W J Staszewski

C Boller1

S Grondel

C Biemans

E O'brien4

C Delebarre

G R Tomlinson

Damage Detection Using Stress and Ultrasonic Waves

INTRODUCTION

Elastic waves and their propagation have been used for many years to analyse impact response problems, mechanical properties of various materials and structural damage. Various types of methods based on sound and ultrasound are applied for nondestructive testing (NDT). Acoustic Emission and Ultrasonic inspection are the most widely used techniques in industrial applications. The first technique is passive and does not require any external signal excitation; stress waves are structure-born and produced internally by defects. The second approach requires high-frequency external excitation. The maturity and proven damage detection applications are the major advantages of these techniques. The Acousto-Ultrasonic approach combines elements of Acoustic Emission and Ultrasonic inspection. Although the method has been around for more than thirty years, it has not been used widely in practice. Recent advancements in this area include applications of guided ultrasonic waves. Lamb waves are particularly attractive for damage detection in aerospace structures; a vast amount of literature has been published over the last ten years.

This chapter briefly discusses damage detection methods based on sound and ultrasound. Various elements related to wave propagation mechanism, monitoring strategy and transducer schemes are discussed. The focus is on definitions, basic principles, similarities and dissimilarities between different techniques. For more details the reader is referred to the appropriate literature. Engineering applications are illustrated using damage detection case studies. Further application examples of Acoustic Emission and Lamb wave based damage detection are given in Chapter 6.

ACOUSTIC EMISSION

Background

Acoustic Emission (AE) is one of the first and most widely used NDT techniques for structural damage detection. This is supported by a number of literature references related to the subject (e.g. [START_REF] Scott | Basic Acoustic Emission[END_REF][START_REF] Muravin | Inspection, Diagnostics and Monitoring of Construction Materials and Structures by the Acoustic Emission Methods[END_REF][START_REF] Holroyd | The Acoustic Emission and Ultrasonic Monitoring Handbook[END_REF]). The technique relies on transient sound waves propagating in the analysed material. Most of these waves are shorttime transient events (burst signals) of significant energy between 100-1000 kHz. The waves can propagate long distances in circles, i.e. in all possible directions (Figure 4.1). Therefore AE testing can cover large, often inaccessible, monitored areas. The distance of propagation depends on material properties, geometry, frequency and environment.

Acoustic events at their origin are high-frequency (in MHz), wideband impacts emitted internally by microcracks and/or inclusion de-cohesion (e.g. metallic inclusions, bubbles) under external loading applied to monitored specimens. These material defects release elastic energy due to rapid local stress redistribution as a result of loading. The energy results from growing cracks, rubbed surfaces of cracks or dislocations. Other sources of AE are also possible such as phase transformation or melting. The AE process is similar to seismic waves propagating as a result of dislocations of various rock formations leading to earthquakes. In contrast to other sound/ultrasound-based techniques it is a passive as well as static method of damage detection, when using the terms mentioned in Chapter 1 before; AE transducers listen to the monitored structure. A relatively small number of sensors is required to cover large monitored areas. of AE transducers such as piezoelectric, electrodynamic, laser and capacitance sensors. Piezoelectric transducers are well-proven and by far the most widely used for AE testing. Figure 4.2 shows examples of classical piezoelectric transducers. The sensitivity of AE transducers is determined by the bandwidth and the resonance frequency. The sensitivity in general is defined using a logarithmic scale given in dB as

Transducers

A = 20 × log U U ref (4.1)
where U is the output voltage from the sensor and U ref is the reference voltage equal to 1 µV. Sensors data need to be amplified before further signal processing. Often preamplifiers are integrated with some types of sensors. Sensors are usually mounted using magnetic holding devices, clamps or glue. Good acoustic contact between the sensor and the monitored surface is essential for AE testing. When the sensor is not permanently bonded on the surface, often special types of coupling agents (e.g. silicon grease, oil) are required. Various signal processing techniques are used to analyse AE signals and extract features related to defects. Usually, a threshold level is defined. The AE threshold needs to be exceeded before further analysis. A number of well-defined standard parameters are then used in order to identify defects. This includes:

Signal Processing

• Peak amplitude -maximum amplitude of the signal;

• Arrival time -absolute time when the signal first crosses the threshold level;

• Duration time -time interval between the first and last crossings of the threshold;

• Rise time -time interval between the first threshold crossing and the peak amplitude;

• Ring down count -number of threshold level crossings. The information obtained from AE tests is displayed using numerical (cumulative or differential characteristics giving total number of events, total energy or pressure) and/or graphical (cluster graphs giving source location) diagrams. Signal waveforms can also be analysed using the time and frequency domains. Source location is possible using arrays

Testing and Calibration

AE testing comprises three major steps: detection of AE activities (damage detection), estimation of the AE burst rate (damage severity) and AE source location (damage location). All steps require the AE equipment to be calibrated. This involves an electronic waveform generator (often called a pulser) which produces pre-defined transient signal or a special device which simulates an AE event using the break of the pencil graphite lead. The later device is commonly known as the Hsu-Nielsen source, after the developers of the technique. The break of a 2H 0.5 mm diameter pencil lead approximately 3 mm from its tip generates an acoustic signal which is similar to a natural AE burst. AE detects damage in the moment of its occurrence. Defect characteristics are unique events and once the event is missed signals cannot be reproduced. Therefore successful damage detection often requires continuous monitoring in service. Also, defects (e.g. cracks) that do not grow are not detected. Burst signals depend on loading and environmental conditions. It is important to monitor these effects for the reference.

The AE testing offers relatively rapid inspection for damage and what is important it can be used for a wide variety of materials. The method offers global monitoring and does not require access to whole monitored areas.

ULTRASONICS

Background

There exist various damage detection techniques based on ultrasound (Buirks et al. 1991;[START_REF] Rose | Ultrasonic Waves in Solid Media[END_REF][START_REF] Schmerr | Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach[END_REF] and/or properties of ultrasonic waves propagating in the material in order to detect defects. It is well known that various types of waves can propagate in solids. Longitudinal (often called compressional, dilatational pressure or P-wave) and shear (often called transverse or S-wave) waves are the most commonly used wave modes for ultrasonic testing. Both types of waves are represented graphically in Figure 4.5. The direction of particle motion is either in the direction (longitudinal waves) or perpendicular (shear waves) to the direction of wave propagation.

The velocity of propagating waves is one of the most important parameters in ultrasonics. The relationship between the velocity of the longitudinal ultrasonic wave and elastic properties of the material is often used to evaluate the condition of the structure. This relationship can be given as

ν 2 ≈ E L ρ (4.2)
where ν is the velocity of the longitudinal ultrasonic wave, E L is the longitudinal modulus of elasticity and ρ is the density of the monitored specimen. Similarly, the square velocity of the shear wave is proportional to the shear modulus E s over density. Ultrasonic waves can travel long distances in solid materials. However, the energy of sound decreases with the distance of propagation. Propagating waves are additionally scattered (or reflected) and absorbed by different material/structural boundaries. As a result, the amplitude of propagating waves is attenuated. When a wave passes between different media, the velocity of propagation changes. Additionally, various mode conversions can occur. Ultrasonic testing utilises wave attenuation, reflection and refraction for damage detection. Successful NDT requires an understanding of the ultrasonic field. This involves not only wave propagation principles but also problems related to pressure variations, directivity analysis and beam angle of divergence, as discussed in [START_REF] Rose | Ultrasonic Waves in Solid Media[END_REF].

It is important for successful damage detection that the wavelength of ultrasound used for testing is of the order of the defect's size. The wavelength λ can be calculated as

λ = c f (4.3)
where c is the wave velocity and f is the input frequency. This formula is valid only for continuous waves (or pulse waves with a large number of oscillations). The wavelength value for short-time pulse waves (pulse wave with a small number of oscillations) varies due to the broadband nature of the pulse signal.

Inspection Modes

Ultrasonic testing procedures are based on two major inspection modes. These are normal beam inspection and angle beam inspection. The ultrasonic pulse travels through a thickness of material in a normal beam inspection. The so-called pulse-echo method requires two transducers. The pulse is generated by the transmitter and captured by the receiver.

In contrast, the pitch-catch mode involves only one transducer. The pulse generated by the transducer passes trough the material, gets reflected by a boundary and is captured by the same transducer. The angle beam inspection mode introduces refracted shear waves to the monitored specimen. The inspection modes are shown graphically in Figure 4.6.

Transducers

There exist various types of ultrasonic transducers. Most acoustic transducers use piezoelectric ceramics which convert electrical energy into the mechanical energy and vice versa. Figure 4.7 shows a diagram of a typical ultrasonic transducer. The type, size and frequency bandwidth of transducers are important for damage detection resolution. There are various types of transducers used in practice. This includes contact and noncontact transducers. Contact transducers require appropriate coupling (gel, water) for the ultrasound energy transfer. Special types of transducers are used when specimens are immersed in water. Noncontact transducers do not require any coupling but show significant attenuation at high frequencies.

The geometry, frequency and size of the transducers determine the ultrasonic beam near field (Figure 4.8). The following formulas can be used in practice [START_REF] Rose | Ultrasonic Waves in Solid Media[END_REF])

sin α 2 = 0.6λ r and N = r 2 f c (4.4)
where α is the angle of divergence, r is the transducer radius and N is the near field distance defined as the point on the axis of the transducer which separates intense oscillations from a smooth decay. A small angle of divergence, required in practice, is a compromise between the transducer radius and the input frequency. Often both parameters are decreased in order to reduce destructive interferences in the near field.

Frequency bandwidth is also important since it affects the penetration of the material and damage detection sensitivity. Low-frequency transducers (below 2 MHz) offer better penetration whereas high-frequency transducers (above 15 MHz) provide better sensitivity to small defects.

Display Modes

Ultrasonic data is presented in a form of ultrasonic scans representing damage detection results. Four different types of scans are used in practice. These are:

• A-scan: pulse amplitude presented as a function of travel time;

• B-scan: two-dimensional representation giving travel time of an ultrasonic pulse versus transducer position;

• C-scans: signal echoes are displayed in a top view of the test surface giving defect location maps;

• D-scans: a modified C-scan in which time-of-flight values are displayed in a top view on the specimen surface. Ultrasonic inspection can be used for a wide range of materials. It is highly sensitive to small defects. However, it requires direct access to inspected surfaces and good inspection skills. Full defects maps (C-scan, D-scan) require scanning and are relatively time consuming. Nevertheless, ultrasonic testing has been used for many years as one of the most successful damage detection techniques.

ACOUSTO-ULTRASONICS

The acousto-ultrasonic technique is based on stress waves introduced to a structure by a probe at one point and sensed by another probe at a different position. The acoustoultrasonic method, introduced in the late 1970s [START_REF] Vary | The acousto-ultrasonic approach[END_REF], combines elements of Ultrasonics, guided wave Ultrasonics and Acoustic Emission. All these four techniques utilise stress waves propagating in structures for damage detection. However, Acoustic Emission is the only technique which does not require any external excitation; stress waves are produced by material defects. The other three techniques require an external source of high-frequency excitation to produce stress waves. However, paths of classical ultrasonic waves are well defined and traceable. Similarly, guided wave Ultrasonics is based on well-defined propagation of waves, as described in Section 4.5. In contrast, propagation of acousto-ultrasonic waves is difficult to analyse. This is due to the fact that the method uses high-frequency (usually above 0.5 MHz) impulse excitation which results in a large number of mixed modes. Alternatively, the broadband excitation (Gaussian white noise or sweep sine) could also be used, as reported in [START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF] and shown in Section 4.8.1. Actuating and receiving transducer probes are often not in a line-of-sight position regarding the damage. As a result, the actual transducer responses include not only directly propagating wave modes but also reflected and scattered modes. The rich content of the stress wave energy of complex modes carries a lot of information regarding possible material and/or structural damage. Figure 4.10 shows an example of an acoustoultrasonic stress wave from damage detection studies in a composite-metal joint [START_REF] Kwon | Monitoring fatigue damage in adhesively bonded composite-metal joints by acoustic methods[END_REF].

There exist a number of different parameters to describe the stress wave energy. Often AE parameters described in Section 4.2.3 are used in practice. Alternatively, the Stress Wave Factor (SWF), based on power spectral density, is applied. The SWF can be defined using spectral moments given as [START_REF] Kiernanand | PC analysis of an acousto-ultrasonic signal[END_REF])

M n = f max 0 S(f )f n df( 4.5)
where S(f ) is the power spectral density, f max is the maximum frequency of the analysed spectrum and n = 1, 2,....N. The M 0 moment is the most widely used parameter in practice. [START_REF] Kwon | Monitoring fatigue damage in adhesively bonded composite-metal joints by acoustic methods[END_REF] 4.5 GUIDED WAVE ULTRASONICS

Background

Guided waves are governed by the same wave equations as bulk waves. However, in contrast to bulk waves, they have an infinite number of modes associated with propagation.

Guided waves are able to interact with defects in structures due to their propagation properties that are highly sensitive to any discontinuities in materials. This section defines various types of guided waves and briefly introduces Lamb waves which are the most widely used guided waves for structural damage detection.

Guided Waves

Propagating wave packets which are superpositions of various modes are often called guided waves. There are various types of guided waves available in practice. Wave packets, resulting from appropriate stress and strain boundary conditions, which travel on the surface of a solid body, are known as surface waves. Surface waves usually exhibit large amplitudes and travel slower than other types of guided waves.

Rayleigh waves are the best known surface waves [START_REF] Viktorov | Rayleigh and Lamb Waves[END_REF]. They are nondispersive for uniform material properties. However, their mechanism of propagation is very complex; waves are polarised and surface particles are moved around an ellipse. The components of Rayleigh waves can couple with a medium surrounding the surface of the body. This coupling affects the amplitude and velocity of the wave. The amplitude of the wave decreases rapidly with depth. The rate of decrease depends on the wavelength. Therefore inspection methods based on Rayleigh waves are used mostly to detect surface defects. Other examples of surface waves include Stonely and Love waves. These waves are not commonly used for damage detection and are better known to seismologists than to damage detection experts. Stonely waves occur at an interface between two media. They are closely related to Rayleigh waves. The existence of Stonely waves depends on the density and shear modulus ratios of the neighbouring media. Love waves are horizontally polarized shear waves which also exist on the surface. However, in contrast to the other two types of surface waves they are highly dispersive. Love waves are the fastest propagating surface waves.

Lamb waves [START_REF] Viktorov | Rayleigh and Lamb Waves[END_REF]) are dispersive plate waves that occur for traction-free forces on both surfaces of the plate. The velocity of these waves depends on the product of frequency of excitation and thickness of the plate. They can propagate long distances and are used for damage detection of plate-like structures. Lamb waves are the most widely used guided waves for damage detection. They are discussed in more details in the next section.

Lamb Waves

Lamb waves refer to elastic perturbations propagating in a solid plate with free boundaries for which the displacements correspond to different basic propagation modes, with symmetric and antisymmetric vibrations. For a given plate thickness d and acoustic frequency f , there exists a finite number of such propagation modes specified by their phase velocities. A complete description of such propagation characteristics for plates is normally given in the form of a set of dispersion curves, illustrating the plate-mode phase velocity as a function of the frequency-thickness product. Each curve represents a specific mode, which is conventionally called A 0 ,S 0 ,A 1 ,S 1 ,A 2 ,S 2 ,etc.whereA n denotes antisymmetric modes and S n symmetric modes.

When an excitation is applied at some point on the plate, the excitation energy encounters the upper and lower surfaces of the plate. Longitudinal waves (P) are polarised parallel to the plate whereas shear horizontal waves (SH) form a series of modes. After some time when longitudinal waves are polarised in the direction perpendicular to the surface, shear vertical waves (SV) form modes in connection with P waves; these combined P+SV guided waves are known as Lamb waves.

The most common approach for solving the Lamb wave problem is the method of potentials. For a solid medium bounded by two parallel planes a distance 2d apart (see Figure 4.12), the equation of motion, which contains only the particle displacement, can be given as [START_REF] Achenbach | Wave Propagation in Elastic Solids[END_REF])

µw i,jj + (λ + µ)w j,ji + ρf i = ρ ẅi (4.6)
where w i are displacements, f i are body forces, λ, µ are Lamé constants and ρ is the density. The boundary conditions for the surface tractions can be defined as

t i = S ij n j (4.7)
where t i are traction forces, S ij are stresses and n j are cosine directions. The longitudinal and shear waves of plain strain are governed by the well-known wave equations that can be obtained from Equation (4.6) using the Helmholtz decomposition as [START_REF] Achenbach | Wave Propagation in Elastic Solids[END_REF])

∂ 2 φ ∂x 2 1 + ∂ 2 φ ∂x 2 3 = 1 c 2 L ∂ 2 φ ∂t 2 (4.8) ∂ 2 ψ ∂x 2 1 + ∂ 2 ψ ∂x 2 3 = 1 c 2 T ∂ 2 ψ ∂t 2 (4.9)
respectively. Here, φ and ψ represent decomposed displacement variables, c L indicates the velocity of longitudinal wave whereas c T is the velocity of shear (transverse) waves. The actual longitudinal and transverse displacements of the plate can be obtained as

w 1 = ∂φ ∂x 1 + ∂ψ ∂x 3 (4.10) w 2 = ∂φ ∂x 3 - ∂ψ ∂x 1 (4.11)
respectively. Similarly, stresses can also be expressed in terms of the field variables but are not analysed here. The solutions of Equations (4.8-9), representing travelling waves in the x 1 direction and standing waves in the x 3 direction, can be assumed in the form

φ = (x 3 ) exp[i(kx 1 -ωt)] (4.12) ψ = (x 3 ) exp[i(kx 1 -ωt)] (4.13)
where ω is the frequency and k is the wavenumber. Substituting Equations (4.12-13) into Equations (4.8-9) gives the unknown amplitude functions and as

(x 3 ) = C 1 sin(px 3 ) + C 2 cos(px 3 ) (4.14) (x 3 ) = D 1 sin(qx 3 ) + D 2 cos(qx 3 ) (4.15)
where

p 2 = ω 2 c 2 L -k 2 and q 2 = ω 2 c 2 T -k 2 (4.16)
and C 1 ,C 2 ,D 1 ,D 2 are arbitrary constants. Since both field variables involve sine and cosine functions, which are odd and even respectively, the solutions are often split into two symmetric and antisymmetric modes. The displacements for the symmetric modes are w 1 = ikC 2 cos(px 3 ) + qD 1 cos(qx 3 ) (4.17)

w 2 =-pC 2 sin(px 3 ) -ikD 1 sin(qx 3 ) (4.18)
whereas the solutions for the antisymmetric modes can be given as

w 1 = ikC 1 sin(px 3 ) -qD 2 sin(qx 3 ) (4.19) w 2 = pC 1 cos(px 3 ) -ikD 2 cos(qx 3 ) (4.20)
These equations, known as Rayleigh-Lamb equations for guided waves in plates, show that Lamb wave propagation is generally complex due to the coexistence of at least two modes at any given frequency and the strongly dispersive nature of these modes at high frequency. The traction-free boundary conditions for the plain strain need to be additionally applied in order to obtain the constants C 1 , C 2 , D 1 and D 2 . This leads to Rayleigh-Lamb frequency relations known as the dispersion equations [START_REF] Achenbach | Wave Propagation in Elastic Solids[END_REF])

tan(qh) tan(ph) =- 4k 2 pq (q 2 -k 2 ) 2 (4.21)
for symmetric modes and

tan(qh) tan(ph) =- (q 2 -k 2 ) 2 4k 2 pq (4.22)
for antisymmetric modes, where h = d/2. The above equations can be solved numerically in order to predict velocities of a propagating Lamb wave of frequency f in a plate of thickness d. The results are presented as a function of the fd frequency-thickness product. Figure 4.13 gives an example of the dispersion characteristics for an aluminium plate. This clearly shows how many complex modes can propagate in the plate. Single S 0 and A 0 Lamb wave modes are possible only for small values of the frequency-thickness product (f d < 2). Furthermore, a single and pure Lamb wave mode may generate a variety of other modes either by interacting with a surface or subsurface flaw or by crossing the interface between two materials of different boundaries. As a result, the problem becomes very difficult to solve analytically. Often numerical analysis is employed to study wave propagation. Figure 4.14 shows a simulated example of the Lamb wave propagating in the aluminium plate with a damage slot positioned in the middle of the plate. Here, the scatter around the damage can be clearly observed in the contour plots of the propagating wave. The study was performed using the Local Interaction Simulation Approach (LISA) [START_REF] Delsanto | Connection machine simulation of ultrasonic wave propagation in materials -I: the one-dimensional case[END_REF][START_REF] Delsanto | Connection machine simulation of ultrasonic wave propagation in materials -II: The two-dimensional case[END_REF][START_REF] Delsanto | Connection machine simulation of ultrasonic wave propagation in materials -III: The three-dimensional case[END_REF]. The method has been proven to perform very well for wave propagation models with complex boundaries, imperfect material interfaces and heterogeneous materials [START_REF] Agostini | Flaw detection in composite plates by means of Lamb waves[END_REF][START_REF] Lee | Modeling of acousto-ultrasonic wave interactions with defects in metallic structures[END_REF], 2003a, 2003b, 2003c).

Monitoring Strategy

Real engineering structures under inspection are usually quite complex when compared with simple plates studied in laboratory conditions and reported in the literature. The complexity is determined by various types of joints, stiffeners, rivets, complicated shapes or varying thickness. This causes the entire analysis to be much more complicated and requires an appropriate monitoring strategy.

The methodology or strategy of monitoring is extremely important for successful damage detection. The basic factors, which determine the Lamb wave based damage detection analysis are related to properties of the structure under inspection, transducer schemes, choice of excitation input signal, and appropriate signal processing, as reported in [START_REF] Wilcox | Mode transducer selection for long range Lamb wave inspection[END_REF][START_REF] Staszewski | Acoustic wave propagation phenomena modelling and damage mechanisms in ageing aircraft[END_REF]. Other elements include various aspects related to transducer coupling methods, optimal sensor locations and sensor validation procedures. The last but not least is the hardware used for monitoring, graphical interface and data storage organisation.

The dispersive nature of Lamb waves and also the finite number of modes at a given frequency makes long-range inspection very difficult. To overcome these problems, low frequency-thickness products are often utilised for damage detection. In this case only two fundamental modes A 0 and S 0 , are used. It is important to limit the bandwidth of the excitation to a range over which there is little dispersion (i.e. the phase velocity does not change significantly with frequency).

The first applications of Lamb waves for damage detection used bulky wedge (anglebeam) transducers. It appears that piezoceramic elements are now the most widely used transducers due to the fact that they can be used as sensors and actuators at the same time. Often piezoceramics can become an integral part of the monitored structure, as shown in Section 4.6.1. Recent advancements in this area include optical fibre sensors [START_REF] Betz | Acousto-ultrasonic sensing using fiber Bragg gratings[END_REF] and MEMS (Micro-Electro-Mechanical Systems) sensors [START_REF] Khuri-Yakub | Silicon micromachined ultrasonic transducers[END_REF]. Both types of sensors can be used not only for Lamb wave detection but also for strain measurements. MEMS devices can additionally generate Lamb waves. A number of different practical aspects need to be considered once choosing transducers for Lamb wave detection. This includes coupling, connectors and environmental protection. The cold bonding is usually preferred to hot bonding. The process of bonding must be as easy (if not easier) as the procedure for bonding strain gauges. Also, bonded sensors are better than embedded due to possible sensor failures and replacements. Reliable connectors and environmental protections are required to prevent sensor failures. Wireless applications are possible with piezoceramic and MEMS sensors. Coupling, connectors and environmental protection are particularly important in the case of optical fibre and piezoceramic sensors.

Different types of signals are used as input excitation. It is considered that the simpler the input signal the simpler the output signal for damage detection. The choice of input excitation is often a compromise between the amplitude and the mode generation. Lowvoltage signals are possible when the input frequencies are within transducer resonance frequencies (see Section 4.6.3). This is often associated with intelligent signal processing to remove undesired modes and extract features related to damage. In practice transducers resonance frequencies do not coincide with single Lamb wave mode frequencies.

Previous studies [START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]Staszewski et al. 1999a) show that even simple input signals can lead to complex output signals due to various attenuation and dispersion effects which are not related to damage. This clearly shows that intelligent signal processing is one of the most important elements of the Lamb wave based damage monitoring strategy. Examples of various signal processing techniques for damage detection are discussed in Chapter 5.

Once the transducers excitation signals are chosen the question is where to put sensors for optimal damage detection. Recent years have shown considerable progress on the problem of determining the number and location of sensors in engineering structures. Some of these techniques are discussed in Chapter 5.

PIEZOELECTRIC TRANSDUCERS

Piezoelectricity and Piezoelectric Materials

Piezoelectric transducers are the most widely used sensors for damage detection based on sound and ultrasound. Piezoelectricity is an electric polarisation effect due to mechanical forces. In other words an electric charge is collected on the surface of the piezoelectric material when it is squeezed.1 Often the converse effect is possible, i.e. the material generates a mechanical strain in response to the applied electric field. Both effects are illustrated graphically in Curie in 1880s. It is an anisotropic property of crystalline materials and results from nonuniform charge distributions within a crystal's cells. There are number of different materials which exhibit these effects. Natural materials include quartz (SiO 2 ), Rochelle salt and tourmaline. The quartz crystal exhibits stiffness comparable to steel and shows high voltage sensitivity. Manufactured piezoelectric ceramics were introduced in the early 1950s. The first material developed with piezoelectric properties was barium titanate (BaTiO 3 ). The two most widely used manufactured materials are lead zirconate titanate (PZT) and polyvinylidene fluoride (PVDF). The former is a ceramic and the later is a polymer film. Piezoelectric ceramics (often called piezoceramics) are quite brittle and need to be handled with care. Piezoelectric polymer films are, in contrast, very flexible and easy to handle for sensor applications. PVDF exhibits the strongest known piezoelectric behaviour of all polymers. PVDF offers better direct piezoelectricity and worse inverse piezoelectricity than the PZT, and therefore it is more often used for sensing applications. Piezoceramic materials have been used for various types of transducers for many years. Recent years have shown extensive progress in developing sensing devices. The adaptation and integration of piezoceramic sensors onto, or into, structures has become more feasible. Low-profile piezoceramic sensors with wrap-around electrodes are available as thin plates and discs which can be bonded, and/or embedded, in monitored structures. Other important developments in this area include piezoceramic paints [START_REF] Egusa | Polling characteristics of PZT/epoxy piezoelectric paints[END_REF], Smart Layers  (Chang 1998) and piezoceramic fibres (Yoshikawa et al.). Smart Layers  are Kapton dielectric films with an embedded network of distributed piezoceramic PZT sensors. These layers can be fabricated in various sizes and shapes, as shown in Figure 4.16.

Piezoceramic sensors are particularly attractive for structure integrated damage detection since they exhibit simultaneous actuator and sensor behaviour. This allows for both passive and active damage detection. In fact the majority of damage detection applications based on guided wave ultrasonic utilise PZT type sensors. There exist a vast amount of literature related the properties and behaviour of piezoceramic elements.

Constitutive Equations

Piezoelectric materials exhibit both mechanical and electrical properties. It is well known that mechanical properties of any linear elastic material can be described by the Hooke's where ε is the permittivity. The electric field E is equivalent to a force field in mechanics and represents the work done against the electric field in order to move a charge. In contrast, the electric displacement D gives the redistribution of charge when the material is subjected to an electric field. Equations (4.23) and (4.24) are often called constitutive mechanical and electrical equations, respectively. Piezoelectric materials exhibit coupled mechanical, electrical and piezoelectric properties. The constitutive equations describing these materials can be given as

S = S E T + d t E D = ε T E + dT (4.25)
where S is the mechanical strain, E is the electric field, T is the mechanical stress, D is the electrical displacement (all these state variables are second-order tensors), d is the piezoelectric coefficient and ε is the permittivity. Here, subscripts E and T indicate that the quantities are under constant (usually zero) electrical and stress fields, respectively. Additionally, superscript t indicates the transpose. Equation (4.25) gives the strain-charge form of the constitutive equations which relate the electric and mechanical fields. The other three forms give rearranged relationships between the state variables. The constitutive equations describe how voltage changes in the material when the charge moves, and the other way around. Piezoelectric coefficients d ij defined as

d ij = strain applie delectric field = charge density applied stress (4.26)
characterises either the strain produced by an applied electric field or the charge density developed from the applied stress field. The subscript i indicates the direction of electric field or displacement, whereas the subscript j gives the direction of strain or stress. The axis notation can be chosen as shown in Figure 4.17; i, j = 1, 2, and 3 indicate the shear strains associated with directions 1, 2 and 3, respectively. Figure 4.18 illustrates the physical interpretation of the piezoelectric coefficients. Here, the electric field E 3 is applied to the transducer of thickness t and width w. The transducer expands vertically (in the direction indicated by the subscript 3) by the factor equal to 1 + ε 3 ; the new thickness of the transducer becomes t(1 + ε 3 ). At the same time the transducer shrinks horizontally (in the direction indicated by the subscript 1) by the factor equal to 1ε 1 ; the new width of the transducer becomes w(1ε 1 ). The piezoelectric coefficients are then equal to

d 33 = ε 3 E 3 and d 31 = ε 1 E 3 (4.27)
Note that d 31 is in this case a negative number.

Often the conversion between the electrical and mechanical (or mechanical and electrical) energies is described using the electromechanical coupling coefficient k ij which is defined as

k 2 ij =
stored electrical energy applied mechanical energy = stored mechanical energy applied electrical energy (4.28)

where the subscript notations are identical to the convention used for piezoelectric coefficients. This parameter describes the efficiency of the energy conversion.

Properties

The manufactured piezoceramics are initially isotropic. They are composed of randomly oriented cells which are polarised when the strong DC current is applied. This process is called poling due to the fact that electric dipoles are permanently aligned by the electrical field. In contrast to natural piezoelectric materials, piezoceramics show high charge sensitivity.

The voltage-strain behaviour of piezoceramics is in general nonlinear. Only the linear part of the voltage-strain curve is used in practice for sensing. Most piezoceramics also exhibit a typical hysteretic behaviour which exhibits the energy loss.

All piezoceramic materials have operating limits associated with temperature and voltage. The piezoelectric properties are exhibited below the so-called Curie temperature. When this temperature is reached or exceeded the material is not piezoelectric anymore, i.e. electric dipoles change their orientations. Some piezoceramics can operate up to 550 • C. The properties of piezoceramic materials also change as a result of temperature. Figure 4.19 shows an example of the experimental peak-to-peak voltage amplitude as a function of the temperature. The excessive voltage (usually above 1000 V) has a similar depolarising effect. Piezoceramic materials are also very brittle. High level of tensile stress can lead to mechanical damage and depolarisation.

The impedance can be used in order to describe dynamic properties of the piezoceramic transducers. The maximum performance of the transducer is achieved when the excitation frequency is chosen as one of the natural resonance frequencies of the transducer. This enables a very efficient conversion from the electrical to mechanical energy. Resonance frequencies are associated with different vibration modes of the transducers. These modes can be used for the generation of different types of waves. Often Frequency Response Functions (FRF) and transfer characteristics are obtained for the transducers using theoretical and experimental studies. Various piezoceramic elements available in the standard Finite Element (FE) codes can be used for the theoretical analysis. embedded or bonded transducers will be influenced by the acoustical loading conditions associated with the boundaries.

PASSIVE DAMAGE DETECTION EXAMPLES

Crack Monitoring Using Acoustic Emission

Acoustic Emission analysis was performed on a full scale Airbus A320 aircraft inner wing as a part of the certification procedure. from background, fretting, etc., were eliminated before any source location studies. The analysis of the data established within 50 cycles showed the conclusive evidence of the presence of cracks and was used to direct conventional NDT to the locations. The growth of the length of the cracks was also monitored and showed that the five cracks that were detected could be monitored contemporaneously.

The results showed that Acoustic Emission could detect stress waves that emanate from growing cracks that occurred during full-scale fatigue tests.

Impact Damage Detection in Composite Materials

This section presents impact damage detection results from the passive monitoring approach utilising piezoceramic sensors (Staszewski et al. 1999b). The experimental tests involved a series of impacts on a simple composite structure shown in Figure 4.24.

The two specimens used in the experimental investigations were rectangular composite panels (530 × 225 mm) fabricated from the T300/914 carbon/epoxy unidirectional prepreg using nominal thickness of 0.15 mm/ply. They were made from 32 plies, except for the area at the root which was 48 plies, and the lay-up sequences were [+45, -45, 0 4 , +45, -45, 0 4 , +45, -45, 0 2 ,9 0 2 ,0 2 , -45, +45, 0 2 ] s and [-45, +45, 0 3 , -45, +45, 0 2 , +45, -45, 90 2 , -45, +45, O] s , respectively. A ply build-up region was fabricated to simulate a reinforcement. A composite stiffener was bonded to the root side of the panel. The padded area allowed for the effects of ply drop-offs on the detection procedure to be investigated. The stiffener allowed for a disbond to be analysed. The composite panels were representative of aircraft wing skin parts. The panels were instrumented with one piezoceramic PZT Sonox P5 plate (15 × 15 × 1 mm) bonded to the root side, as shown on a schematic diagram in Figure 4.25. The composite panel was mechanically fastened to a stiffening aluminium subframe using screws. This allowed for the quick replacement of the analysed plates. The composite panel was attached to the metal loading frame, as shown in Figure 4.26.

The composite/aluminium structure was statically loaded (0.2 kN, displacement equal to 18 mm) as a cantilever beam using a Schenck 250 kN servo-hydraulic test machine. The functionality of the sensor was checked using low energy 2 J impacts at random positions (Staszewski et al. 1999) on the plate. A mobile impactor was used in the tests. This initial study was followed by a series of test impacts at positions A and B (Figure 4.25) with the energy levels equal to: 2, 4, 8, 12, 20 J for impact location A; and 2, 4, 8, 12, 16, 24 J for impact position B. The panel was A-scanned after each impact to check for possible damage. The presence of clear delamination was observed after the 20 and 24 J impact at position B.

The impact strain data were acquired from the piezoceramic sensor using a digital 2-channel HP oscilloscope with a sampling frequency of 400 kHz. An example of the strain data for the 4 J impact at position B is given in Figure 4.27. The absolute maximum values of the strain data are given in Figure 4.28. The results show that in general the characteristics increase with the impact energy. Thus impact energy levels can be estimated analysing the strain data obtained from the piezoceramic sensor. These energy levels can be related to possible damage severities in the composite panels. Further analysis examples using advanced signal processing techniques are given in Chapter 5.

ACTIVE DAMAGE DETECTION EXAMPLES

Crack Monitoring in Metallic Structures Using Broadband Acousto-Ultrasonics

A simple fatigue experiment was performed in order to obtain the strain data from metallic specimens with growing cracks [START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]. The specimens used were two rectangular (400 × 150 × 2 mm) aluminium plates with cracks initiated by spark erosion in the middle of the plates. Figure 4.29 shows the experimental set-up. The fatigue test was performed on a Schenck 250 kN servo-hydraulic test machine running in load control.

The aluminium specimens were subjected to three different load conditions:

• no load -close to 0 kN;

• static load -at 12.5 kN mean load;

• dynamic load -static load plus sinusoidal tension-tension cycling loading with a frequency of 6 Hz and maximum amplitude of 11.5 kN. Crack lengths were determined using a microscope to an accuracy of 0.01 mm. Figure 4.30 shows an example of a crack propagation curve for the analysed specimens.

Each plate was instrumented with six piezoceramics (PZT material Sonox P5 ;d i m e nsions: 15 × 15 × 1 mm) bonded in a symmetrical configuration 20 mm below and above the initiated crack, as shown in Figure 4.31. The piezoceramic no. 2 on each plate was used as an actuator in order to perform the acousto-ultrasonics damage detection.

The study involved two different types of excitation:

1. Gaussian white noise with the maximum frequency of 16 kHz;

2. sine sweep from 10 to 50 kHz with the ramp time of 0.01 s. The maximum amplitude of excitation was equal to 20 V. The signals were generated using the DIFA SCADAS II and Philips PM5138 generators.

The piezoceramics 5 and 6 were used to monitor the growing cracks. The response data, from the piezoceramic sensors no. 5 and 6, were recorded by a DIFA SCADAS II 24 channel measuring system running the LMS 3.4.04 data acquisition software and by a digital 2-channel Tektronix TDS 210 oscilloscope. Figure 4.32 shows an example of the piezoceramic response to the sine sweep excitation for different types of load used in the experiment. Here, a clear drift can be observed for the dynamic load due to the 6 Hz tension-tension cycling loading.

Various signal parameters were estimated for the analysed data. Figure 4.33 gives the results obtained for the minimum amplitude values, for plate no. 5 under the sweep sine excitation. Results, obtained for the static load condition, show a fairly linear behaviour while for the no load condition a similar linear relationship can only be observed after a certain crack length. The results obtained with monitoring under the dynamic load are quite arbitrary. Performing, however, a linear regression over these data may possibly lead to a result similar to the data for the no load and static load conditions. The reason for the relatively good result when monitoring under the static load condition can be seen in a clear situation of the crack being fully open. For the other two load conditions the situation of the crack looks likely to be less clear.

The experimental data were also analysed using the classical Fourier transform approach. Examples of power spectra for the data from sensor no. 5 are given in Figure 4.34. Here, two dominant spectral components at about 1.5 and 2.7 kHz can be observed in Figure 4.34a for the data representing the undamaged plate. The amplitude of these two components decreases when the crack grows in the plate, as shown in Figure 4.34b and 4.34c, whereas the remaining parts of the spectrum are relatively unchanged. The results show that any direct comparison of spectra requires further signal processing analysis. A number of spectral statistical parameters, discussed in Chapter 5, were calculated for the sensor data under the [START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF] This preliminary study shows that satisfactory damage detection results can be obtained under all analysed load conditions. However, this requires the appropriate choice of excitation and signal processing. Also, the best results are mainly obtained under the static load when cracks tend to be clearly open or under the dynamic load when cracks open and close successively due to cycling loading.

Impact Damage Detection in Composite Structures Using Lamb Waves

This section presents damage detection analysis based on Lamb waves. The study involves the composite wing-box structure described in Section 4.7.2. Piezoceramic sensors have been used to introduce the excitation signals and to acquire the Lamb wave responses. The sensors chosen for these tests were 3 × 4 × 1 mm piezoceramic P160 plates. A general rule of ultrasound excitation is to excite the actuator at its natural resonances rather than at any other frequency. This approach enables a very efficient conversion from the electrical to mechanical energy, as discussed in Section 4.6.3. A two-dimensional numerical model was developed [START_REF] Assaad | The finite element code at ISEN (IEMN)[END_REF]) using the ATILA FE code in order to perform the modal and harmonic analysis of the piezoelectric transducers. The analysis resulted in four natural vibration modes for the frequency bandwidth between 100 kHz and 1 MHz. The second mode at 400 kHz was the best electromechanically coupled (k e = 43 %). The computation of the displacement field, shown in Figure 4.20, shows that the analysed mode of vibration is a transverse mode.

The transducers have been positioned in various locations in order to detect different types of damage (e.g. impact damage, delaminations, substructural disbonds). Figure 4.36 shows sensor locations on the composite structure. A series of impacts, as described in Section 4.7.2, was performed in order to introduce various severities of damage to the composite skin. A pair of transducers, i.e. El (actuator) and R1 (sensor), was used to allow for the inspection of the plate near impact location 4 (see Figure 4.36). One more pair of transducers, i.e. E2 (actuator) and R2 (receiver), were positioned in order to monitor the disbond growth between the stiffener and the composite plate under cyclical loading and detect possible delaminations at impact location 3 (see Figure 4.36).

A pulse 5-cycle sine burst signal at 400 kHz used to excite the actuators at their resonance frequencies. The Lamb waves excited by the actuators propagated along the plate and were received by the sensors. The responses were amplified, filtered and transferred to a digital oscilloscope. The data acquisition utilised 64 averages in order to improve the signal-to-noise ratio.

Preliminary tests have been performed on the structure in order to study the influence of external conditions, such as additional vibration and temperature, on Lamb wave testing procedure. This was followed by a sequence of testing steps:

• acquisition of Lamb wave signals before the tests (no damage condition);

• specimen loaded under 0.2 kN: acquisition of Lamb wave signals;

• vibration conditions: acquisition of Lamb wave signal during, the tests;

• impact tests performed with a mobile impactor using different energy levels (2 J, 4 J, 8 J, 12 J, and 24 J) in location 3 and 4: after each impact, inspections of the plate with Lamb waves were carried out;

• cyclic loading with different displacement levels (9 ± 7 mm, 14 ± 9 mm, 20 ± 12 mm, 22 ± 15 mm): acquisition of Lamb wave signals during the tests.

The first set of tests has demonstrated good reproducibility of the Lamb wave responses under external conditions (external vibration and loading). The performance of all the sensors was not affected by varying thermal conditions; Lamb wave responses remained unchanged after the thermal tests.

The second set of tests was performed in order to measure the sensitivity of Lamb wave signals to damage (delamination and disbond). Damage detection analysis utilised the presence of other modes, or modes conversion due to possible discontinuities in the structure.

Figure 4.37a shows the Lamb wave response from sensor R1 in the absence of defects. Here, the first wave packet was identified as the S 1 mode from a measurement of its group velocity using the time-of-flight analysis. When the specimen was loaded under 0.2 kN and submitted to an external vibration excitation, no influence on the shape of the Lamb wave signal was observed, which proves that loading did not have any influence regarding the signal recorded. However, after a damaging impact at location 4, one of the modes is lost, as shown in Figure 4.37b. The presence of delamination was confirmed using a conventional ultrasonic A-scan. Figure 4.38 gives the local minima of the envelope function calculated using the Hilbert transform [START_REF] Randall | Frequency Analysis[END_REF]. These minima, corresponding to mode changes due to damage, can be used for damage detection.

Figure 4.39 shows the time history of the Lamb wave response for the transmission test between the E2 and R2 transducers before the impact tests were performed (no damage condition). The signal exhibits more complex modes because of the geometry of the analysed plate (thickness variations and stiffener). Further testing reveals a clear

SUMMARY

A brief introduction to damage detection methods based on sound and ultrasound has been given in this chapter. The presentation includes AE, Ultrasonics, Acousto-Ultrasonics and guided wave Ultrasonics. The material presented shows that there are various similarities between these techniques. All these techniques utilise piezoceramic transducers for damage monitoring and testing. Therefore a brief description of piezoceramic materials has been given for the sake of completeness. Simple application examples show how the methods work in practice. The potential of the methods for aerospace applications is presented in Chapter 6 where various large-scale and flight test examples are given.

Figure 4 . 1

 41 Figure 4.1 Acoustic Emission principle

Figure 4 . 2

 42 Figure 4.2 Acoustic Emission transducers (reproduced by the permission of MISTRAS Holdings Corp.)

Figure 4 . 3

 43 Figure 4.3 Acoustic Emission burst (a) and continuous; (b) signals[START_REF] Staszewski | Wavelet signal processing for acoustic emission data[END_REF]

  Figure 4.3 gives examples of burst and continuous AE signals.

Figure 4 .

 4 Figure 4.4 summarises AE parameter used for damage detection. Often the amplitude values of the AE signal are integrated over the duration time in order to estimate the energy of the burst signal. The level of the background noise can be estimated calculating the Root Mean Square (RMS) value of the signal before the arrival time. Various clustering techniques of AE signals are used in order to mark and locate the highest AE activities.The information obtained from AE tests is displayed using numerical (cumulative or differential characteristics giving total number of events, total energy or pressure) and/or graphical (cluster graphs giving source location) diagrams. Signal waveforms can also be analysed using the time and frequency domains. Source location is possible using arrays

Figure 4 . 4

 44 Figure 4.4 Summary of Acoustic Emission parameters used for damage detection

  Figure 4.5 Longitudinal (a) and shear (b) bulk waves

Figure 4 . 6 Figure 4 . 7 Figure 4 . 8

 464748 Figure 4.6 Ultrasonic inspection modes: (a) pulse-echo; (b) catch-pitch

Figure 4 . 9

 49 Figure 4.9 Ultrasonic C-scan examples from a composite plate: (a) delamination after the 2 J impact; (b) delamination after 7 J impact[START_REF] Pedemonte | Signal processing for passive impact damage detection in composite structures[END_REF]

Figure 4 .

 4 Figure 4.10 Acousto-ultrasonic stress waves from damage detection in a composite-metal joint: (a) unfatigued specimen; (b) fatigued specimen[START_REF] Kwon | Monitoring fatigue damage in adhesively bonded composite-metal joints by acoustic methods[END_REF]

Figure 4 . 11

 411 Figure 4.11 Stress wave factors for the data presented in Figure 4.10: (a) peak amplitude; (b) M 1 / M 0[START_REF] Kwon | Monitoring fatigue damage in adhesively bonded composite-metal joints by acoustic methods[END_REF]

Figure 4

 4 Figure 4.12 Geometry of the plate for Lamb wave propagation

Figure 4 .Figure 4 . 14

 4414 Figure 4.13 Lamb wave dispersion characteristics for an aluminium plate

Figure 4 .

 4 Figure 4.15 Direct (a) and inverse; (b) piezoelectric effects

Figure 4 .

 4 Figure 4.16 Smart Layer  sensors. (Courtesy of Acellent Technologies Ltd, California)

Figure 4 . 17 Figure 4 .

 4174 Figure 4.17 Axis notation for the strain/stress and electric fields

  Figure 4.20 gives an example of such analysis for a simple disc-type transducer. Here, the strongest radial and thickness modes are identified. It is clear that resonance frequencies of the 3

Figure 4 .Figure 4 . 20

 4420 Figure 4.19 Temperature effect on Lamb wave responses when monitored with piezoceramic sensors

Figure 4 .Figure 4 . 23

 4423 Figure 4.21 Schematic diagram representing the wing specimen on a fatigue test rig

Figure 4 .

 4 Figure 4.24 Composite structure representative of an aircraft wing skin

Figure 4 .

 4 Figure 4.27 Impact strain data for the 4 J impact at position B(Staszewski et al. 1999)

Figure 4 .

 4 Figure 4.28 Absolute maximum values for the impact strain data

Figure 4 .Figure 4 . 30

 4430 Figure 4.29 Experimental set-up for crack monitoring using acousto-ultrasonics[START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]

Figure 4 .

 4 Figure 4.31 Sensor locations on the aluminium specimen used in crack monitoring tests[START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]

Figure 4 .

 4 Figure 4.32 Acousto-ultrasonic responses to sine-sweep excitation for: (a) no load; (b) static load;(c) dynamic load[START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]

  Figure 4.35 shows an example of the Root Mean Square (RMS) of Spectral Difference characteristics for sensor no. 5. Again, results obtained for the static load condition show a monotonic behaviour while for the no load and dynamic load conditions the results are not satisfactory -the curves do not indicate any crack growing in a plate. More examples related to these investigations are given in[START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF].

Figure 4 .

 4 Figure 4.35 Root Mean Square for the Spectral Difference values for the sweep sine excitation[START_REF] Biemans | Crack detection in metallic structures using broadband excitation of acousto-ultrasonics[END_REF]

Figure 4 .

 4 Figure 4.36 Impact positions and locations of Lamb wave transducers on the composite wing-box structure

Figure 4 .

 4 Figure 4.37 Lamb wave responses from sensor R1: (a) undamaged plate; (b) damaged plate

Figure 4 . 38

 438 Figure 4.38 Local minima for the Lamb wave responses from sensor R1: dashed line -before testing, solid line -after damaging impact

Figure 4 .Figure 4 . 40 Figure 4 .

 44404 Figure 4.39 Lamb wave responses from sensor R2 for the undamaged plate

Piezo is in fact a Greek term for to squeeze.


Julien Cucherousset
email: julien.cucherousset@cict.fr

S Bouletreau

A Martino

J.-M Roussel

F Santoul

Using stable isotope analyses to determine the ecological effects of non-native fishes

Keywords: biological invasions, food web, trophic interactions, trophic subsidies

Freshwater fish ecology has greatly benefited from the use of innovative tools such as stable isotope analysis to determine the ecological effects of non-native fishes. Stable isotope analyses are based on the predictable relationship between the isotope composition of a consumer and its prey, and have become increasingly popular amongst aquatic ecologists. In parallel, they have been implemented as a sensitive, cost-effective and temporally integrative tool to analyse the trophic interactions between native and non-native species, and to detect sorne subtle ecological effects of human activities, such as the introduction of non-native freshwater fish species. This review aimed to understand how stable isotope analyses have been used and how they have provided new insights into the ecological impacts of non-native freshwater fishes. Specifically, the published literature (45 articles) was reviewed to establish the current state-of-the-art. The use of stable isotope analyses in the field is still an emerging approach. The majority of studies were conducted on lentic ecosystems in North America targeting three main families of non-native fish species. Measurements were most commonly made with carbon and nitrogen stable isotopes using muscle samples. The most recent theoretical and methodological advances were illustrated by selecting sorne case studies conducted with different non-native species and biotic interactions. Finally, several recommendations for an optimised use of stable isotope analyses for freshwater ecological studies related to trophic interactions of non-native freshwater fish species were established.

Introduction

Biological invasions occur worldwide and induce important ecological and economie consequences (Mack et al. 2000; Lockwood et al. 2007; Davies 2009; Gozlan et al. 2010). Fish are amongst the most frequently introduced species in freshwater ecosystems because of their strong association with human activities [START_REF] Copp | To be, or not to be, a nonnative freshwater fish[END_REF][START_REF] Garcia-Berthou | Introduction pathways and establishment rates of invasive aquatic species in Europe[END_REF]. Although sorne introduced fish species failed to establish self-sustained populations and sorne established populations have no detected effects on the recipient ecosystems [START_REF] Gozlan | Introduction of non-native freshwater fish: is it ali bad?[END_REF], there is a large body of literature demonstrating that non-native freshwater fish can have strong ecological impacts at variable levels of biological organisation, ranging from genes to ecosystems (review in Cucherousset & Olden 2011). The effects of non-native freshwater fish include biodiversity loss and biotic homogenisation (e.g. [START_REF] Rahel | Homogenization of fish faunas across the United States[END_REF][START_REF] Olden | Biotic homogenization: a new research agenda for conservation biogeography[END_REF]), the introduction of pathogens (e.g. [START_REF] Gozlan | Biodiversity -disease threat to European fish[END_REF] and hybridisation (e.g. [START_REF] Allendorf | The problems with hybrids: setting conservation guidelines[END_REF][START_REF] Cucherousset | Do native brown trout and non-native brook trout interact reproductively?[END_REF]. However, there are still many cases where potential ecological effects have not been investigated.

Although unintentional introductions can occur, freshwater fish are principally introduced in new ecosystems for four main reasons, i.e. aquaculture, anglingjsport, fisheries and ornamental [START_REF] Gozlan | Introduction of non-native freshwater fish: is it ali bad?[END_REF]. Introduced fish species are generally larger than native species as they represent a non-random subset of freshwater fish [START_REF] Blanchet | Non-native species disrupt the worldwide patterns of freshwater fish body size: implications for Bergmann's rule[END_REF]) and the most common example is the introduction of large-bodied predatory species (e.g. [START_REF] Eby | Effects of stocking-up freshwater food webs[END_REF]Syvaranta et al. 2010). Although body size might not always be a biological trait that facilitates invasion success [START_REF] Miller | Invasion success: does size really matter?[END_REF], body size plays an important role in determining trophic position (e.g. [START_REF] Woodward | Body-size in ecological networks[END_REF][START_REF] Hildrew | Body Size: The Structure and Function of Aquatic Ecosystems[END_REF]). Because of their functional role in ecosystems, introduced large-bodied species can induce new biological interactions with native species of prey, competitors and/or predators. These new interactions arise from direct (e.g. competition and predation) and indirect (e.g. trophic cascade) effects that can destabilise native communities and food webs (e.g. [START_REF] Baxter | Fish invasion restructures stream and forest food webs by interrupting reciprocal prey subsidies[END_REF][START_REF] Lockwood | Invasion Eco/ogy[END_REF]). As the mechanisms by which food webs can be destabilised by non-native fish species are extremely complex [START_REF] Gozlan | Current knowledge on non-native freshwater fish introductions[END_REF][START_REF] Cucherousset | The ecological impacts of normative freshwater fishes[END_REF], new and integrative tools are needed to determine and quantify the potential ecological effects of non-native species on recipient food webs (e.g. [START_REF] Ehrenfeld | Ecosystem consequences of biological invasions[END_REF]). In addition, differentiating between the effects of introduced fish and other environmental changes induced by human activities on aquatic ecosystems remains challenging [START_REF] Cucherousset | The ecological impacts of normative freshwater fishes[END_REF].

Stable isotope analysis (SIA) has become popular amongst ecologists (reviews in [START_REF] Grey | Stable isotopes in freshwater ecology: current awareness[END_REF][START_REF] Fry | Stable Isotope Eco/ogy[END_REF]). Indeed, this method is based on the predictable relationship between the isotopie composition of consumers and their prey, allowing the investigation of diet composition and energy flow in food webs [START_REF] Fry | Stable Isotope Eco/ogy[END_REF]. Compared with more conventional approaches used in freshwater ecology, such as the analyses of stomach contents that offer only a snapshot of diet composition, SIA is cast-effective and temporally integrative allowing the analyses of long-term diet patterns (e.g. [START_REF] Grey | Stable isotopes in freshwater ecology: current awareness[END_REF][START_REF] Rybczynski | Comparing trophic position of stream fishes using stable isotope and gut contents analyses[END_REF]. The most common stable isotopes used by freshwater ecologists are nitrogen (o 15 N) and carbon (o 13 C) that allow the quantification of trophic position and energy sources of an organism, respectively [START_REF] Peterson | Stable isotopes in ecosystem studies[END_REF][START_REF] Fry | Stable Isotope Eco/ogy[END_REF]. Additionally, other stable isotopes such as sulphur (o 34 S) have been increasingly used [START_REF] Grey | Stable isotopes in freshwater ecology: current awareness[END_REF]. In parallel, mixing models have been developed to determine the contributions of different prey items to the diet of the consumers [START_REF] Phillips | Source partitioning using stable isotopes: coping with too many sources[END_REF]. Therefore, SIA appears as a sensitive and versatile tool to investigate competition and predation processes in food webs [START_REF] Peterson | Stable isotopes in ecosystem studies[END_REF][START_REF] Fry | Stable Isotope Eco/ogy[END_REF] and is a promising approach to quantify the ecological effects of non-native freshwater fish.

This article aimed to understand how SIA has been used and how it has provided new insights into the ecological effects of non-native freshwater fishes. Specifically, (1) the literature published since the year 1999 was reviewed to establish the current state-of-theart in the field; (2) the most recent theoretical and methodological advances were illustrated using case studies; and (3) these findings were discussed in the light of the new challenges facing fish ecologists to establish severa! recommendations for an optimised used of this tool.

State-of-the-art review

The literature published citing Vander [START_REF] Vander Zanden | Stable isotope evidence for the food web consequences of species invasions in lakes[END_REF], the first of its kind to quantify the ecological impacts of non-native freshwater fish on food web structure using SIA, was reviewed. To do so, the ISI Web of Knowledge was used to collect articles. Second, additional articles were collected by analysing the references in the articles from the initial search and by conducting additional searches. Based on these searches, articles that used SIA and that stated and/or discussed the status (i.e. non-native) of the fish species analysed for stable isotopes were selected. For each selected article, the publication year, geographie location, type of aquatic ecosystem, isotope(s) analysed, tissue used, and number, family and trophic position of the non-native fish species were collected. Trophic position was obtained from FishBase (http:/ jwww.fishbase.org). The use of mixing models and stomach contents analyses and the type of study (observationally or experimentally) were also determined.

Forty-five articles were collected (Supplementary Data 1). Between 1999 and 2008, the number of articles remained relatively low, i.e. ranging from 0 to 7 articles per year. The number of articles published thereafter has substantially increased, with 12 articles each year in [START_REF] Davies | Invasion Bio/ogy[END_REF]2010 (Fig. la), showing the recent increased use of SIA in this field. As expected in any review on biological invasions ( e.g. [START_REF] Pysek | Geographical and taxonomie biases in invasion ecology[END_REF], geographie and taxonomie biases in publication patterns were observed. Indeed, 62.2% of the studies were performed in North America and 26.7% in Europe. Only 4.4, 4.4 and 2.2% of the studies were conducted in Asia, Africa and South America, respectively. The greatest numbers of studies were performed on Cyprinidae (20.6%), Salmonidae (20.6%) and Centrarchidae (17.6%) as non-native species in the recipient ecosystem (Fig. 1 b ). These proportions differ from [START_REF] Cucherousset | The ecological impacts of normative freshwater fishes[END_REF] where the most studied families were Salmonidae (35.8%) and Centrarchidae (17.9%). This discrepancy might reflect the commonness of Cyprinidae in many fish assemblages.

The majority of studies (57.8%) were conducted in lentic ecosystems (lakes or ponds), whilst 42.2% were conducted in lotie ecosystems (rivers or streams). Virtually all these studies (95.6%) were conducted observationally (Table 1) and most (91.1 %) used carbon and nitrogen stable isotopes in combination. Only one study used three stable isotopes, i.e. carbon, nitrogen and sulphur. In the majority of cases, stable Vander Zanden et al. (1999) investigated the ecological effects of two non-native predatory species, smallmouth bass, Micropterus dolomieu (Lacepède), and rock bass, Ambloplites rupestris (Rafinesque). The study was conducted in Canadian lakes where lake trout, Salvelinus namaycush (Walbaum), is the native predatory species. They compared the abundance and diversity of prey fish and the trop hic position of native S. namaycush in lakes with and without non-native species.

The authors found that the presence of the nonnative species significantly decreased the diversity and abundance of littoral prey fish. Lakes without and with the non-native species had, on average, 5.3 and 0.6 species of prey fish, respectively; and 5.9 and 0.9 fish per trap per day, respectively. Stable isotopes (ô 15 N and ô 13 C) indicated a lower trophic position and lower ô 13 C values of native S. namaycush in lakes with the non-native species compared with lakes without nonnatives, whilst no changes in prey fish values were measured. Specifically, trophic position averaged 3.28 and 3.90 and ô 13 C averaged -29.20 and -27.48% 0 in invaded and un-invaded lakes, respectively. Using a two-source mixing model, a modification of S. namaycush diet composition was observed whereby the species had, on average, a higher proportion of zooplankton (84 vs 36%) and a lower proportion of littoral prey fish (16 vs 64%) in their diet in invaded lakes compared with lakes with no non-native fish species. A lower contribution of littoral prey and a higher contribution of pelagie prey in invaded lakes were also observed.

Competitive interactions between salmonids species

Salmonids are amongst the most widely introduced species because of their high values for angling and aquaculture, and ecological impacts of non-native salmonids have been reported across the globe (e.g. Townsend 2003; review in Cucherousset & Olden 2011). An insightful example is the reciprocal introductions of European brown trout, Sa/mo trutta L., into North America and North American brook trout, Salvelinus fontinalis (Walbaum), into Europe. When introduced in North America, non-native S. trutta negatively affected native S. fontinalis. Similarly, nonnative S. fontinalis can affect native S. trutta in European rivers. In both cases, impacts included redd superimposition (e.g. Essington et al. 1998; Cucherousset et al. 2008), hybridisation (e.g. [START_REF] Sorensen | Reproductive interactions between sympatric brook and brown trout in a small Minnesota stream[END_REF][START_REF] Cucherousset | Do native brown trout and non-native brook trout interact reproductively?[END_REF] or changes in habitat niche (e.g. Fausch & White 1981; [START_REF] Korsu | Niche characteristics explain the reciprocal invasion success of stream salmonids in different continents[END_REF]; but see [START_REF] Blanchet | Competitive interactions between native and exotic salmonids: a combined field and laboratory demonstration[END_REF]). However, little was known about potential trophic interactions between the two species in Europe.

U sing SIA, [START_REF] Cucherousset | Stable isotope evidence of trophic interactions between introduced brook trout (Salvelinus fontinalis) and native brown trout (Sa/mo trutta) in a mountain stream of southwest France[END_REF] studied the effects of the presence of non-native S. fontinalis on the trophic niche of native S. trutta in a Pyrenean stream of south-west France. The study area consisted ofthree stretches, i.e. one with the non-native species only, one with the native species only and one where both species coexist. Stable isotope analysis revealed that the trophic niche of the non-native S. fontinalis was not affected by the presence of native S. trutta as no significant differences in ô 15 N and ô 13 C values were observed between stretches. Conversely, significant changes in native S. trutta stable isotope values were measured between stretches. Specifically, ù 13 C decreased by 1.6% 0 and ù 15 N increased by 0.6roo when S. trutta coexisted with non-native S. fontinalis; the trophic niche of S. trutta becoming similar to that of S. fontinalis when the two species coexisted. Additionally, mixing models revealed an increased consumption of terrestrial invertebrates by S. trutta when coexisting with S. fontinalis; terrestrial invertebrates being the main prey items consumed by the non-native S. fontinalis in bath stretches. This niche convergence was somewhat surprising as niche divergence or partitioning caused by competition was expected ( e.g. [START_REF] Nakano | Flexible niche partitioning via foraging mode shift: a proposed mechanism for coexistence in stream-dwelling charrs[END_REF]. The authors hypothesised that it might be caused by behavioural interactions between the two species [START_REF] Cucherousset | Stable isotope evidence of trophic interactions between introduced brook trout (Salvelinus fontinalis) and native brown trout (Sa/mo trutta) in a mountain stream of southwest France[END_REF].

Consumption of anadromous species by non-native catfish

Eggs and carcasses from anadromous fish are known to subsidise freshwater ecosystems with marine-derived nutrients [START_REF] Schindler | Pacifie salmon and the ecology of coastal ecosystems[END_REF][START_REF] Macavoy | Anadromous fish as marine nutrient vectors[END_REF]. Many organisms use these nutrients directly and indirectly [START_REF] Bilby | Incorporation of nitrogen and carbon from spawning coho salmon into the trophic system of small streams: evidence from stable isotopes[END_REF], and SIA has been used to determine the importance of marine-derived nutrients for freshwater organisms because marine-derived nutrients have very distinct stable isotope values compared with freshwater-derived nutrients (e.g. [START_REF] Fry | Stable Isotope Eco/ogy[END_REF]. For instance, Gregory-Eaves et al. (2007) demonstrated that ù 15 N and ù 13 C values of periphyton, zooplankton and rainbow trout, Oncorhynchus mykiss (Walbaum), were positively correlated with the density of sockeye salmon, Oncorhynchus nerka (Walbaum), spawners in lakes of western North America. Stable isotope analysis has also allowed tracing of the fluxes of marine-derived nutrients transported from the ocean by anadromous fish to terrestrial organisms such as invertebrates (e.g. [START_REF] Hocking | Consumption and distribution of salmon (Oncorhynchus spp.) nutrients and energy by terrestrial flies[END_REF], birds (e.g. [START_REF] Christie | Tracking salmon nutrients in riparian food webs: isotopie evidence in a ground-foraging passerine[END_REF], wolves [START_REF] Szepanski | Assessment of anadromous salmon resources in the diet of the Alexander Archipelago wolf using stable isotope analysis[END_REF] or bears (e.g. [START_REF] Hilderbrand | The importance of meat, particularly salmon, to body size, population productivity, and conservation of North American brown bears[END_REF].

The presence of non-native fish species can modify the energy, nutrient and organism fluxes between ecosystems (e.g. [START_REF] Baxter | Fish invasion restructures stream and forest food webs by interrupting reciprocal prey subsidies[END_REF]) and two studies have used SIA to quantify predation of non-native catfish on anadromous fish. [START_REF] Macavoy | Marine nutrient contributions to freshwater apex predators[END_REF] quantified stable isotope values (ù 15 N, ù 13 C and ù 34 S) in native anadromous A/osa, native resident freshwater fish and non-native resident catfish, Jeta/urus furcatus (Valenciennes), in the Rappahannock River (Virginia, USA). Irrespective of the stable isotope used, the stable isotope values for non-native I. furcatus differed significantly from those of resident freshwater fish and anadromous A/osa. Using mixing models, the contributions of anadromous A/osa to the diet of nonnative I. furcatus were, on average, 42% ( ± 24 SD) and 43% ( ± 16 SD) using ù 13 C and ù 34 S, respectively. Those values were highly variable amongst individuals, ranging from 0--86% to 10--69% using ù 13 C and ù 34 S, respectively, but no significant correlations between catfish size and the contribution of anadromous prey to the diet were observed [START_REF] Macavoy | Marine nutrient contributions to freshwater apex predators[END_REF]. Syvaranta et al. (2009) examined the contribution of anadromous fish species such as allis shad, A/osa a/osa (L.), to the diet of the non-native European catfish, Si/urus glanis L., in the Garonne River (south-western France) using ù 15 N and ù 13 C. S. glanis was introduced from eastern European waters to western Europe in the 1950s, and the species is considered as having an opportunistic diet, although few studies on its trophic ecology outside ofits native range are available (review in [START_REF] Copp | Voracious invader or benign feline? A review of the environmental biology of European catfish Si/urus glanis in its native and introduced ranges[END_REF]; but see [START_REF] Kopp | Trophic ecology of the pikeperch (Sander lucioperca) in its introduced areas: a stable isotope approach in southwestem France[END_REF]Syvaranta et al. 2010 for studies using SIA). Syvaranta et al. (2009) compared the stable isotope values of European catfish, Si/urus glanis L., and their potential prey fish (A. a/osa and cyprinids) below and above a dam that prevented the migration of anadromous fish. ù 13 C values of freshwater prey fish differed only slightly below and above the dam (mean values = -26.3 and -24.6% 00 respectively) and anadromous shad were highly 13 C-enriched compared with freshwater prey fish (mean value = -19.3% 0 ). S. glanis sampled below the dam were 13 C-enriched compared with individuals sampled above the dam (mean values = -19.8% 0 and -25.2% 0 , respectively). Using a twosource mixing madel, the averaged contribution of anadromous fish to the diet of non-native S. g/anis ranged between 53 and 65% below the dam. As observed by [START_REF] Macavoy | Marine nutrient contributions to freshwater apex predators[END_REF], the contribution of anadromous fish to the diet of non-native S. glanis was highly variable amongst individuals (ranging from 30 to 90%) and not significantly correlated with individual size (Syvaranta et al. 2009).

Consequences of hybridisation between native common carp and non-native domesticated carp

Common carp, Cyprinus carpio L., is a worldwide invader [START_REF] Vilizzi | The common carp, Cyprinus carpio, in the Mediterranean region: origin, distribution, economie benefits, impacts and management[END_REF]) that has established self-sustained populations in many countries, except in sorne temperate areas (e.g. [START_REF] Britton | Non-native fishes and climate change: predicting species responses to warming temperatures in a temperate region[END_REF]). The species can potentially have strong ecological impacts on ecosystem functioning (e.g. [START_REF] Koehn | Carp (Cyprinus carpio) as a powerful invader in Australian waterways[END_REF][START_REF] Cucherousset | The ecological impacts of normative freshwater fishes[END_REF]. A recent meta-analysis demonstrated that this engineer species affects nutrient dynamics and many organisms such as macrophytes, plankton and macro-invertebrates [START_REF] Matsuzaki | Contrasting impacts of invasive engineers on freshwater ecosystems: an experiment and meta-analysis[END_REF]. Using SIA in Lake Navaisha (Kenya), [START_REF] Britton | From introduction to fishery dominance: the initial impacts of the invasive carp Cyprinus carpio in Lake Naivasha, Kenya, 1999 to 2006[END_REF] found C. carpio can also display relatively high ù 15 N values.

To understand further the ecological effects of C. carpio, Matsuzaki et al. (2010) performed a study that was based on the combination of SIA and molecular tools. The aim of the study was to quantify the functional consequences of hybridisation between native C. carpio and non-native domesticated C. carpio in Lake Kasumigaura (Japan). The authors used nuclear DNA markers (i.e. single nucleotide polymorphism) to establish the degree of hybridisation of each individual and SIA (ù 15 N and ù 13 C) with mixing models to determine functional differences amongst individuals with a variable degree of hybridisation. A significant correlation was measured between ù 13 C and the degree of hybridisation, whilst no significant correlation was measured for ù 15 N, indicating that C. carpio with higher levels of hybridisation displayed a more frequent use of littoral habitat compared with individuals with lower levels of hybridisation (Matsuzaki et al. 2010). When using a mixing model, these results translated into a higher proportion in the diet of limnetic primary consumers (as opposed to shrimps, periphyton and detritus) for individuals with lower levels of hybridisation compared with individuals with higher levels of hybridisation [START_REF] Matsuzaki | Stable isotope and molecular analyses indicate that hybridization with nonnative domesticated common carp influence habitat use of native carp[END_REF].

Conclusions

Stable isotope analysis appears as a powerful tool to determine the ecological effects (and their underlying mechanisms) of non-native fish species on recipient ecosystems. Stable isotope analysis has allowed ecologists to detect subtle and/or counter-intuitive ecological effects caused by non-native fish species at different spatial scales because of its ability to integra te ecological processes over a relatively long period of time. As costs of analyses are decreasing and SIA is becoming more popular amongst ecologists, its use should be generalised to many systems and new directions should be taken to strengthen its applicability. Although care should be taken to account for potential constraints associated with the methodology (e.g. scales (e.g. [START_REF] Grey | Stable isotope analysis of archived roach (Rutilus rutilus) scales for retrospective study of shallow lake responses to nutrient reduction[END_REF] or mucus (e.g. [START_REF] Church | Mucus: a new tissue fraction for rapid determination of fish diet switching using stable isotope ana1ysis[END_REF].

-Use of archived tissues to study the long-effects of non-native species and reconstruct historical food webs (e.g. [START_REF] Grey | Stable isotope analysis of archived roach (Rutilus rutilus) scales for retrospective study of shallow lake responses to nutrient reduction[END_REF]. Preliminary work aiming to understand the potential effect of preservation methods of stable isotope values should be conducted in parallel.

-Development of approaches that will use SIA in conjunction with other new methodologies. For instance, [START_REF] Cunjak | Using stable isotope analysis with telemetry or mark-recapture data to identify fish movement and foraging[END_REF] demonstrated that SIA can be used in conjunction with individual fish tagging and tracking. Also the use ofmolecular tools (review in Blanchet 2012) in conjunction with SIA (e.g. [START_REF] Charles | Genetic differentiation between anadromous and freshwater resident brown trout (Sa/mo trutta L.): insights obtained from stable isotope analysis[END_REF]) should help to provide new insights simultaneously into both the ecological and evolutionary effects of non-native freshwater fish.

-Stable isotope data should be analysed using new analytical and statistical approaches developed to obtain a quantitative understanding of changes in stable isotope values and food web structure (e.g. [START_REF] Layman | Can stable isotope ratios provide for communitywide measures of trophic structure?[END_REF][START_REF] Schmidt | Quantitative approaches to the analysis of stable isotope food web data[END_REF]. Mixing models that take into account the variability in trophic fractionation and in stable isotope values of the prey and consumers when estimating diet composition should be preferentially used (e.g. [START_REF] Parnell | Source partitioning using stable isotopes: coping with too much variation[END_REF].

-Stable isotope ecology should help future investigations into the ecological consequences of non-native fish species, notably on their effects on the fluxes of energy, nutrients and ecosystem across ecosystems. For instance, the stable isotope of hydrogen (ùD) appeared useful to investigate the energy fluxes between freshwater and terrestrial ecosystems (e.g. [START_REF] Doucett | Measuring terrestrial subsidies to aquatic food webs using stable isotopes of hydrogen[END_REF].

-Further studies that include SIA should be developed to test whether the functional patterns observed in the field can be tested and validated experimentally. This would allow a generalisation of the findings, helping researchers to understand the applied and theoretical aspects associated with biological invasions.

Figure 1 .

 1 Figure 1. (a) Number of articles using stable isotope analyses in the context of non-native freshwater fish published per year since the year 1999 (bars) and number of citations per year of Vander Zanden et al. (1999) (line). (b) Families of the studied non-native freshwater fish.

Table 1 .

 1 Distribution (number of articles and percentage) of 45 selected articles published since the year 1999 in terms of approach (experimental and observational), isotope analysis (C, carbon; N, nitrogen and S, sulphur), tissue (muscle, fin and scale), stomach contents and mixing models

	Parame ter	Variable	Number of articles Percentage
	Approach	Experimental	2	4.4
		Observational	43	95.6
	Isotope	c	1	2.2
		N	2	4.4
		C&N	41	91.1
		C,N&S		2.2
	Tissue	Muscle	36	80.0
		Fin	6	13.3
		Scale	2	4.4
		n.a.	1	2.2
	Stomach contents Y es	19	42.2
		No	26	57.8
	Mixing models	Y es	16	35.6
		No	29	64.4
	isotopes were measured in muscle samples (80.0%),
	whilst only 17.8% of the studies used non-lethal
	approaches measuring stable isotopes in fin (13.3%)
	or scale (4.4%) samples. Over 42% of studies used
	stomach contents analyses in combination with SIA
	and mixing models were performed in 35.6% of
	studies. Most studies focused their analyses on a low
	number of non-native species (one non-native species:
	46.7%; two non-native species: 20.0%). On average,
	the trophic position of the non-native fish species
	studied was 3.3 (± 0.7 SD), ranging from 2.0 to 4.5.
	These trophic position values are likely to reflect the
	relative importance of Cyprinidae that are usually
	omnivorous.

lnsights from selected case studies lntroduced bass in North American lakes

  [START_REF] Post | Getting to the fat of the matter: models, methods and assumptions for dealing with lipids in stable isotope analyses[END_REF][START_REF] Perga | Laboratory measures of isotope discrimination factors: comments on Caut et al[END_REF], the following research directions are suggested.-Use of non-lethal approaches to collect samples for SIA. Indeed, recent methodological advances perform SIA based on the quantification of stable isotopes in fins (e.g. Jardine et al. 2005; Syvaranta et al. 2010),

Acknowledgments

The authors are grateful to the organising committee of the NeoFishMeD workshop held in Mugla University (Turkey) for the opportunity to present this work.

Supporting information

The following supplementary material is available:

Supporting Data 1: List of 45 selected articles published since 1999 and used in the analyses.

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