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Elastic behaviors of wedge disclination loops near a free surface
163
Materials Science and Engineering, 12 (1973) 163-166 (~ Elsevier Sequoia S~A., Lausanne - Printed in The Netherlands
Elastic Behaviors of Wedge Disclination Loops Near a Free Surface TSU-WEI C H O U and T I E N - L I N LU Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Del. 19711 (U.S.A.) (Received September 12, 1972: in revised form October 30, 1972)
Summao,* The elastic field of a wedge diselination loop near a free surface is studied by using the method of Green's function. The behaviors of the loop are quite different from those of twist disclinations and dislocations. For a h)op situated at a finite distance beneath a free surface, it may be attracted to or repelled from the surface, depending upon its orientation with respect to the surface. When a loop assumes one of the two particular orientations, it can exist in metastable equilibrium. This type of behavior was realized for dislocations only when they approach a surface film plated on an elastic half-plane. 1. I N T R O D U C T I O N
Disclinations are line imperfections in solids and are traditionally known as rotational dislocations. They can be conveniently categorized as twist and wedge types where the rotation axes are normal and parallel to the disclination lines, respectively. A comprehensive review of the state-of-the-art of disclination theory has recently been made by Chou 1. The elastic behaviors of disclinations in nonhomogeneous media were first examined by Chou 2'3. It was found that for a twist disclination loop lying parallel and near to a planar phase boundary, the elastic field can be constructed by an image method. Closed form solutions were also obtained for twist loops situated in the phase boundary. In all these cases, the elastic fields of disclinations involve only shear stress field. However, when the plane of the loop is in some arbitrary orientation with respect to the interface of a two phase system, there is a dilatational field associated with the loop. Consequently, it is likely * R6sum6 en franqais/t la fin de l'article. Deutsche Zusammenfassung am Schlug des Artikels.
to interact not only with screw dislocations and other twist disclinations but also with edge dislocations, point defects and wedge disclinations. To demonstrate this point, Chou and Lu 4 examined the interaction between a free surface and a disclination loop. It was found that if a loop is pinned down beneath a free surface, it can assume an orientation of lowest strain energy. The aim of this paper is to extend the above investigation to a more complicated case, namely, a wedge disclination loop. The term "wedge disclination loop" was first used by Li and Gilman 5 in their study of deformation configurations in polymers. It is used to denote a disclination loop where the rotation axis lies in the plane of the loop. Hence, it is understood that loops of this type bear some twist characteristics. In Fig. 1 we demonstrate the elastic half-plane and the geometry of the loop under consideration. The center of the wedge loop with radius a and angle of rotation co is situated at a distance z0 below the free surface. The orientation of the loop is defined
Fig. 1. Wedge disclination loop in an elastic half-plane.
164
TSU-WEI CHOU, TIEN-LIN LU
by the angle ~ m a d e by the n o r m a l h to the l o o p a n d the z axis. T h e axis of r o t a t i o n is a s s u m e d to be parallel to the y axis. T h e shear m o d u l u s a n d P o i s s o n ' s ratio of the elastic m e d i u m are d e n o t e d by # a n d v, respectively. In view of the c o m p l e x i t y
of the elastic p r o b l e m involved, we c h o o s e only to consider the case where the l o o p d i m e n s i o n is m u c h smaller t h a n its distance f r o m the free surface. H o w e v e r , the solutions o b t a i n e d f r o m this case will u n d o u b t e d l y p r o v i d e us insight of the b e h a v i o r s of finite loops in a two phase system.
2. ELASTIC FIELD 2.1. D i s p l a c e m e n t The d i s p l a c e m e n t c o m p o n e n t s can be m o s t easily o b t a i n e d by the m e t h o d of G r e e n ' s function. The G r e e n ' s function for a point load in a semi-infinite elastic m e d i u m has been o b t a i n e d by M i n d l i n 6. By resolving the stresses due to a p o i n t load o n t o the plane of the l o o p a n d then by considering the balance of virtual work, the d i s p l a c e m e n t at the position of p o i n t load a p p l i c a t i o n can be obtained. In essence, this m e t h o d is equivalent to a s t r a i g h t f o r w a r d application of the reciprocal t h e o r e m in elasticity. Interested readers are referred to the w o r k s of Li a n d G i l m a n 5 a n d H u a n g a n d M u r a v for details of this m e t h o d . The resulting d i s p l a c e m e n t c o m p o n e n t s u, v a n d w a l o n g the x, y a n d z axes, respectively, are given in the following: u = [~oa4/32 ( I - v ) ]
{-(1-2v)
cos ~//~3 + [ - 4 ( 1
-v)(1-2v)
sin2~ cos ct + ( 1 - 2 v ) cos ~]//~ 3
+ 3 [2(1 - 4v) x~ sin sct - (1 - 4v) (x e - ~2) sin 2 ~ cos ct - (1 - 6v) x~ sin + ((1 - 2v)x 2 + ~2) cos ~ ] / R s + 3 [2((5 - 6v + 4v 2) x z + (7 - 14v + 4 v e ) x z o ) sin3~ + ((9 - 20v + 8v 2) x 2 - (1 + 8v)z 2 - 6 z z o - (5 - 8v)z 2) sin2 ct cos ~ + ((1 - 6 v ) x z - (9 - 1 4 v ) x z o ) sin ~ - ((1 - 2v)x 2 + z 2 - 4(1 - v)zz o - (3 - 4v)z 2) cos c~]//~ 5 + 15x [ ( 3 X 2 -- ~2)~ sin s c t - (x 2 - 3~ e) x sin 2 ~ cos c~- ( 2 x 2 - ~2) ~ sin ~ - xY 2 cos c~]//~7 + 15[((1 + 4v)z2~ - 8ZZo~ + ( 3 - 4 v ) z 2 ~ - ( 7 - 4 v ) x 2 z - 3
(3 - 4 v ) x 2 z o ) x
sinS~+((5+4v)x2z:
+ 8 (1 - v) x z z z o + 3 (3 - 4v) x 2 Zoz - (3 - 4v)x ¢ + 6 z z o ~2) sin 2 ~ cos ~ + (2x 2 z + 2(3 - 4v) x : z o - z 2
+2(1 +2v)ZZo~-(3-4v)z~e)x sin~+ (xez2-4(1 -v)xZzzo - (3-4v)xez~-2ZZoe e) cos ~]/~ 7 + 210x [(3x 2 - ~e) z z o ~ sin 3 ~ + (x 2 - 3~ 2) x z z o sin e c~ cos ~ - (2x 2 - ~2) ZZo ~ sin ~ + XZZo~: cos ~]//~9 + 4(1 - v)(1 - 2v)x sin s ~ [ - 3 (2/~ + e)/(/~ 3 (/~ + ~)z) + x 2 (8/~2 + 9 / ~ + 3e 2)/(Ks (K + e)3)3 + 12 (1 - v)(1 - 2v) sin z c~cos ~ [ I/(/~(R + e) 2) - 2 x 2 (3/~ + e)/(K 3 (/~ + e) 3) + x 4 (5/~ 2 + 4/~e + e e) / (g5 ( g + ~)4) ] } ;
(1)
v = [09a4y/32 (i - v)] {3(1 - 2v)(x cos ~ - ~ sin ~ ) / R ~ +3(1-2v)[4(1-v)~sinST+4(1-v)xsin2~cos~+~
sin c t - x cos ~]//~5
+ 15 [(3x e - ~2) ~ sin ~~ _ x (x 2 - 3~ e) sin 2 ~ cos ~ - (2x 2 - ~z)~ sin ~ - x~ e cos ~ ] / ~ 7 + 15 [ ( - (7 - 4v)x 2 z - 3 (3 - 4v)x z z o + (1 + 4v) z 2 e + 4 z z o e + (3 - 4v)z 2 e)sin 3 ~ + ( - (3 - 4v)x 3 + (5 + 4 v ) x z 2 + 2 ( 7 - 4v)XZZo + 3 (3 - 4v) x z 2) sin 2 ~ cos ~ + (2x z z + 2(3 - 4v) x 2 Zo - z 2 -
2 (3 - 2v) ZZo e - (3 - 4v) z 2 ~) sin ~ + x (z 2 - 4(1 - v) ZZo - (3 - 4v)z 2) cos ~] //~7
+ 2 1 0 [ ( 3 x 2 - ~2)ZZo~ sin 3 ~ + (x 2 - 3~ e) x z z o sin e ~ cos c~- (2x 2 - ~2)ZZo~ sin c~+ XZZo ~2 cos ~]//~9 + 4(1 - v)(1 - 2v) sin s ~ [ - (2/~ + e)/(/~s (/~ + e)2) + X 2 (8~2 + 9 / ~ + 3~:)/(/~ 5 (/~ + z)3)] +12(1-v)(1-2v)xsin2~cosct[-(3R+~)/RS(~+~)S)+x2(5R2+4R~+~2)/(.~5(R+~)¢)]};
(2)
165
ELASTIC BEHAVIORS OF WEDGE DISCLINATION LOOPS
w = [ o m 4 / 3 2 (1 - v)] {(1 - 2v)sin ~/R 3 + (1 - 2v)[-8(1 - v)sin 3 ~ - sin a]//~3 + 3 [ (1 - 4v)(x 2 - 22) sin 3 a + 2 (1 - 4v) x2 sin 2 c~cos 2((1 - 2v)x 2 + l~z2) sin c~- (1 - 2v) x~ cos ~ ] / R 5
-
+ 3 [ ( - (9 - 20v + 8v2)x a + (5 - 12v - 8v2)z 2 + 2(3 - 4v -
8v2)zzo
+ (1 + 4v - 8y2)z 2) sin3a + 2x((5 - 10v - 4v2)z + (3 - 2v - 4v2)zo) sin 2 c~cos ~ + 2((1 - 2v)x 2 + sin c~+ (1 -
vz 2 - (1 - 4v)ZZo - (2 - 3v)z 2)
2v)x~ cos c~]/R s
+ 15 [(3x 2 - 22) 22 sin 3 c~- (x 2 - 3~2) x~ sin 2 a cos ~ - (2x 2 - 22) 22 sin a - x~ 3 cos ~ ] / R ~ + 151-(-(7-4v)xZz2-4(1
+2v)xZzzo+3(3-4v) xzzg+ (1 + 4v)z2~2 + 4(1 +2v)zzo ~2
- (3 - 4v) zg~Z)sin 3 c~+ ( - (3 - 4v)xa ~ + (5 + - 3(3 - 4v)
4v)xz 2 ~ + 8(1 + 2v) xzzo~
xz2~) sin 2 ~ cos ~ + (2x 2 z 2 + 8vx 2ZZo
- 2(3 - 4v)x 2 z 2 - z 2 ~2 _ 2(3 +
2v)zzo ~2 + (3 - 4v)zg~ 2) sin ~ + x~(z 2 - 2(1 + 2v)ZZo
+ (3 - 4v) z g) cos ~]//~7 + 210 [ ( 3 x 2 - ~ z) ZZoy2 sin 3 ~ + ( x 2 _ _ (2x 2 _ ~Z)zz0 ~2 sin ~ +
XZZo~3 cos a]//~9 + 4(1 - v)(1 - 2v)x sin 2 c~cos :¢[3 (2/~ + ~)/(/~3 (/~ + ~)2)
(3)
X2 (81~ 2 "-}-9iR~ + 3~2)/(/~5 (*R + ~)3)] } ;
--
.R2=x2 +y2+7-2, /~2 = x2+y2 +,~2,
where
3$2)XZZoy sin 2 a cos a
~=Z__aO ' ~ =
Z+Zo"
2.2. lmage )brces O n the b a s i s of these d i s p l a c e m e n t c o m p o n e n t s , it is a s t r a i g h t f o r w a r d , t h o u g h t e d i o u s , m a t t e r to c o m p u t e the stress field of the d i s c l i n a t i o n l o o p . O n e stress c o m p o n e n t of p a r t i c u l a r interest is the n o r m a l stress ann a c t i n g on the p l a n e of the l o o p . By e m p l o y i n g this stress c o m p o n e n t , the i n t e r a c t i o n e n e r g y of the w e d g e l o o p can be e v a l u a t e d b y c o n s i d e r i n g the t o t a l w o r k d o n e in c r e a t i n g such a l o o p u n d e r the d i s c l i n a t i o n self-stress field : E l = --
f~' io~ io Gnn * r 2 sin OdrdOdoo
= - [ 3 ~ / ~ 2 a8/8192(1 .
+
v)z~] {5 [7(1 + 5v - 12v 2) sin 6
( - 28 + 87v + 116v 2 - 48v 3) sin 4 ~ + (7 - 88v - 12v 2 - 16v 3) sin 2 ~ ] / ( 1 - 2v) + ( - 89 + 1064v) sin 6
+ 16(22 - 57v - 2v 2) sin 4 : ¢ - 5 (67 + 38v + 16v 2) sin 2 a + 168 }.
(4)
• d e n o t e s the n o r m a l stress w h i c h is i n d u c e d in the elastic m e d i u m o w i n g to the i n t r o d u c t i o n of the free rYnn surface. T h e i m a g e force a c t i n g on the l o o p o w i n g to the p r e s e n c e of a flee surface can be c a l c u l a t e d f r o m the interaction energy : ~E, El z
--
(5)
~Z O '
It is o b v i o u s t h a t the d e p e n d e n c e of b o t h E~ a n d F~ on l o o p o r i e n t a t i o n is i d e n t i c a l . T h e v a r i a t i o n of l o o p i n t e r a c t i o n e n e r g y a n d i m a g e force w i t h o r i e n t a t i o n is d e m o n s t r a t e d in Fig. 2. 3. CONCLUSIONS AND DISCUSSION
T h e b e h a v i o r of a w e d g e d i s c l i n a t i o n l o o p n e a r a free surface is q u i t e different f r o m t h a t of a twist disclination loop and dislocations. Some important
features of w e d g e d i s c l i n a t i o n b e h a v i o r s a n d t h e i r c o m p a r i s o n s w i t h d i s l o c a t i o n s are given in the following : a. F o r a w e d g e d i s c l i n a t i o n l o o p s i t u a t e d at a finite d i s t a n c e f r o m the free surface, the s t r a i n e n e r g y is a
166
TSU-WEI CHOU, TIEN-LIN LU 4096
Z
3•Fw20
e
Et
15 yF. wz aS
FI
4096
b. The interaction energy of a wedge disclination loop may be positive or negative depending upon the loop orientation, whereas in the case of a twist disclination loop, the presence of free surface always tends to reduce the total elastic energy. c. The image force on a twist disclination loop always attracts the loop toward the free surface. However, the image effect on a wedge disclination loop is quite different. For a wedge loop at a finite distance beneath a free surface the image force may either attract or repel the loop depending upon its orientation, except at two particular orientations. For these particular orientations, the loop is in a metastable equilibrium with respect to the free surface. It is noted that dislocations may experience zero image force in a bi-material system only when they approach a surface film with elastic properties different from the matrix phase. However, this behavior is now possible near a free surface for disclinations merely due to the orientation effect.
o5
Z
240
- 90 °
-60"
-30 °
0° a
30 e
60*
90 °
REFERENCES
minimum if the loop plane is parallel to the free surface. However, in the case of twist loops, this lowest energy orientation is affected by the Poisson's ratio of the elastic medium.
1 T.W. Chou, Proc. Intern. Syrup. on Defect Interaction in Solids, Indian Institute of Science, Bangalore, India, 1972. 2 T. W. Chou, J. Appl. Phys., 42 (1972) 4092. 3 T. W. Chou, J. Appl. Phys., 42 (1971) 4931. 4 T. W. Chou and T. L. Lu, J. Appl. Phys., 43 (1972) 2562. 5 J. C. M. Li and J, J. Gilman, J. Appl. Phys., 41 (1970) 4248. 6 R. D. Mindlin, Physics, 7 (1936) 195. 7 W. Huang and T. Mura, J. Appl. Phys., 41 (1970) 5175.
Comportements ~lastiques de boucles-coins de disclinations au voisinaoe d'une surface libre
Das elastische Verhalten yon Stufen-Disklinationsringen in der Niihe e&er freien Oberfliiche
Le champ 61astique d'une boucle-coin de disclination situ6e au voisinage d'une surface libre est 6tudi6 ii l'aide de la m6thode de la fonction de Green. Les comportements de la boucle sont totalement diffrrents de ceux de disclinations de torsion ou de dislocations. Une boucle siture Aune distance finie de la surface libre peut ~tre attirre ou repoussre par la surface suivant son orientation par rapport ~l cette surface. Lorsque la boucle possrde l'une ou l'autre de deux orientations particuli~res, elle peut se trouver en 6quilibre m6tastable. Les dislocations n'ont un comportement de ce type que si elles se rapprochent d'un film superficiel drpos6 sur un demi-plan 61astique.
Das elastische Feld eines Stufen-Disklinationsringes in der N~ihe einer freien Oberfl/iche wird mit Hilfe der Methode der Greenschen Funktionen untersucht. Das Verhalten der Ringe unterscheidet sich betr/ichtlich vom Verhalten der SchraubenDisklinationen und Versetzungen. Ein Ring in endlichem Abstand vonder freien Oberfl/iche kann, je nach seiner Orientierung relativ zur Oberfl~iche, v o n d e r Oberfl~iche angezogen oder abgestol3en werden. Hat ein Ring eine von zwei speziellen Orientierungen, so kann er sich in einem metastabilen Gleichgewicht befinden. Dieses Verhalten konnte fiir Versetzungen nur fiJr den Fall realisiert werden, dass sie sich einer Oberfl~ichenschicht n/iherten, die auf eine elastische Halbebene aufgebracht war.
Fig. 2. Variation of interaction energy and image forces with disclination loop orientation.
Materials Science and Engineering, 12 (1973) 163-166 (~ Elsevier Sequoia S~A., Lausanne - Printed in The Netherlands
Elastic Behaviors of Wedge Disclination Loops Near a Free Surface TSU-WEI C H O U and T I E N - L I N LU Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Del. 19711 (U.S.A.) (Received September 12, 1972: in revised form October 30, 1972)
Summao,* The elastic field of a wedge diselination loop near a free surface is studied by using the method of Green's function. The behaviors of the loop are quite different from those of twist disclinations and dislocations. For a h)op situated at a finite distance beneath a free surface, it may be attracted to or repelled from the surface, depending upon its orientation with respect to the surface. When a loop assumes one of the two particular orientations, it can exist in metastable equilibrium. This type of behavior was realized for dislocations only when they approach a surface film plated on an elastic half-plane. 1. I N T R O D U C T I O N
Disclinations are line imperfections in solids and are traditionally known as rotational dislocations. They can be conveniently categorized as twist and wedge types where the rotation axes are normal and parallel to the disclination lines, respectively. A comprehensive review of the state-of-the-art of disclination theory has recently been made by Chou 1. The elastic behaviors of disclinations in nonhomogeneous media were first examined by Chou 2'3. It was found that for a twist disclination loop lying parallel and near to a planar phase boundary, the elastic field can be constructed by an image method. Closed form solutions were also obtained for twist loops situated in the phase boundary. In all these cases, the elastic fields of disclinations involve only shear stress field. However, when the plane of the loop is in some arbitrary orientation with respect to the interface of a two phase system, there is a dilatational field associated with the loop. Consequently, it is likely * R6sum6 en franqais/t la fin de l'article. Deutsche Zusammenfassung am Schlug des Artikels.
to interact not only with screw dislocations and other twist disclinations but also with edge dislocations, point defects and wedge disclinations. To demonstrate this point, Chou and Lu 4 examined the interaction between a free surface and a disclination loop. It was found that if a loop is pinned down beneath a free surface, it can assume an orientation of lowest strain energy. The aim of this paper is to extend the above investigation to a more complicated case, namely, a wedge disclination loop. The term "wedge disclination loop" was first used by Li and Gilman 5 in their study of deformation configurations in polymers. It is used to denote a disclination loop where the rotation axis lies in the plane of the loop. Hence, it is understood that loops of this type bear some twist characteristics. In Fig. 1 we demonstrate the elastic half-plane and the geometry of the loop under consideration. The center of the wedge loop with radius a and angle of rotation co is situated at a distance z0 below the free surface. The orientation of the loop is defined
Fig. 1. Wedge disclination loop in an elastic half-plane.
164
TSU-WEI CHOU, TIEN-LIN LU
by the angle ~ m a d e by the n o r m a l h to the l o o p a n d the z axis. T h e axis of r o t a t i o n is a s s u m e d to be parallel to the y axis. T h e shear m o d u l u s a n d P o i s s o n ' s ratio of the elastic m e d i u m are d e n o t e d by # a n d v, respectively. In view of the c o m p l e x i t y
of the elastic p r o b l e m involved, we c h o o s e only to consider the case where the l o o p d i m e n s i o n is m u c h smaller t h a n its distance f r o m the free surface. H o w e v e r , the solutions o b t a i n e d f r o m this case will u n d o u b t e d l y p r o v i d e us insight of the b e h a v i o r s of finite loops in a two phase system.
2. ELASTIC FIELD 2.1. D i s p l a c e m e n t The d i s p l a c e m e n t c o m p o n e n t s can be m o s t easily o b t a i n e d by the m e t h o d of G r e e n ' s function. The G r e e n ' s function for a point load in a semi-infinite elastic m e d i u m has been o b t a i n e d by M i n d l i n 6. By resolving the stresses due to a p o i n t load o n t o the plane of the l o o p a n d then by considering the balance of virtual work, the d i s p l a c e m e n t at the position of p o i n t load a p p l i c a t i o n can be obtained. In essence, this m e t h o d is equivalent to a s t r a i g h t f o r w a r d application of the reciprocal t h e o r e m in elasticity. Interested readers are referred to the w o r k s of Li a n d G i l m a n 5 a n d H u a n g a n d M u r a v for details of this m e t h o d . The resulting d i s p l a c e m e n t c o m p o n e n t s u, v a n d w a l o n g the x, y a n d z axes, respectively, are given in the following: u = [~oa4/32 ( I - v ) ]
{-(1-2v)
cos ~//~3 + [ - 4 ( 1
-v)(1-2v)
sin2~ cos ct + ( 1 - 2 v ) cos ~]//~ 3
+ 3 [2(1 - 4v) x~ sin sct - (1 - 4v) (x e - ~2) sin 2 ~ cos ct - (1 - 6v) x~ sin + ((1 - 2v)x 2 + ~2) cos ~ ] / R s + 3 [2((5 - 6v + 4v 2) x z + (7 - 14v + 4 v e ) x z o ) sin3~ + ((9 - 20v + 8v 2) x 2 - (1 + 8v)z 2 - 6 z z o - (5 - 8v)z 2) sin2 ct cos ~ + ((1 - 6 v ) x z - (9 - 1 4 v ) x z o ) sin ~ - ((1 - 2v)x 2 + z 2 - 4(1 - v)zz o - (3 - 4v)z 2) cos c~]//~ 5 + 15x [ ( 3 X 2 -- ~2)~ sin s c t - (x 2 - 3~ e) x sin 2 ~ cos c~- ( 2 x 2 - ~2) ~ sin ~ - xY 2 cos c~]//~7 + 15[((1 + 4v)z2~ - 8ZZo~ + ( 3 - 4 v ) z 2 ~ - ( 7 - 4 v ) x 2 z - 3
(3 - 4 v ) x 2 z o ) x
sinS~+((5+4v)x2z:
+ 8 (1 - v) x z z z o + 3 (3 - 4v) x 2 Zoz - (3 - 4v)x ¢ + 6 z z o ~2) sin 2 ~ cos ~ + (2x 2 z + 2(3 - 4v) x : z o - z 2
+2(1 +2v)ZZo~-(3-4v)z~e)x sin~+ (xez2-4(1 -v)xZzzo - (3-4v)xez~-2ZZoe e) cos ~]/~ 7 + 210x [(3x 2 - ~e) z z o ~ sin 3 ~ + (x 2 - 3~ 2) x z z o sin e c~ cos ~ - (2x 2 - ~2) ZZo ~ sin ~ + XZZo~: cos ~]//~9 + 4(1 - v)(1 - 2v)x sin s ~ [ - 3 (2/~ + e)/(/~ 3 (/~ + ~)z) + x 2 (8/~2 + 9 / ~ + 3e 2)/(Ks (K + e)3)3 + 12 (1 - v)(1 - 2v) sin z c~cos ~ [ I/(/~(R + e) 2) - 2 x 2 (3/~ + e)/(K 3 (/~ + e) 3) + x 4 (5/~ 2 + 4/~e + e e) / (g5 ( g + ~)4) ] } ;
(1)
v = [09a4y/32 (i - v)] {3(1 - 2v)(x cos ~ - ~ sin ~ ) / R ~ +3(1-2v)[4(1-v)~sinST+4(1-v)xsin2~cos~+~
sin c t - x cos ~]//~5
+ 15 [(3x e - ~2) ~ sin ~~ _ x (x 2 - 3~ e) sin 2 ~ cos ~ - (2x 2 - ~z)~ sin ~ - x~ e cos ~ ] / ~ 7 + 15 [ ( - (7 - 4v)x 2 z - 3 (3 - 4v)x z z o + (1 + 4v) z 2 e + 4 z z o e + (3 - 4v)z 2 e)sin 3 ~ + ( - (3 - 4v)x 3 + (5 + 4 v ) x z 2 + 2 ( 7 - 4v)XZZo + 3 (3 - 4v) x z 2) sin 2 ~ cos ~ + (2x z z + 2(3 - 4v) x 2 Zo - z 2 -
2 (3 - 2v) ZZo e - (3 - 4v) z 2 ~) sin ~ + x (z 2 - 4(1 - v) ZZo - (3 - 4v)z 2) cos ~] //~7
+ 2 1 0 [ ( 3 x 2 - ~2)ZZo~ sin 3 ~ + (x 2 - 3~ e) x z z o sin e ~ cos c~- (2x 2 - ~2)ZZo~ sin c~+ XZZo ~2 cos ~]//~9 + 4(1 - v)(1 - 2v) sin s ~ [ - (2/~ + e)/(/~s (/~ + e)2) + X 2 (8~2 + 9 / ~ + 3~:)/(/~ 5 (/~ + z)3)] +12(1-v)(1-2v)xsin2~cosct[-(3R+~)/RS(~+~)S)+x2(5R2+4R~+~2)/(.~5(R+~)¢)]};
(2)
165
ELASTIC BEHAVIORS OF WEDGE DISCLINATION LOOPS
w = [ o m 4 / 3 2 (1 - v)] {(1 - 2v)sin ~/R 3 + (1 - 2v)[-8(1 - v)sin 3 ~ - sin a]//~3 + 3 [ (1 - 4v)(x 2 - 22) sin 3 a + 2 (1 - 4v) x2 sin 2 c~cos 2((1 - 2v)x 2 + l~z2) sin c~- (1 - 2v) x~ cos ~ ] / R 5
-
+ 3 [ ( - (9 - 20v + 8v2)x a + (5 - 12v - 8v2)z 2 + 2(3 - 4v -
8v2)zzo
+ (1 + 4v - 8y2)z 2) sin3a + 2x((5 - 10v - 4v2)z + (3 - 2v - 4v2)zo) sin 2 c~cos ~ + 2((1 - 2v)x 2 + sin c~+ (1 -
vz 2 - (1 - 4v)ZZo - (2 - 3v)z 2)
2v)x~ cos c~]/R s
+ 15 [(3x 2 - 22) 22 sin 3 c~- (x 2 - 3~2) x~ sin 2 a cos ~ - (2x 2 - 22) 22 sin a - x~ 3 cos ~ ] / R ~ + 151-(-(7-4v)xZz2-4(1
+2v)xZzzo+3(3-4v) xzzg+ (1 + 4v)z2~2 + 4(1 +2v)zzo ~2
- (3 - 4v) zg~Z)sin 3 c~+ ( - (3 - 4v)xa ~ + (5 + - 3(3 - 4v)
4v)xz 2 ~ + 8(1 + 2v) xzzo~
xz2~) sin 2 ~ cos ~ + (2x 2 z 2 + 8vx 2ZZo
- 2(3 - 4v)x 2 z 2 - z 2 ~2 _ 2(3 +
2v)zzo ~2 + (3 - 4v)zg~ 2) sin ~ + x~(z 2 - 2(1 + 2v)ZZo
+ (3 - 4v) z g) cos ~]//~7 + 210 [ ( 3 x 2 - ~ z) ZZoy2 sin 3 ~ + ( x 2 _ _ (2x 2 _ ~Z)zz0 ~2 sin ~ +
XZZo~3 cos a]//~9 + 4(1 - v)(1 - 2v)x sin 2 c~cos :¢[3 (2/~ + ~)/(/~3 (/~ + ~)2)
(3)
X2 (81~ 2 "-}-9iR~ + 3~2)/(/~5 (*R + ~)3)] } ;
--
.R2=x2 +y2+7-2, /~2 = x2+y2 +,~2,
where
3$2)XZZoy sin 2 a cos a
~=Z__aO ' ~ =
Z+Zo"
2.2. lmage )brces O n the b a s i s of these d i s p l a c e m e n t c o m p o n e n t s , it is a s t r a i g h t f o r w a r d , t h o u g h t e d i o u s , m a t t e r to c o m p u t e the stress field of the d i s c l i n a t i o n l o o p . O n e stress c o m p o n e n t of p a r t i c u l a r interest is the n o r m a l stress ann a c t i n g on the p l a n e of the l o o p . By e m p l o y i n g this stress c o m p o n e n t , the i n t e r a c t i o n e n e r g y of the w e d g e l o o p can be e v a l u a t e d b y c o n s i d e r i n g the t o t a l w o r k d o n e in c r e a t i n g such a l o o p u n d e r the d i s c l i n a t i o n self-stress field : E l = --
f~' io~ io Gnn * r 2 sin OdrdOdoo
= - [ 3 ~ / ~ 2 a8/8192(1 .
+
v)z~] {5 [7(1 + 5v - 12v 2) sin 6
( - 28 + 87v + 116v 2 - 48v 3) sin 4 ~ + (7 - 88v - 12v 2 - 16v 3) sin 2 ~ ] / ( 1 - 2v) + ( - 89 + 1064v) sin 6
+ 16(22 - 57v - 2v 2) sin 4 : ¢ - 5 (67 + 38v + 16v 2) sin 2 a + 168 }.
(4)
• d e n o t e s the n o r m a l stress w h i c h is i n d u c e d in the elastic m e d i u m o w i n g to the i n t r o d u c t i o n of the free rYnn surface. T h e i m a g e force a c t i n g on the l o o p o w i n g to the p r e s e n c e of a flee surface can be c a l c u l a t e d f r o m the interaction energy : ~E, El z
--
(5)
~Z O '
It is o b v i o u s t h a t the d e p e n d e n c e of b o t h E~ a n d F~ on l o o p o r i e n t a t i o n is i d e n t i c a l . T h e v a r i a t i o n of l o o p i n t e r a c t i o n e n e r g y a n d i m a g e force w i t h o r i e n t a t i o n is d e m o n s t r a t e d in Fig. 2. 3. CONCLUSIONS AND DISCUSSION
T h e b e h a v i o r of a w e d g e d i s c l i n a t i o n l o o p n e a r a free surface is q u i t e different f r o m t h a t of a twist disclination loop and dislocations. Some important
features of w e d g e d i s c l i n a t i o n b e h a v i o r s a n d t h e i r c o m p a r i s o n s w i t h d i s l o c a t i o n s are given in the following : a. F o r a w e d g e d i s c l i n a t i o n l o o p s i t u a t e d at a finite d i s t a n c e f r o m the free surface, the s t r a i n e n e r g y is a
166
TSU-WEI CHOU, TIEN-LIN LU 4096
Z
3•Fw20
e
Et
15 yF. wz aS
FI
4096
b. The interaction energy of a wedge disclination loop may be positive or negative depending upon the loop orientation, whereas in the case of a twist disclination loop, the presence of free surface always tends to reduce the total elastic energy. c. The image force on a twist disclination loop always attracts the loop toward the free surface. However, the image effect on a wedge disclination loop is quite different. For a wedge loop at a finite distance beneath a free surface the image force may either attract or repel the loop depending upon its orientation, except at two particular orientations. For these particular orientations, the loop is in a metastable equilibrium with respect to the free surface. It is noted that dislocations may experience zero image force in a bi-material system only when they approach a surface film with elastic properties different from the matrix phase. However, this behavior is now possible near a free surface for disclinations merely due to the orientation effect.
o5
Z
240
- 90 °
-60"
-30 °
0° a
30 e
60*
90 °
REFERENCES
minimum if the loop plane is parallel to the free surface. However, in the case of twist loops, this lowest energy orientation is affected by the Poisson's ratio of the elastic medium.
1 T.W. Chou, Proc. Intern. Syrup. on Defect Interaction in Solids, Indian Institute of Science, Bangalore, India, 1972. 2 T. W. Chou, J. Appl. Phys., 42 (1972) 4092. 3 T. W. Chou, J. Appl. Phys., 42 (1971) 4931. 4 T. W. Chou and T. L. Lu, J. Appl. Phys., 43 (1972) 2562. 5 J. C. M. Li and J, J. Gilman, J. Appl. Phys., 41 (1970) 4248. 6 R. D. Mindlin, Physics, 7 (1936) 195. 7 W. Huang and T. Mura, J. Appl. Phys., 41 (1970) 5175.
Comportements ~lastiques de boucles-coins de disclinations au voisinaoe d'une surface libre
Das elastische Verhalten yon Stufen-Disklinationsringen in der Niihe e&er freien Oberfliiche
Le champ 61astique d'une boucle-coin de disclination situ6e au voisinage d'une surface libre est 6tudi6 ii l'aide de la m6thode de la fonction de Green. Les comportements de la boucle sont totalement diffrrents de ceux de disclinations de torsion ou de dislocations. Une boucle siture Aune distance finie de la surface libre peut ~tre attirre ou repoussre par la surface suivant son orientation par rapport ~l cette surface. Lorsque la boucle possrde l'une ou l'autre de deux orientations particuli~res, elle peut se trouver en 6quilibre m6tastable. Les dislocations n'ont un comportement de ce type que si elles se rapprochent d'un film superficiel drpos6 sur un demi-plan 61astique.
Das elastische Feld eines Stufen-Disklinationsringes in der N~ihe einer freien Oberfl/iche wird mit Hilfe der Methode der Greenschen Funktionen untersucht. Das Verhalten der Ringe unterscheidet sich betr/ichtlich vom Verhalten der SchraubenDisklinationen und Versetzungen. Ein Ring in endlichem Abstand vonder freien Oberfl/iche kann, je nach seiner Orientierung relativ zur Oberfl~iche, v o n d e r Oberfl~iche angezogen oder abgestol3en werden. Hat ein Ring eine von zwei speziellen Orientierungen, so kann er sich in einem metastabilen Gleichgewicht befinden. Dieses Verhalten konnte fiir Versetzungen nur fiJr den Fall realisiert werden, dass sie sich einer Oberfl~ichenschicht n/iherten, die auf eine elastische Halbebene aufgebracht war.
Fig. 2. Variation of interaction energy and image forces with disclination loop orientation.