IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, NO. 4, JULY 1992 1941

Finite Element Modelling and Physics of Remote Field Eddy Current Responses for

Axially Aligned Cracks Yu Shi Sun, Senior Member, IEEE, He Yun Lin, Mei Juan Chen, Chun De Wang,

Xiao Bing Wu, David Atherton, and Thomas R. Schmidt

Abstract-A simplified physical model for two-dimensional (2- D) finite element analysis of axially aligned crack responses for the remote field eddy cu_rrent (RFEC) technique i s proeosed. Two formulations: the A-V formulation and the H or T for- mulation, are applied to model the problems. Reasonable agreement between calculated and some preliminary experi- mental results is achieved. A hypothesis, based on the physical model, is presented. It explains RFEC signals from axially aligned cracks and makes some predictions of possible features for future verification.

INTRODUCTION HE remote field eddy current (RFEC) inspection T technique uses an internal transducer and is capable

of inspecting steel tubulars with equal sensitivity to both internal and external defects including cracks. It is cur- rently receiving considerable interest for a wide range of applications using steel tubes including heat exchangers, boiler and furnace tubes, as well as gas and liquid pipe- lines.

Originally, most of work with RFEC was experimental [l], but with the advent of finite element analysis (FEA) and computer-based analytic techniques, modelling in- vestigations are playing an increasingly important role. These started with FEA of pipes with no defects and even- tually included circumferential slots which could be modelled using 2-D axisymmetric simulations [ 2 ] - [ 6 ] . Axial crack modelling is normally a three-dimensional (3-D) simulation problem, which requires extensive effort and investment. This paper describes a FEA method for reducing a simulated infinite axial crack to a 2-D simu- lation and gives some results.

Since the early development and application of RFEC in the late 1950’s, users have been aware that crack ori- entation plays an important role in detecting defects in ferrous and nonferrous conductors. A number of interest- ing and somewhat unexpected observations in crack and other defect responses using RFEC were noted [ l ] , [7].

Manuscript received November 11, 1991; revised March 20, 1992. Y. S . Sun, H. Y . Lin, M. J . Chen, C. D. Wang, and X. B . Wu are with

the Nanjing Aeronautical Institute, Nanjing, Jiangsu 210016, China. D. L. Atherton and T. R . Schmidt are with the Department of Physics,

Queen’s University, Kingston, ON, K7L 3N6, Canada. IEEE Log Number 9200739.

These can be summarized as follows: 1) The defect dis- turbance of the magnetic field in the interior of the pipe is extended (sometimes referred to as smearing) in the cir- cumferential direction. Fig. 1 [ l ] shows an example of this effect. The responses to the symmetrical hemispher- ical pits shown on the left side of the pipe sample are extended in the circumferential direction, approximately five times to those in the axial direction.

2) In steel an axially oriented crack shows less distur- bance than a circumferentially oriented crack. Fig. 1 shows this effect. The two defects on the right are saw cuts of equal depth but different orientations. The circum- ferentially oriented cut in steel gives a more pronounced indication.

3) Axial cracks have a nonlinear phase-depth relation. Fig. 2 [7] shows this relationship. There are relatively insignificant signals for shallow cracks but these increase dramatically as full penetration is reached. 4) Axial crack indications are dependent on crack

width. Fig. 2 shows an increase in signal with crack width. A 3-D model is nomally required to explore the behav-

ior and interaction of an axial crack with the interrogating electromagnetic field and resultant eddy currents. This contributes considerably to the complexity and computer time requirement for the simulations. By making a few simplifying assumptions, such as an infinitely long axial crack, this simulation can be reduced to a 2-D model. While crack end effects would be of interest and are lost by this assumption, most naturally occurring axial cracks are quite long compared with their depth and are well por- trayed by this model.

SIMPLIFIED PHYSICAL MODEL A simplified 2-D model for approximation of axially

aligned RFEC crack responses is obtained by adopting the following assumptions: 1) It is assumed that there is only an axial component of the magnetic field intensity, H,, in the remote field region outside the pipe wall. 2) The field outside the wall in the remote field region is considered to be uniform in both its circumferential and axial direc- tions.

The first assumption is almost true for nonferromag- netic pipes, because the flux lines are nearly axial in this

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, NO. 4, JULY 1992

oriented crocks with vorious widths would give rise to extra signal response. The second assumption is important for simplifying the

physical model, as it makes the model equivalent to the case of a pipe in an infinitely long solenoid with ampere

,

turns per unit length, & N , where Width of cracks

251pm __ .- 502pm * * * * - 813~m

IsN = Bo. (1)

The resulting 2-D physical model is shown in Fig. 3.

0 0 0

5.1 3.8 2.5 1.3 0.8 3.8 5.1 2.5 2.5 67% 50% 33% 17% 10% 50% 67% 3% 33%

- - I

76" I.D. Pipe. 7.6mm Wall Thickness

Complete Inner-Wall Scan

- 1.8m-

Fig. 1 . RFEC probe responses to flaws in a steel pipe showing their typi- cally wide circumferential extent.

, , .

Fig. 2. Signal amplitude and phase as functions of crack depth for cracks of various widths in steel pipe.

case. It is not quite true for ferromagnetic pipes, because there is a significant normal component (H,) of go in the remote field region. However, it becomes negligible when it enters the pipe wall which has higher permeability be- cause of the continuity of the normal components of the flux density, B.

The second assumption is based on the fact that there is a much smaller field attenuation outside the pipe than in the remainder of the energy transmission path. This underestimates axially aligned crack responses a little, as

and the Maxwell-Faraday law

$ B . z = - j u p ~ S B - Z , (3)

with interface conditions of,

Hr, = Hr2 (4)

4, = 4 2 ( 5 )

Er, = E,, (6) and the constitutive relations,

- ] = uE

B = p M -

where B is the magnetic field intensity and H,,, Hr2 are its tangential components on the interface between media 1 and 2, CZ is the total current included in the closed in- tegration loop, ] is the electric current density, S is the area of the surface defined by the closed loop, E is electric field intensity and E,, , Et2 are its tangential components on the interface between media 1 and 2, U is the conduc- tivity, is the magnetic flux density and, B,,, Bn2 are its

SUN et al . : FINITE ELEMENT MODELLING OF REMOTE FIELD EDDY CURRENT RESPONSES 1943

IY

Fig. 3 . Simplified physical model used for calculation of RFEC axial crack responses.

normal components on an interface of media 1 and 2, w is the angular frequency of the source field, p is the permeability.

Two kinds of formulations, 2 - V and T, are employed for modelling this problem.

A- V FORMULATION A magnetic vector potential 2 and a scalar electric po-

tential V are used in this formulation to represent the field distribution.

The governing equations become,

v x v v X A + a(jwX + V V ) = o v - ( jwX + VV) = 0,

(9)

(10) and elec- where v = 1 / p , and the magnetic flux density

tric field intensity E are functions of A and V as, - B = V x A (1 1) E = - ( j w 2 + V V ) . -

(12) For the model of Fig. 3, in this formulation, 2 has two

components, A, and A,. Obviously, there are three un- knowns per node point in the conducting region and two unknowns per node in air to be solved, although the phys- ical model itself is a two-dimensional one.

The advantage of using this model is that it is one of, the most popular formulations for 3-D modelling of gen- eral eddy current problems. It is recognized and accepted by most scientists in this field. Therefore, results of these calculations could serve as references for other formula- tions.

a or T Formulation In the H formulation the governing equation becomes,

(13) v2Zi - jwupR = o and in the T formulation it becomes,

v2T - jwupT = o (14) where T is electric vector potential which is defined as,

V x T = f (15) As the differential form of the Maxwell-Ampere law

is,

V x H = J (16)

- which is identical in form to that of (15), the solutions of T and a from these two formulations will be almost iden- tical with a difference of only a constant value. Therefore, only the formulation is discussed in this paper.

The T formulation can be considered as a degenerate form of the T-il formulation [8]. It is also one of the most popular formulations for 3-D eddy current problems, for the special case of the simplified physical model where the magnetic scalar potential is a constant value and there- fore eliminated from the governing equation.

The advantages of these two formulations are: 1) Only one component, H, or T,, per node point is to

be computed. This gives a tremendous reduction in the computer resource requirements.

2) Only the conducting region is considered. This gives even greater savings in computer resources.

BOUNDARY CONDITIONS FOR THE H FORMULATION Outer Sugace of the Pipe Wall

If an integration path for (2) is taken along the z-direction at any point “p” outside the pipe wall in Fig. 3, from plus infinity to minus infinity, we will always have

(17) Hp = ZsN = Ho.

This means that the whole outer field, including those points on the wall surface and inside the crack, is uniform with a constant value of Ho. Thus, it can be concluded that a Dirichlet boundary condition with a value of Ho = ZsN should be imposed on all the outer surfaces including those of the crack.

Inner Surface of the Pipe Wall

pipe, we have, If the integration path is taken at any point q inside the

Hq = ZsN + I,, (18) where Ze is the total eddy current flowing inside the pipe wall. It should be a constant value independent of the po- sition of the pipe section where it is defined. However, Z, is an unknown, dependent on the field solution.

If an integration path for (3) is taken along the inner surface loop of the pipe wall, we will get

Considering the interface conditions of (4) and (6) for the particular condition, we have

EAR;) = w n

H,(R;) = K ( R + ) ,

and

where Ri is the inner radius of the pipe. The superscription + and - represent just inside the pipe wall and just inside the air region in the pipe, respectively.

1944 IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, NO. 4, JULY 1992

Then, considering the constitutive relation of (7) and

- aHq J, = -

which is an alternative form of (16) applicable to the pipe inner surface, with the assumption that

an ’

Et = constant, (20)

we get,

- + aHq = 0 an

which is a mixed, or third kind of, boundary condition to be imposed on the inner surface of the pipe wall.

Iterative Steps in the Solution As the assumption of (20) is only an approximation, an

iterative procedure is used. It is as follows: 1) Solve the problem by applying the third boundary condition, (21). 2) Take the average of Hq on the pipe inner surface. Then, solve the problem again by imposing the averaged Hq on the pipe inner surface as a Dirichlet boundary condition. 3) Verify the solution using (19) and repeat the iteration until satisfactory convergence is obtained.

Experience shows that repeating these steps once is normally sufficient for a precise solution.

SOME CALCULATION RESULTS Results from the A-V Formulation

Two problems were examined. The first example used for the x - V formulation calculation is a hard aluminium alloy tube with an outer diameter of 39.9 mm and an inner diameter of 33.75 mm. Its measured conductivity is 1.7 * lo7 S/m. A very long axially aligned slot with a width of 3.00 mm is machined in its outer surface. The excita- tion frequency is 400 Hz.

An identical example is used for the experimental mea- surements introduced later.

The calculated eddy current distributions for different slot depths are illustrated in Fig. 4 and the distribution of the real part of the Poynting vector is shown in Fig. 5.

Tables I and I1 show the field homogeneity of the cal- culated magnetic flux density magnitude and phase distri- butions, for different slot depths.

defect depth: 333% defect depth: 50.0%

defect depth: 66.1% defect depth: 83.3%

Fig. 4 . Calculated eddy current distributions for example 1 using the A-V formulation.

defect depth 33.3% defect depth 50.0%

defect depth 66.7% defect depth: 83.3%

Fig. 5 . Calculated Poynting-vector distributions for example 1 using the A-V formulation.

TABLE I CALCULATED B, USING THE A-V FORMATION: MAGNITUDE

Magnitude

Results from the T Formulation The first calculations were for the same problem as for

the A-V formulation. The second example used for the formulation calculation is an aluminium alloy pipe with an outer diameter of 107.92 mm and an inner diameter of 98.36 mm. Its conductivity is 3.85 x lo7 S/m. An infi- nitively narrow crack, with a width of 0.0 mm, is as-

Error % Depth % Bmax Bm,, Bwer

0.0 13.803 13.459 13.606 0.561 16.7 13.868 13.515 13.669 0.552 33.3 13.973 13.628 13.781 0.582 50.0 14.163 13.820 13.945 0.635 66.1 14.487 14.146 14.296 0.678 83.3 15.284 14.946 15.097 0.833

100.0 27.990 27.790 27.915 0.402

SUN et a l . : FINITE ELEMENT MODELLING OF REMOTE FIELD EDDY CURRENT RESPONSES 1945

TABLE I1 CALCULATED B, USING THE A-V FORMATION: PHASE

TABLE 111 EXPERIMENTALLY MEASURED B, (MAGNITUDE)

Phase (Degrees)

0.0 113.60 112.96 113.41 0.561 16.7 113.80 113.17 113.62 0.552 33.3 114.20 113.53 113.97 0.582 50.0 114.83 114.11 114.54 0.635 66.7 115.88 115.10 115.53 0.671 83.3 118.48 117.50 117.91 0.833

100.0 180.30 179.58 180.00 0.402

DEFECT DEPTH: 75% ot = 0"

1 I

Fig. 6. Calculated results for example 2 using the T formulation: eddy current streamlines for an infinitely narrow crack of 75% penetration.

25% PENETRATION 5.36

5.31

E 5.25

2 5.20

5.14

5.09 53 5.03 v 4.90

h

4.81

0.00 0.13 0.25 0.30 0.51 0.63

X-COORD (CM) Fig. 7. Calculated results for example 2 using T formulation: Poynting

vector distribution for an infinitely narrow crack of 25% penetration.

75% PENETRATION 5.30

5.31 - z 5.25 2 5.20

5.14

5.09 53 5.03 y 4.98

4.92

4.07

0.00 0.13 0.25 0.38 0.51 0.63

X-COORD (CM) Fig. 8. Calculated results for example 2 using the Tformulation: Poynting

vector distribution for an infinitely narrow crack of 75% penetration.

20 40 60 80

upper 27.9 28.1 28.45 29.1 lower 27.9 28.1 28.55 29.2 left 27.9 28.0 28.5 29.1 right 27.9 28.0 28.5 29.2

Clear pipe upper 15.9 16.0 16.2 16.6 lower 15.9 16.0 16.2 16.6 left 15.9 16.0 16.2 16.6 right 15.9 16.0 16.2 16.6

65% depth slot upper 16.9 17.2 17.6 17.6 lower 16.9 17.1 17.4 17.6 left 16.9 17.2 17.5 17.8 right 16.9 17.2 17.5 17.6

No pipe

30

25 X

t m 0 20

Y--

0 U 3

?= f D

2 15

I 10

10 ' 0 2 4 6 a

Crack Depth/Wall Thickness Fig. 9. Comparison of calculated results for example 1 with preliminary

experimental measurements.

sumed on the outer surface. The excitation frequency is 1350 Hz.

The calculated eddy current streamlines from the sec- ond example for a crack with depth equal to 75% of the wall thickness are illustrated in Fig. 6. The calculated dis- tribution of the real part of Poynting vector for cracks with depths of 25% and 75% of wall thickness are shown in Figs. 7 and 8.

PRELIMINARY TEST RESULTS The test sample has exactlythe same parameters as the

calculated example using the A- V formulation. The pipe is enclosed in a long solenoid excited with 400

Hz ac. A Hall probe is used to measure BZ inside the pipe. Four points: upper, lower, left, and right, are measured for each axial position along the solenoid axis. The results are given in Table 111.

A comparison of the calculated results from the first example with the experimentally measured results is given in Fig. 9.

1946 IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, NO. 4 , JULY 1992

t ube wal l eddy current J

t ube wal l flaw eddy c u r r e n t J

(b)

f law

Fig. 10. A crack affects primarily the low eddy current density zone. (a) Eddy current streamlines without defect. (b) Eddy current streamlines per- turbed by groove on inside of tube wall. (c) Eddy current streamlines per- turbed by groove on outside of tube wall.

DISCUSSION OF THE PHYSICAL BEHAVIOR 1) The circumferential extension of a defect indication

is caused by the fact that the field in the interior region of a pipe is a function of the total eddy currents, referring to (1 8), which is continuous circumferentially around the pipe. Even though the disturbance may be highly local- ized, as with a thin axial crack, the perturbation affects the total eddy currents circulating circumferentially around the pipe. In the ideal infinitely long crack case assumed, according to Maxwell-Ampere law, the interior magnetic field should be uniform everywhere. In actual- ity, for defects of finite axial extent the perturbations per- sist in the circumferential dimension for several times their extent in the axial direction.

2) As the signal variation due to an axial crack is caused by the total eddy current change in the pipe wall, a shal- low, nonpenetrating crack produces only a small localized change in the eddy current path causing a relatively insig- nificant change in the total path. As the crack depth ap- proaches and finally penetrates the wall, it completely blocks the eddy current path causing a rapid increase in signal response.

3) In the remote field region the skin effect phenome- non causes the largest eddy current density magnitude at the outer wall surface with an approximately exponential decrease with depth. While the loss of cross section due to a crack on the inside of the wall affects primarily the low eddy current density zone (see Fig. 10(a) and 10(b)), for a crack on the outside of the wall, the concentration of the eddy current is under the root of the axial crack and the cross sectional loss due to the crack affects the low eddy current density zone also (Fig. lO(c)).

Thus, a crack always gives minimum influence on the total eddy current value, no matter whether it is on the outer or inner surfaces of the pipe wall. Therefore, outer cracks and inner cracks give similar signal responses.

4) Although even an infinitely thin crack forces the eddy current to divert under the crack root, crack width affects the circumferential length of the restricted cross section under the crack root and the total circumferential impedance of the eddy current path. Therefore, axial crack responses are functions of crack width.

OBSERVATIONS 1) It is noted that the material properties, p and a, and

frequency, U , determine the second term of (13) as a sin- gle factor, wpa, which is inversely proportional to the square of the skin depth. Therefore, in the ideal case, the signal response of an axially aligned crack is basically a function of crack depth to skin depth ratio, independent of what kind of materials, ferrous or nonferrous, the pipe is made of.

In practical situations, when there are some deviations from the above assumptions, there might be some minor differences between the responses from the two kinds of materials.

2) Axially aligned cracks have a localized effect on the eddy currents in the pipe wall. As the signal response rep- resents an average over the entire length of the eddy cur- rent streamline, this signal must be closely dependent on the ratio of the crack depth to pipe diameter. The greater this ratio, the greater should be the signal response.

CONCLUSION 1) Two simplified mathematical models for 2-D mod-

elling of axially aligned RFEC crack responses have been introduced. They are based on a simplified physical model of the problem with some assumptions taken from prac- tical RFEC field distribution characteristics.

The calculated results are in fairly good agreement with the preliminary experimental results provided in this pa- per.

2) A hypothesis explaining the RFEC signal response behavior of axially aligned cracks has been presented. It is based on the simplified physical model and Maxwell’s laws.

The model explains some of the basic characteristics of the responses and the authors make a few observations which are subject to further verification.

REFERENCES

[ l ] T. R. Schmidt, “Remote field eddy current inspection technique,” Materials Evaluation, vol. 42, no. 2, pp. 225-230, Feb. 1984.

[2] W. Lord, “Finite element model for the remote field eddy current ef- fect,” PR 179-520, American Gas Association, Sept. 1986.

[3] D. L. Atherton, S . Sullivan, and M. Daly, “A remote-field eddy cur- rent tool for inspecting nuclear reactor pressure tubes,” Er. J . Non- destructive Testing, vol. 30, no. 1, pp. 22-28, Jan. 1988.

[4] D. L. Atherton, et a l . , “The application of finite element calculations to the remote field inspection technique,” Mater. Eval., vol. 45, no. 9, pp. 1083-1086, Sept. 1987.

[5] W. Lord, Y. S. Sun, et a l . , “Physics of remote field eddy current effect,” Review of Progress in Quantitative Nondestructive Evalua- tion, D. 0. Thompson and D. E. Chimenti (eds.), Vol. 7A, Plenum, New York, pp. 165-172, 1988.

[6] Y. S. Sun, “A finite element study of diffusion energy flow in low-

SUN et al.: FINITE ELEMENT MODELLING OF REMOTE FIELD EDDY CURRENT RESPONSES 1947

frequency eddy current phenomena,” Mater. Evul., vol. 47, no. 1, pp. 87-92, Jan. 1989.

[7] D. L. Atherton, 0. Klink, and T. R. Schmidt, “Remote field current responses to axial and circumferential slots in ferromagnetic pipe,” Mater. Eval., vol. 49, no. 3, pp. 356-360, 1991.

[8] C. J. Carpenter, “Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power fre- quencies,” Proc. ZEE, vol. 124, no. 11, pp. 1026-1034, Nov. 1977.

Yu Shi Sun (SM’88) graduated from the Harbin Institute of Technology of China in 1957.

He is a professor in the Department of Automatic Control at Nanjing Aeronautical Institute, Nanjing (NAI), People’s Republic of China. He worked as a faculty member of HIT in the Department of Electrical Engi- neering until 1973. He spent the years 1986-88 with the NDE Research Group at Colorado University as a vsiting faculty member. He is currently a visiting professor at Queen’s University, Ontario, Canada working with the Applied Magnetics Group in the Department of Physics.

He Yun Lin received the B.S. and M.S. degrees from NAI in 1985 and 1989, respectively.

He is a Ph.D. student in the Department of Automatic Control at NAI working on 3-dimensional formulation and modelling for eddy current NDE problems. He has presented several papers at international conferences and has also published in lEEE Transactions on Magnetics.

Mei Juan Chen received the B.S., M.S., and Ph.D. degrees from Xi’an Jaotong University of China in 1984, 1987, and 1989, respectively.

She is associate professor and vice-head of the Electrical Engineering Teaching and Research Office, Department of Automatic Control, NAI. Her major research is on numerical analysis of electromagnetic field prob- lems. Currently, she is working on modelling the pulse response of RFEC devices, T formulation, for axial crack modelling and also on wavelike behavior of eddy current phenomena.

Chun De Wang received the B.S. degree from NAI in 1991.

the A-V formulation. His senior project was on 3-dimensional modelling of axial cracks using

Xiao Bing Wu received the B.S. degree from NAI in 1990. She is an M.S. degree student in the Department of Automatic Control,

NAI, working on signal processing and pattern recognition for eddy current NDE problems. Her senior project was on 3-dimensional formulations of eddy current NDE problems.

David Atherton received the B.A. and M.A. degrees from the University of Cambridge, Cambridge, England.

He came to Canada in 1959 and joined Ferranti Packard Electric in To- ronto becoming Head of Superconductivity Research. He moved to Queen’s University in 1971 where he is Professor of Physics and leads the Applied Magnetics Research Group. He has published more than 160 papers, prin- cipally on applied superconductivity and magnetics. His current research is concentrated on magnetic inspection techniques for pipelines, the remote field eddy current technique, ferromagnetic hysteresis theory and the ef- fects of stress on the magnetic properties of steel. He has acted as specialist consultant on magnetics to many international companies.

Thomas R. Schmidt received the M.S. degree in mechanical engineering from the University of WI, Madison, in 1950.

He retired from Shell Development, Houston, TX as Senior Staff Re- search Engineer in 1987. He is currently Adjunct Professor of Physics in the Applied Magnetics Group at Queen’s University, continuing his work with the remote field technique.

Professor Schmidt is a registered Professional Engineer in the State of Texas.