Nonlinear model of flaw detection in steel pipes by magnetic flux leakage

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NDT&E International, Vol. 28, No. 1, pp. 35-40, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0963-8695/95 $10.00 + 0.00

Nonlinear .model of.flaw detection =n steel p=pes by magnetic flux leakage Eduardo Altschuler and Alberto Pignotti* FUDETEC, Center for Industrial Research, L. Alem 1067, (1001) Buenos Aires, Argentina Received 26 April 1994; revised 8 November 1994

A numerical model of crack detection by magnetic flux leakage that takes into account the saturation of the induction flux density with the applied field is developed. The model contains no free parameters and is validated with detailed measurements of material properties and magnetic fields reported by Fbrster. It is then used to discuss the dependence of signals on the tube wall thickness, crack depth and width, and the strength of the applied field. An estimate of the noise level is used to analyse the signal-to-noise ratio and calculate the optimal applied field for crack detection. Difficulties arising in internal crack detection of shallow cracks in thick-walled pipes are also discussed. Keywords: steel pipes, magnetic flux leakage, numerical model

be the source of additional complications. In the following we include the former and neglect the latter effects, knowing that magnetic saturation is essential for a sound description of the process {1°1, and hoping that at high applied fields hysteresis effects may be ignored.

Crack detection by magnetic flux leakage is a standard technique that has been widely used for the inspection of ferromagnetic pipes for many years. Pioneering work in this field was performed by Ftrster [l-q, who reported detailed measurements and analysed the basic physical principles and mechanisms on which this technique is based, using simple mathematical models. Many other researchers have performed model calculations using more sophisticated numerical tools [s-g], but there are still many consequences of practical importance that can be drawn from a model analysis of this process that are not found in the current literature. In the following sections we address some of these questions using a parameterization of the steel magnetic properties taken from Reference 4.

Based on analysis of detailed measurements, Ftrster pointed out that indeed, the leaked field is not proportional to the applied field, and introduced a nonlinear enhancement factor to account for this effect[+1.To obtain such a behaviour from detailed model calculations with no free parameters is the first objective of this work. After the validity of the model is established in this way, we analyse the behaviour of the signal strength with the crack depth and width, and the wall thickness, and discuss what is the optimal applied field for crack detection.

In a linear model, in which the magnetic permeability is assumed to be constant, the solution of the field equations is everywhere proportional to the strength of the applied magnetic field. We know, however, that ferromagnetic materials deviate from such behaviour in more than one respect: on one hand, the saturation of the magnetization of the material is a highly nonlinear phenomenon; on the other hand, hysteresis effects arc also present, and may

Formulation of the model We neglect the effect of the pipe wall curvature and formulate a two-dimensional model of a steel plate of thickness t, that is equal to the pipe wall thickness, with an infinitely long crack in the z direction, with rectangular cross-section of width a and depth b. When compared to measurements performed on steel bars, the thickness of the plate is chosen to be the bar diameter. This choice

* Author to whom correspondence should be addressed

35

E. Altschuler and A. Pignotti

v

For our two-dimensional problem, the only non-vanishing component of A is the z component, which is only a function of x and y, and we denote it by A(x, y).

d

sensor

nm,-~

Boundary conditions for A at infinity derive from the assumed value Ho for the magnetic field. At steel-air interfaces, they follow from the continuity of the tangential component of H (continuity of the normal component of B is automatically satisfied in this formulation).

~

Equation (4) is solved by the finite differences method. A uniform initial value for # is chosen to be:

Y

i

sensor

v --lb-

x

--D*-

B(Ho)

Ho Figure 1 Basic geometry used in the model, showing the crack width, depth, plate or tube thickness and location of sensor coils

and Equations (1), (3) and (4) are iterated until a selfconsistent solution is found.

Model validation

provides the same distance between the root of the crack and the opposite steel surface that is available in the bar, and this distance is of primary importance for determining the amount of leaked magnetic flux. Furthermore, this comparison is made for notches not larger than 4% of the slab thickness, for which the result is rather insensitive to the actual thickness value.

To validate the model we use the measurements reported by F6rstcr [3m of the maximum value of the magnetic field ~ inside the crack for different crack depths, widths and strong applied fidds. They were performed on chromium steel bars for which the reported values of the material parameters are, in our notation, B, = 1.546 T and H~ = 731.8 A m - 1.

The sensing coil used to detect the leaked field is assumed to lie on a plane at a distance d from the steel surface, either on the same side of the crack, for the case of external cracks, or the opposite side, for internal cracks. The applied magnetic field Ho is introduced as the boundary value of the field at infinity. Figure 1 shows the basic geometry used.

Table 1 shows the results of the comparison between the model and the measurements for different crack geometries and applied fields. It should be remarked that even though the comparison was performed at a location that is highly sensitive to the crack geometry, the results arc within 10% of the measured values. These deviations between the model and the published measurements can be attributed to the approximations to the material properties implied by Equation (1), and the approximate representation of the geometry, including the detailed shape of the crack cross-section, which we have taken to be rectangular.

Following a parameterization used in Reference 4 with a slightly different notation, the dependence of the induction flux density B on the magnetic field H inside the steel is assumed to be: =

LH + H~_!

+ #oH = #H

(1)

The conclusion from this analysis is that the model represents adequately the basic relevant physical mechanisms, and should therefore provide a valuable aid in assessing the capabilities and limitations of this NDT technique. We should point out, however, that the approximation of Equation (1) is not supposed to be applicable to weak fields (ie to fields such that H < / ~ ) , but we find that this is no major drawback, because this technique is used with strong applied fields.

where B, and ~ are parameters that depend on the steel grade, and that represent the saturation value of B and the field intensity at which one half of this saturation value is reached, respectively. Because the divergence of the induction field density vanishes: V. B = 0

(2)

B can be derived from a vector potential A, ie: B = V x A

Modelling of the signal and noise

(3)

The normal component of the leaked field is assumed to be detected by an assembly of two flat coils wound in opposite directions, lying on a plane paralld to the steel surface at a fixed distance d from it (see Figure 1). These coils move with respect to the pipe surface, and a signal is generated whenever the sensor sweeps a region in which a non-uniform normal component of the magnetic field

In the absence of electric currents, A satisfies the equation:

v x [Y--~]=O

(4,

Here the permeability p is a function of A through Equations (1) and (3).

36

Nonlinear model of flaw detection Table 1. Comparison of values of maximum field $tr~rtgth in cracks measured by F6rster and calculated by the present model Measured Hg (kAm - t )

Model Hg (kAm -1)

7.96 7.96 7.96 7.96 7.96

333 200 142 107 69

339 212 150 110 74

1.8 6.0 5.6 2.8 7.2

1.01 2.05 3.00 4.00

7.96 7.96 7.96 7.96

150 232 337 426

158 245 355 436

5.3 5.6 5.3 2.3

2.05 2.05 2.05 2.05

4.77 6.37 9.55 11.14

146 193 271 306

140 194 291 333

-4.1 0.5 7.4 8.8

a (mm)

b (mm)

0.285 0.55 0.86 1.27 2.05

2.50 2.50 2.50 2.50 2.50

0.32 0.37 0.32 0.31 0.37 0.37 0.37 0.37

H 0 (kAm -1)

0.04

Deviation %

0.16

signal width 0.12

0.02.

i :

0.08 "d

0.08 I

"~'~0.04.

l -0.02-

o.o% .o.~ 08

o~5

i~o

I~5

2.o

b [mm]

~2

~6

0.020

T/,~/s/

+

t=Snun

~

t=lOnnn

=

t=lSmm

>,

t=2Omm

Figure 2 Typical signal calculated in the model Figure 3 Dependence of external crack signal strength on crack depth, for a = 0.5 rnm, Ho--10 000 A m -1, and different values of pipe wall thickness

is found. Each coil is 12.3 mm long and 2.1 mm wide, and has six turns.

may be different causes of noise, in the following we adopt as a measure of noise the peak-to-peak signal generated by an external crack of 0.1 mm depth and 0.5 mm width (actually, the value thus obtained depends only weakly on the chosen width).

A typical signal obtained by this model is shown in Figure 2. It is characterized by two major peaks, and we represent it by the peak-to-peak voltage, which we call 'signal strength', and the time interval between the occurrence of the two peaks, which we call 'signal width'. The signal voltage represents the expected raw sensor output, prior to any amplification stage. It is a function of d which, in the examples discussed below, is taken to be equal to 1 ram.

Resu Its As an example of possible applications of this model we discuss here the effect of the pipe wall thickness and applied magnetic field intensity on the detection of cracks in steel pipes.

In order to discuss the ability of this technique to detect cracks of a given size, it is necessary to estimate the noise level and to require a signal-to-noise ratio (SNR) larger than some threshold value which, according to usual practice, we choose to be equal to 3. Even though there

Figures 3 and 4 show the dependence of the signal strength on the external crack depth and width, respectively, for different values of the wall thickness t. Whereas

37

E. Altschuler and A. Pignotti 0.12

0.08

"-, 0.09"

"., 0.06

0.06"

0.04'

0.03-

~ 0.02"

0. 0.o

o~s

t---5mm ~

1~o a [m=]

l~S

0.00 0.0

2.0

0'.5

1'.0

115

2.0

a [nun]

t=lOmm ~ t=15mm ~.~: t=2Omm

t=Smm ~

Figure 4 Dependence of external crack signal strength on crack width, for b = 1.0 mm, H0 = 10 000 A m - 1, and different values o f pipe wall thickness

t=lOmm

t=15mm

'

t=2Omm

Figure 6 Dependence of internal crack signal strength on crack width, for b = 1.0 mm, Ho = 10 000 A m - 1 and different values of pipe wall thickness

,0.40

0.05

0"12t

0.04,

o.o9

~.30

0.03

0.06

0.20 0.02"

~ 0.031

~0.10

0.01

,

_

0.0. 000¢~

,

0.5

---

,

,

,

1.0

1.5

2.0

0.00

io

b [ram]

t=5mm ~

t=lOmm ~ t=15mm

io

0.°°

H o [gA/m]

~ t=2Omm

E.~na/

Figure 5 Dependence of internal crack signal strength on crack depth, for a = 0.5 m m , / - ~ = 10 0 0 0 A m - 1, and different values of pipe wall thickness

=

lnterna/ ~

Rat/o

Figure 7 Signal strength as a function of applied field, for external and internal cracks o f a = 0.5 ram, t = 10 mm and b / t = 5%. The ratio o f internal to external crack strengths is also plotted

the dependence on the crack depth is approximately linear, in the case of the crack width it grows sharply for small values of a, but tends to saturate at higher values. As the wall thickness is increased, the signal strength decreases, approaching a non-vanishing limiting value. Similar results are shown in Figures 5 and 6 for internal cracks. Apart from the fact that internal cracks give rise to weaker signals, the main difference concerns the dependence on the wall thickness: signal strength for internal cracks is more sensitive to the wall thickness for small values of t, and it tends to zero as t becomes very large.

0.12"

9.40

0.09-

~.30

~ 0.06-

O.20

0.03-

0.10

0.~

0

I0

~.00

20 Ho [KAIm]

Signal strength is plotted in Figures 7 and 8 as a function of the applied field, for crack depths equal to 5% and 12.5% of the wall thickness, respectively. It is seen to grow more rapidly than the applied field for weak fields, but tends to level off at higher fields. The ratio R of

---w- Externa/

~

Interna/ ~

Rat/o

Figure 8 Signa strength as a function of applied field, for external and internal cracks o f a = 0.5 ram, t = 10 mm and b i t = 12.5%. The ratio of internal to external crack strengths is also plotted

38

Nonlinear model of flaw detection 1.00:

/'0

10

i0

we think of the noise as originating from surface rugosity, and estimate the noise level by equating it to the strength of a signal of a very small external crack. This makes it possible to analyse the dependence of the SNR on the applied field. We find that the value of Ho which maximizes the SNR may vary depending on the nature and size of the crack. In the example considered, a field of 8000 A m - 1 would be ideal for the detection of internal 12.5% cracks and external 5% cracks, but 5% internal cracks cannot satisfy the detectability criterion SNR > 3 for any value of the applied field.

30

Ho [KAIm]

Conclusions 12.5% Ext

~-

12.5% Ira ~

5% Ext

r~ 5% Int

A model was developed in which the only parameters, in addition to those defining the geometry, are those that describe the magnetic properties of the" steel, and have been taken from the existing literature.

Figure 9

Dependence of signal-to-noise ratio on applied field for external and internal cracks of a = 0.5 mm, t = 10 mm and b / t = 12.5% and 5%

The agreement obtained between the model and the measurements reported by Ftrster suggests that the main assumption that hysteresis effects are negligible is adequate for high applied fields. Thus, the model provides a useful tool for the analysis of the M F L technique for the inspection of steel tubes.

internal to external crack signal strengths is also shown. This ratio is very small for weak applied fields, and goes through a maximum value at a field of approximately 20 000 A m - 1 for 5% cracks and 12 000 A m - 1 for 12.5% cracks.

The most safient features exhibited by the model are:

Finally, Figure 9 shows an example of the dependence of the SNR on the applied field, for external and internal cracks and two values of the relative crack depth, b/t = 12.5% and 5%. The SNR increases with Ho until a maximum value is reached, after which it starts to decrease slowly.

1. Signal strength increases with both crack depth and width, but whereas the dependence on the former is almost linear, as a function of the latter it rises sharply for small values of a, but tends to saturate for large values. 2. For a given crack geometry, signal strength increases with decreasing wall thickness, particularly for internal cracks. 3. Signal strength increases more rapidly than the applied field up to moderate values of Ho. Beyond that, the increase subsides and the signal levels off. 4. The inclusion of an adequate description of material properties at high fields is crucial for obtaining realistic results. In particular, the saturation of the induction flux density is essential for the detection of internal cracks. 5. A model of noise level can be introduced to obtain the value of the applied field which maximizes the SNR for cracks of given type and size. 6. In spite of that, examples are given for which internal cracks of small b/t ratio cannot be detected, no matter how strong the applied field.

Discussion The conservation of the induction flux, expressed by Equation (2), is the basis for understanding several results mentioned in the previous section. The finding that, for a given crack depth, the signal is larger for thinner wails, follows from the fact that, when the wall is thinner, larger values of B are required to channel the induction flux dislodged by the presence of the cracks. These higher values of B at the base of the crack imply higher values of H, which, through the boundary conditions, propagate inside the crack and to the location of the detector. The saturation of B at high magnetic fields is also a crucial ingredient for understanding some of our results, such as the dependence of the internal to external crack signal ratio R on the applied field. For weak fields, the increase in B due to the crack presence is localized near the crack base, and the value of B at the opposite wall surface is hardly affected. Because of this reason, for weak fields R is extremely low. As the field increases, however, because of the saturation of the magnetic material, the increase in B is spread across the available cross-section, and the internal crack signal strength becomes comparable to the external one.

Acknowledgements The authors are indebted to Adrian Kohan for his participation in the early stages of this work.

References

Finally, the evaluation of the detectability of cracks requires an assessment of the noise level. In our analysis

1 F~rster, F. 'Nondestructive inspection by the method of magnetic leakage fields' Defektoskopiia 11 (1982) pp 3-25

39

E. Altschuler and A. Pignotti 2 F6rster, F. 'On the way from the 'Know-how' to the 'Know-why' in the magnetic leakage field method of nondestructive testing (part one)' Mater Eva143 (1985) pp 1154-1161 3 F6rster, F. 'On the way from the 'Know-how' to the 'Know-why' in the magnetic leakage field method of nondestructive testing (part two)' Mater Eva143 (1985) pp 1398-1403 4 F6rster, F. 'New findings in the field of nondestructive magnetic leakage field inspection' NDT Intern 19 (1986) pp 1-12 5 Lord, W. and Hwang, J.H. 'Defect characterization from magnetic leakage fields' Brit J N D T 19 (1977) pp 14-18 6 Brmlar, E. 'Magnetic leakage fields calculated by the method of finite differences' NDT Intern 18 (1985) pp 353-357 7 Atherton, D.L and Daly, M.G. 'Finite element calculations of

magnetic flux leakage detector signals' N D T Intern 20 (1987) pp 8

235-238 Athertoa,D . L and Czura, W. 'Finiteelement calculationson the

effects of permeability variation on magnetic flux leakage signals' NDT Intern 20 (1987) pp 239-241 9 Athertoa, D.L. 'Finite element calculations and computer measurements of magnetic flux leakage patterns for pits' Brit .I NDT 30 (1988) pp 159-162 10 Pignotti, A. and Kohan, A. "Importance of magnetic saturation effects in the detection of internal tube cracks by magnetic flux leakage' Proceedings o f the XIIlth World Conference on Nondestructive Testing, Sao Paulo, Brazil. Elsevier, Amsterdam (1992) pp 456-460

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