Analysis of eddy currents in a bar containing an embedded defect

Analysis of eddy currents in a bar containing an embedded defect

J.W. Liaw

, S.L. Chu

, C.S. Yeh

, M.K. Kuo

Materials Research Laboratories, Industrial Technology Research Institute, Bldg. 77, 195-5 Chung Hsing Rd., Section 4 Chutung, Hsinchu, Taiwan, ROC b

Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, ROC Received 25 October 1997; received in revised form 7 May 1998; accepted 14 July 1998

Abstract

The eddy current distribution is studied in a metal bar containing an embedded defect inserted within an encircling coil. Furthermore, the impedance change of the coil regarding the characteristics of the defect is determined. The defects investigated are nonmetallic inclusions and embedded cracks which may occur during the manufacturing processes of bars. To simulate these problems, a two-dimensional transverse-electric model is proposed, and then a set of coupled surface integral equations are formulated systematically. Since the magnetic field is unknown along the boundary of the defect, an additional boundary condition derived from Maxwell’s equations is used. Using the boundary element method (BEM), these integral equations are solved in terms of nodal unknowns-current density. After the current distribution is obtained, the impedance changes of the coil caused by the defects are calculated vs. the configuration of defect for various frequencies. If the inclusion is circular and located at the center of the bar, good agreements are found by comparing the BEM solutions with the analytical ones. An auxiliary surface integral equation is also derived to further determine the currents on both sides of the crack.䉷 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Eddy currents; Cracks; Inclusion; Boundary element method

1. Introduction

Eddy current testing (ECT) has been widely used for product evaluation in steel industries. One important appli-cation is to detect nondestructively the surface and sub-surface defects of steel bar and wire by ECT [1]. The defects could be cracks or nonmetallic inclusions, e.g. oxide. As we know, bars and wires are used as raw material for the manu-facture of various mechanical parts, such as nuts, screws, bolts, shafts, etc. Therefore, these defects are not tolerated, when their lengths exceed a specific value. If the bars are ferromagnetic materials, such as carbon-steel, they always need to be magnetized to saturation to reduce the effect of permeability variation during the off-line inspection. In the on-line inspection of hot rolling, since the temperature of the bar is above the Curie temperature, the texture of carbon-steel is without ferromagnetic property. Therefore we do not consider the hysteresis behavior in this paper for actual application. Another application of the eddy current is using a coil to induce eddy current to generate Joule heat [2,3] in a billet. Due to the presence of defects, the current distribution changes, and the temperature distri-bution on the billet’s surface deviates from normal. Thus, using infra-red sensors to scan the billet, one can detect and locate the surface and sub-surface defects.

The principle of ECT is that a flaw in a conductor, as an

obstacle for eddy currents, perturbs the distribution of current in the conductor and the electromagnetic field in space so as to change the impedance of the coil. This means we can evaluate defects in metals according to impe-dance changes whose values are related to the depth, length and orientation of the defect [4–10]. Hower et al. [11] used the integral equation of magnetic and electric field to solve the three-dimensional embedded defects problem, but the essential limitation is the low frequency assumption. Beiss-ner [12] studied the effect of flaws on eddy currents using the boundary element method (BEM). Kahn [13] used BEM to investigate the current distribution in a half-plane with an inclined surface crack. Bowler [14] solved a boundary inte-gral equation in terms of current dipoles to analyze a half-space with a sub-surface crack. In ECT, the probes, encirc-ling coils, are usually used to detect the surface and sub-surface defects of metal bars at a fixed frequency. Dodd and Deeds [15] gave a close-form solution for the interaction between an encircling coil and a bar, an axial-symmetric problem. Forster [1] used a mercury model, a tube full of mercury and encircled by a coil, to measure experimentally the impedance change of the coil by a crack. The corre-sponding problem generally is a three-dimensional one, due to the configuration of coil, crack, and bar. However if we assume the axial lengths of solenoid coil, bar, and longitudinal defect all are infinitely long, and their axial 0963-8695/99/$ - see front matter䉷 1999 Elsevier Science Ltd. All rights reserved.

orientations are parallel to the z direction, the problem is reduced to a two-dimensional, transverse-electric one. For this problem, Spal and Kahn [16] used an analytical method to study a cylinder containing a vertical surface crack. Later Kahn and Spal [17] formulated an integral equation which is solved numerically by BEM. Chari and Kincaid [18] also studied this problem using the finite element method (FEM). After that, Brudar [19] employed the finite difference method (FDM) to analyze this problem.

In this paper, we study the behavior of eddy currents in a bar with an embedded defect, and utilize the results to calculate the impedance change of a coil, an essential parameter of ECT. The defects investigated in this multiply-connected problem are nonmetallic inclusions and embedded cracks which may occur during the manufacturing processes. This induction problem is

formulated by a set of surface integral equations solved by BEM. The significant difficulty encountered is that

the magnetic field inside the embedded flaw is

unknown, and is proved to be uniform. Hence an addi-tional boundary condition along the defect is derived. The embedded crack investigated is assumed to be closed, i.e. there is no crack opening displacement. This assumption is reasonable, because all the voids will collapse to seams during the rolling of bars. After the eddy current distributions on the surface of the conductor as well as the flaw are obtained, the impedance change of the coil caused by the flaw is determined versus the depth, length, and orientation of flaw for various frequencies. If the shape of the inclu-sion is circular and located at the center of the bar, we provide analytical solutions to compare with BEM solu-tions, and good agreement is found. In addition, an auxiliary surface integral equation is derived to deter-mine the currents on both sides of the crack. To reduce the number of parameters for each problem, all para-meters and variables of numerical results are presented in non-dimensional form.

2. Integral equations for TE mode

In this section, the integral representations of

magnetic field and current density in conductors as well as the integral equations of the surface components

are derived systematically. Conductors with an

embedded flaw, inclusion and crack, are studied.

Consider a metal bar with an arbitrary cross-section placed within a solenoid-type encircling coil. The axial lengths of bar, flaw and coil are infinitely long, and their axial orientations are parallel to the z direction. The cross-sectional views with different flaws are shown in Fig. 1(a,b). These are two-dimensional transverse-elec-tric (TE) mode problems, because the magnetic field is

Hˆ Hzez, where ez is the unit vector of z direction, and

all functions are independent of z. In this paper, we assume that a sinusoidal current with constant amplitude I0 is imposed on the coil to induce an eddy current in the metal bar, and that all magnetic flux is confined within the coil. Hence, the interrogating magnetic strength field inside the coil but outside the conductor is uniform, Hzˆ nI0, where n

is the winding turns per unit length of coil. For simplicity, the common time harmonic factor exp(jvt) is omitted in the following equations. It is assumed that the material proper-ties of the conductor are linear, isotropic, and homogeneous

with constant permeability m and constant electrical

conductivitys. The current density J in conductors obeys Ohm’s law, JˆsE, where E is the electric field, so that

7× H ˆ (s ⫹ jv10)E. Here,10is the dielectric constant of metal. Furthermore, we assume there is no contribution of charge density in conductors. As a result, from Maxwell’s equations, the integral representations of E and magnetic Fig. 1. (a) The cross-sectional view of a bar, with a nonmetallic inclusion,

placed within an encircling coil; (b) Cross-sectional view of a bar, with an embedded crack, placed within an encircling coil (coordinatej僆 [0,d]).

field H in the conductor are given by Hˆ ⫺…s⫹ jv10† Z 2V…n × E†G ds⫺ Z 2V…n × H†7G ds ⫺Z 2V…n × H† ×7G ds; …1† Eˆ jvmZ 2V…n × H†G ds⫺ Z 2V…n × E†7G ds ⫺Z 2V…n × E† ×7G ds; …2†

which are Stratton–Chu formula [20], where no volume integral exists. V represents the region of the conductor

and2V denotes its boundary. In the above equations, n is

the unit normal vector out of the conducting medium. Green’s function G, the singular solution of Helmholtz equation, satisfies

72

G⫹ …v210m⫺jvsm†G ˆ ⫺d…x⫺x0†; …3†

whered (x⫺x0) denotes the Dirac delta function. For the two-dimensional problems with an interior region, Green’s function is given by

G…x; x0† ˆ ⫺1

4Y0…k兩x⫺x

0_{兩†; …4†}

where Y0is the Bessel function of the second kind of order zero. Since, for a quasi-static problem, the displacement current whose order is much less than the order of eddy current can be neglected, the complex wave number k is given by

kˆ 1⫺j

d ; …5†

in terms of the eddy current penetration depthd

dˆ 1=ppsmf;

where fˆv/2p. Moreover, for this 2-D TE-mode, Eq. (1) is reduced to the integral representation for Hzas

Hz…x† ˆ ⫺ Z 2VHzn…x 0_†·70_{G…x; x}0_{† dl}0_⫺Z 2VJsG…x; x 0_{† dl}0_; x僆 V; …6† where the displacement current is neglected and the relation, ⫺(ez× n) ×70ˆ n·70, is used. The 2-D gradient is defined

as70ˆ ex2/2x0⫹ ey2/2y0. If the boundary is insulated, since

no normal component of the current is on boundary, the current density of the surface is Jsˆ sEt. Here, Etis the

tangential component of electric field on the surface, namely Etˆ E·t. The tangential vector, t, is the unit vector along the

surface of cross-section of bar, tˆ ez× n. Since, for

TE-mode problem Hzsatisfies scalar Helmholtz equation, Eq.

(6) can also be directly derived by Green’s theorem [17], with2Hz/2n ˆ ⫺Js, x僆 2V.

Since the surface 2V allows no normal component of

current to pass through, the integral representation of elec-tric field for TE-mode is derived from Eq. (2) as

E…x† ˆ ⫺ Z 2VjvmHzG dl 0_⫺Z 2VEtez×7 0_{G dl}0_{; x 僆 V;} …7† where n × E ˆ 0, n × H ˆ Hzt, (n × E) ×70G ˆ Etez

× 70_{G, are used. Multiplying Eq. (7) by s, the integral}

representation for current density is obtained

J…x† ˆ ⫺ Z 2VjvmsHzG dl 0_⫺Z 2VJsez×7 0_{G dl}0_{; x 僆 V:} …8† To study the effect of embedded flaws, nonmetallic inclu-sions and longitudinal cracks, the integral equations for a conductor containing a flaw are further derived. In particu-lar, for the conductors with a closed crack, the integral equations will reduce to special ones, due to the degenerated geometry. Since the cross-sections of bars with an embedded flaw are multiply-connected domains, one extra condition is also required for the flaw, an inclusion or a crack, as illustrated in the following.

2.1. Bar with a nonmetallic inclusion

The boundary of a bar with a nonmetallic inclusion or void is2V ˆ 2V1傼 2V2, where 2V1denotes the circum-ference of the bar, and2V2the boundary of the inclusion. The cross-sectional view of the bar is depicted in Fig. 1(a). If the boundary is smooth, as x approaches the boundary, the integral equation of magnetic field, from Eq. (6), is

1 2Hz…x† ˆ ⫺ Z 2V1傼2V2 Hzn…x0†·70G dl0⫺ Z 2V1傼2V2 JsG dl0; x僆2V1傼2V2: …9† The first integral of the right-hand side of Eq. (9) takes the Cauchy principal value. The tangential direction of the outer surface2V1follows the counterclockwise direction, and that of the inner surface2V2follows the clockwise direction. As

x approaches the surface, taking the inner product of Eq. (8)

with t, we obtain the surface integral equation for Js, after

some manipulation 1 2Js…x† ˆ ⫺jv Z 2V1傼2V2 smHzGn…x†·n…x0† dl0 ⫺Z 2V1傼2V2 Jsn…x†·70G dl0; x僆2V1傼2V2; …10†

where the relation t·(ez×70)ˆ n·70is used. Eq. (9) is the

Fredholm integral equations of the first kind for unknown Js,

and is that of the second kind for Hz. While Eq. (10) is that of

numerical computations, since Hzis given on2V1, either Eq.

(9) or Eq. (10) can be chosen as the governing equation of Js.

Therefore Eq. (10) is equivalent to Eq. (9) for calculating Js

as x 僆 2V1傼 2V2. Subsequently, we should discuss the

magnetic field in the inclusion. Because the nonmetallic inclusion or void is not a conductor, no current can pass through these defects. Since the displacement current is neglected, the magnetic field in the inclusion satisfies

7 × H ˆ 0, which leads to 2Hz/2x ˆ 0, 2Hz/

2y ˆ 0. Therefore the magnetic field in the inclusion is easily proven a constant, denoted by Hz…2†. However, Eqs. (9) and (10), for a conductor, cannot provide sufficient con-ditions to determine this constant magnetic field within the inclusion. To overcome this problem, we derive an addi-tional condition using the integral form of Maxwell’s equation,7 × E ˆ ⫺jvB, for the inclusion

⫺Z

2V2

Jsdlˆ ⫺jvsmomr2A2Hz…2†; …11†

where A2; mr2; H…2†z correspond to the cross-sectional area of the inclusion, its relative permeability, and the constant magnetic field within it, respectively. Physically, Eq. (11) states that the line integral of current around the flaw is proportional to the net magnetic flux through the area of the flaw perpendicularly. This additional condition can be regarded as the governing equation of the nonmetallic inclu-sion. With the aid of Eq. (11), either Eq. (9) or Eq. (10) is chosen for solving Js; Hz…2†. After the distributions of

Js; H…2†z are determined, the fields Hz, E, and J in the

conductor can be calculated using the integral representa-tion of Eqs. (6–8), respectively.

2.2. Bar with an embedded crack

For a bar with a longitudinal embedded crack, its bound-ary is2V ˆ 2V1傼 2Vc. The embedded crack is assumed to

be a mathematical line submerged in a bar and its boundary is2Vcˆ2V⫹c 傼2V⫺c, as shown in Fig. 1(b), i.e. the crack is closed without crack opening displacement. We can regard this crack as a squeezed void. That is to say the upper surface 2V⫹c coincides with the lower surface 2V⫺c. Moreover, the crack is assumed to be an insulator which allows no current to pass through it, i.e. J·n^ ˆ 0. As x approaches the surface, the integral equation of magnetic field is derived from Eq. (6) as

1 2Hz…x† ˆ ⫺ Z 2V1 Hzn…x0†·70G dl0⫺ Z 2V1 JsG dl0 ⫺Z 2V⫹_c ‰JsŠG dl 0_; x僆2V1; …12a† Hz…x† ˆ ⫺ Z 2V1 Hzn…x0†·70G dl0⫺ Z 2V1 JsG dl0 ⫺Z 2V⫹_c ‰JsŠG dl 0_; x僆2Vc; …12b†

where the discontinuity of current is ‰Js…x0†Š

ˆ J⫹

s …x0† ⫹ J⫺s …x0† ˆs…E⫹·t⫹⫹ E⫺·t⫺†; x0僆2Vc. The superscripts⫹ , and ⫺ denote the upper and the lower surface of the crack, respectively, and t⫺ˆ ⫺t⫹, n⫺ˆ ⫺n⫹. We can also derive the integral equation of current from Eq. (8) as follows 1 2Js…x† ˆ ⫺jv Z 2V1 smHzGn…x†·n…x0† dl0 ⫺Z 2V1 Jsn…x†·70G dl0⫺ Z 2V⫹ c ‰JsŠn…x†·70G dl0; x僆2V1; …13a† 1 2hJs…x†i ˆ ⫺jv Z 2V1 smHzGn⫹…x†·n…x0† dl0 ⫺Z 2V1 Jsn⫹…x†·70G dl0⫺ Z 2V⫹ c ‰JsŠn⫹…x†·70G dl0; x僆2Vc; …13b† where Jh i ˆ Js s⫹⫺Js⫺. For numerical computation, Eq. (13a) is equivalent to Eq. (12a) as x僆 2V1. Regardless of x僆

2V⫺c or x僆2V⫹c, Eqs. (12b) and (13b) are the only forms which we have. In general,2Vcis an arc-curve. If the crack

is a straight line, the third term of the right-hand side of Eq. (13b) vanishes, due to n⫹⬜ x⫺x0, as x,x0僆 2Vc. Moreover,

if2V1is a circle and the crack is a straight line withuˆ 90⬚, the current distribution in the bar is symmetric with respect to the crack, leading to具Js典 ˆ 0.

Subsequently, an addition condition for determining the magnetic field in the crack is derived. First the crack is assumed opened, i.e. the area of crack is not equal to zero. Since this crack can be regarded as a squeezed void, the same as a nonmetallic inclusion, the magnetic field inside the opened crack is also proven a constant, denoted by H…c†z . Furthermore, we assume the magnetic field inside the crack remains bounded, as the area of crack approaches zero to be closed. In the following, we will show the numerical results confirm this assumption that Hz…c†is finite. As a consequence, the additional condition (Eq. 11) is degenerated to

Z

2V⫹_c ‰JsŠ dl ˆ 0; …14†

due to no crack opening displacement. The above equation illustrates that the line integral of current flow around the

embedded crack is equal to zero, i.e. there is no net magnetic flux through it. The additional condition (Eq. 14) can be regarded as the governing equation for the embedded crack. Although this condition (Eq. 14) does not include the magnetic field, Hz…c†, we can use it with the governing equations of the conductor to solve the magnetic field and the current around the crack. Therefore, either Eq. (12a) or Eq. (13a) is chosen to combine with Eqs. (12b) and (14) to solve the surface current density Json2V1and‰JsŠ; H…c†z on

2Vc. Subsequently, if we want to know the current density

on each side of crack in detail, Eq. (13b) supplies an auxili-ary integral equation. Substituting Js; ‰JsŠ; Hz…c† into Eq. (13b), the value具Js典 on 2Vcis calculated. Then the values

of Js⫹; J⫺s can straightforwardly be gained from [Js],具Js典.

3. Impedance of the coil

By the electromagnetic interaction between coil, conduc-tor, and flaw, the distribution of eddy currents will affect the input impedance of the coil at a given frequency. Therefore, once the distribution of surface current is determined, one can calculate the coil’s impedance, which is the measured parameter in ECT, for presenting the inductive behavior between the coil and the medium. Since an alternating current with constant amplitude is applied to the coil, the impedance value Z per unit length of the coil with embedded flaw, by definition, is expressed as

Zˆ V I ˆ ⫺n I Z 2Vo E·dl …15† ˆ ⫺n I ⫺jv Z Vo⫺ V1 B·da⫹Z 2V1 E·dl   ˆ ⫺n Io ⫺jvmonIo…1⫺h†A ⫹ 1 s Z 2V1 Jsdl   ;

where the fill factorh is the ratio of the cross-sectional area of bar to that of coil, namely A. Ifh equals unity, the first term of the above equation vanishes. The region V0, V1, V0⫺V1are the cross-sections of coil, bar, and air, respec-tively. The line integral along2V1 is in the counterclock-wise sense. We define the impedance change as the difference between impedance in the presence and the absence of a flaw. Consequently, the impedance change

DZ ˆ Z⫺Zocaused by the embedded flaw is given by

DZ ˆ ⫺n

sIo

Z 2V1

DJsdl; …16†

whereDJsˆ Js⫺Jso. Here, Zo, Jsocorrespond to the

impe-dance and current density on the surface of an unflawed bar. The above equation is available for any kind of embedded defect. However the distribution of eddy current depends on the position and shape of the flaw. Eq. (16) states that only the change of current,DJs, on the circumference of the bar

contributes to the impedance change. There is a little

difference with a surface crack problem, in which DJsas

well as the additional current, [Js], on the crack surface

contribute to the impedance change [13]. According to Lorentz reciprocity theorem of two-dimensions, we have Z

2V E0× H⫺E × H0

ÿ

·n dlˆ 0; …17†

where E0, H0 are the electric and magnetic field of the unflawed bar, and E, H are that of the flawed bar. The

boundary 2V is 2V ˆ 2V1 傼 2V2 for an inclusion, or

2V ˆ 2V1傼 2Vcfor an embedded crack. Using Eq. (17),

Eq. (16) can be expressed in another form for the inclusion problem DZ ˆ 1 sI2 o Z 2V2 JsHzo⫺…sEo·t†Hz…2†   dl: …18†

Eq. (18) had already been obtained by Auld et al. [21]. For the embedded closed crack problem, Eq. (18) becomes

DZ ˆ 1 sIo2 Z 2V⫹ c ‰JsŠHzodl: …19†

In fact, real measurement systems of ECT apply a source with constant amplitude sinusoidal voltage instead of current to the coil. However, the impedance of the coil, a function of frequency, is an intrinsic property regardless of the exciting method. Therefore the impedance change calculated by Eq. (16) can be utilized to simulate the real measurement.

4. Dimensionless treatment

The above equations are available for arbitrary shape of inclusion and any curve of crack. For simplicity, we only consider the circular inclusion and straight line crack in the following. Bars with circular cross-sections are considered. In order to present the problems clearly, all terms are reduced to the non-dimensional forms, denoted by a super-script ‘*’, as follows fⴱˆ f =fg; dⴱˆ d=R; dⴱpˆ dp=R; Hzⴱˆ Hz=nIo; J_sⴱˆ J_sR=nI_o; Zⴱ ˆ Z=vLo; xⴱˆ x=R;

where the limit frequency is fgˆ (2psmR2)⫺1[1]. Here, R

is the radius of the bar, d corresponds to the diameter of circular inclusion or the length of embedded crack, as

depicted in Fig. 1(a,b) and dpis the depth of inclusion’s or

crack’s tip from the surface. The self-inductance of the coil of per unit length in air without bar is L_oˆm_on2pR2_o, where Rois the radius of the coil’s cross-section. According to the

above expressions, f* can also be written as f* ˆ 2(R/d)2, and the ratio, d/d, is expressed as d=dˆ dⴱpfⴱ=2. The non-dimensional impedance change in terms of the fill factor

h ˆ (R/Ro)2and the relative permeability mrof the metal

bar is expressed as DZⴱ_{ˆ ⫺hm} r Z 2V1 DJⴱ s dlⴱ=pfⴱ: …20† By this treatment, the number of parameters for each problem is reduced to minimum. In calculating Eq. (20), the current density Jsⴱ is determined by BEM, and Js0ⴱ of a

flawless circular bar is given by an analytical solution [1]

Jⴱs0ˆaJ1…ar=R†=J0…a†; …21†

whereaˆ …1⫺j†pfⴱ=2.

In Eq. (21), J0, J1are the Bessel function of the first kind of order zero and order one. According to Eq. (15), the impedance of the coil, with the flawless bar, is written as Z0ⴱˆ j…1⫺h†⫺hmr

2aJ1…a† fⴱJ0…a†

: …22†

5. Numerical results and discussion

The numerical method, BEM, is used to discretize the geometric boundary and physical unknowns for solving these integral equations. In the discretization, the boundary is divided into several segments, and each segment is an element which is a three-node, isoparameter, quadratic one for interpolation. The criterion for discretizing the boundary is that these nodes are distributed on the boundary as evenly as possible. Since the procedure of BEM is a standard one [22], no attempt is made to mention it in detail. For the inclusion, it is easy to deal with the numerical calcu-lation. For the embedded crack, since there exist two crack tips, the behavior of eddy currents at these places can be described using special elements possessing a weak singu-larity [23], 1=pr, for the interpolation and extrapolation of BEM. When the collocation point x coincides with each node of discretization, one corresponding integral equation in the above is utilized. After the discretization and the integration are done, a set of algebraic equations are obtained in terms of the nodal values Jsof 2V1 and [Js],

Hzof 2Vc as unknowns for cracked problems, and can be

easily solved by a numerical method, e.g. LU decomposi-tion. When obtaining the surface current densities, we substitute them into the integral representation of impe-dance (Eq. 16), and use the above procedure of discretiza-tion and integradiscretiza-tion to determine the impedance value of the coil.

5.1. Circular bar with a nonmetallic inclusion

In the following, we only present the numerical results for the circular inclusion withmr2ˆ 1. To solve this problem,

some collocation points are chosen on the boundary

2V ˆ 2V1傼 2V2, and these points also correspond to the

nodes of discretization. No matter if the collocation points are on2V1or on2V2, either Eq. (9) or Eq. (10) can be used to combine with Eq. (11) as the governing equations in terms of Jsⴱ…x†; x 僆2V1傼2V2. Fig. 2(a) shows the

distri-bution of current density on the circumference of a bar with a circular inclusion for f*ˆ 5, d* ˆ 0.2, dpⴱˆ 0:1, and Fig. 2(b) for the distribution on the boundary of inclusion. In Fig. 2, the corresponding result of normalized non-dimensional

impedance change is DZ*/hmr, ˆ (⫺2.403 × 10⫺3,

1.003 × 10⫺2), and the constant magnetic field in the Fig. 2. (a) Distribution of Jⴱs on circumference2V1for f*ˆ 5, d* ˆ 0.2,

dpⴱˆ 0:1. H…2†ⴱz ˆ …0:838; ⫺0:260†, DZ*/hmrˆ (⫺2.403 × 10⫺3, 1.003× 10⫺2). The solid line denotes Re…Jsⴱ†, and the dashed line Im…Jsⴱ† (circular nonmetallic inclusion); (b) Distribution of Jsⴱon the surface of the inclu-sion. The solid line denotes Re…Jsⴱ†, and the dashed line Im…Jⴱs† (circular nonmetallic inclusion).

inclusion is Hz…2†ⴱˆ …0:838; ⫺0:260†. The numerical results for a case with f*ˆ 50, d* ˆ 0.2, dⴱpˆ 0:02 are plotted in Fig. 3(a,b). The current densities almost keep constant along the circumference, but change abruptly in the vicinity of a sub-surface defect, as indicated in Fig. 2(a) and Fig. 3(a). Moreover, the closer the inclusion is to the circumference, the higher the current density is in the gap between them. Generally, a lower frequency is adopted for detecting deeper embedded defects, due to the skin effect of the eddy current. If the circular inclusion is at the center of a circular bar, there is an available analytical solution for this axisym-metric problem. Table 1 lists the results, Jⴱs; Hz…2†ⴱ, of analy-tical and BEM solutions at f*ˆ 5,50 for comparison. From Table 1, one can observe that the results of BEM reasonably agree with analytical solutions. Hence the accuracy of BEM is verified for the bars with a nonmetallic inclusion. Figs. 4 and 5 show the impedance changes of a coil by the circular inclusion for various lengths and depths at

f* ˆ 5,50, respectively.

5.2. Circular bar with an embedded crack

Either Eq. (12a) or Eq. (13a) is used to combine with Eq. (12b) and Eq. (14) or BEM formulation with unknowns— Jsⴱ…x†; x 僆2V1; Jsⴱ…x†

 

; H…c†ⴱ

z x僆2V⫹c. Once the surface currents are gained, substituting them into Eq. (13b) one can further determine Js⫺; Js⫹on the crack surface. Brudar [19] calculated a case, f*ˆ 5, d* ˆ 1/3, dⴱpˆ 1=6; uˆ 90⬚, but only presented the distribution of the magnetic field in the circular bar. For this case, our results for the constant magnetic field at the crack, Hz…c†ⴱˆ …0:7; ⫺0:4†, quite agree with his. In Fig. 6, we depict the current distribution on the circumference and crack surface for f*ˆ 5, d* ˆ 0.2,

dⴱpˆ 0:1; uˆ 90⬚. The results of f* ˆ 50, d* ˆ 0.2,

dⴱpˆ 0:02; uˆ 90⬚ are plotted in Fig. 7. The cases with a perpendicular crack,uˆ 90⬚, are symmetrical problems, so the distributions of surface current on both sides of the crack are the same, i.e. Js⫺ˆ Js⫹. Moreover, when the crack is located at the center of the circular bar, symmetry of current exists not only on the upper and lower side of the crack but also on the right-hand and left-hand side, due to the geometric symmetry. Fig. 8 illustrates this symmetry for f*ˆ 5, d* ˆ 0.6, dⴱpˆ 0:7; uˆ 90⬚. Since there is nearly Fig. 3. (a) Distribution of Jsⴱon circumference2V1for f*ˆ 50, d* ˆ 0.2,

dpⴱˆ 0:02. H…2†ⴱz ˆ …0:692; ⫺0:285†, DZ*/hmrˆ (5.710 × 10⫺3,8.236× 10⫺3). The solid line denotes Re…Jⴱs†, and the dashed line Im…Jsⴱ† (circular nonmetallic inclusion). (b) Distribution of Jsⴱon the surface of the inclu-sion. The solid line denotes Re…Jsⴱ†, and the dashed line Im…Jsⴱ† (circular nonmetallic inclusion).

Table 1

Comparison of analytical and BEM solutions for a circular inclusion at the center of a bar. JsOⴱ: the current density along the circumference of a bar. JsIⴱ: the current density along the circular inclusion

d*ˆ 1.0, dpⴱˆ 0:5 d*ˆ 0.5, dpⴱˆ 0:75

Analytical BEM Analytical BEM

f*ˆ 5 H…2†ⴱz (0.536,⫺0.580) (0.536,⫺0.580) (0.391,⫺0.639) (0.391,⫺0.639)

JsOⴱ (⫺0.936, ⫺1.807) (⫺0.936, ⫺1.807) (⫺0.923, ⫺1.754) (⫺0.923, ⫺1.754)

JsIⴱ (0.725, 0.670) (0.725, 0.669) (0.400, 0.244) (0.400, 0.243)

f*ˆ 50 H…2†ⴱz (⫺8.816E⫺2, ⫺1.597E⫺2) (⫺8.816E⫺2, ⫺1.597E⫺2) (⫺3.331E⫺2, 3.987E⫺2) (⫺3.331E⫺2, 3.987E⫺2)

JsOⴱ (⫺4.446, ⫺5.027) (⫺4.446, ⫺5.027) (⫺4.489, ⫺5.017) (⫺4.489, ⫺5.017)

no perturbation of current density on the circumference, as shown in Fig. 8(a), it implies ECT is not sensitive to a deeper crack, according to Eq. (16). Fig. 9 shows the

asymmetric case, u ˆ 60⬚, for f* ˆ 5, d* ˆ 0.2,

dⴱp ˆ 0:1. In the above cases, the numerical results are consistent with the assumption, H…c†ⴱz is a finite value in a closed crack, leading to Eq. (14). From these figures, we also observe that the surface current along the circumference almost remains constant until near the crack tip, then changes abruptly. In addition, the current density approaches to infinite at the crack tips, in which the asymptotic behavior is 1=pr [23], as indicated in these figures. Here, we have to mention that the

cross-section of metal with contour, 2V ˆ 2V1 傼

2Vc, is simply-connected for a surface-breaking crack

problem [16,17], while is multiply-connected for an embedded crack problem. This leads to some significant differences of current distribution in the two problems.

For the embedded crack, there are two crack tips, so current can flow around the crack. Since the crack is an obstacle for eddy current, there is a stagnation zone, where the amplitude of current density nearly vanishes, in the middle of the embedded crack, as shown in Fig. 6(b) and Fig. 7(b). When the depth of a sub-surface crack decreases, the current density within the narrow gap, between the crack tip and the circumference, increases. Whereas, if the depth of the crack is zero, i.e. the surface-crack problem, the crack connects with the circumference, and there is only one crack tip for this simple-connected problem. Therefore, on the two sides of the mouth of crack, there are two corners which are dead zones for current, so current cannot flow through the mouth, and only along the crack from one side to the other, passing by the crack tip Fig. 4. Impedance change of the coilDZ*/hmrvs. various d* and dⴱpfor

f*ˆ 5 (circular nonmetallic inclusion).

Fig. 5. Impedance change of the coilDZ*/hmrvs. various d* and dⴱpfor f*ˆ 50 (circular nonmetallic inclusion).

Fig. 6. (a) Distribution of Jsⴱon circumference2V1for f*ˆ 5, d* ˆ 0.2, dⴱpˆ 0:1;uˆ 90⬚. Hz…2†ⴱˆ …0:821; ⫺0:274†, DZ*/hmrˆ (⫺9.984 × 10⫺4, 4.759× 10⫺3). The solid line denotes Re…Jsⴱ†, and the dashed line Im…Jsⴱ† (embedded crack); (b) Distribution of‰JsⴱŠ on the surface of the crack. The solid line denotes Re…‰JsⴱŠ†, and the dashed line Im…‰JsⴱŠ† (embedded crack).

[23]. Another difference is that for the embedded crack the line integral of current along the crack equals zero, Eq. (14), while for surface-breaking crack, it is not equal to zero. Under the same conditions, dⴱ; dⴱp; fⴱ, it is also observed that the influence of a nonmetallic inclusion on the eddy current perturbation is more significant than that of a crack by comparing Fig. 2(a) with Fig. 6(a), and Fig. 3(a) with Fig. 7(a).

After surface currents are obtained by BEM, the normal-ized non-dimensional impedance changes,DZ*/hmr, can be

calculated subsequently by using Eq. (20). The results for the perpendicular crack,u ˆ 90⬚, against the crack length and depth are shown in Figs. 10 and 11 for f* ˆ 5, 15, respectively. The tendency that impedance change is located in the second quadrant, when f* is lower, is observed from these figures.

6. Conclusion

In this paper, we had shown the application of BEM for solving surface integral equations is useful for bars with a sub-surface crack or nonmetallic inclusion. Since the magnetic field, a constant, is unknown in the embedded flaw, we employ a necessary integral equation as the inner boundary condition to combine with the governing integral equations of the bar for solving the current and magnetic field. With the aid of an extra auxiliary surface integral equation, the eddy current distribution around the crack can be described clearly. In addition, we also calculate the impedance changes of the coil in terms of surface current vs. the length, depth and orientation of the embedded flaw. The results of the numerical simulation enable us to select an optimal frequency to inspect bars by encircling-type probes, Fig. 7. (a) Distribution of Jⴱs on circumference2V1for f*ˆ 50, d* ˆ 0.2,

dⴱpˆ 0:02; uˆ 90⬚. H…c†ⴱz ˆ …0:593; ⫺0:289†, DZ*/hmrˆ (2.782 × 10⫺3, 2.780× 10⫺3). The solid line denotes Re…Jⴱs†, and the dashed line Im…Jsⴱ† (embedded crack); (b) Distribution of‰JsⴱŠ on the surface of the crack. The solid line denotes Re…‰JⴱsŠ†, and the dashed line Im…‰JsⴱŠ† (embedded crack).

Fig. 8. (a) Distribution of Jⴱs on circumference2V1for f*ˆ 5, d* ˆ 0.6, dpⴱˆ 0:7;uˆ 90⬚. Hz…c†ⴱˆ …0:376; ⫺0:645†, DZ*/hmrˆ (3.538 × 10⫺4, 5.957× 10⫺4). The solid line denotes Re…Jsⴱ†, and the dashed line Im…Jsⴱ† (embedded crack); (b) Distribution of‰JⴱsŠ on the surface of the crack. The solid line denotes Re…‰JsⴱŠ†, and the dashed line Im…‰JⴱsŠ† (embedded crack).

and also offer an interpretation about the characteristics of flaws for ECT measurements. We do not intend to extend these problems to complicated cases in this paper. However, all the above formulations can be applied for 2-D multi-flaw problems.

The foregoing method also has a potential for the appli-cation of infra-red scanning billet, in which the coil induces eddy currents to generate heat [3], and the infra-red sensors scan the temperature difference to detect the defects. When the surface current is determined by BEM, one can use the integral representation of current to calculate the current density field, which is a heat source, in the billet. Then combining FEM to calculate the temperature field on the surface of the billet, one can estimate the results of thermo-graphy for the inspection of the billet.

Acknowledgements

This research is sponsored by the Ministry of Economic Affairs, Taiwan, ROC. All the calculations were performed by the alpha computer of the Materials Research Labs., Industrial Technology Research Institute.

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