Theoretical and Applied Fracture Mechanics 10 (1988) 191-196 191 North-Holland

T R A N S I E N T R E S P O N S E O F AN E L A S T I C C O N D U C T O R W I T H A P E N N Y - S H A P E D C R A C K U N D E R E L E C T R O M A G N E T I C F O R C E

Y. S H I N D O , H. T A M U R A and A. T A K E U C H I

Department of Mechanical Engineering II, Faculty of Engineering, Tohoku Unioersity, Sendai 980, Japan

The axisymmetric dynamic response of a penny-shaped crack in an elastic conductor under an impulsive electric current flow and a constant axial magnetic field is analyzed. The axial current flow is disturbed by the presence of the crack and the torsional shear stresses are caused by the interactions between the magnetic field and the disturbed current. Laplace and Hankel transforms are used to reduce the electromagnetoelastic problem to a Fredholm integral equation of the second kind in the Laplace transform plane. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results on the dynamic stress intensity factor are obtained and are presented in a graphical form.

1. Introduction

The stress intensity factor approach of linear elastic fracture mechanics has proven to be very successful in predicting the unstable fracture of brittle solids. When an electromagnetic load is applied to the structural member, the same ap- p roach is expected to apply, at least as far as the initiation of the crack mot ion is concerned. When an electric current flow is disturbed by diversion a round a crack, there is local intensification of the current density. If an electrically conduct ing cracked medium is used in a strong magnetic field, the initial stress and strain state is caused not only by mechanical forces but also electromagnetic forces. Considerable work has been done on the determinat ion of the static singular stress field a round a penny-shaped crack in an elastic conduc- tor under a uni form axial current flow and a constant axial magnetic field [1,2].

In recent fracture mechanics, it is impor tant to reveal the behavior of dynamic singular stresses in the vicinity of the crack like imperfections to impact loads [3]. Near the crack tip, the magni- tude of the dynamic stress intensity factor is con- siderably larger than the corresponding statical one and, in many instances, may initiate the un- stable mot ion of the crack and eventually the fracture of a structural component . A n impuls ive current flow is induced in the structural compo- nents of the magnetic conf inement fusion reactor such as the Tokamak type by a rapid plasma mot ion or a plasma extinction. The first wall of

the fusion reactor receives the dynamic electro- magnetic forces which occur due to the interaction between the magnetic f i dd and the induced cur- rent. However, the transient response of a cracked conductor under an impulsive electromagnetic force has not been considered yet. In this paper, transient response of an elastic conduc to r contain- ing a penny-shaped crack under an electric current flow and a constant axial magnet ic field is analyzed to show the effect of the impulsive electromag- netic force on the behavior of the cracked solid. The current flow and the magnet ic field are per- pendicular to the crack. The current flow is dis- turbed by the presence of the crack and the tran- sient torsional shear stresses are caused by the interaction between the magnet ic field and the disturbed current. Laplace and Hankel t ransforms are used to reduce the problem to a Fredholm integral equation of the second kind [4]. The cur- rent densities and singular stresses near the crack tip are then expressed in closed forms. A numeri- cal Laplace inversion procedure is used to obtain the time dependence of the solution [5]. Numer ica l results on the dynamic stress intensity factor are obtained and are presented in a graphical form.

2. Statement of the problem and electric current field

Consider an electrically conduct ing elastic solid having a penny-shaped crack of radius a. The center of the crack is taken as the origin o, and the

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192 Y. Shindo et al. / Transient response o f an elastic conductor

Fig. 1. An elastic conductor with a penny-shaped crack.

cylindrical coordinates (r, 0, z) are used with the z-axis coinciding with the axis of the symmetry, as shown in Fig. 1. The cracked conductor is per- meated by a static uniform axial magnetic field of magnetic induction B 0 and an impulsive axial electric current flow passes through the conductor. The undisturbed transient electric current is

J,o(t) =JoY(t), (1)

where J~o is the z component of the undisturbed electric current density vector J0, t is the time, J0 is a constant with the dimension of current density and f ( t ) is a function of t. The current flow is disturbed by the presence of the crack and the transient torsional stresses are caused by the inter- action between the magnetic field and the dis- turbed current.

Neglecting the skin effect of transient electric current, the electric potential ~e(r, z) is

q~e(r, z ) = - J0 t) , (2) - g f ( t ) z + z,

where o is the electrical conductivity. The dis- turbed electric potential function 5P~(r, z, t) is governed by the following Laplace equation:

l(T~e.r),r'At-~e,zz-~-O, (3)

where a comma denotes partial differentiation with respect to the coordinate. The nontrivial compo- nents of the electric current density vector J are

Jr = - -O' l~e , r = - - O ' ~ e , r '

J~ = -oeb~.~ =JoY(t) - oSa~,z, (4)

where Jr and J~ are the r, z components of J.

In the case of electrically insulated crack surfaces, the electrical boundary conditions are given by

~ , : = ~ f ( t ) , z=O,O<~r<a,

, ~ = 0 , z = 0 , a<~r<oo. (5)

The potential 5P~ is required to vanish as

V~F2 + Z 2 ~ 00 .

The Hankel transform is applied on eq. (3) and the result is

~e = [°°sa( s, t) e-S~Jo(sr) ds, (6) J0

where J, ( ) is the n th order Bessel function of the first kind and a(s, t) is the unknown function to be determined later. A simple calculation leads to the current expressions:

Jr = ° fo°°S2a( S, t) e-S:J](sr) ds,

J: =JoY(t) + o[°°sZa(s, t) e-'~Jo(sr) ds. (7) ao

Making use of the boundary conditions (5), we have the dual integral equations:

fo ~s2a(s' t )J°(sr)ds = - J° f (

fo°°Sa(s, t)Jo(sr ) ds = O,

O<~r<a,

a ~ r < oo.

(8)

The solution of the dual integral eqs. (8) may be obtained by using a new function h(~) and the result is

sa(s, t) = foah(& t) sin(s~) d& (9)

where

h(& t )= - ( 2 ) - ~ f ( t ) .

Thus, we obtain

sa(s, t ) - 2 Jo 1 -~ -~ f ( t )-~ {sin(sa) -

(10)

(11)

Y. Shindo et al. / Transient response of an elastic conductor 193

The singular parts of the current densities in the neighborhood of the crack tip are obtained as

Jr -- -- 2 jo a/~-~lf(t) sin ½01,

J~- 2 jo~a/2 q f ( t ) cos ½01, (12)

where (q, 0a) are the polar coordinates defined as

r l=~(r - -a)2+z 2,

01= t a n -1 z . (13) r - a

3. Transient singular stress field

In eq. (16), the time variable may be removed by application of the Laplace transform relations:

f * ( p ) = fo~f(t) e-P' dt,

1 f B f . ( p ) ep tdp, (18) f ( t ) = ~--@v t •

where B r denotes the Bromwich path of integra- tion. Hankel transform is then applied resulting in

5 u~ = sA*(s, p) e-Y:Jl(sr) as

x £~s2a*(~ , p ) { e -'~ - e -'~ } S ~ ( ~ ) d~.

(19/

The disturbed electric current Jr derived in the preceding section and the axial magnetic field B 0 induce the transient torsional electromagnetic body force -BoJ r called the Lorentz force. For the torsional shear problem, the displacement compo- nents (Ur, UO, U~) in the radial, tangential and normal directions can be written as

Ur=Uz=O, ua=uo(r, z, t). (14)

Corresponding to eqs. (14), all stress components except or0 and oz0 vanish and the shear stresses are given by

OrO = l~r( Uo,r ) , r , OzO = lZUO,z , (15)

in which g is the shear modulus of elasticity. Under these conditions, the displacement equation of motion can be written as

1 1 BoJr U O , r r At- - - U o , r - - - " ~ U O "JF U O , z z - - - - r r ~ g

1 =72u0,,, (16)

¢2

in wl~ich c 2 = ~ - ~ is the shear wave speed with O being the mass density of the material. Equation (16) is to be solved by imposing the mixed boundary condition:

o~8 = 0, z=O,O<~r<a,

u a = 0 , z = 0 , a<~r<~. (17)

Here, A*(s, p) is the unknown to be solved and ~,(s, p) is

~(s , p ) = ~/s 2 + ( p / c 2 ) 2 . (20)

The associated stresses are

Or~=gfo~sA*(s, p) e-rZ{SJo(sr)-2jl(sr)} ds

+oBo(~)2fo°°S2a*(s, p ) ( e - VZ- e - 'z }

×(SJo ( s r ) -2 j l ( s r ) }d s ,

°z~ = -gfo sy(s, p)A*(s, p) e-VZJl(sr) ds

p J .'o (s, p)

× {7 e -Yz - s e-Sz}Jl(Sr ) ds.

(21)

From the Laplace transform of the boundary condition (17), there results the following dual integral equations:

fo°°ST(S, p)A*(s, p)Jl(sr) ds

o B o f ~ s ~ - a*(s, p)Jl(sr) ds, g ao V(s, p ) + s

0 ~ < r < a , o o

[ sA*(s, p)J l (sr)ds=O, a~r<oo . (22) go

194 Y. Shindo et al. / Transtent response ()fan elastic conductor

The solution of the dual integral eqs. (22) can be obtained by a procedure described in [4] and the result is

× folff'~g*(~, P)J3/2(sa~) d~,

(23)

in which J3/2( ) is the 3-order Bessel function of the first kind. The second equation of (22) is satisfied identically and the first equation of (22) gives the Fredholm integral equation of the sec- ond kind:

1 * g*(~ ' P) + fo g '(*/' P )K(~ , 77, P) dr/

_ 1 p2 K( ~, 1, P ), (24)

where the kernel K(~, 7, P) and 7'(s, P), P are defined by

K(~, 7, P ) = ~ / ~ fo~{7'( s, P ) - s }

× J3/2(s~)J3/2(sn) ds,

(25)

7, ( s" p) = V~s 2 + p2 , P = ap/c 2. (26)

We note that the kernel function K(~, ~/, P), eq. (25), is a semi-infinite integral which has a rather slow rate of convergence. To improve this prob- lem, we evaluate the kernel. By the use of contour integration technique [4], eq. (25) can be written as

×K3/2(aP~ )da , ~>~/, (27)

×I3/z(aPS) da, 7/> ~,

where 13/2() and K3/2() represent the usual modified Bessel functions.

The singular parts of the stresses in the neigh- borhood of the crack tip are obtained as

K 3 ( T ) sin ½01 , ~r0

/ < ~ ( r ) oz0 - - c o s ½01, (28)

where T = c2t/a is the normalized time. The dy- namic stress intensity factor K3(T) is defined as

K 3 ( T ) = lira ~/2(r-a)o:o(r,O, t) r ~ c /

= 6K3o / p )

where

K30 = ~----~ BoJo a3/2.

× e-er ( -~ 2) dP , (29)

(30)

Similarly, we obtain the crack surface displace- ment uo(r, 0, t ) a s

uo(r, O, t)=6Uoo2-~ £f*(p)g*(1, p)

4. Numerical results and discussion

The three types of transient electric currents which are considered in this paper are as follows:

f ( t ) = H ( t ) ,

1 {e_( l f ( t ) = -~-~

(Case I) (32)

-c , , ) ' ._e-( l -ca)t .}H(t) ,

(Case II) (33)

f ( t ) = (1 - e -2 '~ )H( t ) , (Case III) (34)

where H(t) is the Heaviside unit step function in time, and the time constant a and C. are

a = R /2L , C, = ~1 - 4L/CR 2 . (35)

Case II and Case III correspond to the L C R and L R discharge circuits, and L, C, R are the inductance, the capacitance and the resistance, respectively.

The Fredholm integral equation of the second kind (24) can be solved on an electric computer by the standard techniques. Once this is done, K3(T ) can be found from eq. (29) by using a numerical Laplace inversion scheme described in [5]. Note that letting p ~ 0 in eq. (24) yields the corre- sponding stress intensity factor for the static case.

Y. Shindo et aL/ Transient response of an elastic conductor 195

[ . 5 ,

I.O g

I-

0.5

J

O. 0 20 4 0 6 0 8 0 T

Fig. 2. Dynamic stress intensity factor versus time (Case I).

1.5

1.0

Xt

0.5

/

O. 0 20 4.0 60 80 T / B

Fig. 3. Dynamic stress intensity factor versus time (Case II).

First, we discuss the Case I. As t ~ oo, K3(T ) tends to the static solution K30 [1]. The dynamic stress intensity factor is normalized by K30. Figure 2 exhibits the variation of the normalized dynamic stress intensity factor K3(T) /K3o with the nor- malized time T. A close examinat ion of the result reveals that the dynamic stress intensity factor rises very quickly with time, reaching a peak and then decreases in magni tude and tends to the static solution for sufficiently long time. The in- ertia effect can increase the value of K3(T ) by approximately 32.9% over the corresponding value for the elastostatic case.

Next, we consider the Case I and Case III . As an example, L = 7 I~H, C = 22 mF, R = 50 mf~ are considered. Figures 3 and 4 show the results of Case II and Case III , respectively. The dashed curve is the result for the quasistatic case which is realized by cancelling the term (1/c2)uo, , in eq. (16). The effect of the ratio fl = c2/aa = 2Lc2/aR on the variations of the normalized dynamic stress intensity factor K3(T) /K3o with the time variable T / f l = ta = t R / 2 L is shown in these figures. The peak 'va lue of K3(T) /K3o appears to be higher as fl is decreased and K3(T) /K3o approaches to the quasistatic solution as fl is increased. The dy- namic stress intensity factor K3(T)//K3o for fl = 10 is almost coincident with the quasistatic result and the impact effect for fl > 10 diminishes. The val- ues of the elastic constants for a luminum are

/x = 2.83.101° N / m 2, p = 2.88- 103 k g / m 3, c 2 =

3136 m / s and fl = 1.0, 2.0, 10.0 corresponds to a penny-shaped crack of radius a = 0.88, 0.44, 0.088 m. For the a luminum with the penny-shaped crack of small a, the impact effect diminishes.

In conclusion, the dynamic response of a con- ductor with a penny-shaped crack to impact elec- t romagnetic forces is obtained and the peak am- plitude of the dynamic stress intensity factor is found to depend on the parameter fl = 2 LCE/aR.

1.5

1.0

g

I..-

v

0.5 /

~ quasi stolic

0.0 2.0 4.0 6,0 80

T / ~

Fig. 4. Dynamic stress intensity factor versus time (Case III).

196 Y. Shindo et al. / Transient response of an elastic conductor

A c k n o w l e d g m e n t s

This w o r k was s u p p o r t e d in pa r t by the Scien-

t if ic Resea rch F u n d of the M i n i s t r y o f E d u c a t i o n

for the f iscal yea r 1987. Par t ia l s u p p o r t was also

p r o v i d e d by the K u r a t a F o u n d a t i o n ( the K u r a t a

R e s e a r c h Gran t ) . T h e au tho r s wish to express

the i r s incere a p p r e c i a t i o n to E m e r i t u s Prof. A.

A t s u m i of T o h o k u U n i v e r s i t y for his k ind en-

c o u r a g e m e n t and t h o u g h t f u l gu idance .

constant axial magnetic field", Engrg. Fracture Mech. 23(6), pp. 977-982 (1986).

[2] Y. Shindo and A. Takeuchi, "Electromagnetic twisting of a penny-shaped crack in an elastic conducting cylinder", Theoret. Appl. Fract. Mech. 8(3), pp. 213 217 (1987).

[3] G.C. Sih, ed., Elastodynamic Crack Problems, Noordhoff, Leyden (1977).

[4] Y. Shindo, "Sudden twisting of an infinite elastic conduc- tor with a penny-shaped crack in a constant axial magnetic field", Z. Angew. Math. Mech. 62(11), pp. 599-607 (1982).

[5] A. Papoulis, "A new method of inversion of the Laplace transform", Quart. Appl. Math. 14(4), pp. 405 414 (1956).

R e f e r e n c e s

[1] Y. Shindo, "The linear magnetoelastic problem of a uni- form current flow disturbed by a penny-shaped crack in a