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Interaction between non-parallel dislocations in piezoelectrics
International Journal of Engineering Science 47 (2009) 894–901
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Interaction between non-parallel dislocations in piezoelectrics Vladimir I. Alshits a,b,*, Jerzy P. Nowacki b, Andrzej Radowicz c a b c
Shubnikov Institute of Crystallography RAS, Leninskii pr. 59, 119333 Moscow, Russia Polish-Japanese Institute of Information Technology, ul. Koszykowa 86, 02-008 Warsaw, Poland ´ stwa Polskiego 7, 25-314 Kielce, Poland Kielce University of Technology, Al.1000-lecia Pan
a r t i c l e
i n f o
Article history: Received 7 April 2009 Received in revised form 29 April 2009 Accepted 5 May 2009 Available online 13 June 2009 Communicated by M. Kachanov Keywords: Piezoelectrics Dislocations Stress field Electric field 4D formalism
a b s t r a c t The total interaction force F12 between two crossing (non-intersecting) straight dislocations is found and analyzed for the three types of piezoelectric media of unrestricted anisotropy: an unbounded body, an infinite plate and a half-infinite body. In the latter two cases the dislocations are supposed to be parallel with the surfaces, which are in turn implied to be mechanically free of tractions and electrically closed (metalized). The found force F12 is orthogonal to the parallel planes, P and Q, containing the crossing dislocations. In an unbounded medium the value F12 proves to be independent of the distance between P and Q. On the other hand, it depends on directions of the dislocations and on their Burgers vectors: the force F12 may be either attractive or repulsive. In a plate the interaction becomes sensitive to dislocation positions yð1;2Þ with respect to the surfaces. Only in the situations, when dislocations are much closer to each other than to the both surfaces, their interaction may be approximately described by the solution for an unbounded medium. Otherwise, corrections arising from the image forces due to the plate surfaces become essential. The dislocation in the vicinity of a surface strongly acts on its counterpart only until the latter situates even closer to the same surface than the first one. When the second dislocation leaves this narrow zone, the interaction force on it abruptly decreases to a very small level. With an increase in the thickness of the plate, this behavior becomes more and more pronounced. In a half-infinite medium the interaction between the dislocations is exactly described by a Heaviside step-like dependence F 12 / Hðyð1Þ yð2Þ ) valid for any yð1;2Þ . It is shown that we deal here with an analog of the plane capacitor effect. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Dislocations in piezoelectrics produce in the vicinity of their cores not only high mechanical stresses but also singular electric fields, which might course inadmissible disturbances in functioning of modern supersensitive electronic devices. Interactions between dislocations in piezoelectric crystals and long-range electroelastic fields created by dislocations may be of crucial importance for solid elements working under conditions (e.g. at high temperatures), which might provoke a loss of stability of the real structure of a material due to a sudden relaxation of internal stresses. Such processes are determined by elementary displacements of individual dislocations through random distributions of other crystal defects. Under not very high temperatures, the main role in dislocation pinning is played by point defects (interstitial and impurity atoms, vacancies, etc.). Intersections of a moving dislocation with other dislocations are much more seldom events (a typical distance between point defects in a slip plane is 0.1 lm, whereas an average distance between dislocations is normally 10–100 lm. And the both types of defects create energy barriers for a dislocation motion of comparable height. * Corresponding author. Address: Shubnikov Institute of Crystallography RAS, Leninskii pr. 59, 119333 Moscow, Russia. Tel.: +7 495 3308274; fax: +7 499 1351011. E-mail addresses: [email protected], [email protected] (V.I. Alshits). 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.05.011
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On the other hand, as was proved by Kroupa [1] (for an infinite isotropic medium) and by Orlov and Indenbom [2,3] (for an arbitrary anisotropic medium), the total interaction force between non-parallel dislocations is independent of the distance between them. Thus, in contrast to point defects, which interact with a dislocation in a contact manner, most of dislocations in a crystal simultaneously contribute to the combined force on anyone of them. In the end of this paper we shall demonstrate that this force may be very substantial and at high enough temperatures might cause a motion of even aged dislocations. In strong piezoelectrics an electric contribution to elastic stresses may be comparable with a customary Hooke’s strain influence. Thus, the force of interaction between dislocations found in [1–3] should be further extended for a piezoelectric medium. The first studies of dislocation fields in piezoelectrics were accomplished for straight dislocations in unbounded media [4–6]. Then the results of these works were generalized to piezoelectric plates [7,8] and layer-substrate structures [9]. Considerations in [7–9] of 2D fields of straight dislocations were based on the 8D formalism [10] extending the sextic Stroh theory [11] to a description of media with piezoelectric coupling. In [12] a 4D variant of this formalism was used for a reformulation of a number of classical results of dislocation theory to a description of electroelastic fields of arbitrary curved dislocations in unbounded piezoelectric media of unrestricted anisotropy. In this paper we shall extend for piezoelectrics the other classical result of dislocation theory: the mentioned above theorem of Orlov and Indenbom [2,3] about the integral interaction force between two non-parallel straight dislocations in an infinite anisotropic elastic medium. We shall consider a series of piezoelectric structures: infinite and semi-infinite media and an infinite plate. In all cases the crossing (non-intersecting) dislocations will be supposed to be parallel to surfaces. Though the fields of both straight dislocations are two-dimensional, for non-parallel dislocations the description of their interaction generally requires solving a 3D problem. However, as was shown by Orlov and Indenbom [2], the finding of the integral interaction of crossing dislocations may be reduced to a simple 1D problem saving one from cumbersome direct calculations. Fortunately, the same arguments are equally applicable to all our piezoelectric structures, as well as to their purely elastic analogues not analyzed in [2,3]. 2. Statement of the problem 2.1. Generalized 4D dislocations in piezoelectrics and their modified description Consider a piezoelectric medium with the elastic moduli cijkl , the piezoelectric moduli eikl and the permittivity tensor eik . In such medium the stress tensor rij and the electric displacement Di are related to the elastic distortion ukl and the electric field Ek by the constitutive equations
rij ¼ cijkl ukl ekij Ek ; Di ¼ eikl ukl þ eik Ek ;
ð1Þ
and the corresponding equilibrium equations are given by
rij;i ¼ 0; Di;i ¼ 0:
ð2Þ
In Eqs. (1) and (2) and in the forthcoming, as usual, repeated indices imply summation and the notation . . . ;l @=@xl is accepted. In the presence of a dislocation one should distinguish between elastic (ukl ), plastic (intrinsic) (u0kl ) and total (uTkl ¼ ul;k ) distortions which are related to each other through the Kroener equation [13]
uTkl ¼ ukl þ u0kl :
ð3Þ
By definition,
u0kl ðrÞ ¼ nSk bl d r rS nS ;
ð4Þ
where bl are components of the Burgers vector b of the dislocation, the vector nS is the unit normal at the point rS to the arbitrarily chosen surface S bounded by the dislocation line and dðyÞ is the Dirac delta function. piezoelectrics one can generalize a traditional dislocation defined by a jump of the displacement vector u(r) In þ u S u S ¼ b on the arbitrary surface S bounded by the dislocation line. According to [12], the jump of potential Du at the same cut surface S leads to the source electric field quite similar to (4),
E0 ðrÞ ¼ DunS d r rS nS ;
ð5Þ
which in analogy with (3) determines the total electric field
E ¼ E0 þ E0 ¼ E0 ru:
ð6Þ
Thus, one can introduce the electro-elastic line defect – a 4D dislocation – characterized by a double jump at the surface S of both the displacement, Du ¼ b, and the electric potential, Du, i.e. by the ‘‘Burgers” 4-vector
B¼
b
Du
:
ð7Þ
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In addition to it one can define the corresponding four-dimensional ‘‘displacements” [10,12]
UK ¼
uk ; K ¼ k ¼ 1; 2; 3;
ð8Þ
u; K ¼ 4
and extended ‘‘elastic” and intrinsic distortions, U lK and U 0lK , and ‘‘stresses” RiJ represented as non-square 3 4 matrices (l; i ¼ 1; 2; 3; K; J ¼ 1; . . . ; 4)
U lK ¼
K ¼ k ¼ 1; 2; 3;
ulk ;
El ; K ¼ 4;
( U 0lK
¼
u0lk ; K ¼ k ¼ 1; 2; 3; E0l ;
RiJ ¼
K ¼ 4;
rij ; J ¼ j ¼ 1; 2; 3; Di ;
ð9Þ
J ¼ 4:
In these terms with the extended moduli tensor [10]
C iJKl
8 cijkl ; > > > eikl ; > > : eil ;
J; K ¼ j; k ¼ 1; 2; 3; J ¼ j ¼ 1; 2; 3; K ¼ 4;
ð10Þ
J ¼ 4; K ¼ k ¼ 1; 2; 3; J ¼ 4; K ¼ 4:
constitutive equations (1) take the appearance of the modified Hooke’s law,
RiJ ¼ C iJKl U lK ;
ð11Þ
and equilibrium equations (2) are combined to
RiJ;i ¼ 0:
ð12Þ
Manipulating with Eqs. (3), (6), (8) and (9) we get also the extended Kroener’s relation
U TlK U K;l ¼ U lK þ U 0lK ;
ð13Þ
which, together with (11), allows us to transform Eq. (12) to an inhomogeneous form
C iJKl U K;li ¼ F 0J
ð14Þ
with the effective source force exciting internal stresses due to a dislocation:
F 0J ¼ C iJKl U 0lK;i :
ð15Þ
We should stress that the same 4D formalism remains valid also for a usual (non-generalized) dislocation (i.e. with Du ¼ 0) after a trivial replacement in (7): B ! B0 ¼ ðb; 0Þt , where the superscript t denotes a transposition. 2.2. Reducing the problem to a 1D formulation Let us consider the two straight dislocations with the Burgers vectors Bð1Þ ; Bð2Þ and the directions along the unit vectors s , sð2Þ , respectively (Fig. 1). Choose the Cartesian coordinate system with the xz plane coinciding with the P plane containing dislocation 1 and parallel with dislocation 2, z axis along dislocation 1 (the direction sð1Þ ) and the origin at the intersection of dislocation 1 with the projection of 2 on P (Fig. 1). The directions of the x and y axes are by the unit vectors mð1Þ determined ð1Þ ð1Þ ð1Þ ð1Þ and n, respectively, which are chosen so that the mixed product m ns m n s ¼ 1. In these coordinates the introduced above extended intrinsic distortion (9)2 with (4) and (5) for dislocation 1 is equal ð1Þ
ð1Þ
ð1Þ
U 0lK ðrÞ ¼ nl BK HðxÞdðyÞ nl BK Hðr mð1Þ Þdðr nÞ;
ð16Þ
where the Heaviside step function H(x) is introduced.
Q
m(2)
τ (2)
y
2
n n m(1) P
τ
(1)
α
2′ 1
τ(1)
m(1)
x
z Fig. 1. Two crossing dislocations 1 and 2 in parallel planes P and Q, their geometrical characteristics and the Cartesian coordinate system for an unbounded medium.
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As was shown in [12], the force dF on the element ds of dislocation 2 at the point rð2Þ from the stress field Rð1Þ ðrð2Þ Þ (in our case, created by dislocation 1) is determined by the extended formula of Peach–Koehler 12
ð1Þ
ð2Þ ð2Þ q ds;
dF p ¼ eipq RiJ ðrð2Þ ÞBJ
s
ð17Þ
where eipq is the antisymmetric unit permutation tensor. Accordingly, the total interaction force is given by the integral over the length of dislocation 2:
F 12 p ¼ eipq
Z
1
1
ð2Þ ð2Þ ð2Þ Rð1Þ iJ ðr Þds BJ sq :
ð18Þ
Taking into account that even in an unbounded medium of unrestricted anisotropy the stress field Rð1Þ ðrÞ is expressed in the form of some integrals or implicit sums [12], one cannot expect easy calculations from Eq. (18). Meanwhile, the resulting formula (at least, for an unbounded media) is very simple and can be obtained without much calculations, as it has been done by Orlov and Indenbom [2,3] for a purely elastic problem. The first important observation in a considered geometry is its translational invariance: clearly, one can shift any of dislocations in the plane (P or Q) parallel to the other dislocation (and hence to the surfaces and interfaces when they occur) with no influence on the integral force or the energy of interaction. This means that the force F12 must be parallel to the normal n to the mentioned planes (Fig. 1):
F12 kn:
ð19Þ
The next observation is less trivial but also very understandable. The addition to dislocation 1 in the same plane P of N – 1 dislocations of the same (Bð1Þ , sð1Þ ) type will evidently enhance the force on dislocation 2 by a factor N : F12 ! NF12 . If per each unit length of dislocation 2 we introduce in the plane P one dislocation of type 1, the integral force on 2 will be certainly infinite, however the force per its unit length will be exactly equal to the value F12 , which we are seeking for. And the result will not change if we redistribute the considered discrete set of dislocations by their uniform continuous distribution characterized by the constant density of dislocation in the plane P : alK / dðr nÞ. The discrete dislocation density a0lK ðx; yÞ related to intrinsic distortion (16) is determined by the known differential relation [13] ð1Þ a0lK ðx; yÞ ¼ elpq rp U 0qK ðx; yÞ ¼ sð1Þ l BK dðxÞdðyÞ:
ð20Þ
Now we should find the continuous uniform (along x) density alK which is equivalent in an integral sense to the infinite set of dislocations of type 1 distributed in plane P so that one dislocation falls on a unit length of dislocation 2. This means that the initial dislocation 1 (20) should be spread, say, within the range 12 < s < 12. If dislocation 1 and projection 20 of dislocation 2 on plane P make the angle a, then, as is seen from Fig. 2, the corresponding strip of a decomposed dislocation (grey domain) should have the width
1 1 sin a < x < sin a: 2 2
ð21Þ
Accordingly, the needed density alK is equal
alK ¼
1 sin a
Z
1 sin a 2
12 sin a
a0lK ðx; yÞdx ¼
ð1Þ sð1Þ l BK dðr nÞ: sin a
ð22Þ
It can be checked with (20)1 that this dislocation density corresponds to the intrinsic distortion ð1Þ ð1Þ
U 0lK ¼
ml BK sgnðr nÞ: 2 sin a
ð23Þ
Thus, we obtained the universal 1D source of a one-dimensional stress field Rð1Þ ðr nÞ. However the latter should be different for the mentioned above three types of piezoelectric structures, which are going to be considered. Accordingly, the further analysis should be carried out separately for these boundary problems.
x 1 2
P
α
1
s
τ (2)
sin α 1
2
τ(1)
2′
− 12
z
− 12 sin α
Fig. 2. Continuous decomposition of dislocation 1 into a strip in plane xz (P). The width of the strip 12 sin a < x < 12 sin a relates the unit length of 1 0 1 dislocation 2 2 < s < 2 . Dashed line 2 is a projection of dislocation 2 on plane P (see Fig. 1).
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3. Interaction between dislocations in an unbounded piezoelectric medium As was proved in [12], in this case
RiJ ðr nÞ ¼ C iJKl ðnÞU 0lK ðr nÞ;
ð24Þ
where the planar tensor of our extended ‘‘elasticity” is introduced,
h i C iJKl ðnÞ ¼ C iJNp np ðnnÞ1
NM
nq C qMKl C iJKl :
ð25Þ
Here ðnnÞ1 is the 44 tensor inverse to the tensor with the components ðnnÞJK ¼ ni C iJKl nl . Thus, after substituting (23) into (24) one obtains at r ¼ rð2Þ :
1 2
ð1Þ ð1Þ ð2Þ Rð1Þ nÞ ¼ C iJKl ðnÞml BK iJ ðr
sgnðrð2Þ nÞ : sin a
ð26Þ
In accordance with the above consideration the integral force F12 is determined by a substitution of (26) into (17) instead of 12 12 Rð1Þ iJ and a further replacement of @F p =@s by F p : ð1Þ
ð2Þ ð2Þ q :
ð2Þ F 12 p ¼ eipq RiJ ðr ÞBJ
s
ð27Þ
Bearing in mind (19), it is natural to write F lowing transformation:
12
¼
ðF 12 p np Þn.
Then it is convenient to make in the scalar product
ð2Þ ð2Þ eipq np sð2Þ ¼ n mð2Þ n i ¼ mi ni ðn mð2Þ Þ: q ¼ ns i
In view that identically
ni C iJKl ðnÞ
F12 ¼ Fsgnðrð2Þ nÞ;
¼ 0 and
F ¼ n
C iJKl ðnÞ
¼
C lKJi ðnÞ,
ð1Þ ð1Þ ð2Þ ð2Þ mi BJ C iJKl ðnÞBK ml
2 sin a
:
F 12 p np
the fol-
ð28Þ
the final result takes the form
ð29Þ
Thus, depending on mutual orientations of the involved vector characteristics, the interaction force F12 may be either repulsive ðF n > 0) or attractive ðF n < 0), however its magnitude proves to be independent of the distance between dislocations. The presence in (29)1 of sgn(rð2Þ n) just regulates particular directions of the force F12 for positions of dislocation 2 up or down from dislocation 1 (e.g., for yð2Þ > 0 the repulsive force is directed along n, and for yð2Þ < 0 its direction is opposite). On the other hand, the proportionality F12 / sgn(rð2Þ n) provides in the given case the validity of the 3rd Newton’s law:
F12 ¼ F21 :
ð30Þ
It will be shown below that in media with surfaces not only spatial dispersion returns to the interaction, but also classical property (30) loses its applicability. In order to obtain the interaction force between dislocations in purely elastic medium one should just replace in (29) ð1Þ
ð2Þ
ð1Þ
ð2Þ
BJ C iJKl ðnÞBK ! bj cijkl ðnÞbk ;
ð31Þ
cijkl ðnÞ ¼ cijrp np ½ðnnÞ1 rt nq cqtkl cijkl :
ð32Þ
where
This immediately returns us to the result of Orlov and Indenbom [2,3]. Here we must note that the interaction force in piezoð1;2Þ ; 0Þt is also described by the transformation electrics between ordinary dislocations with the Burgers vectors (7) Bð1;2Þ ¼ ðb ð1Þ ð2Þ ð1Þ ð2Þ in (29) BJ C iJKl ðnÞBK ! bj C ijkl ðnÞbk which looks similar to (31) but is not identical with it because the tensor C ijkl ðnÞ by its definition (25), in contrast to (32), still contains piezoelectric moduli eikl and permittivity eik . The further transition from a purely elastic crystal to an isotropic medium has been done in [2,3]. Naturally, it coincided with the Kroupa result [1]:
F12 ¼
h i n 2m ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ðmð1Þ b Þðmð2Þ b Þ þ ðmð1Þ b Þðmð2Þ b Þ þ ðmð1Þ mð2Þ Þ ðb b Þ ðb nÞðb nÞ ; 2 sin a 1 2m ð33Þ
where
m is the Poisson ratio.
4. Crossing dislocations in a piezoelectric plate Let us now consider the case of an infinite piezoelectric plate with two crossing dislocations, both parallel to the faces of the plate. This time it is convenient to choose the Cartesian coordinate system with the plane xz being a middle plane for the plate, the axis z not coinciding, but parallel with dislocation 1, and the axis y orthogonal to the faces, which thus have
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symmetric coordinates y ¼ h and y ¼ h (Fig. 3a). In contrast, dislocation planes P and Q parallel to the surfaces are not supposed to be symmetric with respect to the plate faces being characterized by the coordinates yð1Þ and yð2Þ . For simplicity, we shall consider only one type of boundary conditions at the plate surfaces: they will be supposed to be free of tractions and electrically closed, i.e. metalized. One can show [14] that for such particular boundary conditions the known solution [3] developed for purely elastic anisotropic plate may be extended to the case of a piezoelectric plate excited by a transverse 1D distribution of ‘‘plastic” distortions U 0lK ðr nÞ. This time the formula (24) must be modified to the form
0 yU ; lK 2
3y
RiJ ðyÞ ¼ C iJKl ðnÞ U 0lK ðyÞ U 0lK
h
ð34Þ
where y ¼ r n and
U 0lK ¼
1 2h
Z
h
h
U 0lK ðyÞdy;
yU 0lK ¼
1 2h
Z
h h
yU 0lK ðyÞdy:
ð35Þ
Bearing in mind that now dislocation 1 does not coincide with axis z of the coordinate system, the function U 0lK ðyÞ in (34) and (35) should differ from (23) by the shift y ! y yð1Þ , i.e. ð1Þ ð1Þ
U 0lK ðyÞ ¼
ml BK sgnðy yð1Þ Þ: 2 sin a
ð36Þ
Substituting (36) into (35) one obtains ð1Þ ð1Þ
U 0lK ¼
ml BK yð1Þ ; 2 sin a h
yU 0lK ¼
ð1Þ 2 ! ð1Þ ð1Þ ml BK h y : 1 2 sin a 2 h
ð37Þ
With (36), (37), equation (34) taken at y ¼ yð2Þ determines the needed ‘‘stress” field of the set of decomposed dislocations of type 1 at the dislocation line 2: ð1Þ ð1Þ
R
ð1Þ ð2Þ iJ ðy Þ
¼
C iJNp ðnÞ
mi BJ
2 sin a
ð2Þ
sgnðy
" ð1Þ 2 #! yð1Þ 3yð2Þ y : y Þþ 1 h 2h h ð1Þ
ð38Þ
The further manipulations are the same as in the previous section after obtaining Eq. (26). However their result is quite different from (29): 12
F
¼ F sgnðy
ð2Þ
" ð1Þ 2 #! yð1Þ 3yð2Þ y ; y Þþ 1 h 2h h ð1Þ
ð39Þ
where the vector F is given by Eq. (29)2. The force (39) can be conveniently decomposed into a sum of two terms:
F12 ¼ F121 þ F 12im :
ð40Þ
The first term with a superscript 1 is identical with the found above force on dislocation 2 in an infinite medium, Eq. (29)1 with rð2Þ n replaced by yð2Þ yð1Þ . The second term in (40)
F12im ¼ F
" ð1Þ 2 #! yð1Þ 3yð2Þ y 1 h 2h h
ð41Þ
is determined by the image fields of dislocation 1 due to the plate surfaces. It is natural that this term is continuous at any variations of yð1;2Þ including the point yð1Þ ¼ yð2Þ , in contrast to the first term in (39) and (40) discontinuous at this point. Of course, the introduced planar sources related to dislocations 1 and 2 being on different distances from the surfaces produce different image components of electroelastic ‘‘stresses” Rð1;2Þ . That is why, in this case F12 – F21 , i.e. we lose the
(a)
(b)
y
y
Q
h y(2)
Q
y(2)
P
y(1)
P
y(1)
-h
0
Fig. 3. Positions of the surfaces and dislocation planes P and Q in the shifted along y coordinate systems for a plate (a) and a semi-infinite medium (b).
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3rd Newton law (30) in the plate. For the same reason, the combined force F12 (39), in contrast to the magnitude of F121 independent of any distances, acquires a dependence on positions yð1;2Þ of dislocations 1 and 2. Certainly, in the limiting case j yð1;2Þ j<< h, when the both dislocations are much closer to each other than to the surfaces, image force term (41) becomes much smaller than F and the interaction is closing to that for an unbounded medium. The other limiting situation, when dislocation 1 falls in the vicinity of one of the surfaces (yð1Þ h or yð1Þ h), proves to be much less trivial and more interesting. Indeed, in such situations this dislocation acts on the other one with a force F12 2F121 only when the latter situates even closer to the same surface. Otherwise, the force F12 practically disappears. For instance, at yð1Þ h to the main order Eq. (39) approximately reduces to
( 12
F
ð1Þ
2FHðy
ð2Þ
y Þ¼
2F; h < yð2Þ < yð1Þ ; h < yð1Þ < yð2Þ :
0;
ð42Þ
As we shall see in the next section, for a semi-infinite medium this relation becomes exact. We stress that Eq. (39) contains piezoelectric characteristics only through the factor F. If to replace the latter by the expression for a purely elastic medium [see transformation (31)], one obtains the result, which appears to be also original (at least, we never met it in the literature).
5. The effect of a plane capacitor in a semi-infinite piezoelectric medium Let us now consider a semi-infinite piezoelectric body with the surface being again free of tractions and metalized (electrically closed). This case may be obtained by a limiting transition from the above plate. However before the transition it is desirable to shift the coordinate system so that the origin would fall at one of the surfaces. Say, let one face of the plate situate at y ¼ 0 and the other one at y ¼ 2h. In this system one should replace y ! y h in (34) and appropriately shift limits of integration in (35). At the following limiting procedure h ! 1 we shall make use of the finiteness of the function U0 ðyÞ at infinity, notating U0 ð1Þ U01 . Then
lim
h!1
3ðy hÞ 2h
Z
3
!
2h
0
ðy hÞU ðyÞdy
¼
0
Z 2 Z 2 3 y 3 ðt 1ÞU0 ðhtÞdt ¼ U01 ðt 1Þdt ¼ 0 lim 1 2 h!1 h 2 0 0
and Eq. (34) is transformed to
RiJ ðyÞ ¼ C iJNp ðnÞ U 0pN ðyÞ U 0pN
ð43Þ
with
U 0pN ¼ lim
h!1
1 2h
Z
2h 0
! U 0pN ðyÞdy :
ð44Þ
Substituting Eq. (36) into (43) and (44) one obtains the ‘‘stress” field ð1Þ ð1Þ
Rð1Þ iJ ðyÞ ¼ 2C iJNp ðnÞ
mi BJ Hðyð1Þ yÞ; 2 sin a
ð45Þ
specifying (38) to the case of the considered semi-infinite medium (Fig. 3b). Accordingly, instead of (39), the interaction force from dislocation 1 to dislocation 2 indeed takes the form similar to (42):
F12 ¼ 2FHðyð1Þ yð2Þ Þ:
ð46Þ
Certainly, Eq. (46) is also valid for a purely elastic medium after transformation (31) in (29)2. The obtained effect, when dislocation 2 suffers the action from dislocation 1 only until the object of the force is closer to the surface than its source, has a clear physical origin analogous to the known effects in electrostatics. The decomposed set of dislocations of type 1 produces in the semi-infinite plate the same field, Eq. (45), as a couple of planar sources in the infinite space. This pair is formed by the identical planar ‘‘dislocation” and its image counterpart situated at the surface and characterized by the opposite topological ‘‘charge” (say, Bð1Þim ¼ Bð1Þ Þ: ð1Þ ð1Þ
ð1Þim Rð1Þ1 ðyÞ þ RiJ ðyÞ C iJKl ðnÞ iJ
ml BK ð1Þ ½ sgnðy yð1Þ Þ sgnðyÞ ¼ RiJ ðyÞ; 2 sin a
ð47Þ
where in the right-hand side one automatically obtains (45). Thus, we have got sort of a plane capacitor where the planar sources produce fields which are doubled between the source planes and are eliminated beyond a ”capacitor”. We also note in passing, that for a half-infinite medium the following relation
F12 þ F21 ¼ 2F replaces the 3rd Newton’s law, Eq. (30).
ð48Þ
V.I. Alshits et al. / International Journal of Engineering Science 47 (2009) 894–901
901
6. Concluding remarks The found expressions for interaction forces between crossing dislocations in all three types of considered piezoelectric media contain in their denominators the factor sin a which provides a singularity at a ! 0. Of course, this singularity is quite expectable, because at a ¼ 0 we obtain two parallel dislocations, which clearly must have an infinite total interaction force between dislocations of infinite length, L ! 1. On the other hand, for a finite size L of a crystal along the dislocations, the above considerations also retain an approximate meaning if L sin a >> d, where d is the distance between the parallel planes, P and Q, containing the dislocations. This provides a physical exclusion of the above singularity from our results, because they remain valid only until
d sin a >> : L
ð49Þ
Let us now evaluate a magnitude of the studied interaction. As we have seen, for reasonable values of the above sina, the ð1Þ ð2Þ scale of interaction between two dislocations is given by the force F 12 BJ C iJKl ðnÞBK . Even for strong piezoelectrics where 2 the electric contribution to this force is substantial, the latter may be roughly estimated as F 12 lb , where l is the shear modulus of the crystal. Now one should take into account that in real crystals there are many dislocations: the typical dislocation density in dielectrics is 104 —106 cm2 . So, in a cube with the line size 1 cm, apart from dislocation 1, a lot of other dislocations (say, the number N 105 ) will act on dislocation 2 with comparable forces. As we know, they can either attract, or repulse dislocation 2. Denote N þ and N the numbers of dislocations acting on dislocation 2 with forces which have, respectively, the same or opposite signs of their projections on n, compared with the force produced by dislocation 1. Of course, the numbers N þ and N should be comparable, but not exactly equal to each other. We expect that the difference DN ¼j N þ N j, being much smaller than N, remains rather large, say DN 101 N 104 . Then the combined magnitude 2 of the interaction force component along n would be F DN lb , which is equivalent to the application to dislocation 2 the stress
r
F DNb l: bL L
ð50Þ
For DN 104 , b 108 cm, L 1 cm this gives a fairly high level of the stress r 104 l. Thus the considered interactions of dislocations may cause quite serious forces at least on some of them. However, one should take into account that normally only freshly introduced dislocations can move in a crystal under the action of external forces or internal stresses. Aged dislocations are surrounded by ‘‘clouds” of point defects (the so-called Cottrell atmospheres) which practically prevent their motions. Still, at the annealing regime under high temperatures even aged dislocations may move together with their atmospheres due to much more intensive processes of diffusion of lattice defects. In this particular regime the studied above interaction between dislocations can lead to a relaxation of the internal stresses and even to a decrease of the dislocation density. Acknowledgements The authors are grateful to Prof. M. Kachanov for useful comments and to Prof. V.N. Lyubimov for helpful discussions and kind assistance with references. The paper was supported by the Polish Foundation MNiSW (Grant # N N501 252334). V.I.A. also acknowledges support from the Kielce University of Technology (Poland). References [1] F. Kroupa, The force between non-parallel dislocations, Czech. J. Phys. B 11 (1961) 847–848. [2] S.S. Orlov, V.L. Indenbom, Interaction of non-parallel dislocations in an anisotropic medium, Sov. Phys. Crystallogr. 14 (1970) 675–677. [3] V.L. Indenbom, Dislocations and internal stresses, in: V.L. Indenbom, J. Lothe (Eds.), Elastic Strain Fields and Dislocation Mobility, North-Holland, Amsterdam, 1992, pp. 1–174 (Chapter 1). [4] A.M. Kosevich, L.A. Pastur, E.P. Feldman, Dislocation and linear charge fields in piezoelectric crystals, Sov. Phys. Crystallogr. 12 (1968) 797–801. [5] G. Saada, Dislocations dans lex cristaux piezoelectriques, Phys. Status Solidi B 44 (1971) 717–731. [6] G. Faivre, G. Saada, Dislocations in piezoelectric semiconductors, Phys. Status Solidi B 52 (1972) 127–140. [7] J.P. Nowacki, V.I. Alshits, A. Radowicz, Green’s functions for an infinite piezoelectric strip with line sources at the surfaces, Int. J. Appl. Electromagn. Mech. 12 (3–4) (2000) 177–202. [8] J.P. Nowacki, V.I. Alshits, A. Radowicz, Green’s function for an infinite piezoelectric strip with a general line defect, J. Tech. Phys. 43 (2) (2002) 133–153. [9] J.P. Nowacki, V.I. Alshits, A. Radowicz, 2D electro-elastic fields in a piezoelectric layer-substrate structure, Int. J. Eng. Sci. 40 (2002) 2057–2076. [10] D.M. Barnett, J. Lothe, Dislocations and line charges in anisotropic piezoelectric insulators, Phys. Status Solidi B 67 (1975) 105–111. [11] A.N. Stroh, Steady-state problems in anisotropic elasticity, J. Math. Phys. 41 (1962) 77–103. [12] J.P. Nowacki, V.I. Alshits, Dislocation fields in piezoelectrics, in: F.R.N. Nabarro, J. Hirth (Eds.), Dislocations in Solids, vol. 13, North Holland, Amsterdam, 2007, pp. 47–79 (Chapter 72). [13] E. Kroener, Physics of defects, in: R. Balian et al. (Eds.), Les Houches Session XXXV, Course 3, North-Holland, Amsterdam, 1980. [14] V.I. Alshits, J.P. Nowacki, A. Radowicz, 1D electroelastic fields in piezoelectrics excited by intrinsic strains, Crystallogr. Rep. 53 (6) (2009), in press.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Interaction between non-parallel dislocations in piezoelectrics Vladimir I. Alshits a,b,*, Jerzy P. Nowacki b, Andrzej Radowicz c a b c
Shubnikov Institute of Crystallography RAS, Leninskii pr. 59, 119333 Moscow, Russia Polish-Japanese Institute of Information Technology, ul. Koszykowa 86, 02-008 Warsaw, Poland ´ stwa Polskiego 7, 25-314 Kielce, Poland Kielce University of Technology, Al.1000-lecia Pan
a r t i c l e
i n f o
Article history: Received 7 April 2009 Received in revised form 29 April 2009 Accepted 5 May 2009 Available online 13 June 2009 Communicated by M. Kachanov Keywords: Piezoelectrics Dislocations Stress field Electric field 4D formalism
a b s t r a c t The total interaction force F12 between two crossing (non-intersecting) straight dislocations is found and analyzed for the three types of piezoelectric media of unrestricted anisotropy: an unbounded body, an infinite plate and a half-infinite body. In the latter two cases the dislocations are supposed to be parallel with the surfaces, which are in turn implied to be mechanically free of tractions and electrically closed (metalized). The found force F12 is orthogonal to the parallel planes, P and Q, containing the crossing dislocations. In an unbounded medium the value F12 proves to be independent of the distance between P and Q. On the other hand, it depends on directions of the dislocations and on their Burgers vectors: the force F12 may be either attractive or repulsive. In a plate the interaction becomes sensitive to dislocation positions yð1;2Þ with respect to the surfaces. Only in the situations, when dislocations are much closer to each other than to the both surfaces, their interaction may be approximately described by the solution for an unbounded medium. Otherwise, corrections arising from the image forces due to the plate surfaces become essential. The dislocation in the vicinity of a surface strongly acts on its counterpart only until the latter situates even closer to the same surface than the first one. When the second dislocation leaves this narrow zone, the interaction force on it abruptly decreases to a very small level. With an increase in the thickness of the plate, this behavior becomes more and more pronounced. In a half-infinite medium the interaction between the dislocations is exactly described by a Heaviside step-like dependence F 12 / Hðyð1Þ yð2Þ ) valid for any yð1;2Þ . It is shown that we deal here with an analog of the plane capacitor effect. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Dislocations in piezoelectrics produce in the vicinity of their cores not only high mechanical stresses but also singular electric fields, which might course inadmissible disturbances in functioning of modern supersensitive electronic devices. Interactions between dislocations in piezoelectric crystals and long-range electroelastic fields created by dislocations may be of crucial importance for solid elements working under conditions (e.g. at high temperatures), which might provoke a loss of stability of the real structure of a material due to a sudden relaxation of internal stresses. Such processes are determined by elementary displacements of individual dislocations through random distributions of other crystal defects. Under not very high temperatures, the main role in dislocation pinning is played by point defects (interstitial and impurity atoms, vacancies, etc.). Intersections of a moving dislocation with other dislocations are much more seldom events (a typical distance between point defects in a slip plane is 0.1 lm, whereas an average distance between dislocations is normally 10–100 lm. And the both types of defects create energy barriers for a dislocation motion of comparable height. * Corresponding author. Address: Shubnikov Institute of Crystallography RAS, Leninskii pr. 59, 119333 Moscow, Russia. Tel.: +7 495 3308274; fax: +7 499 1351011. E-mail addresses: [email protected], [email protected] (V.I. Alshits). 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.05.011
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895
On the other hand, as was proved by Kroupa [1] (for an infinite isotropic medium) and by Orlov and Indenbom [2,3] (for an arbitrary anisotropic medium), the total interaction force between non-parallel dislocations is independent of the distance between them. Thus, in contrast to point defects, which interact with a dislocation in a contact manner, most of dislocations in a crystal simultaneously contribute to the combined force on anyone of them. In the end of this paper we shall demonstrate that this force may be very substantial and at high enough temperatures might cause a motion of even aged dislocations. In strong piezoelectrics an electric contribution to elastic stresses may be comparable with a customary Hooke’s strain influence. Thus, the force of interaction between dislocations found in [1–3] should be further extended for a piezoelectric medium. The first studies of dislocation fields in piezoelectrics were accomplished for straight dislocations in unbounded media [4–6]. Then the results of these works were generalized to piezoelectric plates [7,8] and layer-substrate structures [9]. Considerations in [7–9] of 2D fields of straight dislocations were based on the 8D formalism [10] extending the sextic Stroh theory [11] to a description of media with piezoelectric coupling. In [12] a 4D variant of this formalism was used for a reformulation of a number of classical results of dislocation theory to a description of electroelastic fields of arbitrary curved dislocations in unbounded piezoelectric media of unrestricted anisotropy. In this paper we shall extend for piezoelectrics the other classical result of dislocation theory: the mentioned above theorem of Orlov and Indenbom [2,3] about the integral interaction force between two non-parallel straight dislocations in an infinite anisotropic elastic medium. We shall consider a series of piezoelectric structures: infinite and semi-infinite media and an infinite plate. In all cases the crossing (non-intersecting) dislocations will be supposed to be parallel to surfaces. Though the fields of both straight dislocations are two-dimensional, for non-parallel dislocations the description of their interaction generally requires solving a 3D problem. However, as was shown by Orlov and Indenbom [2], the finding of the integral interaction of crossing dislocations may be reduced to a simple 1D problem saving one from cumbersome direct calculations. Fortunately, the same arguments are equally applicable to all our piezoelectric structures, as well as to their purely elastic analogues not analyzed in [2,3]. 2. Statement of the problem 2.1. Generalized 4D dislocations in piezoelectrics and their modified description Consider a piezoelectric medium with the elastic moduli cijkl , the piezoelectric moduli eikl and the permittivity tensor eik . In such medium the stress tensor rij and the electric displacement Di are related to the elastic distortion ukl and the electric field Ek by the constitutive equations
rij ¼ cijkl ukl ekij Ek ; Di ¼ eikl ukl þ eik Ek ;
ð1Þ
and the corresponding equilibrium equations are given by
rij;i ¼ 0; Di;i ¼ 0:
ð2Þ
In Eqs. (1) and (2) and in the forthcoming, as usual, repeated indices imply summation and the notation . . . ;l @=@xl is accepted. In the presence of a dislocation one should distinguish between elastic (ukl ), plastic (intrinsic) (u0kl ) and total (uTkl ¼ ul;k ) distortions which are related to each other through the Kroener equation [13]
uTkl ¼ ukl þ u0kl :
ð3Þ
By definition,
u0kl ðrÞ ¼ nSk bl d r rS nS ;
ð4Þ
where bl are components of the Burgers vector b of the dislocation, the vector nS is the unit normal at the point rS to the arbitrarily chosen surface S bounded by the dislocation line and dðyÞ is the Dirac delta function. piezoelectrics one can generalize a traditional dislocation defined by a jump of the displacement vector u(r) In þ u S u S ¼ b on the arbitrary surface S bounded by the dislocation line. According to [12], the jump of potential Du at the same cut surface S leads to the source electric field quite similar to (4),
E0 ðrÞ ¼ DunS d r rS nS ;
ð5Þ
which in analogy with (3) determines the total electric field
E ¼ E0 þ E0 ¼ E0 ru:
ð6Þ
Thus, one can introduce the electro-elastic line defect – a 4D dislocation – characterized by a double jump at the surface S of both the displacement, Du ¼ b, and the electric potential, Du, i.e. by the ‘‘Burgers” 4-vector
B¼
b
Du
:
ð7Þ
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In addition to it one can define the corresponding four-dimensional ‘‘displacements” [10,12]
UK ¼
uk ; K ¼ k ¼ 1; 2; 3;
ð8Þ
u; K ¼ 4
and extended ‘‘elastic” and intrinsic distortions, U lK and U 0lK , and ‘‘stresses” RiJ represented as non-square 3 4 matrices (l; i ¼ 1; 2; 3; K; J ¼ 1; . . . ; 4)
U lK ¼
K ¼ k ¼ 1; 2; 3;
ulk ;
El ; K ¼ 4;
( U 0lK
¼
u0lk ; K ¼ k ¼ 1; 2; 3; E0l ;
RiJ ¼
K ¼ 4;
rij ; J ¼ j ¼ 1; 2; 3; Di ;
ð9Þ
J ¼ 4:
In these terms with the extended moduli tensor [10]
C iJKl
8 cijkl ; > > >
J; K ¼ j; k ¼ 1; 2; 3; J ¼ j ¼ 1; 2; 3; K ¼ 4;
ð10Þ
J ¼ 4; K ¼ k ¼ 1; 2; 3; J ¼ 4; K ¼ 4:
constitutive equations (1) take the appearance of the modified Hooke’s law,
RiJ ¼ C iJKl U lK ;
ð11Þ
and equilibrium equations (2) are combined to
RiJ;i ¼ 0:
ð12Þ
Manipulating with Eqs. (3), (6), (8) and (9) we get also the extended Kroener’s relation
U TlK U K;l ¼ U lK þ U 0lK ;
ð13Þ
which, together with (11), allows us to transform Eq. (12) to an inhomogeneous form
C iJKl U K;li ¼ F 0J
ð14Þ
with the effective source force exciting internal stresses due to a dislocation:
F 0J ¼ C iJKl U 0lK;i :
ð15Þ
We should stress that the same 4D formalism remains valid also for a usual (non-generalized) dislocation (i.e. with Du ¼ 0) after a trivial replacement in (7): B ! B0 ¼ ðb; 0Þt , where the superscript t denotes a transposition. 2.2. Reducing the problem to a 1D formulation Let us consider the two straight dislocations with the Burgers vectors Bð1Þ ; Bð2Þ and the directions along the unit vectors s , sð2Þ , respectively (Fig. 1). Choose the Cartesian coordinate system with the xz plane coinciding with the P plane containing dislocation 1 and parallel with dislocation 2, z axis along dislocation 1 (the direction sð1Þ ) and the origin at the intersection of dislocation 1 with the projection of 2 on P (Fig. 1). The directions of the x and y axes are by the unit vectors mð1Þ determined ð1Þ ð1Þ ð1Þ ð1Þ and n, respectively, which are chosen so that the mixed product m ns m n s ¼ 1. In these coordinates the introduced above extended intrinsic distortion (9)2 with (4) and (5) for dislocation 1 is equal ð1Þ
ð1Þ
ð1Þ
U 0lK ðrÞ ¼ nl BK HðxÞdðyÞ nl BK Hðr mð1Þ Þdðr nÞ;
ð16Þ
where the Heaviside step function H(x) is introduced.
Q
m(2)
τ (2)
y
2
n n m(1) P
τ
(1)
α
2′ 1
τ(1)
m(1)
x
z Fig. 1. Two crossing dislocations 1 and 2 in parallel planes P and Q, their geometrical characteristics and the Cartesian coordinate system for an unbounded medium.
V.I. Alshits et al. / International Journal of Engineering Science 47 (2009) 894–901
897
As was shown in [12], the force dF on the element ds of dislocation 2 at the point rð2Þ from the stress field Rð1Þ ðrð2Þ Þ (in our case, created by dislocation 1) is determined by the extended formula of Peach–Koehler 12
ð1Þ
ð2Þ ð2Þ q ds;
dF p ¼ eipq RiJ ðrð2Þ ÞBJ
s
ð17Þ
where eipq is the antisymmetric unit permutation tensor. Accordingly, the total interaction force is given by the integral over the length of dislocation 2:
F 12 p ¼ eipq
Z
1
1
ð2Þ ð2Þ ð2Þ Rð1Þ iJ ðr Þds BJ sq :
ð18Þ
Taking into account that even in an unbounded medium of unrestricted anisotropy the stress field Rð1Þ ðrÞ is expressed in the form of some integrals or implicit sums [12], one cannot expect easy calculations from Eq. (18). Meanwhile, the resulting formula (at least, for an unbounded media) is very simple and can be obtained without much calculations, as it has been done by Orlov and Indenbom [2,3] for a purely elastic problem. The first important observation in a considered geometry is its translational invariance: clearly, one can shift any of dislocations in the plane (P or Q) parallel to the other dislocation (and hence to the surfaces and interfaces when they occur) with no influence on the integral force or the energy of interaction. This means that the force F12 must be parallel to the normal n to the mentioned planes (Fig. 1):
F12 kn:
ð19Þ
The next observation is less trivial but also very understandable. The addition to dislocation 1 in the same plane P of N – 1 dislocations of the same (Bð1Þ , sð1Þ ) type will evidently enhance the force on dislocation 2 by a factor N : F12 ! NF12 . If per each unit length of dislocation 2 we introduce in the plane P one dislocation of type 1, the integral force on 2 will be certainly infinite, however the force per its unit length will be exactly equal to the value F12 , which we are seeking for. And the result will not change if we redistribute the considered discrete set of dislocations by their uniform continuous distribution characterized by the constant density of dislocation in the plane P : alK / dðr nÞ. The discrete dislocation density a0lK ðx; yÞ related to intrinsic distortion (16) is determined by the known differential relation [13] ð1Þ a0lK ðx; yÞ ¼ elpq rp U 0qK ðx; yÞ ¼ sð1Þ l BK dðxÞdðyÞ:
ð20Þ
Now we should find the continuous uniform (along x) density alK which is equivalent in an integral sense to the infinite set of dislocations of type 1 distributed in plane P so that one dislocation falls on a unit length of dislocation 2. This means that the initial dislocation 1 (20) should be spread, say, within the range 12 < s < 12. If dislocation 1 and projection 20 of dislocation 2 on plane P make the angle a, then, as is seen from Fig. 2, the corresponding strip of a decomposed dislocation (grey domain) should have the width
1 1 sin a < x < sin a: 2 2
ð21Þ
Accordingly, the needed density alK is equal
alK ¼
1 sin a
Z
1 sin a 2
12 sin a
a0lK ðx; yÞdx ¼
ð1Þ sð1Þ l BK dðr nÞ: sin a
ð22Þ
It can be checked with (20)1 that this dislocation density corresponds to the intrinsic distortion ð1Þ ð1Þ
U 0lK ¼
ml BK sgnðr nÞ: 2 sin a
ð23Þ
Thus, we obtained the universal 1D source of a one-dimensional stress field Rð1Þ ðr nÞ. However the latter should be different for the mentioned above three types of piezoelectric structures, which are going to be considered. Accordingly, the further analysis should be carried out separately for these boundary problems.
x 1 2
P
α
1
s
τ (2)
sin α 1
2
τ(1)
2′
− 12
z
− 12 sin α
Fig. 2. Continuous decomposition of dislocation 1 into a strip in plane xz (P). The width of the strip 12 sin a < x < 12 sin a relates the unit length of 1 0 1 dislocation 2 2 < s < 2 . Dashed line 2 is a projection of dislocation 2 on plane P (see Fig. 1).
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3. Interaction between dislocations in an unbounded piezoelectric medium As was proved in [12], in this case
RiJ ðr nÞ ¼ C iJKl ðnÞU 0lK ðr nÞ;
ð24Þ
where the planar tensor of our extended ‘‘elasticity” is introduced,
h i C iJKl ðnÞ ¼ C iJNp np ðnnÞ1
NM
nq C qMKl C iJKl :
ð25Þ
Here ðnnÞ1 is the 44 tensor inverse to the tensor with the components ðnnÞJK ¼ ni C iJKl nl . Thus, after substituting (23) into (24) one obtains at r ¼ rð2Þ :
1 2
ð1Þ ð1Þ ð2Þ Rð1Þ nÞ ¼ C iJKl ðnÞml BK iJ ðr
sgnðrð2Þ nÞ : sin a
ð26Þ
In accordance with the above consideration the integral force F12 is determined by a substitution of (26) into (17) instead of 12 12 Rð1Þ iJ and a further replacement of @F p =@s by F p : ð1Þ
ð2Þ ð2Þ q :
ð2Þ F 12 p ¼ eipq RiJ ðr ÞBJ
s
ð27Þ
Bearing in mind (19), it is natural to write F lowing transformation:
12
¼
ðF 12 p np Þn.
Then it is convenient to make in the scalar product
ð2Þ ð2Þ eipq np sð2Þ ¼ n mð2Þ n i ¼ mi ni ðn mð2Þ Þ: q ¼ ns i
In view that identically
ni C iJKl ðnÞ
F12 ¼ Fsgnðrð2Þ nÞ;
¼ 0 and
F ¼ n
C iJKl ðnÞ
¼
C lKJi ðnÞ,
ð1Þ ð1Þ ð2Þ ð2Þ mi BJ C iJKl ðnÞBK ml
2 sin a
:
F 12 p np
the fol-
ð28Þ
the final result takes the form
ð29Þ
Thus, depending on mutual orientations of the involved vector characteristics, the interaction force F12 may be either repulsive ðF n > 0) or attractive ðF n < 0), however its magnitude proves to be independent of the distance between dislocations. The presence in (29)1 of sgn(rð2Þ n) just regulates particular directions of the force F12 for positions of dislocation 2 up or down from dislocation 1 (e.g., for yð2Þ > 0 the repulsive force is directed along n, and for yð2Þ < 0 its direction is opposite). On the other hand, the proportionality F12 / sgn(rð2Þ n) provides in the given case the validity of the 3rd Newton’s law:
F12 ¼ F21 :
ð30Þ
It will be shown below that in media with surfaces not only spatial dispersion returns to the interaction, but also classical property (30) loses its applicability. In order to obtain the interaction force between dislocations in purely elastic medium one should just replace in (29) ð1Þ
ð2Þ
ð1Þ
ð2Þ
BJ C iJKl ðnÞBK ! bj cijkl ðnÞbk ;
ð31Þ
cijkl ðnÞ ¼ cijrp np ½ðnnÞ1 rt nq cqtkl cijkl :
ð32Þ
where
This immediately returns us to the result of Orlov and Indenbom [2,3]. Here we must note that the interaction force in piezoð1;2Þ ; 0Þt is also described by the transformation electrics between ordinary dislocations with the Burgers vectors (7) Bð1;2Þ ¼ ðb ð1Þ ð2Þ ð1Þ ð2Þ in (29) BJ C iJKl ðnÞBK ! bj C ijkl ðnÞbk which looks similar to (31) but is not identical with it because the tensor C ijkl ðnÞ by its definition (25), in contrast to (32), still contains piezoelectric moduli eikl and permittivity eik . The further transition from a purely elastic crystal to an isotropic medium has been done in [2,3]. Naturally, it coincided with the Kroupa result [1]:
F12 ¼
h i n 2m ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ðmð1Þ b Þðmð2Þ b Þ þ ðmð1Þ b Þðmð2Þ b Þ þ ðmð1Þ mð2Þ Þ ðb b Þ ðb nÞðb nÞ ; 2 sin a 1 2m ð33Þ
where
m is the Poisson ratio.
4. Crossing dislocations in a piezoelectric plate Let us now consider the case of an infinite piezoelectric plate with two crossing dislocations, both parallel to the faces of the plate. This time it is convenient to choose the Cartesian coordinate system with the plane xz being a middle plane for the plate, the axis z not coinciding, but parallel with dislocation 1, and the axis y orthogonal to the faces, which thus have
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symmetric coordinates y ¼ h and y ¼ h (Fig. 3a). In contrast, dislocation planes P and Q parallel to the surfaces are not supposed to be symmetric with respect to the plate faces being characterized by the coordinates yð1Þ and yð2Þ . For simplicity, we shall consider only one type of boundary conditions at the plate surfaces: they will be supposed to be free of tractions and electrically closed, i.e. metalized. One can show [14] that for such particular boundary conditions the known solution [3] developed for purely elastic anisotropic plate may be extended to the case of a piezoelectric plate excited by a transverse 1D distribution of ‘‘plastic” distortions U 0lK ðr nÞ. This time the formula (24) must be modified to the form
0 yU ; lK 2
3y
RiJ ðyÞ ¼ C iJKl ðnÞ U 0lK ðyÞ U 0lK
h
ð34Þ
where y ¼ r n and
U 0lK ¼
1 2h
Z
h
h
U 0lK ðyÞdy;
yU 0lK ¼
1 2h
Z
h h
yU 0lK ðyÞdy:
ð35Þ
Bearing in mind that now dislocation 1 does not coincide with axis z of the coordinate system, the function U 0lK ðyÞ in (34) and (35) should differ from (23) by the shift y ! y yð1Þ , i.e. ð1Þ ð1Þ
U 0lK ðyÞ ¼
ml BK sgnðy yð1Þ Þ: 2 sin a
ð36Þ
Substituting (36) into (35) one obtains ð1Þ ð1Þ
U 0lK ¼
ml BK yð1Þ ; 2 sin a h
yU 0lK ¼
ð1Þ 2 ! ð1Þ ð1Þ ml BK h y : 1 2 sin a 2 h
ð37Þ
With (36), (37), equation (34) taken at y ¼ yð2Þ determines the needed ‘‘stress” field of the set of decomposed dislocations of type 1 at the dislocation line 2: ð1Þ ð1Þ
R
ð1Þ ð2Þ iJ ðy Þ
¼
C iJNp ðnÞ
mi BJ
2 sin a
ð2Þ
sgnðy
" ð1Þ 2 #! yð1Þ 3yð2Þ y : y Þþ 1 h 2h h ð1Þ
ð38Þ
The further manipulations are the same as in the previous section after obtaining Eq. (26). However their result is quite different from (29): 12
F
¼ F sgnðy
ð2Þ
" ð1Þ 2 #! yð1Þ 3yð2Þ y ; y Þþ 1 h 2h h ð1Þ
ð39Þ
where the vector F is given by Eq. (29)2. The force (39) can be conveniently decomposed into a sum of two terms:
F12 ¼ F121 þ F 12im :
ð40Þ
The first term with a superscript 1 is identical with the found above force on dislocation 2 in an infinite medium, Eq. (29)1 with rð2Þ n replaced by yð2Þ yð1Þ . The second term in (40)
F12im ¼ F
" ð1Þ 2 #! yð1Þ 3yð2Þ y 1 h 2h h
ð41Þ
is determined by the image fields of dislocation 1 due to the plate surfaces. It is natural that this term is continuous at any variations of yð1;2Þ including the point yð1Þ ¼ yð2Þ , in contrast to the first term in (39) and (40) discontinuous at this point. Of course, the introduced planar sources related to dislocations 1 and 2 being on different distances from the surfaces produce different image components of electroelastic ‘‘stresses” Rð1;2Þ . That is why, in this case F12 – F21 , i.e. we lose the
(a)
(b)
y
y
Q
h y(2)
Q
y(2)
P
y(1)
P
y(1)
-h
0
Fig. 3. Positions of the surfaces and dislocation planes P and Q in the shifted along y coordinate systems for a plate (a) and a semi-infinite medium (b).
900
V.I. Alshits et al. / International Journal of Engineering Science 47 (2009) 894–901
3rd Newton law (30) in the plate. For the same reason, the combined force F12 (39), in contrast to the magnitude of F121 independent of any distances, acquires a dependence on positions yð1;2Þ of dislocations 1 and 2. Certainly, in the limiting case j yð1;2Þ j<< h, when the both dislocations are much closer to each other than to the surfaces, image force term (41) becomes much smaller than F and the interaction is closing to that for an unbounded medium. The other limiting situation, when dislocation 1 falls in the vicinity of one of the surfaces (yð1Þ h or yð1Þ h), proves to be much less trivial and more interesting. Indeed, in such situations this dislocation acts on the other one with a force F12 2F121 only when the latter situates even closer to the same surface. Otherwise, the force F12 practically disappears. For instance, at yð1Þ h to the main order Eq. (39) approximately reduces to
( 12
F
ð1Þ
2FHðy
ð2Þ
y Þ¼
2F; h < yð2Þ < yð1Þ ; h < yð1Þ < yð2Þ :
0;
ð42Þ
As we shall see in the next section, for a semi-infinite medium this relation becomes exact. We stress that Eq. (39) contains piezoelectric characteristics only through the factor F. If to replace the latter by the expression for a purely elastic medium [see transformation (31)], one obtains the result, which appears to be also original (at least, we never met it in the literature).
5. The effect of a plane capacitor in a semi-infinite piezoelectric medium Let us now consider a semi-infinite piezoelectric body with the surface being again free of tractions and metalized (electrically closed). This case may be obtained by a limiting transition from the above plate. However before the transition it is desirable to shift the coordinate system so that the origin would fall at one of the surfaces. Say, let one face of the plate situate at y ¼ 0 and the other one at y ¼ 2h. In this system one should replace y ! y h in (34) and appropriately shift limits of integration in (35). At the following limiting procedure h ! 1 we shall make use of the finiteness of the function U0 ðyÞ at infinity, notating U0 ð1Þ U01 . Then
lim
h!1
3ðy hÞ 2h
Z
3
!
2h
0
ðy hÞU ðyÞdy
¼
0
Z 2 Z 2 3 y 3 ðt 1ÞU0 ðhtÞdt ¼ U01 ðt 1Þdt ¼ 0 lim 1 2 h!1 h 2 0 0
and Eq. (34) is transformed to
RiJ ðyÞ ¼ C iJNp ðnÞ U 0pN ðyÞ U 0pN
ð43Þ
with
U 0pN ¼ lim
h!1
1 2h
Z
2h 0
! U 0pN ðyÞdy :
ð44Þ
Substituting Eq. (36) into (43) and (44) one obtains the ‘‘stress” field ð1Þ ð1Þ
Rð1Þ iJ ðyÞ ¼ 2C iJNp ðnÞ
mi BJ Hðyð1Þ yÞ; 2 sin a
ð45Þ
specifying (38) to the case of the considered semi-infinite medium (Fig. 3b). Accordingly, instead of (39), the interaction force from dislocation 1 to dislocation 2 indeed takes the form similar to (42):
F12 ¼ 2FHðyð1Þ yð2Þ Þ:
ð46Þ
Certainly, Eq. (46) is also valid for a purely elastic medium after transformation (31) in (29)2. The obtained effect, when dislocation 2 suffers the action from dislocation 1 only until the object of the force is closer to the surface than its source, has a clear physical origin analogous to the known effects in electrostatics. The decomposed set of dislocations of type 1 produces in the semi-infinite plate the same field, Eq. (45), as a couple of planar sources in the infinite space. This pair is formed by the identical planar ‘‘dislocation” and its image counterpart situated at the surface and characterized by the opposite topological ‘‘charge” (say, Bð1Þim ¼ Bð1Þ Þ: ð1Þ ð1Þ
ð1Þim Rð1Þ1 ðyÞ þ RiJ ðyÞ C iJKl ðnÞ iJ
ml BK ð1Þ ½ sgnðy yð1Þ Þ sgnðyÞ ¼ RiJ ðyÞ; 2 sin a
ð47Þ
where in the right-hand side one automatically obtains (45). Thus, we have got sort of a plane capacitor where the planar sources produce fields which are doubled between the source planes and are eliminated beyond a ”capacitor”. We also note in passing, that for a half-infinite medium the following relation
F12 þ F21 ¼ 2F replaces the 3rd Newton’s law, Eq. (30).
ð48Þ
V.I. Alshits et al. / International Journal of Engineering Science 47 (2009) 894–901
901
6. Concluding remarks The found expressions for interaction forces between crossing dislocations in all three types of considered piezoelectric media contain in their denominators the factor sin a which provides a singularity at a ! 0. Of course, this singularity is quite expectable, because at a ¼ 0 we obtain two parallel dislocations, which clearly must have an infinite total interaction force between dislocations of infinite length, L ! 1. On the other hand, for a finite size L of a crystal along the dislocations, the above considerations also retain an approximate meaning if L sin a >> d, where d is the distance between the parallel planes, P and Q, containing the dislocations. This provides a physical exclusion of the above singularity from our results, because they remain valid only until
d sin a >> : L
ð49Þ
Let us now evaluate a magnitude of the studied interaction. As we have seen, for reasonable values of the above sina, the ð1Þ ð2Þ scale of interaction between two dislocations is given by the force F 12 BJ C iJKl ðnÞBK . Even for strong piezoelectrics where 2 the electric contribution to this force is substantial, the latter may be roughly estimated as F 12 lb , where l is the shear modulus of the crystal. Now one should take into account that in real crystals there are many dislocations: the typical dislocation density in dielectrics is 104 —106 cm2 . So, in a cube with the line size 1 cm, apart from dislocation 1, a lot of other dislocations (say, the number N 105 ) will act on dislocation 2 with comparable forces. As we know, they can either attract, or repulse dislocation 2. Denote N þ and N the numbers of dislocations acting on dislocation 2 with forces which have, respectively, the same or opposite signs of their projections on n, compared with the force produced by dislocation 1. Of course, the numbers N þ and N should be comparable, but not exactly equal to each other. We expect that the difference DN ¼j N þ N j, being much smaller than N, remains rather large, say DN 101 N 104 . Then the combined magnitude 2 of the interaction force component along n would be F DN lb , which is equivalent to the application to dislocation 2 the stress
r
F DNb l: bL L
ð50Þ
For DN 104 , b 108 cm, L 1 cm this gives a fairly high level of the stress r 104 l. Thus the considered interactions of dislocations may cause quite serious forces at least on some of them. However, one should take into account that normally only freshly introduced dislocations can move in a crystal under the action of external forces or internal stresses. Aged dislocations are surrounded by ‘‘clouds” of point defects (the so-called Cottrell atmospheres) which practically prevent their motions. Still, at the annealing regime under high temperatures even aged dislocations may move together with their atmospheres due to much more intensive processes of diffusion of lattice defects. In this particular regime the studied above interaction between dislocations can lead to a relaxation of the internal stresses and even to a decrease of the dislocation density. Acknowledgements The authors are grateful to Prof. M. Kachanov for useful comments and to Prof. V.N. Lyubimov for helpful discussions and kind assistance with references. The paper was supported by the Polish Foundation MNiSW (Grant # N N501 252334). V.I.A. also acknowledges support from the Kielce University of Technology (Poland). References [1] F. Kroupa, The force between non-parallel dislocations, Czech. J. Phys. B 11 (1961) 847–848. [2] S.S. Orlov, V.L. Indenbom, Interaction of non-parallel dislocations in an anisotropic medium, Sov. Phys. Crystallogr. 14 (1970) 675–677. [3] V.L. Indenbom, Dislocations and internal stresses, in: V.L. Indenbom, J. Lothe (Eds.), Elastic Strain Fields and Dislocation Mobility, North-Holland, Amsterdam, 1992, pp. 1–174 (Chapter 1). [4] A.M. Kosevich, L.A. Pastur, E.P. Feldman, Dislocation and linear charge fields in piezoelectric crystals, Sov. Phys. Crystallogr. 12 (1968) 797–801. [5] G. Saada, Dislocations dans lex cristaux piezoelectriques, Phys. Status Solidi B 44 (1971) 717–731. [6] G. Faivre, G. Saada, Dislocations in piezoelectric semiconductors, Phys. Status Solidi B 52 (1972) 127–140. [7] J.P. Nowacki, V.I. Alshits, A. Radowicz, Green’s functions for an infinite piezoelectric strip with line sources at the surfaces, Int. J. Appl. Electromagn. Mech. 12 (3–4) (2000) 177–202. [8] J.P. Nowacki, V.I. Alshits, A. Radowicz, Green’s function for an infinite piezoelectric strip with a general line defect, J. Tech. Phys. 43 (2) (2002) 133–153. [9] J.P. Nowacki, V.I. Alshits, A. Radowicz, 2D electro-elastic fields in a piezoelectric layer-substrate structure, Int. J. Eng. Sci. 40 (2002) 2057–2076. [10] D.M. Barnett, J. Lothe, Dislocations and line charges in anisotropic piezoelectric insulators, Phys. Status Solidi B 67 (1975) 105–111. [11] A.N. Stroh, Steady-state problems in anisotropic elasticity, J. Math. Phys. 41 (1962) 77–103. [12] J.P. Nowacki, V.I. Alshits, Dislocation fields in piezoelectrics, in: F.R.N. Nabarro, J. Hirth (Eds.), Dislocations in Solids, vol. 13, North Holland, Amsterdam, 2007, pp. 47–79 (Chapter 72). [13] E. Kroener, Physics of defects, in: R. Balian et al. (Eds.), Les Houches Session XXXV, Course 3, North-Holland, Amsterdam, 1980. [14] V.I. Alshits, J.P. Nowacki, A. Radowicz, 1D electroelastic fields in piezoelectrics excited by intrinsic strains, Crystallogr. Rep. 53 (6) (2009), in press.