Simulation of detecting contact nonlinearity in carbon fibre polymer using ultrasonic nonlinear delayed time reversal
SSimulation of detecting contact nonlinearity in carbon fibre polymerusing ultrasonic nonlinear delayed time reversal
Martin Lints , , Andrus Salupere , Serge Dos Santos Tallinn University of Technology, Department of Cybernetics,Akadeemia tee 21, 12618 Tallinn, Estonia. [email protected] INSA Centre Val de Loire, Blois Campus;COMUE ”L´eonard de Vinci”, U930 ”Imagerie et Cerveau” Inserm,3 rue de la Chocolaterie, CS23410, 41034 Blois, France.
Abstract
A finite element method simulation of a carbon fibrereinforced polymer block is used to analyse the non-linearities arising from a contacting delamination gapinside the material. The ultrasonic signal is amplifiedand nonlinearities are analysed by delayed Time Re-versal – Nonlinear Elastic Wave Spectroscopy signalprocessing method. This signal processing methodallows to focus the wave energy onto the receivingtransducer and to modify the focused wave shape,allowing to use several different methods, includingpulse inversion, for detecting the nonlinear signatureof the damage. It is found that the small crack withcontacting acoustic nonlinearity produces a noticeablenonlinear signature when using pulse inversion signalprocessing, and even higher signature with delayedtime reversal, without requiring any baseline infor-mation from an undamaged medium.
In the past, the use of carbon fibre reinforced poly-mer (CFRP) has been limited to non-structural partsof high-tech aeronautical products. In recent times,due to the effort of weight reduction and product life-time enhancement, the application areas of CFRPhave widened to the load-bearing parts of the aero-nautical, automotive and civil engineering products.Due to the increased demands on the strength of theCFRP products and possible complex failure mecha-nisms, the Non-Destructive Testing (NDT) methodsof CFRP have been an important applied and aca-demic problem.The complex failure mechanisms of CFRP includemicrocracking and delamination. Microcracking canoccur at lower loads or due to aging and can bedifficult to examine using ultrasonic NDT. With in-creased loading, the damage can evolve to delami-nations, a very fine cracking between the layers ofthe CFRP. These damages are difficult to detect us- ing ultrasonic methods due to their small thicknesses.The damage can exhibit itself as a contact acousticalnonlinearity [1]. A statistical distribution of microc-racks or delamination damage in the material couldalso be described by hysteresis in a continuum ma-terial model [2, 3, 4]. This can also be applicablefor other materials than CFRP, for example biologi-cal tissues [5, 6]. Nonlinear ultrasonic methods havebeen in development for detecting and localizing fa-tigue and micro-crack damage by their nonlinear ef-fects [7, 8]. The detection of harmonic overtonesis one of the simplest measures of nonlinearities [9].Many nonlinear analysis methods not requiring filter-ing have been developed, for example scaling subtrac-tion method [10, 11] or pulse inversion (PI) with itsgeneralizations [12, 13], and applications of time re-versal using scattering as new sources.This paper proposes a delayed TR-NEWS signalprocessing method [14] for detecting the nonlinearsignature of a single small crack in CFRP as con-tact acoustical nonlinearity. In the Finite ElementMethod (FEM) simulation, the CFRP is modelled asanisotropic, layered medium. The ultrasonic signal isfocused by TR-NEWS to the region of the materialwith the defect. The nonlinear signature of the crackis analysed by PI and compared with the delayed TR-NEWS method, which allows to create arbitrary waveenvelope at the focusing region of TR-NEWS. It isused here to create an interaction of waves near thedamage. The signature of the damage appears as thenonlinear effect of the wave interaction on the con-tacting crack. This signal processing requires onlyone transmitting and one receiving transducer. Theeffectiveness of the delayed TR-NEWS method hasbeen shown in the previous work by physical experi-ments and simulations in undamaged and linear ma-terials [14]. In this paper, the FEM simulation modelis advanced further by including absorbing boundaryconditions and the contacting crack defect in the ma-terial. a r X i v : . [ c s . C E ] F e b ints et al. , p. This section describes the simulation which is basedon a physical experiment, and describes the differ-ences and similarities between the simulation and theexperiment. It shows some important points aboutthe mathematical model, the delayed TR-NEWS sig-nal processing and the FEM simulation. Detailed in-formation about the derivation of mathematical andFEM model is available at [15].
The test object is a CFRP block consisting of 144layers (Fig. 1). It is composed of fabric woven fromyarns of fibre and impregnated with epoxy. The cross-section of the yarns have elliptical shape (Fig. 2) andthe material has inclusions of pure epoxy, so a wavepropagating through the material will encounter yarns(fibres with epoxy) and areas of pure epoxy.Figure 1: (Colour online) CFRP block in the test con-figuration with transmitting transducer on side andreceiving on topThe simulation is in time domain, since the TR-NEWS procedure relies on transient echoes and com-plex wave motion for the wave energy focusing pro-cess. Due to the heavy computational cost of timedomain simulation, a simple laminate model is usedwhere: i) the material consists of homogeneous lay-ers, ii) each layer has its own elasticity properties,and iii) dispersion arises due to the periodical dis-continuity of the material properties. It consists ofCFRP layers with 90 ◦ /0 ◦ weave, 45 ◦ /45 ◦ weave andepoxy layer. The thicknesses of the layers are givenby random variable functions which reflect the actualstructure of the material. The random variable distri-bution, describing the CFRP structure, is measuredfrom a close-up image of the CFRP test object [15].This links the distribution of the microstructure in-side the actual material with the thicknesses of thelayers in the laminate model. It should enable a morerealistic simulated material having dispersion effectsdue to discontinuities.The three different kind of layers have the followingmechanical properties: i) isotropic pure epoxy: E = Figure 2: The layered structure of the CFRP with thefabric yarns in tight packing and epoxy in the voids3 . ν = 0 . ρ = 1200 kg/m ; ii) transverselyisotropic composite with 0/90 ◦ weave: E = E =70 GPa, G = 5 GPa, ν = 0 . ρ = 1600 kg/m ;and iii) transversely isotropic composite with 45 ◦ /45 ◦ weave: E = E = 20 GPa, G = 30 GPa, ν =0 . ρ = 1600 kg/m . For the simulation, a laminatemodel was constructed using 50 pairs of epoxy andcarbon fibre layers, where carbon fibre weave directionalternated between each pair.Figure 3: (Colour online) The laminate materialmodel with layers of stochastic thicknesses and ab-sorbing boundary conditions on bottom and leftboundaries and four fixed degrees of freedomThe boundary conditions of the model (Fig. 3) in-clude Lysmer-Kuhlemeyer absorbing boundary condi-tions [16] so the wave energy would pass through thesimulation region. Four degrees of freedom (DOFs)are fixed, the rest are free. The simulation model in-cludes a contacting delamination defect in the ma-terial near the receiving transducer (Fig. 4). Thetransmitting shear wave transducer can send maxi-mum 50 kPa pulse at 70 ◦ degree angle. This subsection repeats the signal processing methodthat is applied for this problem and has been pub-lished in the previous work [14]. It is included for aself-contained discourse in this paper.In the physical experiments, on which the simula-tion is based on, the CFRP block (Fig. 1) was studiedusing TR-NEWS NDT equipment and signal process-ing methods [14]. The 2D FEM simulations reflect ints et al. , p.
321 4 5 TxRx mm mm . mm . mm . mm mm Figure 4: Schematic (not to scale) of the simulationgeometry, location of crack, transmitter and receiverpoints without the layersit as closely as possible in terms of transducer place-ment, frequencies and signal processing.The roles of the transducers are not changed duringthe experiment: the focusing of the ultrasonic waverelies on the TR-NEWS signal processing. This is atwo-pass method where the receiving and transmit-ting transducers do not change their roles. In thissense the “Time Reversal” describes the signal pro-cessing method which accounts for internal reflectionsof the material as virtual transducers, used for focus-ing the wave in the second pass of the wave transmis-sion. The placement of the transducers is not impor-tant from the signal processing standpoint: in NDTinvestigation they could be placed arbitrarily and theydo not have to be in line with each other, but theconfiguration must remain fixed during the completeTR-NEWS procedure.Figure 5 outlines the TR-NEWS signal processingsteps. The simulation uses the same signal processingsteps as are usually applied to physical experiments.Firstly the chirp-coded excitation c ( t ) is transmittedthrough the medium. c ( t ) = A · sin ( ψ ( t )) , (1)where ψ ( t ) is linearly changing instantaneous phase.In this work, a linear sweep from 0 to 2 MHz wasused. Then the chirp-coded coda response y ( t ) witha time duration T is recorded at the receiver y ( t, T ) = h ( t ) ∗ c ( t ) = (cid:90) R h ( t − t (cid:48) , T ) c ( t (cid:48) ) dt (cid:48) , (2)where h ( t − t (cid:48) , T ) is the impulse response of themedium. The y ( t, T ) is the direct response from thereceiving transducer when the chirp excitation c ( t ) istransmitted through medium. Next the correlationΓ( t ) between the received response y ( t, T ) and chirp-coded excitation c ( t ) is computed during some time period ∆ t Γ( t ) = (cid:90) ∆ t y ( t − t (cid:48) , T ) c ( t (cid:48) ) dt (cid:48) (cid:39) h ( t ) ∗ c ( t ) ∗ c ( T − t, T ) , (3)where the h ( t ) ∗ c ( t ) ∗ c ( T − t, T ) is pseudo-impulse re-sponse which is proportional to the impulse response h ( t ) if using linear chirp excitation for c ( t ) becauseΓ c ( t ) = c ( t ) ∗ c ( T − t ) = δ ( t − T ). Therefore theactual correlation Γ( t ) ∼ h ( t ) contains informationabout the wave propagation paths in complex media. T x x t(2)initial responset(4)side lobes αβ α + β αβαβ R x focusingT x R x x T x(2) t f Figure 5: (Colour online) Schematic process of TR-NEWS with the virtual transducer concept. (1) Theinitial broadband excitation T x ( t ) propagates in amedium. (2) Additional echoes coming from inter-faces and scatterers in its response R x could be associ-ated to a virtual source T (2) x . (3) Applying reciprocityand TR process to R x . (4) The time reversed new ex-citation T x = R x ( − t ) produces a new response R x (the TR-NEWS coda y T R ( t )) with a spatio-temporalfocusing at z = 0; y = 0; t = t f and symmetric sidelobes with respect to the focusing.Time reversing the correlation Γ( t ) from the previ-ous step results in Γ( − t ) used as a new input signal.Re-propagating Γ( − t ) in the same configuration anddirection as the initial chirp yields y T R ( t, T ) = Γ( T − t ) ∗ h ( t ) ∼ δ ( t − T ) , (4)where y T R ∼ δ ( t − T ) is now the focused signal underreceiving transducer where the focusing takes placeat time T . This is because Γ( t ) contains informationabout the internal reflections of the complex media,and transmitting its time reversed version Γ( T − t )will eliminate these reflection delays by the time sig-nal reaches the receiver, resulting in the focused signal y T R (Eq. (4)). The test configuration must remainconstant during all of these steps, otherwise the fo-cusing is lost. The steps of this focusing process in aphysical experiment are shown in Fig. 6. ints et al. , p. A . U (1) 0 320 640 960 12801.00.50.00.51.0 (2)0 320 640 960 1280t, µs1.00.50.00.51.0 A . U . (3) 0 320 640 960 1280t, µs1.00.50.00.51.0 (4) Figure 6: Bi-layered aluminium experimental chirp-coded TR-NEWS signal processing steps: (1) chirpexcitation, (2) output recorded at Rx, (3) cross-correlation between input and output, (4) focusingresulting from taking time-reversed cross-correlationas new input [14].PI is an established method for detecting nonlin-earities [12]. The procedure used here involves con-ducting TR-NEWS measurements with positive andnegative sign for A in Eq. (1) and comparing the fo-cused signals. Differences could indicate the presenceof nonlinearities.Delayed TR-NEWS signal processing considers asingle y T R focusing wave as a new basis which can beused to build arbitrary wave shapes at the focusing.This is done by time-delaying and superimposing n time-reversed correlation Γ( T − t ) signals (Fig. 7 leftcolumn)Γ s ( T − t ) = n (cid:88) i =0 a i Γ( T − t + τ i ) = n (cid:88) i =0 a i Γ( T − t + i ∆ τ ) , (5)where a i is the i -th amplitude coefficient and τ i the i -th time delay. In case of uniform time delay the ∆ τ is the time delay between samples. Upon propagat-ing this Γ s ( t − T ) through the media according to thelast step of TR-NEWS, a delayed and scaled shape ofsignal at the focusing point can be created. The de-layed TR-NEWS signal processing optimization canbe used for amplitude modulation, signal improve-ment and sidelobe reduction [14].It is possible to predict what the delayed TR-NEWSfocusing output would be in a linear material (Fig. 7right column): y dT R ( t ) = (cid:34)(cid:88) i a i Γ c ( T − t + τ i ) (cid:35) ∗ h ( t ) linearity ======== (cid:88) i a i Γ c ( T − t + τ i ) ∗ h ( t ) = (cid:88) i a i y T R ( t − τ i ) . (6)The purpose of the prediction is twofold. Firstly itcan be used to figure out optimal delay and ampli- − − A . U . (1) ⇓ (4) − − A . U . (2) + ⇓ (5) + − − A . U . (3) || ⇓ (6) || Figure 7: Delayed TR-NEWS signal processing stepsin bi-layered aluminium, starting from the cross-correlation step (left column) and prediction of lin-ear superposition of waves (right column): (1) cross-correlation(Eq. (3)), (2) delayed and scaled cross-correlation, (3) linear superposition of two cross-correlations which becomes the new excitation, (4)focusing (Eq. (4)), (5) delayed and scaled focusing,(6) linear superposition of the two focusing peaks.tude parameters a i and τ i beforehand for the de-layed TR-NEWS experiment, using the original fo-cusing peak y T R . Secondly it could be possible toanalyse the differences between the measured delayedTR-NEWS result and its prediction, which acts as abaseline for comparison. The difference could indicatethe magnitude of nonlinearity, because the predictionrelies on the applicability of linear superposition andis found to be accurate in experiments with linear ma-terial [14].
The simulation program considers 2D wave propaga-tion in a solid material with linear elasticity. Thenonlinearity comes from an internal defect, a crackin the computational region (Fig. 4) which can comeinto contact with itself. This contacting nonlinearityhas asymmetric stiffness and is therefore nonclassi-cally nonlinear. Since the CFRP is a complex ma-terial, then in this work it is modelled as a laminatewith anisotropic layers arranged in a periodic manner,described in Section 2.1. Because the physical exper-iment was conducted on the corner of a large CFRPblock, the simulation is also in a semi-infinite quarter-space. The region has two free surfaces for reflectionand two absorbing boundaries for the wave energy toescape.The constitutive equation of the material itself islinear (although anisotropic). The linear plane strainelastodynamics problem is solved ρ ¨ u i − σ ij,j = b i , (7)where ρ is material density, u i is displacement com- ints et al. , p. σ ij is stress component and, b i is body forcecomponent [17]. Einstein summation convention isused and comma in index denotes spatial derivative.The constitutive equation in the variational formula-tion is0 = (cid:90) Ω ( σ ij δε ij + ρ ¨ u i δu i ) dxdy − (cid:90) Ω b i δu i dxdy − (cid:90) Γ t i δu i ds (8)where ε ij is strain and t i is traction component onboundary. In our case the region Ω is a 2D space andboundary Γ surrounding it is a 1D line. The bodyforces are zero in this simulation. Strain is assumedto be small.The matrix formulation of the finite element modelwith damping is M ¨∆ + C ˙∆ + K ∆ = F, (9)where M is mass matrix, K is stiffness matrix, F isexternal forcing and ∆ is displacement vector [15].The damping matrix C is used to apply the Lysmer-Kuhlemeyer absorbing boundary conditions [16] as adiagonal matrix, allowing to take advantage of theexplicit solution scheme.The element matrices are M e = (cid:90) Ω ρ Ψ T Ψ dxdy, (10) K e = (cid:90) Ω B T C e Bdxdy, (11) F = (cid:90) Γ Ψ T f ds, (12)where C e is here the constitutive matrix for the planestrain elasticity.Linear triangular three-node elements (T3), alsoknown as constant strain triangles [17], were chosenfor this problem for the following reasons. Firstly be-cause the epoxy layers in the laminate model can bevery small, therefore small elements are needed any-way, with T3 being computationally cheapest. Sec-ondly, linear elements are well suited for nonlinearproblems: since the strain is constant throughout theelement, the computation of nonlinear constitutive re-lations would also be simple. In this simulation, thematerial itself is linear but future work might includenonlinearity or hysteresis.The T3 element lumped mass matrix [18] is M e = ρA e I , (13)where I is 6 × A e is the areaof the element. The element stiffness matrix is K e = A e B Te C e B e , (14) where matrix B is B = 12 A e β β β γ γ γ γ β γ β γ β , (15)and with x i and y i being the node coordinates [17],then β = y − y , γ = x − x ,β = y − y , γ = x − x ,β = y − y , γ = x − x . (16)The external distributed force is simply divided intorelevant nodes. The Lysmer-Kuhlemeyer absorbingboundary conditions are applied as viscous stresseson the boundaries, which means that they can be ap-plied on DOF basis, making the damping matrix C diagonal. The viscous stresses on the boundary DOFsare c ii = (cid:90) Γ aρV p ds, normal motion DOF , (17) c ii = (cid:90) Γ bρV s ds, shear motion DOF , (18)where Γ is the boundary portion of the element [16].In this work the scaling parameters are a = 1 and b = 1. The wave velocities used for these boundaryconditions are V p = 2972 m/s and V s = 1956 m/s [19].Equation (9) is solved for each timestep ∆ t = 5 · − s by explicit central difference scheme (cid:18) M ∆ t + C t (cid:19) u n +1 = F n − (cid:18) K − M ∆ t (cid:19) u n − (cid:18) M ∆ t − C t (cid:19) u n − . (19)This scheme is solved by dividing the equation by theterm in the first parentheses, which is simple if M and C are diagonal matrices. Each simulation considers a60 µ s time window. There is a single source of nonlinearity in this sim-ulation: the contacting crack fully inside the mate-rial (Fig. 4). If the material is at rest, then the crackis small and straight. In this work, there is neithera preload nor an initial gap in the contacting crack.This simple material defect results in a localised non-classical nonlinearity, which can be analysed by vari-ous signal processing methods.It is known that frictional contact problems can besensitive to timestep length and loading path [20].In this work, it is assumed that the small timesteplength and relatively small forces involved keep theerror small. Therefore an explicit solution methodscheme is utilized, similarly to [21]. A more precisesolution could be expected from an implicit scheme,but that is left for the future. Further refinements ints et al. , p. n sx , n sy ) and on master (higher) ( n mx , n my ), then thenormal contact gap is g N = n sy − n my and the tangen-tial gap (offset) is g T = n sx − n mx . In case of normalpenetration of one surface into another, then g N > g N ≤
0. The coef-ficient of friction is µ = 0 .
6, and the solution aimsto satisfy the Kuhn-Tucker conditions on the cracksurface: g N ≤ ,λ N = σ · n ≤ ,g N · λ N = 0 , (20)where λ N is the normal force on crack, σ is stress and n is the normal vector of the surface. The penaltyplus Lagrange multiplier method is used for normalcontact and the penalty method for friction [23].The contact logic for the node pairs can be summa-rized by following steps. • The initial contact forces are zeroed: normal λ N = 0 and tangential λ T = 0. • System in Eq. (19) is solved. • Vector gap functions are found: g N = n sy − n my and g T = n sx − n mx . • Normal forcing is updated λ N = λ N + g N b where b is some big penalty value and λ N ≥ • Logic diverges to 3 paths:
No force is applied in case of no contact.
Only normal force is applied if preceding stephad no contact or had contact with tangen-tial slip.
Normal and tangential forces are applied ifprevious iteration had non-slip contact. • The normal contact condition is verified by set-ting the penetration value g P = g N where g N ≥
0. Then the L -norm of penetration is evaluated (cid:104) g P | g P (cid:105) < ε where ε is the limiting value for theerror due to contact penetration. If the conditionis not fulfilled, the iteration is repeated, otherwisenew timestep is taken.A more thorough explanation of this contact gap logicis available at [15]. The signal analysis of the time domain simulation re-sults of the damaged and undamaged medium are compared, describing some analysis measures whichcould allow to detect the presence of damage asnonlinearity. The simulation follows ultrasonic TR-NEWS NDT procedures where the transducer datais available as time-series, measured at some specificlocation. The signals are low-pass filtered to keeponly the ultrasonic component. Here five measure-ment points are analysed at various distances from thecrack damage and transmitting transducer (Fig. 4). Avideo of the displacement fields for TR-NEWS focus-ing to point 3 in cracked medium is available at [24].
Figure 8 shows the undamaged CFRP TR-NEWS fo-cusing for the receiver positions 1 to 5 (Fig. 4). It isan ordinary TR-NEWS focusing where at the middleof the signal (30 µ s) is the focusing, surrounded by thesidelobes. There are two aspects to note about thisis figure. Firstly, the sidelobes shift toward the mainfocusing and comparatively decrease in amplitude asthe receiving transducer position shifts toward thetransmitting transducer (from position 1 to position5), indicating lower noise as the signal gets stronger.Secondly, the sidelobes are symmetrical in respect tothe main lobe. This does not happen in nonlinear(damaged) material. The PI results are identical, in-dicating no nonlinearity, and are not shown here. t, ms −1.0−0.50.00.51.01.52.0 u , n m pos. 1pos. 2pos. 3pos. 4pos. 5 Figure 8: (Colour online) Unnormalized TR-NEWSfocusing of undamaged CFRP simulationFigure 9 shows the TR-NEWS results of the crackedCFRP test object simulation for the receiving trans-ducer positions 1 to 5 (Fig. 4). Here the PI signalprocessing is also applied and it shows the nonlin-earity as difference between results from initial chirpsignals with positive and negative sign. This nonlin-ear, damaged case exhibits nonlinearity particularlystrongly in receiving position 3 (near the middle ofthe crack). Also, the sidelobes are unsymmetrical inrespect to the main lobe.Figure 10 shows the envelopes of the PI measure ofnonlinearity across the measuring points. The non- ints et al. , p. −0.8−0.6−0.4−0.20.00.20.40.60.81.0 . A . U . −0.8−0.6−0.4−0.20.00.20.40.60.81.0 . A . U . −0.8−0.6−0.4−0.20.00.20.40.60.81.0 . A . U . −0.8−0.6−0.4−0.20.00.20.40.60.81.0 . A . U . t, ms −0.8−0.6−0.4−0.20.00.20.40.60.81.0 . A . U . differencepositivenegative Figure 9: (Colour online) Normalized TR-NEWS fo-cusing of damaged CFRP simulation with PI appliedto detect nonlinearities as difference between negativeand positive excitationslinearity magnitude depends on the measuring pointlocation in respect to the crack: point 3 near the mid-dle of the crack shows strongest nonlinearity, points 2and 4 show less, and points 1 and 5 show the least. t, ms u , n m pos. 1pos. 2pos. 3pos. 4pos. 5 Figure 10: (Colour online) Envelopes of the PI non-linearity measures from all of the measuring pointsFigure 11 shows the unnormalized focusing signalfor the damaged medium, which can be comparedwith corresponding undamaged result in Fig. 8. Thefocused signals have some interesting properties:1. The highest signal amplitude comes from the t, ms −3−2−101234 u , n m pos. 1pos. 2pos. 3pos. 4pos. 5 Figure 11: (Colour online) Unnormalized TR-NEWSfocusing of damaged CFRP simulationreceiver position closest to the crack midpoint(pos. 3), not from the position closest to thetransmitter (pos. 5).2. Comparing the amplitudes of the positions 2 and4, at far and near side of the crack end respectiveto transmitter: the farther position has larger fo-cusing amplitude than the nearer position. Sincethe simulation region has two absorbing bound-aries, the wave propagation is mostly in one di-rection, therefore the defect between pos. 2 and 4must be capturing the wave energy and the TR-NEWS signal processing is using that energy as anew “virtual transducer” for the pos. 2 focusing.This could be further analysed in future worksfrom the correlation signals which generate thesefocused signals.3. Amplitudes from the measurement positions 1and 5 are “right way” around: the nearer mea-surement point has larger focusing amplitudethan the farther.Figure 12 shows a snapshot of the simulation u dis-placement at a time moment t = 32 . µ s, just after thefocusing. The defect in material is acting as a sourceof new excitation after TR-NEWS focusing. Wave en-ergy is captured between the damage and outside wallof the material and emitted as a wave. Section 2.2 describes the delayed TR-NEWS signalprocessing method which allows to create arbitraryenvelope wave at the focusing (Eq. (5)), instead of thesimple peak of the TR-NEWS. Equation (6) showsthat in linear material, the outcome of the delayedTR-NEWS process can be predicted. Since thismethod with prediction works very well in physicalNDT measurements of linear materials [14], it is nowtested in simulation with the nonlinearity, supposing ints et al. , p. u field attime t = 32 . µ s with a wave emission coming fromthe damaged region. Video available at [24]that the difference between the simulation result andthe linear prediction (Eq. (6)) is due to nonlinear in-teraction of waves in the presence of nonlinearitiesor damage. Figure 13 shows the comparison betweenthe linear superposition prediction and the simulationresult of a simple delayed TR-NEWS process wheretwo focusing peaks are at superposition with 1 µ s timedelay. The difference between the prediction and thesimulation is large and obvious, indicating the pres-ence of nonlinearity. This measure of nonlinearityseems to be stronger than the measure calculated fromPI (Fig. 9), making it a good candidate for further in-vestigation. t, ms −1.0−0.50.00.51.0 A . U . simulationpredictiondifference Figure 13: (Colour online) Delayed TR-NEWS signalprocessing with one delay of amplitude a i = 1 anddelay value τ = 1 µ s (Eq. (6)): comparison betweenthe linear prediction and the nonlinear simulation out-comeThe delayed TR-NEWS signal processing could alsobe used for activating the contacting gap as the en-ergy pocket. This could be done by creating a new fo-cusing wave envelope which would have the resonantfrequency of the defect, permitting higher amplitude waves near the damaged region, which would enhancethe extraction of the nonlinear signature. This studyis left for the future. This paper investigated nonlinear NDT by using asimple FEM simulation model for a crack nonlinear-ity in CFRP. In the laminate model, the damage isa simple horizontal contacting crack near the receiv-ing transducer. The signal processing uses TR-NEWSmethod for focusing the available wave energy nearthe receiving transducer. The magnitude of nonlin-earity due to the damage is measured firstly withPI, secondly with the proposed delayed TR-NEWSsignal processing procedure. While PI indicates thepresence of the nonlinearity, the simple delayed TR-NEWS procedure shows it even more strongly and ispromising for future investigations and further devel-opment due to its signal processing flexibility.Since the delayed TR-NEWS procedure allows togenerate a wave at the focusing with arbitrary en-velope, it could be used in the future to excite thecrack damage by its resonance frequencies, using thedamage as an energy pocket. Other perspectives in-clude a more detailed simulation model for the CFRPin order to take more of its microstructure geometryinto account to have stronger focusing. Additionally,the damage could be modelled either by a collectionof various cracks at various angles or by hysteresis.Moreover, heating from the frictional forces at thedamage could be considered for a more precise simu-lation model.
Acknowledgement
This research has been conducted within the co-tutelle
PhD studies of Martin Lints, between the Tallinn Uni-versity of Technology, Department of Cybernetics inEstonia and the Institut National des Sciences Ap-pliqu´ees Centre Val de Loire at Blois, France. Theresearch is supported by Estonian Research Council(project IUT33-24).
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