Promotion 2013 Année scolaire 2012 - 2013

ECOLE DES MINES DE DOUAI

MOURET Stéphane

ETUDE BIBLIOGRAPHIQUE

APPLICATION DE L’ACCOUSTIQUE NON LINEAIRE POUR LA CARACTERISATION

DES MATERIAUX COMPOSITES A MATRICE POLYMERE

APPLICATION OF THE NON LINEAR ACOUSTIC FOR THE CHARACTERIZATION

OF THE COMPOSITES WITH ORGANIC MATRIX

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THANKS

At first, thank you to Mister Salim CHAKI, Assistant Professor at the Ecole Nationale

Supérieure des Mines de Douai and tutor for this literature survey.

Secondly, thank you to Mister Walid HARIZI, PhD applicant, for his time and help.

Last but not least, thank you to Mister Jean-Louis Cordonnier and all the employees of

the documentation center for their work and availability.

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Table des matières

THANKS .......................................................................................................................................................... 3

ABSTRACT ...................................................................................................................................................... 7

RESUME ......................................................................................................................................................... 9

1. Introduction ......................................................................................................................................... 11

2. Brief history of the nondestructive tests ............................................................................................. 13

3 States of the art ................................................................................................................................... 15

3.1 Theory .......................................................................................................................................... 15

3.1.1 The composites with organic matrix and their possible damages ...................................... 15

3.1.2 The ultrasonic process ......................................................................................................... 18

3.1.3 The acoustic emission (EA) .................................................................................................. 20

3.1.4 The non linear acoustic (NLA) .............................................................................................. 21

3.2 Experimental procedures ............................................................................................................ 23

3.2.1 Use of acoustic and acousto-elastic measurements ........................................................... 23

3.2.2 Nonlinear Wave Modulation Spectroscopy (NWMS) .......................................................... 24

3.2.3 Combines of Time Reversal & Non Elastic Wave Spectroscopy (TR-NEWS) ....................... 25

3.2.4 Nonlinear Resonant Ultrasound Spectroscopy (NRUS) ....................................................... 26

3.2.5 Dynamic Acousto-Elastic Testing (DAET) ............................................................................. 26

4 Technology .......................................................................................................................................... 29

4.1 Materials & software used .......................................................................................................... 29

4.2 Material suppliers ........................................................................................................................ 31

5 Application areas ................................................................................................................................. 33

6 Current activities ................................................................................................................................. 35

6.1 The active laboratories on the nonlinear acoustic subject ......................................................... 35

6.2 Future congress and technical workshops .................................................................................. 37

7 Conclusion ........................................................................................................................................... 39

REFERENCES ................................................................................................................................................ 41

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ABSTRACT

The composite materials with organic matrix are present in many industrial activities.

They are used on the transportation and energy markets. They are mainly chosen for

their good specific mechanical properties and their good corrosion resistance.

By definition, pieces made in composite materials are heterogeneous, have interface

areas and porosity. These discontinuities could be the starting points for a crack when

the structure will be under load. It is fundamental to detect and quantify these

phenomena before breakage of the structure.

From 2000, researches have been in progress in order to develop new nondestructive

tests which could give information about microscopic defects. The recent research axes

use nonlinear acoustic phenomena.

The goal of this literature review is to summarize the last main works done in these

researches. The conclusion gives the scope of the possible applications of these new

techniques and identifies, today, the most relevant procedure for the composite

materials with organic matrix.

Keywords

Composite materials Scattered defects

Non destructive tests Nonlinear acoustic

Damage Detection

Micro cracks Porosity

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RESUME

Les matériaux composites à matrice organiques sont présents dans de nombreux

secteurs industriels. Ils sont utilisés sur le marché des transports et de l’énergie. Ils sont

principalement choisis pour leurs bonnes propriétés mécaniques spécifiques et leur

résistance à la corrosion.

Par définition, les pièces fabriquées en matériaux composites sont hétérogènes,

contiennent des zones d’interfaces et de la porosité. Ces dernières peuvent être le point

de départ d’une fissure lorsque la pièce est sollicitée. Il est fondamental de détecter et

quantifier ces défauts avant la fissuration de la structure.

Depuis les années 2000, des recherches sont en cours pour développer des techniques

de contrôle non destructif qui donneraient des informations concernant des défauts

microscopiques. Les axes de recherches actuels exploitent des phénomènes de non-

linéarité acoustique.

L’objectif de cette étude bibliographique est de synthétiser les derniers travaux réalisés

dans ce domaine. La conclusion permet de statuer sur les applications de ces méthodes

et d’identifier la technique qui semble, à ce jour, la plus pertinente pour les matériaux

composites à matrice organique.

Mots clefs

Matériaux composites Défauts diffus

Contrôles non destructifs Acoustique non-linéaire

Endommagement Détection

Microfissure Porosité

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1. Introduction

Nowadays, the manufacturing companies have a constant interest in cost reduction

and in lighter products. Therefore, design is more and more accurate and the amount

of material used for making products decreases. But, thinner structures generate

bigger stresses in the final piece and that may decrease its lifetime [1]. In these

conditions, the traditional materials sometime cannot achieve the mechanical

requirements. The composite materials, with organic matrix, are one of solutions to

reach these technical challenges. Nevertheless, this type of structures is

heterogeneous, anisotropic, has interface areas and porosity. These discontinuities

could be the starting points for a crack when the structure is under load.

For safety reasons, aeronautic, nuclear or wind energy markets have more and more

very high demand for defect free parts. By consequence, it is fundamental to be able

to detect damage into composite structures. Obviously, in order to prevent or detect

the damage, airplanes wing or wind blades need nondestructive tests. The definition

of nondestructive testing (NDT) or nondestructive evaluation (NDE) is “all the

techniques and processes which can give information about the health of pieces or

structures without creation additional damage afterward “[2]. Currently, most well

known NDT processes are: radiography, ultrasonic sounding, acoustic emission,

holography, infrared scan, and neutronography. These techniques are relevant and

useful for the detection and localization of cracks. Nevertheless, quantification and

localization of scattered damage, micro-cracks or micro-porosity need more accurate

and sensitive process. In that purpose, nonlinear acoustic tests (NLA) have been

rising for around ten years. The aim of this literature review is to summarize the

works done around these NLA techniques, especially those dedicated to composites

material with organic matrix.

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2. Brief history of the nondestructive tests

“As the scientific instrumentation, the nondestructive tests (NDT) are a privileged

field for the physics discoveries. Indeed, the NDT history started with the modern

physics at the end of the 19e century: X rays, Foucault’s current, piezoelectricity,

discoveries…etc. Nevertheless, NDT techniques have mainly evolved within the

industry after the Second World War. The metallurgical industry was precursor for

the steel analyze and the welds radiography. Then, there was a big improvement in

sixties/seventies pulled by the fast growth of civil aeronautics, nuclear industries,

pressure vessels / pipes, gas and oil pipelines and the offshore platforms. During the

next decade, the huge improvement of informatics tools has been allowing

development of new NDT techniques such as optical controls”. [2].

Nowadays, ultrasonic (US) and acoustic processes are widely use for the detection

and description of defects. These methods become uncertain for complex and

heterogeneous structures like concrete or composite materials [3]. By consequence,

in order to improve the sensitivity of US & (linear) acoustic techniques, many teams

of searchers work on the nonlinear acoustic phenomena [4]. The name used here is

the Non Elastic Wave Spectroscopy (NEWS). These processes are more relevant

and useful than the linear acoustic one for the detection of scattered defect, micro-

cracks and the forward damage [4] [5] [6] [7] [8]. The American teams have been

working on this subject from 2000 whereas the first French thesis is from 2005 [9].

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3 States of the art

3.1 Theory

3.1.1 The composites with organic matrix and their possible damages

The composite materials, with organic matrix, are divided in two families: the

thermoplastics and the thermosets. On one hand, the thermoplastic polymers (ex:

polypropylenes, polystyrene, polyamide) are used on big markets such as building,

automotive or electronic devices. These products require mainly low price, low

weight, good visual aspect, chemical resistance and recycling behavior. On the other

hand, the thermoset polymers (ex: unsaturated polyester, epoxy, vinyl-ester) are

used for structural parts on demanding and growing markets like aeronautic (Figure

1.1 & Figure 1.2), aerospace and more recently the wind energy blades (Figure 2).

Figure 1.1 - Composites parts currently on a plane [10]

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Figure 1.2 - Composites parts under development [10]

Figure 2 - Wind Blade design [11]

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These products require high specific mechanical properties (which are the ratio

strength on weight) [5]. In both cases, a reinforcement material is added to the

polymer in order to reach the mechanical properties specifications. These products

are from different chemical nature. It could be mineral such as glass, carbon, silica,

and boron or organic such as aramid, hemp or sisal. The mix of polymer and

reinforcement fibers creates anisotropic structures (Figure 3).

Figure 3 – Composite parts are anisotropic structures [12]

By definition, the composite products have several phases. The manufacturing

process and the thermo-mechanic background have a huge impact on the micro-

mechanic of composite. We speak about thermo-mechanical history. Due to that, the

scale (macroscopic, mesoscopic or microscopic) has always to be taken into account

during technical discussions. It is also fundamental regarding any mathematical

modeling. During its life, or even simply after manufacturing, the product could have

damage. Due to the manufacturing process itself, porosity is always in parts. It

comes from the resin’s cross-linking (which is the chemical reaction between

macromolecules). The porosity is all the gas bubbles entrapped into the matrix

(Figure 4). The amount of porosity (size and repartition) has an impact on the

structural damage occurring after mechanical loading. Indeed, the initial “small”

defects could be the start point for bigger damage.

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Figure 4- Laminate with high porosity level [13]

In a composite structure, the main damage modes are:

- Intra-fiber breaks

- Transversal or longitudinal matrix breaks

- Interfacial fiber / matrix break

- Delimitation

The purpose of the nonlinear acoustic techniques is to detect and localize defect at

microscopic scale in order to prevent from bigger and more dangerous damage.

The nonlinear acoustic, used for the non destructive testing, needs the combination

of the knowledge about ultrasonic and acoustic emission.

3.1.2 The ultrasonic process

“The general principle of the ultrasonic technique is based on ultrasonic wave’s

emission and analyze of its echo. The echo sound is due to change into the physical

properties of the analyzed area. The distance between echoes gives information

about the defect localization in depth. The ratio between the amplitude of the waves

sent and the amplitude of the refracted signal gives an estimate of the size of the

defect.

Ultrasonic waves are mechanical vibrations which come from and flow through all

material support (solid, liquid or gas). The US waves are defined by oscillated

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frequencies which cannot be heard by human being. In fact, the US waves are from

15 kHz (cleaning applications) to over 100 MHz (micro-acoustic and electronic

applications). The range from 1 to 10 MHz represents the biggest area of the

applications used for the nondestructive tests in industry. US waves have some

specific properties which are linked with the elasticity behavior of the support. There

are three types of US waves: longitudinal, transversal and surface waves (Figure 6)”

[2]. The longitudinal and transversal waves are the most widely used for NDT. These

are the volume waves.

Figure 6 –Wave types and propagation modes into a solid [2]

The speed of the volume waves are given by the following equations [2]:

For longitudinal waves:

For transversal waves:

With E = young modulus (Pa), σ = Poisson’s ratio and ρ = density (kg/m3).

The wave length value is given by the equation:

with f = vibration frequency

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During its flow within the support, US waves are attenuated. That is due to

combination of diffusion and absorption. At the interface of two media, the waves are

transmitted and/or reflected. Deeper information is available in the reference [2].

Ultrasonic waves are also a possible way to stimulate / excite a specimen.

3.1.3 The acoustic emission (EA)

Material under solicitations releases energy through transient elastic waves. This

phenomenon is the acoustic emission. Detection and analyze of these signals give

information about the stress field and the potential damage into the piece. Generally

speaking, detection is realized by using a piezoelectric transducer with frequency

between 50 to 500 kHz.

Acoustic emission process is sensitive to damage, but it can only detect that when

the structure is under loaded. This technique gives data about the defect localization

but not its geometry. A large part can be analyzed with only few sensors. The

acoustic emission is more than a NDT; it can have a monitoring use [14]. Indeed, if

the sensor records a signal, it will indicate that the structure starts breaking (Figure

7).

Acoustic emission is used for the NDE (proprieties measurement), NDT (mainly for

under pressure equipments). “AE is one of only few nondestructive techniques which

can detect the initiation defects and their propagation. The main difficulties are the

surrounding sound and the environmental conditions”. [14]

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Figure 7 – Acoustic emission description during cracking [14]

The combination of US and AE process can be used in order to analyze the damage

modes. This has been done at the French Engineering School, Ecole National

Supérieure des Mines de Douai [3].

3.1.4 The non linear acoustic (NLA)

Non linear phenomenon can be big in inhomogeneous materials, such as concrete or

composites fibers/organic matrix, and the presence of defect increases moreover this

status. By consequence, nonlinear techniques could find applications on composite

materials for the detection of macro and micro-cracks or scattered defects [7]. There

are two expressions for nonlinearity behavior: the “classical nonlinearity” (without

hysteresis) and the non-classical nonlinearity (with hysteresis).

Under the mechanical stress F, cracking starts on C. The crack is like an acoustic sound source. The transducer t records a signal.

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3.1.4.1 The classical non linearity

That is described by the addition of the nonlinear term β into the Hooke’s law as

follow (see also Figure.8):

σ is the stress in the direction x from the field of US waves.

ε is the strain the direction x from the field of US waves.

E is the Young’s modulus in the direction x from the field of US waves.

β is the nonlinear in the direction x from the field of US waves.

E is measured through the propagation speed of longitudinal and transversal waves

in anisotropic material such as composites.

β is measured from acousto-elastic measurements. Indeed, β could be described

with the acousto-elastic answers and the waves flow direction [7]. There are 3

technical ways for its determination which are explained in the paragraph 3.2.

Experimental procedures.

3.1.4.2 The non classical non linearity

The nonlinear acoustic techniques interest not only the industrial area but also the

medical one. As an example, the reference [6] explains how nonlinear acoustic

process could be useful for micro-cracks detection into bones. Based on the works of

Van Den Abeele et al, this presentation explains clearly the theory of nonlinearity.

The equation is presented below with the Figure.8. Nevertheless, for more detailed

information, the reader would have to refer to the book “Non linear acoustics”, which

was written by. Hamilton M.F and Blackstock D.T [15].

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Linear case:

Nonlinear case:

With K (ε, έ) = K0 {1 – βε – δε² - … - α [∆ε + ε (t) sign (έ)]}

Figure.8 – Explanation of the non linearity [6]

3.2 Experimental procedures

The goal of these procedures is to find new relevant indicators which give

information about health of the material.

3.2.1 Use of acoustic and acousto-elastic measurements

In the modified Hooke’s law, the young’s modulus E and the nonlinear parameter β

can be obtained from acoustic measurements. The speeds of the longitudinal and

transversal waves give the young’s modulus value. The nonlinear parameter can be

expressed in function of the strain constants (second and third degree). The

measurements are realized in immersion while the specimen is under loaded

(tensile test device for example). A set of tools, needed for the experimentation, is

shown on Figure.9. The trick here is that the device allows the measurements of

speeds with different wave directions.

Classical NL Non classical NL = hysteresis

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Figure.9 – Experimentation set-up for the measurement of β [7]

3.2.2 Nonlinear Wave Modulation Spectroscopy (NWMS)

The second way to characterize the nonlinearity is to use the creation of harmonics.

The sample is exited by two monochromatic waves (f1 ≠ f2). When the material is

damaged, there are creation additional waves with new frequencies [6].

In fact, if a material is purely linear, its response, after a sinusoidal pulse excitation

with a frequency f, will be sinusoidal signal with the same frequency. At the opposite,

if there is a nonlinearity behavior, the answer will be composed of a fundamental

trace with the frequency f combined with the amplitude A1, and others harmonics

with frequencies 2f, 3f…combined with the respective amplitudes A2, A3…In that

case, the nonlinear parameter is given by the following expression:

d is the distance of propagation and k is the number of waves [7].

This procedure, which excites a sample with elastic waves in order to look for the

“new” frequencies, is called non-linear wave modulation spectroscopy (NWMS). It is

one of many NEWS techniques [16]. In order to quantify the evolution of the

damage, a “variable of damage” can be defined by:

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3.2.3 Combines of Time Reversal & Non Elastic Wave Spectroscopy (TR-NEWS)

There are two ways of research around this topic (Figure 10).

Figure.10 – Defect detection methods and nonlinear effects measured [8]

“Parametric interaction of acoustic waves is chosen as the non-linear method for the

extraction of the non-linear signature. The interaction between waves is generated

by two transducers. A 1 MHz transducer (broadband) is excited by a 20 tone-burst

signal at f1 = 490 kHz through a 55 dB amplifier. A second 1 MHz transducer

(narrowband) is excited by a 35 tone-burst signal at f2 = 860 kHz another 55 dB

amplifier. In order to guarantee exact synchronization of TR waves, all generators

are triggered by an external generator. A laser vibrometer Polytec OFV-5000 is used

to measure the surface velocity. The bandwidth of the decoder is about 1,5 MHz […]

“ [8].

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3.2.4 Nonlinear Resonant Ultrasound Spectroscopy (NRUS)

The resonant frequency decreases while the damage density increases and the

hysteresis variable “α” increases [6].

This technique was used in the damage evaluation in bones. The results obtained

are interesting and could be a good base for future works on composite materials

[6].

3.2.5 Dynamic Acousto-Elastic Testing (DAET)

This testing method is based on the interaction between a stabilized low frequency

wave and Ultra Sonic pulsations (High frequencies). These interactions give a

variation of the TOF (Time Of Flight) and modulation the energy. According to the

studies done on bones; the measurements of the TOF and the low frequency

vibration (ε LF) give the determination of the two nonlinear settings β & δ.

L L

This process is very close to the Single Mode Nonlinear Resonant Acoustic

Spectroscopy (SIMONRAS). “That involves a study of the nonlinear response of a

single resonant mode of the material specimen. Resonance frequency shifts, and

harmonics and damping characteristics are analyzed as function of the resonance

peak acceleration“[17]. “Under resonance, conditions, if the excitation is increased, a

resonance frequency shift is observed”[18]. The Figure.11 and the Figure 12 give

illustration of the meaning of the previous descriptions [17]. According to these

graphs, the undamaged specimen is not very sensitive to this test. Its response is

essentially linear. At the opposite, the damaged specimen is very sensitive to this

technique. Its response is highly non linear; it’s illustrated by a significant resonance

frequency shift with amplitude.

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“An analyzer gain-phase” exits the material at one of its resonance mode. The signal

is sinusoidal and its frequency changes during the test. The signal intensity is

controlled by the “analyzer gain-phase” and then increased of 52 dB. Sensors send

and receive the signal.

High precautions must be taken in order to minimize the nonlinearity due to the

electronic devices. The use of a high-power coupler is advised and the calibration

can be performed by using an aluminum specimen. During his thesis, M. Bentahar

has proved that the electronic devices can reach a linear behavior up to strain of 10-

5.. It is enough because, in non homogeneous structures, the nonlinearity appears

from a level of 10-8 – 10-6 strain.

Figure 11 – Response of undamaged sample [17]

Figure 12 - Response of damaged sample [17]

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4 Technology

4.1 Materials & software used

Only few articles give data about the materials, tools and software used for the

non-linearity measurements. An overview of this topic is given below.

Studies have been performed at the French Engineering School “Institut National

Supérieur des Sciences Appliquées” (INSA). They have investigated the Single

Mode Nonlinear Resonant Acoustic Spectroscopy on Sheet Molding Compound

specimen (which is a composite with organic matrix) under a flexural loading.

The general set-up used is shown on the Figure 13. In addition, the software

ABAQUS™ has been use in order model the vibrations. Prior to that, the elastic

constant had been measured by ultrasonic technique [19]. Regarding the

experiment, the material was excited close to its frequency of resonance. The

vibrations of the structures are, at the same time, recorded through an optical

set-up (vibrometer laser and a stepping motor).

Figure.13 – Experimental set-up used for the nonlinearity study of a SMC sample

under flexural loading [7]

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Other studies have been performed by the University of Tours and the

engineering school “Centrale Lille”. They have investigated the TR-NEWS axe of

research. The materials used are presented bellow on the Figure 14 [8] [14].

Figure.14 – TR – NEWS experimental set-up using parametric interaction [8]

The Universities of Cagliari and Sheffield have analyzed the damage detection in

composite laminates using non-linear acoustic. A part of their electronic devices

is presented with the Figure.15 [20].

Figure.15 – Instruments used in impact damage detection in composite

laminates using nonlinear acoustic [20]

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4.2 Material suppliers

By research on internet, the potential suppliers identified are:

-Roga-instruments

-GE

-Olympus

-Time Group INC

-Sonotron NDT

-Votrum

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5 Application areas

Requests for safety and quality are more and more present. Nowadays, the

composite materials, with organic matrix, are used on markets like transport

(aeronautic, railway, trucks and cars), wind-energy, pressure pipes and pressure

vessels.

The linear acoustic and ultrasonic methods are useful testing process for the

localized and millimetric damage but the non linear acoustic processes, still under

investigation in laboratories, could have the following interests:

- The comprehension of damage initiation at microscopic scale

- The detection of scattered defects

- The detection microscopic defects

- The prediction of life time for a composite part

- The maintenance process during the life of the product

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6 Current activities

Through this literature review, we realize that only few studies deal with

composite materials with organic matrix. Most them are dedicated to weld

analyze or corrosion damage on steel plate. It proves that there is interest for a

laboratory to analyze the specificities of the composite materials with organic

matrix. In addition, it seems very interesting to follow the researches about

biomedical.

6.1 The active laboratories on the nonlinear acoustic subject

The information below is based on the literature (Figure 16). Researches may

have changed of laboratories from the writing time. This could especially be true

for the PhD applicants.

Figure 16 – Possible contacts for nonlinear acoustic

LABORATORIES RESEARCHES

BELGIUM

Catholic University Leuven Keon E.A. VAN DEN ABEELE

Catholic University Leuven Jan CAMELIET

FRANCE

INSA Lyon Mourad BENTAHAR

INSA Lyon Rachid EL GUERJOUMA

INSA Lyon T. MONNIER

INSA Lyon L. DEVILLE

INSA Lyon J.C. BADOUX

Université du Maine – LAUM B. CASTAGNEDE

ENSMM Besançon Bernard CRETIN

Ecole Centrale Lille Olivier BOU MATAR

INSERM Mickaël TANTER

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LABORATOTIES RESEARCHES

FRANCE

ENIVL Blois Serge DO SANTOS

Université François Rabelais Tours Frédéric PATAT

Université François Rabelais Tours Jean Pierre REMENIERAS

Université François Rabelais Tours Thomas GOURSOLLE

LUSSI - Université François Rabelais Tours S. CALLE

LIP – UPMC – Paris - France S. HAUPERT

LIP – UPMC – Paris - France J. RIVIERE

LIP – UPMC – Paris - France M. TALMANT

LIP – UPMC – Paris - France P. LAUGIER

ITALY

University of Cagliari - Italy F. AYMERICH

Polytechnico di Torino - Italy Michele GIFFA

Polytechnico di Torino - Italy Marco SCALERANDI

UNITED KINGDOM

University of Sheffield - UK W.J. STRASZEWSKI

Cambridge university – UK G. ZUMPANO

University of Bath - UK M. MEO

UNITED STATES OF AMERICA

Los Alamos National Laboratory - USA Paul. A. JOHNSON

Los Alamos National Laboratory - USA T.J. ULRICH

Stevens Institute of technology - USA Alexander SUTIN

University of Nevada - USA R.A. GUYER

SWEDEN

Blekinge Institute of Technology - Sweden Kristian HALLER

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6.2 Future congress and technical workshops

On the COFREND [21] website, the following events are announced:

IEEE 2012 – International Symposium on US

From 2012/10/07 to 2012/10/10

Dresden – Germany

ECND 2014 – 11e European Conference about NDT

From 2014/10/06 to 2014/10/10

Prague – Czech Republic

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7 Conclusion

The composite materials, with organic matrix, are more and more present in our

life. In the current economical and environmental situation, traditional industry

like automotive, new business like wind energy or growing market like

aeronautic, all, use more and more composite materials. The main reason is that

composites have better specific mechanical properties than traditional raw

materials like steel. The weakness of composite structure comes from its

heterogeneity. By nature, composite part has interfaces and porosity. It is well

known that these areas, inside the part, could be starting points for damage

afterwards.

The current non destructive methods are useful for detection of defect such as

cracks, delamination or impact damages. Unfortunately, even if the maintenance

and control procedure are well defined, these methods could only prevent from

technical hitch or accident. By consequence, more sensitive tests have interest

to detect very small non linearity, before the creation of any millimetric defect.

These new methods could also be an additional tool for searchers and engineers

regarding the prediction of shelf life for the design activity.

According to the scientific studies present into this document, studying the

resonance mode of the structure (SIMONRAS & DAET) is the most promising

way for the composite materials with organic matrix. The studies performed in

biomedical field has also to be considerate as possible ways of ideas.

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[18] ZUMPANO G., MEO M. – A new nonlinear elastic time reversal acoustic method for

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structures – A simulation study. – International Journal of Solids and Structures, 2007,

44, 3666-3684

[20] AYMERICH F., STASZEWSKI W.J. – Impact damage detection in composite

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[17] VAN DEN ABEELE K.E-A., [et al]. – Micro-damage diagnostics using nonlinear

elastic wave spectroscopy. – NDT&E International, 2001, 34, 239-248

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44

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