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Finite element modeling of surface acoustic wave
propagation in polycrystalline aluminium: effective
phase velocity
Martin Ryzy, Tomas Grabec, Istvan Veres
To cite this version:
Martin Ryzy, Tomas Grabec, Istvan Veres. Finite element modeling of surface acoustic wave
prop-agation in polycrystalline aluminium: effective phase velocity. Forum Acusticum, Dec 2020, Lyon,
France. pp.1833-1838, �10.48465/fa.2020.0165�. �hal-03240228�
FINITE ELEMENT MODELING OF SURFACE ACOUSTIC WAVE
PROPAGATION IN POLYCRYSTALLINE ALUMINUM: EFFECTIVE
PHASE VELOCITY
Martin Ryzy
1Tom´aˇs Grabec
2Istv´an A. Veres
3 1Research Center for Non-Destructive Testing GmbH, 4040 Linz, Austria
2
Nuclear Physics Institute, 25068 ˇ
Reˇz, Czech Republic
3Qorvo Inc., Apopka, FL 32703, USA
martin.ryzy@recendt.at
ABSTRACT
In this work we use a finite element method (FEM) to solve the elastodynamic problem of multiple scattering of sur-face acoustic Rayleigh-waves (SAWs) in polycrystalline aluminum. A realistic grain structure is generated by using a Laguerre tessellation algorithm. We calculate an ensem-ble average (coherent) response by repeated simulations and present two methods to extract the effective Rayleigh-wave phase velocity. Both methods give identical results in a frequency range from 12 MHz to 100 MHz. They re-veal a microstructure induced phase velocity transition at a frequency were the Rayleigh-wavelength is approximately one third of the average grain size.
1. INTRODUCTION
Grain boundary scattering in polycrystalline metals is not only an interesting use case of the more general class of problems dealing with wave propagation in random, het-erogeneous media, but also of great interest for nondestruc-tive evaluation (NDE) applications, as the grain scatter-ing is related to the microstructural properties of common engineering metals (e. g. mean grain size). Usually, lon-gitudinal bulk acoustic waves are studied in this context, and a multitude of experimental [1–9], analytical [10–17] and numerical studies [17–23] (mostly based on the finite-element method) exist, which investigate scattering in-duced changes of the effective phase velocity and/or atten-uation. The effect of grain boundary scattering on surface acoustic Rayleigh waves is much less studied [21, 24], al-though this wave-mode is easier experimentally accessible as it propagates along the materials’ surface and may al-low a more accurate determination of phase velocity and attenuation in experiments.
We recently closed this gap by demonstrating a broad-band laser-ultrasound measurement of the effective sur-face acoustic wave phase velocity and attenuation in poly-crystalline aluminum in a frequency range from around 10 MHz to 130 MHz [8]. In an ongoing work, we aim to complement/verify these experiments by modeling the underlying elastodynamic problem with a finite element model (FEM) in a synthetic polycrystal. Here, we present
first results of this study, related to the behavior of the ef-fective phase velocity of Rayleigh-waves.
2. FINITE ELEMENT MODELING OF SURFACE ACOUSTIC WAVE PROPAGATION
Figure 1. Sketch of the synthetic polycrystal used for the finite element SAW-propagation study. The dimensions of the domain are Lx× Ly× Lz= 1 × 3 × 0.45 mm3. The
location of the line-source for wave excitation is indicated as red solid line.
We generated a synthetic polycrystal with the dimen-sions of Lx × Ly × Lz = 1 × 3 × 0.45 mm3 with the
polycrystal generation software NEPER [25] (https:// neper.info) as illustrated in Fig. 1. The large sample-height (Lx) was chosen to avoid any Lamb-like modes
and back reflections from the lower surface. The ob-tained grain structure is a Laguerre-tessellation with a log-normal distribution of equivalent spherical grain diame-ters d (see Fig. 2). By fitting the according probabil-ity densprobabil-ity function p(d) ∝1/(√2πσd) exp −(ln d−µ)2
/2σ2
(see red solid line in Fig. 2) we obtain an expecta-tion value for the equivalent spherical diameter of ¯d =
Figure 2. Distribution of the equivalent spherical grain diameter d of the synthetic polycrystal (blue bins). The red solid line is a fit of a log-normal probability density function with an expectation value of 134.1 µm (indicated as vertical black dashed line) and a standard deviation of 50.4 µm.
exp µ + 0.5σ2 = 134.1 µm and a standard deviation of
sd(d) =pexp (2µ + σ2) (exp (µ2) − 1) = 50.4 µm. The
domain was discretized into a structured cubic mesh with a lattice spacing of 1.25 µm which gives 2.0736 billion de-grees of freedom in total. The structured mesh type has been successfully applied in previous FEM grain scatter-ing studies and was shown to perform well when its spac-ing is smaller than a tenth of the mean grain size and a tenth of the wavelength [18, 20, 23], which is the case for the investigated structure and range of frequencies. A ran-domly rotated stiffness tensor Cijkl with the single
crys-tal elastic constants of aluminum (C11 = 107.6 GPa,
C12 = 61.80 GPa, C44 = 28.40 GPa) [26] was assigned
to each grain by using a method by Arvo [27], as described previously [23], which leads to a uniform distribution of crystallite orientations and thus to an untextured, macro-scopically isotropic polycrystal. Surface acoustic waves were excited by applying a uniform pressure load in y-direction along a line located on the top surface (x = 0) at y = 0 (indicated by the red line Fig. 1). The load had a Gaussian spatial distribution with a full width at half maximum (FWHM) of 18.32 µm, and a temporal distri-bution which corresponds to a laser-source with a Gaus-sian pulse length of 25 ns (FWHM) [23, 28]. Boundary conditions were chosen to sustain plane surface acoustic wave propagation and to maximally suppress unwanted wave-modes (Lamb waves, reflected bulk waves). Thus we used symmetry boundary conditions on the lateral sam-ple faces (z = 0 - plane and z = Lz - plane) and in
the y = 0 - plane, absorbing boundaries at the domain-bottom x = −Lx and at the face opposite to the
excita-tion (y = Ly), and a free boundary in the plane of
wave-propagation x = 0. An overall simulation time of 1050 ns was chosen to allow the Rayleigh-wave to propagate to the opposite sample face and a time step of ∆t ≈ 0.92 ns
was chosen to fulfill the Courant–Friedrichs–Lewy condi-tion [29]. The model was solved with the commercial fi-nite element time-domain software PZFLEX (Weidlinger Associates, Mountain View, CA) on an Intel Xeon E5620 computer (2 × 4 cores, 2.4 GHz) with typical computation times of 50 hours per simulation.
3. MODEL VERIFICATION AND EXTRACTION OF THE EFFECTIVE RAYLEIGH-WAVE PHASE
VELOCITY
To validate the FEM-model we first performed a simula-tion in a homogenized version of the polycrystal by assign-ing the same Voigt-averaged (i. e. isotropic) elastic con-stants (C11V = 112.0 GPa, C44V = 26.2 GPa) to each
crystallite. A sketch of the obtained normal surface dis-placement ux at a time t = 450 ns after wave excitation
is shown in Fig. 3(a). As expected, two wavefronts are visible. The first bipolar pulse at y ≈ 1.3 mm corre-sponds to the Rayleigh surface acoustic wave and has a much higher magnitude as the second one at y ≈ 2.8 mm, which stems from the surface skimming longitudinal wave (SSLW). Both modes have a planar wavefront and the cov-ered distances comply with their Voigt-averaged wave ve-locities of cV
R = 2911.4 ms−1 and cVL = 6440.6 ms−1 (a
density of ρ = 2700 kgm−3 was used), respectively. We thus infer that the FEM-model supports plane Rayleigh-wave propagation sufficiently.
Fig. 3(b) shows the surface displacement uxof a
poly-crystalline simulation at the same instance of time as the homogenized case. As before, Rayleigh-wave and SSLW ballistic wavefronts are clearly visible, but this time they are slightly distorted due to the grain boundary scattering. In addition to the wavefront distortion, the grain scattering causes diffuse wavefield-fluctuations, mainly in the back-ward direction. Differences in the wave speeds (compared to the homogeneous case) are not evident in this view.
In order to analyze the effective phase velocities, the wavefields were spatially averaged along the z-direction in both cases. In addition, the simulations in the poly-crystalline sample were repeated 28 times by generating 4 tessellations with the same grain statistics (equivalent to the one shown in Fig. 1) and running 7 simulations in each tessellation while using different sets of random grain rotations in each run. This approach was previously shown to be sufficient to obtain an ensemble average re-sponse [20]. The resulting averaged surface displacement fields are shown in the center graph of Fig. 4 for the homo-geneous (upper graph) and the polycrystalline case (lower graph). In both cases, the coherent Rayleigh-wave (more pronounced peak) and the SSLW are clearly visible. The effect of grain scattering induces an effective attenuation of the averaged Rayleigh-wave and a cancellation of the grain noise. This is most evident in the amplitude drop of the Rayleigh-wave peak in the polycrystalline case, which is illustrated with the three cross sections at y = 0.75, 1.5 and 2.25 mm. Moreover, the peaks are broadened compared to the homogeneous case. This can be seen in the insets of the center graphs of Fig. 4), which show magnified views
Figure 3. Out-of-plane displacement uxat the sample surface x = 0 for a homogeneous material model (a) and a
poly-crystalline material model (b) at a time t = 450 ns after wave excitation, calculated with a time-domain FEM-model. Two ballistic plane wavefronts are visible in both cases: One belongs to the Rayleigh surface acoustic wave at y ≈ 1.3 mm and the other one to the surface skimming longitudinal wave (SSLW) at y ≈ 2.8 mm. Please note the non-linear scaling of the color bars, which is useful to improve the visibility of the SSLW.
of the Rayleigh-peak time-traces at y = 1.5 mm.
For the extraction of the frequency dependent effective Rayleigh-wave phase velocity cR(f ) we applied two
dif-ferent data processing procedures:
The first one is based on evaluating the phase of the temporal Fourier transform Ft{ux(y, t)} = ˜ux(y, f ) of
the initial signal, whose magnitude |˜ux(y, f )| and phase
arg(˜ux(y, f )) are shown in the left graphs of Fig. 4 for a
frequency of f = 60 MHz. Please note that the magnitude is mainly determined by the Rayleigh-mode and the SSLW appears as a beating of these two wave-modes (see oscilla-tions) and dominantly decays due to geometric spreading, whereas a plane Rayleigh-wave does not suffer geometric attenuation. In contrast to the homogeneous case, the mag-nitude in the polycrystalline case decays in an exponen-tial manner due to the induced effective attenuation. The effective Rayleigh phase velocity cR was obtained by
fit-ting (minimization of the root mean square error (RMSE)) a straight line φ(y, f ) = φ0(f ) + kR(f )y (green lines in
the insets) to the (unwrapped) phase arg(˜ux(y, f )) (blue
lines), and by using the fundamental dispersion relation 2πf = cRkRfor each frequency in a range from 12 MHz to
100 MHz. For the fits a range of y ∈ [1, 2.9]mm was cho-sen to avoid interference from the SSLW for small y and er-rors induced by Fourier-transforming a partially truncated
Rayleigh-pulse at large y.
In the second method, we separated the Rayleigh-wave from the SSLW in the time domain, by restricting ux(y, t)
to regions where both wave-modes are clearly separated (see semi-transparent yellow region in the center graphs of Fig. 4 where y ∈ [0.45, 2.95]mm). Then a Tukey-window with a length of 56.8 ns and a rise time of 4.66 ns (red line in the insets of the center graphs of Fig. 4) was applied around the center of the Rayleigh-peak (blue line) and the obtained traces were zero-padded. Thereafter, fast Fourier transforms were applied along y and t to get the response ˜
ux(k, f ) in the reciprocal space (see right graphs in Fig. 4),
where only the Rayleigh-mode is left. Then, for each frequency f , Lorentzian functions of the form g(k, f ) ∝
α2+ (k R− k)2
−0.5
where fit (RMSE minimization) to the magnitude |˜ux(k, f )| to obtain to Rayleigh-peak
cen-ter wavenumbers kRand the phase velocity cR= 2πf k−1R .
Examples of the fits are shown in the insets of the right graphs of Fig. 4 for a frequency of 60 MHz (indicated by the red line in the main graph). Please note that a broad-ening (see particularly the side-lobes in the homogeneous case) of the Rayleigh-peaks occurs due to the applied zero-padding. However, as we are only interested in the peak-center positions, this will not influence the obtained results.
Figure 4. Averaged surface normal displacement fields of FEM-simulations from a homogeneous sample (upper row of graphs) and from a polycrystalline sample (lower row of graphs). The center graphs show the wavefields in the physical domain, the insets time traces of the the Rayleigh-wave peaks at y = 1.5 mm and the Tukey-window used for its separation from the SSLW. The graphs on the left show the magnitude and (unwrapped) phase (inset) of the temporal Fourier-transform of the initial data at a frequency of 60 MHz. The green line is a linear fit to the phase and overlaps with the data. The graphs on the right side show the magnitude of the signals in the reciprocal space (kf - space) obtained after separation of the Rayleigh-wave from the SSLW by taking data from the semi-transparent yellow areas in the center graphs and applying a Tukey-window (as shown in the inset of the center graph) prior to double Fourier-transformation. The insets in the right graphs show parts of the cross sections around the Rayleigh-wave (blue line) used for fitting Lorentzian functions (green lines) at a frequency of f = 60 MHz.
4. RESULTS AND DISCUSSION
The extracted effective Rayleigh-wave phase velocities are shown in Fig. 5 for a frequency range from 12 MHz to 100 MHz. For the homogeneous case, both data pro-cessing methods (blue and green solid lines) give virtu-ally identical results and maximvirtu-ally deviate by around 10 ms−1(≈ 0.34 %) from the Voigt-averaged phase veloc-ity cV
R= 2911.40 ms−1 (red line). The observed
system-atic deviation is caused by numerical dispersion (‘mesh-scattering’) and is comparable to the numerical phase ve-locity dispersion typically observed in bulk-wave FEM-studies [23]. The ‘oscillations‘ at lower frequencies ob-served in the phase-fitting-method may be caused by re-maining influences from the SSLW, which is more effec-tively rejected at higher frequencies when the width of the excitation pulse starts to exceed its (half) wavelength.
The phase velocity of the polycrystalline sample is somewhat lower than the Voigt-velocity. This can be ex-pected, as Voigt-averaged elastic constants (and so the wave velocities) are upper boundaries for effective elas-tic constants [30]. Again, both fitting methods give virtu-ally identical results, except for small deviations at high frequencies and the previously discussed oscillations
ob-served in the phase-fitting method. Please note that like in the homogeneous case, the numerical dispersion also here leads to a steady drop of the phase velocity with rising frequency. However, in contrast to the homogenized sam-ple, there is a pronounced velocity drop of around 10 ms−1 (≈ 0.35 %) at approximately 70 MHz, where the grain-size to wavelength ratioλ/d¯= 0.31 ≈1/3and the normalized
wavenumber amounts to kV
Rd ≈ 20. This transition is very¯
similar to that predicted by Stanke & Kino [11] for trans-verse waves in polycrystalline aluminum, were the phase velocity drops around 3.3 % at kdmfp ≈ 30 (see Fig. 7
in reference [11])1. For this reason, and because both
of the applied data-processing methods give the same sults, we are confident that the observed transition is re-lated to the grain structure here as well. Hence, an accu-rate measurement of the Rayleigh-wave phase velocity dis-persion may be used for microstructure characterization in future NDE-applications. Further numerical studies with synthetic polycrystals with different microstructures (e. g.
1Please note that d
mfpin [11] is the mean free path length in the
mi-crostructure and not identical with the equivalent spherical grain diameter expectation value ¯d as used here. Moreover, the microstructure morpholo-gies are different, which will affect both dmfpand ¯d. Thus, the values for
20 40 60 80 100
f
(MHz)
2870 2880 2890 2900 2910 2920 2930p
h
a
se
v
el
o
ci
ty
c
R!
m
s
− 1"
Voigt-average |˜ux(k, f)|, Lorentzian arg(˜ux(y, f)), linearfit-type
homogeneous
polycrystalline
Figure 5. Effective Rayleigh-wave phase velocities ex-tracted from finite element simulations of a homogeneous and a polycrystalline material model. Two data processing methods are compared, one based on a linear fit of the sig-nal phase in yf -space (blue solid lines) and the other one on a Lorentzian fit of the signal magnitude in kf -space (green solid lines). The red line indicates the theoretical phase velocity of cV
R= 2911.4 ms−1 obtained from
Voigt-averaged elastic constants.
varying grain size distributions) or the development of ana-lytical mean-field models for the prediction of the effective Rayleigh-wave velocity in polycrystalline media will help to elucidate this issue further.
5. CONCLUSION
In summary, we developed a finite-element model to solve the elastodynamic problem of surface acoustic plane-wave propagation in a polycrystalline aluminum sample by using a Laguerre-tessellation to emulate a realistic microstruc-ture. The model accounts for multiple scattering events of any order, and can be generalized to other materials with lower crystal symmetry, texture, and different grain morphologies. We demonstrated and compared two data-processing methods for the determination of the effective Rayleigh-wave phase velocity. Results from both meth-ods are in excellent agreement and reveal a phase-velocity transition at a frequency were the Rayleigh-wavelength is approximately one third of the grain size. Due to the simi-larity of this transition to that observed for transverse wave, we conclude that it is caused by the grain structure and may thus be used for its characterization in future NDE appli-cations.
6. ACKNOWLEDGEMENTS
This work was funded by the Austrian research funding association (FFG) under the scope of the COMET pro-gram within the research project “Photonic Sensing for Smarter Processes (PSSP)” (Contract No. 871974), by the project “multimodal and in-situ characterization of inho-mogeneous materials” (MiCi) by the federal government of Upper Austria and the European Regional Develop-ment Fund (EFRE) in the framework of the EU-program IWB2020 and by OP RDE, MEYS (ESS-Scandinavia-CZ-OP, CZ.02.1.01/0.0/0.0/16 013/0001794).
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