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Perturbed Magnetic Fields of an Infinite Plate with a Central Crack

F. Qin, Y. Zhang, Y. N. Liu

Abstract: Based on the linearized magnetoelastic theory, analytical solutions o

f the perturbed magnetic field

generated by structural deformation of an infinite ferromagnetic elastic plate containing a central crack were obtained by the Fourier transform method. The results show that the perturbed magnetic field intensity is proportional to the applied tensile stress, and it is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangent intensity component of the perturbed magnetic field close to the crack shows antisymmetric distribution along the crack and inverses its direction sharply at the two faces of the crack, while the normal shows symmetric distribution along the crack and presents singular points at the crack tips.

Key words: magnetoelasticity, crack, perturbed magnetic fields, Fourier transform method

The project supported by the National Natural Science Foundation of China (10472004).

F. Qin(*), Y. Zhang, Y. N. Liu

College of Mechanical Engineering and Applied Electronics Technology,

Beijing University of Technology, Beijing 100124, China

*Email:qfei@bjut.edu.cn

1

1 Introduction

To understand the quantitative relationship between mechanical stress and perturbed magnetic fields of ferromagnetic materials and structures is of great significance for new generation magnetism-base nondestructive test (NDT) technologies and magnetomechanical coupling problems. Based on the Pao-Yeh’s linearized magnetoelastic theory [1], the first author and his coworkers had presented an approach to solve perturbed magnetic fields induced by deformation, and by using of the Fourier transform technique and the variable separation method, magnetic field distortion induced by mechanical stress of a half-plane structure under a point force [2] and an infinite plate with a circle hole [3,4] were analyzed respectively, and the analytical solutions of the perturbed magnetic fields were obtained. With regard to crack problems of ferromagnetic materials and structures, Yeh [5] investigated the induced magnetic fields in an infinite magnetized elastic solid generated by a tension fault, and attempted to predict earthquake according to characteristics of the fields. Shindo et al. [6, 7] investigated effect of a magnetic field on the singular behavior of stresses near a crack tip for various crack geometries. Chun-Bo Lin and Yeh [8] applied the complex variable theory to analyze the magnetoelastic problems of an infinite soft ferromagnetic solid containing a finite plane crack. Liang Wei et al. [9] employed the complex function method and singular integral equations to obtain solutions of a coupling crack problem. Based on the works described above, perturbed magnetic fields of an infinite ferromagnetic plate with an internal crack under external mechanical force were analyzed analytically in this paper, and the main characteristics of the perturbed magnetic fields near the crack were discussed.

2 Interaction Between Deformation and Magnetic Fields

2.1 Rigid Magnetic Fields. Without deformation, the rigid stationary magnetic fields of an isotropic

magnetoelastic solid occupying a spatial domain Ω and subjected to an external magnetic field can be described

2

by a magnetic flux density B, a field intensity H, and a magnetized intensity M in Ω, and B0, H0, and M0=0 in the free space where might be air or vacuum. These magnetic quantities are governed by the Ampere law and the Gauss law, which have forms as

eijk H k , j ? 0 , Bi ,i ? 0

(1)

Where eijk is the permutation symbol and i, j, k=1,2,3 for three-dimensional problems. The “,” in subscripts denotes partial derivative with respect to spatial coordinates, for example, Hk,j=?Hk/?xj. Whether in Ω or in the free space, Eq.(1) must be held. Therefore, the superscripts “0” of the quantities are intentionally omitted for briefness. Constitutive law for the magnetic fields is

M i ? ?H i , Bi ? ?0 ( H i ? M i ) ? ?0 ?r H i , ?r ? 1 ? ?

(2)

Where χ is the magnetic susceptibility of the material, ?0=4π×10-7H/m is the universal constant, and ?r the relative magnetic permeability. On the boundary of Ω, the fields satisfy the following continuity conditions

eijk N j ( H k0 ? H k ) ? 0 ,

Ni ( Bi0 ? Bi ) ? 0

(3)

where Ni are components of a unit vector normal to the boundary. Introducing magnetic scalar potential ? and defining H=???, where ? is the Nabla operator, then Eq. (1) can be transferred into two Laplace equations for the free space and the material space, respectively,

?2? 0 ? 0

,

? 2? ? 0

(4)

2.2 Effect of Magnetic Field on Mechanical Deformation. Deformation of the solid considered in Sec.

2.1 consists of two parts. One is caused by applied mechanical loads and the other by the Maxwell forces from the external magnetic field. More complicated, the deformation of the solid generates perturbed magnetic fields in the material space and in the free space. Similar to the rigid situation discussed in Sec.2.1, the perturbed fields can be described by a magnetic flux density b, a magnetic field intensity h, and a magnetized intensity m in the material space Ω, and

3

b0, h0, and m0=0 in the free space. As the gradient of displacements is assumed to be small, according to the linearlized theory in Ref.[1], the total magnetic fields are superposition of the rigid magnetic fields and the perturbed fields, i.e.

Bitotal ? Bi ? bi , H itotal ? H i ? hi , M itotal ? M i ? mi

(5)

The total magnetic quantities should satisfy Eq.(1), this leads to the governing equations of the perturbed fields

eijk hk , j ? 0

,

bi ,i ? 0

(6)

The equilibrium equation under mechanical loads and magnetic fields is

tij,i ? ?0 (M i H j ,i ? M i h j ,i ? mi H j ,i ) ? fi ? 0

(7)

where tij is the magnetomechanical stress tensor and fj the mechanical body force per unit volume. Neglecting the effect of magnetostriction, the constitutive equations are

tij ?

?0 M i M j ? ? ij ? ?0 ( H j mi ? H i m j ) ?

bi ? ?0 (hi ? mi ) ? ?0 ?r hi

(8) (9)

mi ? ?hi ,

where

? ij ? ?uk , k?ij ? G(ui , j ? u j ,i )

(10)

is the Cauchy stress tensor. λ and G are the Lamèconstants, and ?ij is the Kronecker delta symbol. Substituting Eqs.(8)–(10) into Eq.(7) and using Eqs. (1) and (6) , omitting the mechanical body force, it yields

Gu j ,ii ? (? ? G)ui , ji ? 2?0 ? ( Hi H j ,i ? hi H j ,i ? Hi h j ,i ) ? 0

Considering the condition |bi|/|Bi|<<1 and |hi|/|Hi|<<1, it can be further simplified as

u j ,ii ?

1 2?Bi ui , ji ? h j ,i ? 0 1 ? 2? G?r

(11)

Here, the third term in the left side of the equation represents the effect of the Maxwell forces. It has been shown that for non soft ferromagnetic material when the external magnetic field B is of magnitude of the earth’s magnetic field (about 40 A/m), the effect of the Maxwell forces on displacement can be neglected [2]. Therefore,

4

the third item in Eq.(11) can be neglected and thus Eq.(11) is reduced to the Lamè –Navier’s equation, which is commonly seen in textbooks on theory of elasticity.

2.3 Perturbed Magnetic Fields Induced by Deformation. Under condition of neglecting the effect of

magnetostriction, the boundary conditions which the perturbed magnetic field should satisfy were obtained as followings [4],

eijk N j (hk0 ? hk ) ? eijk N mum, j ( H k0 ? H k ) , Ni (bi0 ? bi ) ? N mum,i ( Bi0 ? Bi )

(12)

Eq.(12) indicates that the displacement gradient um,i plays a key role to initiate the perturbed magnetic field. In other words, the perturbed magnetic field does not arise when the displacement gradient on the boundary is zero. Especially for plane problems boundary condition Eq.(12) can be expressed in a Descartes system as [2]

0 0 0 0 nx (bx ? bx ) ? ny (by ? by ) ? ? x ( Bx ? Bx ) ? ? y ( By ? By )

(13a) (13b)

0 0 0 0 nx (hy ? hy ) ? ny (hx ? hx ) ? ? x ( H y ? H y ) ? ? y ( H x ? H x )

Where

? x ? nxux,x ? nyu y ,x ,

? y ? nxux, y ? nyu y , y

Similar to the rigid magnetic fields in Sec. 2.1, we introduce the perturbed magnetic scalar potentials φ0 and φ for the free space and the material space, respectively. By defining h0 = ??φ0 and h= ??φ, Eq.(6) is reduced to

? 2? 0 ? 0

,

? 2? ? 0

(14)

Eqs (12) and (14) are used to determine the perturbed fields. According to Ref.[4], in summary, there are three main steps to obtain the perturbed magnetic fields. Step 1. Solving Eq.(4) to obtain the rigid magnetic field B and H. Step 2. Solving the Lamè –Navier equation to obtain displacement, u, and its gradient on the boundary. Step 3. Solving Eq.(14) and Eq.(12) to obtain the perturbed magnetic fields, b and h.

5

3 Perturbed Magnetic Fields of an Infinite Plate with a Central Crack

3.1 Solutions of Rigid Magnetic Fields. Fig.1 shows an infinite plate containing a central crack of length

2a. The plate was subjected to tensile stress T and an external magnetic filed specified by magnetic flux density B0. A rectangular Cartesian coordinate system (x, y) is attached to be the center of the crack for reference purposes. The magnetic fields in rigid state can be easily determined as [6]

0 By ? B0 ,

0 Hy ?

B0

?0

,

0 My ? 0

(in the free space) (in the plate)

(15a) (15b)

By ? B0 , H y ?

B0

?0 ?r

, My ?

?B0 ?0 ? r

Besides, all the other components of B, H and M are zero.

3.2 Displacement Solutions. The displacement solution of the problem can be found in many textbooks of

fracture mechanics. Applying complex variable function method, the displacement components of the plate with a crack can be deduced as [10]

2G(u x ? iu y ) ? F (? z 2 ? a 2 ? z 2 ? a 2 ) ? F ( z ? z )

z z ? a2

2

(16)

Here, the complex variable and the complex conjugate function are z=x+iy and z =x?iy, respectively, F=1/2 T, G=E/2(1+ν), E and ν are Young’s modulus and Poisson’s ratio of the plate, κ=(3?ν)/(1+ν)for plane stress, and κ=3?4ν for plane strain. The analytical expression of displacement gradient on the boundary of the crack could be obtained from Eq.(16) as

?u y ?u x ?i ) ? F? ?x ?x z z2 ? a2 ?F z z 2 ? a2 ? F (z ? z) a2 ( z 2 ? a 2 )3

2G (

(17)

In addition,

z ? rei? ,

z ? re ?i? ,

z z ?a

2 2

?

? ? ? ? ? ?? exp?i?? ? 1 2 ?? , 2 ?? r1r2 ?? r

z z ?a

2 2

?

? ? ? ? ? ?? exp?? i?? ? 1 2 ?? 2 ?? r1r2 ? ? r

According to Eq.(17), the displacement gradient of the upper (θ1=π, θ2=0, θ=0 or π, r1=a?x, r2=a+x) and the

6

lower surface (θ1=?π, θ2=0, θ=0 or ?π, r1=a?x, r2=a+x) of the crack can be computed respectively as

? u y ,x ?

? u y ,x

?T (? ? 1) x 2 4G a ? x2 T (? ? 1) x ? 2 4G a ? x2

( x ?a) ( x ?a)

(18a) (18b)

The quantities with superscripts “+” and “?” in Eq. (18) denotes the upper and the lower half-plane along the crack, respectively. Eq.(18) describes the projections of the displacement gradient on the normal of the crack boundary.

3.3 Perturbed Magnetic Fields Induced by Deformation. The perturbed magnetic scalar potentials φ0

and φ are governed by Eq.(14). Substituting Eq.(15) and (18) into Eq.(13), the mixed boundary conditions along the crack (y=0, |x|< a ) in the perturbation state can be expressed as following

0 hx ? hx ? ?u y ,x

?B0 , ?0 ? r

0 by ? by ? 0

Where

?h

and

0 x

? hx

? ? ?h

?

0 x

? hx

?

?

? ?B0 ? ? ? u y ,x ? ?0 ? r ?

? ? ?B0 ? ? ? ? u y ,x ? ? ?0 ? r ? ?

?

? ? , ? ?

?

?b

0 y

? by

? ? ?b

?

0 y

? by

?

?

? 0,

?h ? ? ?h ?

0 ? x

0 ? x

0 0 ? 0 , bx ? bx

? ? ? ?

?

?

?0

So the boundary conditions satisfy

? ? hx? ? hx? ? [u y ,x ? u y ,x ]

?B0 , ?0 ? r

? ? by ? by ? 0

(19)

In terms of the perturbed magnetic potentials, the boundary conditions Eq.(19) can be expressed as

?? ? ?? ? ?B0 T (? ? 1) ? ?? ?x ?x ?0 ?r 2G x a2 ? x2

,

?? ? ?? ? ? ?0 ?y ?y

(20)

We can employ the Fourier transform method to solve the Laplace equation

? 2? ? ? 2? ? ? ?0 , ?x 2 ?y 2 ? 2? ? ? 2? ? ? ?0 ?x 2 ?y 2

For the upper half-plane, taking the Fourier transform with respect to variable x, let

7

~ F ? ? ( x, y) ? ? ? (? , y)

?

?

and use the properties of the Fourier transform operation

F[

? 2? ? ~ ~ ] ? (i? ) 2 ? ? (? , y) ? ?? 2? ? ?x 2 ~ ? 2? ? ? ? 2? ? F [ 2 ] ? 2 F [? ? ] ? ?y ?y ?y 2

Thus the Laplace equation changes into an order differential equation

~ ? ? 2? ? ? ~ ? 2? ? ?0 ?y 2

Its general solution is

~ ? ? (? , y) ? A(? )e?y ? B(? )e??y ~ as y→+∞, φ is limited, hence ? ? (? , y) must be limited. Thus for ?ξ ≠ 0, it can be denoted as [2]

~ ? ? (? , y ) ? C (? )e

?? y

~ where C(ξ) is an arbitrary function of ξ. Hence the reverse Fourier transform of ? ? (? , y) is

? ? ( x, y) ?

Analogically, for ?2φ- =0 (y<0),

1 ?? ~ ? 1 ?? ? ? y ?i?x i?x d? ??? ? (? , y)e d? ? 2π ??? C(? )e 2π

(21)

y→ ?∞ , it can be denoted as

~ ? ? (? , y ) ? D(? )e

? y

where D(ξ) is an arbitrary function of ξ.

? ? ( x, y) ?

1 ?? ~ ? 1 ?? ? y ?i?x i?x ??? ? (? , y)e d? ? 2π ??? D(? )e d? 2π

(22)

Substituting Eqs.(21) and (22) into Eq.(20), the left side of first equation of (20) is

? ?? ? ?? ? ? ? ?x ? ?x ? ? ? ? ? 1 2π

?

y ?0

?

??

??

i? [C (? ) ? D(? )]e i?x d?

(23)

Applying the Fourier transform to its right side, we get

? x F? 2 ? a ? x2 ? ? d ?J 0 (a? )? ? ?iπ ? aJ1 (a? ) ? ? iπ d? ? ?

So its right side is

8

?

?B0 T (? ? 1) 1 ?? ? ? iπ ? aJ1 (a? )ei?x d? ?0 ?r 2G 2π ???

(24)

where, J1 is the first order of the primal Bessel function. Comparing Eqs (23) and (24), we get

C (? ) ? D(? ) ?

For the second part of Eq.(20), its left side is

? ?? ? ?? ? ? ? ?y ? ?y ?

?B0 T (? ? 1) π ? a ? ? ?1 J1 (a? ) ?0 ?r 2G

(25)

? 1 ?? ? ? ? | ? | [C (? ) ? D (? )]ei?x d? ? 2 π ?? ? ? y ?0

Comparing with its right side, zero, leads to

C (? ) ? D(? ) ? 0

Solving Eqs (25) and (26), we get

(26)

C (? ) ?

?B0 T (? ? 1) π ? a ? ? ?1 J1 (a? ) , ?0 ?r 4G

D(? ) ? ?

?B0 T (? ? 1) π ? a ? ? ?1 J1 (a? ) ?0 ?r 4G

According to Bessel function integral formula [11], we have

?

?

0

? ?1 J1 (a? )ei?z d? ? {[a 2 ? (?iz) 2 ]1/ 2 ? iz}

1 a

Substituting C(ξ) into Eq. (21), the potentials are obtained as

? ? ( x, y) ?

?

?? ?B0 T (? ? 1) ? a ? ? ? ?1 J1 (a? )e ?|? | y?i?x d? ?? ?0 ?r 8G

?B0 T (? ? 1) ?0 ?r 8G

? a ? (x ? iy)

2

2

? i( x ? iy) ? a 2 ? (? x ? iy) 2 ? i(? x ? iy)

?

(27)

The intensities of the perturbed magnetic fields of the upper half plate are

? ?? ? ?B0 T (? ? 1) ? ? ( x ? iy ) ? ( x ? iy ) ? ? ?i? ? i? ?x ?0 ? r 8G ? a 2 ? ( x ? iy ) 2 a 2 ? ( x ? iy ) 2 ? ? ?

hx? ?

??

? ? ?B0 T (? ? 1) ? z ?B0 T (? ? 1) ? z ?Re 2 ? ? ?Im 2 ? ?0 ? r 4G ? a ? z 2 ? ?0 ? r 4G ? z ? a2 ? ? ? ? ?

(28a)

? hy ?

? ? i ? ( x ? iy) ?? ?? ? ?B0 T (? ? 1) ?? ? i ? ( x ? iy) ?? ? ? 1? ? ? ? 1?? ?y ?0 ?r 8G ?? a 2 ? ( x ? iy) 2 ? ? a 2 ? ( x ? iy) 2 ?? ? ? ?? ??

? ? ? ?B0 T (? ? 1) ? z ?B0 T (? ? 1) ? z ? 1? ? ? 1? ?Im 2 ?Re 2 ?0 ? r 4G ? ? ?0 ? r 4G ? ? a ? z2 z ? a2 ? ? ? ?

9

(28b)

Where, z denotes the point in the upper half of the plate. Similarly, substituting D(ξ) into Eq.(22), we have

? ? ( x, y) ? ?

?B0 T (? ? 1) ?0 ?r 8G

? a ? (x ? iy) ? i(x ? iy) ?

2 2

a 2 ? ( x ? iy)2 ? i( x ? iy)

?

(29)

Thus intensities of the perturbed magnetic fields of the lower half plate are

hx? ?

?B0 T (? ? 1) ? z ?? Im 2 ?0 ? r 4G ? z ? a2 ?

? ? ? ?

(30a)

? hy ?

? ?B0 T (? ? 1) ? z ? 1? ?Re 2 2 ?0 ?r 4G ? ? z ?a ? ?

(30b)

Where z denotes the point in the lower half plate. To sum up the above arguments, by using of the complex function z to express Eq.(28) and (30), it can be written in a concise form as

hx ?

?B0 T (? ? 1) ? z ?Im 2 ?0 ?r 4G ? z ? a2 ?

? ? ? ?

(31a)

hy ?

? ?B0 T (? ? 1) ? z ? 1? ?Re 2 ?0 ? r 4G ? ? z ? a2 ? ?

(31b)

4 Results and Discussion

Substituting a=1 for half of the crack length, and the results were presented in dimensionless form by factor χB0T(κ+1)/(4μ0μrG), Fig.2 shows the distribution of the perturbed magnetic field intensities components hx? and hy? in the plate respectively, and Fig.3 shows the field distribution near the crack(y=± 0.05). An overview distribution of the perturbed magnetic field near the crack is presented in Fig.2. The tangent magnetic intensity is anti-symmetric along the crack, and inverses its direction sharply at the crack face, and reaches its maximum at the crack tips. The normal magnetic intensity shows symmetric distribution along the crack and presents singular points at two crack tips. Those features can be more clearly and directly observed in Fig.3, in which the perturbed field intensity distribution in planes y=± 0.05 was shown. Fig.4 shows the change

10

of the magnetic intensity along the crack with various distance (y=± 0.05, y=± 0.1, y=± 0.2) from the crack for making them being comparable. It is found that with distance off the crack increasing, the perturbed magnetic field intensities decay rapidly. All of the characteristics show an obvious local feature, although it presents difference distribution patterns between the tangent and the normal intensities. Fig.5 shows the distribution characteristics of the resulted intensity of the perturbed magnetic fields near the crack. Singular points at the crack tips were observed, and intensity became less intense in the central region of the crack. Similarly, with distance off the crack increasing, the perturbed magnetic field intensity decay rapidly. This suggests that it is easier to observe the perturbed magnetic field at the location close to the crack tips.

5

Conclusions

Based on the linearized magnetoelastic theory, following the Fourier transform method, an infinite

ferromagnetic elastic plate containing a central crack was considered under the case of weak external magnetic field, and the perturbed magnetic fields generated by structural deformation under a magnetic field was analyzed . The results show that (1) The perturbed magnetic field intensity component is proportional to the applied tensile stress. (2) The perturbed magnetic field induced by mechanical stress is dominated by the displacement gradient on the boundary of the magnetoelastic solid. (3) As for the magnetic intensity components of the perturbed magnetic field close to the crack, the tangent shows anti-symmetric distribution along the crack and inverses its direction sharply at the two faces of the crack, while the normal shows symmetric distribution along the crack and presents singular points at the crack tips. They both take on stronger local feature.

11

References

[1] Y.H.Pao, C.S.Yeh: A linear theory for soft ferromagnetic elastic solids. International Journal of Engineering Science. 11, 415-436 (1973) [2] Qin Fei, Yan Dongmei, Zhang Xiaofeng: Perturbed Magnetic Fields Generated by Deformation of Structures in Earth Magnetic Field (in Chinese). Chinese Journal of Theoretical and Applied Mechanics. 38, 799-806 (2006) [3] Qin Fei, Yan Dong-mei, Zhang Yang: Analytical solution of the perturbed magnetic field induced in an infinite plate by a circinal hole under tension (in Chinese). Acta Mechanica Solida Sinica. 28, 281-286 (2007) [4] Fei Qin, Dongmei Yan: Analytical Solution of the Perturbed Magnetic Fields of Plates Under Tensile Stress. ASME: Journal of Applied Mechanics. 75, 0310041-6 (2008) DOI: 10.1115/1.2870266 [5] C.S. Yeh: Magnetic fields generated by a tension fault. Bulletin of the College of Engineering. National Taiwan University. 40, 47-56 (1987) [6] Y. Shindo, D. Sekiya, F. Narita, K. Hohiguchi: Tensile testing and analysis of ferromagnetic elastic strip with a central crack in a uniform magnetic field. Acta Materialia. 52, 4677-4684 (2004) DOI: 10.1016/j.actamat.2004.06.029 [7] Y. Shindo, T. Komatsu, F. Narita, K. Horiguchi: Magnetic stress intensity factor for an edge crack in a soft ferromagnetic elastic half-plane under tension. Acta Mechanica. 182, 183-193 (2006) DOI: 10.1007/s00707-005-295-2 [8] Chun-Bo Lin, Chau-Shioung Yeh: The magnetoelastic problem of a crack in a soft ferromagnetic solid. International Journal of Solids and Structures. 39, 1-17 (2002) DOI: 10.1016/S0020-7683(01)00176-7 [9] Liang Wei, Shen Ya-peng, Fang Dai-ning: Coupling field in an infinite soft ferromagnetic elastic plane

12

with a through crack (in Chinese). Acta Mechanic Sinica. 33, 758-766 (2001) [10] Zhang Xing: Fracture and Damage Mechanics (in Chinese). Beijing University of Aeronautics and Astronautics Press, Beijing (2006) [11] Fan Tianyou: Foundation of Fracture Theory (in Chinese). Science Press, Beijing (2003)

13

Figure List

Fig.1 An infinite plate with a central crack subjected to stress T and magnetic field B0 Fig.2 Intensities of the perturbed magnetic field around the crack Fig.3 Intensities of the perturbed magnetic field along the crack length Fig.4 Intensities of the perturbed magnetic field for various distance from the crack Fig.5 Resulted intensity of the perturbed magnetic field near the crack

14

Fig.1 An infinite plate with a central crack subjected to stress T and magnetic field B0

15

hx

(a) tangent magnetic intensity hx

hy

(b) normal magnetic intensity hy

Fig.2 Intensities of the perturbed magnetic field around the crack

16

2.5 2 1.5 1 0.5

hx+ hx-

hx

0 -0.5 -1 -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0

x

(a) tangent magnetic intensity hx

0.5

1

1.5

2

2

hy+ hy1.5

1

hy

0.5

0

-0.5

-1 -2

-1.5

-1

-0.5

0

x

0.5

1

1.5

2

(b) normal magnetic intensity hy

Fig.3 Intensities of the perturbed magnetic field along the crack length

17

2.5 2 1.5 1 0.5 y=0.05 y=0.1 y=0.2 y=-0.05 y=-0.1 y=-0.2

hx

0 -0.5 -1 -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0 x 0.5 1 1.5 2

(a) tangent magnetic intensity hx

2

1.5

y=0.05

1 y=0.1 0.5 y=0.2

hy

0 -0.5 -1 -2

-1.5

-1

-0.5

0 x

0.5

1

1.5

2

(b) Normal magnetic intensity hy

Fig.4 Intensities of the perturbed magnetic field for various distance from the crack

18

Fig.5 Resulted intensity of the perturbed magnetic field near the crack

19

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