Tensile testing and analysis of ferromagnetic elastic strip with a central crack in a uniform magnetic field

Acta Materialia 52 (2004) 4677–4684

www.actamat-journals.com

Tensile testing and analysis of ferromagnetic elastic strip witha central crack in a uniform magnetic field

Y. Shindo *, D. Sekiya, F. Narita, K. Hohiguchi

Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aoba-yama 02, Sendai 980-8579, Japan

Received 28 April 2004; received in revised form 16 June 2004; accepted 16 June 2004

Available online 17 July 2004

Abstract

The effect of magnetic fields on the fracture mechanics parameters such as the stress intensity factor, energy density, etc., is

discussed by analyzing the plane strain and plane stress problems of a soft ferromagnetic strip with a central crack under a uniform

magnetic field. The problem of an infinitely long soft ferromagnetic elastic strip with a central crack is formulated by means of

integral transforms and reduced to the solution of a Fredholm integral equation of the second kind. Numerical values on the

fracture mechanics parameters are obtained. Tensile tests are also conducted on center-cracked soft magnetic plate with strain gage

technique, and the numerical predictions for the plane stress case are compared with the test results. Agreement between theory and

experiment is fair.

� 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Theory and modeling; Tension test; Soft ferromagnetic materials; Fracture; Magneto-elastic interactions

1. Introduction

With the increasing applications of ferromagnetic

materials as fusion reactors, MHD devices, magneti-

cally levitated vehicles, etc., the theme of magneto-

elastic interactions has been addressed in the recent

past. If a ferromagnetic material is used in a magnetic

field, the combination of mechanical and magnetic

forces could produce elevated stresses and strains. Theferromagnetic materials may be degraded in such a

stress level. The strength of the ferromagnetic materials

is also weakened by the presence of defects such as voids

and cracks. It is therefore important to understand the

degradation phenomena of the ferromagnetic materials.

The fracture behavior of the ferromagnetic materials is

thus of much recent interest. The influence of the

magnetic field on the singular stress distributions nearthe crack tip in an infinite body was shown by Shindo

[1,2] based on a linear theory for the soft ferromagnetic

* Corresponding author. Tel./fax: +81-22-217-7341.

E-mail address: [email protected] (Y. Shindo).

1359-6454/$30.00 � 2004 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2004.06.029

elastic materials of multidomain structure [3]. Consid-ered by Shindo [4] is the problem of determining the

stress intensity factors in an infinitely long strip of a soft

ferromagnetic elastic material with a crack parallel to

the edges of the strip.

This paper applies a linear theory for the ferromag-

netic elastic materials of multidomain structure [3] to the

problem of determining the distribution of stress in a

cracked soft ferromagnetic strip permeated by a uniformmagnetostatic field normal to the crack surface. The

plane strain and plane stress problems are considered for

the infinitely long strip containing a central crack nor-

mal to the edges of the strip. The method of solution

involves the use of Fourier transforms to reduce the

mixed boundary value problem to two simultaneous

dual integral equations. The solution is then given in

terms of a single Fredholm integral equation of thesecond kind. Numerical values on the fracture me-

chanics parameters such as stress intensity factor, energy

density, etc. are obtained. Tensile tests are also con-

ducted on soft magnetic plate with a central crack to

obtain the values of the fracture mechanics parameters,

ll rights reserved.

mail to: [email protected]

4678 Y. Shindo et al. / Acta Materialia 52 (2004) 4677–4684

and the numerical results for the plane stress are com-

pared with the experimental values.

2. Problem statement and basic equations

Consider a soft ferromagnetic isotropic linear elastic

strip of width 2h and length 2lðl ! 1Þ which contains a

central crack of length 2a aligned with its plane normal

to the free edges as shown in Fig. 1. A rectangular

Cartesian coordinate system ðx; y; zÞ is attached to be the

center of the crack for reference purposes. The x-axis isdirected along the line of the crack and y-axis along thedirection of the perpendicular bisector of the crack. The

edges of the soft ferromagnetic elastic strip are therefore

the lines with equations x ¼ �h, while the crack occupies

the segment �a < x < a, y ¼ 0. We consider a uniform

normal stress, ryy ¼ r0, applied with a uniform magnetic

field of magnetic induction B0y ¼ B0. Only the first

quadrant with appropriate boundary conditions needs

to be analyzed owing to symmetry.All magnetic quantities are divided into two parts,

those in the rigid body state and those in the perturba-

tion state as follows:

B ¼ B0 þ b;

M ¼ M0 þm;

H ¼ H0 þ h;

ð1Þ

where B, M and H are the magnetic induction, magne-

tization and magnetic intensity vectors, respectively. The

first parts, which are indicated by the subscript 0, are

magnetic quantities in the underformed body. The sec-

ond parts, which are represented by lower case letters,

are corrections to account for the additional changes in

140

mm

2a

2h

Weights

Straingage

P

Crack

B0

B0

Cryocooler-cooledsuperconducting magnet

xy

o

Fig. 1. Testing set-up.

magnetic quantities due to deformations. The magneto-

elastic solution for the rigid body state is

Bec0y ¼ B0; Hec

0y ¼B0

l0

; Mec0y ¼ 0;

06 x < a; y ¼ 0; ð2Þ

Be0y ¼

B0

lr

; He0y ¼

B0

l0lr

; Me0y ¼ 0;

h < x < 1; 0 < y < 1; ð3Þ

B0y ¼ B0; H0y ¼B0

l0lr

; M0y ¼vB0

l0lr

;

06 x6 h; 0 < y < 1; ð4Þ

where B0y , H0y and M0y are, respectively, the y-compo-

nents of B0, H0 and M0. The superscript eðh < x < 1;0 < y < 1Þ and ecð06 x < a; y ¼ 0Þ indicate values

outside the strip. Note that l0 ¼ 4p� 10�7 N/A2 (H/m;

H: Henry) is the magnetic permeability of the vacuum,

lr ¼ 1þ v is the specific magnetic permeability, and v is

the magnetic susceptibility.The effect of the magnetization as induced by the

deformation becomes important for a cracked soft fer-

romagnetic solid in a magnetic field normal to the crack

surface. In this case, the body force of the type

l0M � rH must be considered on account of the sharp

gradient of magnetic field near the crack. For the mag-

netic field in the solid, it is assumed that:

hx;x þ hy;y ¼ 0; ð5Þ

hx;y � hy;x ¼ 0 ð6Þand

bx ¼ l0lrhx;by ¼ l0lrhy ;

�ð7Þ

mx ¼ vhx;my ¼ vhy ;

�ð8Þ

where a comma denotes partial differentiation with re-spect to the coordinate, and ðhx; hyÞ, ðbx; byÞ and ðmx;myÞare the x and y-components of h, b and m. Eqs. (5) and

(6) for the perturbed state are satisfied by introducing a

magnetic potential / such that:

hx ¼ /;x; hy ¼ /;y ; ð9Þ

/;xx þ /;yy ¼ 0: ð10Þ

Using a dipole model for the magnetization, we ob-

tain the equations of equilibrium as

txx;x þ tyx;y þvB0/;xy

lr¼ 0;

txy;x þ tyy;y þvB0/;yy ¼ 0:

)ð11Þ

lr

Y. Shindo et al. / Acta Materialia 52 (2004) 4677–4684 4679

The components of magnetoelastic stresses txx; tyy ; txy ¼tyx and Maxwell stresses tMxx ; t

Myy ; t

Mxy ¼ tMyx are also obtained

as:

txx ¼ rxx;

tyy ¼ ryy þvB2

0

l0l2rþ 2vB0/;y

lr;

txy ¼ tyx ¼ rxy þ vB0/;x

lr;

9>=>; ð12Þ

rxx ¼ 2lux;x þ jðux;x þ uy;yÞ;ryy ¼ 2luy;y þ jðux;x þ uy;yÞ;rxy ¼ ryx ¼ lðux;y þ uy;xÞ;

9=; ð13Þ

tMxx ¼ � B0/;y

lr� B2

0

2l0l2r;

tMyy ¼ ð1þ2vÞB0/;y

lrþ ð1þ2vÞB2

0

2l0l2r;

tMxy ¼ tMyx ¼ B0/;x;

9>>=>>; ð14Þ

where rxx, ryy , rxy ¼ ryx are the elastic stress compo-

nents, ux and uy are the displacement components, j ¼ kfor plane strain, and j ¼ 2kl=ðkþ 2lÞ for plane stress,

and k ¼ 2Gm=ð1� 2mÞ and l ¼ G are the Lam�e constantswith G ¼ E=2ð1þ mÞ being the modulus of rigidity.

Making use of Eq. (12), the two Eqs. (11) become:

ux;xx þ ux;yy þjl

�þ 1

�ðux;x þ uy;yÞ;x þ

2vB0/;xy

llr

¼ 0;

ð15Þ

uy;xx þ uy;yy þjl

�þ 1

�ðux;x þ uy;yÞ;y þ

2vB0/;yy

llr

¼ 0:

ð16ÞThe mixed boundary conditions in the perturbation

state may be expressed as follows:

hecx ðx; 0Þ � hxðx; 0Þ ¼ �ðvB0=l0lrÞuy;xðx; 0Þ ð06 x < aÞ;/ðx; 0Þ ¼ 0 ða6 x6 hÞ; ð17Þ

becy ðx; 0Þ � byðx; 0Þ ¼ 0 ð06 x < aÞ;/ecðx; 0Þ ¼ 0 ð06 x < aÞ;

ð18Þ

ryxðx; 0Þ ¼ �ðvB0=lrÞhxðx; 0Þ ð06 x6 hÞ; ð19Þ

ryyðx; 0Þ ¼ ðvB0=lrÞ ðv�

� 2Þhyðx; 0Þ � B0=l0lr

�ð06 x < aÞ;

uyðx; 0Þ ¼ 0 ða6 x6 hÞ;ð20Þ

heyðh; yÞ � hyðh; yÞ ¼ 0 ðy < 1Þ; ð21Þ

hexðh; yÞ � lrhxðh; yÞ ¼ �ðvB0=l0lrÞux;yðh; yÞ ðy < 1Þ;ð22Þ

rxxðh; yÞ ¼ 0 ðy < 1Þ; ð23Þ

rxyðh; yÞ ¼ 0 ðy < 1Þ: ð24Þ

3. Solution procedure

Using now the Fourier transform, the components of

displacements and magnetic potential may be obtained

as:

ux ¼2

p

Z 1

0

AðaÞ�

þ y�

� jþ 3ljþ l

1

a

�BðaÞ

þ 2vB0

lrðjþ lÞ aðaÞ�e�ay sinðaxÞda

� 2

p

Z 1

0

CðaÞ sinhðaxÞ�

þ x coshðaxÞ�

� jþ 3ljþ l

sinhðaxÞ 1a

�DðaÞ

þ 2vB0

lrðjþ lÞ bðaÞ sinhðaxÞ�cosðayÞda� a0x;

ð25Þ

uy ¼2

p

Z 1

0

AðaÞf þ BðaÞyge�ay cosðaxÞda

þ 2

p

Z 1

0

CðaÞcoshðaxÞf þDðaÞx sinhðaxÞg sinðayÞda

þ b0y; ð26Þ

/ ¼ 2

p

Z 1

0

aðaÞe�ay cosðaxÞda

þ 2

p

Z 1

0

bðaÞ coshðaxÞ sinðayÞda; ð27Þ

where AðaÞ, BðaÞ, CðaÞ, DðaÞ, aðaÞ and bðaÞ are the un-

known functions to be solved, and a0, b0 are the real

constants, which will be determined from the far-field

loading conditions. Applying the Fourier transform to

Eq. (10) yields:

/ec ¼ 2

p

Z 1

0

aeðaÞsinhðayÞcosðaxÞda; y¼ 0; 06x< a;

ð28Þ

/e ¼ 2

p

Z 1

0

beðaÞe�ax sinðayÞda; h < x < 1; ð29Þ

where aeðaÞ and beðaÞ are also unknowns. The magnetic

fields can be obtained by making use of Eqs. (9) and(27). The magnetic fields in the void inside the crack and

outside the strip can also be obtained from Eqs. (9), (28)

and (29).

By applying the far-field loading conditions, the

constants a0 and b0 are obtained as:

a0 ¼j

4lðjþ lÞ r0

�� vB2

0

l0l2r

�;

b0 ¼jþ 2l

4lðjþ lÞ r0

�� vB2

0

l0l2r

�:

ð30Þ


The boundary conditions of Eqs. (21)–(24) lead to the

following relations between unknown functions:

vB0

l0lr

asinhðahÞCðaÞþ vB0

l0lr

ahcoshðahÞ�

�jþ3ljþl

sinhðahÞ�DðaÞ

� lr

�� 2v2B2

0

l0l2r ðjþlÞ

�asinhðahÞbðaÞ�ae�ahbeðaÞ¼ f1ðaÞ;

ð31Þ

a coshðahÞbðaÞ�

� e�ahbeðaÞ�¼ f2ðaÞ; ð32Þ

a sinhðahÞCðaÞ þ ah coshðahÞ�

� ljþ l

sinhðahÞ�DðaÞ

þ vB0ðjþ 3lÞ2llrðjþ lÞ a sinhðahÞbðaÞ ¼ f3ðaÞ; ð33Þ

a coshðahÞCðaÞ þ ah sinhðahÞ�

� jþ 2ljþ l

coshðahÞ�DðaÞ

þ vB0ðjþ 2lÞllrðjþ lÞ a coshðahÞbðaÞ ¼ f4ðaÞ; ð34Þ

where fiðaÞ ði ¼ 1; 2; 3; 4Þ are given in Appendix A.

Application of the mixed boundary conditions in Eqs.

(17) and (20) gives rise to two simultaneous dual integral

equations:Z 1

0

a aðaÞ�

� vB0

l0lr

AðaÞ�sinðaxÞda ¼ 0; 06 x < a;

Z 1

0

aðaÞ cosðaxÞda ¼ 0; a6 x6 h;

Z 1

0

a AðaÞ�

�vB0ðjþ2lÞ2llrðjþlÞ v

�� ljþ2l

�aðaÞ

�cosðaxÞda

�jþ2ljþl

Z 1

0

E1ðahÞcoshðaxÞf þE2ðahÞaxsinhðaxÞgda

¼ pðjþ2lÞ4lðjþlÞr0; 06x< a;

Z 1

0

AðaÞ cosðaxÞda ¼ 0; a6 x6 h; ð35Þ

where EiðahÞ ði ¼ 1; 2Þ are given in Appendix B.

The solution of two simultaneous dual integral

equations (35) may be obtained by using a new functionUðnÞ and the result is

AðaÞ ¼ p2a2

Z 1

0

n1=2UðnÞJ0ðaanÞdn: ð36Þ

The function UðnÞ is governed by the following Fred-

holm integral equation of the second kind:

UðnÞ þZ 1

0

UðgÞKðn; gÞdg ¼ jþ 2l2lðjþ lÞy0

r0n1=2; ð37Þ

y0 ¼ 1� jþ 2l2ðjþ lÞ v

�� ljþ 2l

�vbclr

� �2

; ð38Þ

where b2c ¼ B20=ll0. The kernel function Kðn; gÞ is

Kðn; gÞ ¼ ðngÞ1=2 a2ðjþ 2lÞy0ðjþ lÞ

Z 1

0

ae�ahI0ðaagÞ

� L1ðahÞI0ðaanÞf þ aanL2ðahÞI1ðaanÞgda;ð39Þ

where LiðahÞ ði ¼ 1; 2Þ are given in Appendix C, and I0ðÞand I1ðÞ are the zero and first order modified Bessel

functions of the first kind, respectively.

The displacement components ux, uy and magnetic

potential / near the crack tip are:

ux � kh12l

r2

1=2 1a1cos h

2

2ljþl þ

vbclr

� �2� ��

þ 2þ jþ3ljþl

vbclr

� �2�sin2 h

2

� �;

uy � kh12l

r2

1=2 1a1sin h

2

2ðjþ2lÞjþl

h� 2þ jþ3l

jþlvbclr

� �2�cos2 h

2

� �;

9>>>>>>>>>=>>>>>>>>>;

ð40Þ

/ � kh1a1

r2

� �1=2 vB0ðjþ 2lÞll0lrðjþ lÞ sin

h2

ð41Þ

and the singular parts of the strains, elastic stresses,

magnetoelastic stresses and Maxwell stresses in the

neighborhood of the crack tip are:

exx � kh1a1E0ð2rÞ1=2

cos h2

ð1� m0Þ 1þ 2jþ3l2ðjþlÞ

vbclr

� �2� ��

�ð1þ m0Þ 1þ jþ3l2ðjþlÞ

vbclr

� �2�sin h

2sin 3h

2

� �;

eyy � kh1a1E0ð2rÞ1=2

cos h2

ð1� m0Þ 1þ 2jþ3l2ðjþlÞ

vbclr

� �2� ��

þ ð1þ m0Þ 1þ jþ3l2ðjþlÞ

vbclr

� �2� �

sin h2sin 3h

2

�;

exy � kh12a1Gð2rÞ1=2

sin h2

jþ2l2ðjþlÞ

vbclr

� �2�

� 1þ jþ3l2ðjþlÞ

vbclr

� �2� �

cos h2cos 3h

2

�;

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

ð42Þ

rxx � kh1a1ð2rÞ1=2

cos h2

1þ 2jþ3l2ðjþlÞ

vbclr

� �2� ��


vbclr

� �2� �

sin h2sin 3h

2

�;

ryy � kh1a1ð2rÞ1=2

cos h2

1þ 2jþ3l2ðjþlÞ

vbclr

� �2� ��

þ 1þ jþ3l2ðjþlÞ

vbclr

� �2� �

sin h2sin 3h

2

�;

rxy � kh1a1ð2rÞ1=2

sin h2

jþ2l2ðjþlÞ

h� 1þ jþ3l

2ðjþlÞvbclr

� �2� �

cos h2cos 3h

2

�;

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

ð43Þ


txx � kh1a1ð2rÞ1=2

cos h2

1þ 2jþ3l2ðjþlÞ

vbclr

� �2� ��


vbclr

� �2� �

sin h2sin 3h

2

�;

tyy � kh1a1ð2rÞ1=2

cos h2

1þ 4jþ7l2ðjþlÞ

vbclr

� �2� ��

þ 1þ jþ3l2ðjþlÞ

vbclr

� �2� �

sin h2sin 3h

2

�;

txy � kh1a1ð2rÞ1=2

sin h2

jþ2ljþl

vbclr

� �2�


vbclr

� �2� �

cos h2cos 3h

2

�;

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

ð44Þ

tMxx � kh1a1ð2rÞ1=2

vðjþ2lÞ2ðjþlÞ

bclr

� �2

cos h2;

tMyy � kh1a1ð2rÞ1=2

vð1þ2vÞðjþ2lÞ2ðjþlÞ

bclr

� �2

cos h2;

tMxy �kh1

a1ð2rÞ1=2vðjþ2lÞ2ðjþlÞ

bclr

� �2

sin h2;

9>>>>=>>>>;

ð45Þ

where m0 ¼ m=ð1� mÞ, E0 ¼ E=ð1� m2Þ for plane strain,

and m0 ¼ m, E0 ¼ E for plane stress, and a1 is given by

a1 ¼ 1þ v2ðjþ lÞ ðjf þ 2lÞ þ vð2jþ 5lÞg bc

lr

� �2

:

ð46ÞThe polar coordinates r and h are defined as:

r ¼ ðxn

� aÞ2 þ y2o1=2

; ð47Þ

h ¼ tan�1 yx� a

� �: ð48Þ

The stress intensity factor kh1 is obtained as

kh1 ¼ limx!aþ

2ðxf � aÞg1=2tcyyðx; 0Þ ¼a1y0

r0a1=2Uð1Þ; ð49Þ

where

tcyyðx; 0Þ ¼ tyyðx; 0Þ þ tMyy ðx; 0Þ: ð50Þ

The energy density is expressible in the form

S ¼ r2

tcxxexx�

þ tcxyexy þ tcyxeyx þ tcyyeyy�

ð51Þ

and hence

S ¼ ahk2h1 ð52Þin which the coefficient ah depends on the angle h. For acrack under mode I loading, fracture will always occur

in the normal plane (h ¼ 0). The coefficient ah for h ¼ 0

is given by

ah ¼1� m0

2a21E0 1

(þ 2jþ 3l2ðjþ lÞ

vbclr

� �2)

� 1

"þ v2ðjþ lÞ jf þ 2lþ vð4jþ 7lÞg bc

lr

� �2#:

ð53Þ

4. Experimental procedure

Tensile tests were conducted on nickel–iron soft

magnetic materials with center-cracked plate specimen

geometry in the 100 mm diameter bore of a 10 T (T:Tesla) cryocooler-cooled superconducting magnet at

room temperature (Fig. 1). Three soft magnetic mate-

rials (TMC-V: Young’s modulus E ¼ 182 GPa, Pois-

son’s ratio m ¼ 0:146, specific magnetic permeability

lr ¼ 27,900; TMH-B: E ¼ 203 GPa, m ¼ 0:279, lr ¼10,690; TMB: E ¼ 146 GPa, m ¼ 0:228, lr ¼ 9030) de-

veloped by NEC/Tokin Co. Ltd. were selected. The

specimens are 140 mm length and 1 mm thickness. Thecrack length was varied (2a ¼ 10; 15; 20 mm) while

keeping the specimen width fixed at 2h ¼ 40 mm. Ini-

tial through-the-thickness notches were machined using

electro-discharge machining. The specimens were fa-

tigue precracked and then annealed to obtain the op-

timum magnetic properties. A simple strain gage

method is very suitable to determine the magnetic

stress intensity factor [5]. A five element strip gage(KFG-1-120-D19-16N10C2 from Kyowa Electronic

Instruments Co. Ltd., Japan) was installed along the

90� line and the center point of the element closest to

the crack tip was 2 mm. The strain sensors have an

active length of 1 mm.

Tensile load and a magnetic field were simultaneous

applied to center-cracked plate specimens. A supercon-

ducting magnet was used to create a static uniformmagnetic field of magnetic induction B0 normal to the

crack surface. The specimens were loaded by P ¼ 29:4 Nload that consisted of weights. For each specimen size

five tests were performed. The strains were recorded as a

function of magnetic field.

For the plane stress case, the strain eyy near the cracktip given by Eq. (42) can be rewritten in the form

Eeyy ¼kh1

ð2rÞ1=2v

2ð1þ mÞvþ 2mþð5� mÞvf g vbclr

� �2

� cosh2

ð1"

� mÞ 2ð1(

þ mÞþð3þ mÞ vbclr

� �2)

þð1þ mÞ 2ð1(

þ mÞþð3� mÞ vbclr

� �2)sin

h2sin

3h2

#

þA0þOðr1=2Þ; ð54Þ

where A0 is the unknown coefficient. If all the magnetic

field quantities are made to vanish, then Eq. (54) reduces

to the strain near the crack tip in an elastic plane body

[6]. Setting h ¼ p=2 gives

c0Eeyyr1=2 ¼ kh1 � 2c0A0r1=2 þ � � � ; ð55Þ

where


c0 ¼4

v

2ð1þ mÞvþ 2mþ ð5� mÞvf g vbclr

� �2

2ð3þ 2m� m2Þ þ ð9� 2m� 3m2Þ vbclr

� �2p1=2: ð56Þ

From Eq. (55), a plot of c0Eeyyr1=2 versus r1=2 is linear forsmall values of r and the intercept at r ¼ 0, at the crack

tip, gives the stress intensity factor kh1.

Fig. 3. Stress intensity factor versus magnetic field (h=a ¼ 1:5, 2.0, 5.0).

Fig. 4. Stress intensity factor versus magnetic field (lr ¼ 100, 1000,

10,000).

5. Results and discussion

The determination of the stress intensity factor re-quires the solution of the function of UðnÞ. The solutionof the Fredholm integral equation of the second kind

(37) governing UðnÞ has been computed numerically by

the use of Gaussian quadrature formulas. Once this is

done, kh1 and S can be found from Eqs. (49) and (51).

Fig. 2 provides the normalized stress intensity factor

kh1=r0a1=2 as a function of h=a for m ¼ 0:25, lr ¼ 10,000

and the normalized magnetic field bc ¼ 0, 0.0032, 0.0047obtained from the plane stress and plane strain analyses.

For l ¼ 80 GPa, bc ¼ 0:0032 and 0.0047 correspond to

the magnetic induction of B0 ¼ 1:0 and 1.5 T, respec-

tively. The dashed curves obtained for bc ¼ 0 coincide

with the purely elastic plane stress and plane strain

cases. Comparing the results of these two cases for

bc ¼ 0, little difference is observed (two dashed curves

approximately overlap). The nomarlized stress intensityfactor tends to infinity as h=a ! 1, and decreases slowly

as h=a increases and tends to the result of the infinite

solid as h=a ! 1. Applying the magnetic field increases

the stress intensity factor. The values of kh1=r0a1=2 for

h=a ! 1 are found to be kh1=r0a1=2 ¼ 1:089, 1.215

(bc ¼ 0:0032, 0.0047) for plane stress and kh1=r0a1=2 ¼1:083, 1.199 (bc ¼ 0:0032, 0.0047) for plane strain. Fig. 3exhibits the normalized stress intensity factor kh1=r0a1=2

as a function of bc for m ¼ 0:25, lr ¼ 10,000 and h=a ¼1:5, 2.0, 5.0 obtained from the plane stress analysis. The

magnetic field effect can increase the values of the stress

intensity factor and depends on h=a. Fig. 4 shows the

variation of the normalized stress intensity factor

Fig. 2. Stress intensity factor versus strip-width to crack-length ratio

(bc ¼ 0, 0.0032, 0.0047).

kh1=r0a1=2 against bc for m ¼ 0:25, h=a ¼ 2:0 andlr ¼ 1000, 5000, 10,000 in the plane stress case. The

effect of the magnetic field on the stress intensity factor

is more pronounced with increasing the specific mag-

netic permeability. Fig. 5 shows the energy density S for

h ¼ 0 under different bc for m ¼ 0:25, lr ¼ 10,000 and

h=a ¼ 1:5, 2.0, 5.0, where S has been normalized by the

energy density S0 of infinite ferromagnetic elastic solid

for bc ¼ 0. All the curves increase with increasing bc.The energy density tends to increase with decreasing

h=a.

Fig. 5. Energy density factor versus magnetic field.


(TMC-V).


(TMH-B).


(TMB).

Fig. 9. Stress intensity factor versus magnetic field (h=a ¼ 2:0).


Fig. 6 gives a comparison of the theoretical results of

kh1=r0a1=2 versus h=a with experimental data for TMC-

V. The theoretical results agree very well with the ex-

perimental data. Figs. 7 and 8 show the corresponding

results for TMH-B and TMB. Experimental measure-ments verify the predictions of a theoretical model. The

calculated kh1=r0a1=2 of TMC-V, TMH-B and TMB for

h=a ¼ 2 under various values of bc � 102 are compared

with the measured data in Fig. 9. A large value of bctends to increase the stress intensity factor depending on

the material.

6. Conclusions

The linear magneto-elastic problem for a soft ferro-

magnetic strip with a central crack has been analyzed

theoretically and the effect of the magnetic fields on the

stress intensity factor and energy density has been

summarized in a drawing. An experimental study has

also been conducted in which strain fields were used tocharacterize the stress intensity factor of a central crack

in nickel–iron soft magnetic materials with plate speci-

men geometry under the magnetic field. Both the theo-

retical and experimental results confirm the fact that the

applied magnetic field tends to intensify the fracture

machanics parameters such as stress intensity factor.

The excellent agreement between theoretical calculations

and measurements of stress intensity factor establishesthe validity of the linear theory for magneto-elastic

interactions.

Appendix A

fiðaÞ ði ¼ 1; 2; 3; 4Þ in Eqs. (31)–(34) are given by:

f1ðaÞ ¼R10

F1ðs; aÞ vB0

l0lrsAðsÞ � vB0

l0lr

2ðjþ2lÞjþl � sy

n oBðsÞ

h� lr �

2v2B20

l0l2r ðjþlÞ

n osaðsÞ

ids;

f2ðaÞ ¼R10

F2ðs; aÞsaðsÞds;

f3ðaÞ ¼R10

F1ðs; aÞ sAðsÞ � jþ2ljþl � sy

� �BðsÞ

hþ vB0ðjþ3lÞ

2llrðjþlÞ saðsÞids;

f4ðaÞ ¼R10

F2ðs; aÞ sAðsÞ � 2jþ3ljþl � sy

� �BðsÞ

hþ vB0ðjþ2lÞ

llrðjþlÞ saðsÞids;

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

ðA:1Þ

where

F1ðs; aÞ ¼ 2p

as2þa2 sinðshÞ;

F2ðs; aÞ ¼ 2p

ss2þa2 cosðshÞ:

�ðA:2Þ


Appendix B

The functions EiðahÞ ði ¼ 1; 2Þ in Eq. (35) are given

by:

E1ðahÞ ¼ vB0

2llr

b5ðahÞ coshðahÞþðv�1Þb2ðahÞb1ðahÞ

f1ðaÞ � f2ðaÞf g� v2B2

0

2ll0l2r


b4ðahÞb2ðahÞ

nþ b5ðahÞ coshðahÞ�b2ðahÞ

b2ðahÞ sinhðahÞ

of3ðaÞ

� v2B20ðjþ4lÞ

2ll0l2r ðjþlÞ


n� sinh2ðahÞ

b2ðahÞ� b3ðahÞ

b2ðahÞ

of4ðaÞ;

E2ðahÞ ¼ � vB0

2llr

sinhðahÞ coshðahÞb1ðahÞ

f1ðaÞ � f2ðaÞf gþ 1

b1ðahÞb2ðahÞv2B2

0

2ll0l2rb4ðahÞ sinhðahÞ

nþ b1ðahÞ

ocoshðahÞf3ðahÞ

þ 1b1ðahÞb2ðahÞ

v2B20ðjþ4lÞ

2ll0l2r ðjþlÞ sinh

2ðahÞ coshðahÞn

� b1ðahÞosinhðahÞf4ðahÞ;

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;ðB:1Þ

where

b1ðahÞ ¼ ah lr þv2B2

0ðj�lÞ

2ll0l2r ðjþlÞ

n ohsinhðahÞ þ coshðahÞ�

þ lr þv2B2

0ð2jþlÞ

2ll0l2r ðjþlÞ

n osinhðahÞ

hþ coshðahÞ� sinhðahÞ coshðahÞ;

b2ðahÞ ¼ ahþ sinhðahÞ coshðahÞ;b3ðahÞ ¼ ah coshðahÞ � l

jþl sinhðahÞ;b4ðahÞ ¼ ah� l

jþl sinhðahÞ coshðahÞ;b5ðahÞ ¼ ah coshðahÞ � 2l

jþl sinhðahÞ:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;ðB:2Þ

Appendix C

The functions LiðahÞ ði ¼ 1; 2Þ in Eq. (39) are given

by:

L1ðahÞ ¼ � b5ðahÞ coshðahÞþðv�1Þb2ðahÞ2b1ðahÞ

vbclr

� �2

c1

� b5ðahÞ coshðahÞþðv�1Þb2ðahÞ2b1ðahÞ

b4ðahÞb2ðahÞ

vbclr

� �2�

þ b5ðahÞ coshðahÞ�b2ðahÞb2ðahÞ sinhðahÞ

�c2

� b5ðahÞ coshðahÞþðv�1Þb2ðahÞ2b1ðahÞ

ðjþ4lÞ sinh2ðahÞðjþlÞb2ðahÞ

vbclr

� �2�

� b3ðahÞb2ðahÞ

�c3;

L2ðahÞ ¼ sinhðahÞ coshðahÞ2b1ðahÞ

vbclr

� �2

c1


b4ðahÞ sinhðahÞ vbclr

� �2�

þ 2b1ðahÞ�coshðahÞc2


ðjþ4lÞ sinh2ðahÞ coshðahÞjþl

vbclr

� �2�

� 2b1ðahÞ�sinhðahÞc3;

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ðC:1Þwhere

c1 ¼ lr þ vbclr

� �2

;

c2 ¼ 12

vbclr

� �2

;

c3 ¼ 1jþ2l ðjþ lÞ þ l

2

vbclr

� �2� �

:

9>>>>>=>>>>>;

ðC:2Þ

References

[1] Shindo Y. Trans ASME J Appl Mech 1977;44:47.

[2] Shindo Y. Trans ASME J Appl Mech 1978;45:291.

[3] Pao YH, Yeh CS. Int J Eng Sci 1973;11:415.

[4] Shindo Y. Theor Appl Mech 1981;30:203.

[5] Yoshimura K, Shindo Y, Horiguchi K, Narita F. Fatigue Fract

Eng Mater Struct 2004;27:213.

[6] Dally JW, Sanford RJ. Exper Mech 1987;27:381.

Documents

Tensile testing and analysis of ferromagnetic elastic strip with a central crack in a uniform magnetic field