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Time reversal invariance for a one-dimensional model of contact acoustic nonlinearity
Journal of Sound and Vibration 394 (2017) 515–526
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Time reversal invariance for a one-dimensional model of contact acoustic nonlinearity Philippe Blanloeuil a,n, L.R. Francis Rose a, Martin Veidt b, Chun H. Wang c a b c
Sir Lawrence Wackett Aerospace Research Centre, School of Engineering, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia School of Mechanical & Mining Engineering, University of Queensland, Brisbane, QLD 4072, Australia School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney NSW 2052, Australia
a r t i c l e i n f o
abstract
Article history: Received 13 October 2016 Received in revised form 27 January 2017 Accepted 31 January 2017 Handling editor: L. G. Tham Available online 10 February 2017
The interaction of a one-dimensional (1D) wave packet with a contact interface characterized by a unilateral contact law is investigated analytically and through a finite difference model. It is shown that this interaction leads to the generation of higher harmonic, sub-harmonic and zero-frequency components in the reflected wave, resulting in a pulse distortion that is attributable to contact acoustic nonlinearity. However, the results also show that the re-emission of a time reversed version of this distorted first reflection results in a healing of the distortions and a perfect recovery of the original pulse shape, thereby demonstrating time reversal invariance for this type of contact acoustic nonlinearity. A step-by-step analysis of the contact interaction provides insights into both the distortion arising from the first interaction and the subsequent healing during the second interaction. These findings suggest that time reversal invariance should also apply more generally for scatterers exhibiting non-dissipative contact acoustic nonlinearity. & 2017 Elsevier Ltd. All rights reserved.
Keywords: Contact acoustic nonlinearity Time reversal invariance Finite difference method
1. Introduction The application of nonlinear acoustics for nondestructive evaluation and structural health monitoring has attracted considerable research interest over the past two decades, driven by the prospect that various forms of structural damage may induce a nonlinear response that could lead to earlier damage detection than would be possible through conventional linear ultrasonics [1–3]. It is pertinent to distinguish between cases where the structural damage (and hence the source of nonlinearity) is more-or-less uniformly distributed throughout the structure, so that the response can be adequately modelled by an appropriate nonlinear constitutive equation for the material [4–6], and cases where the damage is localized, with the surrounding material behaving linearly. The present work is concerned with the latter case. In practice, two important forms of localized damage are fatigue cracks in structural alloys and delaminations in composite laminates. For both of cases, the nonlinear response can generally be attributed to contact acoustic nonlinearity (CAN) [7–15], which induces the generation of new frequency components such as higher harmonics, sub-harmonics and zero frequency (DC) response. The mechanisms involved in CAN include clapping between the contacting interfaces [7,9], as well as dissipative mechanisms due to frictional sliding [8,12]. Theoretical models of varying levels of sophistication have been proposed for all of these mechanisms, as comprehensively reviewed in [14]. We note in particular the recent work on the vibrational response of n
Corresponding author. E-mail address: [email protected] (P. Blanloeuil).
http://dx.doi.org/10.1016/j.jsv.2017.01.050 0022-460X/& 2017 Elsevier Ltd. All rights reserved.
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beams with breathing cracks [15–18], of plates with delaminations [19,20], as well as wave scattering by cracks and delaminations [21–25] as examples of the current state of the art for CAN modelling. The time reversal (TR) method [26–28] allows one to reconstruct a source signal at the excitation point from signals recorded and reemitted at some measurement points away from the source, after being reversed in time. This reversibility is based on the spatial reciprocity and time invariance of the linear wave equation, and is referred to as TR invariance. This property has been successfully exploited to derive various TR techniques for detecting and imaging structural damage through a wide variety of distinct approaches [29–42]. The TR invariance has been demonstrated in various propagation media and holds for linear scatterers as amply demonstrated by Fink et al [26,28,43], and hence for damage that can be modelled as linear scatterers, e.g. corrosion thinning or open cracks. TR invariance has also been verified for classical material nonlinearity as long as the propagation distance is shorter than the shock formation distance [44]. In that case, the wave form is progressively recovered during the back propagation and the distortions accumulated during the initial propagation are cancelled. However, to the best of our knowledge, the issue of time reversal invariance in the presence of CAN at a localized nonlinear defect has not been investigated previously. It can be expected on physical grounds that time reversal invariance will not hold in the presence of dissipative mechanisms such as frictional sliding, but it is not immediately obvious whether or not this invariance may still hold for frictionless contact that can generate nonlinear responses such as pure clapping at the contact interface. This is addressed in the present work. Of particular relevance regarding the present work is the suggestion by Sohn et al. [45,46] that the nonlinearity induced by the presence of damage would result in a breakdown of TR invariance, and therefore experimental observation of a breakdown could serve as a baseline-free diagnostic for detecting damage [47]. However, considering the spectral enrichment due to CAN, it is not clear whether the TR breakdown is attributable to the nonlinear nature of the defect, or to the limited bandwidth of the sensors and their inability to capture the full range of frequency information. Accordingly, a numerical modelling is employed here in order to investigate the TR process solely associated with CAN without any possible bandwidth limitations of practical sensors and actuators. A simplified one-dimensional (1D) model of CAN is considered in this work, whereby a longitudinal wave packet interacts with a contact interface between the propagation medium and a rigid boundary acting as a limiter. A similar configuration has recently been used to model the vibrational response of a rod with a breathing crack in [48]. CAN is modelled here with a unilateral contact law, which correctly captures the dynamics of normal impacts between contacting surfaces associated with clapping. A time explicit finite differences (FD) scheme is used to solve the propagation problem. The presentation is organized as follows. The configuration and governing equations are presented in Section 2, as well as an analytical representation for the incident and reflected wave field. A computational approach based on finite difference approximations is outlined in Section 3. Detailed results are presented in Section 4, and Section 5 provides concluding remarks as well as indications for future work.
2. Formulation of a 1D propagation model with CAN A semi-infinite isotropic and homogeneous medium in contact with a rigid plane is considered as shown in Fig. 1. The material is steel, with Young’s modulus E = 210 GPa, Poisson’s coefficient ν = 0.3 and density ρ = 7800 kg/m3. A longitudinal wave packet (or tone burst) is generated at x = 0 mm and propagates along the x -axis. Under these conditions, a 1D propagation model can be adopted. The contact interface is made with the rigid plane located at x = L . A pre-existing compressive stress state σ0≤0 is considered to exist at the contact interface. The incident wave packet propagates and interacts with the contact interface, and the reflected wave contains new frequency components due to CAN. Additionally, a transparent boundary is considered at x = 0 mm, so that the wave travelling back from the contact interface is not reflected at x = 0 mm. The 1D wave equation is given by:
Fig. 1. 1D model configuration. A plane wave is propagating in a semi-infinite medium. The solid is in contact with a rigid boundary where contact laws are considered to model CAN. The reflected wave contains higher harmonics due to CAN.
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∂ 2u ( t , x) ∂t 2
−c2
∂ 2u ( t , x) ∂x2
= 0,
517
(1)
where c is the wave velocity. The solution for the displacement u (t , x ) can be expressed as the sum of forward travelling displacement u+ and a backward travelling displacement u−:
u ( t , x ) = u + ( t − x / c ) + u− ( t + x / c )
(2)
In the case of a longitudinal wave, the incident wave can generate only compression or tension at the contact interface, where tension tends to open the interface. The contact behavior is modelled by a unilateral contact law defined as follows:
where σn is the normal stress computed from the total displacement gradient ∂u/∂x at the interface, and σ0 is a prescribed static compressive stress acting at the interface. The first line in Eq. (3) indicates that the elastic medium cannot penetrate the rigid plane. A negative displacement corresponds to an opening of the contact interface. The second line in Eq. (3) indicates that only compression can be applied at the interface. Finally, the last line is the complementary condition that indicates that the interface is either open or closed. Considering 1D propagation with the solution defined by Eq. (2), the contact problem can be solved by considering separately the closed case and the open case. 2.1. Closed interface The medium is assumed to be initially in contact with the rigid plane. Thus the displacement at x = L is zero:
u ( t , x) = u+( t−x/c )+u−( t +x/c )=0,
x=L
(4)
Thus, the reflected wave is the opposite of the incident wave: u−( t + x/c )= − u+( t − x/c ). The stress generated by the displacement is computed as:
σ ( t , x) = E
∂u (t , x) E = −u+′ ( t−x/c )+u−′ ( t +x/c ) ∂x c
(
)
(5)
where the derivatives are expressed as follow:
⎧ u ′ t −x / c ) = ⎪ ⎪ +( ⎨ ⎪ u ′ ( t +x / c ) = ⎪ ⎩ −
⎞ 1 ⎜⎛ c u̇ ( t , x)− σ ( t , x) ⎟ ⎠ 2⎝ E ⎞ 1 ⎜⎛ c u̇ ( t , x)+ σ ( t , x) ⎟ ⎠ 2⎝ E
(6)
where u′ denotes the derivative with respect to the argument, whereas u̇ denotes the partial derivative with respect to time. The stress at the contact interface is given by σn ( t )=σ ( t , L ). Note that when the interface is closed the normal stress at the interface is up to twice the incident stress. As long as the sum σn (t ) + σ0 is negative, the interface is closed. The opening of the contact interface thus occurs when σn (t ) + σ0=0. 2.2. Open interface It follows from Eqs. (3) and (5) that the interface opens when
E −u+′ ( t−L/c )+u−′ ( t +L/c ) = − σ 0 c
(
)
(7)
Since Eq. (7) holds for x = L , the space dependence is omitted in the following. This relation is integrated in time between the last contact instant τ and the current time t to find the opening displacement and the displacement associated to the backward propagating wave at x = L ,
u−( t ) = u+( t )+( u−( τ )−u+( τ ) )−
cσ 0 ( t −τ ) E
(8)
During the contact stage, Eq. (4) gives u−( τ )= − u+( τ ), so that Eq. (8) can be re-written as:
u−( t ) = u+( t )−2u+( τ )−
cσ 0 (t−τ ) E
(9)
This expression gives the contribution of the backward propagating wave during the opening stage. Finally, the total
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displacement at the interface, or opening gap, is given by:
u ( t ) = 2u+( t )−2u+( τ )−
cσ 0 (t−τ ) E
(10)
This last equation is true as long as the interface is open. When u ( t , L )=0, then contact is activated and we revert to the equations presented in Section 2.1. One can see that for this 1D case it is possible to solve the contact problem analytically with a piecewise resolution. In this work, the problem is solved numerically with the FD scheme presented in the next section. Nevertheless, the analytical results presented in this section will prove useful for understanding the mechanisms involved at the contact interface. This will be especially valuable for explaining the observed results when the incident wave is the TR of an initially reflected wave.
3. Computational procedure The wave equation is solved in time using the classic second order finite difference approach which allows one to obtain the solution at a new time step by a simple matrix-vector product [49]. The novel step in the present work is the implementation of the unilateral contact law defined by Eq. (3). A transparent boundary condition is used at x = 0 to reduce the size of the computational domain. Thus, at x = 0 mm, only the incident wave, or excitation, is considered:
u+′ ( t−x/c ) = f ( t ),
x =0
(11)
where u+′ ( t − x/c ) is defined by Eq. (6) and f (t ) is the source term, equivalent to an imposed particle velocity. This equation defines the incident wave and is transparent to the reflected wave. This condition is rewritten into a strain formulation:
2f ( t ) ∂u (t , x) u̇ ( t , x) − = − , ∂x c c
x =0
(12)
Eq. (12) is then discretized using a right off-centered differentiation for the strain, as follows:
2f ( t ) −3u1m +4u2m −u3m 1 u1m +1−u1m − = − δt 2δ x c c
(13)
where δx and δt denote respectively the space step and the time step. The superscript m corresponds to the time step whereas the subscript k corresponds to the space discretization, with k = 1 in Eq. (13) to implement the input condition given by Eq. (12). Eq. (13) can be rearranged to obtain u1m + 1, which is the new value of displacement at the first node k = 1 at the time step m + 1. The wave equation Eq. (1) is discretized using the Euler finite difference scheme [49]:
ukm +1−2ukm +ukm −1 δt 2
−c2
ukm+1−2ukm +ukm−1 = 0, δx2
∀k ∈ ⎡⎣ 2,K −2⎤⎦
(14)
where K is the total number of nodes. This second order general scheme is conditionally stable under the Courant- Friedrichs-Lewy (CFL) condition: cδt /δx<1 [49]. Eq. (14) can be rearranged to obtain ukm + 1, the new value at node k, in terms of known quantities from the previous time steps. Finally, the last two nodes have to be considered, the discretization being directly linked to the contact condition. Thus, two cases must be considered separately. 3.1. Closed interface A contact state is always assumed first. During contact, the displacement of the last node k = K at the interface is zero: uKm=0. This value is injected in the general scheme Eq. (14) to compute uKm−+11. The normal stress at the contact interface is computed by a left off-centered differentiation of Eq. (5):
σn = E
−4uKm−1+uKm−2 +3uKm 2δ x
The stress value σn+σ0 is then evaluated. If the value is negative, the interface remains closed and positive, that means that the interface is under tension and must open.
(15) uKm + 1=uKm .
If the value is
3.2. Open interface When the computed value of σn+σ0 is positive, the interface opens, and Eq. (3b) must be invoked to set σn+σ0=0, where σn is given by Eq. (15). Thus, the displacement of node K is now given by:
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⎛ 2σ δ x ⎞ uKm = ⎜ − 0 + 4uKm− 1 − uKm− 2 ⎟/3 ⎝ ⎠ E
519
(16)
The displacement at the interface uKm given by Eq. (16) is then injected in Eq. (14) to obtain uKm−+11. Finally the new displacement at the interface is derived as uKm + 1=uKm . The open stage lasts until the computed value of displacement at the interface is positive, in which case Eq. (3a) must be invoked. The vector of displacements U m + 1at time step m + 1 is computed from the vector of displacements at the previous time step by a matrix-vector product U m + 1=A c, oU m+B , where A c and A o are the matrices respectively written for contact condition and open condition, as outlined above, and where B contains the excitation source term and the contribution of the static stress σ0 in case of open state only.
4. Numerical investigation of TR invariance with CAN The preceding computational procedure is used to solve two distinct nonlinear propagation problems, namely the forward propagation step labelled E1, and the back-propagation step labelled E2. The reflected wave obtained in E1, which contains new frequency components due to CAN, is selected, time-reversed, and used as the new excitation in E2 . The objective is to verify whether the new reflected wave obtained in E2 recovers the original shape of the incident wave in E1, hence demonstrating TR invariance. 4.1. TR invariance example The incident wave packet is defined as a 1 MHz 10 cycles tone burst windowed by a Hann window. The size of the computation domain is L = 0.05 m and the domain is discretized in space by 800 nodes, so that the wavelength at the incident frequency is divided by 96, thus ensuring that all non-negligible higher harmonics are sufficiently discretized. The time step is computed from the CFL condition, with a 0.5 multiplicative factor. The amplitude of the incident wave is set at 5 nm, which corresponds to a maximum stress of 1.1 MPa in steel. The static stress at the contact interface is specified as σ0= − 1.2 MPa. Fig. 2a shows the time signal recorded at x = 0, for this first propagation E1. One can see first the incident wave packet followed by the reflected wave. The reflected wave is distorted due to CAN. As a consequence, its spectrum contains higher harmonics and a DC component, as shown in Fig. 2b. The DC component reflects the non-zero average of the reflected signal, which can be inferred from the time trace in Fig. 2a. It is interesting to note that the maximum stress created by the incident wave in the medium (i.e. 1.1 MPa) is smaller than the prescribed static stress at the interface (1.2 MPa), so that one might expect that the stress is too low to open the interface. However, when the interface is closed, the displacement at the
Fig. 2. Displacement history and spectrum for the first propagation step E1. (a) Time signals at x ¼0, showing the incident pulse followed by a distorted reflected pulse. (b) spectrum of the reflected pulse.
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Fig. 3. Displacement and stress history at the contact interface during the first propagation step E1. (a) opening displacement at the contact interface and (b) contact stress.
interface is zero and the stress is doubled, so that the stress generated by the incident wave at the interface has actually a peak value of 2.2 MPa if the interface remains closed, hence large enough to open the interface. The time history of the displacement and contact stress at the interface are shown in Fig. 3. One can see that the interface does open, as indicated by the negative displacements in Fig. 3a, and that during the open phase, the stress is truncated at a maximum value of 1.2 MPa so that the total contact stress σn+σ0 is never positive, in accordance with the contact law Eq. (3). This truncation of the contact stress profile is at the origin of the new frequency components as well as the DC component. However, this truncation does not mean that the reflected stress profile exhibits a similar truncation, contrary to what might be expected intuitively. To clarify this point, Fig. 4 shows the stress profiles corresponding to the displacement of Fig. 2a at x = 0. One can see that the incident pulse exhibits a maximal stress of 1.1 MPa, whereas the reflected pulse exhibits strong distortions and has a maximal stress value that exceeds the incident maximal stress value, but no evidence of truncation. However, higher stress values are generated over shorter periods of time so that energy is actually conserved.
Fig. 4. Time history of the stress at x ¼ 0 for the first propagation step E1.
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Fig. 5. Displacement history and spectrum for the first propagation step E2 . (a) Time signals at x¼ 0, showing the incident distorted pulse followed by a reflected pulse free of distortions. (b) spectrum of the reflected pulse.
The reflected pulse at x = 0 from step E1 is selected by applying a time gating and then time reversed to be used as the new excitation for the second step E2. Fig. 5a shows the corresponding displacements at the origin, where the incident pulse can be observed to be the time reversed version of the reflected wave obtained in E1, which is shown in Fig. 2a. It can be seen that the new reflected wave in Fig. 5a is identical to the first incident wave of E1. This is confirmed by considering its spectrum shown in Fig. 5b, where only the fundamental frequency component is present. Considering the stress profile shown in Fig. 4, one can see that the new input is again able to activate a loss of contact at the interface, as indicated in Fig. 6. However, this second interaction does not result in the generation of additional higher harmonics, but on the contrary leads to a “healing” of the distortions, and thereby to a recovery of the original signal. Thus, the TR process has cancelled the distortions generated by the first interaction with the contact interface, and TR invariance is confirmed for this 1D model of CAN.
Fig. 6. Displacement and stress history at the contact interface during the first propagation step E2 . (a) opening displacement at the contact interface and (b) contact stress.
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Fig. 7. Displacement profiles for a single-cycle input pulse. (a) displacement for E1 and (b) displacement for E2 , in the case of a one cycle pulse. The incident and reflected pulses are shown in more detail in the rectangular insets.
It is emphasized that this demonstration of TR invariance depends crucially on employing a sufficiently fine time interval for modelling and sampling, so as to capture adequately all the higher harmonics generated by the pulse distortion, as indicated in Fig. 2b. If instead the reflected signal from the first propagation step E1 is filtered to retain only the contribution around the fundamental frequency prior to time reversal and re-emission, as proposed in [39,45–47], the new reflected pulse would not constitute an exact reconstruction of the original input pulse, and TR invariance would therefore appear to have been lost. However, this apparent loss of TR invariance would be attributable in the present case to an incorrect implementation of the TR protocol. Following this remark on the need to retain the full frequency spectrum shown in Fig. 2b, it is also recalled that the linear superposition principle does not hold in the presence of nonlinearity. This means that TR invariance will again appear to be violated if the contributions from the various frequency components in Fig. 2b, (i.e. the DC, fundamental and higher harmonic components), are separately time reversed and re-emitted sequentially, with the reflection of these separate TR signals being subsequently summed to generate the reconstructed pulse. Instead, it is essential that the TR signals corresponding to the various frequency components should be re-emitted simultaneously, to ensure the perfect reconstruction shown in Fig. 5. 4.2. Generation and cancellation of distortions in nonlinear TR To provide a deeper insight into the detailed contact dynamics leading to TR invariance, the FD simulation is repeated in this section for a single-cycle pulse windowed by a Hann window. The incident pulse now has an amplitude of 3.2 nm due to the windowing. The static stress at the interface is again set to σ0= − 1.2 MPa. The TR invariance is still observed after the propagation E2, as shown in Fig. 7. Note that the reflected pulse in Fig. 7b corresponds exactly to the incident pulse in Fig. 7a, the apparent time reversal being due to the reflected pulse travelling in the opposite direction of propagation. A step by step analysis of the displacement and stress is first carried out in the case of the propagation E1, based on the development of Section 2. This aims to detail the generation of the distortions during E1. Then, the same analysis is carried out for E2. Fig. 8a-b show respectively the displacement and stress profiles of the input at x = 0, in the case of the propagation E1. This displacement corresponds to the first pulse observed in Fig. 7a. This excitation will propagate towards the interface and the same waveform arrives at the contact interface because the propagation is non-dispersive. Therefore, the waveform at x = 0 is used as the incident wave at the contact interface x = L where the incident and reflected waves co-exist for the duration of the input pulse. The interface is initially closed, with a prescribed static stress σ0= − 1.2 MPa. The displacement associated with the incident wave packet is first positive as shown by the solid red line in Fig. 8a, which corresponds to a compression. The interface remains closed as shown in Fig. 8c, and the displacement of the reflected wave is simply the opposite of the displacement of the incident wave, as indicated by Eq. (4) and shown in Fig. 8e. The stress associated with this reflected wave is again compressive, as shown in Fig. 8f, so that the dynamic stress at the interface is twice the stress generated by the incident wave, as indicated by Eq. (5) and shown in Fig. 8d.
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Fig. 8. Displacement - stress analysis for the generation of the distortions during E1. (a) incident displacement u+(t − x/c ) at x = 0 ; (b) incident stress σ+(t − x/c ) at x = 0 ; (c) opening displacement u (t , x ) at x = L ; (d) contact stress σn (t ) at the interface; (e) displacement for the reflected wave u−(t + x/c ) at x = L ; (f) stress for the reflected wave σ−(t + x/c ) at x = L ; (g) numerical solution for the displacement of the reflected wave u−(t + x/c ) at x = 0 ; and (h) numerical solution for the stress of the reflected wave σ−(t + x/c ) at x = 0 .
During the second half of the input cycle, the stress at the interface is positive and eventually attains the value −σ0=1.2 MPa, whereupon the interface opens, according to the contact laws of Eq. (3). The stress σn at x = L remains constant during the open stage, in accordance with the requirement σn+σ0=0 in Eq. (3). The opening displacement during that stage is given by Eq. (10): u ( t )=2u+( t )−2u+( τ )−cσ0/E (t − τ ). This portion of the solution is plotted as a dashed blue line in Fig. 8c. Meanwhile, the displacement corresponding to the reflected wave at the interface is given by Eq. (9), which is recalled here: cσ u−( t )=u+( t )−2u+( τ )− E0 (t − τ ), where τ is the last instant of contact. This part of the solution is plotted in Fig. 8e. One can clearly see that this part of the solution differs from the incident sinusoidal wave form. It creates the distortion in the received signal. The stress corresponding to the reflected wave is σ−( t )=σn ( t )−σ+(t ) and is plotted in Fig. 8f. The interface is open as long as the opening displacement u (t , L ) is negative, which is found from Fig. 8c. After this instant, the interface is closed again, which means that the displacement of the reflected wave becomes once again the opposite of the incident wave displacement, and the stress corresponding to reflected wave is equal to the stress generated by the incident wave. This completes the plots in Fig. 8e and Fig. 8f respectively. It is noted that the transition from open to closed corresponds to an impact, which introduces a slope discontinuity in the displacement history in Fig. 8e, and a jump discontinuity in the stress history in Fig. 8f. The occurrence of such impacts have also been reported in previous finiteelement simulations of breathing cracks [16] and delaminations [20] involving the clapping mechanism of CAN. Finally, the numerical solution for the displacement and the stress signals of the reflected wave received at x = 0 are shown in Fig. 8g and Fig. 8h respectively. The wave forms correspond to the ones predicted analytically by the preceding step-by-step approach, with the difference that the discontinuities generate oscillations in the numerical signals attributable to contact instabilities. The distortions are directly generated by the CAN, as expected, and are in agreement with the theoretical contact analysis. During the second propagation step E2 , the received signal is time reversed and used as the new source. Therefore, the input displacement is the TR of the signal shown in Fig. 8g, with the associated input stress being the TR of the signal shown
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Fig. 9. Displacement - stress analysis for the generation of the distortions during E2 (a) incident displacement u+(t − x/c ) at x = 0 ; (b) incident stress σ+(t − x/c ) at x = 0 ; (c) displacement for the reflected wave u−(t + x/c ) at x = L ; (d) stress for the reflected wave σ−(t + x/c ) at x = L ; (e) numerical solution for the displacement of the reflected wave u−(t + x/c ) at x = 0 ; and (f) numerical solution for the stress of the reflected wave σ−(t + x/c ) at x = 0 .
in Fig. 8h. Ideally, the displacement and stress signals should be the TR of those given in Fig. 8e-f. Therefore, these last two signals are considered to analyze the interaction of TR pulse and the contact interface during E2. The new incident displacement and stress are shown in Fig. 9a-b respectively. Considering the incident stress, it can be seen that the interface will open at the instant of the stress jump in Fig. 9b. During the open stage, the stress corresponding to the reflected wave is again σ−( t )=σn ( t )−σ+(t ), where σn ( t )= − σ0 . Therefore, the initial incident stress of E1 is recovered, as shown in Fig. 9d as the dashed blue line. The displacement corresponding to the reflected wave at the interface is again computed using Eq. (9). This leads to the part of the solution plotted as the dashed blue line in Fig. 9c, which clearly corresponds to the sinusoidal waveform part of the incident wave of E1. Finally, when the opening displacement is equal to 0, the interface closes and the reflected displacement is equal and opposite to the incident displacement, whereas the reflected stress is equal to the incident stress, as shown in Fig. 9c-d. The waveform has thus recovered its sinusoidal shape and corresponds to the incident wave of E1, keeping in mind that the reflected wave is travelling in the opposite direction to the incident pulse. The corresponding numerical results for displacement and stress are shown in Fig. 9e-f and correspond to the step-by-step approach. It can be seen that the analytical stress discontinuity in Fig. 9b leads to oscillatory behavior in the numerical computations in Fig. 9f. The preceding step-by-step analysis provides a detailed insight into the contact dynamics due to clapping, and the mechanisms leading to a full recovery of an original pulse shape during a correct implementation of the TR protocol.
5. Conclusions The time reversal invariance involving CAN has been investigated in the particular case of frictionless contact. A 1D FD numerical model has been used to solve the propagation problem of longitudinal waves that interact with a contact interface where unilateral contact is considered. The forward propagation step leads to the generation of new frequency components that reflect the distortions of the received reflected wave. The latter is selected and reversed in time before being re-emitted, in accordance with the TR protocol. The numerical results show that the distortions generated at the contact interface during the first interaction are locally cancelled during the second interaction with the interface, which demonstrates that TR invariance holds for the present type of contact nonlinearity. This result is not immediately obvious, but both the generation and cancellation of the distortions at the contact interface have been explained by a step-by-step analysis of the contact dynamics. This detailed analysis suggests that TR invariance should hold more generally for 3D scatterers exhibiting non-dissipative CAN. This theoretical study brings some insights regarding the retro-propagation of ultrasonic waves in case of CAN. It appears
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that the nonlinear clapping mechanism is not sufficient to break the TR invariance, contrary to the assumption in [39,45–47] that the presence of damage would necessarily lead to a breakdown of TR invariance. However, it is also noted that common forms of CAN are almost invariably associated with dissipative mechanisms [8,12,13], which would lead to a breakdown of TR invariance, so that the diagnostic test proposed in [45–47] may be still be useful in practice. The point here is that this breakdown of TR invariance is not intrinsically associated with CAN, as demonstrated by the present model, but it is due to the concomitant dissipative mechanisms, which have not been modelled in the present work. Furthermore, in practice, ultrasonic transducers necessarily have a limited bandwidth. This practical limitation may also compromise the practical demonstration of TR invariance, as noted earlier in Section 4.1. However, referring to the spectrum of the reflected wave shown in Fig. 2b, it is reasonable to expect that TR invariance would be verified experimentally provided that the available bandwidth is sufficient to capture contributions up to, say, the fourth harmonic, i.e. a 5 MHz bandwidth should suffice for the input pulse used in the present work. In this regard, it is also worth noting that the configuration of two glass rods in contact that was used by Solodov [7] is very similar to the 1D model depicted in Fig. 1, except that the contact interaction leads to a transmitted wave in addition to the reflected wave. Thus, for an experimental verification of TR invariance using that configuration, one would need to time reverse and re-emit both the transmitted and the reflected pulses. A finiteelement computational model involving transmission and reflection, and also incorporating varying levels of frictional dissipation, is currently being investigated, and the results will be reported elsewhere.
Acknowledgment This study was supported in part by an Australian Research Council (ARC) Discovery Grant (DP150101899).
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Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Time reversal invariance for a one-dimensional model of contact acoustic nonlinearity Philippe Blanloeuil a,n, L.R. Francis Rose a, Martin Veidt b, Chun H. Wang c a b c
Sir Lawrence Wackett Aerospace Research Centre, School of Engineering, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia School of Mechanical & Mining Engineering, University of Queensland, Brisbane, QLD 4072, Australia School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney NSW 2052, Australia
a r t i c l e i n f o
abstract
Article history: Received 13 October 2016 Received in revised form 27 January 2017 Accepted 31 January 2017 Handling editor: L. G. Tham Available online 10 February 2017
The interaction of a one-dimensional (1D) wave packet with a contact interface characterized by a unilateral contact law is investigated analytically and through a finite difference model. It is shown that this interaction leads to the generation of higher harmonic, sub-harmonic and zero-frequency components in the reflected wave, resulting in a pulse distortion that is attributable to contact acoustic nonlinearity. However, the results also show that the re-emission of a time reversed version of this distorted first reflection results in a healing of the distortions and a perfect recovery of the original pulse shape, thereby demonstrating time reversal invariance for this type of contact acoustic nonlinearity. A step-by-step analysis of the contact interaction provides insights into both the distortion arising from the first interaction and the subsequent healing during the second interaction. These findings suggest that time reversal invariance should also apply more generally for scatterers exhibiting non-dissipative contact acoustic nonlinearity. & 2017 Elsevier Ltd. All rights reserved.
Keywords: Contact acoustic nonlinearity Time reversal invariance Finite difference method
1. Introduction The application of nonlinear acoustics for nondestructive evaluation and structural health monitoring has attracted considerable research interest over the past two decades, driven by the prospect that various forms of structural damage may induce a nonlinear response that could lead to earlier damage detection than would be possible through conventional linear ultrasonics [1–3]. It is pertinent to distinguish between cases where the structural damage (and hence the source of nonlinearity) is more-or-less uniformly distributed throughout the structure, so that the response can be adequately modelled by an appropriate nonlinear constitutive equation for the material [4–6], and cases where the damage is localized, with the surrounding material behaving linearly. The present work is concerned with the latter case. In practice, two important forms of localized damage are fatigue cracks in structural alloys and delaminations in composite laminates. For both of cases, the nonlinear response can generally be attributed to contact acoustic nonlinearity (CAN) [7–15], which induces the generation of new frequency components such as higher harmonics, sub-harmonics and zero frequency (DC) response. The mechanisms involved in CAN include clapping between the contacting interfaces [7,9], as well as dissipative mechanisms due to frictional sliding [8,12]. Theoretical models of varying levels of sophistication have been proposed for all of these mechanisms, as comprehensively reviewed in [14]. We note in particular the recent work on the vibrational response of n
Corresponding author. E-mail address: [email protected] (P. Blanloeuil).
http://dx.doi.org/10.1016/j.jsv.2017.01.050 0022-460X/& 2017 Elsevier Ltd. All rights reserved.
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beams with breathing cracks [15–18], of plates with delaminations [19,20], as well as wave scattering by cracks and delaminations [21–25] as examples of the current state of the art for CAN modelling. The time reversal (TR) method [26–28] allows one to reconstruct a source signal at the excitation point from signals recorded and reemitted at some measurement points away from the source, after being reversed in time. This reversibility is based on the spatial reciprocity and time invariance of the linear wave equation, and is referred to as TR invariance. This property has been successfully exploited to derive various TR techniques for detecting and imaging structural damage through a wide variety of distinct approaches [29–42]. The TR invariance has been demonstrated in various propagation media and holds for linear scatterers as amply demonstrated by Fink et al [26,28,43], and hence for damage that can be modelled as linear scatterers, e.g. corrosion thinning or open cracks. TR invariance has also been verified for classical material nonlinearity as long as the propagation distance is shorter than the shock formation distance [44]. In that case, the wave form is progressively recovered during the back propagation and the distortions accumulated during the initial propagation are cancelled. However, to the best of our knowledge, the issue of time reversal invariance in the presence of CAN at a localized nonlinear defect has not been investigated previously. It can be expected on physical grounds that time reversal invariance will not hold in the presence of dissipative mechanisms such as frictional sliding, but it is not immediately obvious whether or not this invariance may still hold for frictionless contact that can generate nonlinear responses such as pure clapping at the contact interface. This is addressed in the present work. Of particular relevance regarding the present work is the suggestion by Sohn et al. [45,46] that the nonlinearity induced by the presence of damage would result in a breakdown of TR invariance, and therefore experimental observation of a breakdown could serve as a baseline-free diagnostic for detecting damage [47]. However, considering the spectral enrichment due to CAN, it is not clear whether the TR breakdown is attributable to the nonlinear nature of the defect, or to the limited bandwidth of the sensors and their inability to capture the full range of frequency information. Accordingly, a numerical modelling is employed here in order to investigate the TR process solely associated with CAN without any possible bandwidth limitations of practical sensors and actuators. A simplified one-dimensional (1D) model of CAN is considered in this work, whereby a longitudinal wave packet interacts with a contact interface between the propagation medium and a rigid boundary acting as a limiter. A similar configuration has recently been used to model the vibrational response of a rod with a breathing crack in [48]. CAN is modelled here with a unilateral contact law, which correctly captures the dynamics of normal impacts between contacting surfaces associated with clapping. A time explicit finite differences (FD) scheme is used to solve the propagation problem. The presentation is organized as follows. The configuration and governing equations are presented in Section 2, as well as an analytical representation for the incident and reflected wave field. A computational approach based on finite difference approximations is outlined in Section 3. Detailed results are presented in Section 4, and Section 5 provides concluding remarks as well as indications for future work.
2. Formulation of a 1D propagation model with CAN A semi-infinite isotropic and homogeneous medium in contact with a rigid plane is considered as shown in Fig. 1. The material is steel, with Young’s modulus E = 210 GPa, Poisson’s coefficient ν = 0.3 and density ρ = 7800 kg/m3. A longitudinal wave packet (or tone burst) is generated at x = 0 mm and propagates along the x -axis. Under these conditions, a 1D propagation model can be adopted. The contact interface is made with the rigid plane located at x = L . A pre-existing compressive stress state σ0≤0 is considered to exist at the contact interface. The incident wave packet propagates and interacts with the contact interface, and the reflected wave contains new frequency components due to CAN. Additionally, a transparent boundary is considered at x = 0 mm, so that the wave travelling back from the contact interface is not reflected at x = 0 mm. The 1D wave equation is given by:
Fig. 1. 1D model configuration. A plane wave is propagating in a semi-infinite medium. The solid is in contact with a rigid boundary where contact laws are considered to model CAN. The reflected wave contains higher harmonics due to CAN.
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∂ 2u ( t , x) ∂t 2
−c2
∂ 2u ( t , x) ∂x2
= 0,
517
(1)
where c is the wave velocity. The solution for the displacement u (t , x ) can be expressed as the sum of forward travelling displacement u+ and a backward travelling displacement u−:
u ( t , x ) = u + ( t − x / c ) + u− ( t + x / c )
(2)
In the case of a longitudinal wave, the incident wave can generate only compression or tension at the contact interface, where tension tends to open the interface. The contact behavior is modelled by a unilateral contact law defined as follows:
where σn is the normal stress computed from the total displacement gradient ∂u/∂x at the interface, and σ0 is a prescribed static compressive stress acting at the interface. The first line in Eq. (3) indicates that the elastic medium cannot penetrate the rigid plane. A negative displacement corresponds to an opening of the contact interface. The second line in Eq. (3) indicates that only compression can be applied at the interface. Finally, the last line is the complementary condition that indicates that the interface is either open or closed. Considering 1D propagation with the solution defined by Eq. (2), the contact problem can be solved by considering separately the closed case and the open case. 2.1. Closed interface The medium is assumed to be initially in contact with the rigid plane. Thus the displacement at x = L is zero:
u ( t , x) = u+( t−x/c )+u−( t +x/c )=0,
x=L
(4)
Thus, the reflected wave is the opposite of the incident wave: u−( t + x/c )= − u+( t − x/c ). The stress generated by the displacement is computed as:
σ ( t , x) = E
∂u (t , x) E = −u+′ ( t−x/c )+u−′ ( t +x/c ) ∂x c
(
)
(5)
where the derivatives are expressed as follow:
⎧ u ′ t −x / c ) = ⎪ ⎪ +( ⎨ ⎪ u ′ ( t +x / c ) = ⎪ ⎩ −
⎞ 1 ⎜⎛ c u̇ ( t , x)− σ ( t , x) ⎟ ⎠ 2⎝ E ⎞ 1 ⎜⎛ c u̇ ( t , x)+ σ ( t , x) ⎟ ⎠ 2⎝ E
(6)
where u′ denotes the derivative with respect to the argument, whereas u̇ denotes the partial derivative with respect to time. The stress at the contact interface is given by σn ( t )=σ ( t , L ). Note that when the interface is closed the normal stress at the interface is up to twice the incident stress. As long as the sum σn (t ) + σ0 is negative, the interface is closed. The opening of the contact interface thus occurs when σn (t ) + σ0=0. 2.2. Open interface It follows from Eqs. (3) and (5) that the interface opens when
E −u+′ ( t−L/c )+u−′ ( t +L/c ) = − σ 0 c
(
)
(7)
Since Eq. (7) holds for x = L , the space dependence is omitted in the following. This relation is integrated in time between the last contact instant τ and the current time t to find the opening displacement and the displacement associated to the backward propagating wave at x = L ,
u−( t ) = u+( t )+( u−( τ )−u+( τ ) )−
cσ 0 ( t −τ ) E
(8)
During the contact stage, Eq. (4) gives u−( τ )= − u+( τ ), so that Eq. (8) can be re-written as:
u−( t ) = u+( t )−2u+( τ )−
cσ 0 (t−τ ) E
(9)
This expression gives the contribution of the backward propagating wave during the opening stage. Finally, the total
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displacement at the interface, or opening gap, is given by:
u ( t ) = 2u+( t )−2u+( τ )−
cσ 0 (t−τ ) E
(10)
This last equation is true as long as the interface is open. When u ( t , L )=0, then contact is activated and we revert to the equations presented in Section 2.1. One can see that for this 1D case it is possible to solve the contact problem analytically with a piecewise resolution. In this work, the problem is solved numerically with the FD scheme presented in the next section. Nevertheless, the analytical results presented in this section will prove useful for understanding the mechanisms involved at the contact interface. This will be especially valuable for explaining the observed results when the incident wave is the TR of an initially reflected wave.
3. Computational procedure The wave equation is solved in time using the classic second order finite difference approach which allows one to obtain the solution at a new time step by a simple matrix-vector product [49]. The novel step in the present work is the implementation of the unilateral contact law defined by Eq. (3). A transparent boundary condition is used at x = 0 to reduce the size of the computational domain. Thus, at x = 0 mm, only the incident wave, or excitation, is considered:
u+′ ( t−x/c ) = f ( t ),
x =0
(11)
where u+′ ( t − x/c ) is defined by Eq. (6) and f (t ) is the source term, equivalent to an imposed particle velocity. This equation defines the incident wave and is transparent to the reflected wave. This condition is rewritten into a strain formulation:
2f ( t ) ∂u (t , x) u̇ ( t , x) − = − , ∂x c c
x =0
(12)
Eq. (12) is then discretized using a right off-centered differentiation for the strain, as follows:
2f ( t ) −3u1m +4u2m −u3m 1 u1m +1−u1m − = − δt 2δ x c c
(13)
where δx and δt denote respectively the space step and the time step. The superscript m corresponds to the time step whereas the subscript k corresponds to the space discretization, with k = 1 in Eq. (13) to implement the input condition given by Eq. (12). Eq. (13) can be rearranged to obtain u1m + 1, which is the new value of displacement at the first node k = 1 at the time step m + 1. The wave equation Eq. (1) is discretized using the Euler finite difference scheme [49]:
ukm +1−2ukm +ukm −1 δt 2
−c2
ukm+1−2ukm +ukm−1 = 0, δx2
∀k ∈ ⎡⎣ 2,K −2⎤⎦
(14)
where K is the total number of nodes. This second order general scheme is conditionally stable under the Courant- Friedrichs-Lewy (CFL) condition: cδt /δx<1 [49]. Eq. (14) can be rearranged to obtain ukm + 1, the new value at node k, in terms of known quantities from the previous time steps. Finally, the last two nodes have to be considered, the discretization being directly linked to the contact condition. Thus, two cases must be considered separately. 3.1. Closed interface A contact state is always assumed first. During contact, the displacement of the last node k = K at the interface is zero: uKm=0. This value is injected in the general scheme Eq. (14) to compute uKm−+11. The normal stress at the contact interface is computed by a left off-centered differentiation of Eq. (5):
σn = E
−4uKm−1+uKm−2 +3uKm 2δ x
The stress value σn+σ0 is then evaluated. If the value is negative, the interface remains closed and positive, that means that the interface is under tension and must open.
(15) uKm + 1=uKm .
If the value is
3.2. Open interface When the computed value of σn+σ0 is positive, the interface opens, and Eq. (3b) must be invoked to set σn+σ0=0, where σn is given by Eq. (15). Thus, the displacement of node K is now given by:
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⎛ 2σ δ x ⎞ uKm = ⎜ − 0 + 4uKm− 1 − uKm− 2 ⎟/3 ⎝ ⎠ E
519
(16)
The displacement at the interface uKm given by Eq. (16) is then injected in Eq. (14) to obtain uKm−+11. Finally the new displacement at the interface is derived as uKm + 1=uKm . The open stage lasts until the computed value of displacement at the interface is positive, in which case Eq. (3a) must be invoked. The vector of displacements U m + 1at time step m + 1 is computed from the vector of displacements at the previous time step by a matrix-vector product U m + 1=A c, oU m+B , where A c and A o are the matrices respectively written for contact condition and open condition, as outlined above, and where B contains the excitation source term and the contribution of the static stress σ0 in case of open state only.
4. Numerical investigation of TR invariance with CAN The preceding computational procedure is used to solve two distinct nonlinear propagation problems, namely the forward propagation step labelled E1, and the back-propagation step labelled E2. The reflected wave obtained in E1, which contains new frequency components due to CAN, is selected, time-reversed, and used as the new excitation in E2 . The objective is to verify whether the new reflected wave obtained in E2 recovers the original shape of the incident wave in E1, hence demonstrating TR invariance. 4.1. TR invariance example The incident wave packet is defined as a 1 MHz 10 cycles tone burst windowed by a Hann window. The size of the computation domain is L = 0.05 m and the domain is discretized in space by 800 nodes, so that the wavelength at the incident frequency is divided by 96, thus ensuring that all non-negligible higher harmonics are sufficiently discretized. The time step is computed from the CFL condition, with a 0.5 multiplicative factor. The amplitude of the incident wave is set at 5 nm, which corresponds to a maximum stress of 1.1 MPa in steel. The static stress at the contact interface is specified as σ0= − 1.2 MPa. Fig. 2a shows the time signal recorded at x = 0, for this first propagation E1. One can see first the incident wave packet followed by the reflected wave. The reflected wave is distorted due to CAN. As a consequence, its spectrum contains higher harmonics and a DC component, as shown in Fig. 2b. The DC component reflects the non-zero average of the reflected signal, which can be inferred from the time trace in Fig. 2a. It is interesting to note that the maximum stress created by the incident wave in the medium (i.e. 1.1 MPa) is smaller than the prescribed static stress at the interface (1.2 MPa), so that one might expect that the stress is too low to open the interface. However, when the interface is closed, the displacement at the
Fig. 2. Displacement history and spectrum for the first propagation step E1. (a) Time signals at x ¼0, showing the incident pulse followed by a distorted reflected pulse. (b) spectrum of the reflected pulse.
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Fig. 3. Displacement and stress history at the contact interface during the first propagation step E1. (a) opening displacement at the contact interface and (b) contact stress.
interface is zero and the stress is doubled, so that the stress generated by the incident wave at the interface has actually a peak value of 2.2 MPa if the interface remains closed, hence large enough to open the interface. The time history of the displacement and contact stress at the interface are shown in Fig. 3. One can see that the interface does open, as indicated by the negative displacements in Fig. 3a, and that during the open phase, the stress is truncated at a maximum value of 1.2 MPa so that the total contact stress σn+σ0 is never positive, in accordance with the contact law Eq. (3). This truncation of the contact stress profile is at the origin of the new frequency components as well as the DC component. However, this truncation does not mean that the reflected stress profile exhibits a similar truncation, contrary to what might be expected intuitively. To clarify this point, Fig. 4 shows the stress profiles corresponding to the displacement of Fig. 2a at x = 0. One can see that the incident pulse exhibits a maximal stress of 1.1 MPa, whereas the reflected pulse exhibits strong distortions and has a maximal stress value that exceeds the incident maximal stress value, but no evidence of truncation. However, higher stress values are generated over shorter periods of time so that energy is actually conserved.
Fig. 4. Time history of the stress at x ¼ 0 for the first propagation step E1.
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Fig. 5. Displacement history and spectrum for the first propagation step E2 . (a) Time signals at x¼ 0, showing the incident distorted pulse followed by a reflected pulse free of distortions. (b) spectrum of the reflected pulse.
The reflected pulse at x = 0 from step E1 is selected by applying a time gating and then time reversed to be used as the new excitation for the second step E2. Fig. 5a shows the corresponding displacements at the origin, where the incident pulse can be observed to be the time reversed version of the reflected wave obtained in E1, which is shown in Fig. 2a. It can be seen that the new reflected wave in Fig. 5a is identical to the first incident wave of E1. This is confirmed by considering its spectrum shown in Fig. 5b, where only the fundamental frequency component is present. Considering the stress profile shown in Fig. 4, one can see that the new input is again able to activate a loss of contact at the interface, as indicated in Fig. 6. However, this second interaction does not result in the generation of additional higher harmonics, but on the contrary leads to a “healing” of the distortions, and thereby to a recovery of the original signal. Thus, the TR process has cancelled the distortions generated by the first interaction with the contact interface, and TR invariance is confirmed for this 1D model of CAN.
Fig. 6. Displacement and stress history at the contact interface during the first propagation step E2 . (a) opening displacement at the contact interface and (b) contact stress.
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Fig. 7. Displacement profiles for a single-cycle input pulse. (a) displacement for E1 and (b) displacement for E2 , in the case of a one cycle pulse. The incident and reflected pulses are shown in more detail in the rectangular insets.
It is emphasized that this demonstration of TR invariance depends crucially on employing a sufficiently fine time interval for modelling and sampling, so as to capture adequately all the higher harmonics generated by the pulse distortion, as indicated in Fig. 2b. If instead the reflected signal from the first propagation step E1 is filtered to retain only the contribution around the fundamental frequency prior to time reversal and re-emission, as proposed in [39,45–47], the new reflected pulse would not constitute an exact reconstruction of the original input pulse, and TR invariance would therefore appear to have been lost. However, this apparent loss of TR invariance would be attributable in the present case to an incorrect implementation of the TR protocol. Following this remark on the need to retain the full frequency spectrum shown in Fig. 2b, it is also recalled that the linear superposition principle does not hold in the presence of nonlinearity. This means that TR invariance will again appear to be violated if the contributions from the various frequency components in Fig. 2b, (i.e. the DC, fundamental and higher harmonic components), are separately time reversed and re-emitted sequentially, with the reflection of these separate TR signals being subsequently summed to generate the reconstructed pulse. Instead, it is essential that the TR signals corresponding to the various frequency components should be re-emitted simultaneously, to ensure the perfect reconstruction shown in Fig. 5. 4.2. Generation and cancellation of distortions in nonlinear TR To provide a deeper insight into the detailed contact dynamics leading to TR invariance, the FD simulation is repeated in this section for a single-cycle pulse windowed by a Hann window. The incident pulse now has an amplitude of 3.2 nm due to the windowing. The static stress at the interface is again set to σ0= − 1.2 MPa. The TR invariance is still observed after the propagation E2, as shown in Fig. 7. Note that the reflected pulse in Fig. 7b corresponds exactly to the incident pulse in Fig. 7a, the apparent time reversal being due to the reflected pulse travelling in the opposite direction of propagation. A step by step analysis of the displacement and stress is first carried out in the case of the propagation E1, based on the development of Section 2. This aims to detail the generation of the distortions during E1. Then, the same analysis is carried out for E2. Fig. 8a-b show respectively the displacement and stress profiles of the input at x = 0, in the case of the propagation E1. This displacement corresponds to the first pulse observed in Fig. 7a. This excitation will propagate towards the interface and the same waveform arrives at the contact interface because the propagation is non-dispersive. Therefore, the waveform at x = 0 is used as the incident wave at the contact interface x = L where the incident and reflected waves co-exist for the duration of the input pulse. The interface is initially closed, with a prescribed static stress σ0= − 1.2 MPa. The displacement associated with the incident wave packet is first positive as shown by the solid red line in Fig. 8a, which corresponds to a compression. The interface remains closed as shown in Fig. 8c, and the displacement of the reflected wave is simply the opposite of the displacement of the incident wave, as indicated by Eq. (4) and shown in Fig. 8e. The stress associated with this reflected wave is again compressive, as shown in Fig. 8f, so that the dynamic stress at the interface is twice the stress generated by the incident wave, as indicated by Eq. (5) and shown in Fig. 8d.
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Fig. 8. Displacement - stress analysis for the generation of the distortions during E1. (a) incident displacement u+(t − x/c ) at x = 0 ; (b) incident stress σ+(t − x/c ) at x = 0 ; (c) opening displacement u (t , x ) at x = L ; (d) contact stress σn (t ) at the interface; (e) displacement for the reflected wave u−(t + x/c ) at x = L ; (f) stress for the reflected wave σ−(t + x/c ) at x = L ; (g) numerical solution for the displacement of the reflected wave u−(t + x/c ) at x = 0 ; and (h) numerical solution for the stress of the reflected wave σ−(t + x/c ) at x = 0 .
During the second half of the input cycle, the stress at the interface is positive and eventually attains the value −σ0=1.2 MPa, whereupon the interface opens, according to the contact laws of Eq. (3). The stress σn at x = L remains constant during the open stage, in accordance with the requirement σn+σ0=0 in Eq. (3). The opening displacement during that stage is given by Eq. (10): u ( t )=2u+( t )−2u+( τ )−cσ0/E (t − τ ). This portion of the solution is plotted as a dashed blue line in Fig. 8c. Meanwhile, the displacement corresponding to the reflected wave at the interface is given by Eq. (9), which is recalled here: cσ u−( t )=u+( t )−2u+( τ )− E0 (t − τ ), where τ is the last instant of contact. This part of the solution is plotted in Fig. 8e. One can clearly see that this part of the solution differs from the incident sinusoidal wave form. It creates the distortion in the received signal. The stress corresponding to the reflected wave is σ−( t )=σn ( t )−σ+(t ) and is plotted in Fig. 8f. The interface is open as long as the opening displacement u (t , L ) is negative, which is found from Fig. 8c. After this instant, the interface is closed again, which means that the displacement of the reflected wave becomes once again the opposite of the incident wave displacement, and the stress corresponding to reflected wave is equal to the stress generated by the incident wave. This completes the plots in Fig. 8e and Fig. 8f respectively. It is noted that the transition from open to closed corresponds to an impact, which introduces a slope discontinuity in the displacement history in Fig. 8e, and a jump discontinuity in the stress history in Fig. 8f. The occurrence of such impacts have also been reported in previous finiteelement simulations of breathing cracks [16] and delaminations [20] involving the clapping mechanism of CAN. Finally, the numerical solution for the displacement and the stress signals of the reflected wave received at x = 0 are shown in Fig. 8g and Fig. 8h respectively. The wave forms correspond to the ones predicted analytically by the preceding step-by-step approach, with the difference that the discontinuities generate oscillations in the numerical signals attributable to contact instabilities. The distortions are directly generated by the CAN, as expected, and are in agreement with the theoretical contact analysis. During the second propagation step E2 , the received signal is time reversed and used as the new source. Therefore, the input displacement is the TR of the signal shown in Fig. 8g, with the associated input stress being the TR of the signal shown
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Fig. 9. Displacement - stress analysis for the generation of the distortions during E2 (a) incident displacement u+(t − x/c ) at x = 0 ; (b) incident stress σ+(t − x/c ) at x = 0 ; (c) displacement for the reflected wave u−(t + x/c ) at x = L ; (d) stress for the reflected wave σ−(t + x/c ) at x = L ; (e) numerical solution for the displacement of the reflected wave u−(t + x/c ) at x = 0 ; and (f) numerical solution for the stress of the reflected wave σ−(t + x/c ) at x = 0 .
in Fig. 8h. Ideally, the displacement and stress signals should be the TR of those given in Fig. 8e-f. Therefore, these last two signals are considered to analyze the interaction of TR pulse and the contact interface during E2. The new incident displacement and stress are shown in Fig. 9a-b respectively. Considering the incident stress, it can be seen that the interface will open at the instant of the stress jump in Fig. 9b. During the open stage, the stress corresponding to the reflected wave is again σ−( t )=σn ( t )−σ+(t ), where σn ( t )= − σ0 . Therefore, the initial incident stress of E1 is recovered, as shown in Fig. 9d as the dashed blue line. The displacement corresponding to the reflected wave at the interface is again computed using Eq. (9). This leads to the part of the solution plotted as the dashed blue line in Fig. 9c, which clearly corresponds to the sinusoidal waveform part of the incident wave of E1. Finally, when the opening displacement is equal to 0, the interface closes and the reflected displacement is equal and opposite to the incident displacement, whereas the reflected stress is equal to the incident stress, as shown in Fig. 9c-d. The waveform has thus recovered its sinusoidal shape and corresponds to the incident wave of E1, keeping in mind that the reflected wave is travelling in the opposite direction to the incident pulse. The corresponding numerical results for displacement and stress are shown in Fig. 9e-f and correspond to the step-by-step approach. It can be seen that the analytical stress discontinuity in Fig. 9b leads to oscillatory behavior in the numerical computations in Fig. 9f. The preceding step-by-step analysis provides a detailed insight into the contact dynamics due to clapping, and the mechanisms leading to a full recovery of an original pulse shape during a correct implementation of the TR protocol.
5. Conclusions The time reversal invariance involving CAN has been investigated in the particular case of frictionless contact. A 1D FD numerical model has been used to solve the propagation problem of longitudinal waves that interact with a contact interface where unilateral contact is considered. The forward propagation step leads to the generation of new frequency components that reflect the distortions of the received reflected wave. The latter is selected and reversed in time before being re-emitted, in accordance with the TR protocol. The numerical results show that the distortions generated at the contact interface during the first interaction are locally cancelled during the second interaction with the interface, which demonstrates that TR invariance holds for the present type of contact nonlinearity. This result is not immediately obvious, but both the generation and cancellation of the distortions at the contact interface have been explained by a step-by-step analysis of the contact dynamics. This detailed analysis suggests that TR invariance should hold more generally for 3D scatterers exhibiting non-dissipative CAN. This theoretical study brings some insights regarding the retro-propagation of ultrasonic waves in case of CAN. It appears
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that the nonlinear clapping mechanism is not sufficient to break the TR invariance, contrary to the assumption in [39,45–47] that the presence of damage would necessarily lead to a breakdown of TR invariance. However, it is also noted that common forms of CAN are almost invariably associated with dissipative mechanisms [8,12,13], which would lead to a breakdown of TR invariance, so that the diagnostic test proposed in [45–47] may be still be useful in practice. The point here is that this breakdown of TR invariance is not intrinsically associated with CAN, as demonstrated by the present model, but it is due to the concomitant dissipative mechanisms, which have not been modelled in the present work. Furthermore, in practice, ultrasonic transducers necessarily have a limited bandwidth. This practical limitation may also compromise the practical demonstration of TR invariance, as noted earlier in Section 4.1. However, referring to the spectrum of the reflected wave shown in Fig. 2b, it is reasonable to expect that TR invariance would be verified experimentally provided that the available bandwidth is sufficient to capture contributions up to, say, the fourth harmonic, i.e. a 5 MHz bandwidth should suffice for the input pulse used in the present work. In this regard, it is also worth noting that the configuration of two glass rods in contact that was used by Solodov [7] is very similar to the 1D model depicted in Fig. 1, except that the contact interaction leads to a transmitted wave in addition to the reflected wave. Thus, for an experimental verification of TR invariance using that configuration, one would need to time reverse and re-emit both the transmitted and the reflected pulses. A finiteelement computational model involving transmission and reflection, and also incorporating varying levels of frictional dissipation, is currently being investigated, and the results will be reported elsewhere.
Acknowledgment This study was supported in part by an Australian Research Council (ARC) Discovery Grant (DP150101899).
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