A formula for the orientational derivative of dislocation energy factor

Scripta METALLURGICA

Vol. 3, pp. 317-320, 1969 Printed in the U n i t e d States

Pergamon

Press,

Inc.

A FORMULA FOR THE ORIENTATIONAL DERIVATIVE OF DISLOCATION ENERGY FACTOR M.M. Shukla and J. Lothe Institute of Physics, University of Oslo, Norway.

(Received M a r c h

ii,

1969)

Consider a dislocation in the x-z plane, and let 8 be the -nEle of d/slocation orientation relative to the z- axxs.

The energy factor E is a function of dislocation orientation 8.

A

formula for ~E/98 will be developed. With the dislocation along the z-axis, the Eshelby-Read-Shockley (i) prescription for determination of displacements and stresses is

I{aik}l

: o

(l)

aik = Cilkl + (Cilk2 + ci2kl) Pn + ci2k2 Pn 2

aik(n) Ak(n) = 0

(2~

(3)

6 1 [ i Ak(n) Den) bk = ~ n=l

(h)

6

o = [ "- Bi2k(n) Ak(n) D(n)

(5)

n=l

B i j k (n) = C i j k l + Cijk2 Pn

(6)

6 1 [ Ak(n) D(n) Znn n = -~T;Tn= 1

(7)

(8)

n n = x + pny

317

318

ORIENTATIONAL

DERIVATIVE

6 1 °iJ = " ~ = ~ 1 B i j k ( n ) ~ ( n )

OF D I S L O C A T I O N

ENERGY

Vol.

3, No.

-1 D(n) nn

(9)

Here u~ and b k are elastic displacements and Burgers vector components, respectively, and sij are the dislocation stresses. roots of Eq. (i).

Cijkl are the elastic coefficients, and Pn are the six

The other parameters are defined by the equations.

The plus and minus

signs in Eqs. (~) and (5) are used when the imaginary part of Pn is positive or negative, respectively.

The six roots Pn will occur in pairs of complex conjugates.

Let the q-th root be the

complex conjugate of the m-th one,

(10) It follows from the structure of the above equations, Eqs. (i) - (6), that

(ZZ)

=

and

D(q) = - D*(m) (12) Note the minus sign i n Eq. (12). I t i s now easy to see that the above extended forms of the equations, with summations over all six roots, are completely equivalent with the usual forms (see Hirth and Lothe (2)) involving s~-.m-tions over 3 fundamental roots and the operations of taking real and imaginary parts. convenient.

In the present development, the extended forms are

In the same way, Forem--'s (3) formula for the energy factor becomes b. 1

~'

6

= ~;T

X Bi2k(n) ~ ( n )

D(n)

(13)

n=l

Lothe (~) has developed an energy flow theorem which can be written in the form i

8E/~8 = - 2r 2

o3k ~xK d¢

(1~)

o The integral is along a semicircle of constant radius r. x=

r

cos ¢

(15)

y = r sin ¢ . Substituting Eqs. (7) and (9) into Eq. (i~), one obtains w

8E/88 = 1 ~ ( ! 8W 2

'

d¢ BnB n '

) D(n)D(n') ~ B3kq(n)Aq(n)Ak(n')

(16)

kq

where ~n = cos ~ + Pn sin ¢

By the substitution ~ = cotan ¢, the integrals simply become

(z?)

5

Vol.

3, No.

5

ORIENTATIONAL

DERIVATIVE

OF D I S L O C A T I O N

ENERGY

319

(18) 8nSn O

(~ + Pn )(~ + Pn _Qo

Closing the contour of integration in the upper half plane and taking residues, one sees that W

(19) n n

Pn Pn

0

when Pn and Pn' are on different sides of the real axis, and zero otherwise.

The plus and

minus sign applies according to whether Pn is above or below the real axis, respectively.

Thus

(20)

i ±Pn'Pn i ' D(n) D(n' )kq[ B3kq(n) Aq(n) Ak(n' ) , BE/Be = 1~n!'

the m~mation only including pairs (n,n') such that Pn and Pn' are on different sides of the real axis.

Finally, for the sake of consistency with the more usual formalism we also give the result in terms of three fundamental roots Pn' n = 1,2,3, chosen to be those with positive imaginary part.

The result is

1 BE/Be = ~

Imnn, ~ ~

1

D(n) D*(n')kq~

B3kq(n) AqCn) Ak*Cn')

In Eq. (21) the term n = n' must also be included in the m~mation. of".

n = 1,2,3.

(21)

Im means "imaginary part

n' = 1,2,3.

Eqs. (20) or (21) only involve those factors which have to be determined in calculations of stresses and energy factors by the Eshelby-Read-Shockley-Foreman theory.

Thus, when

stresses and energy factors are calculated, DE/Be may readily be calculated as well.

For all

those high symmetry directions for which explicit calculations are possible (Foreman (3), Duncan and Wilsdorf (5)), explicit formula for BE/Be can also be developed. calculations of E and BE/Be for the twelve

Thus, with

< l ~ O > a n d directions in the (lll) plane

of f.CoC., a quite good continous plot of E(8) should be obtainable by extrapolation.

A more extensive discussion with applications will be published elsewhere later. One of us (M.M.S.) acknowledges the fellowship support of Norwegian Agency for International Development.

320

ORIENTATIONAL

DERIVATIVE

OF D I S L O C A T I O N

Reference

ENERGY

Vol.

s

i.

J.D. Eshelby, W.T. Head, and W. Shockley, Acta Met. I, 251 (1953).

2.

J.P. Hirth and J. Lothe, Theory of Dislocations, McGraw-Hill, New York (1968).

3.

A.J.E. Foreman, Acta Met. 3, 322 (1955).

~.

J. Lothe, Phil. Mag. 15, 9 (1967).

5.

T.R. Duncan and D. Kuhlmann-Wilsdorf, J. Appl. Phys. 38, 313 (1967).

3, No.

5