nonlinear generation of a zero group velocity mode in an
TRANSCRIPT
Ultrasonics 119 (2022) 106589
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Nonlinear generation of a zero group velocity mode in an elastic plate bynon-collinear mixingPierric Mora a,b,โ, Mathieu Chekroun a, Samuel Raetz a, Vincent Tournat a
a Laboratoire dโAcoustique de lโUniversitรฉ du Mans (LAUM), UMR 6613, Institut dโAcoustique - Graduate School (IA-GS), CNRS, Le Mans Universitรฉ, Franceb GERS-GeoEND, Univ Gustave Eiffel, IFSTTAR, F-44344 Bouguenais, France
A R T I C L E I N F O
Keywords:Lamb wavesNon-collinear interactionQuadratic non-linearityHarmonic generation
A B S T R A C T
Zero Group Velocity (ZGV) modes are peculiar guided waves that can exist in elastic plates or cylinders,and have proved to be very sensitive tools in characterizing materials or thickness variations with sub-percent accuracy at space resolutions of about the plate thickness. In this article we show theoretically andexperimentally how such a mode can be generated as the sum-frequency interaction of two high amplitudeprimary waves, and then serve as a local probe of material non-linearity. The solutions to the phase matchingcondition, i.e. condition for a constructive non-linear effect, are obtained numerically in the mark of classical,quadratic non-linearity. The coupling coefficients that measure the transfer rate of energy as a function oftime from primary to secondary modes are derived. Experiments are conducted on an aluminum plate usingpiezo-electric transducers and a laser interferometer, and explore the interaction for incident symmetric andanti-symmetric fundamental Lamb modes. In an experiment operated without voltage amplifier we demonstratethat the resonant nature of these ZGV modes can be leveraged to accumulate energy from long excitationsand produce detectable effects at extraordinarily low input power even in such weakly non-linear material.
1. Introduction
Nonlinear ultrasonics in non-destructive testing has received con-siderable interest in the last decades owing to its promise to detectmaterial degradation at much earlier stages than linear techniquesdo [1].
As one among other non-linear effects that have shown great sen-sitivity to micro-structural changes, harmonic generation is often theground of a strategy to interrogate a medium [2]. In this context,appealing guided wave based techniques have emerged [3โ6], althoughtheir development has been hindered by the complexity of wave prop-agation in these structures which adds to the other difficulties that anynon-linear based method must cope with. Indeed, due to dispersion, thephase matching condition without which no significant magnitudes areattained is in general not fulfilled for collinear interaction, except atsome very specific combinations of modes and frequencies that have tobe carefully identified [7,8], and among which only few are at reachin practice given the state of the art in transduction.
In an attempt to broaden the design space of this class of methods,recent works have begun to explore the new possibilities offered bynon-collinear mixing [9โ13]. Non-collinear mixing is the name givento the phenomenon of harmonic generation when the two primarywaves come from different directions. Although well known [14โ17]
โ Corresponding author at: GERS-GeoEND, Univ Gustave Eiffel, IFSTTAR, F-44344 Bouguenais, France.E-mail address: [email protected] (P. Mora).
and with identified advantages applied in non-destructive evaluationin the case of bulk waves (see e.g. [18,19]), and despite pioneerexperimental demonstrations in plates in the 70s [20,21], this effect hasremained out of focus for Lamb waves until the articles cited above.The benefits of non-collinear mixing are in general that it allows aneasier separation between system and sample non-linearity. In the caseof guided waves, the other benefit is that the solutions to the phasematching condition are tremendously more numerous, allowing mostmodes to be generated by a well chosen pair of primary waves. Weshow in this article how a zero group velocity mode can be excited thisway.
Waves having zero group velocity (ZGV) and non-zero wavenumbercan exist in elastic waveguides. They behave like thickness resonancesalthough, due to their finite wavelength (about the plate thickness),they remain strongly localized in space and can serve as local probesof sample properties (e.g. defects [22], thickness changes [23โ26],mechanical constants [27โ30], bonding [31โ33]) with a single pointmeasurement. Owing to their high quality factor, these modes are usu-ally relatively easy to generate and can be detected at great precisionwith non-contact setups. For instance, ๐ โ 15000 was reported [23] inaluminum alloys, which enabled detecting and mapping [23,24] rela-tive thickness variations as small as 2 ร 10โ4. Successful applications
vailable online 22 September 2021041-624X/ยฉ 2021 Elsevier B.V. All rights reserved.
https://doi.org/10.1016/j.ultras.2021.106589Received 11 June 2021; Received in revised form 14 September 2021; Accepted 15
September 2021Ultrasonics 119 (2022) 106589P. Mora et al.
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in non-destructive testing were reported with air-coupled transducers inthe 150 โ 250 kHz range [22], or with laser ultrasonics from MHz [23โ1,33โ36] to GHz [32] frequencies, covering thicknesses down toonded nanostructures [32]. One noticeable exception to the contact-ess detection that is usually required is found in civil engineeringthe โโimpact-echoโโ technique [37]) where, due to the small size ofhe sensors regarding the thickness of the walls being probed, contactetection can be meaningful without significantly affecting the modes.astly, let us also mention that ZGV modes have led to the designf high-Q micro resonators, either optical [38] or mechanical [39]or electroacoustic components. Insights on the ranges of existence ofGV modes as a function of Poissonโs ratio can be found in Ref. [34].ther aspects have been discussed such as the influence of boundaryonditions and layering [40], anisotropy [28], or their time decayehavior due to dispersion and material damping [35].
Altogether, these qualities make ZGV modes attractive candidateso achieve new performances in evaluating the non-linear propertiesf a medium or a bonded assembly [41โ43]. However โ except fromef. [44] which explores theoretically self-action effects in the mark ofysteretic non-linearity โ reported works cover only linear properties.ne obstacle regarding harmonic generation is that it is in general notossible to find a primary mode that collinearly satisfies the phaseatching condition with a ZGV mode as target. The natural way to
vercome this difficulty is to resort to non-collinear mixing. This is thetarting point of the present article.
Let us give a brief illustration before going into details. Fig. 1 showssetup in which two narrow-banded, high amplitude wave trains are
aunched in an aluminum plate in such a way that their wavevectorsnd frequencies sum to a ZGV point of the plate. After an interactionime of half a hundred cycles, the remaining field has a clear componentt the expected ZGV frequency.
This article is organized as follows. We first obtain the solutionso the phase matching condition considering the โโsum frequencyโโnteraction and the so-called S1S2 ZGV mode as target. This mode
is the lowest frequency and most widely known and exploited ZGVmode that is supported by plates in a very wide range of Poissonโsratio, covering most engineering isotropic materials (metals, concrete,plastics, etc.). We also derive and discuss the coefficients that quantifythe interaction between selected modes. We then describe experimentalrealizations, the first two of which giving evidence of the phenomenonfor different choices in the primary modes. Finally, a third experimentemphasizes one of the unique benefits that a ZGV mode can provide bydemonstrating detectable effects obtained at low driving amplitudes.
2. Theoretical background
2.1. General considerations
We recall here a few background results of non-collinear harmonicgeneration in hyperelastic materials showing classical, quadratic non-linearity. These results can be obtained by considering the equationsof motion for finite amplitudes in the perturbative approximation.They were first derived for bulk waves [14โ17] and are qualitativelysimilar for guided waves [9,10] as the methodology to establish themextends naturally. Non-collinear mixing is also possible for other typesof non-linearities, although it involves other angles, admissible sym-metry pairings and harmonic orders โ see Ref. [45] for bulk wavesand hysteretic quadratic non-linearity. Contact acoustics non-linearity,another family of non-classical non-linearities, could also be relevant inthe future in the case of the non-linear and non-collinear interactionsat a crack [13,46].
Let us consider two primary displacement fields ๐ฎ1 and ๐ฎ2 propa-gating in a plate made of such a material. ๐ฎ1 and ๐ฎ2 are assumed to bemonochromatic and composed of a single Lamb mode each:
๐ฎ = Re(
๐ด ๐ (๐ง)ei(๐1๐กโ๐ค๐1 ๐ฑ))
, (1a)
2
1 1 1 d
able 1Material constants representative of several aluminum alloys. The third order moduliere determined by R. T. Smith et al. [47] from acoustoelastic measurements.Density Lamรฉโs Murnaghanโs
๐ ๐ ๐ ๐ ๐ ๐
2.7 gโcm3 58 24 โ311 โ401 โ408 GPaโ224 โ237 โ276โ202 โ305 โ300โ388 โ358 โ320โ337 โ395 โ436
๐ฎ2 = Re(
๐ด2๐2(๐ง)ei(๐2๐กโ๐ค๐2 ๐ฑ)
)
, (1b)
where the coordinate system ๐ฑ = (๐ฅ, ๐ฆ, ๐ง)๐ is such that the free surfacesof the plate are parallel to the (๐ฅ, ๐ฆ) plane, and with the convention๐1 โฅ ๐2 > 0. The wave vectors ๐ค๐ are contained in the (๐ฅ, ๐ฆ) plane.Both pairs (๐๐,๐ค๐) satisfy the dispersion relations of the plate andare associated with mode shapes ๐๐. From the non-linear interactionbetween ๐ฎ1 and ๐ฎ2 in the region where they intersect, small secondarywaves are produced whose amplitudes are proportional to the product๐ด1๐ด2. If furthermore (๐1,๐ค1) and (๐2,๐ค2) satisfy the phase-matchingcondition, that is, if their sum or difference
๐3 = ๐1 ยฑ ๐2, (2a)๐ค3 = ๐ค1 ยฑ ๐ค2, (2b)
is such that (๐3,๐ค3) satisfies the dispersion relations of the plate, thenone mode,1 called ๐ฎ3, emerges from the secondary field. Its amplitudegrows in proportion with the interaction time ๐ก:
๐ฎ3 = Re(
๐ด3(๐ก)๐3(๐ง)ei(๐3๐กโ๐ค๐3 ๐ฑ)
)
, (3a)
๐ด3(๐ก) = ๐ด1๐ด2 ๐พ(๐ก)1,2,3 ๐ก, (3b)
with ๐3 the mode shape related to (๐3,๐ค3), and ๐พ (๐ก)1,2,3 a coupling constantwhich depends on the triplet of modes and on the elastic constants ofthe material โ a derivation is given in Appendix A. In Eqs. (2), โโ+โโand โโโโโ signs correspond respectively to the โโsumโโ and โโdifferenceโโinteractions (๐ด2 must be replaced by its conjugate ๐ดโ
2 for the โโโโโ case).Eq. (3b) is often written as a function of the propagation distance. Herethe choice of the time variable is made necessary by the ZGV modes inwhich we are ultimately interested, which, because of their zero powerflux, would otherwise yield infinite terms.
The coupling constants ๐พ (๐ก)123 are zero when triplets of inadequatesymmetries are involved, prohibiting several families of interactions.A complete discussion can be found in [9,10]. For our specific case ofa target of P-SV symmetric type, the possible combinations are those oftwo antisymmetric or two symmetric modes, regardless their belongingto the P-SV or SH families.
2.2. System under interest
From here on in this theoretical section we consider a plate made ofa homogeneous and isotropic material, of thickness โ (see Fig. 2-d). Thematerial is modeled using elastic constants characteristic of aluminum(see Table 1). For such Poissonโs ratio (๐ = 0.35), the S1S2 ZGV mode (ormerely โโZGV modeโโ for short) is around (๐๐๐บ๐ , ๐๐๐บ๐ ) โ (0.93๐๐ , 0.25)ร2๐โโ, with ๐๐ the shear wave velocity.
1 Note that there could in principle be more than one mode satisfyinghe phase-matching condition simultaneously. Such exceptional cases are notiscussed here.
Ultrasonics 119 (2022) 106589P. Mora et al.
Fig. 1. (a) Schematic of the non-collinear interaction. (b) Photograph of the experimental setup. (c) Snapshot of the primary field arriving in the interaction region. (d)Fourier-transform of the field remaining after the interaction at the ZGV frequency. (e) Spectrum for piezos #1 and #2 activated alone or simultaneously.
Table 2Frequency ranges and minimum interaction angles for mode pairs of admissible types
giving solutions to the phase-matching condition for the generation of the S1S2 ZGVmode as a sum interaction.
Primary modes Frequency range(in units of 1
2๐๐๐บ๐ )
Min. angle
S0 โ S0 63%โ137% 127โฆ
A0 โ A0 73%โ127% 155โฆ
SH0 โ SH0 73%โ127% 149โฆ
S0 โ SH0 S0: 92%โ150%SH0: 50%โ108%
139โฆ
A0 โ A1 A0: 16%โ70%A1: 130%โ184%
132โฆ
A0 โ SH1 A0: 41%โ81%SH1: 119%โ159%
144โฆ
2.3. Phase matching condition
Here we seek the solutions to Eqs. (2) for the sum interaction whenthe target is the ZGV mode, i.e. (๐3, โ๐ค3โ) = (๐๐๐บ๐ , ๐๐๐บ๐ ), and givena phase direction that is arbitrarily set along the ๐ฆ axis, i.e. ๐ค3โโ๐ค3โ =(0, 1, 0)๐ .
Methodology. We first numerically obtain (see e.g. [48]) the dispersionrelations of the plate ๐๐(๐) and identify the modal branches that wewant to be considered as primary fields: e.g. the S0 branch for theS0 โ S0 โ ZGV case. We vary ๐ค1 on a square grid [โ๐๐๐๐ฅ, ๐๐๐๐ฅ] ร[โ๐๐๐๐ฅ, ๐๐๐๐ฅ] whose bounds ๐๐๐๐ฅ are empirically set large enough toobtain all solutions. At each test point of the grid, ๐ค2 = ๐ค3 โ ๐ค1 isdeduced from Eq. (2b), and the sum angular frequency ๐+ = ๐1(โ๐ค1โ)+๐2(โ๐ค2(๐ค1)โ) is stored. Finally, a contour finder [49] (the Marchingsquares algorithm) is called to find the iso-values ๐+(๐ค1) โ ๐3 = 0and returns the admissible ๐ค1 vectors, from which the correspondingadmissible ๐ค2, angular frequencies ๐1 and ๐2, and interaction anglesarccos (๐ค๐1 ๐ค2โโ๐ค1โ โ๐ค2โ) can be post-treated.
Results. The methodology is applied to all possible primary pairings.The results for the three most canonical cases A0โA0 โ ZGV, S0โS0 โZGV and SH0โSH0 โ ZGV are represented in Fig. 2. Solutions are alsofound to the more exotic cases S0 โ SH0 โ ZGV, A0 โ A1 โ ZGV andA0 โ SH1 โ ZGV and are summarized in Table 2. Other combinationsare either of wrong symmetry parity (e.g. A0 โ S0 cannot generate asymmetric mode) or start at frequencies above the target.
Fig. 2-a represents (thick lines) the portions of the fundamentalmodal branches where the phase-matching condition can be satisfiedby a non-collinear arrangement. Triangles mark the lowest and highestpossible frequencies and correspond to a 180โฆ interaction angle, whilecircles mark the case ๐1 = ๐2 where the interaction angle is minimal.Note that the interaction angle in this symmetric case can be easily
3
obtained by geometric considerations โ it equals 2 arccos(๐1โ๐3), with๐1 and ๐3 the phase velocities of primary and secondary waves. Thetwo ticks (blue line) correspond to ๐1, ๐2 = 1
2๐๐๐บ๐ ยฑ 20%. Fig. 2-brepresents the complete set of admissible primary wavevectors โ it isactually the raw output of the contour finder. To help understanding,the wavevectors corresponding to the points marked in Fig. 2-a aredrawn for the S0 โ S0 โ ZGV case. Finally, Fig. 2-c represents theangle that must be set between ๐ค1 and ๐ค2, as a function of the drivingfrequency ๐1.
Closed-form approximate solution. Note that, because the dispersion ofthe A0 and S0 modes is small in the allowed frequency range, relativelygood approximate results can be obtained by using the closed-formsolution to non-collinear interaction of bulk waves of speed ๐1 and ๐3with ๐1 โค ๐3 โ this solution is exact for the case of SH0 โ SH0 โ ZGV.The solution space of wavevectors is an ellipse with foci ๐ and ๐ค3, withextreme values ๐ค1 =
12๐ค3(1ยฑ
๐3๐1) at ๐1 =
12๐3(1ยฑ
๐1๐3). Using this formula
with the phase velocity of A0 or S0 taken at half the ZGV frequencypredicts bounds of 1
2๐๐๐บ๐ ยฑ 22% (exact: ยฑ27%) for the A0 โA0 โ ZGV
interaction, and of 12๐๐๐บ๐ ยฑ 44% (exact: ยฑ37%) for the S0 โ S0 โ ZGV
interaction.
2.4. Interaction coefficients
Here we give numerical values of the interaction coefficients ๐พ (๐ก)1,2,3appearing in Eqs. (3b) and (A.12).
Normalization of the mode shapes. The coefficients ๐พ (๐ก)1,2,3 are definedup to a factor that depends on the normalization of the mode shapes๐๐. We choose to adopt a definition that allows a comparison withthe well-known, scalar case of harmonic generation caused by twocollinear longitudinal bulk waves. A detailed discussion of this choiceis given in Appendix B. In a nutshell, we require the Lamb modeRe
(
๐ด๐๐๐(๐ง)ei(๐๐กโ๐ค๐ ๐ฑ)
)
to carry the same thickness-average mechanicalenergy as a plane bulk wave ๐ด๐ cos๐(๐กโ๐ฅโ๐๐), leading to 1
โ โซ โ0 โ๐๐โ
2d๐ง =1 for a homogeneous plate. We set the phase by requiring in-planecomponents to be pure real, with a non-negative value at ๐ง = 0 alongthe radial and azimuthal directions: ๐๐,๐ฅ|๐ง=0 โฅ 0 (P-SV) and ๐๐,๐ฆ|๐ง=0 > 0(SH) for ๐คโโ๐คโ = (1, 0, 0)๐ .
Generalized acoustic non-linear parameter. We introduce the follow-ing dimensionless coefficients to characterize the non-linear couplingwithin a given mode triplet:
๐ฝ1,2,3 =2๐ด3๐๐
๐ด1๐ด2๐1๐2๐ก, (4)
with ๐๐ the longitudinal wave velocity. Eq. (4) is meant to be appliedto measure ๐ฝ1,2,3 from a time signal, under a plane wave assumption.When applied to bulk longitudinal waves with ๐ โ ๐ , ๐ฝ coincides
1 2 1,2,3Ultrasonics 119 (2022) 106589P. Mora et al.
r
wssu
Zginw
wi
t๐ฝ
m
Fig. 2. Space of pairs ((๐1 ,๐ค1); (๐2 ,๐ค2)) solutions to the phase matching condition for S0 โ S0 โ ZGV, SH0 โ SH0 โ ZGV and A0 โA0 โ ZGV interactions. (a) Admissible frequencyranges (thick lines) superimposed on dispersion curves (thin lines). (b) Admissible wavevectors. Examples of pairs (๐ค1 ,๐ค2)S0 are drawn and the reflection across the point 1
2๐ค๐๐บ๐
elating them is represented by a thin dotted line. (c) Interaction angle as a function of the driving frequency of one of the two modes.
dZtwtiittn
3
(tt
ith the acoustic parameter of quadratic non-linearity ๐ฝ defined as atrain-hardening2 factor ๐ = ๐๐2๐ (๐ + ๐ฝ๐2), where ๐ is the stress, ๐ thetrain, and ๐ฝ = 3
2 + ๐+2๐๐+2๐ . Eq. (4) can be related to material parameters
sing Eq. (3b), giving ๐ฝ1,2,3 = 2๐พ (๐ก)1,2,3๐๐โ๐1๐2. Furthermore, as ๐พ (๐ก)1,2,3depends linearly on the Murnaghanโs constants ๐, ๐, ๐, we can write:
๐ฝ1,2,3 = ๐ฝ(0)1,2,3 +๐๐ฝ1,2,3๐๐
๐ +๐๐ฝ1,2,3๐๐
๐ +๐๐ฝ1,2,3๐๐
๐. (5)
The four coefficients that appear in the right-hand side of Eq. (5)depend only on the linear material constants (๐, ๐, ๐) and on the tripletof modes. Comparing again to bulk longitudinal waves, one obtains๐ฝ(0) = 3โ2, ๐๐๐ฝ = 1โ(๐ + 2๐), ๐๐๐ฝ = 2โ(๐ + 2๐) and ๐๐๐ฝ = 0.
Conversion factor for measurements of the out-of-plane component. In thiswork the observable is the surface out-of-plane displacement ๐ข๐ง|๐ง=0. Itis therefore convenient to use the alternative normalization (superscript(๐ง)) ๐ฎ = Re
(
๐ด(๐ง)๐ ๐(๐ง)
๐ (๐ง)ei(๐๐กโ๐ค๐ ๐ฑ))
such that ๐ (๐ง)๐,๐ง |๐ง=0 = 1 and ๐ด(๐ง)
๐ is themeasured amplitude. We call
๐ผ1,2,3 =๐1,๐ง๐2,๐ง
๐3,๐ง
|
|
|
|
|๐ง=0(6)
the conversion factor to the former normalization such that Eq. (4)expresses as:
๐ฝ1,2,3 = ๐ผ1,2,32๐ด(๐ง)
3 ๐๐
๐ด(๐ง)1 ๐ด(๐ง)
2 ๐1๐2๐ก. (7)
Results. The coefficients appearing in Eq. (5) are represented in Fig. 3for the interactions S0 โ S0 โ ZGV, A0 โ A0 โ ZGV and SH0 โ SH0 โ
GV, in the allowed frequency ranges. Selected numerical values areiven in Table 3, along with the conversion factor (note that ๐ผ1,2,3s pure imaginary). The derivatives with respect to ๐, ๐ and ๐ areormalized by the P-wave modulus ๐ + 2๐ to facilitate a comparisonith ๐ฝ (๐ฝ = โ5.1 to โ9.1 with our set of constants).
Let us explain on an example how to read Fig. 3. Suppose oneants to calculate the theoretical value of ๐ฝ1,2,3 for the S0 โ S0 โ ZGV
nteraction, with equal primary frequencies ๐1 = ๐2 = 12๐๐๐บ๐ . Then,
one has to read on Fig. 3-a at abscissa 1 the values of the solid (โ1.44)and dashed curves (โ2.40, โ1.61, 0.31) โ also reported in Table 3.These three latter values must then be multiplied respectively by ๐, ๐and ๐, and divided by ๐ + 2๐. Finally, ๐ฝ1,2,3 is obtained by summinghese four values according to Eq. (5). Now, if one wants to calculate1,2,3 for another combination of primary frequencies, the procedure is
similar but the curves must be read at another abscissa by taking either๐ = ๐1 or ๐ = ๐2.
2 Note that several definitions coexist in the literature and often include ainus sign (i.e. measuring softening for positive strains) and/or a ร2 factor.
4
e
Fig. 3. Normalized non-linear coefficients for the three canonical combinations. Thenon-linear acoustic parameter ๐ฝ1,2,3 can be obtained from a set of curves by applyingEq. (5).
As expected, these three interactions exhibit complementary sensi-tivities to ๐, ๐, ๐. As Murnaghanโs constants are usually of the samesign, the SH0 โ SH0 โ ZGV interaction is the one with weakesttransfer rate, because it depends on ๐ and ๐ with similar magnitudesbut opposite signs and has zero component along ๐. On the other hand,the S0 โ S0 โ ZGV interaction sums contributions of ๐ and ๐ while itepends weakly on ๐, and is therefore the strongest one. The A0โA0 โ
GV interaction, although somewhat weaker, is in turn interesting inhat its relative sensitivities to the three constants change significantlyith the frequency, such that, in principle, it seems possible to infer
hem all from experiments repeated at different driving frequencies andnteraction angles. Another noteworthy characteristic of this interactions that the geometric non-linear contribution ๐ฝ(0)1,2,3 is quite small athe extreme limits of the frequency range, i.e. for a 180โฆ interac-ion angle: this point is the most sensitive choice to weak materialon-linearities.
. Experiments
Three experiments performed on a 1.5 mm thick aluminum platealloy Al 5754-H111, lateral dimensions 20 ร 30 cm2) are reported inhe following sections. Prior to these experiments, the exact value ofhe S1S2 ZGV resonance in the plate was determined using a classicalxperimental setup โ we measured ๐ = 1.98 MHz.
๐๐บ๐Ultrasonics 119 (2022) 106589P. Mora et al.
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Table 3Selected theoretical values of the coefficients from which the non-linear acoustic parameter ๐ฝ1,2,3 can be calculated, for the three canonical combinations, and of the conversionactor ๐ผ1,2,3 for measurements of the out-of-plane displacement (see Eqs. (5) and (6)). Ranges for ๐ฝ1,2,3 are minโmax values from the sets of third order moduli given in Table 1hich represent different alloys.
Primary modes Configuration ๐ฝ(0)1,2,3 (๐ + 2๐)ร{
๐๐ฝ1,2,3๐๐
๐๐ฝ1,2,3๐๐
๐๐ฝ1,2,3๐๐
}
๐ผ1,2,3 Typical ๐ฝ1,2,3 in aluminum
S0 โ S0๐1 =
12๐๐๐บ๐
โจ๐ค1 ,๐ค2โฉ = 127โฆโ1.44 โ2.40 โ1.61 0.31 โ0.39 i 6.4 to 11.9
๐1 =12๐๐๐บ๐ ยฑ 37%
โจ๐ค1 ,๐ค2โฉ = 180โฆโ2.64 โ2.80 โ3.01 0 โ0.36 i 10.0 to 17.8
A0 โ A0๐ฟ๐1 =
12๐๐๐บ๐
โจ๐ค1 ,๐ค2โฉ = 155โฆโ1.15 0.66 0.10 โ1.23 โ0.99 i โ0.2 to 1.4
๐1 =12๐๐๐บ๐ ยฑ 27%
โจ๐ค1 ,๐ค2โฉ = 180โฆโ0.19 0.68 0.73 0 โ1.0 i โ3.3 to โ5.2
SH0 โ SH0๐ฟ๐1 =
12๐๐๐บ๐
โจ๐ค1 ,๐ค2โฉ = 149โฆโ0.28 0 โ0.43 0.28 0.0 to 0.3
๐1 =12๐๐๐บ๐ ยฑ 27%
โจ๐ค1 ,๐ค2โฉ = 180โฆโ0.36 0 โ0.90 0.60 0.1 to 0.9
Fig. 4. Schematic of the experimental setup common to Exps. #1 and #2. The primaryfield is generated by two contact transducers fed with high voltage. The surfaceout-of-plane displacement is recorded as a point detection over a 2D grid.
Setup. Fig. 4 represents the experimental setup that is common to Exps.#1 and #2, and of which Exp. #3 only slightly differs. Two independenttonebursts at frequencies ๐1 and ๐2 close to 0.99 MHz are sent by asignal generator (Tektronix AWG-3022C) into a high power gated am-plifier (RITEC SNAP 5000), and then fed into piezoelectric disks (25 mmdiameter, 2 mm thickness) glued on heads. The heads (wedges for Exps.#1 and #3, metallic footprints for Exp. #2, see Figs. 6-a and 7-a) aredesigned to select a desired Lamb mode and are themselves glued ontothe aluminum plate. The plate is mounted on a 2D translation stage(Newport UTS50CC and SMC100CC) that can span 5 cm on each axis atmicrometric precision. Waves are detected with a laser interferometer(Tempo 1D, Bossa Nova Technologies; bandwidth of 20 kHz โ 1 GHz)that measures the absolute out-of-plane displacement on a focused spotthat is pointwise regarding the millimetric wavelenghts involved. Thisdevice is chosen for its very good sensitivity as well as for allowingnon-contact detection, which is essential to detect a ZGV mode withoutdramatically affecting it. It has however a drawback in that it saturatesfor peak amplitudes above 5 โ 6 nm, levels that are easily reachedwith our transducers when driven at amplified voltages. Digitization(PicoScope 5444D) is done on 15 bits at a 15 MHz sampling rate andsignals are averaged 40 (for 2D scans) to 80 times (at single points)to improve the signal to noise ratio (SNR). The piezoelectric disks areoperated near a thickness resonance and can hence be considered asnarrow-band transducers. Experiments #1 and #2 operate at voltagesabout 200 Vpp (i.e. 100 Wโchannel) on 50 cycles, while Exp. #3 doesnot use the amplifier and operates at 20 Vpp (i.e. 1 Wโchannel) on 2000
5
cycles.
3.1. Experiment #1: S0 โ S0 โ ZGV
We report in this section a first experiment in which the ZGV modeis generated using the S0 โ S0 โ ZGV interaction.
Methodology. Wedges made of Nylon (type PA6) are manufactured toselectively excite the S0 mode. SnellโDescartesโ law (2.7 mmโ๐s for thelongitudinal wave in Nylon, and 5.0 mmโ๐s phase velocity for the S0Lamb mode at 1 MHz) predicts an optimal angle of incidence of 33โฆ.According to the predictions of Section 2.3, the wedges are glued on theplate to form an angle of 127โฆ. The driving frequencies are set with asmall difference to facilitate the distinction between input and samplenon-linearity: ๐1 = 1.09 MHz and ๐2 = 0.89 MHz (i.e. ยฑ10%). This small|๐1 โ ๐2|โ2 does not affect significantly the interaction angle which isalmost constant around |๐1 โ ๐2|โ2 = 0 (see Fig. 2-c).
Results and discussion. Results are shown in Figs. 5 and 6.A typical signal recorded at the center of the region where beams #1
and #2 cross is represented in Fig. 5-a. It consists of two parts. Duringthe first tens of ฮผs the interferometer detects the intense, primaryS0 field, and saturates โ the inset shows in thinner line the actualamplitude reconstructed from an acquisition at lower driving voltage.For harmonics analysis, this first part is blind. Once the 50 cyclestrains are gone, the interferometer detects a much lower field, mostof which is linear and composed of unwanted, direct A0 modes, andof a diffuse A0 + S0 field due to edges reflections. Fig. 5-b showsa timeโfrequency analysis of the signal. Different spectral contribu-tions appear after the saturated part. Of those, 2๐1, 2๐2, 3๐1 and 3๐2are thought to be mainly due to the wedge/plate and wedge/piezoadhesive bonds because they can vary from absent to strong whenunmounting/remounting the wedges and repeating the measurement.The component at ๐1 + ๐2 = 1.98 MHz = ๐๐๐บ๐ distinguishes fromall others in that, besides being detectable only when both piezo areactivated, it is much more monochromatic and of almost constantamplitude over long times โ remember that 1000 ฮผs represent nearly2000 periods. This clearly indicates that we are observing the expectedphenomenon. Fig. 5-c shows a spectrum of the non-saturated part ofthe signal (the time window is depicted in black in Fig. 5-a) togetherwith spectra of #1 and #2 alone, allowing the previous remark to beappreciated more quantitatively. Then, Figs. 5-d to -g show how thepeak amplitude at ๐1+๐2 varies along with several control parameters.One indeed observes a quadratic trend with the driving voltage (Fig. 5-d) and a linear trend with the number of cycles (Fig. 5-e). Fig. 5-f givesan idea of how sharp the effect is in matching ๐1 + ๐2 with ๐๐๐บ๐ ,although strictly speaking the peak width depends also critically onthe number of input cycles. The fact that no harmonics is generatedfor ๐1 + ๐2 < ๐๐๐บ๐ is not surprising because the S1 โ S2 branch isnot propagative โ and as such cannot accumulate energy over long
excitations, and other interactions S0โS0 โ S0,SH0,A0,SH1,A1 are notUltrasonics 119 (2022) 106589P. Mora et al.
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possible due to wrong symmetry pairings or absence of solution to thephase matching condition. However, as emphasized by the asymmetryof the peak, the side ๐1 + ๐2 > ๐๐๐บ๐ requires another explanation.We think that several adverse features combine to this absence ofdetection, namely a window effect and two physical reasons to whyhigher frequencies should be generated with much less efficiency:
โข as we are blind during forcing, we only observe whatever remainswithout traveling (the ZGV mode) or comes back as a smaller,diffuse field,
โข the group velocity becomes significant, so the secondary wavesoon leaves the interaction area without accumulating much en-ergy,
โข there is a strong mismatch in the phase matching condition:because the dispersion relation ๐ (๐) is flat at ๐๐๐บ๐ , increasingslightly ๐1 + ๐2 means a large change on โ๐ค3โ, and hence on theinteraction angle.
Fig. 5-g shows a study on the influence of the difference |๐1 โ ๐2|โ2.It can be seen that despite the relatively high level of input non-linearity observed at |๐1 โ ๐2|โ2 = 0 (for which 2๐1 = 2๐2 = ๐1 +๐2, thus comprising second harmonic generation at 2๐1 and 2๐2 fromeach excited beam, and the non-collinear wave mixing effect ๐1 + ๐2exciting the ZGV mode), the subtraction step (#1) โ #1 โ #2 (redsquares) is effective in recovering at good precision the true amplituderesulting only from the non-linear and non-collinear wave-mixing effectat ๐1+๐2 and allows to isolate the sample non linearity. We rely on thisconclusion for Exp. #2 described in the following section.
Finally, (๐ฅ, ๐ฆ) scans (4 ร 4 cm2 sampled in 64 ร 64 points) areresented in Fig. 6 as space and wavenumber analyses. Fig. 6-b showssnapshot of the primary field at early times, when arriving in the
nteraction region. Fig. 6-c shows the wavenumber-frequency contentf the primary field at the driving frequencies, reconstructed fromower driving voltages to avoid saturation biases, and time windowedo select only the incident train. Theoretical dispersion relations areuperimposed in thin colored circles to facilitate interpretation. Then,igs. 6-d and -e show the late field (i.e. for ๐ก > 120 ฮผs to excludeaturation) filtered at the ZGV frequency. When both piezos are active,he observed field is indeed dominated by the expected ZGV modeith an almost purely space-harmonic content, and is maximal at the
ocation where the interaction occurred.
.2. Experiment #2: A0 โ A0 โ ZGV
We report in this section a second experiment that is similar to therevious one except that it explores the A0 โ A0 โ ZGV interaction.
ethodology. Comb type heads consisting of a smooth (top) and aachined (bottom) faces are manufactured in brass disks (see Fig. 7-
). Piezoelectric disks are glued on the smooth faces and operatedear a thickness resonance. The machined faces are glued to the platend transmit an out-of-plane excitation distributed in a pattern thatatches a A0 planar wavefront (2.6 mm at 1 MHz), thus selectively
xciting this mode (see Fig. 7-b,c). The heads are oriented to formn angle of 155โฆ between the preferential directions of the waveseing generated, according to the predictions of Section 2.3. The rearurroundings of each transducer are covered with adhesive paste withhe twofold purpose to dampen the strong specular rays coming fromhe neighboring edges, and to reduce the resonant behavior of theeads. Relying on the good promises of the subtraction processing (seeig. 5-g), the driving frequencies are set equal: ๐1 = ๐2 = 0.99 MHz.
esults. Fig. 7 shows (๐ฅ, ๐ฆ) scans of the wave field (4 ร 4 cm2 sampled in0 ร 80 points). For details in the interpretation, the reader is referred
to the previous section and to the very similar Fig. 6. Here again, thedetection of the ZGV mode resulting from the non-collinear interactionis demonstrated.
6
.3. Measured non-linear acoustic parameters
The theoretical framework presented in Section 2.4 and Appendix As restricted to plane waves. However, the experiments involve beams ofinite width. The S1S2 ZGV mode is quite dispersive, and hence prone toiffraction attenuation. When generated by a brief point or line sourceuch as a laser impact, diffraction dominates over material dampingt early times, resulting in a fast ๐กโ1โ2 decay [35]. Here the context isomewhat different in that the interaction zone acts as a distributedource with a moving phase, but we do notice a dependence on theime window used to extract the ZGV amplitude. This dependencean be well observed in Fig. 8-f on the part labeled free decay โ seeection 3.4.
There are two known ways to cope with diffraction and damping.he first and best way is to fit the observed spaceโtime variationsith a propagation model. It is well documented for collinear gener-tion [2], however, methodological developments are still needed foron-collinear mixing. The second way is to use only the slope of theariations, taken as close to the source as possible. This is the approachetained here. Going back to Fig. 8-f, and also to Fig. 5-e, variationsppear well linear over time segments of at least 50 ฮผs, as the planeave model predicts it. In a collinear setup this is a good indication
hat attenuation biases are limited on these scales.
ethodology. The ZGV amplitude is measured by time-windowing over0 ฮผs just after the interaction has ceased. For instance, the blackindow drawn in Fig. 5-a is replaced with a window going from 80 ฮผs
o 130 ฮผs. The signals are then 3D-Fourier-transformed and |๐ด3| isead at the ZGV frequency as the max value in the (๐๐ฅ, ๐๐ฆ) space. Themplitudes |๐ด1| and |๐ด2| are obtained similarly by time-windowing toelect the interaction (e.g. from 30 ฮผs to 70 ฮผs in Fig. 5-a), 3D-Fourier-ransforming, and reading the max values at the driving frequencies ๐1nd ๐2. Finally, Eq. (7) is applied by taking ๐๐ = 6270 mโs, ๐ก = 50 ฮผss excitation duration; ๐1 = ๐2 = 2๐ ร 0.99 MHz and |๐ผ1,2,3| = 0.99 forhe A0 โ A0 โ ZGV case; ๐1 = 2๐ ร 0.89 MHz, ๐2 = 2๐ ร 1.09 MHz and๐ผ1,2,3| = 0.39 for the S0 โ S0 โ ZGV case.
esults. We obtained |๐ด1| = 8.9 nm, |๐ด2| = 5.4 nm and |๐ด3| = 13.4 ร0โ3 nm for the S0 โ S0 โ ZGV experiment, giving |๐ฝexpS0 ,S0 ,ZGV
| = 0.7,nd |๐ด1| = 11.1 nm, |๐ด2| = 10.9 nm and |๐ด3| = 19.8 ร 10โ3 nm for the0 โ A0 โ ZGV experiment, giving |๐ฝexpA0 ,A0 ,ZGV
| = 1.1. Compared to thelausible values estimated in the theoretical analysis ๐ฝ thS0 ,S0 ,ZGV = 6.4o 11.9 and ๐ฝ thA0 ,A0 ,ZGV
= โ0.2 to 1.4 (see Table 3), the agreements very good for the A0 โ A0 โ ZGV experiment while it is bad forhe S0 โ S0 โ ZGV one. We currently lack a satisfactory explanation,lthough finite-beam effects could have been naively discarded despitehe justification given above, and although this comparison is onlyndicative because the model material constants (๐, ๐, ๐) represent otherluminum alloys.
.4. Experiment #3: generation at low amplitudes
ethodology. We go back to the S0 โ S0 โ ZGV interaction and usethe same setup as for Exp. #1, except that the amplifier is now removedsee Fig. 8-a). The outputs of the signal generator are directly connectedo the piezoelectric disks (with 50 ฮฉ loads) glued on the wedges. Theriving frequencies ๐1 and ๐2 are set with a small difference ๐2 โ ๐1 =
10 kHz (i.e. ยฑ0.5%) around half the ZGV frequency. The generator isable to deliver up to 20 Vpp into 50 ฮฉ on each channel. Compared toExp. #1, the input voltages are lower by a factor of roughly 10. Non-linearly generated amplitudes are then expected to be reduced by afactor of โ 100. To compensate these lower levels, we increase thenumber of cycles from a few tens to 2000 (i.e. โ ร40). This clearly doesnot balance the drop entirely, but proves to generate amplitudes highenough to be unambiguously detected with a few tens averages.
Ultrasonics 119 (2022) 106589P. Mora et al.
Fig. 5. Typical signals at the center of the crossing region for the S0 โ S0 โ ZGV interaction. (a) Time signal. (b) Timeโfrequency analysis. (c) Spectrum of the non-saturatedpart (๐ก > 120 ฮผs) for piezos #1 and #2 activated alone or simultaneously. (d, e, f, g) Peak amplitude at the sum frequency when varying (d) the driving voltage of both piezos,(e) the number of cycles of the tonebursts, (f) the sum frequency ๐1 + ๐2 while keeping constant (๐1 โ ๐2)โ2 = 0.1 MHz, (g) the difference |๐1 โ ๐2|โ2 while keeping constant๐1 + ๐2 = 1.98 MHz.
Fig. 6. 2D view of the S0 โ S0 โ ZGV interaction. (a) Photograph of the setup showing the wedge transducers and the laser interferometer. (b) Snapshot of the primary fieldarriving in the interaction region. (c) Wavenumber content of the primary field at the driving frequencies. (d) Real part of the field at the ZGV frequency (windowed to ๐ก > 120 ฮผsto exclude saturation, see Fig. 5-a). (e) Wavenumber content of (d).
Fig. 7. 2D view of the A0 โ A0 โ ZGV interaction. (a) Photograph of the setup showing the brass heads with A0-like footprints, with red arrows representing outgoing waves.(b) Snapshot of the primary field arriving in the interaction region. (c) Wavenumber content of the primary field at the driving frequencies. (d) Real part of the field at the ZGVfrequency (windowed to ๐ก > 120 ฮผs to exclude saturation, see Fig. 5-a). (e) Wavenumber content of (d).
Results and discussion. The results presented in Fig. 8 were recorded
7
with 50 averages. Because of the lower amplitudes, the interferometer
no longer saturates during excitation. This makes this early stage fully observable and enables a more complete picture.Ultrasonics 119 (2022) 106589P. Mora et al.
Fig. 8-b shows a few snapshots of the field (real part) duringand after excitation with a timeโfrequency processing centered at theZGV frequency. One can clearly see that a wave pattern progressivelyappears until ๐ก = 2000 ฮผs, that is, when the excitation stops, and thenvanishes. Fig. 8-c shows a typical signal recorded at the center of theobservation window, with a peak amplitude about 0.8 nm when bothtransducers are emitting. The beatings in the first 2000 ฮผs are due tothe small difference between the driving frequencies. The spectrumof this entire signal is shown in Fig. 8-d: as emphasized by the insetaround 2 MHz no peaks are measured at 2๐1 or 2๐2, meaning that thenon-linearity of the receiving system is now indeed too small to beobserved. The peak at ๐1 + ๐2 (1.8 10โ3 nm peak amplitude) has, inturn, a SNR of about 20 dB. The same signal is then shown in Fig. 8-ein a timeโfrequency representation, with a focus on the ZGV frequencyโ the magnitude at the driving frequencies is also plotted in graylines. A tendency of linear increase with time can indeed be observedduring forcing, followed by a slow, free decay. Finally, Fig. 8-f showsthe same timeโfrequency analysis applied to the data transformed intothe wavenumber domain (๐ฅ, ๐ฆ) โ (๐๐ฅ, ๐๐ฆ). The black line represents themagnitude of the point (๐๐๐บ๐ ,๐ค๐๐บ๐ ) as a function of time, while graylines show the values at neighboring points (the solid/dashed curvesstand for minus/plus a given percentage in the wavenumber ๐). Thetime dependence can this way be observed more accurately โ theFourier transform on 64 ร 64 points can improve the SNR by up to+36 dB for a pure harmonic pattern. One can now observe a trendtowards saturation in the growth. It is also noteworthy that the decayrate results faster for greater (dashed lines) than for smaller (solid lines)wavenumbers. An explication could be that greater wavenumbers areindeed more dispersive than smaller ones, although it is not clear tous whether viscous attenuation also contributes quantitatively to thisdifference.
Before closing this section, we should not disregard the fact thatthe very attractive prospects offered by this low power setup comeat the cost of technical difficulties that are common to resonanceexperiments. Indeed, the price to pay is that the driving frequenciesmust be adjusted with extreme care, because a target error of one partover 2000 (the number of cycles) now significantly degrades the phasematching condition. Such required accuracy starts to be sensitive totemperature effects: this was not an issue for scans running over acouple hours, but we did observe variations with overnight fluctuations(about ยฑ2โฆ in the room). Another limitation could be attenuation. Herethe large number of cycles was still well below the quality factor of theZGV mode, which often exceeds 104 in metal plates, so that even morecycles would have yielded higher amplitudes. But in more dissipativemedia, compensating low amplitudes with long excitations is obviouslya more limited strategy.
4. Conclusion
The non-linear generation of the S1S2 ZGV Lamb mode at theintersection of two high amplitude beams of lower frequency has beenanalyzed theoretically and experimentally in an aluminum plate. Themethodology followed can be transposed to other materials, to thedifference interaction, and to other targeted modes. A new derivationof the coupling coefficients was given that is suited to an observationof the interaction as a function of time, and is thereby adapted tozero group velocity modes. Theoretical values of these coefficients werediscussed for the most canonical cases. The experimental realizationsinvolved the fundamental symmetric and antisymmetric Lamb modes,which are relatively easy to excite selectively and efficiently. A com-plete spaceโtime observation of the phenomenon was achieved in anexperiment operated at low power, from the emergence of the modeduring forcing to its free decay. The main features of the interactionwere observed, such as non-trivial interaction angle, quadratic depen-dence with the driving voltage, or linear dependence with the numberof excitation cycles.
8
In the context of harmonic generation, the specificity of zero groupvelocity modes is that energy can be accumulated by simply launchinglonger primary waves into an intersection area whose diameter can beas small as a few plate thicknesses. In contrast, in configurations thatproduce propagating modes, accumulating energy requires to increasethe interaction area, which dilutes information and soon faces triviallimitations related to the geometrical constraints of the actual system.Taking advantage of this appealing property, we showed that targetingsuch a mode enables to generate detectable non-linear effects usinginput powers up to two orders of magnitude lower than when targetingother modes.
Zero group velocity modes appear as promising tools based onwhich novel non-destructive characterization methods can be devel-oped to assess locally the non-linear properties of a plate. Amongpossible applications, we believe that they could be relevant to evaluatethe bonding quality of an adhesive joint. They could also be a route toachieve harmonic generation with a fully non-contact system using atime-modulated laser excitation.
Codes availability
Python programs for finding solutions to the phase-matching con-dition and for obtaining theoretical values of the non-linear parameterare available upon request.
Declaration of competing interest
The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
Acknowledgments
P. Mora acknowledges funding from the EUR-IAGS post-doctoralprogram. We thank H. Mรฉziere, J. Marudai and E. Egon for manufac-turing the transducers. Thanks to J. Blondeau and S. Letourneur fortheir support in setting up the experiments, and to R. Hodรฉ for runningpreliminary characterization measurements.
Appendix A. Derivation of the interaction coefficients
The interaction coefficients appearing in Eq. (3b) are usually de-rived in the literature in the frequency domain, by expanding in amodal series the small wave field resulting from the non-linear inter-action. Here, we take a slightly different approach in that the modalseries is expressed in the wavenumber domain to allow for an analyticaltreatment of the time variable. Indeed, this time-domain resolutionis necessary because Zero Group Velocity modes yield singularitieswhen viewed in the frequency domain โ their forced response growsunbounded with time as energy accumulates without traveling.
We start by briefly recalling the equations of motion for finiteamplitudes in a waveguide as they can be established in the pertur-bative approximation. More details can be found in e.g. [50โ52] forthe collinear case, and in [6,9] for a formalism more suited to thenon-collinear case.
Let ๐ฑ = (๐ฅ, ๐ฆ, ๐ง)๐ refer to the so called โโnaturalโโ coordinates, i.e.attached to the system at rest. The system under interest is a traction-free plate that is unbounded in the (๐ฅ, ๐ฆ) plane and comprised between๐ง = 0 and ๐ง = โ (see Fig. 2-d). Two wave trains of finite amplitude,labeled #1 and #2 further on, are generated within the plate. We call๐ฎ(๐ฑ) the total displacement field produced by these waves and theirinteraction. Perturbation theory starts by expanding ๐ฎ as a sum of termsof different order:
๐ฎ = ๐ฎ1 + ๐ฎ2 + ๐ฎ11 + ๐ฎ22 + ๐ฎ12, (A.1)
Ultrasonics 119 (2022) 106589P. Mora et al.
Fig. 8. Low power generation of the S0 โS0 โ ZGV. (a) Schematic of the experimental setup (Exp. #3). (b) Snapshots of the field (real part) at the ZGV frequency processed witha timeโfrequency filter. (c) Typical signal (out-of-plane displacement). (d) Spectrum of (c) (entire signal). (e) Timeโfrequency analysis of (c) at the ZGV frequency (blue) and at thedriving frequencies (grays). (f) Timeโfrequency analysis in the wavenumber domain at the ZGV frequency: at ๐ค = ๐ค๐๐บ๐ (black), ๐ค = ๐ค๐๐บ๐ (1โpct) (solid grays) and ๐ค = ๐ค๐๐บ๐ (1+pct)(dashed grays).
where ๐ฎ1 and ๐ฎ2 (order 1) denote the primary fields, and ๐ฎ11, ๐ฎ22 and๐ฎ12 (order 2) denote fields resulting from non-linear interactions: ๐ฎ11refers to the self-interaction of wave #1, ๐ฎ22 to the self-interaction ofwave #2, and ๐ฎ12 to the mutual interaction of waves #1 and #2. Note that๐ฎ12 is the same as ๐ฎ3 in Section 2.1 when the phase matching conditionis satisfied and a single mode dominates. We shall focus only on thislatter term further on. For ๐ฎ12, the balance of forces and boundaryconditions read, in absence of external load:
๐๐2๐ก ๐ฎ12 โ ๐๐ข๐ฏ ๐๐ฟ(๐ฎ12) = ๐๐ฃ๐๐ , (A.2a)
๐๐ฟ(๐ฎ12) โ ๐ง๐ง = ๐๐ ๐ข๐๐ for ๐ง = 0, โ. (A.2b)
In Eqs. (A.2) ๐ง๐ง is the unit vector along the ๐ง axis, oriented up-wards, and ๐๐ฟ(๐ฎ12) = ๐ โถ ๐๐ฎ12 is the linear part of the first PiolaโKirchhoff stress tensor referring to ๐ฎ12, with ๐ the 4th-rank stiffnesstensor. The source terms appearing in the right hand side are ๐๐ฃ๐๐ =๐๐ข๐ฏ ๐๐๐ฟ(๐ฎ1,๐ฎ2, 2) and ๐๐ ๐ข๐๐ = โ๐๐๐ฟ(๐ฎ1,๐ฎ2, 2) โ ๐ง๐ง, with ๐๐๐ฟ(๐ฎ1,๐ฎ2, 2) thenon-linear part of the first PiolaโKirchhoff stress tensor of the total field๐ฎ that gathers products between ๐๐ฎ1 and ๐๐ฎ2 (the last argument ฮต2ฮตemphasizes the truncation order).
We assume that the material is hyperelastic, with a classical,quadratic non-linear component. In the case of an isotropic medium,the elastic energy density function can be written as:
= ๐2tr(๐)2 + ๐tr(๐2) + ๐ด
3tr(๐3) + ๐ตtr(๐)tr(๐2)
+ ๐ถ3tr(๐3) + ๐(๐4). (A.3)
In Eq. (A.3) ๐ = 12 [๐๐ฎ + ๐๐ฎ๐ + ๐๐ฎ๐๐๐ฎ] is the Lagrangian strain
tensor, ๐ and ๐ are Lamรฉโs constants, and ๐ด, ๐ต and ๐ถ are Landau andLifshitzโs third order elastic constants โ we recall the relations withMurnaghanโs constants: ๐ = ๐ต + ๐ถ, ๐ = 1
2๐ด + ๐ต and ๐ = ๐ด. In thiscontext, it can be established [52] that:
9
๐๐๐ฟ(๐ฎ1,๐ฎ2, 2) = ๐tr(๐๐ฎ2)๐๐ฎ1 + ๐๐๐ฎ1(๐๐ฎ2 + ๐๐ฎ๐2 )+ ๐tr(๐๐ฎ1)๐๐ฎ2 + ๐๐๐ฎ2(๐๐ฎ1 + ๐๐ฎ๐1 )+ ๐tr(๐๐ฎ๐1 ๐๐ฎ2)๐+๐(๐๐ฎ๐1 ๐๐ฎ2 + ๐๐ฎ๐2 ๐๐ฎ1)+ 2๐ถtr(๐๐ฎ1)tr(๐๐ฎ2)๐+๐ตtr(๐๐ฎ1)(๐๐ฎ2 + ๐๐ฎ๐2 )+๐ตtr(๐๐ฎ2)(๐๐ฎ1 + ๐๐ฎ๐1 )
+ ๐ต2tr(๐๐ฎ1๐๐ฎ2 + ๐๐ฎ2๐๐ฎ1 + 2๐๐ฎ๐1 ๐๐ฎ2)๐
+ ๐ด4(๐๐ฎ1๐๐ฎ2 + ๐๐ฎ2๐๐ฎ1 + ๐๐ฎ๐1 ๐๐ฎ
๐2
+๐๐ฎ๐2 ๐๐ฎ๐1 + ๐๐ฎ๐1 ๐๐ฎ2 + ๐๐ฎ๐2 ๐๐ฎ1
+๐๐ฎ1๐๐ฎ๐2 + ๐๐ฎ2๐๐ฎ๐1 ), (A.4)
in which ๐ is the identity tensor of order 2 and rank 3. Eq. (A.4) andthe results below are also valid for arbitrary variations of the elasticconstants with depth. The case of an anisotropic medium requires tomodify Eqs. (A.3) and (A.4) accordingly, but the further derivation stillstands.
We now take the primary fields to be monochromatic and composedof a single mode each:
๐ฎ1 = Re(
๐ด1๐1(๐ง)ei(๐1๐กโ๐ค๐1 ๐ฑ)
)
, (A.5a)
๐ฎ2 = Re(
๐ด2๐2(๐ง)ei(๐2๐กโ๐ค๐2 ๐ฑ)
)
, (A.5b)
in which we assume ๐1 โฅ ๐2 > 0. In Eqs. (A.5) the amplitudes๐ด1 and ๐ด2 are defined depending on the normalization chosen forthe modeshapes ๐1 and ๐2. This choice is postponed to Appendix B.The driving sources are composed of two parts, labeled with super-script ยฑ to indicate respectively sum and difference interactions: ๐๐ฃ๐๐ =Re
(
12๐ด1๐ด2๐ ยฑ
๐ฃ๐๐ei[(๐1ยฑ๐2)๐กโ(๐ค1ยฑ๐ค2)๐ ๐ฑ]
)
(๐ด2 must be replaced by its con-jugate ๐ดโ for the โโโโโ case), and similarly for ๐ . We recall (see
2 ๐ ๐ข๐๐Ultrasonics 119 (2022) 106589P. Mora et al.
m
๐
fp
๐น
Twโaln
at
w
wtee
Itt
A
Atโ
swdc
ita๐
e.g. [53,54]) that when Fourier-transformed into the wavenumber do-ain (๐ฅ, ๐ฆ) โ (๐๐ฅ, ๐๐ฆ), the causal Greenโs tensor of Eqs. (A.2) reads:
(๐ง, ๐งโฒ, ๐ก) =โ
๐๐๐(๐ก)
๐๐(๐ง)๐๐ป๐ (๐งโฒ)
โซ โ0 ๐โ๐๐โ
2d๐ง, (A.6a)
๐๐(๐ก) = (๐ก)sin ๐๐๐ก๐๐
, (A.6b)
where ๐งโฒ is the depth location of the impulsive source, ๐ป is the her-mitian (or conjugate-transpose) operator, is the Heaviside unitstepfunction, and ๐๐(๐ค, ๐ง) is the modeshape of the ๐th Lamb mode, asso-ciated with the eigen-angular frequency ๐๐(๐ค) โฅ 0. Expressed over theLamb modes of the plate, the resulting field is then of the form:
๐ฎ12 = Re
(
โ
๐๐ดยฑ๐ (๐ก)๐
ยฑ๐ (๐ง)e
โi(๐ค1ยฑ๐ค2)๐ ๐ฑ
)
, (A.7)
with ๐ยฑ๐ = ๐๐(๐ค1ยฑ๐ค2). The superscripts ยฑ and the substitution ๐ด2 โ ๐ดโ
2or the โโโโโ case are from now on considered as implicit. The modalarticipation factors divided by the primary amplitudes are:
๐(๐1,๐2,๐๐) = ๐น ๐ฃ๐๐๐ + ๐น ๐ ๐ข๐๐
๐ , (A.8a)
๐น ๐ฃ๐๐๐ = 1
2 โซ
โ
0๐๐ป๐ ๐ ๐ฃ๐๐(๐1,๐2)d๐ง, (A.8b)
๐น ๐ ๐ข๐๐๐ = 1
2๐๐ป๐ ๐ ๐ ๐ข๐๐ (๐1,๐2)|โ0 . (A.8c)
As we are not interested in rigid body motions ๐๐ = 0, the normalizingterm in Eq. (A.6a) can be more conveniently written as the mean modalenergy by multiplying by the halved squared angular frequency:
๐(๐๐, ๐๐) =12๐2๐ โซ
โ
0๐โ๐๐โ
2d๐ง. (A.9)
The temporal dependence is obtained by convolving the driving func-tion ei(๐1ยฑ๐2)๐ก (assumed to be zero for ๐ก < 0) with the modal propagator๐๐(๐ก). Combining terms, we arrive at:
๐ด๐(๐ก)๐ด1๐ด2
=๐น๐2๐ โซ
๐ก
0ei(๐1ยฑ๐2)๐ sin (๐๐(๐ก โ ๐))๐๐d๐. (A.10)
he sine function can be written as a sum of two exponentials, a formhich exhibits two separate contributions: a first one associated witheโi๐๐(๐กโ๐)โ2i which oscillates rapidly and cannot grow significantly,nd a second one associated with ei๐๐(๐กโ๐)โ2i which can grow withoutimit if the resonance condition ๐๐ = ๐1 ยฑ๐2 is matched. We shall noweglect the former and approximate ๐ด๐(๐ก) by keeping only the latter:
๐ด๐(๐ก)๐ด1๐ด2
โ
โง
โช
โจ
โช
โฉ
i๐พ (๐ก)1,2,๐ei(๐1ยฑ๐2)๐กโei๐๐๐ก๐๐โ(๐1ยฑ๐2)
if ๐๐ โ ๐1 ยฑ ๐2,
๐พ (๐ก)1,2,๐ ๐ก ei(๐1ยฑ๐2)๐ก if ๐๐ = ๐1 ยฑ ๐2,
(A.11)
where we defined:
๐พ (๐ก)1,2,๐ =๐๐๐น๐4i๐
. (A.12)
Finally, if we focus only on the resonance condition and slightly changethe definition of ๐ด๐(๐ก) by pulling out the time-harmonic term ei(๐1ยฑ๐2)๐ก
(i.e. we adopt the definition of Eq. (3a)), then Eq. (A.11) becomesEq. (3b).
The coefficients ๐พ (๐ก)1,2,๐ express how fast in time two given primarymodes transfer energy into a third mode, and are defined up to anarbitrary factor that reflects the normalization chosen for the modes.They are analogous to what is often called โโmixing powerโโ in theliterature, although referring to a transfer rate measured in space, alongthe propagation axis. The link between both forms is discussed below.
10
๐
Correspondence with a representation in space. Let us compare the cou-pling coefficients as they appear in time and space representations. Wenow write the secondary field as:
๐ฎ12 = Re
(
โ
๐๐ด๐(๐ฑ)๐๐(๐ง)ei(๐1ยฑ๐2)๐ก
)
. (A.13)
When the phase-matching condition is satisfied, it can be established [6,9] that:๐ด๐(๐ฑ)๐ด1๐ด2
= ๐พ (๐ฑ)1,2,๐ ๐ช๐ ๐ฑ eโi(๐ค1ยฑ๐ค2)๐ ๐ฑ , (A.14)
where
๐พ (๐ฑ)1,2,๐ =โi(๐1 ยฑ ๐2)๐น๐
4๐๐, (A.15)
nd ๐ช = (๐ค1 ยฑ ๐ค2)โโ๐ค1 ยฑ ๐ค2โ is the phase direction. In Eq. (A.15) ๐๐ ishe mean power flux of mode ๐ along ๐ช:
๐๐ = ๐ช๐ โซ
โ
0๐๐๐ d๐ง, (A.16)
ith ๐๐๐ = โ 12Re
(
๐๐ป๐ ๐๐ฟ(๐๐)
)
the Poynting vector and ๐๐ = i(๐1ยฑ๐2)๐๐
the velocity of the mode. Using the identity โซ โ0 ๐๐๐ d๐ง = ๐๐,๐๐๐ช๐,๐,
here ๐๐,๐ is the energy (or group) velocity, ๐ช๐,๐ is a unit vector inhe direction of the energy flux, and ๐ is the time-average mechanicalnergy defined in Eq. (A.9), the ratio between both rates of transfer ofnergy can be obtained:
๐พ (๐ก)1,2,๐
๐พ (๐ฑ)1,2,๐
=๐๐๐
= ๐๐,๐๐ช๐ ๐ช๐,๐. (A.17)
n other words, the transfer rate per unit space measured in the direc-ion of energy flux, i.e. by setting ๐ฑ = ๐ฅ๐ช๐,๐ in Eq. (A.14), is equal to theransfer rate per unit time divided by the group velocity.
ppendix B. On the choice of normalization of the modeshapes
mplitude. In scalar models of harmonic generation, the amplitudes ofhe components of the displacement field ๐ข are generally defined as ๐ข =๐ ๐ด๐ cos๐๐(๐ก โ ๐ฅโ๐). If we stretch this representation into a transverse
pace dimension ๐ง (call it depth) over a distance โ (call it thickness),e can introduce a trivial modeshape ๐๐(๐ง) = 1 without changing theefinition of ๐ด๐. The time-average and ๐ง-integrated mechanical energyarried by such harmonic wave with unit amplitude ๐ด๐ = 1 is ๐ =12๐๐
2๐ โซ
โ0 |๐๐(๐ง)|
2 d๐ง = 12๐๐
2๐ โ, where ๐ is the density of the medium.
Now, in the case of an elastic plate of density ๐(๐ง), with the purpose ofdealing with coupling constants that have a similar meaning, it seemsnatural to require the modeshapes ๐๐(๐ง) to be normalized such thattheir mechanical energy extends the scalar case:
๐ =12๐2๐ โซ
โ
0๐(๐ง)โ๐๐(๐ง)โ2 d๐ง = 1
2โจ๐โฉ๐2
๐ โ, (B.1)
where โจ๐โฉ = 1โ โซ โ
0 ๐d๐ง is the average density. Note that if the medium ishomogeneous ๐(๐ง) = ๐, Eq. (B.1) simplifies to requiring 1
โ โซ โ0 โ๐๐(๐ง)โ2 d๐ง
= 1.
Phase. Defining the phase of ๐๐ is also necessary, although the choiceis somewhat more arbitrary. In an isotropic plate, the in-plane andout-of-plane components of the modeshapes have a phase difference ofยฑ๐โ2, and either of them can be chosen to be pure real โ this propertyapplies to any layering of materials belonging to a class of anisotropythat includes orthotropy [53]. We then chose to require the in-planecomponents ๐๐,๐ฅ and ๐๐,๐ฆ to be pure-real. As for the ยฑ sign that remainsto be set, imposing a condition on the ๐๐,๐ฅ and ๐๐,๐ฆ components at ๐ง = 0s convenient in that they can be simultaneously null only for ๐ค = ๐ โhis is not the case for ๐๐,๐ง which is null for SH modes and cancelst ๐โ๐ = ๐๐ for P-SV modes. We then decide to require ๐๐,๐ฅ|๐ง=0 and๐,๐ฆ|๐ง=0 to be non-negative in the radial and azimuthal directions, i.e.
๐
๐,๐ฅ|๐ง=0 โฅ 0 and ๐๐,๐ฆ|๐ง=0 โฅ 0 for ๐คโโ๐คโ = (1 0 0) .Ultrasonics 119 (2022) 106589P. Mora et al.
๐
SNco๐ฑ
m
R
An example. Applied to the SH modes of an isotropic and homogeneousplate in the ๐คโโ๐คโ = (1 0 0)๐ phase direction, this definition gives0 = (0 1 0)๐ and ๐๐ =
โ
2 cos(๐ ๐ง ๐โโ) (0 1 0)๐ for ๐ > 0.
caling of the coupling constants ๐พ (๐ก)1,2,3 and ๐ฝ1,2,3 with the plate thickness.ormalized as described upper, ๐๐ does not depend on โ. Then, onean see from Eq. (A.9) that ๐โ๐๐ scales as ๐๐โ, that is, does not dependn โ either. Furthermore, by changing to dimensionless coordinatesโ ๏ฟฝฬ๏ฟฝ = ๐ฑโโ and ๐ โ 1
โ๐๏ฟฝฬ๏ฟฝ, we see from Eqs. (A.4) and (A.8) that๐น๐ is proportional to 1โโ2. We then see from Eqs. (A.12) and (4) that๐พ (๐ก)1,2,3 scales as 1โโ2 while ๐ฝ1,2,3 is a constant that only depends on the
ode triplet.
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