# Modeling of the far-field acoustic emission from a crack under stress.

1. IntroductionAcoustic emission (AE) in a thick structure is dominated by the presence of Rayleigh wave in the far field. Indeed, Rayleigh wave decays as 1/[square root of r] whereas bulk waves decay as 1/r. In a thin structure, the energy released by a defect under stress propagates as guided waves.

Different models have been developed to simulate AE from buried crack under stress. For instance, Harris and Pott [1] developed a Rayleigh wave AE model from a buried crack that predicts Rayleigh waves excited by the starting of a faulting event. The surface wave is expressed by an integral formulation relying on the elastodynamic reciprocity theorem. This formulation combines bulk waves emitted by the starting event and the Rayleigh wave components of the Green's tensor, calculated in [2] from the coupling between the P-wave and S-wave component (SV) polarized in the plane of incidence. The Rayleigh wave is then evaluated by means of the stationary phase technique and the particle velocity of the emitted wave is approximated near the Rayleigh wave arrival time.

A method using a seismic moment tensor approach [3] has been also adopted in the AE field to describe the damage processes since micro-damage and seismic rupture share the same source mechanism, at different time and length scales. The moment tensor method was used for describing different types of point sources and equivalent body forces for displacement discontinuities. Aki and Richards [3] summarized studies of elastic waves generated by seismic sources, where surface wave terms of Green's tensor have been calculated in cylindrical coordinates in the case of a vertically heterogeneous half space.

In the case of a thin structure, the prediction of guided waves emitted by a crack under stress was calculated by reciprocity considerations [7], a moment tensor method [4] or a finite element method [5]. Bogert [4] calculated the AE in plates using a seismic moment tensor method and Green's function. He compared different crack types and different propagation models. In addition, he studied the effect of the use of first and third order plate theory solutions on the AE signal. Achenbach [7] calculated the AE from a surface-breaking crack in a plate. The amplitudes of symmetric and antisymmetric modes were calculated separately by an integral formulation as a function of the crack opening displacement.

In this paper, AE from a crack under stress is obtained by applying the elastodynamic reciprocity principle. In the first part, we present a 3D AE model of Rayleigh wave emitted by a defect under stress. This model couples a fracture-mechanics source model and a Rayleigh-wave Green's function. Green's functions are obtained by applying the elastodynamic reciprocity principle following a method presented by Achenbach in [6]. The AE source model used in the coupling formulation is determined from the exact solution of the crack opening displacement as a function of the initial and final crack length and the crack propagation velocity.

In the second part we present a model for the AE of guided waves using an integral formulation method proposed by Achenbach [7]. This formulation combines a guided wave propagation model and a fracture mechanics source model. The modal solution is calculated by the SAFE method (semi analytical finite elements method).

The originality of the method lies in the fact that this formulation combines a propagation model and a fracture-mechanics source model with an arbitrary crack orientation. The work presented in this paper will be integrated in CIVA, a commercial NDE simulation package developed at CEA-LIST.

2. Rayleigh wave AE from buried crack in a thick structure

2.1 Application of the reciprocity theorem for computing the emitted Rayleigh wave

In a 3D geometry, the particle displacement associated to the Rayleigh wave emitted by a growing crack under stress can be calculated by applying the elastodynamic reciprocity principle. In the frequency domain, the elastodynamic reciprocity theorem for a body of volume V and surface S is written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [n.sub.i] are the components of the outward normal to S, [f.sub.j], [u.sub.j] and [[tau].sub.ij] are the components of body forces, displacements and stresses.

[FIGURE 1 OMITTED]

We define state A as the solution of the acoustic emission problem and state B as the Rayleigh wave emitted by a point source applied in the [x.sub.k] direction ([x.sub.k] = [x.sub.1], [x.sub.2] or z). We apply the reciprocity equation to the region of the half space defined in Fig. 1 where Z is the surface of the crack located in the ([x.sub.2], z) plane. The integrals over the free surface and the hemisphere of radius R as R [right arrow] [infinity] vanish and the reciprocity equation leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [u.sup.G.sub.i,k] and [[tau].sup.G.sub.ij,k] denote the displacement and stress components of the Rayleigh Green's tensor, respectively. X = (r, [theta], z) and [xi] = ([r.sub.0], [[theta].sub.0], [z.sub.0]) are the positions of the observation point and of the source, respectively.

2.1.1 Rayleigh Green's function

The particle displacement of the Rayleigh wave generated by a point load in a cylindrical coordinate system can be calculated by applying the elastodynamic reciprocity theorem [6],

[FIGURE 2 OMITTED]

The displacement components of Rayleigh wave generated by a point load of magnitude Q (Figure 2) applied at z=[z.sub.0] in the [x.sub.1] direction are:

[u.sub.r,1] = [[k.sub.R]/4i] [Q[V.sub.R]([z.sub.0]/I] [V.sub.R] (z)[PHI]'([k.sub.R]r)cos[theta] (3)

[u.sub.[theta],1] = [[k.sub.R]/4i] [Q[V.sub.R]([z.sub.0]/I] [V.sub.R] (z) (-1/[rk.sub.R])[PHI]([k.sub.R]r)sin[theta] (4)

[u.sub.z,1] = [[k.sub.R]/4i] [Q[V.sub.R]([z.sub.0]/I] [W.sub.R] (z)[PHI]([k.sub.R]r)cos[theta] (5)

where ([PHI])'(x) = [d[PHI]/dx]. In the case of a point load of magnitude M in the [x.sub.2] direction, we have:

[u.sub.r,2] = [[k.sub.R]/4i] [M[V.sub.R]([z.sub.0]/I] [V.sub.R] (z)[PHI]'([k.sub.R]r)sin[theta] (6)

[u.sub.[theta],2] = [[k.sub.R]/4i] [M[V.sub.R]([z.sub.0]/I] [V.sub.R] (z)(-1/[rk.sub.R])[PHI]([k.sub.R]r)cos[theta] (7)

[u.sub.z,2] = [[k.sub.R]/4i] [M[V.sub.R]([z.sub.0]/I] [W.sub.R] (z)[PHI]([k.sub.R]r)sin[theta] (8)

In the case of a point load of magnitude P in the z direction, we have:

[u.sub.r,z] = [[k.sub.R]/4i] [P[W.sub.R]([z.sub.0]/I] [V.sub.R] (z)[[PHI]'.sub.0]([k.sub.R]r) (9)

[u.sub.[theta],z] = 0 (10)

[u.sub.z,z] = [[k.sub.R]/4i] [P[W.sub.R]([z.sub.0]/I] [W.sub.R] (z)[[PHI].sub.0]([k.sub.R]r) (11)

where

I = [[integral].sup.[infinity].sub.0][[T.sup.R.sub.rr](z)[V.sup.R](z)[V.sup.R](z) - [T.sup.R.sub.rz](z)[W.sup.R](z)]dz. (12)

Expressions of [V.sup.R], [W.sup.R], [T.sub.rr.sup.R] and [T.sub.rz.sup.R] are available in [6] and

[PHI] ([k.sub.R]r) = [H.sup.(1).sub.1]([k.sub.R]r) (13)

[[PHI].sub.0] ([k.sub.R]r) = [H.sup.(1).sub.0]([k.sub.R]r) (14)

[H.sup.(1).sub.n] is the first kind Hankel function of order n. The Rayleigh wave number is defined as:

[k.sub.R] = [k.sub.t]/[[eta].sub.R] (15)

where [[eta].sub.R] is given using a standard approximate formula for the Rayleigh wave velocity by

[[eta].sub.R] = [0.87 + 1.12v]/[1 + v] (16)

where [k.sub.t] is the shear wave number:

[k.sub.t] = [omega][([rho]/[mu]).sup.1/2] (17)

[rho], v, [lambda] and [mu] denote the density of the medium, the Poisson ratio and the elastic Lame constants defined by: [lambda] = [Ev/(1 + v)(1 - 2v] and [mu] = [E/2(1 + v)], where E is the Young's modulus.

We express the Green tensor in cylindrical coordinates using the superposition principle:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where

[u.sub.r,r] = [[k.sub.R]/4i] [[V.sub.R]([z.sub.0])/I] [V.sub.R](z)[PHI]'([k.sub.R]r), (19)

[u.sub.[theta],r] = 0 (20)

[u.sub.z,r] = [[k.sub.R]/4i] [[V.sub.R]([z.sub.0])/I] [W.sub.R](z)[PHI]([k.sub.R]r), (21)

[u.sub.r,[theta]] = 0, (22)

[u.sub.[theta],[theta]] = [[k.sub.R]/4i] [[V.sub.R]([z.sub.0])/I] V(z) (-1/r[k.sub.R])[PHI]([k.sub.R]r)(1 - 2[(sin[theta]).sup.2]) (23)

[u.sub.z,[theta]] = 0, (24)

We neglect terms attenuating with distance more rapidly than 1/[square root of r] and we use the asymptotic expansion for Hankel functions in the next calculations. These approximations were used by Aki and Richards in [3].

2.1.2 Acoustic emission source

We assume that the crack is a surface of displacement discontinuity. In the case of tensile circular crack of radius a loaded by uniform pressure a on its faces, the crack opening displacement (COD), expressed from the complex solution issued from fracture mechanics in the local cylindrical coordinate system ([rho], [phi], [x.sub.1]) can be looked up in text books. One has:

[DELTA][u.sub.1]([rho]) = 2 [4(1 - [v.sup.2])[sigma]/[pi]E] [square root of ([a.sup.2] - [[rho].sup.2])] (25)

2.1.3 AE from tensile crack

Equation (2) can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[E.sup.+] and [E.sup.-] are respectively the crack surface on [x.sub.1] = [0.sup.+] and [x.sub.1] = [0.sup.-].

In the case of a tensile stress (mode I), the displacement at the surface of the crack in the global cylindrical coordinate system is:

[u.sup.A.[theta]] = [u.sup.A.sub.1] (3)

Eq. (26) can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The integral over the surface [[SIGMA].sup.+] can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

We simplify the expression by considering a single dipole acting at the center of the crack, i.e. at X = (0, 0, h) (see Fig. 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

2.2 Simulation and comparison with literature

We assume that the crack diameter evolves from [l.sub.0] = 1 mm to l = 5 mm at a velocity V = 2000 m/s during T = 0.8 [micro]s, and we consider a sampling frequency [F.sub.e] = 50 MHz. We define [r.sub.j] and [t.sub.n] as:

[r.sub.j] = jV[DELTA]t (33)

[t.sub.n] = [t.sub.0] + n [1/Fe] (34)

j varies from 0 to J/2, and n varies from 0 to N where:

[DELTA]t = 1/[F.sub.e] (35)

N = T/[DELTA]t (36)

J = l/V[DELTA]t (37)

Eq. (25) can be discretized as follows:

[DELTA][u.sub.1] ([r.sub.j], [t.sub.n]) = 2[sigma] [4(1 - [v.sup.2])/[pi]E] [square root of (([([l.sub.0] + nV[DELTA]t)/2).sup.2] - [r.sup.2.sub.j])], for [r.sub.j], [less than or equal to] ([l.sub.0] + nV[DELTA]t)/2 (38)

and

[DELTA][u.sub.1]([r.sub.j], [t.sub.n]) = 0, for ([l.sub.0] + nV[DELTA]t)/2 [less than or equal to] [r.sub.j] [less than or equal to] l/2 (7)

We take the Fourier transform of the COD given by eq. (38) to obtain the displacement field of the emitted Rayleigh wave in the frequency domain given by eq. (32). The displacement field in the time domain is then obtained using the inverse Fourier transform of eq. (32).

We have simulated the displacement field of the Rayleigh wave emitted by a defect evolving at different velocities V. Figure 3 presents the particle velocity of the emitted wave in the time domain obtained by deriving eq. (32) in the time domain.

The amplitude of the emitted wave increases and the arrival time decreases when the crack propagation velocity increases. This observation can be explained by analyzing eq. (40). The displacement of the emitted wave in the time domain is obtained from the convolution of the spatial derivative of the Green's function with the integral of the COD over the crack faces, as

S = [[integral].sup.a.sub.0] [[integral].sup.2[pi].sub.0] -[DELTA][u.sup.A.sub.1] [rho] d [rho] d [phi] (40)

[FIGURE 3 OMITTED]

Figure 4 presents the quantity S in the time domain. The main contribution arises at a time corresponding to the end of the emitting process, which is inversely proportional to V.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

We have compared our results with a result from literature [1] on Fig. 5. There is a good agreement between the two results. Differences can be explained by the assumptions used in the two models. Our integral formulation combines directly the COD and the Rayleigh wave Green's functions. This is not the case for Harris and Pott model [1]. Their integral formulation combines the body waves emitted from the crack approximated at the Rayleigh wave arrival time and Rayleigh wave Green's functions.

Our results presented in Fig. 5(a) are obtained by using the crack opening displacement at all points of the crack obtained from a fracture-mechanics exact model whereas results from the literature [1] in Fig. 5(b) present the velocity of the emitted wave from the starting of faulting event considering only the crack tip velocity.

3. Simulation of the reception by a piston-like transducer model

We have simulated the delivered voltage by an AE sensor using a piston-like model. In fact, only the sensitivity to the normal component of the particle velocity of the emitted wave has been considered. In this case, the delivered voltage can be written as [8]:

E = 2[pi][[integral].sub.s]m([r.sub.0])[v.sub.z]([r.sub.0])d[r.sub.0]. (8)

We consider a crack propagating at a velocity V = 2500 m/s. The crack diameter is evolving from 1 mm to 5 mm under 200 MPa. The receiver is an AE sensor of radius R=5 mm and a constant sensitivity m([r.sub.0]) = [m.sub.0] = 600 V/m/s. The AE sensor is positioned at a distance r = 100 mm away from the epicenter of the crack. Figure 6 illustrates the geometry of the AE problem and the reception by an AE sensor.

[FIGURE 6 OMITTED]

The delivered voltage has been simulated using the method described in Sec. 2 and integrating the particle velocity over the sensor surface. Figure 7 presents the received waveform by the sensor, which results from the integration of the Rayleigh normal particle velocity over the sensor area.

[FIGURE 7 OMITTED]

4. Guided waves AE from buried crack in thin structure

This paragraph describes the AE in a plate by a crack of arbitrary orientation. The geometry of the problem is shown on Fig. 8. Guided modes generated by a crack under stress are calculated again by applying the elastodynamic reciprocity theorem following a method presented by Achenbach [7]. The acoustic field emitted by a crack under stress is now composed of both symmetric and antisymmetric modes [7], whose displacement and stress fields can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [k.sub.m] is the wave number of the mth guided mode. Expressions of [V.sup.m.sub.a], [V.sup.m.sub.s], [W.sup.m.sub.a], [W.sup.m.sub.s], [T.sup.am.sub.xx], [T.sup.sm.sub.xx], [T.sup.am.sub.xz] and [T.sup.sm.sub.xz] are available in [7].

[FIGURE 8 OMITTED]

4.1 Application of the reciprocity theorem for the calculation of the emitted guided waves

We select state A as the solution of the acoustic emission problem presented by equations (42) to (45) and state B, as a virtual symmetric or antisymmetric wave.

4.1.1 Amplitude of symmetric modes:

We select state B as the nth symmetric mode propagating in the positive [x.sub.1] direction. Displacement and stress components can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

We apply the reciprocity relation to the region surrounding the crack defined in Fig. 9. In the absence of body forces, equation (1) may be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[FIGURE 9 OMITTED]

We define [J.sub.1] as the integral along the line -h [less than or equal to] z [less than or equal to] h, x = a, [J.sub.2] along -h [less than or equal to] z [less than or equal to] h, x = b and [J.sub.3], over [SIGMA]. Integrals along the free surfaces z = [+ or -] h vanish. [J.sub.1] may be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

We replace displacement and stress components by their expressions from equations (42) to (45) and (46) to (50), so that integral J1 can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

As the summation produces non zero result only for n = m, the integral can be simplified as follows:

[J.sub.1] = 2i[A.sup.s.sub.n][I.sup.s.sub.nn] (23)

The contribution from x = b leads to:

[J.sub.2] = 0 (24)

Integration over the crack surface can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[J.sub.3] is calculated from the contribution of the two line elements that define [SIGMA], as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

We express the integral over the crack faces as a function of the displacement and stress discontinuities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where

[[??].sup.n.sub.s](0, v) = [u.sup.B.sub.u](0, v) = (28)

[[??].sup.n.sub.s](0, v) = [u.sup.B.sub.v](0, v) = (29)

[[??].sup.sn.sub.uu](0, v) = [[tau].sup.B.sub.uu](0, v) = (30)

[[??].sup.sn.sub.uv](0, v) = [[tau].sup.B.sub.uv](0, v) = (31)

The amplitude of symmetric waves can be obtained from the condition that the total contour integral must vanish [eq. (51)]. One gets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

4.1.2 Amplitude of antisymmetric modes:

Similarly, the amplitude of antisymmetric modes can be obtained by selecting the nth antisymmetric mode as a virtual wave for state B.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

where

[I.sup.A.sub.mn] = [[integral].sup.h.sub.-h][[T.sup.am.sub.xx](z)[V.sup.n.sub.a](z) - [T.sup.am.sub.xz](z)[W.sup.n.sub.a](z)]dz (33)

[[??].sup.n.sub.a](0, v) = [u.sup.B.sub.u](0, v) = (34)

[[??].sup.n.sub.a](0, v) = [u.sup.B.sub.v](0, v) = (35)

[[??].sup.an.sub.uu](0, v) = [[tau].sup.B.sub.uu](0, v) = (36)

[[??].sup.an.sub.uv](0, v) = [[tau].sup.B.sub.uv](0, v) = (37)

4.2 Simulation and comparison with literature

We consider an isotropic plate of 3 mm thickness and Poisson's ratio v = 0.33. We have simulated the surface strain of the first symmetric ([S.sub.0]) and antisymmetric mode ([A.sub.0]) emitted by a vertical surface-breaking tensile crack as shown in Fig. 10 and compared our simulation with a result from literature [5] obtained from a finite element method (). We assume that the time dependence of the crack opening displacement is a unit Heaviside step function.

[FIGURE 10 OMITTED]

The modal solutions were calculated by the semi-analytical finite element method (SAFE) [9]. Figure 11(a) presents the filtered surface strain for modes [S.sub.0] and [A.sub.0] generated by the surface-breaking tensile crack using a Gaussian window with 250 kHz center frequency and bandwidth of 350 kHz (measured at -40dB points). The comparison of our model with the finite element results [5] shows a good agreement.

We have also performed a parametric study giving the mode amplitudes as a function of the crack depth. Results obtained with our model have also been compared with finite element results taken from [5]. A qualitative agreement between both models is observed as [A.sub.0] shows the highest amplitude for the most of the considered crack depths and [S.sub.0] amplitude increases slowly with the crack depth. Nevertheless, our results seem to underestimate the amplitude of the [S.sub.0] mode in comparison to finite element ones.

[FIGURE 11 OMITTED]

5. Conclusion

A formulation coupling an acoustic emission source model and a propagation model has been proposed in the case of thick structures in order to predict the acoustic emission from the propagation of a crack under stress and to quantify the influence of the crack propagation velocity on the Rayleigh wave AE signals. Comparisons of our simulated results with results from the literature showed satisfying agreement even if the two models are different and not based on the same approximations and hypotheses. A similar formulation has been derived for guided wave and compared with results from the literature. In addition, the reception by an AE sensor has been simulated using a piston-like model considering only the sensitivity to the normal component of the emitted wave particle velocity.

[FIGURE 12 OMITTED]

Acknowledgements

The work presented in this paper was partly supported by ANR (French National Research Agency) for the project MACSIM--ANR-08-COSI-005-01.'

References

[1.] G P Harris and J Pott, 'Surface Motion Excited by Acoustic Emission from a Buried Crack', J. Appl. Mech. Vol 51, pp 77-83, March 1984.

[2.] J D Achenbach , A K Gautesen and H McMaken, 'Ray methods for waves in Elastic solids', Pitman publishing INC, 1982.

[3.] K Aki and P G Richards, 'Quantitative Seismology', University Science Books, 2002.

[4.] Ph B Bogert, 'Transient Waves from Acoustic Emission Sources in Isotropic Plates Using a Higher Order Extensional and Bending Theory', Doctoral Thesis, North Carolina State University, 2010.

[5.] C K Lee, J J Scholey, P D Wilcox, M R Wisnom, M I Friswell, B W Drinkwater, 'Guided Wave Acoustic Emission from Fatigue crack growth in Aluminium Plate', Advenced Material Research. Vol 13-14, pp 23-28, 2006.

[6.] J D Achenbach, 'Reciprocity in elastodynamics', Cambridge University Press, 2003.

[7.] J D Achenbach, 'Acoustic Emission from a Surface-breaking Crack in a Layer under Cyclic Loading', J. Mech. Mat. and Struct., Vol 4, No 4, pp 649-657, 2009.

[8.] L Goujon and J C Baboux, 'Behaviour of acoustic emission sensors using broadband calibration techniques', Meas. Sci. Technol. Vol 14, pp 903-908, 2003.

[9.] K Jezzine, 'Approche modale pour la simulation de controle non destructifs par ondes elastiques guidees', doctoral thesis, Universite de Bordeaux I, 2007.

Warida Ben Khalifa (1), Karim Jezzine (1), Sebastien Grondel (2), Gaetan Hello (3) and Alain Lhemery (1)

(1) CEA, LIST, centre de Saclay, bat. 611, point courrier 120, 91191 Gif-sur-Yvette cedex, France

(2) Departement OAE, IEMN, UMR CNRS 8520, Universite de Valenciennes et du Hainaut Cambresis, Le Mont Houy, 59313 Valenciennes Cedex 9, France

(3) IUP GSI-GEII-GM, LMEE, 40 rue du Pelvoux, CE1455 Courcouronnes, 91020 Evry Cedex, France

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Author: | Khalifa, Warida Ben; Jezzine, Karim; Grondel, Sebastien; Hello, Gaetan; Lhemery, Alain |
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Publication: | Journal of Acoustic Emission |

Date: | Jan 1, 2012 |

Words: | 4036 |

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