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

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

Contents lists available at SciVerse ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Two-dimensional problems in a soft ferromagnetic solid with an elliptic hole or a crack Chao Chang, Cun-Fa Gao ⇑, Yan Shi State 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 2011 Received in revised form 8 December 2011 Accepted 15 December 2011 Available online 3 January 2012 Keywords: Soft ferromagnetic solid Elliptic hole Crack Maxwell stress Intensity factors of fields

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 solid under the magnetic field at infinity. First, based on the simplified versions of the linear theory of Pao and Yeh (1973), the general solution of an elliptical hole is obtained according to exact boundary conditions at the rim of the hole. Then, when the hole degenerates into a crack, explicit solutions are given for potential functions and intensity factors of total stresses. In the above analysis, three kinds of magnetic boundary conditions, that is, magnetically permeable, impermeable and conducting boundary condition, are considered on the surface of the hole or the crack, respectively. It is found that in general, the total stresses always have the classical singularity of the r1=2 -type at the crack tips for considered three crack models, and that the applied magnetic field may either enhance or retard crack growth depending on the magnetic boundary conditions adopted on the crack faces, and the Maxwell stresses on the crack faces and at infinity. Since the present solutions for a crack are given in explicit form, they can also serve as a benchmark to test the validity of various analysis approaches or assumptions to more complicated crack problems in a soft ferromagnetic solid. Ó 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 parallel in the rigid body state (Hasanyan & Harutyunyan, 2009). For example, Nickel–iron alloys are a typical soft ferromagnetic material, and they have been widely used as core materials for transformers, generators, induction coils and electric motors due to their inherent coupled magnetoelastic behavior. Hence, it is of both theoretical and practical importance to study the magnetoelastic coupling problem of soft ferromagnetic materials. Pioneering works on magnetoelastic theory have been given by Dunkin and Eringen (1963), Tiersten (1964) and Brown (1966). Since these early theories took the effects of elastic deformation on magnetic fields into account, they are nonlinear and very difficult to use in theoretical analysis. However, for a soft ferromagnetic material, since it has small hysteresis losses (narrow hysterisis loop for H–M curves) and low remnant magnetization, 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 possible to 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 magnetically permeable crack using Pao and Yeh’s theory and the integral transform technique. In contrast, Fil’shtinskii (1993) solved a plane problem in a soft ferromagnetic medium containing a magnetically impermeable crack. Liang, Shen, and Zhao (2000) ⇑ Corresponding author. Tel.: +86 25 8489 6237. E-mail address: [email protected] (C.-F. Gao). 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.12.007

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 in soft ferromagnetic solids. Recently, Gao, Mai, and Wang (2008) re-visited the crack problem in a soft ferromagnetic solid based on the Pao and Yeh’s theory with considering the effects of Maxwell stresses not only on the crack faces but also at infinity. More recently, Chen (2009) developed a nonlinear field theory of fracture mechanics for crack propagation in paramagnetic 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 results of magnetoelastic coupling problems, since the elastic deformation has been taken into account in the boundary conditions of magnetic fields. Thus, Lin and Yeh (2002) and Wan, Fang, Soh, and Hwang (2002) proposed the simplified versions of 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 soft ferromagnetic 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, the magnetic field can be obtained directly from the theory of magnetic fields, and then the stress and deformation can be given with the help of the known magnetic fields. This is similar to solving the problems of thermal stress. Since the solutions obtained from the simplified linear theory are the first-order approximate ones, they can capture a clear physical picture about the magnetoelastic coupling effects in soft ferromagnetic materials. Especially, the simplified linear theory makes it easy to explore 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 the surface of the crack and the remainder surface of the solid. If the Maxwell stress is neglected at the outside surface of the solid, the applied magnetic loads will result in singular stress fields in the solid. Similarly, if the crack is assumed to be impermeable, 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 on the 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 firstorder approximate linear theory of a soft ferromagnetic solid with a crack, which are respectively assumed to be magnetically permeable, impermeable and conducting to investigate the effects of applied magnetic fields on cracks. This paper is arranged 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 presented in Section 3 based on three kinds of magnetic boundary conditions, respectively. When the hole degenerates into a crack, the intensity factors of total stresses are obtained in a closed and explicit form in Section 4. To discuss the effect of the media inside the hole/or crack and at the surrounding space at infinity on fracture behavior, given in Section 5 are the numerical results of the stress distribution around the hole and the stress intensity factor at the crack tip Finally, Section 6 concludes the present work. 2. Outline of basic equations Consider an isotropic soft ferromagnetic solid under small deformation. The static magnetic field satisfies the following equations:

eijk Hk;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 partial differentiation. The constitutive law of magnetic field is:

Bi ¼ l0 ðHi þ M i Þ ¼ l0 lr Hi ¼ lHi ;

ð2Þ

where l0 is the absolute permeability of vacuum; l0 ¼ 4p  107 T  m=A; M is the magnetization; Mi ¼ vHi ; lr stands for the relative magnetic permeability; lr ¼ 1 þ v; l is the magnetic permeability; l ¼ l0 lr ; v is the magnetic susceptibility of the medium. For linear soft ferromagnetic materials one has v  ð102 ! 105 Þ  1. The total stress tensors, T ij , can be expressed as

T ij ¼ t ij þ t M ij ; tij ¼ rij þ

ð3Þ

l0 M M ; rij ¼ kdij uk;k þ Gðui;j þ uj;i Þ; v i j

tM ij ¼ lH i H j 

1 l Hk Hk dij ; 2 0

ð4Þ ð5Þ

where tij is the magnetoelastic stress tensors, t M ij is the Maxwell stress tensors, rij is the elastic stress tensors, dij is the Kronecker 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

T ij;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 used in later analysis as:

H1 þ iH2 ¼ 2w0 ðzÞ;

ð7Þ

B1 þ iB2 ¼ 2lw0 ðzÞ;

ð8Þ

T 11 þ T 22 ¼ 2ðu0 ðzÞ þ u0 ðzÞÞ þ m2 w0 ðzÞw0 ðzÞ;

ð9Þ

T 22  T 11 þ 2iT 12 ¼ 2ðzu00 ðzÞ þ w0 ðzÞÞ þ m2 w00 ðzÞwðzÞ þ m3 w0 ðzÞ2 ;

ð10Þ

where

m2 ¼ 4l0 vð1  tÞ;

m3 ¼ 4l0 ð2v þ 1Þ:

Stress boundary condition is



uðzÞ þ zu0 ðzÞ þ wðzÞ þ

m2 m3 wðzÞw0 ðzÞ þ 2 2

Z

 Z e þ iY e Þds: w0 ðzÞ2 dz ¼ i ð X

ð11Þ

s

Displacement boundary condition can be expressed as

2Gðu1 þ iu2 Þ ¼ juðzÞ  zu0 ðzÞ  wðzÞ 

m2 wðzÞw0 ðzÞ; 2

ð12Þ

where

G¼

E ; 2ð1 þ tÞ

j¼

3t : 1þt

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 and semi-minor axes of the ellipse, respectively. It is assumed that lc is the magnetic permeability of the medium inside the cavity with lc ¼ l0 ð1 þ vc Þ; lm is the magnetic permeability of soft ferromagnetic solid with lm ¼ l0 ð1 þ vm Þ, and l1 is the magnetic permeability at the surrounding space at infinity with l1 ¼ l0 ð1 þ v1 Þ. The solid is subjected to the remote magnetic load B1 2 along the positive direction of the axis x2 . Below, we derive the complex potentials of magnetic field and magnetic-elastic field based on three boundary conditions, respectively. 3.1. Complex potential of magnetic field 3.1.1. Permeable boundary condition For this case, it can be shown that the magnetic field inside the hole is uniform, which means that the complex potential of magnetic field, wc ðzÞ, is a linear function of z. Thus, we can get

wc ¼ c0 z;

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 c

Hc1 þ iH2 ¼ 2c0 ;

ð14Þ

c Bc1 þ iB2 ¼ 2lc c0 ;

ð15Þ

which gives: c

c0 ¼ 

iB2

lc

ð16Þ

:

Now, introduce the conformal mapping function, xð1Þ, as

z ¼ xð1Þ ¼ R



1þ

 m ;

1

R¼

aþb ; 2

m¼

ab ; aþb

ð17Þ

4

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

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 relation between the polar components of H and its rectangular components in the z-plane can be expressed as c

Hcn þ iHt ¼

c

Bcn þ iBt ¼

1 x0 ð1Þ  c c H þ iH2 ; q jx0 ð1Þj 1

1 x0 ð1Þ  c c B þ iB2 : q jx0 ð1Þj 1

ð18Þ

ð19Þ

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

 c x0 ðrÞ  c c c Hn þ iHt s ¼ H þ iH2 s ; rjx0 ðrÞj 1

ð20Þ

 c x0 ðrÞ  c c c B þ iB2 s ; Bn þ iBt s ¼ rjx0 ðrÞj 1

ð21Þ

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

wðzÞ ¼ c1 z þ w0 ðzÞ;

ð22Þ

1

where c is the constant to be determined from remote magnetic loads, and w0 ðzÞ is a analytical function outside the hole up to infinity. Substituting Eq. (22) into Eqs. (7) and (8) and then let z ! 1, we can get 1

c1 ¼ 

iB2 : 2lm

ð23Þ

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

wð1Þ ¼ c1 xð1Þ þ w0 ð1Þ; w0 ð1Þ ¼ c1 x0 ð1Þ þ w00 ð1Þ:

ð24Þ ð25Þ

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

H1 þ iH2 ¼ 2w0 ðzÞ;

ð26Þ

B1 þ iB2 ¼ 2lm w0 ðzÞ;

ð27Þ

1 x 1Þ ðH þ iH2 Þ; q jx0 ð1Þj 1 1 x0 ð1Þ ðB þ iB2 Þ: Bn þ iBt ¼ q jx0 ð1Þj 1 Hn þ iHt ¼

0ð

ð28Þ ð29Þ

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

5

On the boundary of the hole, one has

ðHn þ iHt Þs ¼

2w0 ðrÞ ðH þ iH2 Þs ; rjx0 ðrÞj 1

ð30Þ

2w0 ðrÞ ðB þ iB2 Þs : rjx0 ðrÞj 1

ðBn þ iBt Þs ¼

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

"

#





x0 ðrÞ  c 2 c lc Re w0 ðrÞ ; H1 þ iH2 ¼ lm Re 0 rjx ðrÞj rjx0 ðrÞj " Im

#



ð33Þ



x0 ðrÞ  c 2 c w0 ðrÞ : H þ iH2 ¼ Im rjx0 ðrÞj 1 rjx0 ðrÞj

ð34Þ

Eq. (33) can be re-expressed as

2

2w0 ðrÞ þ

r2

w0 ðrÞ ¼

lc x0 ðrÞ l ð2CÞ þ c x0 ðrÞð2CÞ: lm r2 lm

ð35Þ

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

2w00 ðrÞ þ where

f0m ðrÞ ¼

2

r2



w00 ðrÞ ¼ fom ðrÞ; 

ð36Þ 



lc x0 ðrÞ lc  ð2CÞ  2c1 þ 2C  2c1 x0 ðrÞ: 2 lm r lm

ð37Þ

Multiplying both sides of Eq. (36) with 21pi  rdr1 where 1 stands for an arbitrary point outside the unit circle, and then calculating the Cauchy integral along the circle (Muskhelishivili, 1953), we get

w00 ð1Þ ¼

  1 R

iðm þ 1Þ Bc2  B1 ; 2 2lm 12

ð38Þ

which leads to

w0 ð1Þ ¼ 

  1 R

iðm þ 1Þ Bc2  B1 : 2 2lm 1

ð39Þ

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

Bc2 ¼ B1 2 þ

1  llm b 1 c B ; 1 þ ba llm a 2

ð40Þ

c

which implies that the magnetic induction inside the hole depends on the remotely applied magnetic load, the magnetic permeability 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 permeable boundary condition as

  P ; wð1Þ ¼ c1 R 1 þ

1

ð41Þ

where

P¼

2a Bc2  1: a þ b B1 2

ð42Þ

3.1.2. Impermeable boundary condition In 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

6

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

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

wð1Þ ¼ c1 R



1

1

1

 :

ð44Þ

3.1.3. Conductive boundary condition In 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 c1 R



1þ

1

1

 :

ð46Þ

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

wk ð1Þ ¼ c1 R



1þ

Pk

1

 ;

ð47Þ

where

8 > < P; k ¼ 1; for permeable boundary Pk ¼ 1; k ¼ 2; for impermeable boundary > : þ1; k ¼ 3; for conducting boundary:

ð48Þ

3.2. Complex potential of magnetic-elastic field 3.2.1. Permeable boundary condition Denoting the stress potential function by u and w, we can express them as

uðzÞ ¼ C1 z þ u0 ðzÞ;

ð49aÞ

wðzÞ ¼ C2 z þ w0 ðzÞ;

ð49bÞ

where u0 ðzÞ and w0 ðzÞ are two analytic complex-variable function outside the hole up to infinity, respectively. When B1 2 is applied only at infinity, the total stress at the outside surface of the solid can be calculated according to Eqs. (3)–(5) as

T1 11 ¼  T1 22 ¼

l0 B12 2 ; 2l21

l0 2

ð4v1 þ 1Þ

ð50Þ B12 2

l21

ð51Þ

;

1 T1 12 ¼ T 21 ¼ 0;

ð52Þ

where l1 is the magnetic permeability of the surrounding space and l1 ¼ l0 ð1 þ v1 Þ. After inserting Eqs. (49) and (41) into Eqs. (9) and (10), letting z ! 1 and then using Eqs. (50)–(52) we have

! 1 2l0 v1 B12 12 2 ; C1 ¼ þ m2 c 4 l21 " # 1 l0 ð2v1 þ 1ÞB12 12 2 : C2 ¼  m3 c 2 l21

ð53aÞ ð53bÞ

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

T c11 ¼ 

Hc2 2 ;

ð54Þ

ð4vc þ 1ÞHc2 2 ; 2 c ¼ T 21 ¼ 0:

ð55Þ

T c22 ¼ T c12

l0

l0

2

ð56Þ

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



uðzÞ þ zu0 ðzÞ þ wðzÞ þ

m2 m3 wk ðzÞw0k ðzÞ þ 2 2

Z

 Z e c þ iY e c Þds; w0k ðzÞ2 dz ¼ i ð X s

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

ð57Þ

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

7

e c and Y e c , which can be expressed as Now, let us calculate the value of X

e c ¼ n1 T c þ n2 T c ; X 11 21 e c ¼ n1 T c þ n2 T c ; Y 12 22

ð58Þ ð59Þ

where

dx2 ; ds dx1 n2 ¼ cosðN; x2 Þ ¼  ; ds

n1 ¼ cosðN; x1 Þ ¼

ð60Þ ð61Þ

and N stands for the normal direction of the boundary. Using Eqs. (58)–(61), we obtain

 e c þ iY e c ¼ n1 T c þ n2 T c þ i n1 T c þ n2 T c : X 11 21 12 22

ð62Þ

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

i

Z

c2

e c þ iY e c Þds ¼ l0 H2 ½2v z þ ð2v þ 1Þz: ðX c c 2

ð63Þ

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



uðzÞ þ zu0 ðzÞ þ wðzÞ þ

m2 m3 wk ðzÞw0k ðzÞ þ 2 2

Z

 l Hc2 w0k ðzÞ2 dz ¼ 0 2 ½2vc z þ ð2vc þ 1Þz: 2 s

ð64Þ

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

uðrÞ þ

xðrÞ 0 m w ðrÞw0 ðrÞ m u ðrÞ þ wðrÞ þ 2 k 0 k þ 3 0 2 2 x ð rÞ x ðrÞ

Z

w0k ðrÞ2

x rÞ 0ð

¼ dr

l0 Hc2 2 2

½2vc xðrÞ þ ð2vc þ 1ÞxðrÞ:

ð65Þ

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

u0 ðrÞ þ

xðrÞ 0 u ðrÞ þ w0 ðrÞ ¼ f0 ðrÞ; x0 ðrÞ 0

ð66Þ

where

f0 ðrÞ ¼ 

m2 wk ðrÞw0k ðrÞ m3  2 2 x0 ðrÞ

Z

w0k ðrÞ2

x0 ðrÞ

" þ dr

l0 Hc2 2 2

# ð2vc þ 1Þ  C2 xðrÞ þ ðl0 vc Hc2 2  2C1 ÞxðrÞ:

ð67Þ

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

I 1 f 0 ð rÞ dr; 2pi r r  1 I 1 f0 ðrÞ 1 þ m12 0 dr  1 2 u ð1Þ: w0 ð1Þ ¼  2pi r r  1 1 m 0

u0 ð1Þ ¼ 

ð68Þ ð69Þ

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

m2 Rðc1 c1  P2k c1 c1 þ Pk c12 mÞ1  RPk c1 c1 r3 1  mr2 "2 # m2 m l0 Hc2 1 3 c2 1 1 2 2 1 þ  RPk c c  Rc þ Rð2vc þ 1Þ  RC2 þ Rml0 vc H2  2RmC1 2 2 2 r " # m3 RP2k c1 2 l0 Hc2 c2 2 þ  þ Rmð2vc þ 1Þ  RmC2 þ Rl0 vc H2  2RC1 r 2 m 2 pffiffiffiffiffi R 1  mr pffiffiffiffiffi : þ pffiffiffiffiffiffiffi ðc1 m  Pk c1 Þ2 ln 1 þ mr 2 m3

f0 ðrÞ ¼ 

ð70Þ

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

u0 ð1Þ ¼

k1

1

;

pffiffiffiffiffi k2 1 k3 1 m pffiffiffiffiffi ; w0 ð1Þ ¼ 2 þ þ k4 ln 1 m 1 1þ m

ð71Þ ð72Þ

8

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

where

  m2 m3 12 l0 ð2mvc þ 2vc þ 1Þ c2 H2 þ C2 þ 2mC1 ; k1 ¼ R  Pk c12 þ c  2 2 2 k2 ¼ 

  m2 R c12 Pk k1  mc12 Pk þ P2k c12  c12 þ þ mk1 ; 2 m m

" #   m3 P2k c12 m2 c12 Pk ml0 2vc k1 c2 k3 ¼ R   2vc þ þ 1 H2 þ mC2 þ 2C1  ; 2m 2m 2 m m Rm3 k4 ¼  pffiffiffiffiffiffiffi ðc1 m  Pk c1 Þ2 : 4 m3

ð73aÞ

ð73bÞ

ð73cÞ

ð73dÞ

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

T h þ T q ¼ 4Re½Uð1Þ þ m2 Wð1ÞWð1Þ; i 12 h  T h  T q þ 2iT qh ¼ 2 f ð1ÞU0 ð1Þ þ f 0 ð1ÞWð1Þ þ m2 W 0 ð1Þxð1Þ þ m3 f 0 ð1ÞW 2 ð1Þ ; 2 0 q f ð1Þ

ð74Þ ð75Þ

where

Uð1Þ ¼ u0 ð1Þ=x0 ð1Þ;

Wð1Þ ¼ w0 ð1Þ=x0 ð1Þ;

Wð1Þ ¼ w0 ð1Þ=x0 ð1Þ:

3.2.2. Impermeable boundary condition In Eq. (74), letting k ¼ 2; Pk ¼ 1 and Hc2 ¼ 0, we have

m m3 12 2 þ c þ C2 þ 2mC1 ; 2 2    m2 R 1 12 1 c þ mþ k1 ; m k2 ¼  2 m m   ðm2 þ m3 Þc12 k1 k3 ¼ R þ mC2 þ 2C1  ; 2m m Rm3 k4 ¼  pffiffiffiffiffiffiffi ðm þ 1Þ2 c12 : 4 m3 k1 ¼ R

ð76aÞ ð76bÞ ð76cÞ 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 condition In Eq. (74), if we let k ¼ 3; P3 ¼ þ1 and Hc2 ¼ 0, we have

m  m 3 2 12 c þ C2 þ 2mC1 ; 2   m2 R 1 k1  m c12 þ þ mk1 ; k2 ¼  2 m m   ðm3  m2 Þc12 k1 k3 ¼ R þ mC2 þ 2C1  ; 2m m Rm3 2 12 k4 ¼  pffiffiffiffiffiffiffi ðm  1Þ c : 4 m3 k1 ¼ R

ð77aÞ ð77bÞ ð77cÞ 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 permeable crack, P1 ¼ 1, and Eq. (73) becomes

k1 ¼ 

 a 1 T  T c22 ; 2 22

k2 ¼ 2k1 ;

k3 ¼ 0;

k4 ¼ 0:

ð78Þ

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

9

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

Finally, we have from Eq. (47) as

wðzÞ ¼ c1 z:

ð79Þ

And thus

B1 2

Hc2 ¼

lc l0

T c22 ¼

ð80Þ

;

2

 1 2 B ð4vc þ 1Þ 2 :

ð81Þ

lc

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

 1     T 22  T c22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c uðzÞ ¼ z; z2  a2 þ C1  T 1 22  T 22 2 2  a2  1 1 wðzÞ ¼  T  T c22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C2 z: 2 2 22 z  a2

ð82Þ ð83Þ

It is shown from Eqs. (9), (10), (82) and (83) that for the magnetic permeable crack, the total stress field has singularity of the pffiffiffi 1= r at the crack tip. Thus, the total stress intensity factor can be defined as T

T

ðkI ; kII Þ ¼ lim

x1 !a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðx1  aÞðT 22 ; T 12 Þ:

ð84Þ

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

  x1 c c T 22  iT 12 ¼ T 1 22  T 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ T 22 ; x21  a2

ð85Þ

c where T 1 22 and T 22 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

T

kI ¼ T kII

pffiffiffiffiffiffi 1  pa T 22  T c22 ;

ð86Þ

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 based on 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 T

kI ¼

l0  2

B1 2

 2 4v 1 þ 1

l21



 4vc þ 1 pffiffiffiffiffiffi pa: 2

ð88Þ

lc

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

k1 ¼ 

 a 1 T þ m2 c12 ; k2 ¼ 2k1 ; 2 22

k3 ¼ 0;

a k4 ¼  m3 c12 : 2

ð89Þ

10

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

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

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ ¼ c1 z2  a2 ;  1     T þ m2 c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 12 uðzÞ ¼ 22 z2  a2 þ C1  T 1 z; 22 þ m2 c 2 2  a2  1 1 am3 c12 z  a þ C2 z: T þ m2 c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ln wðzÞ ¼  2 2 zþa 2 22 4 z a

ð90Þ ð91Þ ð92Þ

Additionally, we have

  x1 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  m2 c12 ; T 22  iT 12 ¼ T 1 22 þ m2 c x21  a2

ð93Þ

pffiffiffi which means that for the magnetic impermeable crack, the total stress field has still singularity of the 1= r at the crack tip. Inserting Eq. (93) into Eq. (84), we get T

kI ¼

pffiffiffiffiffiffi 1  pa T 22 þ m2 c12 ;

T

kII ¼ 0:

ð94Þ

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

kI ¼

l0  2

B1 2

 2 4v1 þ 1

l21



 1 m2 pffiffiffiffiffiffi pa: l2m 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 ¼

 b 1 T þ m2 c12 ; 2 11

k2 ¼ 2k1 ;

k3 ¼ 0;

k4 ¼

ib m3 c12 : 2

ð96Þ

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 wðzÞ ¼ c1 z2 þ b ;  1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    T 11 þ m2 c12 1 2 12 z2 þ b þ C1  T 1 uðzÞ ¼ z; 11 þ m2 c 2 2

ð97Þ ð98Þ

2

wðzÞ ¼ 

 b  1 1 ibm3 c12 z  ib þ C2 z T þ m2 c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln 2 z þ ib 2 11 4 2 z þb

ð99Þ

and

  x2 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  m2 c12 : T 22  iT 12 ¼ T 1 11 þ m2 c 2 2 x2  b

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

ð100Þ

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

11

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  T T kI ; kII ¼ lim 2pðx2  bÞðT 11 ; T 21 Þ:

ð101Þ

x2 !b

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

kI ¼

pffiffiffiffiffiffi 1  pb T 11 þ m2 c12 ;

T

kII ¼ 0:

ð102Þ

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

kI ¼ 

B12 2 2



 l0 m2 pffiffiffiffiffiffi þ 2 pa: 2 l1 2lm

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  107 T  m=A, and the semi-major axis of the elliptic hole is 0.01 m. Shown in Figs. 4 and 5 (here vm ¼ 103 and v1 ¼ vc Þ is the distribution of total hoop stress T h at the hole rim based on three magnetic boundary conditions, and it is found that the hoop stress for a magnetically impermeable hole is lager than that for a magnetically permeable hole, but as the vc =vm becomes smaller, the effects of magnetic boundary conditions on

0.8

permeable model impermeable model conductive model

∞

0.7

B2 =2T

0.6

a/b=5

0.5

Tθ (MPa)

∞

B2

0.4 0.3

b

a

0.2 0.1 0.0 -0.1 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

θ (rad) Fig. 4. Effects of magnetic boundary condition on the total hoop stress T h for

vc =vm ¼ 0:1.

7 6

permeable model impermeable model conductive model

∞

B2 =2T a/b=5

5

Tθ (MPa)

4 ∞

B2

3

b

2

a

1 0 -1 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

θ (rad) Fig. 5. Effects of magnetic boundary condition on the total hoop stress T h for

vc =vm ¼ 0:01.

12

C. Chang et al. / International Journal of Engineering Science 52 (2012) 1–21 ×10

3

3.5

permeable model impermeable model conductive model

∞

B2 =2T

3.0

-3

b/a=10

Tθ (MPa)

2.5 ∞

B2

2.0

b

1.5

a

1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

θ (π /1000) Fig. 6. Effects of the hole geometry on the total hoop stress T h for b=a ¼ 103 .

×10

5

2.5

permeable model impermeable model conductive model

∞

B2 =2T -4

b/a=10

Tθ (MPa)

2.0

∞

1.5

B2 b

1.0

a

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

θ (π /1000) Fig. 7. Effects of the hole geometry on the total hoop stress T h for b=a ¼ 104 .

6 5

∞

B2 ∞

B2 =0.5T

4

b

Tθ (MPa)

∞

B2 =1T

3

a

∞

B2 =2T

2 1 0 -1 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

θ (rad) Fig. 8. Effects of the applied magnetic load on T h .

the hoop stress become weaker. In Figs. 6 and 7 (here it is taken that vm ¼ 103 ; vc ¼ v1 ; vc =vm ¼ 102 Þ, the total hoop stress T h 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

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

13

25

20

∞

B2 χm/χc=1000

Tθ (MPa)

15

b

χm/χc=100

a

χm/χc=10

10

5

0

-5 0.0

0.2

0.4

0.8

0.6

1.0

1.2

1.4

θ (rad) Fig. 9. Effects of magnetic susceptibility of the media on T h when the hole and the surrounding space at infinity are filled with the same medium ðv1 ¼ vc Þ.

0.8 ∞

B2

Tθ (MPa)

0.6

χc/χ∞=0.01

b

χc/χ∞=0.1

a

χc/χ∞=0.5

0.4

0.2

0.0

-0.2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

θ (rad) Fig. 10. Effects of magnetic susceptibility of the media on T h when the hole and the surrounding space at infinity are filled with different media ðv1 > vc Þ.

25

20 ∞

B2 χc/χ∞=100

Tθ (MPa)

15

χc/χ∞=10

b

a

χc/χ∞=2

10

5

0

-5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

θ (rad) Fig. 11. Effects of magnetic susceptibility of the media on T h when the hole and the surrounding space at infinity are filled with different media ðv1 < vc Þ.

tip is approaching to a constant for a permeable or conductive crack (Fig. 7), while it is singular for an impermeable crack. 3 2 Given in Fig. 8 (here B1 2 ¼ 2 T; vm ¼ 10 ; vc ¼ v1 ; vc =vm ¼ 10 Þ is the total hoop stress T h under different loadings, and it is found that as the applied load increases, T h increases for three magnetic boundary conditions.

14

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

0.35

χc/χ∞=100 χc/χ∞=10

0.30

χc/χ∞=2

)

0.25

∞

B2

0.15

T

kI (MNm

-3/2

0.20

0.10 0.05 0.00

-0.05 0.0

0.5

1.0

1.5

2.0

∞

B2 (T) T

Fig. 12. Effects of magnetic susceptibility of the media on the intensity factor of total stress kI when the crack and the surrounding space at infinity are filled with different media ðv1 < vc Þ.

0.05 0.00

-0.10

T

k I (MNm

-3/2

)

-0.05

-0.15 -0.20

χc/χ∞=0.01 χc/χ∞=0.1 χc/χ∞=0.5 ∞

B2

-0.25 -0.30 -0.35 0.0

0.5

1.0

1.5

2.0

∞

B2 (T) T

Fig. 13. Effects of magnetic susceptibility of the media on the intensity factor of total stress kI when the crack and the surrounding space at infinity are filled with different media ðv1 > vc Þ.

To discuss the effects of the magnetic susceptibility on the stress distribution, plotted in Figs. 9–11 are the variation of the total hoop stress at the rim of a permeable hole. It is shown in Fig. 9 that when the surrounding space at infinity is filled with the same medium as that inside the hole, i.e., v1 ¼ vc , as vm =vc increases, the total hoop stress becomes larger. In addition, it can be found from Figs. 10 and 11 that the stress concentration is greater for the case of v1 < vc than that for the case of v1 > 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 the case of v1 – qvc , it can be seen from Figs. 12 and 13 that when v1 < vc , the applied magnetic load may enhance the crack open, 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 potential method. Based on three kinds of magnetic boundary condition on the surface of the hole, the general solutions for complex potentials are presented in exact and explicit form when the solid is subjected to the remote uniform magnetic fields. When the hole degenerates into a crack, more concise results are obtained for intensity factors of the total stresses. It is found that the effects of magnetic fields on cracks in the soft ferromagnetic solid are dependent on the adopted boundary conditions along the crack surface and at infinity. For a magnetically permeable crack, the Maxwell stresses in general have to be considered on the surface of the crack and the remainder surface of the solid. Otherwise, if the Maxwell stress is neglected on the surface of the crack or at infinity, the applied magnetic loads may lead to singular stresses in the solid. However, it is now difficult to conclude which crack model is more reasonable since such a conclusion must await further experimental data and additional theoretical and/or computational results. Even so, the results in the present work can still serve as the fun-

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

15

damental solutions to test the correctness of other solutions for more complicated crack problems in soft ferromagnetic materials, 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. Programs Foundation of Ministry of Education of China (20093218110004), and the Program for Changjiang Scholars and Innovative 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 equations can be found in previous publications, it is also necessary to re-derive and check them for the sake of self-containing and reliability. A.1. Expressions of magnetic fields For the 2D problem of a magnetic solid in a rectangular-coordinate system ðx1  x2 Þ, the magnetic equilibrium Eq. (1) can be reduced to:

H2;1  H1;2 ¼ 0;

ðA1Þ

B1;1 þ B2;2 ¼ 0;

ðA2Þ

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

@n ; @x1 @n : H2 ¼ @x2 H1 ¼

ðA3Þ ðA4Þ

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

r2 n ¼ 0;

ðA5Þ

2

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

r2 ¼

@2 @2 þ : @x21 @x22

ðA6Þ

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

@2n ¼ 0: @z@z

ðA7Þ

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

n ¼ wðzÞ þ wðzÞ;

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 w0 ðzÞ þ w0 ðzÞ;

ðA9aÞ

H2 ¼ iðw0 ðzÞ  w0 ðzÞÞ:

ðA9bÞ

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

H1 þ iH2 ¼ 2w0 ðzÞ;

ðA10Þ

M1 þ iM2 ¼ 2vw0 ðzÞ;

ðA11Þ

B1 þ iB2 ¼ 2lw0 ðzÞ:

ðA12Þ

A.2. Expressions of stress fields From Eqs. (3)–(5), we have the total stress as:

T ij ¼ rij þ

l0 1 M M þ lHi Hj  l0 Hk Hk dij : v i j 2

ðA13Þ

16

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

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

rij;i þ l0 ð2v þ 1ÞHi Hj;i  l0 Hi;j Hi ¼ 0:

ðA14Þ

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

r11;1 þ r21;2 þ 2l0 vðH1 H1;1 þ H2 H1;2 Þ þ l0 H2 ðH1;2  H2;1 Þ ¼ 0; r12;1 þ r22;2 þ 2l0 vðH1 H2;1 þ H2 H2;2 Þ þ l0 H1 ðH2;1  H1;2 Þ ¼ 0:

ðA15Þ ðA16Þ

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

r11;1 þ r21;2 þ 2l0 vðH1 H1;1 þ H2 H2;1 Þ ¼ 0; r12;1 þ r22;2 þ 2l0 vðH1 H1;2 þ H2 H2;2 Þ ¼ 0: 

Define a new function V ¼ l0 v 0

V ¼ 4l 0 v w

H21

þ

H22

ðA17Þ

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

ðzÞw0 ðzÞ:

ðA18Þ

ðA19Þ

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

@V ¼ 0; @x1 @V r12;1 þ r22;2 þ ¼ 0: @x2

r11;1 þ r21;2 þ

ðA20Þ ðA21Þ

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

@u1 @u2 @u1 @u2 ; e22 ¼ ; e12 ¼ e21 ¼ þ ; @x1 @x2 @x2 @x1 1 1 2ð1 þ tÞ e11 ¼ ðr11  tr22 Þ; e22 ¼ ðr22  tr11 Þ; e12 ¼ e21 ¼ r12 ; E E E 2 2 2 @ e11 @ e22 @ e12 þ ¼ ; @x1 @x2 @x22 @x21

e11 ¼

where eij stand for the elastic strain, E is the modulus of elasticity, and Substituting Eqs. (A22) and (A23) into Eq. (A24) leads to

@ 2 r11 @ 2 r22 @ 2 r11 @ 2 r22 þ t þ 2 2 @x2 @x1 @x21 @x22

! ¼

ðA22Þ ðA23Þ ðA24Þ

t is the Poisson’s ratio.

2ð1 þ tÞ@ 2 r12 : @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; r12;12 þ r22;22 þ V ;22 ¼ 0:

ðA26Þ ðA27Þ

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

2ð1 þ tÞ@ 2 r12 ð1 þ tÞ@ 2 r12 2ð1 þ tÞ@ 2 r12 ¼ þ ¼ ð1 þ tÞðr11;11 þ r22;22 þ V ;11 þ V ;22 Þ: @x1 @x2 @x1 @x2 @x1 @x2

ðA28Þ

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

r2 ðr11 þ r22 Þ ¼ ð1 þ tÞr2 V:

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

r2 ½r2 U  ð1  tÞV ¼ 0:

ðA31Þ

Due to

w0 ðzÞw0 ðzÞ ¼

@ 2 ðwðzÞwðzÞÞ : @z@z

ðA32Þ

Eq. (A31) can be written as

i @4 h m2 U  ¼ 0; wðzÞwðzÞ 2 @z @z2 4

ðA33Þ

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

17

where

m2 ¼ 4l0 vð1  tÞ:

ðA34Þ

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

U¼

m 1 2 hðzÞ þ hðzÞ þ zuðzÞ þ zuðzÞ þ wðzÞwðzÞ: 2 4

ðA35Þ

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

r11 þ r22 ¼ r2 U  2V ¼ 2ðu0 ðzÞ þ u0 ðzÞÞ þ m1 w0 ðzÞw0 ðzÞ; r22  r11 þ 2ir12 ¼ U ;11  U ;22 þ 2iU ;12



@ @ i ¼ @x @y

2

ðA36Þ

@2U U ¼ 4 2 ¼ 2ðzu00 ðzÞ þ w0 ðzÞÞ þ m2 w00 ðzÞwðzÞ; @z

ðA37Þ

where

m1 ¼ 4l0 vð1 þ tÞ: On the other hand, the Maxwell stress can be expressed, by using Eqs. (5) and (A10), as M 0 0 tM 11 þ t 22 ¼ 4l0 vm w ðzÞw ðzÞ;

tM 22



tM 11

þ

M 2it 12

0

ðA38Þ 2

¼ 4lm w ðzÞ :

ðA39Þ

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

T 11 þ T 22 ¼ 2ðu0 ðzÞ þ u0 ðzÞÞ þ m2 w0 ðzÞw0 ðzÞ; 00

0

ðA40Þ 00

0

2

T 22  T 11 þ 2iT 12 ¼ 2ðzu ðzÞ þ w ðzÞÞ þ m2 w ðzÞwðzÞ þ m3 w ðzÞ ;

ðA41Þ

0

where wðzÞ ¼ h ðzÞ, and

m3 ¼ 4l0 ð2v þ 1Þ:

ðA42Þ

A.3. Expressions of displacement fields From Eq. (A23), one has

E

@u1 ¼ ðr11 þ r22 Þ  ð1 þ tÞr22 ; @x1

ðA43Þ

@u2 ¼ ðr11 þ r22 Þ  ð1 þ tÞr11 ; @x2   E @u2 @u1 ¼ r12 : þ 2ð1 þ tÞ @x1 @x2

E

ðA44Þ ðA45Þ

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

@u1 ¼ 2ðu0 ðzÞ þ u0 ðzÞÞ þ m1 w0 ðzÞw0 ðzÞ  ð1 þ tÞðU ;11  VÞ; @x1 @u2 ¼ 2ðu0 ðzÞ þ u0 ðzÞÞ þ m1 w0 ðzÞw0 ðzÞ  ð1 þ tÞðU ;22  VÞ: E @x2

E

ðA46Þ ðA47Þ

Considering the result such that m1 x0 ðzÞx0 ðzÞ þ ð1 þ tÞV ¼ 0, Eqs. (A46) and (A47) can be re-written as

Eu1 ¼ 2ðuðzÞ þ uðzÞÞ  ð1 þ tÞU ;1 þ g 1 ðx2 Þ;

ðA48Þ

Eu2 ¼ 2iðuðzÞ  uðzÞÞ  ð1 þ tÞU ;2 þ g 2 ðx1 Þ;

ðA49Þ

where g 1 ðx2 Þ and g 2 ðx1 Þ are arbitrary real functions. Substituting Eqs. (A48) and (A49) into Eq. (A45), and using

r12 ¼ U ;12 , we get

dg 2 ðx1 Þ dg ðx2 Þ ¼ 1 : dx1 dx2

ðA50Þ

From Eq. (A50), one has

g 1 ðx2 Þ ¼ u01  bx2 ;

ðA51Þ

g 2 ðx1 Þ ¼ u02  bx1 ;

ðA52Þ

which means that g 1 ðx2 Þ and g 2 ðx1 Þ represent the rigid body displacements, and they can be neglected.

18

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

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

2Gðu1 þ iu2 Þ ¼ 4uðzÞ  ð1 þ tÞðU ;1 þ iU ;2 Þ;

ðA53Þ

where

U ;1 þ iU ;2 ¼ 2

@U m2 ¼ uðzÞ þ wðzÞ þ zu0 ðzÞ þ wðzÞw0 ðzÞ: @z 2

ðA54Þ

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

2Gðu1 þ iu2 Þ ¼ juðzÞ  zu0 ðzÞ  wðzÞ 

m2 wðzÞw0 ðzÞ; 2

ðA55Þ

where

G¼

E ; 2ð1 þ tÞ

j¼

3t : 1þt

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 boundary condition as follows: Consider an arc AB which represents an arbitrary part on the solid boundary, as shown in Fig. 14, where e and Y e are the surface total s is the length of the arc from point A to point B, N stands for the outside normal direction, and X loads. Using the following equations:

dx2 ; ds dx1 n2 ¼ cosðN; x2 Þ ¼  : ds

n1 ¼ cosðN; x1 Þ ¼

ðA56Þ ðA57Þ

e and Y e can be expressed as X

e ¼ n1 ðT 11 Þ þ n2 ðT 21 Þ ; X s s e ¼ n1 ðT 12 Þ þ n2 ðT 22 Þ : Y s s

ðA58Þ ðA59Þ

According to Eq. (A13), we have

H2 ; 2 1 ¼ r12 þ l0 ð2vm þ 1ÞH1 H2 :

ðA61Þ

2

l0 2

ð4vm þ 1ÞH21 

l0

ðA60Þ

T 22 ¼ r22 þ T 12 ¼ T 21

l0

H22 ;

T 11 ¼ r11 þ

ð4vm þ 1ÞH22 

2

l0

ðA62Þ

Using Eqs. (A56)–(A62), we obtain

  e þ iY e ¼ n1 ðT 11 Þ þ n2 ðT 21 Þ þ i½n1 ðT 12 Þ þ n2 ðT 22 Þ  ¼ dx2 ðT 11 Þ  dx1 ðT 21 Þ þ i dx2 ðT 12 Þ  dx1 ðT 22 Þ X s s s s s s s s ds ds ds ds   Z d @U m3 ¼ i 2 þ w0 ðzÞ2 dz : ds @z 2

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

ðA63Þ

19

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

From Eq. (A63) we have

2

@U m3 þ @z 2

Z

w0 ðzÞ2 dz ¼ i

Z

e þ iY e Þds: ðX

ðA64Þ

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

uðzÞ þ zu0 ðzÞ þ wðzÞ þ

m2 m3 wðzÞw0 ðzÞ þ 2 2

Z

w0 ðzÞ2 dz ¼ i

Z

e þ iY e Þds; ðX

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) which are obtained based on the elliptic-hole-method. To this end, consider a permeable crack along the x1 axis in a soft ferromagnetic solid subjected to the remote magnetic load B1 2 along the x2 axis at infinity, as shown in Fig. 2. It is also assumed that the magnetic permeabilities inside the cracks, in ferromagnetic medium and at the surrounding space at infinity are different, and they are lc ¼ l0 ð1 þ vc Þ, lm ¼ l0 ð1 þ vm Þ and l1 ¼ l0 ð1 þ v1 Þ, respectively. For the 2D problem of deformation, the magnetic boundary conditions at the crack faces are

Bþ2 ¼ B2 ;

Hþ1 ¼ H1 ;

on L;

ðB1Þ

where L stands for the crack. Using Eqs. (7), (8) and (22) one has 0 B2 ¼ 2Im½lm w0 ðzÞ ¼ B1 2 þ 2lm Im½w0 ðzÞ;

H1 ¼

2Re½w0 ðzÞ

¼

H1 1

þ

ðB2Þ

2Re½w00 ðzÞ:

ðB3Þ

Substituting Eqs. (B2) and (B3) into Eq. (B1) leads to 0þ 0  0þ  0 w 0 ðx1 Þ  w0 ðx1 Þ ¼ w0 ðx1 Þ  w0 ðx1 Þ; 0 0þ 0þ  0 ðx1 Þ þ w0  0 ðx1 Þ þ w0 ðx1 Þ ¼ w w 0 ðx1 Þ;

1 < x1 < þ1;

ðB4Þ

1 < x1 < þ1:

ðB5Þ

From Eqs. (B4) and (B5), we get 0 w0þ 0 ðx1 Þ  w0 ðx1 Þ ¼ 0;

1 < x1 < þ1:

ðB6Þ

The solution of Eq. (B6) is

w00 ðzÞ ¼ w00 ð1Þ ¼ 0:

ðB7Þ

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

w0 ðzÞ ¼ c1

and wðzÞ ¼ c1 z:

ðB8Þ

It is indicated from Eq. (B8) that the magnetic field inside the medium is uniform which means field inside the crack are also uniform such that Bc2 ¼ B1 2 . On the other hand, we have from Eqs. (9) and (10) that

T 22  iT 12 ¼ u0 ðzÞ þ u0 ðzÞ þ zu00 ðzÞ þ w0 ðzÞ þ

m2 0 m2 00 m3 0 2 w ðzÞwðzÞ þ w ðzÞ : w ðzÞw0 ðzÞ þ 2 2 2

1 B¼ 2 B2 ,

and that the magnetic

ðB9Þ

Substituting Eq. (B8) into (B9) gives

T 22  iT 12 ¼ u0 ðzÞ þ u0 ðzÞ þ zu00 ðzÞ þ w0 ðzÞ þ 2V 0 ;

ðB10Þ

where V 0 is a constant given by

V0 ¼

1 1 m2 jc1 j2 þ m3 ðc1 Þ2 : 4 4

Define a new function, XðzÞ, as

 0 ðzÞ:  0 ðzÞ þ zu  00 ðzÞ þ w XðzÞ ¼ u

ðB11Þ

Then, Eq. (B10) can be rewritten as

T 22  iT 12 ¼ UðzÞ þ XðzÞ þ ðz  zÞU0 ðzÞ þ 2V 0 ; where UðzÞ ¼ u ðzÞ. On the surfaces of the crack, Eq. (B12) becomes to 0

ðB12Þ

20

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

Uþv ðx1 Þ þ Xv ðx1 Þ ¼ T þ22  iT þ12 ; Uv ðx1 Þ þ Xþv ðx1 Þ ¼ T 22  iT 12 ;

ðB13Þ ðB14Þ

where

Uv ðzÞ ¼ UðzÞ þ V 0 ;

Xv ðzÞ ¼ XðzÞ þ V 0 :

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

T þ22 ¼ T 22 ¼ T c22 ; where have

T c22

and

T c22 ¼

l0 2

T c12

T þ12 ¼ T 12 ¼ T c12 ¼ 0;

ðB15Þ

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

ð4vc þ 1ÞHc2 2 ;

T c12 ¼ T c21 ¼ 0;

ðB16Þ

where Hc2 ¼ Bc2 =lc and Bc2 ¼ B1 2 . On the other hand, using Eqs. (B13)–(B16), we obtain

½Uv ðx1 Þ þ Xv ðx1 Þþ þ ½Uv ðx1 Þ þ Xv ðx1 Þ ¼ 2q1 ðx1 Þ;

ðB17Þ

½Uv ðx1 Þ  Xv ðx1 Þþ  ½Uv ðx1 Þ  Xv ðx1 Þ ¼ 2q2 ðx1 Þ;

ðB18Þ

where

 1 þ T þ T 22  iðT þ12 þ T 12 Þ ¼ T c12 ; 2 22  1

q2 ðx1 Þ ¼ T þ22  T 22  iðT þ12  T 12 Þ ¼ 0: 2

q1 ðx1 Þ ¼

ðB19Þ ðB20Þ

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



z



Uv ðzÞ þ Xv ðzÞ ¼ T c22 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 z2  a2 Uv ðzÞ  Xv ðzÞ ¼ Uv ð1Þ  Xv ð1Þ;

C1z þ C0 ; XðzÞ

ðB21Þ ðB22Þ

where C 1 and C 0 are two constants. In Eq. (B21), letting z ! 1 leads to

2C 1 ¼ Uv ð1Þ þ Xv ð1Þ:

ðB23Þ

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

T1 22 ¼ Uv ð1Þ þ Xv ð1Þ:

ðB24Þ

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

2C 1 ¼ T 1 22 :

ðB25Þ

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

 1  T  Tc z

22 : Uv ðzÞ þ Xv ðzÞ ¼ T c22 þ p22ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2  a2

ðB26Þ

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

T 22  iT 12 ¼ Uv ðxÞ þ Xv ðxÞ:

ðB27Þ

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

kI

¼

pffiffiffiffiffiffi 1  pa T 22  T c22 ;

per

kII 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 elliptichole-method in the present work. References Brown, W. F. Jr., (1966). Magnetoelastic interactions. New York: Springer. Chen, X. H. (2009). Nonlinear field theory of fracture mechanics for paramagnetic and ferromagnetic materials. ASME Journal of Applied Mechanics, 76, 041016. 1–7.

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

21

Dunkin, J. W., & Eringen, A. C. (1963). On the propagation of waves in an electromagnetic elastic solid. International Journal of Engineering Science, 1, 461–495. Fil’shtinskii, L. A. (1993). A plane problem of magneto-elasticity for a ferromagnetic medium weakened by cuts. Journal of Applied Mathamatical Mechanics, 57, 903–912. Gao, C. F., Mai, Y.-W., & Wang, B. L. (2008). Effects of magnetic fields on cracks in a soft ferromagnetic material. Engineering Fracture Mechanics, 75, 4863–4875. Hasanyan, D. J., & Harutyunyan, S. (2009). Magnetoelastic interactions in a soft ferromagnetic body with a nonlinear law of magnetization: Some applications. International Journal of Solids and Structures, 46, 2172–2185. Liang, W., Shen, Y. P., & Fang, D. N. (2002). Magnetoelastic coupling on soft ferromagnetic solids with an interface crack. Acta Mechanica, 154, 1–9. Liang, W., Shen, Y. P., & Zhao, M. H. (2000). Magnetoelastic formulation of soft ferromagnetic elastic problems with collinear cracks: Energy density fracture criterion. Theoretical and Applied Fracture Mechanics, 34, 49–60. Lin, C. B., Chen, S. C., & Lee, J. L. (2009). Explicit solutions of magnetoelastic fields in a soft ferromagnetic solid with curvilinear cracks. Engineering Fracture Mechanics, 76, 1846–1865. Lin, C. B., & Lin, H. M. (2002). The magnetoelastic problem of cracks in bonded dissimilar materials. International Journal of Solid Structures, 39, 2807–2826. Lin, C. B., & Yeh, C. S. (2002). The magnetoelastic problem of a crack in a soft ferromagnetic solid. International Journal of Solid Structures, 39, 1–17. Muskhelishivili, N. I. (1953). Some basic problems of the mathematical theory of elasticity. Gronnogen: Noordhoff. Pao, Y. H., & Yeh, C. S. (1973). A linear theory for soft ferromagnetic elastic solids. Interantional Journal of Engineering Science, 11, 415–436. Shindo, Y. (1977). The linear magnetoelastic problem for a soft ferromagnetic elastic solid with a finite crack. ASME Journal of Applied Mechanics, 44, 47–50. Tiersten, H. F. (1964). Coupled magnetomechanical equations for magnetically saturated insulators. Journal on Mathamatical Physics, 5, 1298–1318. Wan, Y. P., Fang, D. N., Soh, A. K., & Hwang, K. C. (2002). Effect of magnetostriction on fracture of a soft ferromagnetic medium with a crack-like flaw. Fatigue & Fracture of Engineering Materials & Structures, 26, 1091–1102.