Interaction between an edge dislocation and a semi-circular cut on straight boundary

f&t. Appl. .&pg Sci. Vol. 22, No. 3. pp. 333341, Printed in Gnat .Bntain.

INTERACTION

1984 0

002iS7225/84 $3.00 + .OO 1984 Pergamon Press Ltd.

BETWEEN AN EDGE DISLOCATION AND A SEMI-CIRCULAR CUT ON STRAIGHT BOUNDARY

S. Shioya Department of Mechanical Engineering The National Defense Academy Yokosuka, Japan

and A. Akiyama Department of Mechanical Engineering Graduate Course The National Defense Academy Yokosuka, Japan (RevisedmanuscriptreceivedSeptember10, 1983)

ABSTRACT

:

Interaction between a semi-circular cut on straight boundary and an edge dislocation, placed on the axis of symmetry, in a semi-infinite medium is investigated on the basis of Airy's stress function. The solution is constructed by considering the relation between the real edge dislocation and the imaginary one. Numerical calculations are worked out in some detail and these results are presented graphically in order to clarify the effect of above mentioned interaction.

1.

Since knowledge vacancies properties

of how the dislocations

or outer boundary of materials,

is necessary

medium.

Now two kinds of the theoretical

elastic interaction between the dislocations and the matrix.

action arises from the mutual difference

by Head

One

Ill is the problem of

and eigen strain caused by the Another is that the inter-

of elastic constants even in the

case of no misfit between the inhomogeneities was first investigated

of the mechanical

to study the effect of a semi-

of the interaction were developed.

misfit of the inhomogeneities

interact with inhomogeneities,

for understanding

it is interesting

circular cut in a semi-infinite treatment

INTRODUCTION

and the matrix.

121, who considered

The latter

an infinite elastic

medium of shear modulus Cl, for xX!, and modulus GZ for xc0 containing a straight dislocation

running parallel to the interface.

this model, the dislocation IJES Vol. 22, No. 3-J

On the basis of

is simply either repelled or attracted by the 333

334

S.SHIOYA and A. AKIYAMA Later Dundurs and Mura

inhomogeneity.

131, and Dundurs and Sendeckyj

[4]

discussed the interaction between a circular inclusion and an edge dislocation in an infinite medium.

They gave the force on the dislocation

and

showed that for certain combinations of material constants, the dislocation has a stable equilibrium position in the infinite medium near the interface. Matsuoka and K. Saito

[51, [61 also discussed the interaction between a

circular inclusion and an edge dislocation in a semi-infinite

medium.

Above all the problems are discussed as two dimensional plane strain problems.

Moreover Weeks

Lin and Mura

[?'I,Willis

[81, Comninou and Dundurs

[91, and

1101 discussed the interaction between a finite length in-

clusion and a screw dislocation.

The present study discusses the inter-

action between an edge dislocation, placed on the axis of symmetry, and a semi-circular

cut on the straight boundary, treating the problems within

the frame-work of the plane strain problem on the basis of Airy's stress function.

Results obtained by numerical calculation

certain region the interaction

indicate that for a

is very affected by the existence of a semi-

circular cut, other region is similar to the interaction between the surface or a circular hole and an edge dislocation.

2.

BASIC EQUATION

The real edge dislocation

and the image edge dislocation with Burgers

vector b, are placed at distance ih from the center of the semi-circular cut on the straight boundary as shown in Fig. 1, respectively. coordinate z and two auxiliary coordinates 1.

The relations of coordinates

z=x

z’=

4

+

iu

re iB +

=re

ie

,

The main

~1, 22, are also defined in Fig.

are given as

2,

zz

d

x.

,j

-I- iYj

=

rje’% (j=l,Zf

(-l+h

I

(1)

In the present problem, it is convenient to use Airy's stress function represented in polar coordinates.

In the absense of body forces, the Airy

stress function X must satisfy the biharmonic equation.

v4x =

0

(2)

335

Interaction betweenan edgedislocation and a semi-circular cuton straight boundary Y

1

FIG.

Coordinate

ur ,

The stress components

Systems

us and ~rswith reference to the coordinates

(r,tl)

are expressed by the formulas.

,

ur = z&+"'" r2ae2

a2x ue = z I (3)

-erg=

-

a _ar

ax

( ) rae

The boundary conditions (i)

I

t

are given as follows

at 8 = *n/2 cle=O

,

Tre= 0

(4)

I

're= 0

(5)

3.

DETERMINATION

(ii) at r = U ur = 0

By considering dislocation

OF

STRESS

FUNCTION

the relation between the edge dislocation and the image

as shown in Fig. 1 and the symmetry with respect to the x-axis,

the desired Airy stress function is given as follows

x=

DA (X0 +

xl)=

rlsin8llogrl

x, = h

c

(6)

X,) - rzsin0210gr2

+ h(sin2e2 -

i=l D,P-" sin ne + i=2 Dnp -II+* sin ne]

2h
(‘3)

S.SHIOYA and A, AKIYAMA

336 where p =T/h,

DA = -ZG.b,/(X+l).

G and $ are the shear modulus and the Poisson's x=3-4$for Equation

plane strain andX=(3-t))/fl+J)

ratio, respectively, and

for plane stress.

(7) is rewritten by using coordinate system (hn + d,p') p" sin n8

x0= h ;=1

for p
1

Considering

(9)

1

(-lfn(2n-31-l + Glrnr d in which 61,n is Kronecker's

(p,8) as follows

n =2n+2

l+f-1)"(2n+3) [ 1

delta.

that equation

(7) satisfies the boundary condition

the relations of unknown coefficients Bn and Dn in equation from equation

(8) are given

(4)

BZn_l = D2n_l , 2n BZn = (Zn+2) D2n+2 Further, considering equation

(nil) _

(10)

(lo), and using Fourier's expansion of cos2n8

and sin2n8 in the interval -n/2~8~a/2

:

4" cos2n0 C--C r p=l

(_l)P+n+l (2p_1) (2p+2n-1) (2p-2n-1)

sin2nB =- ;1&l

p+n+l f;;3-2n_l)(2P-;:-l)

cos(2p-1)8 i

L

(11)

I

the equations of determining ry condition

(41,

sin(2P-l)e

unknown coefficients

i

are given from the bounda-

(ii) as follows

(2m-2)h2m-3 + 4mh2m-l f (2m+2)A2m+l

(-l)n+m 2n(2n+2) h2n I. (2n+2m+l) (2n-2m-1) (2n-2m+l)(2n-2m+3)

++i=

(-l)n+m (2nil) (2nt2) D2n+2X + -$ I= 1

-(2n+2)

(2n+2m+3)(2n+2m+l) (2n+2m-1) (2n-2m+ll = O

(rnhl) (12)

D2m-lh

-(2m-1) = (2m+2)h2m+l + (2m+l)X2m-l

++$I

C-l)“+” 2n(2n+2)X2" (2n+2m+l)(2n-2m-l) (2n-2m+l)

+aE=,

C-1) n+m*l (2n+l) (2n+2) D2n+2h-'2n+2) (2n+2m+3)(2n+2m+l) (2n-2m+l)

To determine the unknown coefficient Dn from equations it is advantageous perturbation

to apply a method of perturbation

parameter.

(13) in the form

(rnzl) (13)

(12) and (131,

in which X=&/h is the

Now we shall seek a solution to equations

(12) and

Interaction betweenan edgedislocation and a semi-circular cuton straight boundary Dn

,E,Dn(t)

=

Substituting

Xn+t

(14)

the equation

equating the coefficients

331

(14) into equations

(12) and (13), and then

of same power of X on both sides, we obtain

(-l)n+m+1(2n+l) (2n+2)Dizi2 Y n=l

(2n+2m+3) (2n+2m+l) (2n+2m-1) (2n-2m+l)

=

5 i (2m-2)6,,2m-3+

+

4m&,,,_, +

(2m+2) gt,3mt,)

(-l)p+m 2p(2p+2) (2p+2m+1)(2p-2m-1) (2p-2m+l) (2p-2m+3) 'VP

(15)

(t) = (2m+2)6*,,,,,+ (2m+l)6t,,,-1 D2m+l

(-l)p+m 2p(2p+2) +8 71 (2p+2m+l) (2p-2m-1) (2p-2m+l) '*j2P

(-l)n+m+l(2n+l)D~~~2 +L? According

n n-1

to these relations the coefficient

determined

successively

Substituting and considering calculated

(16)

(2n+2m+3) (2n+2m+l) (2n-2m+l) Dn(t) in series

(14) can be

for t=1,2..............

the value for the unknown coefficient

the formulas

(3) the distributions

Dn into equation

of stress components

(8) are

directly.

4.

FORCE ON DISLOCATION

In the absence of applied tractions,

the interaction energy is equal

to the part of the strain energy that depends on the position of the dislocation.

In calculating the strain-energy,

evaluate

it as the work required to inject the dislocation.

it is much more convenient to Thus, the

interaction

energy for the edge dislocation with Burgers vector bx in the

x-direction

is

(17) where h is the distance between the center of cut and the dislocation, is the radius of core of dislocation,

and T Xlr

reference to the coordinates

(x,y).

is shearing stress with

6+0

S.SHIOYA and A. AKIYAMA

338

T,.#is composed of two parts as (18) The first term in equation

(18) is the eigen shear stress for an edge dislo-

cation in an infinite medium which is independent of the position of dislocut, and the second term expresses

cation with respect to the semi-circular

the interaction effected by the existence of the semi-circular

The contribution by the first term in equation

straight boundary.

should be omitted from equation

m J

W' =+b,

cut and the

(17).

Equation

(18)

(17) is rewritten as

(19)

r$dx

h+6

The force tending to move the dislocation

in the radial direction by glide

is F =

(20)

-awl/ah

It may be noted that positive F corresponds to repelling the dislocation and F less than zero signifies attracting the dislocation.

The force on

dislocation in the case of this problem is given from equation

F=-

(20)

2G.bL '1 2 2(2m+l) (2m+2)D2m+2 + 8m D2m+l (?c+l)ndhIT + 2D2 + i=,I (21)

Substituting equation

(141 to equation

(21)

the force F on dis location is

expressed as follows

F = FA

(t) At+1 3

I

(22)

in which

F, = -

2G b': (x+l)lTa

Mw

'

2 4m

D2 = O

(2t-2m-1) + E-" D2m+l m=l

N(2t+l) = 2 x 2 (2t-2m) + (2m+l) (2m+2)D2m+2 4m D2m+l m=l

5.

NUMERICAL RESULTS

The foregoing solution will be worked out in some detail.

Fig. 2 shows

interaction

Mween

an edge disfocation

and a semi-circular

Gilt

on straight baundary

the hoop stress 08 on the circular cut for various values of A. along x-and y-axes

distributions

are

shown in Figs.

339

The stress

3 and 4, respectively.

Fig. 4 also indicates that the stress converges to zero at far from the origin of cut. Now we can be evaluate the force F on dislocation and (23) using the coefficients

from equations

M ft) indicated in Table 1.

6.0 i

$ 4.0

2.0

c

-2C ~-

60

180

120 **__

FIG. 2

FIG. 3

The stress distribution circular cut

on

the

The stress distribution along x-axis

FIG. 4 The stress distribution along y-axis

(22)

340

S. SHIOYA and A. AKIYAMA TABLE 1 The values of coefficient in equation

.(t)

t 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.46225 - 0.15126 0.27066 - 0.46745 0.92969 - 0.16163 0.23501 - 0.33331 0.49699 - 0.70901 0.93100 - 0.12048 0.16002 - 0.20795

x x x x x x x x x x x x x x

10' lo3 lo3 lo3 lo3 lo4 10' lo4 lo4 104 lo4 10' lo5 105

(22)

t

.(t)

18 19 20 21 22 23 24 25 26 27 28 29 30

0.25758 - 0.31604 0.39402 - 0.48489 0.57838 - 0.68530 0.82082 - 0.97471 0.11323 - 0.13089 0.15249 - 0.17657 0.20115

x x x x x x x x x x x x x

lo5 lo5 105 10' lo5 lo5 lo5 lo5 lo6 lo6 106 lo6 lo6

FIG. 5 The force on dislocation. _-----_.

the force in the case containing an

* edge dislocation in a semi-infinite medium --

-:

the force in the case containing an edge dislocation and a circular hole in an infinite medium [3]

Fig. 5 shows that the existence of semi-circular

cut affects the force on

the dislocation for h/a<6. In particular, when the dislocation

is far away from the cut (that is h/&>6)

Interaction betweenan edgedislocation and a semi-circular cuton straight boundary

341

it converges to the case of containing an edge dislocation in a semiinfinite medium.

On the other hand, when the dislocation

the cut, the behaivior of the dislocation

is very near

is governed by the cut,

REFERENCES 1.

Cottrell, A-H., Dislocation Univ. Press (1953).

2.

Head, A.K., The Interaction of Dislocations Nag., 44, 92 (1953).

3.

Dundurs, J. and Mura, T., Interaction Between an Edge Dislocation and a Circular Inclusion, J. Mech. Phys. Solids, 12, 177 (1964)

4.

Dundurs, J. and Sendeckyj, G-P., Edge Dislocation Inside a Circular Inclusion, J. Mech. Phys. Solid, 13, 141 (1965).

5.

Matsuoka, A., Shioya, S. and Saito, K., The Elastic Interaction of an Edge Dislocation with a Circular Inclusion in a Semi-Infinite Medium (1st Report, In the Order of a Free Surface, an Inclusion and a Dislocation), Trans. Japan Sot. Mech. Engrs., 42, 3051 (1976).

5.

Saito, K., Matsuoka, A., Kuwata, T. and Shioya, S., The Elastic Interaction with a Circular Inclusion in a Semi-Infinite Medium (2nd Report, In the order of a Free Surface, a Dislocation and an Inclusion) Trans. Japan Sot. Mech. Engrs., 43, 3492 (1977).

7.

Weeks, R.W., Pati, S.R., Ashby, M.F. and Barrand, P., The Elastic Interaction between a Straight Dislocation and a Bubble or a Particle, Acta Metallury, 17, 1403 (1969).

8.

Willis, J.R., Hayns, M.R. and Bullough, R., The Dislocation Void Interaction, Proc. Roy. Sot. Lond., ser. A, 325, 121 (1972).

9.

Comninou, M. and Dundurs, J., Long-Range Interaction Between a Screw Dislocation and a Spherical Inclusion, J. Appl. Phys., 43, 2461 (1972).

10.

and Plastic Flow in Crystalls, Oxford and Boundaries, Phil.

Lin, S.C. and Mura, T., Long-Ranqe Elastic Interaction Between a Dislocation and Ellipsoidal-Inclusion in Cubic Crystals, J. Appl. Phys., 44, 1508 (1973).