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A consistent sign of the burgers vector for materials science and continuum mechanics
Scripta
METALLURGICA
Vol. Printed
5, p p . 2 9 5 - 2 9 6 , 1971 in the United States
Pergamon
Press,
Inc.
A CONSISTENT SIGN OF THE BURGERS VECTOR FOR MATERIALS SCIENCE AND CONTINUUM MECHANICS
Albert G. Guy Professor Department of Metallurgical and Materials Engineering University of Florida Gainesville, Florida 32601
(Peceived DeWit
February
3,
1971)
(i) has discussed some disadvantages of the existence of two opposite conventions
for the sign of the Burgers vector, b,
of a dislocation line.
The purpose of this letter is
to point out that one of the two conventions leads to stresses that are consistent with those used in modern continuum mechanics.
The other convention necessitates
negative sign, which can be (and has been)
the source of confusion,
the introduction of a
annoyance and error.
The
present brief treatment is based on that given by Hirth and Lothe (2). In deriving the elastic stresses associated with a screw dislocation, ties are chosen as positive.
In particular,
chosen so that the displacement the +0 direction,
Fig. i.
all of the quanti-
the shear-type displacement along the cut is
in the z-direction increases by +b as a circuit is made in
[b is a component of b and also has a sense.]
Consequently,
when
an analysis is made to determine a stress, for example (2),
xg
(1)
27T
the negative sign is an essential part of the result. If the usual right-hand convention is employed to construct a Burgers circuit around the screw dislocation in Fig. i, the consistent sign for the Burgers vector
BURGERS CIRCUIT Y
is obtained by drawing the vector from the starting point (S) to the finishing point (F) of the circuit. The reference circuit is one that closes in a perfect crystal.
This is the convention denoted S F / R H
Hirth and Lothe
(3).
by
The advantage of the adoption of
this convention by teachers and scientists in materials science is the posibility of using directly the results from continuum mechanics,
such as Eq.
(i).
FIG. 1 No such methodological argument supports the choice of the opposite convention.
It is simply
a convention employed by certain early workers in
295
A V o l t e r r a screw d i s l o c a t i o n employed i n continuum mechanics and a c o r r e s ponding r i g h t - h a n d Burgers c i r c u i t . (Adapted from R e f e r e n c e 2)
296
CONSISTENT
the field.
SIGN
OF T H E
BURGERS
VECTOR
Vol.
5, N o . 4
An unfortunate result is the fact that the dislocation of Fig. i, for the same
choice of reference axes and line direction, has the stress component (4), =
Oxz
+
ub
~
(2)
This result is of course consistent with Eq. (i) since b is negative in this case, but it is ambiguous in the sense of implying the opposite-sign dislocation unless the defining convention is also specified.
A similar ambiguity in sign occurs for the tensile and com-
pressive stresses associated with an edge dislocation. These pitfalls may cause relatively little inconvenience to those already familiar with the field, but would it not be desirable to eliminate them for those now beginning their study of dislocations?
The adoption of the convention consistent with the methods
of continuum mechanics would benefit students and scientists in both areas. References i.
R. deWit, Aeta Met. 13, 1210 (1965).
2.
J. P. Hirth and J. Lothe, Theory of Dislocations, p. 58f, McGraw-Hill, New York, 1968.
3.
Reference 2, p. 21.
4.
J. Weertman and J. R. Weertman, Elementary Dislocation Theory, p. 33, Macmillan, New York, 1964.
METALLURGICA
Vol. Printed
5, p p . 2 9 5 - 2 9 6 , 1971 in the United States
Pergamon
Press,
Inc.
A CONSISTENT SIGN OF THE BURGERS VECTOR FOR MATERIALS SCIENCE AND CONTINUUM MECHANICS
Albert G. Guy Professor Department of Metallurgical and Materials Engineering University of Florida Gainesville, Florida 32601
(Peceived DeWit
February
3,
1971)
(i) has discussed some disadvantages of the existence of two opposite conventions
for the sign of the Burgers vector, b,
of a dislocation line.
The purpose of this letter is
to point out that one of the two conventions leads to stresses that are consistent with those used in modern continuum mechanics.
The other convention necessitates
negative sign, which can be (and has been)
the source of confusion,
the introduction of a
annoyance and error.
The
present brief treatment is based on that given by Hirth and Lothe (2). In deriving the elastic stresses associated with a screw dislocation, ties are chosen as positive.
In particular,
chosen so that the displacement the +0 direction,
Fig. i.
all of the quanti-
the shear-type displacement along the cut is
in the z-direction increases by +b as a circuit is made in
[b is a component of b and also has a sense.]
Consequently,
when
an analysis is made to determine a stress, for example (2),
xg
(1)
27T
the negative sign is an essential part of the result. If the usual right-hand convention is employed to construct a Burgers circuit around the screw dislocation in Fig. i, the consistent sign for the Burgers vector
BURGERS CIRCUIT Y
is obtained by drawing the vector from the starting point (S) to the finishing point (F) of the circuit. The reference circuit is one that closes in a perfect crystal.
This is the convention denoted S F / R H
Hirth and Lothe
(3).
by
The advantage of the adoption of
this convention by teachers and scientists in materials science is the posibility of using directly the results from continuum mechanics,
such as Eq.
(i).
FIG. 1 No such methodological argument supports the choice of the opposite convention.
It is simply
a convention employed by certain early workers in
295
A V o l t e r r a screw d i s l o c a t i o n employed i n continuum mechanics and a c o r r e s ponding r i g h t - h a n d Burgers c i r c u i t . (Adapted from R e f e r e n c e 2)
296
CONSISTENT
the field.
SIGN
OF T H E
BURGERS
VECTOR
Vol.
5, N o . 4
An unfortunate result is the fact that the dislocation of Fig. i, for the same
choice of reference axes and line direction, has the stress component (4), =
Oxz
+
ub
~
(2)
This result is of course consistent with Eq. (i) since b is negative in this case, but it is ambiguous in the sense of implying the opposite-sign dislocation unless the defining convention is also specified.
A similar ambiguity in sign occurs for the tensile and com-
pressive stresses associated with an edge dislocation. These pitfalls may cause relatively little inconvenience to those already familiar with the field, but would it not be desirable to eliminate them for those now beginning their study of dislocations?
The adoption of the convention consistent with the methods
of continuum mechanics would benefit students and scientists in both areas. References i.
R. deWit, Aeta Met. 13, 1210 (1965).
2.
J. P. Hirth and J. Lothe, Theory of Dislocations, p. 58f, McGraw-Hill, New York, 1968.
3.
Reference 2, p. 21.
4.
J. Weertman and J. R. Weertman, Elementary Dislocation Theory, p. 33, Macmillan, New York, 1964.