Nonlinear effects in the displacement field of a dislocation

Scripta

METALLURGICA

Vol. 6, pp. 4 4 5 - 4 4 6 , 1972 P r i n t e d in the U n i t e d S t a t e s

Pergamon

Press,

Inc.

NONLINEAR EFFECTS IN THE DISPLACEMENT FIELD OF A DISLOCATION

P. C. Gehlen, R. G. Hoagland, and M. F. Kanninen Battelle, Columbus Laboratories, Columbus, Ohio 43201 and J. P. Hirth Metallurgical Engineering Department, Ohio State University, Columbus, Ohio

~Received

April

5,

43201

1972)

Computer simulations have been performed to determine the elastic field of an edge dislocation with Burgers vector a[100] in n-iron as modeled by the Johnson potential (i). The approach used to relax the discrete region has been described previously (2).

An inno-

vation, however, is the use of flexible boundary conditions to allow the elastic medium bounding the crystallite to adjust in response to atomic readjustments within the crystallite.

This in effect gives a solution for the displacement field at the boundary of the

crystallite which includes both the Volterra

linear elastic

contributions arising from nonlinearities at the core. be published at a later date (3).

field of the dislocation and

A complete account of this work will

However, because the unpublished results of this work have

already stimulated a number of studies of the ramifications in dislocation theory,

it was felt

worthwhile to briefly summarize them. For the above dislocation,

the nonlinear effects lead to a net volume change of

0.25 b 2 per unit dislocation length, in agreement with expectation from previous work (4). Importantly,

in addition, the nonlinear displacement field at distances ~ 12A from the dislo-

cation can be described in terms of two normal, unequal, line force dipoles parallel to the dislocation but displaced by about 2 b to the tension side of the dislocation, i.e., lying 2 b below the extra half-plane of the dislocation. in nature.

The field of the dipoles is thus elliptical

Although the effect arises from highly nonlinear behavior near the core [it cannot

be explained by third order perturbation theory (4)], the long-range field can be generated by the linear elastic elliptical center. In the isotropic elastic case, the displacement field of the line force dipole is derived by integration of the Green's function displacement field of a point force (5). result, in cylindrical coordinates fixed on the dipole center in the usual manner are

445

The

446

NONLINEAR

EFFECTS

u

r

IN D I S P L A C E M E N T

=

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

Vol.

6, No.6

M2

4~'~r

[(5 + 1)(1-~)

+ (5 - 1) cos 2e]

M2 "

us = 4~.r u

z

(1

- 5)(1

- ~)

sin

2e

=0

Here ~ m Ok + ~ ) / ~ + 2~), with N and ~ Lame's constants, 5 = MI/M 2, and M I and M 2 are the strengths of the line forces in the @ - 0 and @ - ~/2 directions, respectively.

The remainder

of the elastic field can be derived from the displacements by differentiation and the use of Hooke's law. The elastic field and elastic interactions produced between the elliptical field and dislocations have implications with respect to line tension, precipitation on dislocations, grain boundary-dislocation interactions, pileups, Peierls barriers and point defect interactions. The computer simulation experiments were supported by the Office of Naval Research and the analytic development of the dipole center by the Air Force Office of Scientific Research.

References

I.

R. A. Johnson, Phys. Rev. 145, 423 (1966).

2.

P. C. Gehlen, J. Appl. Phys. 13, 5165 (1970).

3.

P. C. Gehlen, J. P. Hirth, R. G. Hoagland and M. F. Kanninen, to be published.

4.

F.R.N. Nabarro, Theory of Crystal Dislocations, p. 165, Oxford Univ. Press (1967).

5.

J. P. Hirth and J. Lothe, Theory of Dislocations, p. 48, McGraw-Hill, New York (1968).