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Nonlinear behaviour of glass fibre reinforced composites with delamination
Accepted Manuscript Non linear behaviour of glass fibre reinforced composites with delamination M. Hammami, A. El Mahi, C. Karra, M.Haddar PII:
S1359-8368(16)00138-4
DOI:
10.1016/j.compositesb.2016.02.031
Reference:
JCOMB 4069
To appear in:
Composites Part B
Received Date: 31 July 2015 Accepted Date: 14 February 2016
Please cite this article as: Hammami M, El Mahi A, Karra C, M.Haddar Non linear behaviour of glass fibre reinforced composites with delamination, Composites Part B (2016), doi: 10.1016/ j.compositesb.2016.02.031. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Non linear behaviour of glass fibre reinforced composites with delamination M. Hammamia,b*, A. El Mahia, C. Karrab, M.Haddarb a
Université du Maine, LAUM, UMR CNRS n°6613, Av. O. Messiaen, 72085 Le Mans cedex 9, France b
Laboratoire Modelisation Mecanique et Production, Ecole Nationale d’Ingenieur de Sfax, Tunisie
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Abstract
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The aim is to assess the nonlinear behaviour of composite having controlled delaminations. The experimental analysis focused on the variation of nonlinear parameters as function of delamination size. Through a set of increasing resonance amplitudes applied on delaminated glass fibre reinforced composite, the frequency shift and the specific damping coefficient shift were evaluated as a function of strain amplitude. The non linear elastic and dissipative parameters related respectively to the frequency and damping, were determined for different bending modes and different delamination length. Those parameters were compared with linear vibration parameter. The results show that nonlinear parameters are more sensitive to the presence of delamination in composite.
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Keywords: Composite, Delamination, Nonlinear vibration, Frequency, Foss factor, Nonlinear vibration.
1. Introduction
Delaminations are the major defects in composite laminates since it appear at weak interlaminar stress. Delaminations are barely visible from external view and may arise during manufacturing or during service. It is well known that delamination detection in composites
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using vibration analysis is a challenging task. This is mainly attribute to the fact that delamination has no comparable damaged mechanism in other material [1]. The presence of delamination may cause changes in the vibration characteristics of the
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composite structures. It leads to changes in natural frequencies, modal shapes and damping ratios [2].The vibration method testing investigates those changes in the structure to detect
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internal damages, as reviewed in [1,3]. The shift in frequencies is generally investigated to identify the presence of delamination. This method provides reliable and accurate data [4-5]. Natural frequencies can be measured accurately with a single sensor [6-8].An excellent overview of the vibration of delaminated composite can be found in the review of Della [9]. It is important to note that after an impact; more than one delamination is usually present in damaged composite laminates. It was approved that multiple delaminations significantly affect the dynamic characteristics of composite beams [10-12] Damping is of prime importance in the study of composite structures vibration behavior. However, the dissipative properties of composite materials have not yet been completely ________________ * Corresponding author. E-mail address: [email protected]
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT identified. Saravanos [13] elaborated an analytical model to evaluate the effect of the delamination size on the modal damping of fiber reinforced composites. Experimental results were performed on a beam with a single delamination. Recently, an experimental study was developed in order to evaluate the modal damping of fiber reinforced composites provided by [14-17].
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It is usually anticipated that the sensitivity of non linear experiment methods are much more sensitive to the presence to delamination in composites than the linear approach [18]. Experiments based on nonlinear resonance technique were used by Meo [19] to detect the micro-damage and the barely visible crakes in composites. The technique aimed to induce the
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specimen around one of its bending resonance modes with increasing excitation amplitude. The resonance frequency shift and loss factor variation are analyzed as a function of the
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excitation amplitude. Nonlinear dissipative and nonlinear elastic parameters are determined from the down shift of the resonance frequency as well as the augmentation of the loss factor as function of strain amplitude is illustrated by Idriss [20] for sandwich materials. It is often proved that the measurement of nonlinear parameters using nonlinear processes is more sensitive to the presence of internal damage than the linear parameters measured through linear method [20-22]. Therefore this method was applied for complexes shape structures by
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[23]. Nonlinear methods are mostly applied on cracked metal material [24-25] granular material or rock, concrete, ceramic, synthetic slate [26]. The investigation of nonlinear methods in composite structures is still limited [27]. In the present study nonlinear resonance vibration in laminate composite having two
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delaminations was investigated to examine the effect of delaminations size on the linear and nonlinear dynamic behaviour of composites. Experimental findings of linear vibration and
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nonlinear vibration are presented in order to characterize the elastic and dissipative behaviour of delaminated composites. Then, the evolution of nonlinear elastic and dissipative parameters for variable delamination length is presented and discussed for several bending modes. Finally the sensitivity of this method is discussed. 1. Experimental set up The composites were prepared to investigate the effects of delamination and curvature on the stiffness and natural frequency and modal damping.Composite material made of unidirectional layers of E-glass fibres and epoxy matrix, with the stacking sequence [02/902]s. The fibre volume ratio is in the range of 58% to 60%. Hand lay-up procedure was adopted to
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ACCEPTED MANUSCRIPT create laminates using SR 1500 epoxy resin, SD 2505 curing agent and unidirectional glass fibres. Composite plates are carried out under 30kPa as pressure, at room temperature and using vacuum moulding process. Double superposed delaminations are artificially made using Teflon tape during manufacturing of the composites at the interfaces between plies having different fibre directions. Obtained specimen has 250 mm length, 20 mm width and 2.5 mm
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beam. Figure 1 is a schematic presentation of the specimen.
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depth. The centres of the cracks are always located at 95 mm from the clamped end of the
Figure 1 : composite specimen with delamination The resonance method was performed by inducing the specimen to a flexural wave. The experimental equipment employed is shown in figure 2. The test specimen is supported in a cantilever configuration with a clamping block. An analyzer, Stanford Research Systems
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SR785, generated swept-sine signals. Then, the signal was amplified by a Power Amplifier PA25E, with constant gain. The shaker BK480 induced the beam with flexural vibrations in its clamped end. The beam response was detected using an accelerometer with practically negligible mass, BK 352c22 fixed at the free end of the specimen. The accelerometer
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[9.75mV/g] was connected to a conditioner with frequency band. The excitation was controlled and the response was digitalized using GPIB card via an interface LABVIEW. The
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data was saved for post processing. The beam responses were identified in the frequency domain using analysis and fitting the experimental frequency responses using Matlab toolbox. Then, the identification procedure allows us to obtain the values of the natural frequencies f and the loss factor. This test was carried out according to the ASTM C 393 standard [28] A chirp sweep was used to induce the specimen on a frequency range around a given bending modes of the specimen. The frequency sweep was then repeated for 10 increasing amplitudes from 30 mV to 300 mV. This procedure was repeated for the first six bending modes for undamaged specimen and specimen with increasing length of delamination from 10 until 130mm every 10mm [20, 29].
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Non linear elastic behaviour of glass fibre reinforced composites with delamination
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Figure 2 : Experimental setup.
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2. Linear vibration 3.1 Frequency
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To examine the effect of delamination length on frequency of the composite beam in linear vibration, it is important to be aware of the possibility of entering to the nonlinear elastic behaviour domain. The specimens were tested at low amplitude excitation to satisfy this condition.
Figure 3 show the variation of the natural frequencies as a function of delamination parameter for the six first modes. Where, delamination parameter is the ratio of delamination length by
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the free length of the specimen.
In this graph, natural frequencies decrease with delamination lengths, for all bending modes. Delamination reduces the natural frequencies, due to the reduction of specimen stiffness.
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However, reduction of frequencies as function of delamination lengths is not the same for all bending modes. In fact, the frequency of intact composite specimen for the first mode is 26 Hz and is about 21 Hz for the specimen having 0.6 delamination parameter. As a result, for
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the first bending mode, the percentage of the frequency shift is 15%. For the sixth bending mode, however, the frequency shift percentage reaches 45%. The sixth bending mode is much more sensitive than other modes for the presence of delamination. For this mode, a significant frequency shift provides a reliable indicator for delamination damage and growth. For the lowest bending modes particularly, when the frequency value is low, the reduction in frequency is caused by the decrease of the stiffness. However for the highest mode the decrease of frequency is caused by the reduction of stiffness and the position of delaminations with respect to modal deformation [13].
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Figure 3: Evolution of frequency V.s. delamination parameter
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3.2 Loss factor
The loss factors are determined for several modes and different delamination parameter using the bandwidth method. Fig. 4 illustrates the evolution of the loss factor a function of delamination length for different bending modes. Generally, loss factors increase when delamination length increases. For composites with small delamination damping change is due to the effect of the reduction in the flexural rigidity while the friction between plies seems secondary. Friction effects of delamination interfaces dominates the damping at a high delamination parameter [13] There was problem to determine the damping measurements in the first mode which could not be shown, possibly because of imperfection and friction at the clamped end. In addition, the
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ACCEPTED MANUSCRIPT limited number of samples used to analyze at low frequencies can be the cause of the scatter in damping value for the first mode. Mode 3
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Figure 4 : Evolution of loss factor V.s. delamination parameter
3. Nonlinear vibration 4.1 Theoretical backgrounds The linear theory of elasticity is applied in the case of elastic or isotropic material expressed by the Hook’s law which is a linear stress-strain relation. For heterogeneous and damaged materials this linear low can’t describe the nonlinear behaviour of those materials. In fact, the stress train relation of those materials becomes non linear [30]. There are two 6
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT different types on non linearity: the classical non linearity and the hysteretic or the nonclassical non linearity. The classical non linearity is applicable in atomic scale. It has been proved that some damaged materials have a complex compliance and have a nonlinear behaviour that cannot be explained with the classical nonlinear methods [25, 31-32] The general theoretical description of non linear behaviour which include the classical and the
σ (ε ) = k 0 1 + βε + δε 2 + α ( ∆ε + sign(ε& )ε (t ) + ...)ε
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hysteretic nonlinearity is presented in [33]. This can be expressed as: (1)
Where k0 is the linear modulus, “1” presents the linear behaviour of the material, (βε +δε2) presents the classical non linear behaviour of the material and The term α(∆ε+ sign( ε& )ε(t)+...)
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describe the hysteresis in the stress-strain relation. Where, α is the hysteretic coefficient, which describes the hysteretic nonlinear relationship between σ and ε. ∆ε is the local strain
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amplitude, sign( ε& ) equals +1 if the strain rate is positive and −1 if it is negative. The ε is the strain amplitude
The hysteresis nonlinear behaviour of the damaged material can be analyzed by monitoring the fundamental resonance amplitude, the resonance frequency and the loss factor. In fact, damaged materials exhibit a linear decrease of the natural frequency and linear increase of the loss factor as a function of the driving amplitude. The elastic parameters,
α f,
reference [34]
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and the dissipative parameter αη, may be computed by using the following equations cited in
(2)
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∆( f ) f − f 0 = = −εα f f0 f0
Where f is the resonance frequency at the driven amplitude and f0 is the resonance frequency
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at low amplitude, ∆f is the frequency shift and ε is the detected strain amplitude. ∆ (η )
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=
η − η0 = εαη η0
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Where η is the loss factor at the driven amplitude and η 0 is the loss factor at low amplitude,
∆ η is the loss factor shift and ε is the detected strain amplitude. Composite materials have a complex structure and a linear behaviour at undamaged state, but when damages appear, they exhibit nonlinear hysteretic behaviour. In the following, the non-classical nonlinear behaviour of composites with increasing delaminations length will be studied. Then, the sensitivity of the nonlinear resonance method to the presence of delamination will be presented. 7
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT 2.1 Nonlinear resonance curves Ten resonance curves were measured with increasing excitation for an undamaged specimen and for specimens having delamination length from 10 to 130 mm using a length step of 10 mm. This experiment was repeated three times for every delamination length for the six first bending modes. The first mode is difficult to excite with high amplitude in low
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frequencies. We therefore exited the specimen only with the first four excitations. The relaxation and the conditioning time were considered in the investigation.
Fig.5 shows an example of resonant curves for undamaged composite specimens (a), specimen having 0.3 (b) and 0.6 of delamination parameter (c).Those figures plot amplitudes
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response as a function of frequency for increasing driven amplitudes. We present in this figure the first, the third and the fifth bending modes as example. Each curve corresponds to a
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fixed constant drive voltage. For the undamaged specimen, presented in fig.5 (a), 10 amplitude/frequency curves have approximately the same shape for a sequence of drive voltage. The resonance curves show no change when excitation level increases for all bending mode.
For a specimen having 0.3 and 0.6 of delamination parameter, presented in fig.5 (b) and (c),
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the resonance peak is shifting to lower frequency when the drive amplitude increases for all bending modes. Furthermore, the curves tend to flatten and become asymmetric for large amplitudes. As delamination length increase the resonance peaks move to left indicating the nonlinearity. We can say that when delamination increases, the nonlinearity of composites
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amplitudes.
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increases. Material degradation can be observed when frequencies changes with increasing
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Non linear elastic behaviour of glass fibre reinforced composites with delamination
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Fig. 5. Resonance curves at 10 different excitation levels of undamaged specimen (a) specimen having 0.3 delamination parameter (b) and specimen having 0.6 delamination parameter for the 1st,the 3rd and the 5th bending parameter.
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4.2 Frequency and loss factor shift
Figs.6 shows the variation of the frequency shift (f-f0)/f0 as function of strain amplitude for specimens having different delamination lengths. The measured amplitude excitation is converted to the acceleration ü. Then the strain values are obtained using the frequency f and the following relation cited in [32].
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Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT Where L is the specimen length. The strain values obtained by converting the acceleration are in the order 10-7 <ε<10-3. This strain range corresponds to nonlinear material [30] Curves of the Fig. 6 were extracted from the resonance curves for the first six bending modes. They show that the frequency shift decreases as a function of strain amplitude, for different debonding lengths. Furthermore, the decrease of frequency shift is linear with the strain.
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As we said in the section (3.2), there was problem to determine the damping measurements in the first mode which could not be presented. So we present in the fig. 7 the loss factor shift
(η−η0)/η0 V.s strain amplitude (for the 2nd - 6th bending modes). The loss factor shift increases as function of resonance amplitude, for different debonding length. The linear
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increase loss factor shift is also observed in Fig. 7.
This linear variation of the loss factor shift and frequency shift agree with the linear relationship between the frequency shift and the strain amplitude as described in Eq (2)and
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between the loss factor and the strain as described in Eq (3). We can also observe that for the undamaged specimen, there is no significant variation. In fact, the linear variation of frequency shift and the loss factor shift as function of the strain amplitude (ε) is caused by the hysteretic behaviour of delaminated composites.
In this work we have a large range of strain amplitudes. Obtained results were described
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with a linear law. However, a linear law has not fitted the entire curves accurately. This deviation from linear law is may be caused by a combination of linear, classical and the hysteretic nonlinear behaviour as found in [32]. A power law or polynomial law function
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Fig. 6. Evolution of the frequency shift vs. strain amplitude for specimen having different debonding lengths and for the first six bending modes.
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Fig. 7. Evolution of the loss factor shift vs. strain amplitude for specimen having different debonding lengths and for the first six bending modes.
4.3 Non linear parameter The frequency shift and the loss factor shift presented as a function of the strain amplitude are fitted by a linear law for all bending modes as expected from hysteretic nonlinearity. The slops values of these curves are determined. The obtained values correspond to the hysteretic nonlinear elastic parameters αf related to the frequency and the hysteretic non linear 12
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT dissipative parameter αη related to the loss factor, describe the nonlinear hysteretic coefficient in the equation (2) [35]. The evolution of the normalised hysteretic elastic parameters (αf/αf0) and the normalised hysteretic non linear dissipative parameter (αη/αη0) as function of increasing delamination length for the first several bending modes, are presented in the Fig. 8
differences. Those parameters depend on the modes.
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and Fig. 9. The nonlinear parameters (αf and αη) for those modes show significant Comparing the result of the intact specimen with those of Haupert [25] who work with PVC, PMMA, chalk and bone, it is proved that nonlinearity in the (GFRP) is not very significant. The decrease of frequency is caused essentially by the presence of delaminations.
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nonlinear hysteretic parameters αf and αη.
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For excitations of high amplitudes, the nonlinear elastic behaviour can be characterized by the
It should be noticed that for the second mode the result shows that the nonlinear parameters (αf and αη) depend on the delamination length, i.e. the lower nonlinear parameter corresponding the lower debonding length. For the other modes, the nonlinear elastic (αf and
αη) parameters do not have a regular variation trend for increasing delamination length. It appears that the hysteretic parameters are only sensitive to delamination for the second
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structure.
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bending mode. The second bending mode is the sensitive mode to delaminations in composite
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Fig. 8. Evolution of nonlinear elastic parameter for different bending modes as a function of increasing delamination.
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Fig. 9. Evolution of nonlinear dissipative parameter for different bending modes as a function of increasing delamination.
4. Sensitivity
In order to compare the sensitivity of the linear with the nonlinear resonance method, we present in the Fig. 10 the linear normalized parameter (f/f0) and the nonlinear normalized parameter (αf /αf0) for the second bending modes. These parameters are normalized respectively with values obtained from the undamaged specimen. Normalized values of nonlinear elastic parameters increase as a function of the delamination length while normalized values of the linear elastic parameter are almost constant. For the lowest 15
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT bending modes, the nonlinear elastic parameter is more sensitive than the linear elastic parameter. Fig. 11 compares the linear normalized parameter (η/η0) and the nonlinear normalized parameter (αη /αη0) for mode 2. The linear and nonlinear dissipative parameters increase when the delamination parameter increases. Globally the nonlinear dissipative parameter is
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more sensible than linear. Mode 2 80
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Nomalized value
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Fig. 10. Normalized values for linear frequency and nonlinear elastic parameter vs. delamination parameter
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Fig. 11. Normalized values for linear loss factor and nonlinear dissipative parameter vs. delamination parameter
5. Conclusion The effect of increasing delamination length on the linear and nonlinear behaviour of glass fibres reinforced plastics (GFRP) is studied by linear and nonlinear resonance experiments. Results of linear vibration analysis show that the natural frequencies decreases and the loss actor increases with the debonding length. Nonlinear vibration analysis shows a resonance 16
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT frequency and loss factor shift V.s increasing excitation amplitude. The nonlinear parameters were evaluated as a function of the delamination length and compared to the linear parameters. It appears that nonlinear parameters are more sensitive to the presence of delamination in composite structure than the linear parameter. Indeed, the nonlinear resonance method appears to be sensitive to the presence of delamination. The advantage of this method
with unknown natural frequencies of the undamaged structure.
References
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(non linear method) is that it allows the detection of delamination in composite structure even
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1. Zou Y, Tong L, Steven GP. Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures — A Review. J Sound Vib. 2000 Feb;230(2):357–78.
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2. Fritzen CP. Vibration-based structural health monitoring–concepts and applications. Key Eng Mater. 2005;293:3–20. 3. Friswell M I. Damage identification using inverse methods. Philos Trans R soc. 2007; A 365(1851):393-410.
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4. Yang Z, Chen X, Yu J, Liu R, Liu Z, He Z. A damage identification approach for plate structures based on frequency measurements. Nondestruct Test Eval. 2013;28(4):321–41. 5. Kessler SS, Spearing SM, Atalla MJ, Cesnik CES, Soutis C. Damage detection in composite materials using frequency response methods. Compos Part B Eng. 2002 Jan;33(1):87–95.
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6. Salawu OS. Detection of structural damage through changes in frequency: a review. Eng Struct. 1997 Sep;19(9):718–23.
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7. Klepka A, Stra˛czkiewicz M, Pieczonka L, Staszewski W J, Gelman L, Aymerich F, Uhl T. Triple correlation for detection of damage-related nonlinearities in composite structures. Non line dyn. 2015; 81(2-1):453–468. 8. Frieden J, Cugnoni J, Botsis J, GmürT. Vibration-based characterization of impact induced delamination in composite plates using embedded FBG sensors and numerical modeling Compos Part B. 2011; 42(4): 607–613. 9. Della CN, Shu D. Vibration of Delaminated Composite Laminates: A Review. Appl Mech Rev. 2007;60(1):1. 10. Nguyen SN, Lee J, Cho M. Efficient higher-order zig-zag theory for viscoelastic laminated composite plates. Int J Sol Struct.2015; 62 (1):174–185.
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ACCEPTED MANUSCRIPT 11. Sunghee L, Tachyon P, George ZV. Vibration analysis of multi-delaminated beams. Compos: PartB. 2003; 34 (7): 647–659. 12. Della C N. Shu D. Vibration of beams with double delaminations. J Sound Vib. 2004; 282 (2005): 919–935.
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13. Saravanos DA, Hopkins DA. Effect of delaminations on the damped dynamic characteristics of composite laminates: Analysis and experiments. J sound vib. 1995; 192(5): 977–993. 14. Montalva˜o1 D, Karanatsisl D MR, Ribeiro A, Arina J. Baxter R. An experimental study on the evolution of modal damping with damage in carbon fiber laminates. J Compos Mat. 2014;0(0):1–11.
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15. Kiral Z, Ic¸ten B M, Kiral B G. Effect of impact failure on the damping characteristics of beam-like composite structures. Compos Part B: Eng. 2012; 8(43): 3053–3060.
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16. Yesilyurt I, Gursoy H. Estimation of elastic and modal parameters in composites using vibration analysis. J Vib Cont. 2013; 0(0): 1–16. 17. Hammami M, El Mahi A, Karra C, Haddar M. Vibration behavior of composite material with two overlapping delamination. Int J Appl Mech. 2015; in press.
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18. Klepka A, Stra˛czkiewicz M, Pieczonka L, Staszewski W J, Gelman L, Aymerich F, Uhl T. Triple correlation for detection of damage-related nonlinearities in composite structures. Non line dyn.2015; 81(2-1): 453–468. 19. Meo M, Polimeno U, Zumpano G. Detecting Damage in Composite Material Using Nonlinear Elastic Wave Spectroscopy Methods. Appl Compos Mater. 2008 May;15(3):115–26.
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20. Idriss M, El Mahi A, El Guerjouma R. Characterization of sandwich beams with debonding by linear and nonlinear vibration method. Compos Struct. 2015 Feb;120:200–7.
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21. TenCate JA, Pasqualini D, Habib S, Heitmann K, Higdon D, Johnson PA. Nonlinear and nonequilibrium dynamics in geomaterials. Phys Rev Lett. 2004;93(6):065501. 22. Bentahar M, Guerjouma RE. Monitoring progressive damage in polymer-based composite using nonlinear dynamics and acoustic emission. J Acoust Soc Am. 2009;125(1):EL39. 23. Polimeno U., Meo M. Detecting barely visible impact damage detection on aircraft composites structures. Compos Struct. 2009 Dec;91(4):398–402. 24. Dutta D, Sohn H, Harries KA, Rizzo P. A Nonlinear Acoustic Technique for Crack Detection in Metallic Structures. Struct Health Monit. 2009 May 1;8(3):251–62. 25. Haupert S, Renaud G, Riviere J, Talmant M, Johnson PA, Laugier P. High-accuracy acoustic detection of nonclassical component of material nonlinearity. J Acoust Soc Am. 2011;130(5):2654–61. 18
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ACCEPTED MANUSCRIPT 26. Van Den Abeele KE, Sutin A, Carmeliet J, Johnson PA. Micro-damage diagnostics using nonlinear elastic wave spectroscopy (NEWS). Ndt E Int. 2001;34(4):239–48. 27. Aymerich F, Staszewski WJ. Experimental study of impact-damage detection in composite laminates using a cross-modulation vibro-acoustic technique. Struct Health Monit (Internet). 2010 (cited 2015 Mar 4); Available from: http://shm.sagepub.com/content/early/2010/03/16/1475921710365433.abstract
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28. ASTM E 756-98 international “Standard test method for measuring vibration damping properties of materials Designation” E 756_98 29. Okutan Baba B, Thoppul S. Experimental evaluation of the vibration behavior of flat and curved sandwich composite beams with face/core debond. Compos Struct 2009;91:110–9.
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30. Landau L. D , Lifshitz E M .Theory of Elasticity (Internet). (cited 2015 Feb 19). Available from: http://archive.org/details/TheoryOfElasticity
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33. Ostrovsky LA, Johnson PA. Dynamic nonlinear elasticity in geomaterials. Riv Nuovo Cimento. 2001;24(7):1–46.
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34. Johnson P, Sutin A. Slow dynamics and anomalous nonlinear fast dynamics in diverse solids. J Acoust Soc Am. 2005;117(1):124.
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35. Guyer R, McCall K, Boitnott G.Hysteresis, discrete memory, and nonlinear wave propagation in rock: A new paradigm.Phys.Rev.Lett. 1995;74,3491_3493
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S1359-8368(16)00138-4
DOI:
10.1016/j.compositesb.2016.02.031
Reference:
JCOMB 4069
To appear in:
Composites Part B
Received Date: 31 July 2015 Accepted Date: 14 February 2016
Please cite this article as: Hammami M, El Mahi A, Karra C, M.Haddar Non linear behaviour of glass fibre reinforced composites with delamination, Composites Part B (2016), doi: 10.1016/ j.compositesb.2016.02.031. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Non linear behaviour of glass fibre reinforced composites with delamination M. Hammamia,b*, A. El Mahia, C. Karrab, M.Haddarb a
Université du Maine, LAUM, UMR CNRS n°6613, Av. O. Messiaen, 72085 Le Mans cedex 9, France b
Laboratoire Modelisation Mecanique et Production, Ecole Nationale d’Ingenieur de Sfax, Tunisie
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Abstract
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The aim is to assess the nonlinear behaviour of composite having controlled delaminations. The experimental analysis focused on the variation of nonlinear parameters as function of delamination size. Through a set of increasing resonance amplitudes applied on delaminated glass fibre reinforced composite, the frequency shift and the specific damping coefficient shift were evaluated as a function of strain amplitude. The non linear elastic and dissipative parameters related respectively to the frequency and damping, were determined for different bending modes and different delamination length. Those parameters were compared with linear vibration parameter. The results show that nonlinear parameters are more sensitive to the presence of delamination in composite.
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Keywords: Composite, Delamination, Nonlinear vibration, Frequency, Foss factor, Nonlinear vibration.
1. Introduction
Delaminations are the major defects in composite laminates since it appear at weak interlaminar stress. Delaminations are barely visible from external view and may arise during manufacturing or during service. It is well known that delamination detection in composites
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using vibration analysis is a challenging task. This is mainly attribute to the fact that delamination has no comparable damaged mechanism in other material [1]. The presence of delamination may cause changes in the vibration characteristics of the
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composite structures. It leads to changes in natural frequencies, modal shapes and damping ratios [2].The vibration method testing investigates those changes in the structure to detect
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internal damages, as reviewed in [1,3]. The shift in frequencies is generally investigated to identify the presence of delamination. This method provides reliable and accurate data [4-5]. Natural frequencies can be measured accurately with a single sensor [6-8].An excellent overview of the vibration of delaminated composite can be found in the review of Della [9]. It is important to note that after an impact; more than one delamination is usually present in damaged composite laminates. It was approved that multiple delaminations significantly affect the dynamic characteristics of composite beams [10-12] Damping is of prime importance in the study of composite structures vibration behavior. However, the dissipative properties of composite materials have not yet been completely ________________ * Corresponding author. E-mail address: [email protected]
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT identified. Saravanos [13] elaborated an analytical model to evaluate the effect of the delamination size on the modal damping of fiber reinforced composites. Experimental results were performed on a beam with a single delamination. Recently, an experimental study was developed in order to evaluate the modal damping of fiber reinforced composites provided by [14-17].
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It is usually anticipated that the sensitivity of non linear experiment methods are much more sensitive to the presence to delamination in composites than the linear approach [18]. Experiments based on nonlinear resonance technique were used by Meo [19] to detect the micro-damage and the barely visible crakes in composites. The technique aimed to induce the
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specimen around one of its bending resonance modes with increasing excitation amplitude. The resonance frequency shift and loss factor variation are analyzed as a function of the
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excitation amplitude. Nonlinear dissipative and nonlinear elastic parameters are determined from the down shift of the resonance frequency as well as the augmentation of the loss factor as function of strain amplitude is illustrated by Idriss [20] for sandwich materials. It is often proved that the measurement of nonlinear parameters using nonlinear processes is more sensitive to the presence of internal damage than the linear parameters measured through linear method [20-22]. Therefore this method was applied for complexes shape structures by
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[23]. Nonlinear methods are mostly applied on cracked metal material [24-25] granular material or rock, concrete, ceramic, synthetic slate [26]. The investigation of nonlinear methods in composite structures is still limited [27]. In the present study nonlinear resonance vibration in laminate composite having two
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delaminations was investigated to examine the effect of delaminations size on the linear and nonlinear dynamic behaviour of composites. Experimental findings of linear vibration and
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nonlinear vibration are presented in order to characterize the elastic and dissipative behaviour of delaminated composites. Then, the evolution of nonlinear elastic and dissipative parameters for variable delamination length is presented and discussed for several bending modes. Finally the sensitivity of this method is discussed. 1. Experimental set up The composites were prepared to investigate the effects of delamination and curvature on the stiffness and natural frequency and modal damping.Composite material made of unidirectional layers of E-glass fibres and epoxy matrix, with the stacking sequence [02/902]s. The fibre volume ratio is in the range of 58% to 60%. Hand lay-up procedure was adopted to
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ACCEPTED MANUSCRIPT create laminates using SR 1500 epoxy resin, SD 2505 curing agent and unidirectional glass fibres. Composite plates are carried out under 30kPa as pressure, at room temperature and using vacuum moulding process. Double superposed delaminations are artificially made using Teflon tape during manufacturing of the composites at the interfaces between plies having different fibre directions. Obtained specimen has 250 mm length, 20 mm width and 2.5 mm
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beam. Figure 1 is a schematic presentation of the specimen.
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depth. The centres of the cracks are always located at 95 mm from the clamped end of the
Figure 1 : composite specimen with delamination The resonance method was performed by inducing the specimen to a flexural wave. The experimental equipment employed is shown in figure 2. The test specimen is supported in a cantilever configuration with a clamping block. An analyzer, Stanford Research Systems
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SR785, generated swept-sine signals. Then, the signal was amplified by a Power Amplifier PA25E, with constant gain. The shaker BK480 induced the beam with flexural vibrations in its clamped end. The beam response was detected using an accelerometer with practically negligible mass, BK 352c22 fixed at the free end of the specimen. The accelerometer
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[9.75mV/g] was connected to a conditioner with frequency band. The excitation was controlled and the response was digitalized using GPIB card via an interface LABVIEW. The
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data was saved for post processing. The beam responses were identified in the frequency domain using analysis and fitting the experimental frequency responses using Matlab toolbox. Then, the identification procedure allows us to obtain the values of the natural frequencies f and the loss factor. This test was carried out according to the ASTM C 393 standard [28] A chirp sweep was used to induce the specimen on a frequency range around a given bending modes of the specimen. The frequency sweep was then repeated for 10 increasing amplitudes from 30 mV to 300 mV. This procedure was repeated for the first six bending modes for undamaged specimen and specimen with increasing length of delamination from 10 until 130mm every 10mm [20, 29].
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Figure 2 : Experimental setup.
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2. Linear vibration 3.1 Frequency
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To examine the effect of delamination length on frequency of the composite beam in linear vibration, it is important to be aware of the possibility of entering to the nonlinear elastic behaviour domain. The specimens were tested at low amplitude excitation to satisfy this condition.
Figure 3 show the variation of the natural frequencies as a function of delamination parameter for the six first modes. Where, delamination parameter is the ratio of delamination length by
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the free length of the specimen.
In this graph, natural frequencies decrease with delamination lengths, for all bending modes. Delamination reduces the natural frequencies, due to the reduction of specimen stiffness.
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However, reduction of frequencies as function of delamination lengths is not the same for all bending modes. In fact, the frequency of intact composite specimen for the first mode is 26 Hz and is about 21 Hz for the specimen having 0.6 delamination parameter. As a result, for
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the first bending mode, the percentage of the frequency shift is 15%. For the sixth bending mode, however, the frequency shift percentage reaches 45%. The sixth bending mode is much more sensitive than other modes for the presence of delamination. For this mode, a significant frequency shift provides a reliable indicator for delamination damage and growth. For the lowest bending modes particularly, when the frequency value is low, the reduction in frequency is caused by the decrease of the stiffness. However for the highest mode the decrease of frequency is caused by the reduction of stiffness and the position of delaminations with respect to modal deformation [13].
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3.2 Loss factor
The loss factors are determined for several modes and different delamination parameter using the bandwidth method. Fig. 4 illustrates the evolution of the loss factor a function of delamination length for different bending modes. Generally, loss factors increase when delamination length increases. For composites with small delamination damping change is due to the effect of the reduction in the flexural rigidity while the friction between plies seems secondary. Friction effects of delamination interfaces dominates the damping at a high delamination parameter [13] There was problem to determine the damping measurements in the first mode which could not be shown, possibly because of imperfection and friction at the clamped end. In addition, the
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3. Nonlinear vibration 4.1 Theoretical backgrounds The linear theory of elasticity is applied in the case of elastic or isotropic material expressed by the Hook’s law which is a linear stress-strain relation. For heterogeneous and damaged materials this linear low can’t describe the nonlinear behaviour of those materials. In fact, the stress train relation of those materials becomes non linear [30]. There are two 6
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ACCEPTED MANUSCRIPT different types on non linearity: the classical non linearity and the hysteretic or the nonclassical non linearity. The classical non linearity is applicable in atomic scale. It has been proved that some damaged materials have a complex compliance and have a nonlinear behaviour that cannot be explained with the classical nonlinear methods [25, 31-32] The general theoretical description of non linear behaviour which include the classical and the
σ (ε ) = k 0 1 + βε + δε 2 + α ( ∆ε + sign(ε& )ε (t ) + ...)ε
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hysteretic nonlinearity is presented in [33]. This can be expressed as: (1)
Where k0 is the linear modulus, “1” presents the linear behaviour of the material, (βε +δε2) presents the classical non linear behaviour of the material and The term α(∆ε+ sign( ε& )ε(t)+...)
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amplitude, sign( ε& ) equals +1 if the strain rate is positive and −1 if it is negative. The ε is the strain amplitude
The hysteresis nonlinear behaviour of the damaged material can be analyzed by monitoring the fundamental resonance amplitude, the resonance frequency and the loss factor. In fact, damaged materials exhibit a linear decrease of the natural frequency and linear increase of the loss factor as a function of the driving amplitude. The elastic parameters,
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reference [34]
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and the dissipative parameter αη, may be computed by using the following equations cited in
(2)
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∆( f ) f − f 0 = = −εα f f0 f0
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at low amplitude, ∆f is the frequency shift and ε is the detected strain amplitude. ∆ (η )
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η − η0 = εαη η0
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Where η is the loss factor at the driven amplitude and η 0 is the loss factor at low amplitude,
∆ η is the loss factor shift and ε is the detected strain amplitude. Composite materials have a complex structure and a linear behaviour at undamaged state, but when damages appear, they exhibit nonlinear hysteretic behaviour. In the following, the non-classical nonlinear behaviour of composites with increasing delaminations length will be studied. Then, the sensitivity of the nonlinear resonance method to the presence of delamination will be presented. 7
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT 2.1 Nonlinear resonance curves Ten resonance curves were measured with increasing excitation for an undamaged specimen and for specimens having delamination length from 10 to 130 mm using a length step of 10 mm. This experiment was repeated three times for every delamination length for the six first bending modes. The first mode is difficult to excite with high amplitude in low
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frequencies. We therefore exited the specimen only with the first four excitations. The relaxation and the conditioning time were considered in the investigation.
Fig.5 shows an example of resonant curves for undamaged composite specimens (a), specimen having 0.3 (b) and 0.6 of delamination parameter (c).Those figures plot amplitudes
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response as a function of frequency for increasing driven amplitudes. We present in this figure the first, the third and the fifth bending modes as example. Each curve corresponds to a
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fixed constant drive voltage. For the undamaged specimen, presented in fig.5 (a), 10 amplitude/frequency curves have approximately the same shape for a sequence of drive voltage. The resonance curves show no change when excitation level increases for all bending mode.
For a specimen having 0.3 and 0.6 of delamination parameter, presented in fig.5 (b) and (c),
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the resonance peak is shifting to lower frequency when the drive amplitude increases for all bending modes. Furthermore, the curves tend to flatten and become asymmetric for large amplitudes. As delamination length increase the resonance peaks move to left indicating the nonlinearity. We can say that when delamination increases, the nonlinearity of composites
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amplitudes.
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increases. Material degradation can be observed when frequencies changes with increasing
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Fig. 5. Resonance curves at 10 different excitation levels of undamaged specimen (a) specimen having 0.3 delamination parameter (b) and specimen having 0.6 delamination parameter for the 1st,the 3rd and the 5th bending parameter.
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4.2 Frequency and loss factor shift
Figs.6 shows the variation of the frequency shift (f-f0)/f0 as function of strain amplitude for specimens having different delamination lengths. The measured amplitude excitation is converted to the acceleration ü. Then the strain values are obtained using the frequency f and the following relation cited in [32].
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ACCEPTED MANUSCRIPT Where L is the specimen length. The strain values obtained by converting the acceleration are in the order 10-7 <ε<10-3. This strain range corresponds to nonlinear material [30] Curves of the Fig. 6 were extracted from the resonance curves for the first six bending modes. They show that the frequency shift decreases as a function of strain amplitude, for different debonding lengths. Furthermore, the decrease of frequency shift is linear with the strain.
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As we said in the section (3.2), there was problem to determine the damping measurements in the first mode which could not be presented. So we present in the fig. 7 the loss factor shift
(η−η0)/η0 V.s strain amplitude (for the 2nd - 6th bending modes). The loss factor shift increases as function of resonance amplitude, for different debonding length. The linear
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increase loss factor shift is also observed in Fig. 7.
This linear variation of the loss factor shift and frequency shift agree with the linear relationship between the frequency shift and the strain amplitude as described in Eq (2)and
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between the loss factor and the strain as described in Eq (3). We can also observe that for the undamaged specimen, there is no significant variation. In fact, the linear variation of frequency shift and the loss factor shift as function of the strain amplitude (ε) is caused by the hysteretic behaviour of delaminated composites.
In this work we have a large range of strain amplitudes. Obtained results were described
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with a linear law. However, a linear law has not fitted the entire curves accurately. This deviation from linear law is may be caused by a combination of linear, classical and the hysteretic nonlinear behaviour as found in [32]. A power law or polynomial law function
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Strain amplitude ε (10-5)
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Mode 6
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Fig. 7. Evolution of the loss factor shift vs. strain amplitude for specimen having different debonding lengths and for the first six bending modes.
4.3 Non linear parameter The frequency shift and the loss factor shift presented as a function of the strain amplitude are fitted by a linear law for all bending modes as expected from hysteretic nonlinearity. The slops values of these curves are determined. The obtained values correspond to the hysteretic nonlinear elastic parameters αf related to the frequency and the hysteretic non linear 12
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT dissipative parameter αη related to the loss factor, describe the nonlinear hysteretic coefficient in the equation (2) [35]. The evolution of the normalised hysteretic elastic parameters (αf/αf0) and the normalised hysteretic non linear dissipative parameter (αη/αη0) as function of increasing delamination length for the first several bending modes, are presented in the Fig. 8
differences. Those parameters depend on the modes.
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and Fig. 9. The nonlinear parameters (αf and αη) for those modes show significant Comparing the result of the intact specimen with those of Haupert [25] who work with PVC, PMMA, chalk and bone, it is proved that nonlinearity in the (GFRP) is not very significant. The decrease of frequency is caused essentially by the presence of delaminations.
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nonlinear hysteretic parameters αf and αη.
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For excitations of high amplitudes, the nonlinear elastic behaviour can be characterized by the
It should be noticed that for the second mode the result shows that the nonlinear parameters (αf and αη) depend on the delamination length, i.e. the lower nonlinear parameter corresponding the lower debonding length. For the other modes, the nonlinear elastic (αf and
αη) parameters do not have a regular variation trend for increasing delamination length. It appears that the hysteretic parameters are only sensitive to delamination for the second
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structure.
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bending mode. The second bending mode is the sensitive mode to delaminations in composite
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Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT Mode 2
Mode 3
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αf/αf0
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Delamination parameter
Delamination parameter
Mode 5
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Delamination parameter
Mode 6
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Delamination parameter
Fig. 8. Evolution of nonlinear elastic parameter for different bending modes as a function of increasing delamination.
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Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT Mode 3
Mode 2 70
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60 15
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Delamination parameter
Delamination parameter
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Mode 4 14
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Delamination parameter
Mode 6
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12 10
8
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6 4 2
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Delamination parameter
Fig. 9. Evolution of nonlinear dissipative parameter for different bending modes as a function of increasing delamination.
4. Sensitivity
In order to compare the sensitivity of the linear with the nonlinear resonance method, we present in the Fig. 10 the linear normalized parameter (f/f0) and the nonlinear normalized parameter (αf /αf0) for the second bending modes. These parameters are normalized respectively with values obtained from the undamaged specimen. Normalized values of nonlinear elastic parameters increase as a function of the delamination length while normalized values of the linear elastic parameter are almost constant. For the lowest 15
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT bending modes, the nonlinear elastic parameter is more sensitive than the linear elastic parameter. Fig. 11 compares the linear normalized parameter (η/η0) and the nonlinear normalized parameter (αη /αη0) for mode 2. The linear and nonlinear dissipative parameters increase when the delamination parameter increases. Globally the nonlinear dissipative parameter is
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more sensible than linear. Mode 2 80
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αf/αf0
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20
0 0,0
0,1
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Nomalized value
60
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f/f0
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Delamination parameter
Fig. 10. Normalized values for linear frequency and nonlinear elastic parameter vs. delamination parameter
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Mode 2
20
αη/αη0
10
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Normalized value
15
η/η0
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Delamination parameter
Fig. 11. Normalized values for linear loss factor and nonlinear dissipative parameter vs. delamination parameter
5. Conclusion The effect of increasing delamination length on the linear and nonlinear behaviour of glass fibres reinforced plastics (GFRP) is studied by linear and nonlinear resonance experiments. Results of linear vibration analysis show that the natural frequencies decreases and the loss actor increases with the debonding length. Nonlinear vibration analysis shows a resonance 16
Non linear elastic behaviour of glass fibre reinforced composites with delamination
ACCEPTED MANUSCRIPT frequency and loss factor shift V.s increasing excitation amplitude. The nonlinear parameters were evaluated as a function of the delamination length and compared to the linear parameters. It appears that nonlinear parameters are more sensitive to the presence of delamination in composite structure than the linear parameter. Indeed, the nonlinear resonance method appears to be sensitive to the presence of delamination. The advantage of this method
with unknown natural frequencies of the undamaged structure.
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