Acoustic emission wave propagation in a viscoelastic plate

Composites Science and Technology 59 (1999) 1735±1743

Acoustic emission wave propagation in a viscoelastic plate M. Giordano a, L. Condelli b, L. Nicolais b,* a Institute for the Composite Materials Technology, National Research Council, P.le Tecchio 80125 Naples, Italy Department of Materials and Production Engineering, University of Naples ``Federico II'', P.le Tecchio 80125 Naples, Italy

b

Received 13 October 1998; received in revised form 20 January 1999; accepted 5 February 1999

Abstract In loaded materials the strain-energy release due to microstructural changes results in stress-wave propagation. Acoustic emission deals with the recognition of such waves over the material surfaces and for this reason the application of acoustic emission techniques to the monitoring of the mechanical integrity of materials and structures is appropriate. Nevertheless, the use of such techniques in the case of polymer-based composite materials needs a clearer understanding of the relationships between the recorded signal, the damage process and the structure geometry. In this work a model is proposed of the propagation in a viscoelastic plate of waves due to a damage event. The microfailure event has been represented as a point source and ray theory has been applied to develop the physical model of wave propagation. The spectral analysis technique has been applied to the solution of the wave equations. Model calculations in the frequency domain have been compared with experimental spectra resulting from ®ber breakage in di€erent ®ber/matrix systems from single-®ber fragmentation tests. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; C. Damage mechanics; D. Acoustic emission; Wave propagation

1. Introduction Several types of microfailure event occur when polymer composite structures are loaded. These include ®ber pull-out, ®ber breakage, matrix cracking and interfacial debonding. The damage process leads to a degradation of the mechanical properties and to catastrophic failure of the whole structure. Acoustic emission is the only `nondestructive' technique able to monitor the damage growth and accumulation continuously and, concurrently, to provide information on the nature of the damage. The acoustic emission (AE) process deals with: a microfailure event (source) generating transient waves, the propagation of such waves in a ®nite medium and their detection at the transducer position. In particular, a microfailure event within a loaded material induces a sudden redistribution of the strain energy around the damaged area, generating transient dynamical waves. Waves propagate through the medium to the surfaces, where AE is detected by a transducer. The recorded signals bring information about either the source * Corresponding author. Tel.:+39-81-768-2400; fax:+39-81-7682404. E-mail address: [email protected] (L. Nicolais)

mechanism, i.e. the failure mechanism, or the wave propagation through a ®nite medium. A theoretical model needs to elucidate the in¯uence of the failure dynamics and the propagation medium upon the received acoustic signal features. Several attempts have been made in AE process modeling. Starting from the models conceived for the earthquakes, Otsu and Ono [1] developed a generalized theory of AE by using the integral formulation of the elastodynamics. A key role in their formulation is played by a Green function of the second kind in a half space. They have shown that some Green functions in the half space are applicable to the problem of a plate: that is the most realistic case. The same authors [2] have developed a source-representation model investigating tensile and shear cracks. Suzuki et al. [3] extended the analysis to a dissipative medium including a relaxation function. Simulations were compared with experimental results from few-®bers model composites. By contrast, the generalized ray theory [4] has been used to model AE in a elastic plate [5±7]. The wave motion is decomposed into disturbances travelling along a multitude of ray-paths. Laplace transforms were used in developing solutions for several types of sources placed in any location of the plate. No viscoelastic e€ects have been accounted for. Moreover, transient

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266 -3 538(99)00035 -4

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Nomenclature a c C C" ce E f h Hg Hm Ht i k kI kR L P Rn s t T Te U x n " "0 l     ! Ã Ä

attenuation (mÿ1) phase velocity (m/s) source spectrumÐdisplacement (m s) source spectrumÐstrain (s) elastic velocity (m/s) tensile modulus (Pa) frequency (sÿ1) plate half-thickness (m) geometry transfer function material transfer function transducer transfer function imaginary unit wave number (mÿ1) imaginary part of wave number (mÿ1) real part of wave number (mÿ1) source-transducer horizontal distance (m) power per unit volume (W/m3) distance covered by the nth ray (m) displacement (m) time (s) experimental signal duration (s) emission time (s) strain energy per unit volume (J/m3) space variable (m) impact angle of the nth ray strain strain before cracking attenuation length (m) viscosity (Pa s) density (kg/m3) stress (Pa) relaxation time (s) angular frequency (rad/s) Fourier transform symbol (s) dimensionless variable symbol

ultrasonic waves in a viscoelastic plate have been studied by Weaver et al. [8,9]. Double transform technique has been used to solve wave equation. A closed form in the frequency domain for the high-frequency response to a step load has been obtained. In this work a model of the AE process in a ®nite viscoelastic medium has been developed. The strainenergy release at the failure event has been analyzed to recover the source model. The ray theory has been applied to model the propagation problem in a linearly viscoelastic plate with reference to a Maxwellian medium. As spectral analysis technique has been used in solving the wave equation. Model results have been compared with experimental data from single-®ber fragmentation tests, taking into account the transducer transfer function. The paper has been organized as follows: a

general model of the AE process in a viscoelastic plate is stated in Section 2; Section 3 deals with the application of the physical model drawn in Section 2; the model testing through comparison with experimental results and the main remarks are in Section 4. 2. Model statement 2.1. Wave equation and spectral solution Wave motion in a homogeneous medium is described by the momentum balance equation, in the case of negligible Lagrangian inertia that is: 

@2 s ˆ r
 @t2

…1†

where s is the displacement and  is the stress tensor. The closure equation needed to solve the problem must be provided by the constitutive equation of the medium. It relates the stress  with the strain history experienced by the material. In the plane wave case, the most general wave motion equation for a linearly viscoelastic material is: s‡a

@s @s @2 s @2 s @2 s ‡b ‡c 2‡d ‡ e 2 ‡


ˆ 0 @x @t @x @x@t @t

…2†

The spectral analysis technique [10] can be used to evaluate the general solution of Eq. (2), by taking the Fourier Transform: A1 …!†^s ‡ A2 …!†

d^s d2 s^ ‡ A3 …!† 2 ‡


ˆ 0 dx dx

…3†

Equations of this kind have solutions of the form eÿikx and this leads to the characteristic algebraic equation A1 ‡ …ÿik†A2 ‡ …ÿik†2 A3 ‡


ˆ 0

…4†

Eq. (4) de®nes the wavenumber k and its relationship with frequency ! is known as the spectrum relation. Each value of k that satis®es Eq. (4) represents a mode of propagation of the wave and the general solution, in the frequency domain, is given by the superposition of the di€erent modes. s^…x; !† ˆ

X m

Cm …!†eÿikm …!†x

…5†

The time, space solution s…x; t† can be obtained by taking the inverse transform Eq. (5) … …6† s…x; t† ˆ s^…x; !†ei!t d!

M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

The same result is achieved by using the Fourier series representation of s…x; t† X …7† s…x; t† ˆ s^…x; !n †ei!n t where s^…x; !n † are the Fourier coecients, that can be evaluated from Eq. (5) at ! ˆ !n . Finally XX …8† s…x; t† ˆ Cm …!n †ei!n t eÿikm …!n †x The coecients Cm …!n † must be evaluated by imposing the appropriate boundary conditions. 2.2. Linear viscoelastic media Analysis of the wave propagation has been restricted to linear viscoelastic media: in this case the spectrum relationship k ˆ k…!† is derived from Eq. (4) once the constitutive equation has been assigned. The wave number, k, de®nes the phase velocity, c, and the attenuation, a, of a propagating wave. k…!† ˆ kR …!† ‡ ikI …!† ˆ

! ‡ ia…!† c…!†

…9†

In the case of an elastic solid theqwave number is real  and the wave phase velocity c ˆ E is independent on the frequency. No dissipation occurs in an elastic solid; as a consequence the wave frequency content and amplitude do not change during propagation. In the case of a Newtonian ¯uid both attenuation and wave speed change with frequency [11]; s r ! 2! cˆ …10† jaj ˆ 2  Energy is dissipated and the wave is smoothed out changing its frequency content during propagation. As we move to viscoelastic materials, for which there are at least three parameters, E;  and , one can de®ne an elastic velocity ce and an intrinsic frequency ! (or relaxation time ): p …11† ce ˆ E= ! ˆ E= ˆ 1=

…12†

The existence of an intrinsic frequency characterizes viscoelastic materials; for such materials (but not for elastic solids or Newtonian ¯uids) there is an objective criterion for distinguishing between high frequencies, !  ! , and low frequencies, !  ! . Of course, a real material is endowed with a spectrum of relaxation times, spanning characteristic frequencies from some minimum frequency !0 to some maximum one !00 ; hence perhaps by `high frequency' one should understand

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!  !0 , and by `low frequency' !  !00 , with a wide range of frequencies which are comparable with some of the characteristic ones. We assume that the viscoelastic behavior of the medium is represented by the Maxwell model accordingly with experimental evidence of an elastic behavior at high frequencies: d d"  ˆE ÿ dt dt 

…13†

@s . where " ˆ @x The wave equation [Eq. (1)] becomes the well known telegraph equation:

@2 s 1 @s E @2 s ÿ ‡ ˆ0 @t2  @t  @x2 and the wave number k results in s    i 1 p !…! ÿ i† ! !ÿ ˆ k…!† ˆ E  ce 

…14†

…15†

The phase speed depends on the frequency leading to a dispersive medium. In fact di€erent frequency components of the wave travel at di€erent speeds changing wave shape as it propagates. The Maxwell constitutive equation shows two limiting behaviors: . ! ! 0 viscous behavior Both phase velocity c and attenuation a increase with the square root of frequency. . ! ! 1 elastic behavior Phase velocity c is not dependent on frequency while attenuation attains a constant value becoming negligible as relaxation time  increases. Fig. 1 shows the dependence of the wave number on frequency. 2.3. Source model The boundary conditions must be speci®ed at the damage localization within the material. During failure, the strain energy stored in the material is dissipated in the creation of new surfaces, in heat generation and in the generation of stress waves propagating through the medium. However, we assumed a point defect, neglecting also the heat generation. Let us consider the strain-energy release caused by the failure, that is: 1
U ˆ ÿ E"2o 2

…16†

where "0 is the strain before cracking that is fully released at the end of failure process.

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M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

We assume that the failure process is characterized by an emission time Te and all the stored strain energy 1 2 2 E"o is released during the failure. The accurate dynamics of damage process is described by the power P as a function of time. In fact, the failure process brings the material from an equilibrium state (before the damage) to a new equilibrium state (damaged), when the released power is exactly zero. All the information regarding the source mechanism are collected in the expression of the power P during the emission time Te (Fig. 2). Considering no dissipative mechanism acting during failure: …t …"
1 ÿ …17† E "d" ˆ E "2 ÿ "20 ˆ P…t†dt 2 "o 0 The strain history at failure location can be evaluated from the released power. Eq. (17) results: s  …t "…t† P…t† ˆ 1 ‡ 1 2 dt …18† "0 0 2 E"0 2.4. Wave propagation in a plate In this section a model of the wave propagation based on the ray theory has been developed. Acoustical rays have been described as plane waves and total re¯ection

of longitudinal waves on the boundary surfaces has been assumed. Fig. 3 shows the plate geometry The model aims to evaluate the control point P…L; h† displacement in response of a strain longitudinal wave generated in the source point P(0,0). It is based on the consideration that the signal detected at the control point P…L; h† is the superposition of the di€erent rays reaching this position after their re¯ection at the boundary surfaces. The free surface boundary condition yy ˆ 0

wy ˆ 0

…19†

leads to the generation of two re¯ected waves at each ray impact point. A wave is re¯ected in a wave of the same type and with the same angle plus a wave of different type with an angle given by the Snell law [12]. For our analysis the secondary waves have been neglected considering their slowness. Fig. 4 shows the picture of this mechanism for the primary waves. It can be easily shown that the total distance covered by the nth ray is: q …20† Rn ˆ h2 …2n ÿ 1†2 ‡ L2 In addition it must be considered that the nth ray impacts the control point with an angle n n ˆ arctan

L …2n ÿ 1†h

…21†

where n is the angle between the nth ray and the y axis. Total re¯ection hypothesis allows to evaluate the control point response as the superposition of rays travelling distances Rn in a pseudo-in®nite medium. The y component of the displacement in the frequency domain is: s^y …RL;h ; !† ˆ

Fig. 1. Dimensionless phase speed co ˆ c=ce (solid line) and attenuation koI ˆ kI ce  (dotted line) as functions of frequency.

X n

s^n …Rn ; !† cos n

…22†

3. Model calculations In general terms, the relation between the frequency spectrum of the signal detected by the transducer and the source spectrum can be expressed [13] as:

Fig. 2. Strain energy release rate due to a failure event.

Fig. 3. Plate geometry.

M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

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Fig. 4. Re¯ection of the rays on the boundary surfaces.

s…!† ˆ C…!†Hm …!†Hg …!†Ht …!†

…23†

where Hm …!† is the transfer function of the material, Hg …!† is the transfer function of the specimen geometry, Ht …!† is the frequency response of the transducer and C…!† is the spectrum of the source event. It is worth noting that each term in Eq. (23) can be, in principle, separately evaluated. 3.1. Plane wave propagation in an in®nite Maxwell medium The plane wave propagation in an in®nite medium is described by Eq. (5), where the wave number is speci®ed by the Maxwell constitutive equation [Eq. (13)]. In this case an unique mode of propagation applies. The source spectrum must be speci®ed. According to the consideration in Section 2.3, the power release P…t† during the emission time has been assumed to be:   E"2  t …24† jP…t†j ˆ 0 sin  2Te 2 Te From Eq. (18) we evaluate the strain at the source point:   "…t†  t ˆ cos …25† "0 2 Te Fig. 5 shows dimensionless power and normalized strain. The source spectrum in terms of strain is: ÿi!Te

e ‡ 2i!Te C" …!; Te † ˆ ÿ2"0 Te 2 2 4! Te ÿ 2

…26†

~ !† ~ ˆ C… Fig. 6 shows the dimensionless amplitude C~ " …!† "0 T e ~ † Im…C…!† and phase  ˆ arctan Re…C…!† ~ † spectra of the source as a

Fig. 5. Dimensionless power and normalized strain versus dimensionless time.

function of the dimensionless frequency !~ ˆ !Te . It is worth noting that the phase spectrum of the source has a smooth behavior at low frequency !Te  1. On the contrary, a severe fragmentation of the curve, identi®ed by sudden changes of the phase values versus frequency, characterizes the phase spectrum for frequencies !Te  1. Strain at x position within the in®nite medium is: ^ x; Te ; ; ce † ˆ C" …!; Te †eÿik…!;;ce †x "…!;

…27†

In terms of dimensionless variables: ~ ˆ Te x~ ˆ

x l

!~ ˆ !Te k~ ˆ kl

…28†

where the dimensionless number ^ is the Deborah number, i.e. the ratio between the characteristic relaxation time of the viscoelastic medium  and the characteristic

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M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

Fig. 6. Dimensionless amplitude and phase spectra of the source.

time of the damage phenomenon, the emission time Te and l ˆ ce  is the attenuation length. Eq. (27) becomes: ~ !~ † ~ x~ k… ^~ !; ~ x† ~ ˆ C~ e …!† ~ ~ † ~ ÿi ~ m …!~ ; ~ x; ˆ C~ e …!†H "… e

…29†

Fig. 7 shows the dimensionless amplitude spectra for di€erent Deborah number, di€erent dimensionless dis~ tances as a function of the dimensionless frequency !. The dimensionless amplitude spectrum of the source is reported for comparison. Fig. 7a shows the case of ^  1. The viscous behavior is characterized by the lowering of the high frequency components, the energy content of the wave is continuously dissipated as the wave propagates through the medium. In the case of ^  1 (Fig. 7b) the dimensionless amplitude spectrum does not change with propagation, no dissipation mechanism acts. Fig. 7c shows the intermediate case ^  1 where viscoelasticity allows a reduction of the amplitude of the spectrum but the spectrum shape is almost retained. 3.2. Wave propagation in a Maxwellian plate The plane wave propagation in a plate is described by Eq. (22). A new dimensionless variable must be introduced:

Fig. 7. Dimensionless strain amplitude spectra for di€erent dimensionless distances within an in®nite medium at di€erent Deborah numbers.

h~ ˆ Lh plate shape factor. Dimensionless spectrum equation at control point position is:   X   ~ ~ ~ ~ ~ ~ ˆ ~ h; ~ L; C~ …!~ †eÿik…!~ ~†Rn …L;h† cos n h~ s~^y !; n …30†   ~ ~ ~ ~ L; h ˆ C…!~ †Hmg !~ ; or equivalently in terms of strain:

M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

^~ !; ~ ~ † ~ h† ~ h; ~ L; ~ ˆ C~ " …!†H ~ mg …!~ ; ~ L; "…

…31†

where L~ ˆ Ll . A ®rst consideration is that it is not possible to separate the e€ect of geometry from the e€ect of the material on the wave propagation. The transfer functions Hm …!† and Hg …!† from Eq. (23) are grouped into the transfer function Hmg …!†. Fig. 8 shows the dimensionless strain amplitude spec~ for di€erent tra for a ®xed value of the shape factor h,

1741

^ as a function of dimensionless freDeborah numbers , ~ quency !~ for di€erent plate sizes L. In the viscous case, ^  1, (Fig. 8a) the dimensionless strain amplitude spectrum is smooth at low frequency. It fragments for frequencies !Te 51 showing sudden changes of the amplitude values versus frequency. The fragmentation of the amplitude spectrum spreads toward lower frequencies as the plate size increases. In the elastic case, ^  1, (Fig. 8b) the main e€ect of the boundary re¯ection is the fragmentation of the dimensionless strain amplitude spectrum, both in low and high frequency zone. The viscoelastic case, ^  1; (Fig. 8c) shows characteristics of both the limiting cases. A clear explanation of the sudden changes of the amplitude values can be provided. The di€erent arrival times of each ray induces phase modi®cation in fre~ ~ R~ n …L; ~ ~ h† of quency domain according with the term eÿik…!~ † the transfer function. So the summation over rays corresponds to adding components with di€erent phase. This interference phenomenon produces the fragmentation of the amplitude spectra. The more severe fragmentation of the high frequency components can be explained with reference to the phase spectrum of the source. In this region the phase spectrum is very irregular, as a consequence a time shift produces larger e€ect on the amplitude evaluation. On the other hand in the viscous region of the spectrum (!~  1) the interference is reduced due to the slowness of the waves. 4. Results and discussion 4.1. Model testing

Fig. 8. Dimensionless strain amplitude spectra for ®xed shape factor ~ of the plate h=0.033 for di€erent Deborah numbers as a function of the plate size.

The model has been tested by comparison with experimental data. Acoustic emission signals due to ®ber failure propagating in a plate have been obtained from single-®ber fragmentation tests for three di€erent ®ber-matrix systems: polyester-matrix/carbon-®ber (6 mm diameter) (PCA), epoxy-matrix/carbon-®ber (6 mm diameter) (ECA) and epoxy-matrix/carbon-®ber (12 mm diameter) (ECB). Fig. 9 shows a scheme of the test. A two transducers (resonant-wide band) based system has been used to identify and record the AE event. The resonant probe has been used as a trigger for ®ber-failure identi®cation, while the AE signal has been recorded by the wide-band probe. The apparatus allows the source localization along the ®ber through the di€erence of the signal arrival times at the two transducers. More detailed information on the experimental set-up can be found in a previous paper [14]. Fig. 10 shows representative signals for (PCA), (ECA) and (ECB) systems. Each spectrum spans from 10 to 1000 kHz because of the acquisition window of the wide band transducer. The spectra were obtained by

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M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

Fig. 9. Scheme of the ®ber fragmentation test. Table 1 Experimental data from Single Fiber Fragmentation Tests for di€erent matrix/®ber systemsa

PCA ECA ECB


f (kHz)

ce (m/s)

L (mm)

h (mm)

3 12 6

2300 1960 1960

6.5 35 18

1 1 1

a

PCA, polyester/carbon, 6 mm; ECA, epoxy/carbon, 6 mm; ECB, epoxy/carbon, 12 mm.

sampling
f ˆ 1=T, the phase velocity ce as measured, the source position L and the half-thickness of the plate h for each test. On the other hand, the strain amplitude spectrum ^ j"…!†j of the signal recorded at the transducer position is the result of the source event C" …!†, the propagation of the signal through the specimen Hmg …!† and the transfer function of the transducer Ht …!†: ^ j"…!†j ˆ jC" …!†Hmg …!†Ht …!†j

Fig. 10. Experimental (arbitrary units) versus calculated dimensionless strain amplitude spectra for the di€erent matrix/®ber systems.

discrete Fourier Transform (via Fast Fourier Transform) of the time-domain recorded signals. The discretization of the frequencies
! ˆ 2 T is set by the duration T of the recorded signal [10]. Di€erent signals have di€erent durations. Table 1 reports the frequency

…32†

The synthesis of the signal as recorded at the transducer position needs the knowledge of the transfer function of the transducer Ht …!† itself. Fig. 11 shows the transducer transfer function as provided by the manufacturer. Theoretical predictions have been performed by using the spectrum equation [Eq. (32)] in the dimensionless form. In principle, the ®tting parameters are the source emission time Te and the relaxation time  for each ®ber/matrix system, but we assumed the same source emission time for PCA and ECA systems having the 6 mm diameter ®ber and the same matrix relaxation time for ECA and ECB systems having the epoxy matrix. Fig. 10a shows the comparison between PCA experimental amplitude spectrum and the calculated one. In this case both the relaxation time  and the emission time Te have been used as the ®tting parameters. Fig. 10b shows the comparison between ECA experimental amplitude spectrum and the calculated one. In this case the emission time Te has been assumed the same of the PCA case, due to the same ®ber used, the relaxation time  was the only ®tting parameter. Fig. 10c shows the comparison between ECB experimental amplitude spectrum and the calculated one. In the last case the relaxation time  was the same of the ECA case, due to

M. Giordano et al. / Composites Science and Technology 59 (1999) 1735±1743

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. The Maxwell model accounts for the propagation medium viscoelasticity, but the whole formulation holds for a generic linearly viscoelastic medium.

Fig. 11. Transfer function of the WD transducer as provided by the manufacturer, PAC.

Table 2 Model parameters used in simulations

PCA ECA ECB

Te (ms)

 (ms)

~

L~

h~

1.0 1.0 1.3

108 80 80

108 80 61.5

2.8 17.9 7.1

0.15 0.03 0.06

In spite of those assumptions, the model seems to predict amplitude spectra whose shapes are very close to experimental signals. The ®tting parameters in wave synthesis are the emission time Te describing the failure event and the relaxation time of the polymer  accounting for the matrix viscoelasticity. As well, the estimated values of these parameters have physical coherence. The relaxation time  for the polyester resin is of the same order (=100 ms) of that in Felix et al. [15]. Moreover, the ®ber having the biggest size (12 mm) shows the greatest emission time as expected. The calculated Deborah numbers suggest that viscoelasticity plays a key role in the acoustic emission signals due to the ®ber breakage in the structure we analyzed. In conclusion, a simple tool to investigate both the failure dynamic and the role of viscoelasticity in waves propagation through a plate has been developed. References

the same propagation medium, while the emission time Te is the ®tting parameter. Table 2 summarizes the estimated parameters together with the values of the ~ ~ h. ~ L; dimensionless variables ; 4.2. Conclusions The proposed model is able to reproduce the main features of the acoustic emission signals from single ®ber fragmentation tests varying the ®ber size and the propagation medium. In the following the main assumptions of the model have been recalled and criticized. . The acoustic emission point source has been modeled by means of simple considerations on the strain energy release due to the failure event. At this elementary stage, the source is characterized by a single parameter, the source emission time Te . In the real cases failure involves an extended damaged area. So, a more precise source model should consider a deeper analysis of the failure micromechanics. . Wave propagation in a plate has been analyzed with reference to the ray theory. Response at the transducer position is due to the superposition of rays re¯ected on the plate surfaces. Rays have been described as plane waves but, in principle, the same formulation can be extended to spherical waves. In addition the secondary waves has been neglected according to their slowness: the last hypothesis released leads only to a cumbersome problem.

[1] Ohtsu M, Onu K. A generalized theory of acoustic emission and Green's functions in a half space. J Acoustic Emission 1984;3:27±40. [2] Ohtsu M, Onu K. A generalized theory and source representations of acoustic emission. J Acoustic Emission 1986;5:124±33. [3] Suzuki H, Takemoto M, Onu K. The fracture dynamics in a dissipative glass-®ber/epoxy model composite with AE source simulation analysis. J Acoustic Emission 1996;14:35±50. [4] Cagniard L. Re¯ection and refraction of progressive seismic waves. New York: McGraw±Hill, 1962. [5] Ceranoglu AN, Pao YH. Propagation of elastic pulses and acoustic emission in a plate Part 1: theory. J Appl Mech 1981;48:125±32. [6] Ceranoglu AN, Pao YH. Propagation of elastic pulses and acoustic emission in a plate Part 2: epicentral responses. J Appl Mech 1981;48:133±8. [7] Ceranoglu AN, Pao YH. Propagation of elastic pulses and acoustic emission in a plate. Part 3: general responses. J Appl Mech 1981;48:139±47. [8] Weaver RL, Sachse W, Niu L. Transient ultrasonic waves in a viscoelastic plate: theory. J Acoust Soc Am 1989;85:2255±61. [9] Weaver RL, Sachse W, Niu L. Transient ultrasonic waves in a viscoelastic plate: applications to materials characterization. J Acoust Soc Am 1989;85:2262±7. [10] Doyle JF. Wave propagation in structures. New York: Springer, 1997. [11] Landau LD, Lifshitz EM. Fluid mechanics. Oxford: Pergamon Press, 1987. [12] Kolsky H. Stress waves in solids. New York: Dover, 1963. [13] Berthelot JM, Souda MB, Robert JL. Frequency analysis of acoustic emission signals in concrete. J Acoustic Emission 1993;11:11±18. [14] Giordano M, Calabro A, Esposito C, D'Amore A, Nicolais L. An acoustic emission characterization of the failure modes in polymer-composite materials. Comp Sci Tech 1999;58:1923±8. [15] Felix MP. Attenuation and dispersion characteristics of various plastics in the frequency range 1±10 MHz. J Comp Mater 1974;8:273±87.