6 Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents

This chapter summarizes a work which we have conducted over the period 2005-2010 [Bettaieb et al., 2008; Bettaieb, 2009]. The aim was to better understand the Non Destructive Evaluation (NDE) method based on the use of eddy currents. A very simple example, well adapted for beginners, was the evaluation of a homogenous rectangular aluminum plate of constant thickness, in case of a calibrated crack. Such a choice had a great advantage: it allowed the development of simple theoretical analysis, and therefore a very good understanding of the physics of the problem.


Introduction
This chapter summarizes a work which we have conducted over the period 2005-2010[Bettaieb et al., 2008;;Bettaieb, 2009].The aim was to better understand the Non Destructive Evaluation (NDE) method based on the use of eddy currents.A very simple example, well adapted for beginners, was the evaluation of a homogenous rectangular aluminum plate of constant thickness, in case of a calibrated crack.Such a choice had a great advantage: it allowed the development of simple theoretical analysis, and therefore a very good understanding of the physics of the problem.
Naturally, theoretical results must be checked experimentally.To this end, we built a Helmholtz arrangement of two circular coils, to create a convenient uniform alternating field within the plate.Then, to measure the reaction field of the plate, we had to use a sensor.We used alternately a SQUID and a Hall sensor.The choice between those is not obvious, as will be shown later.Indeed, the SQUID is about times more sensitive than the Hall sensor, but its sensitive part is more remote than the Hall sensor, and this may reduce significantly its advantage.We found this comparison extremely interesting [Bettaieb et al., 2010].
Indeed, if we cut a rectangular plate, the cut being parallel to one side and normal to the excitation field, it seems natural that the plate is equivalent to a homogeneous one.Therefore, the cut cannot be detected.Just in case, we made the experiment and there was an unexpected signal!!We rapidly realized the reason of this result: the cut through the plate did not interrupt any current, but the cutting tool had damaged the microstructure of the aluminum over a small part of each half plate.Therefore, none of the two half parts had remained homogeneous.We realized that the microstructure of a small part of the half plates had been modified.In fact, the two halves of the original plate were equivalent to a homogeneous plate, except along the cut.Therefore, we did not detect the cut, but the metallic dislocations around it.Some more theoretical investigations allowed to evaluate the importance of the change in the electrical resistivity associated with the dislocations, and their extension.Then, we started to investigate the dislocations due to hammer shocks and to mechanical flexions.

Theoretical study of a defect free rectangular aluminium plate
We consider the case of a homogeneous rectangular aluminum plate (resistivity .Ω. ).Its dimensions are , , in the , , directions respectively (figure 1).This plate is immersed in an alternative uniform magnetic induction field parallel to the direction, varying at the frequency, with an amplitude of .
if we use a SQUID and .
if we use a Hall probe sensor.We shall study the induced current density using the more appropriate simplifying assumptions, in order to develop a good physical understanding of the phenomena.
In a first step, we assume that all the induced currents lines are straight lines parallel to the direction from one side to the other (figure 1).If this is so, the external field is easy to evaluate by Biot and Savart law.An interesting remark is that the field of the induced currents, inside the plate, is quite much smaller than the excitation field.
In a second step, we improve the evaluation of the induced current lines in a plane parallel to the plane (figure 2).
In such planes, the value of is given under the form of a double Fourier expansion and the values of the amplitude of the components and of the current density are respectively.It may be noted that and are the derivative of a "current function": so that , and , .

Fig. 2. Rectangular aluminium plate under consideration showing the cross section plane for the evaluation of the induced currents lines
The important point is to understand that, in the case of figure 2 ( , ), the current density is essentially parallel to the axis, and that this justifies the above first approximation: the current lines are parallel to (figure 3).
Fig. 3. Lines of induced currents for a defect free rectangular aluminium plate ( , ), the current density is essentially parallel to the axis

Theoretical study of a rectangular aluminum plate with a calibrated crack
Consider now the above plate, where we have made a calibrated crack, as shown in figure 4. We analyze the induced currents produced by the alternating excitation field , considering that the plate is equivalent to 3 sane plates of widths , and ( ) and of thickness , and respectively.The above analysis stands for each of those three plates; the numerical analysis is straight forward, and is given below (figures 8 and 9).

Experimental set-up
In our laboratory, we have established an experimental apparatus based on Helmholtz coils arrangement for excitation, and fixed magnetic sensors (SQUID and Hall probe) (figure 5).The structure has been made mainly of Plexiglas and other non magnetic materials (brass, copper, Al, wood…) in order to avoid any magnetic noise.The samples are moved underneath the sensor by means of an x-y scanning stage.The stage and the data acquisition are controlled with a LABVIEW program which is optimized to automate the entire measuring process.The sensor (SQUID or Hall probe) measures the magnetic field perpendicular to the excitation field and to the scanning direction.A lock-in amplifier has been used to achieve a synchronous detection.
The two versions of the equipment are shown in figure 6.The sensor is located in a fixed position.In the case of the SQUID, mechanical distance between upper side of the plate and the Dewar is .
, and the sensor itself is .higher.In the case of the Hall probe (right part of the figure), the total distance between the plate and the active part of the sensor is .
The difference between the two sensors appears immediately.Indeed, the excitation field, when the SQUID is in use, is limited to . to avoid the saturation.When the Hall 131 probe is in use, this limit reaches .with our generator; but it might be higher with better equipment.In both cases, the frequency is .The sensitivity of the SQUID measuring sensor is .
/µ , and the sensitivity of the Hall sensor is /µ .

Experimental verification
Figure 7 shows the plate with a calibrated crack immersed in the uniform excitation field described above ( , . if we use a SQUID and .if we use a Hall probe sensor).(measured with the SQUID).Note that the difference between theoretical values and experimental values is so small that they cannot be distinguished.

Case of a crack with zero width showing metallic dislocations
We endeavored to repeat the same kind of experiment as above, but with a slot of zero width ( , ) (figure 10). of the plate, at least over small depth.Therefore, the intuitive representation of the situation is shown in figure 12.In this figure, the aluminum parts are themselves divided into two parts: one where the resistivity is normal ( .Ω. ), and one where the resistivity is higher: ′ .Fig. 11.Measured values of , with a Hall probe at above the plate, between and ′ : (a) in the case of a defect free rectangular aluminium plate, (b) in the case of a rectangular plate with a slot of zero width In figure 12, the thickness of the higher resistivity zone is considered ′ at the edge of each half plate, so the total damaged edge area width is .′ or ′ to decrease and repeat this process until is as small as possible.This is developed in [Poloujadoff et al., 1994] and in the reference [Bettaieb et al., 2010].
This led us to determine a best value of ′ .Ω. .) and of ′ ( ′ µ ) which provided a theoretical curve of shown in figure 13 (a).Since the publication of this reference, Dr. Denis GRATIAS suggested a further verification of this approach.This consisted in annealing the two half plates, then reputing the measurements.The annealing cycle is shown in figure 14.Then, we place again the two half plates in the same uniform magnetic excitation field.The curve of the vertical component of the induction field still shows a variation of ′ and ′ , but much smaller than previously; this show that annealing has been very effective, but has not been long enough (figure 15).

Metallic dislocations created by a shock
The above study of dislocations can be carried on by creating small damaged zones by a direct or indirect shock with a hammer (figure 16).We have already reported a preliminary study of this phenomenon [Bettaieb et al., 2010], and we report some more recent progress below.In a second step, we assumed that the deformation of the plate was negligible, but the metallic microstructure has been changed over a depth as large as 1mm.We have considered that in this region the resistivity has been changed by 10% (figure 20 (A)).This model also accounts for the experimental results (figure 20 (B)).
Therefore, this experiment and the corresponding models show that the effect of the shock is partially explained by a deformation of the square aluminum plate and partially by dislocations created in the metal over same depth.We have proved that the effects of the dislocations are much more important [Bettaieb, 2009].

Metallic dislocations created by flexions 8.1 Experimental study
Consider a long aluminum strip (150mm) shown in figure 21 which has been bent by hand, so as to create a fatigue zone in the middle.In the already quoted previous paper [Bettaieb et al., 2010], we had shown that the fatigue of the metallic bent part could be detected.To this end, we had placed the strip in a uniform alternating horizontal field , and we have scanned its upper surface with a Hall probe.However, this first publication was limited to the feasibility of the detection.Since that time, we have greatly improved our experimental process, and have developed an inverse problem method for a better mathematical analysis of the results.

Fig. 1 .
Fig. 1.Defect free rectangular aluminium plate under consideration This excitation frequency has been chosen because it is one of those which are the less present in the general environment in and around our laboratory.

Fig. 4 .
Fig. 4. Rectangular aluminium plate with a calibrated crack having a width () and a depth ( )

Fig. 7 .Figure 9
Fig. 7. Rectangular aluminium plate with a calibrated crack immersed in a uniform excitation field created by Helmholtz coils The vertical component of the resulting induction field, measured between and ′ is given at a vertical distance of (measured with the Hall probe), or at a vertical distance of .(measured with the SQUID).It is clear that the computed values and the measured values tally in two very different cases (figures 8).
Fig. 9. Computed (a) and measured (b) values of between and ′ in the case of a defect free rectangular aluminium plate: (A) measurement with a Hall probe at above the plate, (B) measurement with a SQUID at .above the plate

Fig. 10 .
Fig. 10.Rectangular aluminium plate with a slot of zero width under consideration We expected to find, along the line ′ , a variation of the vertical component of the induced field depicted by curve (a) of figure 11, because all current lines are parallel to the direction.In other words, we did not expect to detect the cut.We were extremely surprised to find the variation depicted by curve (b) of the same figure.In fact, we had considered that the two halves of the plate were equivalent to a homogeneous plate.What we neglected was that, when you cut a metallic plate, the cutting tool may damage the metallic microstructure

Fig. 12 .
Fig. 12. Representation of the two half plates laid near each other, and of two assumed fatigued zones of resistivity Therefore, the next problem is to determine the values of ′ and ′ which account conveniently for the experimental results.This is a simple classical optimization problem.In our case, we pick at random one arbitrary value of ′ and ′ .This allows to evaluate the component of the induced field.It is then necessary to define a global error ′ , ′between the observed field, and the field corresponding to ′ and ′ .Then we modify either

Fig. 13 .
Fig. 13.(a) Theoretical curve with optimized values of ′ µ and ′ ., (b) experimental curve established with a Hall sensor at d=1mm above the aluminium plate in the case of a zero width crack

Fig. 14 .
Fig. 14.Annealing cycle of the two half plates after cutting a plate

Fig. 15 .
Fig. 15.Experimental curve established with a Hall sensor at d=1mm above the aluminium plate in the case of a zero width crack before and after annealing

Fig. 16 .
Fig. 16.(a) Creation of an impact zone by stroking a steel ball, (b) impact upon the square aluminium plate, (c) zoom of the shock print zone with modelling dimensions (D, d)

Fig. 21 .
Fig. 21.Fatigued strip, the length of the damaged (bent) zone is approximately equal to 3mm Our first effort has been to flatten the long strip carefully before measurement, as shown by figures 22 (figure22 (A) is a side photo, and figure 22 (B) an upside down view of the strip during flattening).As a result, the signal obtained along the sensor track is symmetrical about its center (figure23 (A)).