Numerical investigation of non-collinear guided wave mixing in the

case of contact nonlinearity

Philippe Blanloeuil, Chun H. Wang

School of Mechanical and Manufacturing Engineering, University of New South Wales,

Sydney, NSW 2052, Australia

p.blanloeuil@unsw.edu.au

L. R. Francis Rose

Defence Science and Technology Group,

Melbourne, VIC 3207, Australia

Martin Veidt

School of Mechanical & Mining Engineering, University of Queensland,

Brisbane, QLD 4072, Australia

Abstract

This work investigates the feasibility of using guided waves to perform a non-collinear

wave mixing in the presence of a closed crack. Based on the analogy with previous

work on bulk shear waves, it is shown that the mixing of shear horizontal (SH0) guided

waves generate a symmetrical (S0) mode propagating in a different direction and having

a frequency equal to the sum of the frequencies of the incident waves. The conditions

for this guided wave mixing are determined based on theoretical considerations related

to phase velocity matching and demonstrated numerically using a Finite Element model.

The results show that the interaction leads to the generation of a S0 mode at the sum

frequency that is the strongest for a particular angle of interaction between the incident

waves. The non-collinear mixing under imperfect phase velocity matching conditions is

also investigated.

1. Introduction

The evaluation of small damage that may precipitate catastrophic fracture is very

important in many safety-critical structures and systems such as nuclear power plants,

aircraft and engines. Ultrasonic methods based on linear wave scattering are efficient

for detecting defects and characterizing material elasticity, but are less sensitive to

micro-cracks or closed cracks. Recent research reveals that if the excitation amplitude is

sufficient, these defects can behave nonlinearly because of contact dynamics. This

phenomenon, also called contact acoustic nonlinearity [1] (CAN), is of great interest for

nonlinear ultrasonic non-destructive testing. CAN generates higher harmonics [1–3] and

has been largely studied on contacting interfaces and closed cracks [3–6]. However,

higher harmonics methods have several drawbacks. Specifically, the harmonics often

have other potential sources such as the amplifiers, transducers, couplant or the bulk

materials themselves [7,8] which makes it difficult to relate the nonlinear harmonics to

structural damage. To overcome these problems different techniques that are less

sensitive to system-related nonlinearities are required.

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In the presence of a nonlinear strain–stress response, such as the one induced by

material nonlinearity or CAN, the superposition principle no longer holds, and

interaction between waves occurs. This is the fundamental mechanism for the non-

collinear wave mixing, first reported by Jones and Kobett [9], and Rollins [10] in the

1960s. In non-collinear wave mixing, two incident waves propagating in different

directions and intersecting at an area of interest and can interact with each other to

generate a new wave, provided that some conditions regarding the ratio of input

frequencies, the angles of incidence and the wave polarizations are met [9]. In the case

of two incident shear waves, the scattered wave is a longitudinal wave travelling in a

direction parallel with the sum of the incident wave vectors, and with a frequency equal

to the sum of the incident wave frequencies. This phenomenon provides the basis for a

sensitive method to detect nonlinear damage, benefiting from the frequency, modal and

spatial selectivity of the nonlinear waves. Such a method has been successfully applied

to evaluate the state of various materials. Examples of application include the evaluation

of fatigue damage in aluminium [11], physical aging of PVC [12], evaluation of a

diffusion welding interface in titanium [13], epoxy curing [14] and oxidative aging of

asphalt [15]. Regarding CAN, Demcenko et al. conducted testing on contacting

interface between two PVC plates [16] into order to evaluate the mixed response under

imperfect resonance conditions.

Research on nonlinear propagation of guided waves has focussed on the generation of

cumulative second harmonic [17,18]. To generate cumulative second harmonic, the

phase and group velocities of wave modes at the fundamental and second harmonic

frequencies need to match. In addition, a non-zero power flux between the mode pair is

also required. In a recent paper [19], Hasanian et al extended the theoretical

investigation to the nonlinear guided wave mixing, establishing the resonance and

power flux conditions necessary for the guided wave mixing. The resonance condition

also imposes a phase velocity matching between the incident waves and the scattered

wave. Although the theoretical framework in [19] allows one to consider any type of

mixing configuration, particular examples were presented for collinear and orthogonal

incident wave configurations. Considering the previous work performed with bulk shear

waves [20,21], the analogous configuration for guided waves would suggest that two

non-collinear incident SH0 waves would interact with a region of nonlinear response to

produce a scattered S0 wave mode at the sum of the incident frequencies. This particular

phenomenon is investigated in the present work, with a focus on the feasibility of this

non-collinear guided wave mixing in detecting closed cracks.

The paper is organised as follows. Theoretical considerations regarding phase velocity

matching are first presented in Section 2, and a Finite Element (FE) model considering a

closed crack exhibiting CAN is developed in Section 3 to demonstrate the guided wave

mixing. Results are presented and discussed in Section 3 before addressing some

concluding remarks in Section 4.

2. Theoretical considerations

The non-collinear mixing of two incident SH0 waves is considered here, whereby the

two incident waves are travelling along two distinct directions defined by the wave

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vectors �! and �!, and their frequencies are respectively �! and �!, as indicated

schematically in Figure 1. It is assumed that the incident beams intersect each other, and

the angle between the two incident beams is denoted �. The scattered wave, when

existing, is assumed to propagate along a direction defined by the wave vector �� which

forms an angle � with the y axis. A new coordinate system is thus defined, where the

axis y' is collinear with the direction of propagation of the scattered wave. The

configuration is shown in Figure 1.

Figure 1: angle configuration for non-collinear guided wave mixing

Previously, guided wave mixing has been considered theoretically for material

nonlinearity only [19] following the work of de Lima et al [17]. In comparison the

present work focuses on guided wave mixing due to CAN. The theoretical development

obtained for material nonlinearity is recalled here to provide some insights regarding the

conditions required for nonlinear guided wave mixing. The derivations presented in [19]

lead to the definition of a resonance condition for the generation of the mixed modes at

frequency �± = �! ± �! along the direction �′. This condition is written using wave

vectors as follows:

��.�!= �! ± �! .�

! (1)

where �� is the wave vector of a n-th guided wave mode. The resonance condition also

requires a non-zero power flux condition [17,19]. The direction of propagation of the

scattered guided wave is thus given by

tan� =

−�! sin�! ± �! sin�!

�! cos�! ± �! cos�!,

(2)

and the wave number of the scattered wave is given by

�! = �! cos(�! + �)± �! cos(�! − �). (3)

The resonance condition expressed by Eqs (1-3) is also referred to as a phase velocity

matching condition for the scattered wave in [17,19]. This implies that the phase

velocity (or wave number) of the scattered wave �� at the frequency �± has to match a

phase velocity (wave number) of one of the guided modes in the plate at the same

frequency in order to generate a propagating mode. This is always difficult for collinear

wave mixing (considering for example second harmonic guided waves), as only a few

mode pairs can be found to fulfil the phase velocity matching. However, it is much

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easier to meet this condition for non-collinear guided waves due to the possibility to

choose the incident angle appropriately.

To illustrate this, the particular case of two incident SH0 guided waves of identical

frequencies propagating in a 3 mm thick aluminium plate is considered. The frequency

the two incident waves are �! = �! = 150 kHz, and the angle of incidence is assumed

to be the same �! = �!. For this symmetric configuration, the scattering angle is � = 0

which indicates that the scattered wave will propagate along the y axis. Thus, the

resonance condition (3) becomes simply

�! = 2�! cos�! ,�! = 2�! (4)

To generate a propagating mode, this wave number / frequency pair (�!,�!) must

match one of the guided (Lamb) modes in this plate. Figure 2 gives the dispersion

curves of the guided modes in a 3 mm thick aluminium plate. These curves have been

computed with the SAFE method [22]. Consider first the possibility of satisfying the

resonance conditions in Eqs. (1-3) for collinear waves, with the incident wave being the

SH0 mode at 150 kHz, for which the corresponding wavenumber is around 302 rad/m. It

can be seen from Figure 2 that only the SH0 provides a phase matching at 300 kHz. This

is the SH0 second harmonic. Considering next a non-collinear configuration, then the

wave number of the scattered wave is defined as �! = 2�! cos�!, and it is thus

possible to choose the angle of incidence �! = �! so that the wave number of the

scattered wave match the wave number of S0 at the sum of frequencies 300 kHz. The

wave number of the S0 mode at the sum frequency is around 354 rad/m. Therefore the

necessary incident angles to achieve the matching condition can be determined from the

following equation:

�! = 2�! cos�! = 354 rad/m. (5)

For the given incident wave number, this gives �! = �! = 54°, and thus an interaction

angle � = 108° is required to satisfy the resonance condition for wave mixing. This is

the optimal angle of interaction when using the same incident frequencies, with the

same angle of incidence. The effect of small departures from this optimal angle is

investigated numerically in the next section.

Thus, it can be seen that the non-collinear configuration offers considerably more

flexibility for satisfying the resonance conditions in Eq. (1-3), because the angle of

incidence can be tuned to fulfil the phase velocity matching, which was not possible

with the collinear approach. The power flux from SH0 to S0 is non-zero [17] so that this

pair SH0 - S0 is possible. Note that because the direction of propagation is different from

those of the incident waves, the group velocity matching is not required here, contrary

to the requirement for cumulative second harmonic.

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Figure 2: Dispersion curves for a 3 mm thick aluminium plate computed by the

SAFE method

3. Finite Element modelling

In order to investigate the conditions for non-collinear mixing of two SH0 guided waves

interacting with a contact interface, a FE model has been developed using the Abaqus

package. The propagation medium is a 3 mm thick aluminium plate with a through

thickness crack of length 40 mm at the centre of the plate. The material is assumed to be

linear elastic with Young’s modulus � = 70 GPa, � = 0.33 and � = 2700 kg/m3. The

geometry of the FE model is shown in Figure 3. The incident SH0 waves A and B are

generated from in-plane shear displacements imposed on the cross-section of two

chamfered corners of length 33 mm and located at 254 mm from the centre of the crack.

The chamfer geometry (position and angle), is designed to generate the incident waves

with the desired angles of incidence �! = �!. The position and orientation of the

chamfers are altered to consider different angle of interaction � = �! + �!, however,

the distance between the excitation sources and the centre of the crack is always

maintained equal to 254 mm. The frequency of both excitations is �! = �! = 150 kHz

and the amplitude 10 nm. To account for CAN, unilateral contact with Coulomb friction

(coefficient of friction � = 0.6) boundary conditions are defined between the two

opposite crack faces, which models both clapping and frictional sliding mechanisms.

The Abaqus time explicit solver is used for the resolution. The mesh is made of linear

3D stress elements with reduced integration C3D8R, and the element in-plane

dimension is 1.5 mm which provides at least 10 elements per wavelength for the

considered modes and frequencies, whereas 4 elements are used through the plate

thickness.

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Figure 3: FE model geometry, with SH0 source A and B located at 254 mm from

the centre of the crack, with an interaction angle � = �� + ��, where �� = ��.

Three simulations are run: both excitations generated concurrently, wave A only and

wave B only, respectively denoted by AB, A and B. Figure 4 shows the magnitude of

the displacement field for single wave excitations A and B, and for simultaneous wave

excitations AB, before and after interaction with the crack. The interaction angle is

chosen as � = 108° as it was shown previously that this angle fulfils the phase velocity

matching condition. One can see in Figure 4 (d-f) the incident SH0 waves are reflected

and transmitted from the contact interface and that a converted mode is propagating

faster near the y-axis. This wave propagating along the y-axis is enhanced when both

excitations are used as shown in Figure 4 (f) and the higher velocity suggests that this

wave corresponds to a S0 mode. However, the mode nature and the frequency content of

the scattered wave travelling along the y-axis need to be verified.

(a) (b) (c)

(d) (e) (f)

Figure 4: displacement magnitude field before (a-c) and after (d-e) the interaction

with the crack, for wave A only (a,d), wave B only (b,e) and both (c,f). The

interaction angle is 108°.

nm

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A 2D FFT is performed on the normal displacement time history signals recorded at

each node along the y-axis to convert the output in the frequency-wave number domain

and compare it with the theoretical dispersion curves. The 2D FFT is computed for the

sum of signals obtained for the individual excitation A and B, for signals obtained for

the dual excitation AB, and for the difference signals between those two, computed as

AB-A-B. The results are shown respectively in Figure 5 (a-c) for the previous case

where the interaction angle is � = 108°, where the amplitude has been normalised by

one common value, being the maximum peak amplitude obtained for the sum of the

individual excitation.

Figure 5: Wave number-frequency data from 2D FFT along the propagation

direction y obtained from the FE modelling (� = ���°) together with theoretical

dispersion curves. (a) sum of individual excitations A+B, (b) dual excitation AB, (c)

difference AB-A-B.

One can see on Figure 5 that the wave propagating along the y-axis does correspond to

the S0 mode. More precisely, for single wave excitation it can be seen in Figure 5 (a)

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that the S0 mode is mainly generated at 150 kHz, with a lower amplitude peak at 300

kHz. This corresponds to a linear mode conversion to S0, along with the generation of

the second harmonic. When both excitations are used together, the nonlinear response at

300 kHz is enhanced as shown in Figure 5 (b). Comparing Figure 5 (a) and (b) it is clear

that the superposition principle no longer holds due to the presence of nonlinearity.

Therefore, the difference AB-A-B between the dual excitation AB and the sum of the

individual responses A and B is not zero as shown in Figure 5 (c). Computing the

difference AB-A-B allows to remove the contribution due to mode conversion and

spurious second harmonics as proposed in [11] for non-collinear mixing of bulk shear

waves, and thus provides the nonlinear response corresponding to the mixed scattered

wave. One can see in Figure 5 (c) that the S0 mode at the frequency corresponding to the

sum of the incident frequencies remains after subtraction.

This first result demonstrates that the non-collinear interaction of two SH0 waves leads

to the scattering of a S0 mode at the sum frequency when the interaction angle is chosen

to fulfil the phase velocity matching. However, this condition has been established

based on theoretical ground considering a material nonlinearity. The question thus arises

as to whether this condition is also required in the case of CAN. Numerical simulations

are thus conducted for different angles of interaction to investigate the non-collinear

mixing under imperfect phase velocity matching conditions. The amplitude of the S0

mode at 300 kHz is evaluated in the frequency/wavenumber domain for each interaction

angle and the result is shown in Figure 6. Using data collected for � > 0 and � < 0, it is

shown in Figure 6 that the amplitude of the scattered S0 wave is similar on both sides of

crack, as a common value was used for normalisation. Maximal amplitude is observed

for an interaction angle of 108°, i.e. when the phase matching condition is met.

However, it appears that the amplitude of the mixed S0 mode decreases slowly as the

interaction angle moves away from that optimal value, especially for smaller interaction

angles.

Figure 6: Evolution of the amplitude of the scattered S0 at 300 kHz as a function of

the interaction angle.

This result suggests that although the phase velocity matching condition provides the

optimal configuration, good results can still be achieved for imperfect conditions. In

practice, this means that i) inaccuracy in the excitation source positioning can be

tolerated, and ii) a given fixed configuration designed for a particular plate may still be

applied to a plate of slightly different thickness or mechanical properties. More

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importantly, this result suggests that the wave mixing approach for detecting CAN is

robust relative to changes in environmental and operating conditions (EOC), which can

lead to variations in the dispersion relations, and hence in the optimal angle for wave

mixing.

4. Conclusions

The non-collinear guided wave mixing between two SH0 waves interacting with a

closed crack exhibiting CAN has been investigated numerically. It was shown that in

presence of contact nonlinearity, the interaction of two incident SH0 waves leads to the

scattering of a S0 mode which frequency is equal to the sum of the incident frequencies.

The scattered mixed S0 wave can be identified more clearly after subtraction of the

responses obtained for single source excitations. It is shown that the non-collinear

mixing is optimal for a particular interaction angle between the two incident waves.

Theoretical considerations showed that this optimal angle ensures a phase velocity

matching between the scattered S0 mode at the sum frequency and the incident SH0

modes. However, the numerical results suggest that the scattering of the S0 mode can

still be achieved for imperfect matching conditions in case of CAN. This work

demonstrates that the non-collinear guided wave mixing can thus be applied for the

baseline free detection of closed crack or fatigue crack in plates as an alternative to

higher harmonic generation, with the advantage that the nonlinear mixed scattered wave

propagates in a different direction.

Acknowledgements

This study was supported in part by an Australian Research Council (ARC) Discovery

Grant (DP180102658).

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