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

Bettaieb Laroussi, Kokabi Hamid and Poloujadoff Michel *Université Pierre et Marie Curie (UPMC), Laboratoire d'Electronique et Electromagnétisme (L2E), Paris France* 

### **1. Introduction**

126 Nondestructive Testing Methods and New Applications

Teboul, S. et al. (1998). M. Barlaud. Variational approach for edge-preserving regularization

Tillack, G. et al. (2000). X-ray modeling for industrial applications, NDT & E International,

You,Y. et al. (1995). On ill-posed anisotropic diffusion models. *Proceedings of International* 

You,Y. et al.(1996). Behavioral analysis of anisotropic diffusion in image processing. *IEEE* 

*Conference on Image Processing*, 1995, No.2, pp. 268-271

387-397, ISSN: 1057-7149

Vol.33, No.7, (2000), pp. 481-488

using coupled PDEs. *IEEE transactions on Image Processing*, (1998), Vol 7, No.3, pp.

*Transactions on Image Processing*. (1996), Vol.5, No.11, pp. 1539-1553, ISSN: 1057-7149

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 1000 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.

$$B(\mathbf{x}, \mathbf{z}) = \sum\_{m,n} a\_{mn} \cos\left(\frac{m\pi x}{a}\right) \cos(\frac{n\pi z}{c}) \tag{1}$$

$$f\_{\chi} = -\frac{\omega}{\rho} \frac{16ac^2}{\pi} B \sum\_{m,n} \frac{1}{n} \frac{\sin\left(\frac{m\pi}{2}\right) \sin\left(\frac{n\pi}{2}\right)}{(m\pi a)^2 + (n\pi c)^2} \sin\left(\frac{m\pi x}{a}\right) \cos(\frac{n\pi x}{c})\tag{2}$$

$$f\_{\mathbf{z}} = \frac{\omega}{\rho} \frac{16ca^2}{\pi} B \sum\_{m,n} \frac{1}{m} \frac{\sin\left(\frac{m\pi}{2}\right) \sin(\frac{n\pi}{2})}{(m\pi a)^2 + (n\pi c)^2} \cos\left(\frac{m\pi x}{a}\right) \sin(\frac{n\pi x}{c})\tag{3}$$

$$U(\mathbf{x}, \mathbf{z}) = \sum\_{m,n} \frac{1}{\rho} (\rho \omega) (\frac{\mathbf{a}ac}{\pi})^2 \frac{B}{mn} \frac{\sin(\frac{m\pi}{2}) \sin(\frac{n\pi}{2})}{(m\pi a)^2 + (n\pi c)^2} \cos\left(\frac{m\pi x}{a}\right) \cos(\frac{n\pi x}{c}) \tag{4}$$

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 131

probe is in use, this limit reaches ʹǤͶͷ ൈ ͳͲିସܶ with our generator; but it might be higher with better equipment. In both cases, the frequency is ͳͺͲݖܪ. The sensitivity of the SQUID

Fig. 6. The two versions of the experiment set-up: on the left is a SQUID sensor and on the

measuring sensor is ͶǤʹͷܸȀ*µ*ܶ, and the sensitivity of the Hall sensor is ͷܸ݉Ȁ*µ*ܶ.

Fig. 5. High Tc SQUID NDE instrumentation set-up

right a Hall probe

### **3. 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.

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

We analyze the induced currents produced by the alternating excitation field ܤሬԦ, considering that the plate is equivalent to 3 sane plates of widths ܾଵ, ܾଶ ൌ ݓ and ܾଷ (ܾଵ ܾଶ ܾଷ ൌ ͳͲͲ݉݉) and of thickness ܿ, ܿ െ ݀݁ݐ݄ 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).

### **4. 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 ͲǤͷ݉݉, and the sensor itself is ͳʹǤ͵݉݉ 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 ͳ݉݉.

The difference between the two sensors appears immediately. Indeed, the excitation field, when the SQUID is in use, is limited toͲǤͳͷ ൈ ͳͲିସܶ to avoid the saturation. When the Hall

**3. 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.

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

We analyze the induced currents produced by the alternating excitation field ܤሬԦ, considering that the plate is equivalent to 3 sane plates of widths ܾଵ, ܾଶ ൌ ݓ and ܾଷ (ܾଵ ܾଶ ܾଷ ൌ ͳͲͲ݉݉) and of thickness ܿ, ܿ െ ݀݁ݐ݄ and ܿ respectively. The above analysis stands for each of those three plates; the numerical analysis is straight forward, and is given

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

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 ͲǤͷ݉݉, and the sensor itself is ͳʹǤ͵݉݉ 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

The difference between the two sensors appears immediately. Indeed, the excitation field, when the SQUID is in use, is limited toͲǤͳͷ ൈ ͳͲିସܶ to avoid the saturation. When the Hall

(݄ݐ݀݁) depth

sensor is ͳ݉݉.

below (figures 8 and 9).

**4. Experimental set-up** 

been used to achieve a synchronous detection.

probe is in use, this limit reaches ʹǤͶͷ ൈ ͳͲିସܶ with our generator; but it might be higher with better equipment. In both cases, the frequency is ͳͺͲݖܪ. The sensitivity of the SQUID measuring sensor is ͶǤʹͷܸȀ*µ*ܶ, and the sensitivity of the Hall sensor is ͷܸ݉Ȁ*µ*ܶ.

Fig. 5. High Tc SQUID NDE instrumentation set-up

Fig. 6. The two versions of the experiment set-up: on the left is a SQUID sensor and on the right a Hall probe

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 133

(B)

**-20 -10 0 10 20**

**y(mm)**

Figure 9 represents the case of a defect free rectangular aluminum plate. The vertical

vertical distance ݀ of ͳ݉݉ (measured with the Hall probe), and at a vertical distance ݀ of ͳʹǤͺ݉݉ (measured with the SQUID). Note that the difference between theoretical values

(A)

**-20 -10 0 10 20**

 (a) (b)

**y(mm)**

rectangular aluminium plate with a calibrated crack (width ݓ ൌ ͳ݉݉, ݀݁ݐ݄ ൌ ͳ݉݉):

component ܤ௭ of the resulting induction field, measured between ܣ and ܣ*′*

in the case of a

 (a) (b)

is given at a

Fig. 8.B. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′*

and experimental values is so small that they cannot be distinguished.

measurement with a SQUID at ݀ ൌ ͳʹǤͺ݉݉ above the plate

**-4,0x10-5**

**-2,0x10-5**

െǡ ൈ ିૠ

െǡ ൈ ିૠ

**Bz(Tesla)**

**2,0x10-5**

െǡ ൈ ିૠ

ǡ ൈ ିૠ

**4,0x10-5**

**0,0**

െǡ

**-1,5x10-8**

**-1,0x10-8**

**-5,0x10-9**

**Bz(Tesla)**

**5,0x10-9**

**1,0x10-8**

**1,5x10-8**

**0,0**

### **5. Experimental verification**

Figure 7 shows the plate with a calibrated crack immersed in the uniform excitation field described above (ͳͺͲݖܪǡ ͲǤͳͷ ൈ ͳͲିସܶ if we use a SQUID and ʹǤͶͷ ൈ ͳͲିସܶ if we use a Hall probe sensor).

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 ܣ and ܣ*′* is given at a vertical distance ݀ of ͳ݉݉ (measured with the Hall probe), or at a vertical distance ݀ of ͳʹǤͺ݉݉ (measured with the SQUID). It is clear that the computed values and the measured values tally in two very different cases (figures 8).

Fig. 8.A. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′* in the case of a rectangular aluminium plate with a calibrated crack (width ݓ ൌ ͳ݉݉, ݀݁ݐ݄ ൌ ͳ݉݉): measurement with a Hall probe at ݀ ൌ ͳ݉݉ above the plate

Figure 7 shows the plate with a calibrated crack immersed in the uniform excitation field described above (ͳͺͲݖܪǡ ͲǤͳͷ ൈ ͳͲିସܶ if we use a SQUID and ʹǤͶͷ ൈ ͳͲିସܶ if we use a Hall

Fig. 7. Rectangular aluminium plate with a calibrated crack immersed in a uniform

The vertical component of the resulting induction field, measured between ܣ and ܣ*′*

at a vertical distance ݀ of ͳ݉݉ (measured with the Hall probe), or at a vertical distance ݀ of ͳʹǤͺ݉݉ (measured with the SQUID). It is clear that the computed values and the measured

(A)

**-20 -10 0 10 20**

 (a) (b)

**y(mm)**

rectangular aluminium plate with a calibrated crack (width ݓ ൌ ͳ݉݉, ݀݁ݐ݄ ൌ ͳ݉݉):

Fig. 8.A. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′*

measurement with a Hall probe at ݀ ൌ ͳ݉݉ above the plate

**-6x10-7**

**-3x10-7**

**Bz(Tesla)**

**3x10-7**

**6x10-7**

**0**

is given

in the case of a

**5. Experimental verification** 

excitation field created by Helmholtz coils

values tally in two very different cases (figures 8).

probe sensor).

(B)

Fig. 8.B. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′* in the case of a rectangular aluminium plate with a calibrated crack (width ݓ ൌ ͳ݉݉, ݀݁ݐ݄ ൌ ͳ݉݉): measurement with a SQUID at ݀ ൌ ͳʹǤͺ݉݉ above the plate

Figure 9 represents the case of a defect free rectangular aluminum plate. The vertical component ܤ௭ of the resulting induction field, measured between ܣ and ܣ*′* is given at a vertical distance ݀ of ͳ݉݉ (measured with the Hall probe), and at a vertical distance ݀ of ͳʹǤͺ݉݉ (measured with the SQUID). Note that the difference between theoretical values and experimental values is so small that they cannot be distinguished.

(A)

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 135

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 (ߩ ൌ ͷǤͺʹ ൈ ͳͲି଼ΩǤ ݉), and one where the

Fig. 11. Measured values of ܤ௭, with a Hall probe at ݀ ൌ ͳ݉݉ above the plate, between ܣ

Fig. 12. Representation of the two half plates laid near each other, and of two assumed

conveniently for the experimental results. This is a simple classical optimization problem. In

component of the induced field. It is then necessary to define a global error ߳ሺݓ*′*

*′*ߩ and

*′*ߩ and

Therefore, the next problem is to determine the values of ݓ*′*

our case, we pick at random one arbitrary value of ݓ*′*

between the observed field, and the field corresponding to ݓ*′*

In figure 12, the thickness of the higher resistivity zone is considered ௪*′*

: (a) in the case of a defect free rectangular aluminium plate, (b) in the case of a

.

ଶ at the edge of each

*′*ߩ and

. This allows to evaluate the ݖ

. Then we modify either

which account

*′*ߩ ǡ ሻ

resistivity is higher: ߩ*′* ߩ.

rectangular plate with a slot of zero width

fatigued zones of resistivity ߩ<sup>ᇱ</sup>

half plate, so the total damaged edge area width is ݓ<sup>ᇱ</sup>

*′*ܣ and

Fig. 9. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′* in the case of a defect free rectangular aluminium plate: (A) measurement with a Hall probe at ݀ ൌ ͳ݉݉ above the plate, (B) measurement with a SQUID at ݀ ൌ ͳʹǤͺ݉݉ above the plate

### **6. 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 (ܾଶ ൌ ݓ ൌ Ͳ݉݉, ݀݁ݐ݄ ൌ ܿ) (figure 10).

Fig. 10. Rectangular aluminium plate with a slot of zero width under consideration

We expected to find, along the line ܣܣ*′* , 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 ܱݔ 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

(B)

**-20 -10 0 10 20**

 (a) (b)

**y(mm)**

We endeavored to repeat the same kind of experiment as above, but with a slot of zero

free rectangular aluminium plate: (A) measurement with a Hall probe at ݀ ൌ ͳ݉݉ above

in the case of a defect

Fig. 9. Computed (a) and measured (b) values of ܤ௭ between ܣ and ܣ*′*

width (ܾଶ ൌ ݓ ൌ Ͳ݉݉, ݀݁ݐ݄ ൌ ܿ) (figure 10).

**-1x10-8**

**Bz(Tesla)**

**0**

**1x10-8**

**2x10-8**

We expected to find, along the line ܣܣ*′*

the plate, (B) measurement with a SQUID at ݀ ൌ ͳʹǤͺ݉݉ above the plate

**6. Case of a crack with zero width showing metallic dislocations** 

Fig. 10. Rectangular aluminium plate with a slot of zero width under consideration

field depicted by curve (a) of figure 11, because all current lines are parallel to the ܱݔ 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

, a variation of the vertical component of the induced

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 (ߩ ൌ ͷǤͺʹ ൈ ͳͲି଼ΩǤ ݉), and one where the resistivity is higher: ߩ*′* ߩ.

Fig. 11. Measured values of ܤ௭, with a Hall probe at ݀ ൌ ͳ݉݉ above the plate, between ܣ *′*ܣ 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 ௪*′* ଶ at the edge of each half plate, so the total damaged edge area width is ݓ<sup>ᇱ</sup> .

Fig. 12. Representation of the two half plates laid near each other, and of two assumed fatigued zones of resistivity ߩ<sup>ᇱ</sup>

Therefore, the next problem is to determine the values of ݓ*′ ′*ߩ 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 ݓ*′ ′*ߩ and . This allows to evaluate the ݖ component of the induced field. It is then necessary to define a global error ߳ሺݓ*′ ′*ߩ ǡ ሻ between the observed field, and the field corresponding to ݓ*′ ′*ߩ and . Then we modify either

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 137

 before annealing after annealing

Fig. 15. Experimental curve established with a Hall sensor at d=1mm above the aluminium

**-30 -20 -10 0 10 20 30**

**y(mm)**

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

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)

plate in the case of a zero width crack before and after annealing

**7. Metallic dislocations created by a shock** 

**-8,0x10-7**

**-4,0x10-7**

**Bz(Tesla)**

**4,0x10-7**

**8,0x10-7**

**1,2x10-6**

**0,0**

below.

�*′* 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 �*′* � � 5�8� � �0���� � � ��5 � �) and of �*′* (�*′* � 50*µ*�) which provided a theoretical curve of �� shown in figure 13 (a).

Fig. 13. (a) Theoretical curve with optimized values of �*′* � 50*µ*� and �*′* � ��5 � �, (b) experimental curve established with a Hall sensor at d=1mm above the aluminium plate in the case of a zero width crack

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.

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

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).

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

### **7. Metallic dislocations created by a shock**

136 Nondestructive Testing Methods and New Applications

This led us to determine a best value of �*′* � � 5�8� � �0���� � � ��5 � �) and of �*′*

Fig. 13. (a) Theoretical curve with optimized values of �*′* � 50*µ*� and �*′* � ��5 � �, (b) experimental curve established with a Hall sensor at d=1mm above the aluminium plate in

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

3�

**-30 -20 -10 0 10 20 30**

**y(mm)**

Then, we place again the two half plates in the same uniform magnetic excitation field. The

**0 100 200 300 400 500 600 700**

��0��� 380���

**Temps (min)**

but much smaller than previously; this show *that annealing has been very effective, but has not* 

curve of the vertical component of the induction field still shows a variation of �*′*

in [Poloujadoff et al., 1994] and in the reference [Bettaieb et al., 2010].

50*µ*�) which provided a theoretical curve of �� shown in figure 13 (a).

to decrease � and repeat this process until � is as small as possible. This is developed

 (a) (b) (�*′* �

 and �*′* ,

�*′* or �*′*

the case of a zero width crack

*been long enough* (figure 15).

measurements. The annealing cycle is shown in figure 14.

**-4x10-7**

**-2x10-7**

**Bz(Tesla)**

**2x10-7**

**4x10-7**

**0**

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

**0**

**100**

**200**

**Temperature (°C)**

**300**

**400**

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.

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)

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 139

Considering the second impact, we found similar results, but the signal varies within a smaller interval (േͲǤ ൈ ͳͲିହܶ) (see figures 18 (A) and 18 (B)). As for the third impact, it is

(A)

**-10 -5 0 5 10**

**y(mm)**

 (B) Fig. 18. (A) Measured values of ܤ௭ along the ܱݕ axis for ݔ ൌ Ͳ݉݉, (B) matrix representation of the values of ܤ௭ across a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉). Measurement with a

It is extremely important to consider that these results may be assigned to two different causes: the change of the shape of the aluminum square plate and/or a modification of the metallic microstructure inducing a change of the electric resistivity. To clarify this question, we have assumed that the effects of these two causes can be separated. Unfortunately, a rigorous mathematical analysis of such a structure would involve a cylindrical geometry and a uniform field, resulting in a very complicated 3D analysis. For this reason, we made the following approximations which happen to give a sufficient feeling of what happens.

This means that we first replaced the portion of a sphere, in figure 16 (c) by an empty parallelepiped with a square basis of side ݓൌܦ and height ݄ൌ݀ chosen to have the same volume than the portion of the half sphere (figure 19 (A)). This model accounts for the

Hall probe at d=1mm above the square aluminium plate (ܦ ൌ ͵݉݉, ݀ ൌ ͲǤͲͻ݉݉)

**-8x10-6**

**-4x10-6**

**Bz(Tesla)**

**0**

**4x10-6**

**8x10-6**

How may we interpret these results?

observed results (figure 19 (B)).

not really detectable.

In our latest experiments, we stroke a steel ball in the middle of a square aluminum plate with a hammer. The dimensions of the damaged zones depend naturally on the radius of the steel ball and on the violence of the hammer shocks. We have considered only three cases; the depth ݀ and the diameter ܦ of some shock prints are given in the table I.


Table 1. Dimensions of the damaged zones

Consider the first impact, produced near the middle of the square plate. The Aluminum plate is laid upon the stage which has been already described in the figure 6. The excitation field being again at ͳͺͲݖܪ with an amplitude of ͲǤͳͷ ൈ ͳͲିସܶ if we use a SQUID and ʹǤͶͷ ൈ ͳͲିସܶ if we use a Hall probe sensor. The stage is moved as described in §4, the variation of the vertical induction field component, measured by the Hall probe is shown in figure 17 (A). This proves that the impact may be easily detected, since the perturbation signal varies between േͳͲିହܶ. It is also possible to scan a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉) around the center of the impact and to represent it as in figure 17 (B).

Fig. 17. (A) Measured values of ܤ௭ along the ܱݕ axis for ݔ ൌ Ͳ݉݉, (B) matrix representation of the values of ܤ௭ across a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉). Measurement with a Hall probe at d=1mm above the square aluminium plate (ܦ ൌ Ͷ݉݉, ݀ ൌ ͲǤͳͳ݉݉)

Considering the second impact, we found similar results, but the signal varies within a smaller interval (േͲǤ ൈ ͳͲିହܶ) (see figures 18 (A) and 18 (B)). As for the third impact, it is not really detectable.

Fig. 18. (A) Measured values of ܤ௭ along the ܱݕ axis for ݔ ൌ Ͳ݉݉, (B) matrix representation of the values of ܤ௭ across a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉). Measurement with a Hall probe at d=1mm above the square aluminium plate (ܦ ൌ ͵݉݉, ݀ ൌ ͲǤͲͻ݉݉)

How may we interpret these results?

138 Nondestructive Testing Methods and New Applications

In our latest experiments, we stroke a steel ball in the middle of a square aluminum plate with a hammer. The dimensions of the damaged zones depend naturally on the radius of the steel ball and on the violence of the hammer shocks. We have considered only three

Consider the first impact, produced near the middle of the square plate. The Aluminum plate is laid upon the stage which has been already described in the figure 6. The excitation field being again at ͳͺͲݖܪ with an amplitude of ͲǤͳͷ ൈ ͳͲିସܶ if we use a SQUID and ʹǤͶͷ ൈ ͳͲିସܶ if we use a Hall probe sensor. The stage is moved as described in §4, the variation of the vertical induction field component, measured by the Hall probe is shown in figure 17 (A). This proves that the impact may be easily detected, since the perturbation signal varies between േͳͲିହܶ. It is also possible to scan a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉) around the center of the impact and to represent it as in

(A)

**-10 -5 0 5 10 -1x10-5**

**y(mm)**

Fig. 17. (A) Measured values of ܤ௭ along the ܱݕ axis for ݔ ൌ Ͳ݉݉, (B) matrix representation of the values of ܤ௭ across a square domain (െͳͲ݉݉ ݔǡ ݕ ͳͲ݉݉). Measurement with a

(B)

**-1x10-5 -5x10-6**

**Bz(Tesla)**

**0 5x10-6 1x10-5**

Hall probe at d=1mm above the square aluminium plate (ܦ ൌ Ͷ݉݉, ݀ ൌ ͲǤͳͳ݉݉)

cases; the depth ݀ and the diameter ܦ of some shock prints are given in the table I.

Table 1. Dimensions of the damaged zones

figure 17 (B).

 Dimeter ܦ depth ݀ Impact 1 4mm 0.11mm Impact 2 3mm 0.09mm Impact 3 2mm 0.07mm

> It is extremely important to consider that these results may be assigned to two different causes: the change of the shape of the aluminum square plate and/or a modification of the metallic microstructure inducing a change of the electric resistivity. To clarify this question, we have assumed that the effects of these two causes can be separated. Unfortunately, a rigorous mathematical analysis of such a structure would involve a cylindrical geometry and a uniform field, resulting in a very complicated 3D analysis. For this reason, we made the following approximations which happen to give a sufficient feeling of what happens.

> This means that we first replaced the portion of a sphere, in figure 16 (c) by an empty parallelepiped with a square basis of side ݓൌܦ and height ݄ൌ݀ chosen to have the same volume than the portion of the half sphere (figure 19 (A)). This model accounts for the observed results (figure 19 (B)).

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 141

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

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

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 ܤሬԦ, and we have scanned its upper surface with a Hall probe. However, this first publication was limited to

Fig. 21. Fatigued strip, the length of the damaged (bent) zone is approximately equal to 3mm

inverse problem method for a better mathematical analysis of the results.

Since that time, we have greatly improved our experimental process, and have developed an

Our first effort has been to flatten the long strip carefully before measurement, as shown by figures 22 (figure 22 (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

model also accounts for the experimental results (figure 20 (B)).

dislocations are much more important [Bettaieb, 2009].

**8. Metallic dislocations created by flexions** 

**8.1 Experimental study** 

the feasibility of the detection.

about its center (figure 23 (A)).

Fig. 19.A. Square aluminium plate with an empty space of same volume that the one in figure 16 (c)

Fig. 19.B. Theoretical values of ܤ௭ corresponding to the figure 19 (A)

Fig. 20. (A) Square aluminium plate with an inner portion of resistivity ߩ*′* ߩ,) B) theoretical values of ܤ௭ corresponding to the figure 20 (A).

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].

### **8. Metallic dislocations created by flexions**

### **8.1 Experimental study**

140 Nondestructive Testing Methods and New Applications

(A)

(B)

**-20 -10 0 10 20**

**y(mm)**

(A)

(B) Fig. 20. (A) Square aluminium plate with an inner portion of resistivity ߩ*′* ߩ,) B) theoretical

**-20 -10 0 10 20**

**y(mm)**

Fig. 19.B. Theoretical values of ܤ௭ corresponding to the figure 19 (A)

**-1,5x10-5 -1,0x10-5 -5,0x10-6 0,0 5,0x10-6 1,0x10-5 1,5x10-5**

**Bz(Tesla)**

values of ܤ௭ corresponding to the figure 20 (A).

**-3x10-5 -2x10-5 -1x10-5 0 1x10-5 2x10-5 3x10-5**

**Bz(Tesla)**

Fig. 19.A. Square aluminium plate with an empty space of same volume that the one in

figure 16 (c)

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 ܤሬԦ, and we have scanned its upper surface with a Hall probe. However, this first publication was limited to the feasibility of the detection.

Fig. 21. Fatigued strip, the length of the damaged (bent) zone is approximately equal to 3mm

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.

Our first effort has been to flatten the long strip carefully before measurement, as shown by figures 22 (figure 22 (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 (figure 23 (A)).

Study of Metallic Dislocations by Methods of Nondestructive Evaluation Using Eddy Currents 143

(A)

**-20 -10 0 10 20**

**y(mm)**

(B) Fig. 23. (A) Values of ܤ௭ induced by the eddy currents within a flattened fatigued long strip (measurement with a Hall sensor at ݀ ൌ ͳ݉݉ above the strip), (B) Corresponding values of

**-20 -10 0 10 20**

**y(mm)**

The principle of the inverse problem is well known. The relationship between the induction field and the induced currents in the long strip is linear, the coefficients being the conductivity. Simultaneously, the relationship between the strip induced current and the outside induced induction field is also linear. Therefore, if the exciting induction field and the induced induction field are known, the resistivity can be determined [Bettaieb et al., 2010]. It is what has been done to estimate the resistivity (figure 23 (B)) along the strip. This

In this very original chapter, we have shown that a methodology generally used to detect cracks can be also used to measure the effects of a mechanical fatigue created by cutting tools, impact shocks or flexions. The natural continuation should be to study quantitatively

the estimated local resistivity along the damaged zone using the inverse solution

latter result clearly shows the effect of the fatigue where the strip has been bent.

**8.2 Theoretical evaluation of resistivity along the strip** 

**-6x10-5**

**6,0x10-8**

**6,5x10-8**

**7,0x10-8**

**Résistivité (Ohm.m)**

**7,5x10-8**

**8,0x10-8**

**-3x10-5**

**Bz(Tesla)**

**3x10-5**

**6x10-5**

**9x10-5**

**0**

**9. Conclusion** 

(A)

Fig. 22. Bent long strip being flattened in a shaper to eliminate the geometrical deformation before eddy current measurement (A: side view; B: upper view)

Fig. 23. (A) Values of ܤ௭ induced by the eddy currents within a flattened fatigued long strip (measurement with a Hall sensor at ݀ ൌ ͳ݉݉ above the strip), (B) Corresponding values of the estimated local resistivity along the damaged zone using the inverse solution

### **8.2 Theoretical evaluation of resistivity along the strip**

The principle of the inverse problem is well known. The relationship between the induction field and the induced currents in the long strip is linear, the coefficients being the conductivity. Simultaneously, the relationship between the strip induced current and the outside induced induction field is also linear. Therefore, if the exciting induction field and the induced induction field are known, the resistivity can be determined [Bettaieb et al., 2010]. It is what has been done to estimate the resistivity (figure 23 (B)) along the strip. This latter result clearly shows the effect of the fatigue where the strip has been bent.

### **9. Conclusion**

142 Nondestructive Testing Methods and New Applications

(A)

(B)

Fig. 22. Bent long strip being flattened in a shaper to eliminate the geometrical deformation

before eddy current measurement (A: side view; B: upper view)

In this very original chapter, we have shown that a methodology generally used to detect cracks can be also used to measure the effects of a mechanical fatigue created by cutting tools, impact shocks or flexions. The natural continuation should be to study quantitatively

**7** 

**Magnetic Adaptive Testing** 

*2Research Institute for Technical Physics and Materials Science, Budapest* 

Tests of impending material degradation of engineering systems due to their industrial service make an indispensable part of any modern technological processes. *Destructive tests* are extremely important. Their considerable advantage consists in their *straightforwardness*. In great majority of cases they *directly test the very property*, which is at stake. For instance, the limiting endurable stress before the material yields is directly measured by a mechanical loading test, or the number of bending cycles before the system breaks is directly counted to find the fatigue limits, and so on. Besides, the destructive inspections are regularly used in situations of any materialized failure, when knowledge of the *final condition* of the material at the moment of the breakdown helps to avoid any next malfunction of the same or similar systems under equivalent circumstances. However, the destructive tests cannot be used on the finished or half-finished parts in process of their industrial production, simply because those parts – after having been destructively tested – cannot be used for their primary purpose any more. The only possible application of destructive tests in industry is to examine destructively every nth produced piece only, which is not sufficient for reliability of

*Nondestructive tests,* do not suffer by those problems. Causing no harm, they can be used at each produced object, and they can be periodically applied even on systems in service. Judging by the vivid interest in the presently observed improvement of traditional nondestructive tests and in development of some recently discovered ones, the nondestructive evaluation of material objects attracts currently attention perhaps even more than the destructive assessment does. Evaluation of nondestructive tests keeps the user informed about the *actual condition* of the system, and should ensure avoidance of any failure *before* it ever happens. There are numerous methods of nondestructive tests, based on optical, acoustic, electrical, magnetic and other properties of the materials, which can be *correlated* with the watched quality of the whole system. The necessity of the *unambiguous correlation* between the nondestructively measured physical property and the guarded property of the system at stake is an *unreservedly required* claim. It calls for the nondestructive tests to be examined and re-examined to their *one hundred percent reliability* before they can be really applied in crucial cases. Multi-parametric output of a nondestructive testing method is therefore an extremely valuable and welcomed property,

**1. Introduction** 

goods presently required.

Ivan Tomáš1 and Gábor Vértesy2

*1Institute of Physics, Praha,* 

*1Czech Republic 2Hungary* 

most of the microstructure properties which influence the resistivity of the alloy. These include irradiations defects and phase transitions. We would certainly welcome collaboration offers from colleagues.

### **10. Acknowledgment**

We would like to thank warmly Dr. Denis GRATIAS from ONERA/CNRS for several useful advices and discussions notably related to the nature of dislocations. He also suggested the check of our ideas by the annealing procedure.

### **11. References**

