative approach to validate the analytical models in this study and the simulated attenuation in 3D FEM even includes multiple scattering effect; however, no attenuation or scattering data for any polycrystal with triclinic equiaxed grains are available except for cubic polycrystals. A recent study[25] not only determined volumetric grain size distribution functions and TPCFs for five grain microstructures with different types of Voronoi grains, but also simulated the longitudinal attenuation for cubic polycrystals, aluminum and Inconel. Thus, Ref.[25] provides sufficient simulation data for a comparison between analytical attenuation model in this paper and 3D FEM in Ref.[25]. Firstly, two aluminum Voronoi polycrystals created in Ref.[25] that have relatively weak scattering are used to validate attenuation model in Eq. (15). One is composed of 8K Voronoi tessellation grains (8K, nonuniform case in Ref. [25]) and the other is of 8K CVT grains (8K, regular case in Ref. [25]). The single crystal elastic constants for aluminum and 3D grain size distributions for both grain microstructures are given in Ref.[25]. The fitted Gaussian distribution function for 8k nonuniform grains possesses a mean grain diameter 96.62μm and a distribution 11
width 19.88μm, while 8k CVT grains correspond to a mean diameter 97.76μm and a distribution width 12.88μm[25]. After calculating the longitudinal attenuation coefficients for both examples using Eq.(16a) through their Gaussian distribution functions, the theoretical attenuation coefficients are compared with available 3D FEM results from Ref. [25]. The comparisons of attenuation results for these two polycrystals are shown in Figure 2(a) and Figure 2(b), respectively. The wavenumber and attenuation in Figure 2 are normalized by the mean grain diameter 2R, where is the expected value of a 3D grain size distribution, R
1 p ( x )xdx . It is 2 0
worthy to mention that similar plots in Ref.[25] were normalized by an effective grain diameter, which follows a different definition from the mean grain diameter in this study. In addition, 3D grain size distribution could alter the attenuation behavior significantly, for example, the attenuation curve for 8k CVT grains (see Figure 2(b)) has an oscillation in transition region as discussed in Ref.[25]. More demonstration and discussion about the impact of 3D grain size distribution on attenuation can be found in Ref.[25]. From Figure 2, quantitative comparison between predicted attenuation by this work and available 3D FEM in Ref.[25] indicates the mean relative deviation of analytical model from 3D FEM is 9% in Figure 2 (a) for 8k nonuniform grains and 8% in Figure 2 (b) for 8k CVT grains. Compared with Ref.[25] that incorporated the measured TPCF numerically into Kube&Turner model[14], the deviation of the analytical model in this study from 3D FEM is relatively larger. This may be due to the discrepancy of predicted TPCF from 3D grain size distribution from measured TPCF in 3D FEM. From the examination of both examples, it can be concluded that the analytical attenuation model has reasonable agreement with 3D FEM.
12
Figure 2 Comparisson betweenn analytical attenuation model in thhis work andd 3D FEM ppublished in Ref.[[25] for two aluminum ppolycrystalss with 8k noonuniform V Voronoi graiins (a) and 8k 8 regular C CVT grains ((b).
Figure 3 Comparisson betweenn analytical attenuation model in thhis work andd 3D FEM ppublished in R Ref.[25] for a Inconel polycrystals p with 8k reggular CVT ggrains. An Incoonel Voronooi polycrystaal of 8k CVT grains stuudied by Reff.[25], whicch has strongger scatterinng strength, is taken as an additionnal material ssystem to exxamine the applicabilityy of the analyticcal model inn Eq.(15). Baased on the fitted grain size distribution for Innconel polyccrystal 13
with 8k CVT grains in Ref.[25], attenuation curve from analytical model Eq.(15) is computed and plotted in non-dimensional domain (see Figure 3). The predicted curve is also compared in Figure 3 with 3D FEM data obtained by Ref.[25]. The comparison on the aggregate of Inconel Voronoi grains indicates the analytical model agrees well with 3D FEM, although the mean relative deviation (15%) is larger than aluminum Voronoi polycrystals shown in Figure 4. This larger discrepancy is due to the inaccuracy of Born approximation, since it neglects high order scattering events[4] and deviates from 3D FEM, which includes multiple scattering, increasingly with material anisotropy[25]. If one integrates the TPCF in Eq. (2) into some high order attenuation models like Stanke&Kino model[3], it would minimize the discrepancy between analytical model and 3D FEM; however, this development is out of the scope of this paper. In a word, the analytical attenuation model developed in this paper is reasonable, which implies the developed scattering model in Eq. (16) in terms of the analytical attenuation model should be also consistent with 3D FEM if both scattering coefficient and 3D grain size distribution from 3D FEM are available for a comparison.
5. Conclusions Based on existing TPCF for polycrystals with spherical grains, this study proposes a relation between TPCF and 3D grain size distribution for a polycrystal with statistically equiaxed grain. The distribution functions of grain diameters can be any distribution function, such as Gaussian, gamma and lognormal distribution. This correlation is also verified on Voronio polycrystals created by others with known TPCF and 3D grain size distribution. Based on this correlation, one can determine the TPCF of a polycrystal directly from 3D grain diameter distribution instead of extra numerical computation.
14
From the established relation between TPCF and 3D grain size distribution, the explicit attenuation and scattering coefficients are derived via Born approximation to account for various grain size distributions for a polycrystal of equiaxed triclinic grains. The attenuation coefficient in a polycrystal with a grain size distribution is a weighted average of a series of polycrystals of uniform equiaxed grains, and so is the scattering coefficient. From these analytical models, one can directly obtain wave attenuation and scattering coefficients for a polycrystal from its 3D grain size distribution. Finally, the analytical attenuation model with a 3D grain size distribution is verified by comparison with available 3D FEM data in the literature. Since this study establishes the relation between wave scattering and 3D grain size distribution, it has practical application to the ultrasonic characterization of volumetric grain microstructure, and relevant technical development will be addressed in the future.
Acknowledgement The author would like to thank Martin Ryzy for sharing the 3D FEM results published in Ref.[25].
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[37] E. P. Papadakis, “Grain Size Distribution in Metals and Its Influence on Ultrasonic Attenuation Measurements,” J. Acoust. Soc. Am., vol. 33, no. 11, pp. 1616–1621, 1961. [38] R. L. Smith, “The effect of grain size distribution on the frequency dependence of the ultrasonic attenuation in polycrystalline materials,” Ultrasonics, vol. 20, no. 5, pp. 211–214, 1982. [39] S. Kumar, S. K. Kurtz, J. R. Banavar, and M. G. Sharma, “Properties of a threedimensional Poisson-Voronoi tesselation: A Monte Carlo study,” J. Stat. Phys., vol. 67, no. 3–4, pp. 523–551, 1992. [40] M. F. Vaz and M. A. Fortes, “Grain size distribution: The lognormal and the gamma distribution functions,” Scr. Metall., vol. 22, no. 1, pp. 35–40, 1988. [41] B. Roebuck, “Measurement of grain size and size distribution in engineering materials,” Mater. Sci. Technol., vol. 16, no. 10, pp. 1167–1174, 2000. [42] S. I. Rokhlin, J. Li, and G. Sha, “Far-field scattering model for wave propagation in random media,” J. Acoust. Soc. Am., vol. 137, no. 5, pp. 2655–2669, 2015. [43] L. Yang and S. I. Rokhlin, “On Comparison of Experiment and Theory for Ultrasonic Attenuation in Polycrystalline Niobium,” J. Nondestruct. Eval., vol. 31, no. 1, pp. 77–79, Mar. 2012.
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Relation between spatial two-point correlation function and 3D grain size distribution Correlation of wave attenuation and scattering with 3D grain size distribution Comparison between analytical model and existing 3D finite element simulation
PII: DOI: Reference:
S0165-2125(18)30064-7 https://doi.org/10.1016/j.wavemoti.2018.08.012 WAMOT 2274
To appear in:
Wave Motion
Received date : 23 February 2018 Revised date : 7 July 2018 Accepted date : 23 August 2018 Please cite this article as: G. Sha, Correlation of elastic wave attenuation and scattering with three-dimensional grain size distribution for polycrystals of statistically equiaxed grains, Wave Motion (2018), https://doi.org/10.1016/j.wavemoti.2018.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Correlation of elastic wave attenuation and scattering with three-dimensional grain size distribution for polycrystals of statistically equiaxed grains Gaofeng Sha* Department of Mechanical and Aerospace Engineering 201 W 19th Ave, Columbus, Ohio 43210 [email protected] Abstract: This study establishes an explicit relation between spatial two-point correlation function (TPCF) and 3D grain size distribution for aggregates of statistically equiaxed grains. This relation is further validated by applying it to available TPCF and 3D grain size distribution in the literature. Based on this relation, analytical attenuation coefficients for longitudinal and transverse waves, accounting for grain size distribution, are derived from Born approximation for macroscopically isotropic polycrystals of equiaxed triclinic grains. These attenuation models are applicable for whole frequency range except geometric region. Moreover, scattering coefficients for a polycrystal of equiaxed triclinic grains with a 3D grain size distribution are obtained. Finally, the analytical attenuation model for the longitudinal wave is verified by comparison with existing 3D finite element simulation results in the literature. This theoretic study has practical applications to the inverse determination of 3D grain size distribution from ultrasonic measurements. Keywords: Grain size distribution, two-point correlation function, scattering, equiaxed grains
1.Introduction Modeling of elastic wave attenuation and scattering in polycrystalline materials has practical applications in nondestructive evaluation and seismology, thus extensive studies have been reported for wave attenuation and scattering in various polycrystalline media[1]–[20]. A series of
1
analytical attenuation and scattering models[3]–[20] have adopted a spatial two-point correlation function (TPCF) in exponential form, which is governed by one parameter, mean intercept length. The exponential form TPCF assumes that the line intercepts or cord length obeys a Poisson distortion[21], and the mean intercept length can be experimentally determined from many 2D cross-sections of a polycrystal[21]–[23]. However, the exponential form spatial TPCF does not always agree well with the realistic TPCF of a polycrystalline metal (see Ref.[23]) or the spatial TPCF of a Voronoi polycrystal (see Refs.[24], [25]), and it may result in a large discrepancy between predicted attenuation with 3D finite element method (FEM)[25]. Studies[24], [25] developed alternative approaches to incorporate the realistic spatial TPCF into the analytical attenuation models, and both found that analytical models with realistic TPCFs are in better agreement with 3D FEM than those with exponential TPCFs. Physically, the spatial TPCF addresses the probability of two points falling into the same grains[26], hence it is the grains microstructure that controls the spatial TPCF of a polycrystal. For a single phase polycrystal of equiaxed grains, a key factor is the 3D grain size distribution, which is of great importance to the grain boundary engineering[27]. Unlike the line intercept distribution, sophisticated techniques are necessary for the reconstruction of 3D grain size distribution because a direct measurement is difficult [28]–[30]. Note that chord length distribution from 2D cross-sections and 3D grain size distribution are correlated[31]–[33], but relevant theories are complicated. Although Ref.[34] proposed a general relation between spatial TPCF and line intercept distribution (2D grain size distribution), the correlation between 3D grain size distribution and TPCF is still unknown. Virtual Voronoi tessellation grains can represent natural polycrystalline materials and they are widely used in stereology, materials science and mechanics. Moreover, recent progress in 3D FEM modeling of wave scattering in
2
Voronoi grains makes it possible to numerically determine both 3D grain size distribution and wave attenuation coefficient simultaneously[24], [25], [35]. Although some studies[36]–[38] correlated asymptotic attenuation coefficients with 2D grain size distribution, no study attempted to develop analytical attenuation and scattering models with a 3D grain size distribution. Based on the observations in Refs.[24], [25], this study proposes a general relation between 3D grain size distribution and TPCF for a polycrystal of statistically equiaxed grains. Different grain size distribution functions including Gaussian, gamma and lognormal distribution functions are applicable to this relation. This relation is validated by applying it to available 3D grain size distribution and TPCF data in Ref.[25]. Furthermore, analytical attenuation coefficients applicable from Rayleigh region to stochastic region are derived to correlate attenuation with 3D grain size distribution. Besides, the attenuation model for longitudinal wave is verified by comparisons with available attenuation results in Ref.[25]. The scattering coefficients for different mode are also obtained to account for the 3D grain size distribution. These attenuation and scattering models can be used to inversely determine 3D grain size distribution, if experimental attenuation coefficient or scattering amplitude are given. This paper is structured as follows. The proposed relation between TPCF and 3D grain size distribution (probability density function) is addressed in Section 2, and this relation also gets verified in Section 2. The analytical attenuation and scattering models with 3D grain size distribution are derived in Section 3. It is followed by Section 4, validation of the attenuation model. Finally, conclusions are provided.
2. Relation between TPCF and 3D grain size distribution For a polycrystal of uniform equiaxed grains with a grain diameter d , the 3D grain size distribution function or the probability density function of grain diameter would be a Delta
3
function, r d . For such a polycrystal, if treating equiaxed grains as spheres, the TPCF has a simple expression as in Ref.[21]
3r r3 1 w(r , d ) 2d 2d 3 0
0rd
(1)
rd
where w(r , d ) denotes the probability of two points with a distance r falling into the same grain. When equiaxed grains in an aggregate obeys an arbitrary probability density function p(r ) , due
to this relation p(r ) r x p x dx the TPCF for a polycrystal of equiaxed grains with 0
a distribution can be written as:
W (r ) w r , x p( x)dx ,
(2)
0
where the 3D grain size distribution function p ( x ) is obtained from the grain volume distribution assuming spherical grains with equivalent volume as the actual grains. In the literature different 3D grains distribution functions have been reported, including gamma distribution[31], [32], [39]–[41], lognormal distribution[39]–[41] and Gaussian distribution[25]. Thus, in this study the probability density function p ( x) is not confined to a specific function and any form is applicable. Here three common grain size distribution functions are provided. The Gaussian form grain size distribution function is: p (d )
exp[(d ) 2 / 2 / 2 ]
(3)
2
where is the mean grain diameter and stands for distribution width. The lognormal grain diameter distribution function is [39]–[41]:
4
p ( x) where
ln 2 ( x / ) exp 2 d2 d x 2 1
(4)
is the median of the distribution and d stands for the distribution width. The mean 2
grain diameter of this distribution is e d 2 . The two-parameter gamma distribution function is:
xa 1 x p ( x) a exp , b (a) b
(5)
where a, b are two positive variables and ( a ) is a gamma function. The mean grain diameter from Eq. (5) is ab . More details about these three distributions can be found in the textbooks. There are also other distribution functions like Weibull distribution[41] and one parameter gamma distribution[31]; however, it is not necessary to list all of them. Again, any form probability density function can be substitute into Eq. (2), if this distribution function well presents the actual 3D grain size distribution of a polycrystal with statistically equiaxed grains. Following the definition of a spatial Fourier transform in Ref.[4], the spatial Fourier transform of the TPCF in Eq.(1) is:
w (q ,d )
1
2
3
3 dq cos dq 2 2sin dq 2 q r r ( , ) exp w r d d 2 d 3 q6
2
(6)
where bold letters stand for vectors, q = q and r = r . The spatial Fourier transform of the TPCF of a polycrystal with a grain size distribution p ( x ) can be written as: W (q) w q, x p( x)dx ,
(7)
0
where p ( x ) is the same grain size distribution function in Eq. (2). When 3D grain size distribution and TPCF are given, one can obtain some information about 2D grain size distribution from 3D grain size distribution. One study [23] has rigorously derived the 5
relation between mean cord length LC and spatial TPCF, and for a polycrystal with equiaxed grains this relation can be written as[23]:
LC 1 W ' r 0 ,
(8)
where W ' r 0 means the slope of a TPCF near the origin. After substituting Eqs. (1) and (2) into Eq. (8), one can obtain this expression: 1
2 1 LC p ( x)dx . 0 3 x
(9)
For a polycrystal with uniform equiaxed grains with a dimeter d, the mean cord length is
LC
2 d , which is consistent with the result in Ref.[21]. 3
Two examples are taken to validate the relation between TPCF and 3D grain size distribution. One is a polycrystal with 8k equiaxed Voronoi grains created by centroidal Voronoi tessellation (CVT) technique in Ref.[25], and the other is a Voronoi polycrystal with 2k CVT grains as introduced in Ref.[25]. The data of 3D grain size distribution and TPCF were numerically determined for both cases in Ref.[25]. As stated in Ref.[25], the numerical result for 3D grain diameter distribution calculated using equivalent spheres was well fitted by a Gaussian distribution. The fitted Gaussian distribution functions for 8k CVT grains and 2k CVT grains (see regular 8k grains and regular 2k grains in Table II of Ref.[25]) were [25]: p (d )
p (d )
exp[(d 97.76) 2 / 2 / 12.82 ] 12.8 2
exp[(d 154.7) 2 / 2 / 18.512 ] 18.51 2
(8k, regular)
(2k, regular)
(10a)
(10b)
where the unit for grain size is μm. After calculating the TPCFs for both examples through Eq. (10) and Eq. (2), one can compare them with available numerical TPCF data determined in 6
Ref.[25]]. The compparison of prredicted TP PCFs (lines) and measurred TPCFs iin Ref.[25] ((symbols) for thesee two exampples is show wn in Figuree 1. From Fiigure 1, in eeach examplle the prediccted TPCF by b Eq. (2) reeasonably aggrees with numerically n measured T TPCF in Reff.[25] exceppt for a small diiscrepancy ((relative diff fference 15% %) near the ttail. Such a discrepancyy may be caaused by the fittinng of 3D graain size disttribution. Ovverall, the aagreement between theooretical preddiction and 3D FEM is reasonable, thuus the propoosed relationn in Eq. (2) bbetween TP PCF and 3D grain size disttribution is jjustified.
Figuree 1 Compariison betweeen predictedd TPCFs throough 3D graain size disttribution (thhis work) and meeasured TPC CFs in 3D FEM by Ref. f.[25] for twoo Voronoi polycrystals p with 8k CV VT grains and 2k CV VT grains, respectively r y.
Therefoore, this secction establishes a reaasonable coorrelation between b TP PCF and 3D D grain size disstribution. T The TPCF with a graiin size distrribution wiill be used for attenuaation and scatteriing modelinng in the flowing secttion.
7
3. Analytical models with 3D grain size distribution As mentioned above, this study is focused on macroscopically isotropic polycrystals of equiaxed grains but with arbitrary grain symmetry. Recent studies[14], [42] have developed attenuation models for texture free polycrystals of triclinic grains using an exponential form TPCF, and those models only account for scattering induced attenuation. Here following the Born approximation as Ref.[14], the spectral TPCF with a 3D grain size distribution, Eq.(7), will be incorporated into the model for scattering induced attenuation. Based on the derivation in Refs. [14], [42], the general expressions of attenuation coefficients under Born approximation for longitudinal and transverse waves in a polycrystal of triclinic grains are:
2 k04L L 2 2V04L 2 k04T T 4 2V04T
0
0
2 k0 L k03T S IPLL W ( k L k L ) sin d 2 2V02LV02T 2 k03L k0T S IPTT W ( kT kT ) sin d 4 2V02LV02T
0
0
IPLT W ( k L kTS ) sin d ;
(11a)
IPLT W ( kT k LS ) sin d ;
(11b)
where , V0 L , V0T , k0 L , k0T are grain or material density, longitudinal velocity, transverse velocity, longitudinal wavenumber and transverse wavenumber. Here IPLL , IPLT , IPTT means inner products, and a subscript stands for a specific scattering mode, for example, L-L scattering. k L ,kT are wavenumber vectors of incidence wave, k LS ,kTS are wavenumber vectors of scattered
wave, and is the angle between incidence wave vector and scatter wave vector. Two terms exist in each attenuation equation, Eq.(11a) and Eq.(11b), due to scattering mode conversion[4], [14], [42]. The wave velocities in statistically isotropic polycrystal of triclinic grains can be calculated through single crystal elastic constants[14], [42]:
V02L
1 2 4 c11 c22 c33 c12 c23 c31 c44 c55 c66 5 15 15 8
(12a)
V02T
1 1 1 c11 c22 c33 c12 c23 c31 c44 c55 c66 15 15 5
(12b)
In terms of the notations in Ref.[42], expressions for inner products in Eq. (11) are: 1 L0 L1 cos 2 L2 cos 4 1575 1 , IPLT M 0 L0 ( M 1 L1 ) cos 2 L2 cos 4 1575 1 IPTT N 0 2 M 0 L0 ( N1 2 M 1 L1 ) cos 2 L2 cos 4 1575
IPLL
(13)
where inner product coefficients L0 , L1 , L2 , M 0 , M 1 , N 0 , N1 for arbitrary symmetry grains are given through single crystal elastic constants in Ref.[14]. Hence theses inner product coefficients are not repeated here. Considering wave attenuation in a polycrystal of uniform equiaxed grains (diameter d), the spectral TPCF in Eq. (6) should be substituted into the attenuation coefficients, Eq. (11), and finally we obtain: L (d )
k04L 1050 2V04L
3 0 L 0T 2 2 2 0 L 0T
k k 1050 V V
T (d)
0
0
0
L1 cos 2 L2 cos 4
[dqLL cos(dqLL / 2) 2sin(dqLL / 2)]2 sin d 6 d 3 qLL
M 0 L0 ( M 1 L1 ) cos 2 L2 cos 4
0
k04T 21002V04T
k03L k0T 21002V02LV02T
L
0
(14a)
[dqLT cos(dqLT / 2) sin(dqLT / 2)] sin d 6 d 3 qLT
N0 2M0 L0 (N1 2M1 L1 )cos2 L2 cos4
2
[dqTT cos(dqTT / 2) sin(RqTT / 2)]2 sin d 6 d 3qTT
(14b)
[dqLT cos(dqLT / 2) sin(dqLT / 2)]2 M0 L0 (M1 L1 )cos2 L2 cos4 sin d 6 d3qLT
2 where qLL k0L 2 2cos , q LT k 0 L 1 V02L V 0T 2V0 L / V0T cos , qTT k0T 2 2cos .
When the 3D grain size obeys a distribution p ( x ) , the spectral TPCF in Eq. (7) should be substituted into general Eq. (11) to derive the attenuation coefficients for longitudinal wave and transverse wave. There would be a double integral in the attenuation coefficient. Since the order
9
of integration for r and are interchangeable, the longitudinal and transverse attenuation coefficients for an aggregate of triclinic equiaxed grains with a distribution p ( x ) are:
L L r p r dr ,
(15a)
0
T T r p r dr ,
(15b)
0
where L r , T r are given in Eq. (14). Like the 3D grain size distribution function in Eq. (2),
p ( x) in Eq. (15) can be any form, such as Gaussian distribution, gamma distribution and lognormal distribution. Hence the total attenuation coefficient in a polycrystal with a grain size distribution is a weighted average of these for polycrystals with different grain sizes. Since the scattering induced attenuation physically results from wave scattering over all angles, scattering coefficients for different scattering modes can be obtained from analytical attenuation models derived by the Born approximation[10], [15]. The respective scattering coefficients (differential cross-section from a unit volume) for longitudinal and transverse waves in a polycrystal of triclinic equiaxed grains with a grain size distribution are:
LL
0
LT
0
k 04L L0 L1 cos 2 L2 cos 4 [ rq LL cos( rq LL / 2) 2 sin( rq LL / 2)]2 p r dr 6 1050 2V04L r 3 q LL
k0L k03T M0 L0 (M1 L1 )cos2 L2 cos4 [rqLT cos(rqLT / 2) 2sin(rqLT / 2)]2 p r dr 6 1050 2V02LV02T r3 qLT
V02T LT 2V02L
TL TT
0
(16a)
(16b)
(16c)
k04T N0 2M0 L0 ( N1 2M1 L1 )cos2 L2 cos4 [rqTT cos(rqTT / 2) 2sin(rqTT / 2)]2 p r dr (16d) 6 r3 qTT 2100 2V04T
where all the symbols have the same definition as those in Eqs. (14) and (15). Similarly, the total scattering coefficient in a polycrystal with a grain size distribution is a weighted average of scattering coefficients for polycrystals with different grain sizes. For a special case, a polycrystal 10
of uniform equiaxed grains, the probability density function p ( x) would become a Delta function ( x d ) and the scattering coefficients is related to grain diameter d.
4. Verification of the analytical model The attenuation and scattering models in Eq. (15) and Eq. (16) are applicable for polycrystals of triclinic grains with a grain size distribution; however, these models need to be verified by other independent methods. Experimental measurements of 3D grain size distribution, attenuation and scattering for a polycrystal of triclinic equiaxed grains are laborious and inaccurate (experimental attenuation may deviate from theory by a factor of 2 [43]), so experimental validation of analytical models in Eq. (15) and Eq. (16) is not feasible. 3D FEM simulation is an alternative approach to validate the analytical models in this study and the simulated attenuation in 3D FEM even includes multiple scattering effect; however, no attenuation or scattering data for any polycrystal with triclinic equiaxed grains are available except for cubic polycrystals. A recent study[25] not only determined volumetric grain size distribution functions and TPCFs for five grain microstructures with different types of Voronoi grains, but also simulated the longitudinal attenuation for cubic polycrystals, aluminum and Inconel. Thus, Ref.[25] provides sufficient simulation data for a comparison between analytical attenuation model in this paper and 3D FEM in Ref.[25]. Firstly, two aluminum Voronoi polycrystals created in Ref.[25] that have relatively weak scattering are used to validate attenuation model in Eq. (15). One is composed of 8K Voronoi tessellation grains (8K, nonuniform case in Ref. [25]) and the other is of 8K CVT grains (8K, regular case in Ref. [25]). The single crystal elastic constants for aluminum and 3D grain size distributions for both grain microstructures are given in Ref.[25]. The fitted Gaussian distribution function for 8k nonuniform grains possesses a mean grain diameter 96.62μm and a distribution 11
width 19.88μm, while 8k CVT grains correspond to a mean diameter 97.76μm and a distribution width 12.88μm[25]. After calculating the longitudinal attenuation coefficients for both examples using Eq.(16a) through their Gaussian distribution functions, the theoretical attenuation coefficients are compared with available 3D FEM results from Ref. [25]. The comparisons of attenuation results for these two polycrystals are shown in Figure 2(a) and Figure 2(b), respectively. The wavenumber and attenuation in Figure 2 are normalized by the mean grain diameter 2R, where is the expected value of a 3D grain size distribution, R
1 p ( x )xdx . It is 2 0
worthy to mention that similar plots in Ref.[25] were normalized by an effective grain diameter, which follows a different definition from the mean grain diameter in this study. In addition, 3D grain size distribution could alter the attenuation behavior significantly, for example, the attenuation curve for 8k CVT grains (see Figure 2(b)) has an oscillation in transition region as discussed in Ref.[25]. More demonstration and discussion about the impact of 3D grain size distribution on attenuation can be found in Ref.[25]. From Figure 2, quantitative comparison between predicted attenuation by this work and available 3D FEM in Ref.[25] indicates the mean relative deviation of analytical model from 3D FEM is 9% in Figure 2 (a) for 8k nonuniform grains and 8% in Figure 2 (b) for 8k CVT grains. Compared with Ref.[25] that incorporated the measured TPCF numerically into Kube&Turner model[14], the deviation of the analytical model in this study from 3D FEM is relatively larger. This may be due to the discrepancy of predicted TPCF from 3D grain size distribution from measured TPCF in 3D FEM. From the examination of both examples, it can be concluded that the analytical attenuation model has reasonable agreement with 3D FEM.
12
Figure 2 Comparisson betweenn analytical attenuation model in thhis work andd 3D FEM ppublished in Ref.[[25] for two aluminum ppolycrystalss with 8k noonuniform V Voronoi graiins (a) and 8k 8 regular C CVT grains ((b).
Figure 3 Comparisson betweenn analytical attenuation model in thhis work andd 3D FEM ppublished in R Ref.[25] for a Inconel polycrystals p with 8k reggular CVT ggrains. An Incoonel Voronooi polycrystaal of 8k CVT grains stuudied by Reff.[25], whicch has strongger scatterinng strength, is taken as an additionnal material ssystem to exxamine the applicabilityy of the analyticcal model inn Eq.(15). Baased on the fitted grain size distribution for Innconel polyccrystal 13
with 8k CVT grains in Ref.[25], attenuation curve from analytical model Eq.(15) is computed and plotted in non-dimensional domain (see Figure 3). The predicted curve is also compared in Figure 3 with 3D FEM data obtained by Ref.[25]. The comparison on the aggregate of Inconel Voronoi grains indicates the analytical model agrees well with 3D FEM, although the mean relative deviation (15%) is larger than aluminum Voronoi polycrystals shown in Figure 4. This larger discrepancy is due to the inaccuracy of Born approximation, since it neglects high order scattering events[4] and deviates from 3D FEM, which includes multiple scattering, increasingly with material anisotropy[25]. If one integrates the TPCF in Eq. (2) into some high order attenuation models like Stanke&Kino model[3], it would minimize the discrepancy between analytical model and 3D FEM; however, this development is out of the scope of this paper. In a word, the analytical attenuation model developed in this paper is reasonable, which implies the developed scattering model in Eq. (16) in terms of the analytical attenuation model should be also consistent with 3D FEM if both scattering coefficient and 3D grain size distribution from 3D FEM are available for a comparison.
5. Conclusions Based on existing TPCF for polycrystals with spherical grains, this study proposes a relation between TPCF and 3D grain size distribution for a polycrystal with statistically equiaxed grain. The distribution functions of grain diameters can be any distribution function, such as Gaussian, gamma and lognormal distribution. This correlation is also verified on Voronio polycrystals created by others with known TPCF and 3D grain size distribution. Based on this correlation, one can determine the TPCF of a polycrystal directly from 3D grain diameter distribution instead of extra numerical computation.
14
From the established relation between TPCF and 3D grain size distribution, the explicit attenuation and scattering coefficients are derived via Born approximation to account for various grain size distributions for a polycrystal of equiaxed triclinic grains. The attenuation coefficient in a polycrystal with a grain size distribution is a weighted average of a series of polycrystals of uniform equiaxed grains, and so is the scattering coefficient. From these analytical models, one can directly obtain wave attenuation and scattering coefficients for a polycrystal from its 3D grain size distribution. Finally, the analytical attenuation model with a 3D grain size distribution is verified by comparison with available 3D FEM data in the literature. Since this study establishes the relation between wave scattering and 3D grain size distribution, it has practical application to the ultrasonic characterization of volumetric grain microstructure, and relevant technical development will be addressed in the future.
Acknowledgement The author would like to thank Martin Ryzy for sharing the 3D FEM results published in Ref.[25].
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Relation between spatial two-point correlation function and 3D grain size distribution Correlation of wave attenuation and scattering with 3D grain size distribution Comparison between analytical model and existing 3D finite element simulation