two-dimensional problems in a soft ferromagnetic solid with an elliptic hole or a crack

International Journal of Engineering Science 52 (2012) 1–21

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International Journal of Engineering Science

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Two-dimensional problems in a soft ferromagnetic solidwith an elliptic hole or a crack

Chao Chang, Cun-Fa Gao ⇑, Yan ShiState Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China

a r t i c l e i n f o

Article history:Received 14 October 2011Received in revised form 8 December 2011Accepted 15 December 2011Available online 3 January 2012

Keywords:Soft ferromagnetic solidElliptic holeCrackMaxwell stressIntensity factors of fields

0020-7225/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.ijengsci.2011.12.007

⇑ Corresponding author. Tel.: +86 25 8489 6237.E-mail address: [email protected] (C.-F. Gao).

a b s t r a c t

This paper deals with the problem of an elliptic hole or a crack in a soft ferromagnetic solidunder the magnetic field at infinity. First, based on the simplified versions of the linear the-ory of Pao and Yeh (1973), the general solution of an elliptical hole is obtained according toexact boundary conditions at the rim of the hole. Then, when the hole degenerates into acrack, explicit solutions are given for potential functions and intensity factors of total stres-ses. In the above analysis, three kinds of magnetic boundary conditions, that is, magneti-cally permeable, impermeable and conducting boundary condition, are considered on thesurface of the hole or the crack, respectively. It is found that in general, the total stressesalways have the classical singularity of the r�1=2-type at the crack tips for considered threecrack models, and that the applied magnetic field may either enhance or retard crackgrowth depending on the magnetic boundary conditions adopted on the crack faces, andthe Maxwell stresses on the crack faces and at infinity. Since the present solutions for acrack are given in explicit form, they can also serve as a benchmark to test the validityof various analysis approaches or assumptions to more complicated crack problems in asoft ferromagnetic solid.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

A material is called soft ferromagnetic when the magnetic field intensity vector H and magnetization vector M are parallelin the rigid body state (Hasanyan & Harutyunyan, 2009). For example, Nickel–iron alloys are a typical soft ferromagneticmaterial, and they have been widely used as core materials for transformers, generators, induction coils and electric motorsdue to their inherent coupled magnetoelastic behavior. Hence, it is of both theoretical and practical importance to study themagnetoelastic coupling problem of soft ferromagnetic materials. Pioneering works on magnetoelastic theory have beengiven by Dunkin and Eringen (1963), Tiersten (1964) and Brown (1966). Since these early theories took the effects of elasticdeformation on magnetic fields into account, they are nonlinear and very difficult to use in theoretical analysis. However, fora soft ferromagnetic material, since it has small hysteresis losses (narrow hysterisis loop for H–M curves) and low remnantmagnetization, the linear theories can be used to obtain the approximate solutions for some classical problems. Pao and Yeh(1973) developed a linear version of Brown’s theory by using the perturbation method. The linear theory has made it possibleto obtain analytical results for the boundary-value problems of magnetoelastic interaction. Based on Pao and Yeh’s work,Shindo (1977) was first to study the linear magnetoelastic problem of a soft ferromagnetic elastic solid with a magneticallypermeable crack using Pao and Yeh’s theory and the integral transform technique. In contrast, Fil’shtinskii (1993) solved aplane problem in a soft ferromagnetic medium containing a magnetically impermeable crack. Liang, Shen, and Zhao (2000)

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2 C. Chang et al. / International Journal of Engineering Science 52 (2012) 1–21

and Liang, Shen, and Fang (2002) developed a complex potential method to solve the 2D problems of permeable cracks insoft ferromagnetic solids. Recently, Gao, Mai, and Wang (2008) re-visited the crack problem in a soft ferromagnetic solidbased on the Pao and Yeh’s theory with considering the effects of Maxwell stresses not only on the crack faces but also atinfinity. More recently, Chen (2009) developed a nonlinear field theory of fracture mechanics for crack propagation in para-magnetic and ferromagnetic materials.

However, it should be noted that even based on the linear theory of Pao and Yeh, it is still not easy to obtain explicit re-sults of magnetoelastic coupling problems, since the elastic deformation has been taken into account in the boundary con-ditions of magnetic fields. Thus, Lin and Yeh (2002) and Wan, Fang, Soh, and Hwang (2002) proposed the simplified versionsof the linear theory of Pao and Yeh (1973), respectively, by neglecting the effects of elastic deformation on magnetic fields.Lin and Lin (2002) and Lin, Chen, and Lee (2009) adopted the simplified theory to study the magnetoelastic fields in a softferromagnetic solid with impermeable straight or curvilinear cracks.

In these simplified versions, it is assumed that the strain is so small that its back-coupling can be neglected, that is, themagnetic field can be obtained directly from the theory of magnetic fields, and then the stress and deformation can be givenwith the help of the known magnetic fields. This is similar to solving the problems of thermal stress. Since the solutions ob-tained from the simplified linear theory are the first-order approximate ones, they can capture a clear physical picture aboutthe magnetoelastic coupling effects in soft ferromagnetic materials. Especially, the simplified linear theory makes it easy toexplore the effects of Maxwell stresses on the fracture behavior of ferromagnetic materials.

In fact, when a ferromagnetic solid with a crack is placed in a magnetic field, the Maxwell stresses will be induced on thesurface of the crack and the remainder surface of the solid. If the Maxwell stress is neglected at the outside surface of thesolid, the applied magnetic loads will result in singular stress fields in the solid. Similarly, if the crack is assumed to be imper-meable, that is, the Maxwell stress along the crack surface is assumed to be zero, the induced stress fields are also singular.This implies that the singular structure of field variables is dependent on if the Maxwell stresses are taken into account onthe surface of the ferromagnetic solid. However, this issue has not been strictly considered before.

Motivated by the above, in the present work we re-visit a mode-I crack in a soft ferromagnetic solid based on the first-order approximate linear theory of a soft ferromagnetic solid with a crack, which are respectively assumed to be magneti-cally permeable, impermeable and conducting to investigate the effects of applied magnetic fields on cracks. This paper isarranged as follows: the first-order approximate linear equations of a soft ferromagnetic solid are outlined in Section 2,and then for the case of an elliptical hole, the potential functions for magnetic fields and magnetic-elastic fields are presentedin Section 3 based on three kinds of magnetic boundary conditions, respectively. When the hole degenerates into a crack, theintensity factors of total stresses are obtained in a closed and explicit form in Section 4. To discuss the effect of the mediainside the hole/or crack and at the surrounding space at infinity on fracture behavior, given in Section 5 are the numericalresults of the stress distribution around the hole and the stress intensity factor at the crack tip Finally, Section 6 concludesthe present work.

2. Outline of basic equations

Consider an isotropic soft ferromagnetic solid under small deformation. The static magnetic field satisfies the followingequations:

eijkHk;j ¼ 0; Bi;i ¼ 0; ð1Þ

where B stands for the magnetic induction, H is the magnetic intensity, eijk is the permutation tensor, and ‘‘,’’ means partialdifferentiation.

The constitutive law of magnetic field is:

Bi ¼ l0ðHi þMiÞ ¼ l0lrHi ¼ lHi; ð2Þ

where l0 is the absolute permeability of vacuum; l0 ¼ 4p� 10�7T �m=A; M is the magnetization; Mi ¼ vHi; lr stands for therelative magnetic permeability; lr ¼ 1þ v; l is the magnetic permeability; l ¼ l0lr; v is the magnetic susceptibility of themedium. For linear soft ferromagnetic materials one has v � ð102 ! 105Þ � 1.

The total stress tensors, Tij, can be expressed as

Tij ¼ tij þ tMij ; ð3Þ

tij ¼ rij þl0

v MiMj; rij ¼ kdijuk;k þ Gðui;j þ uj;iÞ; ð4Þ

tMij ¼ lHiHj �

12l0HkHkdij; ð5Þ

where tij is the magnetoelastic stress tensors, tMij is the Maxwell stress tensors, rij is the elastic stress tensors, dij is the Kro-

necker delta, k and G are the Lame constants, and ui stand for the displacement components.

C. Chang et al. / International Journal of Engineering Science 52 (2012) 1–21 3

The equilibrium equation is

Tij;i ¼ 0: ð6Þ

For the case of 2D deformation, according to the Appendix A, we can outline the following key equations which will be usedin later analysis as:

H1 þ iH2 ¼ 2w0ðzÞ; ð7Þ

B1 þ iB2 ¼ 2lw0ðzÞ; ð8Þ

T11 þ T22 ¼ 2ðu0ðzÞ þu0ðzÞÞ þm2w0ðzÞw0ðzÞ; ð9Þ

T22 � T11 þ 2iT12 ¼ 2ð�zu00ðzÞ þ w0ðzÞÞ þm2w00ðzÞwðzÞ þm3w0ðzÞ2; ð10Þ

where

m2 ¼ 4l0vð1� tÞ; m3 ¼ �4l0ð2vþ 1Þ:

Stress boundary condition is

uðzÞ þ zu0ðzÞ þ wðzÞ þm2

2wðzÞw0ðzÞ þm3

2

Zw0ðzÞ2d�z

� �s

¼ iZðeX þ ieY Þds: ð11Þ

Displacement boundary condition can be expressed as

2Gðu1 þ iu2Þ ¼ juðzÞ � zu0ðzÞ � wðzÞ �m2

2wðzÞw0ðzÞ; ð12Þ

where

G ¼ E2ð1þ tÞ ; j ¼ 3� t

1þ t:

3. Solution for an elliptic hole

As shown in Fig. 1, there is an elliptic hole in an infinite soft ferromagnetic solid where a and b are the semi-major andsemi-minor axes of the ellipse, respectively. It is assumed that lc is the magnetic permeability of the medium inside the cav-ity with lc ¼ l0ð1þ vcÞ;lm is the magnetic permeability of soft ferromagnetic solid with lm ¼ l0ð1þ vmÞ, and l1 is themagnetic permeability at the surrounding space at infinity with l1 ¼ l0ð1þ v1Þ. The solid is subjected to the remote mag-netic load B12 along the positive direction of the axis x2. Below, we derive the complex potentials of magnetic field and mag-netic-elastic field based on three boundary conditions, respectively.

3.1. Complex potential of magnetic field

3.1.1. Permeable boundary conditionFor this case, it can be shown that the magnetic field inside the hole is uniform, which means that the complex potential

of magnetic field, wcðzÞ, is a linear function of z. Thus, we can get

wc ¼ c0z; ð13Þ

where c0 is a complex constant to be determined.Substituting Eq. (13) into Eqs. (7) and (8) leads to the magnetic components inside the hole as

Hc1 þ iHc

2 ¼ 2�c0; ð14Þ

Bc1 þ iBc

2 ¼ 2lc�c0; ð15Þ

which gives:

�c0 ¼ �iBc

2

lc: ð16Þ

Now, introduce the conformal mapping function, xð1Þ, as

z ¼ xð1Þ ¼ R 1þm1

� �; R ¼ aþ b

2; m ¼ a� b

aþ b; ð17Þ

Fig. 1. An elliptic hole in a soft ferromagnetic solid under the remote magnetic load.


which transforms the outside of elliptic hole in z-plane to the outside of unit circle in 1-plane, and in the 1-plane, the relationbetween the polar components of H and its rectangular components in the z-plane can be expressed as

Hcn þ iHc

t ¼�1q

x0ð1Þjx0ð1Þj Hc

1 þ iHc2

� �; ð18Þ

Bcn þ iBc

t ¼�1q

x0ð1Þjx0ð1Þj Bc

1 þ iBc2

� �: ð19Þ

On the boundary of the hole, one has q ¼ 1; 1 ¼ r ¼ eih and �r ¼ 1=r. In this case, Eqs. (18) and (19) become

Hcn þ iHc

t

� �s ¼

x0ðrÞrjx0ðrÞj Hc

1 þ iHc2

� �s; ð20Þ

Bcn þ iBc

t

� �s ¼

x0ðrÞrjx0ðrÞj Bc

1 þ iBc2

� �s; ð21Þ

where the subscript ‘‘s’’ denotes the points along the boundary.Outside the hole, the complex potential, wðzÞ, can be expressed as

wðzÞ ¼ c1zþw0ðzÞ; ð22Þ

where c1 is the constant to be determined from remote magnetic loads, and w0ðzÞ is a analytical function outside the hole upto infinity.

Substituting Eq. (22) into Eqs. (7) and (8) and then let z!1, we can get

c1 ¼ � iB122lm

: ð23Þ

On the other hand, in the 1-plane, one has form Eqs. (22) and (17) that

wð1Þ ¼ c1xð1Þ þw0ð1Þ; ð24Þw0ð1Þ ¼ c1x0ð1Þ þw00ð1Þ: ð25Þ

Similarly, the polar and rectangular components of magnetic variables in the solid can be expressed, respectively, as

H1 þ iH2 ¼ 2w0ðzÞ; ð26ÞB1 þ iB2 ¼ 2lmw0ðzÞ; ð27Þ

Hn þ iHt ¼�1q

x0ð1Þjx0ð1Þj H1 þ iH2ð Þ; ð28Þ

Bn þ iBt ¼�1q

x0ð1Þjx0ð1Þj B1 þ iB2ð Þ: ð29Þ


On the boundary of the hole, one has

ðHn þ iHtÞs ¼2w0ðrÞrjx0ðrÞj ðH1 þ iH2Þs; ð30Þ

ðBn þ iBtÞs ¼2w0ðrÞrjx0ðrÞj ðB1 þ iB2Þs: ð31Þ

The continuous conditions along the hole boundary requires

Bcn ¼ Bm

n ; Hct ¼ Hm

t : ð32Þ

Using Eqs. (20), (21) and (30)–(32), we obtain

lcRex0ðrÞ

rjx0ðrÞj Hc1 þ iHc

2

� �" #¼ lmRe

2rjx0ðrÞjw

0ðrÞ� �

; ð33Þ

Imx0ðrÞ

rjx0ðrÞj Hc1 þ iHc

2

� �" #¼ Im

2rjx0ðrÞjw

0ðrÞ� �

: ð34Þ

Eq. (33) can be re-expressed as

2w0ðrÞ þ 2r2 w0ðrÞ ¼ lc

lm

x0ðrÞr2 ð2CÞ þ lc

lmx0ðrÞð2CÞ: ð35Þ

Substituting Eq. (25) into Eq. (35) leads to

2w00ðrÞ þ2r2 w00ðrÞ ¼ fomðrÞ; ð36Þ

where

f0mðrÞ ¼lc

lmð2CÞ � 2c1

� �x0ðrÞr2 þ lc

lm2C�

� 2c1� �

x0ðrÞ: ð37Þ

Multiplying both sides of Eq. (36) with 12pi � dr

r�1 where 1 stands for an arbitrary point outside the unit circle, and then cal-culating the Cauchy integral along the circle (Muskhelishivili, 1953), we get

w00ð1Þ ¼R

2lmiðmþ 1Þ Bc

2 � B12� � � 1

12 ; ð38Þ

which leads to

w0ð1Þ ¼ �R

2lmiðmþ 1Þ Bc

2 � B12� � �1

1: ð39Þ

Inserting Eq. (25) with (38) into Eq. (34), we finally obtain

Bc2 ¼ B12 þ

1� lmlc

1þ ba

lmlc

ba

B12 ; ð40Þ

which implies that the magnetic induction inside the hole depends on the remotely applied magnetic load, the magnetic per-meability ratio between the soft ferromagnetic solid and the medium inside the hole and the hole geometries.

Finally, inserting Eq. (39) into Eq. (24), we have the complete solution of the magnetic complex potential under the per-meable boundary condition as

wð1Þ ¼ c1R 1þP1

� �; ð41Þ

where

P ¼ 2aaþ b

Bc2

B12� 1: ð42Þ

3.1.2. Impermeable boundary conditionIn this case, the magnetic fields inside the hole are assumed to be zero, and thus the boundary condition along the hole

boundary is

Bcn ¼ Bm

n ¼ 0: ð43Þ


In Eq. (42), just letting Bc2 ¼ 0 and then using Eq. (41) we have

wð1Þ ¼ c1R 1� 11

� �: ð44Þ

3.1.3. Conductive boundary conditionIn this case, the boundary condition along the hole boundary is

Hct ¼ Hm

t ¼ 0: ð45Þ

Similarly, we can get the magnetic complex potential function as

wð1Þ ¼ c1R 1þ 11

� �: ð46Þ

Finally, observing Eqs. (41), (44) and (46) we can give the united expression of the complex potentials as

wkð1Þ ¼ c1R 1þPk

1

� �; ð47Þ

where

Pk ¼P; k ¼ 1; for permeable boundary�1; k ¼ 2; for impermeable boundaryþ1; k ¼ 3; for conducting boundary:

8><>: ð48Þ

3.2. Complex potential of magnetic-elastic field

3.2.1. Permeable boundary conditionDenoting the stress potential function by u and w, we can express them as

uðzÞ ¼ C1zþu0ðzÞ; ð49aÞwðzÞ ¼ C2zþ w0ðzÞ; ð49bÞ

where u0ðzÞ and w0ðzÞ are two analytic complex-variable function outside the hole up to infinity, respectively.When B12 is applied only at infinity, the total stress at the outside surface of the solid can be calculated according to Eqs.

(3)–(5) as

T111 ¼ �l0B12

2

2l21; ð50Þ

T122 ¼l0

2ð4v1 þ 1Þ B

122

l21; ð51Þ

T112 ¼ T121 ¼ 0; ð52Þ

where l1 is the magnetic permeability of the surrounding space and l1 ¼ l0ð1þ v1Þ.After inserting Eqs. (49) and (41) into Eqs. (9) and (10), letting z!1 and then using Eqs. (50)–(52) we have

C1 ¼14

2l0v1B122

l21

þm2c12

!; ð53aÞ

C2 ¼12

l0ð2v1 þ 1ÞB122

l21

�m3c12

" #: ð53bÞ

On the other hand, the total stress inside the hole can be expressed as

Tc11 ¼ �

l0

2Hc2

2 ; ð54Þ

Tc22 ¼

l0

2ð4vc þ 1ÞHc2

2 ; ð55Þ

Tc12 ¼ Tc

21 ¼ 0: ð56Þ

In general, the boundary condition along the hole boundary can be written as


2wkðzÞw0kðzÞ þ

m3

2

Zw0kðzÞ

2d�z� �

s¼ iZðeXc þ ieY cÞds; ð57Þ

which is valid to three boundary condition when wkðzÞ is given by Eq. (47).


Now, let us calculate the value of eXc and eY c , which can be expressed as
eXc ¼ n1Tc11 þ n2Tc
21; ð58ÞeY c ¼ n1Tc12 þ n2Tc

22; ð59Þ

where

n1 ¼ cosðN; x1Þ ¼dx2

ds; ð60Þ

n2 ¼ cosðN; x2Þ ¼ �dx1

ds; ð61Þ

and N stands for the normal direction of the boundary.Using Eqs. (58)–(61), we obtain
eXc þ ieY c ¼ n1Tc
11 þ n2Tc21 þ i n1Tc

12 þ n2Tc22

�: ð62Þ

Inserting Eqs. (54)–(56), (60) and (61) into Eq. (62), we get

iZðeXc þ ieY cÞds ¼ l0Hc2

2

2½2vczþ ð2vc þ 1Þ�z�: ð63Þ

Substituting Eq. (63) into Eq. (57), we have the condition boundary along the hole rim as


2wkðzÞw0kðzÞ þ

m3

2

Zw0kðzÞ

2d�z� �

s

¼ l0Hc22

2½2vczþ ð2vc þ 1Þ�z�: ð64Þ

In the-1 plane, Eq. (64) is changed into

uðrÞ þ xðrÞx0ðrÞ

u0ðrÞ þ wðrÞ þm2

2wkðrÞw0kðrÞ

x0ðrÞþm3

2

Zw0kðrÞ2

x0ðrÞd�r ¼ l0Hc2

2

2½2vcxðrÞ þ ð2vc þ 1ÞxðrÞ�: ð65Þ

Inserting Eq. (49) into Eq. (65), we have

u0ðrÞ þxðrÞx0ðrÞ

u00ðrÞ þ w0ðrÞ ¼ f0ðrÞ; ð66Þ

where

f0ðrÞ ¼ �m2

2wkðrÞw0kðrÞ

x0ðrÞ�m3

2

Zw0kðrÞ2

x0ðrÞd�rþ l0Hc2

2

2ð2vc þ 1Þ � C2

" #xðrÞ þ ðl0vcHc2

2 � 2C1ÞxðrÞ: ð67Þ

The general solution of Eq. (66) has been given by Muskhelishivili (1953) as

u0ð1Þ ¼ �1

2pi

Ir

f0ðrÞr� 1

dr; ð68Þ

w0ð1Þ ¼ �1

2pi

Ir

f0ðrÞr� 1

dr� 11þm12

12 �mu00ð1Þ: ð69Þ

In addition, substituting Eq. (47) into Eq. (67) leads to

f0ðrÞ ¼ �m2

2Rðc1c1 �P2

kc1c1 þPkc12mÞ1� RPkc1c1r3

1�mr2

þ �m2

2RPkc1c1 �m3

2Rc12 þ l0Hc2

2

2Rð2vc þ 1Þ � RC2 þ Rml0vcHc2

2 � 2RmC1

" #1r

þ �m3

2RP2

kc12

mþ l0Hc2

2

2Rmð2vc þ 1Þ � RmC2 þ Rl0vcHc2

2 � 2RC1

" #r

þ R

2ffiffiffiffiffiffiffim3p ðc1m�Pkc1Þ2 ln

1�ffiffiffiffiffimp

r1þ

ffiffiffiffiffimp

r: ð70Þ

Substituting Eq. (70) into Eqs. (68) and (69) and completing the Cauchy integral, we obtain

u0ð1Þ ¼k1

1; ð71Þ

w0ð1Þ ¼k21

12 �mþ k3

1þ k4 ln

1�ffiffiffiffiffimp

1þffiffiffiffiffimp ; ð72Þ


where

k1 ¼ �R �m2

2Pkc12 þm3

2c12 � l0ð2mvc þ 2vc þ 1Þ

2Hc2

2 þ C2 þ 2mC1

� �; ð73aÞ

k2 ¼ �m2R

2c12Pk

m�mc12Pk þP2

k c12 � c12� �

þ k1

mþmk1; ð73bÞ

k3 ¼ �Rm3P

2kc12

2m�m2c12Pk

2m�ml0

22vc þ

2vc

mþ 1

� �Hc2

2 þmC2 þ 2C1

" #� k1

m; ð73cÞ

k4 ¼ �Rm3

4ffiffiffiffiffiffiffim3p ðc1m�Pkc1Þ2: ð73dÞ

Up to here, we have obtained the general solution for complex potentials by using Eqs. (49), (71)–(73) for three boundaryconditions. Then, the total stress components in the 1 plane can be calculated by

Th þ Tq ¼ 4Re½Uð1Þ� þm2Wð1ÞWð1Þ; ð74Þ

Th � Tq þ 2iTqh ¼12

q2f 0ð1Þ2 f ð1ÞU0ð1Þ þ f 0ð1ÞWð1Þ�

þm2W 0ð1Þxð1Þ þm3f 0ð1ÞW2ð1Þh i

; ð75Þ

where

Uð1Þ ¼ u0ð1Þ=x0ð1Þ; Wð1Þ ¼ w0ð1Þ=x0ð1Þ; Wð1Þ ¼ w0ð1Þ=x0ð1Þ:

3.2.2. Impermeable boundary conditionIn Eq. (74), letting k ¼ 2;Pk ¼ �1 and Hc

2 ¼ 0, we have

k1 ¼ �Rm2

2þm3

2

� c12 þ C2 þ 2mC1

� ; ð76aÞ

k2 ¼ �m2R

2m� 1

m

� �c12 þ mþ 1

m

� �k1; ð76bÞ

k3 ¼ �Rm2 þm3ð Þc12

2mþmC2 þ 2C1

� �� k1

m; ð76cÞ

k4 ¼ �Rm3

4ffiffiffiffiffiffiffim3p ðmþ 1Þ2c12: ð76dÞ

Inserting Eq. (76) into Eqs. (71) and (72), one shall have the complex potentials for the impermeable boundary condition.

3.2.3. Conducting boundary conditionIn Eq. (74), if we let k ¼ 3;P3 ¼ þ1 and Hc

2 ¼ 0, we have

k1 ¼ �Rm3 �m2

2c12 þ C2 þ 2mC1

� ; ð77aÞ

k2 ¼ �m2R

21m�m

� �c12

� �þ k1

mþmk1; ð77bÞ

k3 ¼ �Rðm3 �m2Þc12

2mþmC2 þ 2C1

� �� k1

m; ð77cÞ

k4 ¼ �Rm3

4ffiffiffiffiffiffiffim3p ðm� 1Þ2c12: ð77dÞ

Similarly, the complex potentials can be written out according to Eqs. (71), (72) and (77).

4. Solution for a crack

4.1. Magnetic permeable crack

Letting b ¼ 0 and m ¼ 1, the elliptic hole becomes a crack along x1 axis, as shown in Fig. 2. For a magnetic permeablecrack, P1 ¼ 1, and Eq. (73) becomes

k1 ¼ �a2

T122 � Tc22

� �; k2 ¼ 2k1; k3 ¼ 0; k4 ¼ 0: ð78Þ

Fig. 2. A crack degenerated from the elliptical hole along x1 direction.


Finally, we have from Eq. (47) as

wðzÞ ¼ c1z: ð79Þ

And thus

Hc2 ¼

B12lc

; ð80Þ

Tc22 ¼

l0

2ð4vc þ 1Þ B12

lc

� �2

: ð81Þ

Using Eqs. (49), (71), (72) and (78), we obtain

uðzÞ ¼T122 � Tc

22

� �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 � a2p

þ C1 �12

T122 � Tc22

� �� z; ð82Þ

wðzÞ ¼ � a2

2T122 � Tc

22

� � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 � a2p þ C2z: ð83Þ

It is shown from Eqs. (9), (10), (82) and (83) that for the magnetic permeable crack, the total stress field has singularity of the1=

ffiffiffirp

at the crack tip. Thus, the total stress intensity factor can be defined as

ðkTI ; k

TIIÞ ¼ lim

x1!a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðx1 � aÞ

pðT22; T12Þ: ð84Þ

Using Eqs. (9), (10), (79), (82) and (83), we have

T22 � iT12 ¼ T122 � Tc22

� � x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 � a2q þ Tc

22; ð85Þ

where T122 and Tc22 stand for the total stresses at infinity and inside the crack, respectively, and they can be calculated from

Eqs. (51) and (81).Inserting Eq. (85) into Eq. (84), we get

kTI ¼

ffiffiffiffiffiffipap

T122 � Tc22

� �; ð86Þ

kTII ¼ 0: ð87Þ

In order to prove the rightness of Eqs. (86) and (87), the solutions for a permeable crack is re-derived in the Appendix B basedon a different method, and it is found that the results based on two different approaches are consistent.

Finally, substituting Eqs. (51) and (81) into Eq. (86), we obtain

kTI ¼

l0

2B12� �2 4v1 þ 1

l21

� 4vc þ 1l2

c

� � ffiffiffiffiffiffipap

: ð88Þ

4.2. Magnetic impermeable crack

When the crack is assumed to be magnetic impermeable, Eq. (76) is changed to

k1 ¼ �a2

T122 þm2c12� �; k2 ¼ 2k1; k3 ¼ 0; k4 ¼ �

a2

m3c12: ð89Þ


Similarly, in this case, the final solutions can be written out as follows:

wðzÞ ¼ c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 � a2p

; ð90Þ

uðzÞ ¼T122 þm2c12� �

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 � a2p

þ C1 �12

T122 þm2c12� �� z; ð91Þ

wðzÞ ¼ � a2

2T122 þm2c12� � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 � a2p � am3c12

4ln

z� azþ a

þ C2z: ð92Þ

Additionally, we have

T22 � iT12 ¼ T122 þm2c12� � x1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 � a2q �m2c12; ð93Þ

which means that for the magnetic impermeable crack, the total stress field has still singularity of the 1=ffiffiffirp

at the crack tip.Inserting Eq. (93) into Eq. (84), we get

kTI ¼


T122 þm2c12� �; kT

II ¼ 0: ð94Þ

Substituting Eqs. (51) and (23) into Eq. (94) leads to

kTI ¼

l0

2B12� �2 4v1 þ 1

l21

� 1l2

m

m2

2l0


: ð95Þ

4.3. Magnetic conductive crack

If letting a ¼ 0 and m ¼ �1, the elliptic hole in Fig. 1 becomes a crack along the x2 axis, as shown in Fig. 3. In this case, Eq.(77) becomes

k1 ¼b2

T111 þm2c12� �; k2 ¼ �2k1; k3 ¼ 0; k4 ¼

ib2

m3c12: ð96Þ

Similarly, we can give the solutions for the magnetic conductive crack as follows:

wðzÞ ¼ c1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ b2

q; ð97Þ

uðzÞ ¼T111 þm2c12� �

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ b2

qþ C1 �

12

T111 þm2c12� �� z; ð98Þ

wðzÞ ¼ � b2

2T111 þm2c12� � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 þ b2p þ ibm3c12

4ln

z� ibzþ ib

þ C2z ð99Þ

and

T22 � iT12 ¼ T111 þm2c12� � x2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

2 � b2q �m2c12: ð100Þ

Fig. 3. A crack degenerated from the elliptical hole along x2 direction.


In this case, the structure of singular fields is the same as the above two case, and thus the intensity factor of total stressfield can be expressed as

kTI ; k

TII

� ¼ lim

x2!b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðx2 � bÞ

qðT11; T21Þ: ð101Þ

Using Eqs. (100) and (101), we have

kTI ¼

ffiffiffiffiffiffipbp

T111 þm2c12� �; kT

II ¼ 0: ð102Þ

Inserting Eqs. (50) and (23) into Eq. (102) leads to

kTI ¼ �

B122

2l0

l21þ m2

2l2m


: ð103Þ

5. Numerical examples

Choose Perm alloy 1J50 as a model medium which has the elastic modulus and Poisson’s ratio E ¼ 180 GPa and t ¼ 0:27.It is taken that the absolute permeability of vacuum is l0 ¼ 4p� 10�7 T �m=A, and the semi-major axis of the elliptic hole is0.01 m.

Shown in Figs. 4 and 5 (here vm ¼ 103 and v1 ¼ vcÞ is the distribution of total hoop stress Th at the hole rim based onthree magnetic boundary conditions, and it is found that the hoop stress for a magnetically impermeable hole is lager thanthat for a magnetically permeable hole, but as the vc=vm becomes smaller, the effects of magnetic boundary conditions on

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Tθ (

MP

a)

θ (rad)

permeable model impermeable model conductive model

B∞

2=2T

a/b=5

ab

B∞

2

Fig. 4. Effects of magnetic boundary condition on the total hoop stress Th for vc=vm ¼ 0:1.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1

0

1

2

3

4

5

6

7

B∞

2=2T

a/b=5


Tθ (

MP

a)

θ (rad)

ab

B∞

2

Fig. 5. Effects of magnetic boundary condition on the total hoop stress Th for vc=vm ¼ 0:01.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

B∞

2=2T

b/a=10-3


T θ (

MP

a)

θ (π /1000)

×103

ab

B∞

2

Fig. 6. Effects of the hole geometry on the total hoop stress Th for b=a ¼ 10�3.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

×105

B∞

2=2T

b/a=10-4


T θ (

MP

a)

θ (π /1000)

ab

B∞

2

Fig. 7. Effects of the hole geometry on the total hoop stress Th for b=a ¼ 10�4.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1

0

1

2

3

4

5

6

B∞

2=0.5T

B∞

2=1T

B∞

2=2T

Tθ (

MP

a)

θ (rad)

ab

B∞

2

Fig. 8. Effects of the applied magnetic load on Th .


the hoop stress become weaker. In Figs. 6 and 7 (here it is taken that vm ¼ 103;vc ¼ v1;vc=vm ¼ 10�2Þ, the total hoop stressTh is given for different hole sizes, and it can be seen that when the hole becomes a crack-like defect, the stress at the crack

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-5

0

5

10

15

20

25

χm/χ

c=1000

χm/χ

c=100

χm/χ

c=10

T θ (M

Pa)

θ (rad)

ab

B∞

2

Fig. 9. Effects of magnetic susceptibility of the media on Th when the hole and the surrounding space at infinity are filled with the same medium ðv1 ¼ vcÞ.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.2

0.0

0.2

0.4

0.6

0.8

χc/χ

∞=0.01

χc/χ

∞=0.1

χc/χ

∞=0.5

Tθ (

MP

a)

θ (rad)

ab

B∞

2

Fig. 10. Effects of magnetic susceptibility of the media on Th when the hole and the surrounding space at infinity are filled with different media ðv1 > vcÞ.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-5

0

5

10

15

20

25

χc/χ

∞=100

χc/χ

∞=10

χc/χ

∞=2

Tθ (

MP

a)

θ (rad)

ab

B∞

2

Fig. 11. Effects of magnetic susceptibility of the media on Th when the hole and the surrounding space at infinity are filled with different media ðv1 < vcÞ.


tip is approaching to a constant for a permeable or conductive crack (Fig. 7), while it is singular for an impermeable crack.Given in Fig. 8 (here B12 ¼ 2 T;vm ¼ 103;vc ¼ v1;vc=vm ¼ 10�2Þ is the total hoop stress Th under different loadings, and it isfound that as the applied load increases, Th increases for three magnetic boundary conditions.

0.0 0.5 1.0 1.5 2.0-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35χ

c/χ

∞=100

χc/χ

∞=10

χc/χ

∞=2

kT I (

MN

m-3

/2)

B∞

2 (T)

B∞

2

Fig. 12. Effects of magnetic susceptibility of the media on the intensity factor of total stress kTI when the crack and the surrounding space at infinity are

filled with different media ðv1 < vcÞ.

0.0 0.5 1.0 1.5 2.0

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

χc/χ

∞=0.01

χc/χ

∞=0.1

χc/χ

∞=0.5

kT I (

MN

m-3

/2)

B∞

2 (T)

B∞

2

Fig. 13. Effects of magnetic susceptibility of the media on the intensity factor of total stress kTI when the crack and the surrounding space at infinity are

filled with different media ðv1 > vcÞ.


To discuss the effects of the magnetic susceptibility on the stress distribution, plotted in Figs. 9–11 are the variation of thetotal hoop stress at the rim of a permeable hole. It is shown in Fig. 9 that when the surrounding space at infinity is filled withthe same medium as that inside the hole, i.e., v1 ¼ vc , as vm=vc increases, the total hoop stress becomes larger. In addition, itcan be found from Figs. 10 and 11 that the stress concentration is greater for the case of v1 < vc than that for the case ofv1 > vc .

For the case of a permeable crack, it can be found from Fig. 7 that for the case of v1 ¼ vc , no singularities exist, but for thecase of v1 – qvc , it can be seen from Figs. 12 and 13 that when v1 < vc , the applied magnetic load may enhance the crackopen, while it may retard its growth for the case of v1 > vc.

6. Conclusions

We analyze the 2D problem of a soft ferromagnetic solid with an elliptic hole by using Muskhelishivili’s complex potentialmethod. Based on three kinds of magnetic boundary condition on the surface of the hole, the general solutions for complexpotentials are presented in exact and explicit form when the solid is subjected to the remote uniform magnetic fields. Whenthe hole degenerates into a crack, more concise results are obtained for intensity factors of the total stresses. It is found thatthe effects of magnetic fields on cracks in the soft ferromagnetic solid are dependent on the adopted boundary conditionsalong the crack surface and at infinity. For a magnetically permeable crack, the Maxwell stresses in general have to be con-sidered on the surface of the crack and the remainder surface of the solid. Otherwise, if the Maxwell stress is neglected on thesurface of the crack or at infinity, the applied magnetic loads may lead to singular stresses in the solid. However, it is nowdifficult to conclude which crack model is more reasonable since such a conclusion must await further experimental dataand additional theoretical and/or computational results. Even so, the results in the present work can still serve as the fun-


damental solutions to test the correctness of other solutions for more complicated crack problems in soft ferromagneticmaterials, since these results are not only concise, but also explicit.

Acknowledgements

The authors thank the financial support from the National Natural Science Foundation of China (10972103), the Ph.D. Pro-grams Foundation of Ministry of Education of China (20093218110004), and the Program for Changjiang Scholars and Inno-vative Research Team in University (Grant No. IRT0968).

Appendix A

In the Appendix A, we give the detailed derivation of equations outlined in Section 2. Although some of these equationscan be found in previous publications, it is also necessary to re-derive and check them for the sake of self-containing andreliability.

A.1. Expressions of magnetic fields

For the 2D problem of a magnetic solid in a rectangular-coordinate system ðx1 � x2Þ, the magnetic equilibrium Eq. (1) canbe reduced to:

H2;1 � H1;2 ¼ 0; ðA1ÞB1;1 þ B2;2 ¼ 0; ðA2Þ

where the subscripts‘‘1’’ and ‘‘2’’ represent the x1 and x2.Define a new function nðx1; x2Þ which satisfies:

H1 ¼@n@x1

; ðA3Þ

H2 ¼@n@x2

: ðA4Þ

Then, Eq. (A1) is automatically satisfied, and Eq. (A2) becomes, by using Eqs. (2), (A3) and (A4), to

r2n ¼ 0; ðA5Þ

where r2 is Laplace operator, and it can be expressed as:

r2 ¼ @2

@x21

þ @2

@x22

: ðA6Þ

By using z ¼ x1 þ ix2 and its conjugation �z ¼ x1 � ix2, Eq. (A5) can be written as

@2n@z@�z

¼ 0: ðA7Þ

In general, the solution for Eq. (A7) is:

n ¼ wðzÞ þwðzÞ; ðA8Þ

where wðzÞ is called as the magnetic potential function.Inserting Eq. (A8) into Eqs. (A3) and (A4) one obtains the components of magnetic field as

H1 ¼ w0ðzÞ þw0ðzÞ; ðA9aÞH2 ¼ iðw0ðzÞ �w0ðzÞÞ: ðA9bÞ

From Eqs. (A9) and (2), we finally have

H1 þ iH2 ¼ 2w0ðzÞ; ðA10ÞM1 þ iM2 ¼ 2vw0ðzÞ; ðA11ÞB1 þ iB2 ¼ 2lw0ðzÞ: ðA12Þ

A.2. Expressions of stress fields

From Eqs. (3)–(5), we have the total stress as:

Tij ¼ rij þl0

v MiMj þ lHiHj �12l0HkHkdij: ðA13Þ


By substituting Eq. (A13) into Eq. (6), we obtain

rij;i þ l0ð2vþ 1ÞHiHj;i � l0Hi;jHi ¼ 0: ðA14Þ

For the present 2D problem, Eq. (A14) can be written as:

r11;1 þ r21;2 þ 2l0vðH1H1;1 þ H2H1;2Þ þ l0H2ðH1;2 � H2;1Þ ¼ 0; ðA15Þr12;1 þ r22;2 þ 2l0vðH1H2;1 þ H2H2;2Þ þ l0H1ðH2;1 � H1;2Þ ¼ 0: ðA16Þ

Using H2;1 ¼ H1;2, Eqs. (A15) and (A16) are reduced to

r11;1 þ r21;2 þ 2l0vðH1H1;1 þ H2H2;1Þ ¼ 0; ðA17Þr12;1 þ r22;2 þ 2l0vðH1H1;2 þ H2H2;2Þ ¼ 0: ðA18Þ

Define a new function V ¼ l0v H21 þ H2

2

� , which can be re-written, by using Eq. (A9), as

V ¼ 4l0vw0ðzÞw0ðzÞ: ðA19Þ

Then, Eqs. (A17) and (A18) can be reduced to

r11;1 þ r21;2 þ@V@x1¼ 0; ðA20Þ

r12;1 þ r22;2 þ@V@x2¼ 0: ðA21Þ

For the case of plane stress problem, one can list the following known equations:

e11 ¼@u1

@x1; e22 ¼

@u2

@x2; e12 ¼ e21 ¼

@u1

@x2þ @u2

@x1; ðA22Þ

e11 ¼1Eðr11 � tr22Þ; e22 ¼

1Eðr22 � tr11Þ; e12 ¼ e21 ¼

2ð1þ tÞE

r12; ðA23Þ

@2e11

@x22

þ @2e22

@x21

¼ @2e12

@x1@x2; ðA24Þ

where eij stand for the elastic strain, E is the modulus of elasticity, and t is the Poisson’s ratio.Substituting Eqs. (A22) and (A23) into Eq. (A24) leads to

@2r11

@x22

þ @2r22

@x21

� t@2r11

@x21

þ @2r22

@x22

!¼ 2ð1þ tÞ@2r12

@x1@x2: ðA25Þ

On the other hand, one has from Eqs. (A20) and (A21) that

r11;11 þ r21;21 þ V ;11 ¼ 0; ðA26Þr12;12 þ r22;22 þ V ;22 ¼ 0: ðA27Þ

Substituting Eqs. (A26) and (A27) into the left term of Eq. (A29) leads to

2ð1þ tÞ@2r12

@x1@x2¼ ð1þ tÞ@2r12

@x1@x2þ 2ð1þ tÞ@2r12

@x1@x2¼ �ð1þ tÞðr11;11 þ r22;22 þ V ;11 þ V ;22Þ: ðA28Þ

Substituting Eq. (A28) into Eq. (A25), we finally have

r2ðr11 þ r22Þ ¼ �ð1þ tÞr2V : ðA29Þ

Define a new function Uand let it satisfy:

r11 ¼ U;22 � V ; r22 ¼ U;11 � V ; r12 ¼ r21 ¼ �U;12: ðA30Þ

Then, Eqs. (A20) and (A21) are automatically satisfied, and then substituting Eq. (A30) into Eq. (A29), becomes

r2½r2U � ð1� tÞV � ¼ 0: ðA31Þ

Due to

w0ðzÞw0ðzÞ ¼ @2ðwðzÞwðzÞÞ

@z@�z: ðA32Þ

Eq. (A31) can be written as

@4

@z2@�z2 U �m2

4wðzÞwðzÞ

h i¼ 0; ðA33Þ


where

m2 ¼ 4l0vð1� tÞ: ðA34Þ

The general solution of Eq. (A33) can be expressed as

U ¼ 12

hðzÞ þ hðzÞ þ �zuðzÞ þ zuðzÞ�

þm2

4wðzÞwðzÞ: ðA35Þ

With Eqs. (A35) and (A30), we can obtain

r11 þ r22 ¼ r2U � 2V ¼ 2ðu0ðzÞ þu0ðzÞÞ þm1w0ðzÞw0ðzÞ; ðA36Þ

r22 � r11 þ 2ir12 ¼ U;11 � U;22 þ 2iU;12 ¼@

@x� i

@

@y

� �2

U ¼ 4@2U@z2 ¼ 2ð�zu00ðzÞ þ w0ðzÞÞ þm2w00ðzÞwðzÞ; ðA37Þ

where

m1 ¼ �4l0vð1þ tÞ:

On the other hand, the Maxwell stress can be expressed, by using Eqs. (5) and (A10), as

tM11 þ tM

22 ¼ 4l0vmw0ðzÞw0ðzÞ; ðA38ÞtM

22 � tM11 þ 2itM

12 ¼ �4lmw0ðzÞ2: ðA39Þ

Using Eqs. (A13), (A37), (A10) and (A11), we can finally the expressions of total stress components as

T11 þ T22 ¼ 2ðu0ðzÞ þu0ðzÞÞ þm2w0ðzÞw0ðzÞ; ðA40ÞT22 � T11 þ 2iT12 ¼ 2ð�zu00ðzÞ þ w0ðzÞÞ þm2w00ðzÞwðzÞ þm3w0ðzÞ2; ðA41Þ

where wðzÞ ¼ h0ðzÞ, and

m3 ¼ �4l0ð2vþ 1Þ: ðA42Þ

A.3. Expressions of displacement fields

From Eq. (A23), one has

E@u1

@x1¼ ðr11 þ r22Þ � ð1þ tÞr22; ðA43Þ

E@u2

@x2¼ ðr11 þ r22Þ � ð1þ tÞr11; ðA44Þ

E2ð1þ tÞ

@u2

@x1þ @u1

@x2

� �¼ r12: ðA45Þ

Using Eqs. (A30), (A36), (A37), (A43) and (A44) we get

E@u1

@x1¼ 2ðu0ðzÞ þu0ðzÞÞ þm1w0ðzÞw0ðzÞ � ð1þ tÞðU;11 � VÞ; ðA46Þ

E@u2

@x2¼ 2ðu0ðzÞ þu0ðzÞÞ þm1w0ðzÞw0ðzÞ � ð1þ tÞðU;22 � VÞ: ðA47Þ

Considering the result such that m1x0ðzÞx0ðzÞ þ ð1þ tÞV ¼ 0, Eqs. (A46) and (A47) can be re-written as

Eu1 ¼ 2ðuðzÞ þuðzÞÞ � ð1þ tÞU;1 þ g1ðx2Þ; ðA48ÞEu2 ¼ �2iðuðzÞ �uðzÞÞ � ð1þ tÞU;2 þ g2ðx1Þ; ðA49Þ

where g1ðx2Þ and g2ðx1Þ are arbitrary real functions.Substituting Eqs. (A48) and (A49) into Eq. (A45), and using r12 ¼ �U;12, we get

dg2ðx1Þdx1

¼ � dg1ðx2Þdx2

: ðA50Þ

From Eq. (A50), one has

g1ðx2Þ ¼ u01 � bx2; ðA51Þ

g2ðx1Þ ¼ u02 � bx1; ðA52Þ

which means that g1ðx2Þ and g2ðx1Þ represent the rigid body displacements, and they can be neglected.


Thus, Eqs. (A48) and (A49) becomes

2Gðu1 þ iu2Þ ¼ 4uðzÞ � ð1þ tÞðU;1 þ iU;2Þ; ðA53Þ

where

U;1 þ iU;2 ¼ 2@U@�z¼ uðzÞ þ wðzÞ þ zu0ðzÞ þm2

2wðzÞw0ðzÞ: ðA54Þ

Substituting Eq. (A54) into Eq. (A53), we obtain the final expression of displacement field as

2Gðu1 þ iu2Þ ¼ juðzÞ � zu0ðzÞ � wðzÞ �m2

2wðzÞw0ðzÞ; ðA55Þ

where

G ¼ E2ð1þ tÞ ; j ¼ 3� t

1þ t:

A.4. Expressions of boundary conditions

Since the boundary condition of displacement can be given by Eq. (A55), we now derive the expression of stress boundarycondition as follows: Consider an arc AB which represents an arbitrary part on the solid boundary, as shown in Fig. 14, wheres is the length of the arc from point A to point B, N stands for the outside normal direction, and eX and eY are the surface totalloads.

Using the following equations:

n1 ¼ cosðN; x1Þ ¼dx2

ds; ðA56Þ

n2 ¼ cosðN; x2Þ ¼ �dx1

ds: ðA57Þ

eX and eY can be expressed as
eX ¼ n1ðT11Þs þ n2ðT21Þs; ðA58ÞeY ¼ n1ðT12Þs þ n2ðT22Þs: ðA59Þ
According to Eq. (A13), we have

T11 ¼ r11 þl0

2ð4vm þ 1ÞH2

1 �l0

2H2

2; ðA60Þ

T22 ¼ r22 þl0

2ð4vm þ 1ÞH2

2 �l0

2H2

1; ðA61Þ

T12 ¼ T21 ¼ r12 þ l0ð2vm þ 1ÞH1H2: ðA62Þ

Using Eqs. (A56)–(A62), we obtain

eX þ ieY ¼ n1ðT11Þs þ n2ðT21Þs þ i½n1ðT12Þs þ n2ðT22Þs� ¼dx2

dsðT11Þs �

dx1

dsðT21Þs þ i

dx2

dsðT12Þs �

dx1

dsðT22Þs

� �¼ �i

dds

2@U@�zþm3

2

Zw0ðzÞ2d�z

� �: ðA63Þ

Fig. 14. Surface force at the boundary of a ferromagnetic solid.


From Eq. (A63) we have

2@U@�zþm3

2

Zw0ðzÞ2d�z ¼ i

ZðeX þ ieY Þds: ðA64Þ

Substituting Eq. (A35) into Eq. (A64) leads to


2wðzÞw0ðzÞ þm3

2

Zw0ðzÞ2d�z ¼ i

ZðeX þ ieY Þds; ðA65Þ

which is the expression of stress boundary condition.

Appendix B

In the Appendix B we directly derive the solutions for a permeable crack to show the rightness of Eqs. (86) and (87) whichare obtained based on the elliptic-hole-method. To this end, consider a permeable crack along the x1 axis in a soft ferromag-netic solid subjected to the remote magnetic load B12 along the x2 axis at infinity, as shown in Fig. 2. It is also assumed thatthe magnetic permeabilities inside the cracks, in ferromagnetic medium and at the surrounding space at infinity are differ-ent, and they are lc ¼ l0ð1þ vcÞ, lm ¼ l0ð1þ vmÞ and l1 ¼ l0ð1þ v1Þ, respectively.

For the 2D problem of deformation, the magnetic boundary conditions at the crack faces are

Bþ2 ¼ B�2 ; Hþ1 ¼ H�1 ; on L; ðB1Þ

where L stands for the crack.Using Eqs. (7), (8) and (22) one has

B2 ¼ 2Im½lmw0ðzÞ� ¼ B12 þ 2lmIm½w00ðzÞ�; ðB2ÞH1 ¼ 2Re½w0ðzÞ� ¼ H11 þ 2Re½w00ðzÞ�: ðB3Þ

Substituting Eqs. (B2) and (B3) into Eq. (B1) leads to

�w0�0 ðx1Þ �w0þ0 ðx1Þ ¼ �w0þ0 ðx1Þ �w0�0 ðx1Þ; �1 < x1 < þ1; ðB4Þ�w0�0 ðx1Þ þw0þ0 ðx1Þ ¼ �w0þ0 ðx1Þ þw0�0 ðx1Þ; �1 < x1 < þ1: ðB5Þ

From Eqs. (B4) and (B5), we get

w0þ0 ðx1Þ �w0�0 ðx1Þ ¼ 0; �1 < x1 < þ1: ðB6Þ

The solution of Eq. (B6) is

w00ðzÞ ¼ w00ð1Þ ¼ 0: ðB7Þ

Thus, we have from Eqs. (22) and (B7) that

w0ðzÞ ¼ c1 and wðzÞ ¼ c1z: ðB8Þ

It is indicated from Eq. (B8) that the magnetic field inside the medium is uniform which means B¼2 B12 , and that the magneticfield inside the crack are also uniform such that Bc

2 ¼ B12 .On the other hand, we have from Eqs. (9) and (10) that

T22 � iT12 ¼ u0ðzÞ þu0ðzÞ þ zu00ðzÞ þ w0ðzÞ þm2

2w0ðzÞw0ðzÞ þm2

2w00ðzÞwðzÞ þm3

2w0ðzÞ2: ðB9Þ

Substituting Eq. (B8) into (B9) gives

T22 � iT12 ¼ u0ðzÞ þu0ðzÞ þ zu00ðzÞ þ w0ðzÞ þ 2V0; ðB10Þ

where V0 is a constant given by

V0 ¼14

m2jc1j2 þ14

m3ðc1Þ2:

Define a new function, XðzÞ, as

XðzÞ ¼ �u0ðzÞ þ z �u00ðzÞ þ �w0ðzÞ: ðB11Þ

Then, Eq. (B10) can be rewritten as

T22 � iT12 ¼ UðzÞ þXð�zÞ þ ðz� �zÞU0ðzÞ þ 2V0; ðB12Þ

where UðzÞ ¼ u0ðzÞ.On the surfaces of the crack, Eq. (B12) becomes to


Uþv ðx1Þ þX�v ðx1Þ ¼ Tþ22 � iTþ12; ðB13ÞU�v ðx1Þ þXþv ðx1Þ ¼ T�22 � iT�12; ðB14Þ

where

UvðzÞ ¼ UðzÞ þ V0; XvðzÞ ¼ XðzÞ þ V0:

For the magnetic permeable crack, the stress boundary conditions at the crack faces are

Tþ22 ¼ T�22 ¼ Tc22; Tþ12 ¼ T�12 ¼ Tc

12 ¼ 0; ðB15Þ

where Tc22 and Tc

12 are the total stress applied at the crack faces, and they are equal to the Maxwell stress. Using Eq. (B8) wehave

Tc22 ¼

l0

2ð4vc þ 1ÞHc2

2 ; Tc12 ¼ Tc

21 ¼ 0; ðB16Þ

where Hc2 ¼ Bc

2=lc and Bc2 ¼ B12 .

On the other hand, using Eqs. (B13)–(B16), we obtain

½Uvðx1Þ þXvðx1Þ�þ þ ½Uvðx1Þ þXvðx1Þ�� ¼ 2q1ðx1Þ; ðB17Þ½Uvðx1Þ �Xvðx1Þ�þ � ½Uvðx1Þ �Xvðx1Þ�� ¼ 2q2ðx1Þ; ðB18Þ

where

q1ðx1Þ ¼12

Tþ22 þ T�22 � iðTþ12 þ T�12Þ �

¼ Tc12; ðB19Þ

q2ðx1Þ ¼12

Tþ22 � T�22 � iðTþ12 � T�12Þ �

¼ 0: ðB20Þ

From Eqs. (B17)–(B20) one has the following general solutions as (Muskhelishivili, 1953)

UvðzÞ þXvðzÞ ¼ Tc22 1� zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 � a2p

� �þ 2

C1zþ C0

XðzÞ ; ðB21Þ

UvðzÞ �XvðzÞ ¼ Uvð1Þ �Xvð1Þ; ðB22Þ

where C1 and C0 are two constants.In Eq. (B21), letting z!1 leads to

2C1 ¼ Uvð1Þ þXvð1Þ: ðB23Þ

On the other hand, in Eq. (B12) letting z!1, one has

T122 ¼ Uvð1Þ þXvð1Þ: ðB24Þ

From Eqs. (B23) and (B24) we have

2C1 ¼ T122: ðB25Þ

Using the single-valued condition of displacement, it can be shown (omitting some details) that C0 ¼ 0. Thus, Eq. (B21) re-sults in

UvðzÞ þXvðzÞ ¼ Tc22 þ

T122 � Tc22

� �zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 � a2p : ðB26Þ

Ahead of the crack tip one has from Eq. (B12) that

T22 � iT12 ¼ UvðxÞ þXvðxÞ: ðB27Þ

Using Eqs. (B26), (B27) and (84), we finally have

kperI ¼


T122 � Tc22

� �; kper

II ¼ 0; ðB28Þ

which is consistent with Eqs. (86) and (87).Similar to the case of the permeable crack, the solutions for the cases of an impermeable and conducting crack can also be

derived based on the above approach, and the obtained results can be shown to be the same as those based on the elliptic-hole-method in the present work.

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two-dimensional problems in a soft ferromagnetic solid with an elliptic hole or a crack

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