ULTRASONIC NON DESTRUCTIVE EVALUATION AND NON DESTRUCTIVE TESTING OF ANISOTROPIC COMPOSITE MATERIALS     

Version française



Work performed at or in collaboration with Laboratory Roberval of Université de Technologie de Compiègne




Université de Technologie de Compiègne (UTC)




                     





 
In order to model in a correct manner the bounded nature of transducers, the propagation model with plane waves in anisotropic multilayered media has been extended to bounded beams. The used method relies on the decomposition of the beam in monochromatic plane waves, using the angular spectrum method  [A7, A13, O'2, B4, B5, C4]. It allows to obtain the reflected and transmitted beams through an immersed composite plate, for a 3D geometry. 
At a characteristic pair (angle theta of the acoustic axis, frequency), the acoustic beam generates locally a modal wave (Lamb or Rayleigh wave for instance) while the bounded nature of the beam rather provokes a modal wave beam in the structure. This modal beam re-radiates in the fluid. In the case of an anisotropic structure, the direction of the modal wave beam is deviated with respect to the sagittal plane, determined by the incident bounded beam. In the case of a unidirectionnal composite plate, following the chosen Lamb mode, the modal beam may be deviated in the direction of the fibers.   More...
 
Lamb waves are propagation modes of an isotropic plate in vacuum.
         More about modal waves...
 Symmetric mode S0


  Antisymmetric mode A0
When the ultrasonic testing configuration is such as the ultrasonic field (bounded beams) generates a Lamb wave beam in a structure, the reflected beam comprises a specular part, and another part, corresponding to the radiation of the Lamb waves in the surrounding fluid, called nonspecular part (leaky waves in English because the energy leaks in the fluid). This reflected field presents a minimum between the specular and nonspecular parts, related to a phenomenon of interferences.

The figure presents the reflected field of pressure in water (parallel to the plate) of an unidirectional  carbon/epoxy plate (thickness 0.59 mm, incident angle of the acoustic axis of the transducer 9.8 °, frequency 1.35 MHz).

More about Lamb wave beams... 
As far as defect detection in composite plates is concerned, industrial testing of big composite structures is, at the present time, very slow due to an exploration point by point of them. The use of Lamb waves permits to save considerable time. Indeed, by scanning following one line, a strip of about ten centimeters can be tested.

Dispersion curves of Lamb waves for a 0°/45°/90°/135° carbon/epoxy composite which consists of 8 plies
 In the case of homogeneous isotropic plates, analytical formula are available in order to draw the dispersion curves of Lamb waves. 

In the case of anisotropic multilayered media, equations are to complex and a propagation model in such media is necessary. The model we have developed permits, for any given frequency, incident angles to be determined in order to generate a Lamb wave in the medium.

Reflection or transmission set-up
 Experimentally, a classical method for generating Lamb waves consists in putting two transducers with the same nominal frequency, on the same side of the plate. The transducers are inclined at the angle corresponding to the propagation of a Lamb wave, for the nominal frequency of the transducer.

Echographic set-up


0°/90 carbon/epoxy plate with mirror symmetry, which consists of 8 plies.  f = 1 MHz ; q = 10°
blue : no defect
red : defect in the plate

 Another experimental set-up consists in putting the transmitting transducer and the receiver on a same plane. When the two transducers are inclined with an angle corresponding to a Lamb mode, a signal will be received by the receiver only if it corresponds to a defect. 

A cartography using Lamb waves (L-scan) can thus be drawn [C3,O'2].
   

When the structure is anisotropic, the bounded nature of the incident beam creates a modal wave beam in the structure. This modal beam re-radiates in the surrounding fluid. Due to anisotropic effects, the main axis of the modal beam is no longer contained in the sagittal plane of the incident beam, which creates a deviation with respect to this sagittal plane. In the case of a unidirectionnal composite plate, following the chosen Lamb mode, the modal beam may be deviated in the direction of the fibers.  More...
 

vacuum / semi-infinite multilayered medium structure
 By analogy with the propagation of Rayleigh waves at the interface of a vacuum / semi-infinite isotropic medium structure, the propagation of surface waves that have been termed multilayered Rayleigh waves can be defined [A4, A5, A8, B2, B3, B7].

More about modal waves...

Properties : 
  • A multilayered Rayleigh wave is a linear combination of 3 Floquet waves which are all inhomogeneous.
  • As the Floquet waves are dispersive, the multilayered Rayleigh wave is also dispersive : contrary to the case of isotropic media, the propagation of such a wave occurs for a given (angle,frequency) couple.
  • The writing of boundary conditions leads to the cancellation of a (3x3) determinant. By an angular and frequency scanning, it is possible to obtain the dispersion curves of multilayered Rayleigh modes.
Here, the plate behaves as if it were semi-infinite. When the medium is not lossy, there is no radiation in the medium opposed to the insonification.There is total reflection, and thus the reflection coefficient is equal to 1. Indeed, the drawing of the displacement amplitudes as a function of the thickness, at the angle and at the frequency for which a multilayered Rayleigh wave propagates, shows that these amplitudes are negligible at the end of the medium.









Experimentally: 

Experiments have been done on a 0°/45°/90°/135° carbon/epoxy plate, which consisted of 24 plies. The green curve presents the magnitude of the experimental reflection coefficient. About 3 MHz, a trough is observed. This trough is similar to the one observed for isotropic plates at the Rayleigh angle. The blue curve presents the magnitude of the modelised reflection coefficient when the medium is considered as a lossy one, whereas the red curve corresponds to a non lossy medium. It can be observed that for a non lossy medium, the magnitude of the reflection coefficient is equal to 1. 

The multilayered Rayleigh wave has been experimentally found, using an acousto-optic technic developed by the University of Leuven (Courtrai, Belgium). This technic is based upon the interaction of a laser beam with ultrasounds [A8, B11]. 

The Schlieren images of the reflected field by the composite plate at 3 MHz, for two different incident angles (20° on the left and 16° on the right) permit to observe (on the left) the incident and reflected field. It can be seen that when there is a critical angle, there is a "null" region in the reflected field (on the left, at 20°), whereas there is no null region when the angle is not a critical angle (on the right, at 16°).

This is the difference between the two types of reflected fields which gives experimentally the angle and the frequency for which a critical angle occurs. Here, at 3 MHz and 20° (on the left), a multilayered Rayleigh wave occurs, which is not the case at 16° (on the right).
 
 

Fig. 1
According to the testing configuration (frequency, incident angle), the incident beam (emitted by the emitter transducer), due to its bounded nature, creates a specular reflected beam but also, according to the position of the sagittal plane (or incident plane), to a modal wave beam in the structure (because of the excitation of modal waves) which re-radiates in the surrounding medium.
Fig. 2
If (for instance), the structure is a unidirectional composite plate, and if the trace of the sagittal plane, which is the plane perpendicular to the plane of the plate (and which contains the acoustic axis of the beam), is not parallele to the direction of the fibers, due to the anisotropy of the medium, the direction of the bounded modal wave beam will be deviated with respect to th sagittal plane of the incident bounded beam. Following the chosen mode, the modal beam may be deviated in the direction of the fibers.

Fig. 3
In order to determine the experimental configuration which permits to generate a modal wave (here a Lamb wave) in a structure, it is usefull to draw the dispersion curves for Lamb waves, representing for instance, the incident angle as a function of the frequency (it would be the same to represent the phase velocity instead of the incident angle). Such curves depend on the position of the sagittal plane with respect to the plate. The orientation of this plane with respect to the x1-axis linked to the plate, is given by the azimuthal angle . Therefore, the dispersion curves (Fig. 3) are drawn for a given value of  . In facts, the emitter produces a (bounded) beam, which is centered on an acoustic axis, parallele to the main wave number of the beam (maximum of energy in the beam). The sagittal plane is the plane which contains the acoustic axis of the transducer.

For a given frequency and for a particular mode, the incident field will generate the main Lamb mode in the direction of the acoustique axis, but it will also generate the neighbouring waves, corresponding to the same mode, out of the sagittal plane. Thus, the evolution of the chosen mode has to be represented as a function of the angle  of the sagittal plane. It is usefull to draw the inverse of the phase velocity (termed slowness), as a function of the azimuthal angle .
Fig. 4
Fig. 4 presents two example of slowness curves for Lamb waves, for a fiven frequency. On the left, the slowness curves for a unidirectional carbon/epoxy layer, the fibers being parallel to the x1-axis. On the right, the slowness curves for two perpendicular layers.

These dispersion curves are obtained by writing the boundary conditions (plane wave reasoning for a structure in vacuum), which leads to a dispersion relation of the form   = F(kx1 , kx2).
Fig. 5
click on the image for a zoom
The direction of deviation of the modal beam can be predicted by means of an asymptotic analysis, in the far field, and using a stationary phase argument.  Consider the point corresponding, on the slowness curve, to the projection of the main wave vector of the acoustic axis of the transducer on the plane of the plate. The incident beam also generates waves close to this point on the curve, which participate to the phenomenon of quasi resonance. Consequently, a bounded modal wave beam is generated in the structure. This beam is centered on the main group direction, i.e. the most energetic part of this modal beam will be found along the group direction corresponding to the main wave vector of the acoustic axis, i.e. following the direction of the normal to the curve of slownesses at this point.
As the slowness curve is not circular, due to anisotropic effects, this direction of the normal is different from that of the modal wave number. Thus, the modal beam is deviated in the group direction associated to the modal wave number of the acoustic axis.
It should be noted that here, the phenomenon is a monochromatic one; thus, the angular dispersion is the only one which occurs. As a consequence, this is the group direction which is here important, and not the group velocity.
Fig. 6
click on the image for a zoom
Numerical description of the modal beam deviation phenomenon. Example of a unidirectional carbon/epoxy layer
The testing configuration is chosen such as a symmetric Lamb mode S0 is generated in the plate. On the left of Fig. 6, the direction of the fibers is contained in the sagittal plane, whereas on the right of the figure, this direction is a 45° from the trace of the sagittal plane. 
In both cases, the specular and non specular areas (reflection) corresponding to the radiation of the Lamb wave in the surrounding fluid can be observed.

In the first case (on the left), when =0°, the modal beam is not deviated with  to the sagittal plane, whereas it is the case when the direction of fibers is no more contained in the sagittal plane (on the right).
Fig. 7
Oblique plane for re-radiation

The Lamb wave beam re-radiates into the external fluid, no more in the sagittal plane, but in an oblique plane corresponding to the specular reflection angle of the acoustic axis. The intersection of this oblique plane with the interface direction is the main group direction of the modal beam. Therefore, all the nonspecular effects are to be searched for in this oblique plane and not in the sagittal plane.
Fig. 8
The oblique character of this re-radiation plane has been numerically highlighted. Let   be the angle between the oblique plane and the plate, and  the distance between this plane and the Ox3-axis, for a given height h. If the reflected field is examined at the height h from the plate, the maximum of the magnitude of the pressure is found in the oblique plane, and its projection onto the plane of the plate is located at the distance from the main group direction.
Knowing the incident angle  and the deviation angle of the modal beam, the angle  can be deduced by a simple geometric calculation (here, for the chosen configuration,  =74.7°).
Numerically, Fig. 8 presents the reflected field of pressure for a unidirectional carbon/epoxy plate when h=0 (surface of the plate, on the left) and when h=200 mm (on the right). The red line corresponds to the maximum of pressure at the surface of the plate.  It can be well observed that the two Lamb wave beams are shifted. The measurement of the distance  permits to determine the angle  (75.3°, which is in very good agreement with the value calculated geometrically).
It should be noted that there is an increase of the Lamb wave beam size as a function of the height h, which is due to the spreading of the beam.       

Fig. 9
Experiments have been made on a 0.59 mm thick unidirectional carbon/epoxy plate. The emmitting transducer used to generate the ultrasonic field has a nominal frequency equal to 1 MHz and diameter 3/4 inch (19.05 mm), the duration of excitation has been adjusted in order to obtain a ten cycles excitation. An hydrophone, positioned as close as possible from the surface of the plate (roughly 1 mm), is moved in a plane parallel to the plate, in order to measure the reflected pressure.
Fig. 10Fig. 11

Figures 10 and 11 present comparisons between numerical and experimental results for a carbon/epoxy unidirection plate, the fiber direction being respectively contained and not contained in the sagittal plane.

Nota Bene. The absorption of the material has not been taken into account for the numerical cartographies.


Fig. 1
A modal wave [see W.D. Hayes, "Conservation of action and modal wave action", Proc. Roy. Soc. Lond. A., 320,  187-208, (1970)] is a wave for which the acoustical energy propagates along the layers, whereas it remains bounded in a direction (z) perpendicular to the layer (guided waves, surface waves, interface waves).

Modal waves need one or more boundaries to construct themselves: the acoustic energy propagates in the directions of some subspace of the physical space, whereas these waves present a stationary character in the supplementary subspace (in the sense of the linear space theory).


Periodic diatomic chain of atoms
SH waves in isotropic layers
Longitudinal waves in fluid layers
Fig. 2
The Floquet waves which propagate in the periodic structure are linked to the eigenvalues lambda (and to the eigenvectors) of the transfer matrix tau, which are solutions of the characteristic equation  with s=(a+d)/2.  When lambda is complex (modulus equal to unity), the associated Floquet wave is propagative, whereas it is evanescent when lambda is real.

When the Floquet waves are evanescent, the corresponding domain in the frequency/angle plane (K0,Kx) is called stopping band (or stop band). When the Floquet waves are propagative, the corresponding domain is called passing band (or pass band). These names correspond to the fact that, for a semi-infinite periodic structure, the acoustic energy is totally reflected in the external medium for stopping bands, whereas, for passing bands, a part of the energy goes into the semi-infinite periodic structure and radiates to the infinity. The stopping and passing bands of the structure correspond respectively to the case |s|>1 and |s|<1. These zones are separated in the plane (K0,Kx) by curves s(K0,Kx)=+1 or s(K0,Kx)=-1 (see Fig. 2 in the case of 2 layers in a period). 
Fig. 3
Guided modes

Periodically multilayered medium made up of P periods (see Fig. 3). The transfer matrix tauP of the whole medium can be calculated in terms of tau, by means of Tchebychev polynomials of the second kind Up(s), using Cayleigh-Hamilton theorem:
coefficients of tau P
If the periodic finite structure is surrounded by vacuum, the boundary conditions at the external interfaces imply that the stresses vanishes (TP=0 and T0=0), which leads to a factorized eigenmode equation:   cP(K0,Kx)=0   i.e.  c(K0,Kx) UP-1(s)=0  , which leads to two families of modal waves.
Fig. 4
- The first family of modal waves corresponds to what may be termed structure modal waves. It is given by the condition UP-1(s)=0. From the properties of Tchebychev polynomials, the (P-1) roots of this condition are such that |s|<1, which amounts to saying that these modes do occur in passing bands. For the periodically fluid medium of Fig. 4 with P=3 periods, it can be seen that there are 2 structure modes in each passing band (dashed orange line).
- The second family of modal waves corresponds to what may be termed period modal waves. It is given by the condition c(K0,Kx)=0, which is independent of the number P of periods and thus corresponds to modal waves which are directly linked to the period cell and which are such as |s|>=1. As a consequence, the period modes are located in the stopping bands or on their boundaries (see Fig. 4, solid black line).
Fig. 5 Surface modal waves in a semi-infinite structure in contact with vacuum

- Such a surface modal wave may exist  in a semi-infinite fluid periodic structure in contact with vacuum, whereas it does not exist in a single fluid medium.
- Under conditon c(K0,Kx)=0, when the first interface of the structure is in contact with vacuum, the condition T0=0 implies the condition T1=0 and so on for other period interfaces. In particular, afterp periods, the normal displacement is such as wp=ap w0. 

- Since the structure is semi-infinite, the integer p grows indefinitely as z tends to infinity. Thus, if |a|<1 then the amplitude of the wave, observed at each period interface, will decrease as a function of z, whereas if |a|>1 it will increase (see Fig. 5). 
Fig. 6

- The first case corresponds to a surface modal waves (which has been called a "multilayered Rayleigh mode").

- The second one corresponds to what may be called an "anti-modal wave": such a modal solution has poor physical meaning since its amplitude is not bounded at infinity.

- It should be noted the crucial role that plays here the periodic character of the structure: actually, in the case of semi-infinite homogeneous fluid media, there is no surface modal wave, similar to that which has just been described for a periodically structure.

- When the layers are stacked in inverse order, the surface modal wave and the "anti-modal wave" are exchanged.

Boundary conditions
  Eigenmode equation (dispersion equation) :
 
dzeta0 +dzetaP = 0 : factorized eigenmode equation dzeta0 =dzetaP  : non factorized eigenmode equation
Rigid wall: Zw tends to infinity
or vacuum: Zw = 0
Pure reactive impedance
Propagative waves
in the same external fluid medium
Total transmission
Generalized modal waves (kx is not real)

Total transmission
Anti-generalized modal waves (kx is not real)
Evanescent waves
in the same external fluid medium
Anti-modal waves
Anti-modal waves

Anti-modal waves
Osborne and Hart modal waves




 

A pultruded glass fiber reinforced plastic composite beam has been impacted. The considered profile stacking sequence is an alternating of 5 continuous mat and roving layers, with mat on the outer surfaces (contact : Thierry Chotard)
A C-scan ultrasonic cartography at different positions relative to the impact point permits classical results to be observed : below the impact point (on the rear surface), the zone of damage is larger than that on the surface of impact.
By taking into account the real response of the transducer, the echographic response of the medium [A6, B4], thanks to the model.

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