This article was downloaded by: [Ohio State University Libraries]
On: 05 May 2012, At: 07:50
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954
Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Philosophical Magazine Series 7
Publication details, including instructions for authors and
subscription information:
http://www.tandfonline.com/loi/tphm18
CXL. Dislocations in thin plates
a a
J.D. Eshelby & A.N. Stroh
a
H. H. Wills Physical Laboratory, University of Bristol
Available online: 15 Jul 2009
To cite this article: J.D. Eshelby & A.N. Stroh (1951): CXL. Dislocations in thin plates,
Philosophical Magazine Series 7, 42:335, 1401-1405
To link to this article: http://dx.doi.org/10.1080/14786445108560958
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-
conditions
This article may be used for research, teaching, and private study purposes. Any
substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,
systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any
representation that the contents will be complete or accurate or up to date. The
accuracy of any instructions, formulae, and drug doses should be independently
verified with primary sources. The publisher shall not be liable for any loss, actions,
claims, proceedings, demand, or costs or damages whatsoever or howsoever
caused arising directly or indirectly in connection with or arising out of the use of
this material.
[ 140~ ]
CXL. Dislocations in Thin Plates.
By J. D. ESttELB¥ and A. N. STROH,
H. H. Wills Physical Laboratory, University of Bristol*.
[Received August 15, 1951.]
ABSTRACT.
The stress due to a screw disloeati6n passing normally through a.n
Downloaded by [Ohio State University Libraries] at 07:50 05 May 2012
infinite plate or a disc is largely confined to the neighbourhood of the
dislocation line, in contrast to the case of a dislocation in an infinite
medium. Two screw dislocations in a plate attract or repel one another
with a short-range force in place of the inverse first power law for infinite
parallel dislocations. The stress due to an edge dislocation is not
essentially different in the plate and infinite body so long as the plate
remains flat, but in some circumstances the stress may be largely relieved
by buckling of the plate.
§ 1. INTRODUCTIO~-.
FORTY (1951) has studied the growth of tabular crystals containing one
or more dislocations. Dawson and Vand (1950, 1951) have published
photographs of thin crystals with growth spirals terminating on
dislocations. It seems worth while, therefore, to contrast the properties
of a dislocation in a thin plate and in an infinite body.
We discuss in detail the following configurations of a straight screw
dislocation : (i.) meeting normally the surface of a semi-infinite body,
(ii.) running normally through an infinite plate and (iii.) along the axis of
a disc, with or without a stress-free hole excluding the core of the
dislocation. Case (i.) m a y be of interest in connection with the detailed
topography of the surface near the point of emergence of a dislocation
(Frank 1951). Fig. 25 of Forty's (1951) paper corresponds almost
exactly to ease (iii.).
The corresponding problem for an edge dislocation is only briefly
touched on.
It is interesting to note t h a t the steps in Forty's growth spirals m a y
be an arbitrary number of lattice spacings (of the order of 100), so that we
have to deal with effectively the classical Volterra dislocation with
arbitrary Burgers vector, in contrast with the theory of metals w h e r e
the Burgers vector is assumed to be limited to one of a few simple lattice
displacements.
* Communicated by Professor N. F. Mott, F.R.S.
SER. 7, VOL. 42, ~O. 335.--DEC. 1951 5B
1402 J . D . Eshelby and A. N. Stroh on the
§ 2. A SCREW DISLOCATION NORMAL TO THE SURFACE
OF A SEMI-INFINITE BODY.
For a screw dislocation in an infinite body along the z-axis of cylindrical
coordinates (r, ~, z) the non-vanishing components of displacement and
stress are
b t~b 1
u~= ~-~0, ~ = 2~ r . . . . . . . (1)
The elastic image field which annuls the traction due to the dislocation
on the plane z = 0 is
b r i~b r I~b( z~ R---~)
Downloaded by [Ohio State University Libraries] at 07:50 05 May 2012
u°-~ 2~rR+z' "r°z~-- 2~rR(R~-z)' ~0----~ ~ q - , (2)
with I ~ - ~ r ~ + z 2 and the other components zero. I t is produced by a
distribution of couples along the negative z-axis, twisting about the
z-axis and with density proportional to distance from the o r i g i n . This
may be verified by integrating twice with respect to z the expression
given by Love (1927, p. 187) for a point-couple. The sum of t h e states (1)
and (2) thus gives the elastic field about a screw dislocation perpendicular
to the free surface of the semi-infinite solid z > 0.
§ 3. A SCREW DISLOCATION IN A PLATE OR Disc.
To solve the problem of a screw dislocation in an infinite plate we must
annul the traction on the planes z = d:d. It is easy to see t h a t this can
be done by introducing an infinite series of images of the type (2). A
useful expression is, however, more easily found as follows. We take
from the image representation the fact that only uo, ~'o~ and rr0 do not
vanish. Then
Voz=l~Ouo/Oz, Tro=t~(3uo/Or--uo/r)
and the equilibrium condition is
32uo 1 Ouo uo 02uo ~)
~-~ -~ r Or r 2 + Oz2 - - v .
A simple solution is u 0 : e x p (±/cz)Jl(kr). Multiplication by a flmction
of ]c and integration yields a more general solution. The solution of our
problem is easily found to be
b "~ sinh jl(/cr ) (3)
u s = - - ~ 0 eosh ~7., - . . . .
for then
27r J0 27r r"
Since
oo
sech x = 2 E ( - - 1 ) n exp {--(2nq-1)x}
0
and
j'~o e- k z J i ( k r ) Telk= x q _ V ( x 2 f fr_ r 2 )
Dislocations in Thin Plates 1403
(3) becomes, with d,~-=(2n4c l) d,
b ~_ {d~_z~[~d,~_z)~ -
d,~q-zq-~f[~,~q-z)2+ } ,
exhibiting u 0 as the sum of a set of images of the t y p e (2). Returning
to (3), u0 can be expressed as a series of modified Bessel functions K 1 by
contour integration or by following the analysis of R i e m a n n (1855) in a
similar problem. In this w a y the elastic field of a screw dislocation
passing perpendicularly through an infinite plate of thickness 2d is found
to be
b b z b 2 [nTrr~ . nlrz
u ~ = ~ 0, u o = -- ~ ;. q- ~r Z (sin ½nTr)G K~ ~2-d-) sm ~ - ,
Downloaded by [Ohio State University Libraries] at 07:50 05 May 2012
r0~=- ~-~ Z (sin ½n~r) K 1 \ = ~ - ] cos 2 ~ ' (4)
tzb z ~ [nrrr~ n~rz
r ' ° - - rr r 2 ~ (sin ½nn) K~ \-:~-] sin 2--d- '
n r = O, Trr ~-- TOO~ T z z : Trz: 0.
with s u m m a t i o n over integral n.
The case of a screw dislocation in the annular disc bounded by the
surfaces z= i d , r=ri, ro (ri<ro) can be t r e a t e d by replacing each K I in
the uo of (4) by a linear combination of K 1 and 11 and adjusting the
constants to annul the traction on the cylindrical surfaces. The result is
b b z_ ( (nrrr'~ {nTrr~'~ sin"nrrz~,
tzTr ~ (n~rr~ (nTrr'~\ n~rz
~o,= b~--dXn~AnI ~ \ = ~ - ] + B . K ~ \--~Td-d] J cos 2--d-' (5)
t~bz tzlr f (nlrr'~ (nrrr'~\ nrrz
T,0= ~- ~ ~- ~ 2:n .A'~I 2 \~-//-] - - B ~ K 2 \ 2d ] J sin 2--d'
II,r~-----O~ Trr=TOO=TZZ~--Trz=O~
where
4b 1 (2d/n~rri)eK~(n~rro/2d)--(2d/nwro)eK~(n~rrd2d)
A,~= ~r-5 n-5 sin ½n~rI~(nrrro/2d)K~(nrrri/24)--Ie(n~rrd2d)Ke(nvro/2d )
a n d B,, is obtained from A,~ by writing I~ for K~ in the numerator.
The energy required to form the dislocation in the annulus is
rd -r 0
W=½bj dzj dr.o~ . . . . . . . . (6)
--d ri
When the outer radius ro is infinite
8~b~d Ko(n~rri/2d)
W= ~a n Zo d d n~(n~ri/2d)~K~(nTrri/2d),
~- 4~ " , ~. , r~>>d,
=~-£.2d.ln
(d) ~ . , r~d . . . . . . (7)
5B2
1404 J . D . E s h e l b y and A. N. S t r o h on the
W h e n r,.=0, r o = co the dislocation at the origin exerts a force
f (8)
F = b ' j _ d %z d z = ~~r
X n nodd
- 1 K 1 \ 2d ] . . . .
on a n o t h e r screw dislocation with Burgers v e c t o r b' distant r from it.
Since when x > l Im(x ) and Km(x ) respectively increase a n d decrease
exponentially with x the elastic s t a t e is given closely b y
b b z i~bz
uz~- ~' u°= 2~ r ' "r"°~-~- ¢r r 2' "r°z=O' (9)
if the point r is a few multiples of d a w a y from b o t h the inner and o u t e r
edges of the annulus. (9) represents e x a c t l y the combined effect o f (1)
Downloaded by [Ohio State University Libraries] at 07:50 05 May 2012
together with (2) and its reflection in the plane z--~0, with couples o f
opposite hand. I n (4) t h e t e r m s in K annul the effect of t h a t p a r t of
the couple distribution which would lie inside the plate.
§ 4. SCREW DISLOCATION : DISCUSSION.
I t will be seen t h a t the elastic state of the plate (equations (4)) is quite
different from t h a t of a slab of width 2d m a r k e d out perpendicular to
a screw dislocation in an infinite b o d y (equations (1)). The shear stress
~0z is confined to the n e i g h b o u r h o o d of the dislocation. T h a t this m u s t
be so is clear : ~0~ vanishes at the surface of the plate, and far f r o m the
dislocation this state of aft'airs must persist t h r o u g h o u t the thickness o f
the plate. T h e energy of a screw dislocation in an infinite plate, with
its core excluded b y ~ stress-free hole, is finite, while for a dis]ocation in
an infinite cylinder the integral (6), (#b~/47r). 2d. In (ro/r~), diverges as
r o -+ ~ . Again, in an infinite b o d y two parallel screw dislocations a t t r a c t
like electrostatic line-charges, with a force (t&b'/27r) . 2d/r per length 2d,
whilst each t e r m in (8) is the force due to a line-charge m a d e up o f
particles a t t r a c t i n g with the Y u k a w a p o t e n t i a l const, r -1 exp {--nnr/2d}.
Because ~'vo does n o t vanish the screw dislocation will also i n t e r a c t w i t h
an edge dislocation running parallel to the z-axis. Since %.0 is an odd
function of z tl~ere will be no net force between them, b u t only a couple.
I t is also clear t h a t the image-force a t t r a c t i n g the dislocation t o w a r d s
the edge of a lamina will be v e r y small unless the dislocation is o n l y a
few multiples of d from the edge. I n the simple case where the dislocation
is distant D from the edge of a semi-infinite plate this force is given b y (8)
with b'--~b, r----2D. F o r comparison t h e image force on a length 2d o f
a screw dislocation running at a distance D parallel to the free surface o f
a semi-infinite solid is Ixb~d/D.
§ 5. EDGE DISLOCATIO~N-S.
The stresses ~rr, ~00, T~0 p r o d u c e d b y a n edge dislocation along
the z-axis in an infinite b o d y are derived from the A i r y f u n c t i o n
X----const. r In r sin 0, while ~'~:v(-r~+-roo ) and the remaining c o m p o n e n t s
are zero. To solve the problem of an edge dislocation t r a v e r s i n g a plate
Dislocations in Thin Plates 1405
we must find an image stress system giving rr:--~0 and % : = c o n s t . sin O/r
on the planes z----± d . The necessary analysis would be similar to t h a t of
Sneddon (1946). We should have to evaluate integrals like those in his
§ 5, but with Besset funetion~ of higher order on account of the angular
dependence. The results would not be simple, and the case of an edge
dislocation in an annular disc would be quite intractable. The general
nature of the result can, however, be made out quite simply. I f in the
solution (Love, op. cit. p. 225) for an infinite hollow cylinder with an edge
dislocation we replace ~ b y 2~/()~+2~) the resulting expressions give the
stresses and displacements averaged across the thickness of a disc cut from
the cylinder and having surfaces free of stress (generalized plane stress :
Downloaded by [Ohio State University Libraries] at 07:50 05 May 2012
Love, op. cir., p. 207). The average stresses will fall off as 1/r and the
elastic state will not be v e r y different from what it was when the disc
formed part of the infinite cylinder.
As long as the plate containing the edge dislocation remains fiat there
is thus no widespread relaxation of stress, as there is for the screw
dislocation. However, the energy m a y be reduced by buckling of the
plate. Take a sheet of paper with a hole in it and make a tuck of constant
width b (small compared with the size of the hole) running from the hole
to the edge of the sheet. The paper will form a surface given roughly
b y the equation z~-(bv~2 sin ~)/2~. I f the paper is flattened out to form
a plane dislocated lamina a large amount of strain-energy will be
introduced. On the other hand we should not expect buckling to occur
in a thick dislocated disc.
To find out the relations between the Burgers vector and the thickness
and inner and outer radii of the annular disc for which buckling is
energetically favourable we should have to use the theory of plates with
strain in the middle surface (v. K~rm~n (1910)), modified to take account
of initial stress. These equations are non-linear and it is difficult to
solve t h e m with the necessary boundary conditions. I t seems clear,
however, t h a t in certain circumstances an edge dislocation will be able
to relieve most of its stress by slight buckling of the plate, except within
a few multiples of d from its centre, leading to a state of affairs similar
to t h a t discussed for the screw dislocation.
~EFEI%ENCES.
DAwson, I. M., and VAND, V., 1950, Nature, 165, 295 ; 1951, Proc. Roy. Soc.
206A 555.
FORTY, A. J., 1951, Phil. Mag., [7], 42, 670.
FRANK, F. C., 1951, Acta Cryst. In the press.
v. KiRMi~, TH., 1910, Ency. der Math. Wiss. IV, [4], 348, (Leipzig : Teubner).
LOVE, A. E. H., 1927, Mathematical Theory of Elasticity, (Cambridge : University
Press).
RIE~, B., 1855, Ann. der Phys., 95, 130.
S~EDDO/% I. N., 1946, Proe. Camb. Phil. Soc., 42, 260.
