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The edge dislocation inside an elliptical inclusion
Mechanics of Materials 2 (1983) 319-330 North-Honland
319
THE EDGE DISLOCATION INSIDE AN ELLIPTICAL INCLUSION * William E. WARREN Sandia National laboratories, Albuquerque, N M 87185, U.S.A. Received 29 August 1983
The plane elasticity problem of an edge dislocation located within an elliptical inclusion in an unbounded matrix is considered. A general solution to this problem is obtained in terms of complex potential functions and all coefficients in the series representation of these potential functions are explicitly obtained. Convergence of these series is thus assured. Two specific cases of this general solution are considered in detail. The first case considers a very long, thin inclusion corresponding physically to the geometry of typical crazes in glassy polymers. The second case considers the almost circular inclusion representative of a number of crystal deYects and imperfections. For the second case, expressions for the resultant force on the dislocation agree with results previously obtained for the limiting circular inclusion.
1. Introduction
The study of dislocations has received considerable interest over the years because of their close relationship to crystal defects and imperfections (see Hirth and Lothe (1968) and Nabarro (1967)). The mobility of crystal defects plays an important role in the macroscopic mechanical properties of crystalline materials, and this mobility is dependent upon the internal forces acting on the dislocation. Study of the interaction of dislocations with material inhomogeneities has provided information about certain strengthening or hardening mechanisms in a number of alloyed materials. A survey of theoretical investigations of the elastic interactions between dislocations and inhomogeneities has been provided by Dundurs (1969). The relationship between continuously distributed dislocations and macroscopic plasticity is discussed by Mura (1969) and exploited in a classical analysis by Bilby, Cottrell and Swinden (1963). In this paper we consider the plane elasticity problem of an edge dislocation located at an arbitrary point within an elastic elliptical inclusion which is imbedded in an unbounded matrix with different elastic properties from the inclusion. This problem is solved using the complex potential functions of Muskhelishvili (1953) and the properties of conformal maps. All coefficients in the series representation of the complex potential functions are explicitly obtained and convergence of the respective series is assured. As examples of this analysis, two specific cases of this general solution are considered in detail. First, the case where the ratio of minor-to-major axis c of the ellipse is very small is considered and the solution is expanded in powers of c. This long thin inclusion corresponds physically to the geometry oT typical crazes in glassy polymers (Kambour, 1973; Kramer, 1979), the study of which is important since crazes increase the ductility of the material and also serve as precursors to the actual fracture of the polymer. The stress field around the craze tip is investigated in detail by establishing a polar coordinate system (r, 0) around a loci of the elliptical inclusion and considering small r around the craze tip. Results show that a bounded, very localized stress field exists at the tip of the craze which is independent of the craze tip radius, and this enhancement depends upon the strength and location of the defect within the craze, and upon the relative *
This work performed at Sandia National Laboratories has been supported by the U.S. Department of Enen~,y under Contract DE-AC04-76-DP00789.
0167-6636/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland)
W.E. Warren / Edge dislocation inside inclusion
320
material properties of the craze and bulk polymer. This stress field is similar to that which exists at the tip of a homogeneous craze under uniform loading conditions in the far field (Warren, 1983) and should be of interest in determining the effect of defects resulting from small cracks or voids within the craze on craze growth and eventual fracture of the bulk polymer. A second case is considered in which c approaches 1 (¢ ~ 1), the circular inclusion, and a first order correction to the potential functions of the matrix material are obtained for small deviations from the circular geometry. The force on a dislocation located on the major axis and directed tangentially to this axis is obtained. The limiting circular geometry provides the potential functions and the resultant force on a dislocation located in a circular inclusion and agrees with results obtained by Dundurs and Sendeckyj (1965). These results should be of interest in determining the effect of small geometrical eccentricities of circular inclusions on the mobility of defects within the inclusion. The solution presented here may also be used as a kernel or Green's function to formulate the singular integral equations of Hilbert-type associated with crack problems for this geometry. Of particular interest in this regard is the fracture of glassy polymers where crack growth occurs within previously crazed regions of the material. Under these conditions, the crack is always propagating into a material substantially altered from that of the bulk polymer and provides a significant example of nonhomogeneous fracture.
2. Elastic analysis In this section we consider the plane elasticity problem of an infinite elastic region containing an elliptical-shaped inclusion with different elastic properties. An edge dislocation with complex Burger's vector 7 = bx + iby acts at an arbitrary point z 0 within the inclusion. The geometry of this problem is shown in Fig. 1. The conformal map
l/t),
R>O,
1 ~" : 2"R (z + V/z2 - 4 R 2 ),
(1)
takes the entire z-plane into the exterior region of the unit circle in the ~'-plane, the unit circle representing a cut from - 2R to + 2 R in the z-plane while the elliptical-shaped interface maps into a circle of radius
N1
0 =
zn
=i
I~ r.
Z-PLANE
J ~-PLANE
Fig. 1. Inclusion geometry with mapped region.
321
W.E. Warren / Edge dislocation inside inclusion
a > 1. The point z 0 maps into ~o- In terms of the parameter a, the ratio of minor to major axis of the elliptical inclusion c is given by , = ( a 2 - 1 ) / ( a 2 + 1)
(2)
As a ~ 1 the inclusion degenerates to a thin ribbon, while a ---, 00 provides the circular inclusion. The stress and displacement fields in both regions due to the edge dislocation may be most readily obtained in terms of the complex potentials #)(~') and tk(~') as developed by Muskhelishvili (1953). In terms of these complex potentials, the stress and displacement fields are given by Oxx + ffvy= 2[ ~ ( ~ ) + ~ (~)],
%-
+ 2iaxy =
+
]
'
(3)
2G(u + i o ) = ~ ( ~ ' ) - t o ( ~ ' ) ~ - ( ~ ) - t k (~r), where oxx, O.vy,q~y are cartesian components of the stress tensor; a, v are the displacement components in the x and y directions respectively; G is the material shear modulus; x = ( 3 - 4v) for plane strain, K = ( 3 - v)/(1 + v) for generalized plane stress, where v is Poisson's ratio; and ~(~)=gi'(~')/~'(~'), ff'(~) = ~'(~)/to'(~). A further result which will be of use in establishing continuity conditions across the elliptical interface and for obtaining the force acting on the dislocation is the expression for the resultant force X + i Y on any arc A - B in the material. This resultant force is given by B
X + iY--- - i [ #)(~') + oJ(~)~ (~) + ff (~)] A,
(4)
where [ ]Andenotes the change in the bracketed function in going from point A to point B in the body. We denote the region exterior to the elliptical inclusion as Region 1 with elastic material properties G], KI, and denote the elliptical inclusion as Region 2 with elastic properties G2, ~:2-The potential functions for Region 1 are #h(~'), ~1(~') taken in the form gl] (~) = ~ri( K1 + 1) In ~"+ nY' ~ a_,,~-", =0
: + ~ri(K, + 1) In ~ ' - • i(s~ + 1)a2(~ " - 1/~') (~" - 1/~" ) ,,=o b
~'(~') = -
(5)
oo
G,y(~" + a4/~ ")
-"
where the series terms represent functions holomorphic in the region I '1 > a. The potential functions for Region 2 are ~2(~), tk2(~:) taken in the form G2"/ [ ( ~ 1 - ~10 )~] ]+ #)2(~')= ri(l¢2+l) in ( ~ ' - ~ ' 0 ) ) - " -
G2"Y
[
(
~2(~')= - ~ri(r 2 + 1) In (~'-~'o) 1 1
1)]
n=0
c,,(~"+~-"),
G2Y
(~o + 1/~ro)~" ~'~'o - ~ r i ( r 2 + 1) (~'-~'o)(~'- 1/~'o)
(6)
oo
+ ( ~ ' - 1/~') Y~l ' -d"')('~=: ' and the series terms here represent functions holomorphic in the annular region 1 < ]~l < a. This representation has been taken to insure continuity of the functions across the cut in the z-plane. The constants a_,,, b_,, c., d,, are evaluated from the conditions of continuity of tractions and displacements across the elliptical interface between Regions 1 and 2. Formally, these conditions become X] + iY1 =
X 2 -4-
iY2 on ~' = ae i0,
u I + iv] = u 2 + iv 2 on ~"= ae i°.
(7)
322
W.E. Warren / Edge dislocation inside inclusion
Substituting from (4) into (7)~ gives 6
,t,~(~')-~ ~'~(~;) ' ( ( ) #?,(() + ~ , ( ~ ) =#'2(~') + ~
(f) +
72(f),
~ ' = a e i°
(8)
and (3) 3 into (7)2 gives
(FK, + 1)¢i(~')- (~:2 + 1 )¢,(~')+. ( 1 - / ' ) [ ~,(~:) ~(~" ) ~ ] ( # ) + ¢7, (()] =0, ~=ae i°,
(9)
where F = G 2 / G l, and use has been made of boundary condition (8). Equating corresponding coefficients of e i'°, n = 0, _+1, _+2, .... in (8) and (9) provides a consistent but non-trivial system of relations for obtaining explicit representation of all unknown coefficients. The details of this lengthy reduction will not be given here and we only present the final results of the evaluation. Elimiaating the b_~ and d,, terms from (8) and (9) and invoking convergence requirements on the respective series for a ,,, c,, leads to the two equations
(F~, + 1)a_,,-(K 2 + 1)c,, +(~2 + 1)K" 1A,, = 0
(10)
and (x 2 + 1)a2~,,+1)c,, -2~,,-1~ a (a2~,,+i~ (l-r) -a _.-
a-2~,,-1))c,,-- n ( a 4 - -
1)~,,
-Ka-2("-"l A,, + Ka2A,, tl
IZ
K
(11)
- (~7o_ 1/~o)[(a4B"-' + B,,,+,) - a2(~'o + 1/~'o)B,, ] = 0 , n = 1, 2, ..., for determining a_,, and c,, for n = 1, 2, .... In (10) and (11) K=
G2Y ~i(x2 + 1 ) '
A" = (~'g + ~'°")'
(12)
B" = (~'~ - ~'°")"
With the explicit expression for a_,, and c,, determined from (10) and (11), the b_,, and d,, are evaluated from the relations a 2b
--n
+ 1)a2"[a2cn-a-2c,,i 2] _,,+(n-2) a4a_n_2+ (x2 (l-r)
= na
for n = 3, 4, 5 . . . . .
(13)
and
a2"d,, = a 2 a ~ n _ l ) - a - 2 a _ ( , , + l ) - a 2 c n _ m[ +K
a2
_
(n _ 1) A,,_I
°2
1 + a-2c,,+1 -
(n-
1) c,,_i a 2 0 ' - l ) - ( n + !)c,,+ia 2(''+1~
]
(14)
for n = 2, 3, 4, .... The coefficients b_ 2 and b_ 1 are given by
(x 2 + 1)
a2(b_2 + b o ) = 2a_ 2 + (1 - F) a6~2'
a2b
(K2 + 1) --i = a - 1 -4- (1 -- /") a41~1"
(15)
W.E. Warren / Edge dislocation inside inclusion
323
The remaining coefficients a 0, b 0, c o, d~ correspond to rigid body translations of the two regions and satisfy the additional two conditions
1)ao-2(K2+')Co+(1-F)Go=O,
(FK, +
a4(ao + Go - 2 c o - d l ) = a _ 2 - c 2
(16)
+ 2a6e2 +½KA 2 + K a z.
Thus, if a rigid body translation of Region 1 is prescribed, the constants a o and bo are determiaed and c 0, d] are evaluated from (16). Similarly, if a rigid body translation of Region 2 is prescribed, the constants c o and d] are determined and a 0, bo are evaluated from (16). Since the function ~ ( ~ ' ) vanishes at infinity, there is no rigid body rotation associated with this problem. We also note that the imaginary part of b_ ! is proportional to an externally applied resultant moment acting throughout Region 1, and since this is zero, the constant b_ ] is real. Before proceeding to the evaluation of these potential functions for the two specific geometries when the elliptical inclusion is either very thin or almost circular, we obtain explicit expressions for the coefficients for the case when the Burger's vector 3' is in the y-direction, 3' = i h,,, and z o is on the x-axis. Then we note from (12) that K = K is real, and A , , B , / ( ~ o - 1/~ o) are real since ~'0 is either real or located on the unit circle. Under these conditions, the a _ , , b , , c., d. are all real and (10) and (11) provide
('-F-KKI+']-)"(-I"---~' +a-2"
a_n=
~
( a 2 ~
K
a
1) 2a-2('+1'
B"-I
2nil
1 + (F~1 + 1) a
n "-(a2-1)a-2"A"
/
(17)
j'
2n]A n
"
+(l_a_Z)A,_a2(l_a_2)
2
B,,_, (~'0- 1/~'0)
/
for n = 1, 2, ..., and where ~,,- (r 2+F)
(1- r)
n ( a 4 _ 1)a_20,+1,
(x2--FKl)
(1 +
-4,,
(18)
F~:,) a
The b , , and d,, may then be obtained from a_,. and c, of (17) by using (13), (14) and (15). However, it is much easier to sum the respective b_,, and d,, series in terms of the a , , and c,, than to actually evaluate the individual terms. For example, the series involving the b_,, terms of (5) takes the form ~'
Eb
7o
,,~.-,, = bo +
(~:2 + 1 ) oo,
(l-F)
E'a2nCn~-n+ (: + aZ(~'-
,,=1
OO
E
(19)
1/~ ') n=]
A similar result holds for the series involving the d, terms of (6). We now consider in detail two cases where the elliptical inclusion is either very thin with c ~ 0, a - ~ 1, or where the elliptical inclusion is almost circular with c ~ 1, a -* 0o. For the latter case, the solution reduces in the limit to the solution for a circular inclusion obtained by Dundurs and Sendeckyj (1965).
3. Thin elliptical inclusion
In this section we consider the case where the elliptical inclusion is very thin corresponding to c << 1 and a ---, 1. We take the dislocation to lie along the real axis so that z o is real, and the Burger's vector is in the y
324
W.E. Warren / Edge dislocation inside inclusion
direction so that -/= i by. Then all coefficients a _ . , b_ n, c., dn for n = 1, 2, ... are real, with the a_,,, c,, given by (17). Expanding these coefficients in powers of c and retaining terms up to first-order provides
(l-r) a_2,,[1X . -
c . = - M ( K 2+ 1)
,(N~
-
2)A,,],
(20)
for n = 1, 2, ..., and where
M = G , b , , / { ~ ( K , + 1)}, No = 4 [ ( g 2 - 1)-F(K,- 1)] (1 - r ) , F(K 2 + I)(~, + I)
Nl
+
+ i)
i)] (i +
FK,).
(21)
Evaluation of the potential functions for Region 1 using the coefficients of (20) provide, after transferring to the z-plane ---"wm~ "0 - -'o real, cb,(z) = M [ l n ( z - Xo) + o N o
~,(z)=M(ln(z-xo)
V/z2 -- 4R2 1
(z-xO) ]"
Xo
(Z-Xo)
, [ x0¢z 2 - 4R2
+ cN 1
~/z 2 - 4R ~
(22)
(Z-Xo)
4R2(1 + 2¢ 2) Q
Of particular interest to the investigation of craze growth in glassy polymers is the nature of the stress field at the tip of the thin elliptical inclusion. To investigate this we now establish a polar coordinate system in the z-plane with origin at the fight-hand foci z = 2R. Accordingly, we set z = 2R + re iO, r << R, and introduce a scale factor ro equal to the minimum distance from the foci to the material interface. In terms of the parameter c, the minimum distance ro and the radius of curvature rc at the tip of the inclusion are
"-° = , , ( i + R
+ . . . ),
r~=2c2(1+½( 2+ "") R
(23)
and we note lhat for the thin ellipse, rc = 2ro. The stress field around the tip then becomes Oyy
(! ~ x 0) 1 + No
cos 0 / 2 , (24)
o Jy' ) - - o;~ ( ' ) + 2 i o,(, 'y)--' ( l _ 2x M o)
{( N , - N o )
~ r ° e-i#/2 + N o ~ r ° e-i3°/2 [2(-~) + i sin 0]},
where I --- 2R is the half craze length, and x 0 < I. This stress field has the same form as the stress field at the tip of a simple craze in a glassy polymer subjected to a uniform stress in the far field as obtained by Warren (1983). In particular, we note that since r >/r 0, the stress field remains bounded around the craze tip. Figs. 2 and 3 show typical contour lines of constant dimensionless hydrostatic stress S = ,tr(! - xo)S/Glby and constant dimensionless maximum shear stress :r = ,rr(l - Xo)'r/Glby around the craze tip, where S is the hydrostatic stress and ~- the maximum shear stress. In obtaining these results, we have taken G~/G2 = 4, v~ = -~, v2 = 0 which are material properties representative of crazes in glassy polymers.
W.E. Warren/ Edgedislocationinsideinclusion
325
The hydrostatic stress S in Region 1 for conditions of plane strain is given by S =-~(1 + lq)(o.:?~) + O.v(yl'),
(25)
while the m a x i m u m shear stress ~- in this region is given by
~.__½{(_(t) _(,,±T 2io(tv, _,t, - o.~.~ _,t, - 2:_(,,)} '/2. o>,), - "x.~ . )( "YV _,U.~y
(26)
These two quantities are of interest since they relate to numerous proposed criteria for craze growth in glassy polymers. Figs. 2 a n d 3 s h o w the highly localized nature of the stress e n h a n c e m e n t at the craze tip. The m a x i m u m values of S and T occur at the tip r = r0, 0 = 0, and for this example,
Smax = 1.53,
"rmax =
1.75.
Along the craze extension line, the shear stress • decays much more rapidly than S although it spreads out considerably in directions normal to the craze extension line. In certain polymers this condition may assist in the formation of shear bands which have been observed running at - 45 ° to the plane of the craze. In the absence of a craze, the stresses at the point corresponding to the craze tip x = ! for a dislocation located at x 0 < 1 are S = 0.55 and ~r= 0, so the stress enhancing effect of the craze is significant. The thin elliptical interface occurring at r = p(0) is defined by m
P( g ) = [c°s2 0/2] -'[ 1 - ~'2 ( 1 - c°s O)
70
-6 ¥
]
+
(27)
and suppressing terms of order c 2, the stress c o m b i n a t i o n s of (24) at the b o u n d a r y b e c o m e
[_(~) ± _(1)1 4M ox.~ T%.y l h - ( I - - x 0 ) [ 1
+ ½N0(1 + cos O)],
(28) _it) _(]) . (,) 2M [No( 1 + c o s 0 ) e _ i O + ½ ( N l _ N o ) ( 1 + e _ i a ) ] " 0,'.,' - Oxx + 2,O~y ] b - ( l _ xo)
CRAZE
r/~ 0
5
10
15
20
C
R
25
Fig. 2. Lines of constant dimensionless hydrostatic stress, g = [{ ~ ( I - Xo)}/(Glby)lS.
A O
~ 5
r/ro
10
15
Fig. 3. Lines of constant dimensionless maximum shear stress,
-- {~(i- Xo)}/(ciby),.
326
W.E. Warren / Edge dislocation inside inclusion
Fig. 4 shows the dimensionless stresses S and ~ around the boundary and we note here again that the shear stress ~"is more localized than the hydrostatic stress S at the craze tip. The potential functions for the inclusion Region 2 for this case are not uniformly convergent for small c and for all z in Region 2. These potential functions contain singular points in Region 1 which collapse down into Region 2 in the limit as c--* 0. For z sufficiently removed from x o, these potential functions provide
02(')- (~2M+ ]) [2 +(~,- ~)r] ln(z-xo) +O(t2),
M ( Xo / +o(C), ¢2(z)-(r2+1) [2~2-(K'-l)r]In(z-x°)-[2+(K'-l)r](z-xo)
(29)
which cannot be considered valid for all z in Region 2. In a neighborhood of the craze tip the stresses within the inclusion are effectively those due to a resultant force
4Glby X2 +iY2= - (K1 + 1)(K 2 + 1 ) [ ( K 2 - 1 ) - r ( , , -
])1
(30)
and a dislocation with complex Burger's vector lby, where
b. = by{l + 2 [(r2- r~,) + r2(1- r)] I -
r ( ~ , + I)(K 2 + i)
(31)
'
both acting at x 0 within the inclusion. The stresses near the craze tip are uniform and given by 4M
[2 + ( ~ , - 1 ) r l °~' + °Y(~)= ( l - xo ) (K2+ 1) (32)
and --
(2,
4M (i-xo)
_ ( 2 ) _ _ ( 2 ) 4. 2 1 0 x v =
%,y
~'x~
[(~2- 1)-(~,(~2 + 1)
1)r]
and satisfy the appropriate boundary conditions to first order in c around the craze tip. 2.5
I
I
I
I
lr(£-Xo) s
2.0
G I by
m Ill i-m
'a'(J~- Xo)
' - ""-1.5
m Ill _J Z _o 1.o
.
"
Z III
g Q
%
0.5
%
0 0
I 30
I 60
| 90 ANGLE
Fig.
4.
Boundary stress combinations S and ~.
I 120
150
327
W.E. Warren / Edge dislocation inside inclusion
4. Almost circular inclusion We consider in this section the case where the elliptical inclusion is almost circular corresponding to large values of a, and c close to 1. The results we are after are most readily obtained by mapping the ~'-plane into a t-plane with a simple scale change, ~"= at. With this transformation, the unit circle in the ~'-plane maps into a circle of radius 1 / a in the t-plane, and the interface boundary of radius a in the ~'-plane goes into the unit circle in the t-plane. The map from the z-plane to the t-plane then becomes =
=
Ro( +
(33)
where R 0 - R a , 0 <.%m = 1 / a < 1. The ratio of minor to major axis c is now given by c = (1 - m2)/(1 + m 2)
(34)
and m = 0 corresponds to the circular inclusion. We are interested in the solution for small m (large a), and * andc,,-~ c,,, * wherea*,,, for large a and n, the coefficients a ,, -~ a_,, _ c,* are given by (1¢2 + 1 )
1
a*-" = - K (r~, + 1) n A ' ' ¢1
a2"c * =
'"
P~
(35)
[ 1
_
*' In
- K ( K2 +'1"3
--A. +
(1
- a-
_
2)A.
(a2
a2
1)2
T) o)
"
of a _ . . c,, respectively in the series terms for n > N. where N is some integer, the Using a * . and c.* in vlace . potential functions for Region 1 become *l(t)=r
(K 2 + 1) {
(/'1¢ 1 + 1) l n [ ( t - t ° ) ( 1 - m 2 / ~ t 5 ° ) ]
(1 -
r)
+ F(K, + 1)
ln )
N
(36)
+ ~ . ( a_,, -- a*,, ) r e " t - " + ao + O ( m 2N ), n=l
(~2 + 1) (
r)
ln[(~-t°)(1-
( g 2 -- /'K1)
m2/tt°)]+
F-(~+ij
(1 - m2)(1 - t 2 ) lnt+t(t-to)(1-m2/tto )
+ (l-r) (1 + FK,) t 2 ( 1 - m Z / t 2) ( t - to)(1 - m21tto ) r ( + 1) (~ + r)
(1 + m 2 t 2 )
(K 2 + 1) N+l + (i-r) E [a2"c.-a2"c*]m"t
t(1 - m21t 2 )
-"
n=l
(1 + m2t z) +t2(1-m2/t
N
Y'. n[a_,,-a*_.]m"U" + bo + O(m2N), z) .=,
(37)
where we have made use of (19). With this representation, the function ~ l ( t ) and ~t,~(t) may be evaluated to any accuracy desired. Similar expressions may be obtained for ~2(t), ~2(t) of Region 2. It can be readily shown that the series terms of (36) and (37) have the order relations (38) for n = 1, 2, ..., N. We note that for the circular inclusion, m = 0, the n = 1 term of (38) 2 contributes a
W.E. Warren / Edge dislocation inside inclusion
328
non-zero term to the solution. For small eccentricities from a circular inclusion, rn << 1, and retaining terms to oider m 2, the potential functions of (36) and (37) become
K((1- B)ln(~-~0)-
(
[
~,, (~) = K (1 - A-) ln(~ - ~o)
-(H-A-)
(H- B)In ~-m2(1-
,o
(/j 5 ~o ) -
B)(1 + Q~02)~-~/, so~ /
1
Q~o/~
l n ~ + f~(f~-f~o) (B-A-) +(H-B)-~ 1 -m2
-m2(1- A-")['o('2'o)-
(39)
~0 1 A--(3'~ ~ 2 ) + Q(~-3 + ~o~)]
~o2~(~_ ~o) +
~
-~o~: -+
Q
1+ ~
'
where -
F-
A - ~F ~+
1
-
'
B=
F~ - K2 FK! + 1 '
~2 + 1 (1 - H)= r(,, + 1)
2(1 - F) Q = ( 2 F + K2- 1)
(40)
are the material coefficients defined by Dundurs and Sendeckyj (1965) in their investigation of the circular inclusion. For the circular inclusion of radius R o, m = O, ~ = z / R o , ~o = x o / R o and (39) provides , , ( z ) = K { (1 - H) In( z - Xo) - ( H - B) In z },
(
[
ff,(z)=K (l-A-)ln(z-xo) -(H-A-)Iag+(H-A-)R~
(Z_Xo)
~(Z-~o)
Qz
+(H-B)
1
- R!)
(41)
,
which agree with the results of Dundurs and Sendeckyj (1965) who obtain the solution in terms of the real Airy stress function. The effect of small eccentricities of the inclusion shape on the resultant force acting on an edge dislocation at some point x 0 on the major axis of the inclusion and directed perpendicular to this axis may be obtained directly from these results. The strain energy due to the dislocation is equal to the work done in forming the dislocation. The displacement along the dislocation is just the Burger's vector "t = bx + ib.v, and the total force on the dislocation line may be obtained directly from (4). The strain energy W is given by W = R e { ~ r' ~ ,"~ - i Y )} ,
(42)
where X - iF is the total force along the dislocation discontinuity running through Regions 1 and 2. Thus ~_.,,
,o'(~)
j~=, (43)
and from the boundary condition (8), the values of these terms cancel each other at ~ = 1. It remains only to evaluate the first bracketed term of (43) as ~--, N, a large constant, and the second term at the
W.E. Warren / Edge dislocation inside inclusion
329
dislocation core radius 8, or at ~--4o for terms which remain bounded at ~--40- Carrying out this evaluation, the strain energy due to the dislocation is G2bf ( 2 1 n ( N / 8 ) _ 2 H ln N + ( , 4 + B ) l n ( l _ /~2)(l _ m 4 / ~ 2 ) W - - 2,lr(1¢2 + 1) -- 2A-"Q(~o + m 2 / ~ o ) 2 + 9m2A2(~o + m 4 / ~ o ) 4
+ m212A- - 12A-'z + (1 + 3A-- 2 B ) Q ] ( 4o + m2/~o )2 + Coast.},
(44)
which is real since, for z o real, 4o is either real or on the circle of radius m. In t e ~ s of z o = x o, the strain energy W becomes . 2 C2b;
W = 2~r(1¢2 + 1) { 2 l n ( N / 8 ) - 2 H In N + Coast. + ( A-+ B ) In[ (1 + m 2 )2 _ x ~ / R 2 ] - 2 A Q x 2 o / R 2 + 9m2A'Zx4/R~ + m 2 1 2 A - 12A-2 +(1 + 3 ~ , 4 - 2 B ) Q ] x 2 / R 2 } .
(45)
The force F acting on the dislocation is defined as the negative gradient with respect to position of the dislocation of the elastic strain energy W. Thus, F = - a W / a x o,
(46)
and direct evaluation from (45) shows that
F=
G b;Xo
2(1,R0 .
.
.
[('.-1+.
(
. 2-
x /R o]
+ 2iQ-
-m212A---12A'2 + ( 1 + 3A---2B)Q] }.
A x o2/ R 2o 18 m 2,-2
(47)
The force on the dislocation becomes arbitrarily large as x o approaches the interface at the ellipse n, ajor axis R0(1 + m2). As in the circular case m = 0, the center of the ellipse x 0 = 0 is an equilibrium position and the stability of this equilibrium position for the circular case is discussed extensively by Dundurs and Sendeckyj (1965). For a given eccentricity m, the force expression of (47) indicates the possible presence of two other equilibrium positions in the interval 0 < x o < R0(1 + m2), one of which may occur in the circular case and has been found to be unstable if it exists. The effect of small eccentricities on the equilibrium positions for particular material combinations may be investigated from these results. Expressions for the force on a dislocation with the more general Burger's vector , / = bx + ib,, located at an arbitrary point z o within the inclusion may be obtained from the exact solution presented earlier, but such an investigation is beyond the scope of the present work.
References Bilby, B.A., A.H. Cottrell and K.H. Swinden (1963), "The spread of plastic yield from a notch", Proc. Roy. Soc. Ser. A272, 304. Dundurs, J. (1969), "Elastic interactions of dislocations with inclusions", in: T. Mura, ed., Mathematical Theory of Dislocations, ASME, 70.
Dundurs, J. and G.P. Sendeckyj (1965), "'Edge dislocation inside a circular inclusion", J. Mech. Phys. Solids 13, 141. Hirth, J.P. and J. Lothe (1968), Theory of Dislocations, McGraw-Hill, New York. Kambour, R.P. (1973), "A review of crazing and fracture in thermoplastics", J. Polymer Sci.: Macromolecular Rev. 7, !. Kramer, E.J. (1979), "Environmental cracking of polymers", in E.H. Andrews, ed., : Developments in Polymer Fracture 1, Applied Science Pub., London, .~.
330
W.E. Warren / Edge dislocation inside inclusion
Mura, T. (1969), "Method of continuously distributed dislocations", in: T. Mura, ed., Mathematical Theory of Dislocations, ASME, 25. Muskhelishvili, N.I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, 3rd ed., Noordhoff, Groningen.
Nabarro, F.R.N. (1967), Theory of Crystal Dislocations, Oxford University Press, London. Warren, W.E. (1983), "The stress and displacement fields at the tip of crazes in glassy polymers", Polymer., to appear.
319
THE EDGE DISLOCATION INSIDE AN ELLIPTICAL INCLUSION * William E. WARREN Sandia National laboratories, Albuquerque, N M 87185, U.S.A. Received 29 August 1983
The plane elasticity problem of an edge dislocation located within an elliptical inclusion in an unbounded matrix is considered. A general solution to this problem is obtained in terms of complex potential functions and all coefficients in the series representation of these potential functions are explicitly obtained. Convergence of these series is thus assured. Two specific cases of this general solution are considered in detail. The first case considers a very long, thin inclusion corresponding physically to the geometry of typical crazes in glassy polymers. The second case considers the almost circular inclusion representative of a number of crystal deYects and imperfections. For the second case, expressions for the resultant force on the dislocation agree with results previously obtained for the limiting circular inclusion.
1. Introduction
The study of dislocations has received considerable interest over the years because of their close relationship to crystal defects and imperfections (see Hirth and Lothe (1968) and Nabarro (1967)). The mobility of crystal defects plays an important role in the macroscopic mechanical properties of crystalline materials, and this mobility is dependent upon the internal forces acting on the dislocation. Study of the interaction of dislocations with material inhomogeneities has provided information about certain strengthening or hardening mechanisms in a number of alloyed materials. A survey of theoretical investigations of the elastic interactions between dislocations and inhomogeneities has been provided by Dundurs (1969). The relationship between continuously distributed dislocations and macroscopic plasticity is discussed by Mura (1969) and exploited in a classical analysis by Bilby, Cottrell and Swinden (1963). In this paper we consider the plane elasticity problem of an edge dislocation located at an arbitrary point within an elastic elliptical inclusion which is imbedded in an unbounded matrix with different elastic properties from the inclusion. This problem is solved using the complex potential functions of Muskhelishvili (1953) and the properties of conformal maps. All coefficients in the series representation of the complex potential functions are explicitly obtained and convergence of the respective series is assured. As examples of this analysis, two specific cases of this general solution are considered in detail. First, the case where the ratio of minor-to-major axis c of the ellipse is very small is considered and the solution is expanded in powers of c. This long thin inclusion corresponds physically to the geometry oT typical crazes in glassy polymers (Kambour, 1973; Kramer, 1979), the study of which is important since crazes increase the ductility of the material and also serve as precursors to the actual fracture of the polymer. The stress field around the craze tip is investigated in detail by establishing a polar coordinate system (r, 0) around a loci of the elliptical inclusion and considering small r around the craze tip. Results show that a bounded, very localized stress field exists at the tip of the craze which is independent of the craze tip radius, and this enhancement depends upon the strength and location of the defect within the craze, and upon the relative *
This work performed at Sandia National Laboratories has been supported by the U.S. Department of Enen~,y under Contract DE-AC04-76-DP00789.
0167-6636/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland)
W.E. Warren / Edge dislocation inside inclusion
320
material properties of the craze and bulk polymer. This stress field is similar to that which exists at the tip of a homogeneous craze under uniform loading conditions in the far field (Warren, 1983) and should be of interest in determining the effect of defects resulting from small cracks or voids within the craze on craze growth and eventual fracture of the bulk polymer. A second case is considered in which c approaches 1 (¢ ~ 1), the circular inclusion, and a first order correction to the potential functions of the matrix material are obtained for small deviations from the circular geometry. The force on a dislocation located on the major axis and directed tangentially to this axis is obtained. The limiting circular geometry provides the potential functions and the resultant force on a dislocation located in a circular inclusion and agrees with results obtained by Dundurs and Sendeckyj (1965). These results should be of interest in determining the effect of small geometrical eccentricities of circular inclusions on the mobility of defects within the inclusion. The solution presented here may also be used as a kernel or Green's function to formulate the singular integral equations of Hilbert-type associated with crack problems for this geometry. Of particular interest in this regard is the fracture of glassy polymers where crack growth occurs within previously crazed regions of the material. Under these conditions, the crack is always propagating into a material substantially altered from that of the bulk polymer and provides a significant example of nonhomogeneous fracture.
2. Elastic analysis In this section we consider the plane elasticity problem of an infinite elastic region containing an elliptical-shaped inclusion with different elastic properties. An edge dislocation with complex Burger's vector 7 = bx + iby acts at an arbitrary point z 0 within the inclusion. The geometry of this problem is shown in Fig. 1. The conformal map
l/t),
R>O,
1 ~" : 2"R (z + V/z2 - 4 R 2 ),
(1)
takes the entire z-plane into the exterior region of the unit circle in the ~'-plane, the unit circle representing a cut from - 2R to + 2 R in the z-plane while the elliptical-shaped interface maps into a circle of radius
N1
0 =
zn
=i
I~ r.
Z-PLANE
J ~-PLANE
Fig. 1. Inclusion geometry with mapped region.
321
W.E. Warren / Edge dislocation inside inclusion
a > 1. The point z 0 maps into ~o- In terms of the parameter a, the ratio of minor to major axis of the elliptical inclusion c is given by , = ( a 2 - 1 ) / ( a 2 + 1)
(2)
As a ~ 1 the inclusion degenerates to a thin ribbon, while a ---, 00 provides the circular inclusion. The stress and displacement fields in both regions due to the edge dislocation may be most readily obtained in terms of the complex potentials #)(~') and tk(~') as developed by Muskhelishvili (1953). In terms of these complex potentials, the stress and displacement fields are given by Oxx + ffvy= 2[ ~ ( ~ ) + ~ (~)],
%-
+ 2iaxy =
+
]
'
(3)
2G(u + i o ) = ~ ( ~ ' ) - t o ( ~ ' ) ~ - ( ~ ) - t k (~r), where oxx, O.vy,q~y are cartesian components of the stress tensor; a, v are the displacement components in the x and y directions respectively; G is the material shear modulus; x = ( 3 - 4v) for plane strain, K = ( 3 - v)/(1 + v) for generalized plane stress, where v is Poisson's ratio; and ~(~)=gi'(~')/~'(~'), ff'(~) = ~'(~)/to'(~). A further result which will be of use in establishing continuity conditions across the elliptical interface and for obtaining the force acting on the dislocation is the expression for the resultant force X + i Y on any arc A - B in the material. This resultant force is given by B
X + iY--- - i [ #)(~') + oJ(~)~ (~) + ff (~)] A,
(4)
where [ ]Andenotes the change in the bracketed function in going from point A to point B in the body. We denote the region exterior to the elliptical inclusion as Region 1 with elastic material properties G], KI, and denote the elliptical inclusion as Region 2 with elastic properties G2, ~:2-The potential functions for Region 1 are #h(~'), ~1(~') taken in the form gl] (~) = ~ri( K1 + 1) In ~"+ nY' ~ a_,,~-", =0
: + ~ri(K, + 1) In ~ ' - • i(s~ + 1)a2(~ " - 1/~') (~" - 1/~" ) ,,=o b
~'(~') = -
(5)
oo
G,y(~" + a4/~ ")
-"
where the series terms represent functions holomorphic in the region I '1 > a. The potential functions for Region 2 are ~2(~), tk2(~:) taken in the form G2"/ [ ( ~ 1 - ~10 )~] ]+ #)2(~')= ri(l¢2+l) in ( ~ ' - ~ ' 0 ) ) - " -
G2"Y
[
(
~2(~')= - ~ri(r 2 + 1) In (~'-~'o) 1 1
1)]
n=0
c,,(~"+~-"),
G2Y
(~o + 1/~ro)~" ~'~'o - ~ r i ( r 2 + 1) (~'-~'o)(~'- 1/~'o)
(6)
oo
+ ( ~ ' - 1/~') Y~l ' -d"')('~=: ' and the series terms here represent functions holomorphic in the annular region 1 < ]~l < a. This representation has been taken to insure continuity of the functions across the cut in the z-plane. The constants a_,,, b_,, c., d,, are evaluated from the conditions of continuity of tractions and displacements across the elliptical interface between Regions 1 and 2. Formally, these conditions become X] + iY1 =
X 2 -4-
iY2 on ~' = ae i0,
u I + iv] = u 2 + iv 2 on ~"= ae i°.
(7)
322
W.E. Warren / Edge dislocation inside inclusion
Substituting from (4) into (7)~ gives 6
,t,~(~')-~ ~'~(~;) ' ( ( ) #?,(() + ~ , ( ~ ) =#'2(~') + ~
(f) +
72(f),
~ ' = a e i°
(8)
and (3) 3 into (7)2 gives
(FK, + 1)¢i(~')- (~:2 + 1 )¢,(~')+. ( 1 - / ' ) [ ~,(~:) ~(~" ) ~ ] ( # ) + ¢7, (()] =0, ~=ae i°,
(9)
where F = G 2 / G l, and use has been made of boundary condition (8). Equating corresponding coefficients of e i'°, n = 0, _+1, _+2, .... in (8) and (9) provides a consistent but non-trivial system of relations for obtaining explicit representation of all unknown coefficients. The details of this lengthy reduction will not be given here and we only present the final results of the evaluation. Elimiaating the b_~ and d,, terms from (8) and (9) and invoking convergence requirements on the respective series for a ,,, c,, leads to the two equations
(F~, + 1)a_,,-(K 2 + 1)c,, +(~2 + 1)K" 1A,, = 0
(10)
and (x 2 + 1)a2~,,+1)c,, -2~,,-1~ a (a2~,,+i~ (l-r) -a _.-
a-2~,,-1))c,,-- n ( a 4 - -
1)~,,
-Ka-2("-"l A,, + Ka2A,, tl
IZ
K
(11)
- (~7o_ 1/~o)[(a4B"-' + B,,,+,) - a2(~'o + 1/~'o)B,, ] = 0 , n = 1, 2, ..., for determining a_,, and c,, for n = 1, 2, .... In (10) and (11) K=
G2Y ~i(x2 + 1 ) '
A" = (~'g + ~'°")'
(12)
B" = (~'~ - ~'°")"
With the explicit expression for a_,, and c,, determined from (10) and (11), the b_,, and d,, are evaluated from the relations a 2b
--n
+ 1)a2"[a2cn-a-2c,,i 2] _,,+(n-2) a4a_n_2+ (x2 (l-r)
= na
for n = 3, 4, 5 . . . . .
(13)
and
a2"d,, = a 2 a ~ n _ l ) - a - 2 a _ ( , , + l ) - a 2 c n _ m[ +K
a2
_
(n _ 1) A,,_I
°2
1 + a-2c,,+1 -
(n-
1) c,,_i a 2 0 ' - l ) - ( n + !)c,,+ia 2(''+1~
]
(14)
for n = 2, 3, 4, .... The coefficients b_ 2 and b_ 1 are given by
(x 2 + 1)
a2(b_2 + b o ) = 2a_ 2 + (1 - F) a6~2'
a2b
(K2 + 1) --i = a - 1 -4- (1 -- /") a41~1"
(15)
W.E. Warren / Edge dislocation inside inclusion
323
The remaining coefficients a 0, b 0, c o, d~ correspond to rigid body translations of the two regions and satisfy the additional two conditions
1)ao-2(K2+')Co+(1-F)Go=O,
(FK, +
a4(ao + Go - 2 c o - d l ) = a _ 2 - c 2
(16)
+ 2a6e2 +½KA 2 + K a z.
Thus, if a rigid body translation of Region 1 is prescribed, the constants a o and bo are determiaed and c 0, d] are evaluated from (16). Similarly, if a rigid body translation of Region 2 is prescribed, the constants c o and d] are determined and a 0, bo are evaluated from (16). Since the function ~ ( ~ ' ) vanishes at infinity, there is no rigid body rotation associated with this problem. We also note that the imaginary part of b_ ! is proportional to an externally applied resultant moment acting throughout Region 1, and since this is zero, the constant b_ ] is real. Before proceeding to the evaluation of these potential functions for the two specific geometries when the elliptical inclusion is either very thin or almost circular, we obtain explicit expressions for the coefficients for the case when the Burger's vector 3' is in the y-direction, 3' = i h,,, and z o is on the x-axis. Then we note from (12) that K = K is real, and A , , B , / ( ~ o - 1/~ o) are real since ~'0 is either real or located on the unit circle. Under these conditions, the a _ , , b , , c., d. are all real and (10) and (11) provide
('-F-KKI+']-)"(-I"---~' +a-2"
a_n=
~
( a 2 ~
K
a
1) 2a-2('+1'
B"-I
2nil
1 + (F~1 + 1) a
n "-(a2-1)a-2"A"
/
(17)
j'
2n]A n
"
+(l_a_Z)A,_a2(l_a_2)
2
B,,_, (~'0- 1/~'0)
/
for n = 1, 2, ..., and where ~,,- (r 2+F)
(1- r)
n ( a 4 _ 1)a_20,+1,
(x2--FKl)
(1 +
-4,,
(18)
F~:,) a
The b , , and d,, may then be obtained from a_,. and c, of (17) by using (13), (14) and (15). However, it is much easier to sum the respective b_,, and d,, series in terms of the a , , and c,, than to actually evaluate the individual terms. For example, the series involving the b_,, terms of (5) takes the form ~'
Eb
7o
,,~.-,, = bo +
(~:2 + 1 ) oo,
(l-F)
E'a2nCn~-n+ (: + aZ(~'-
,,=1
OO
E
(19)
1/~ ') n=]
A similar result holds for the series involving the d, terms of (6). We now consider in detail two cases where the elliptical inclusion is either very thin with c ~ 0, a - ~ 1, or where the elliptical inclusion is almost circular with c ~ 1, a -* 0o. For the latter case, the solution reduces in the limit to the solution for a circular inclusion obtained by Dundurs and Sendeckyj (1965).
3. Thin elliptical inclusion
In this section we consider the case where the elliptical inclusion is very thin corresponding to c << 1 and a ---, 1. We take the dislocation to lie along the real axis so that z o is real, and the Burger's vector is in the y
324
W.E. Warren / Edge dislocation inside inclusion
direction so that -/= i by. Then all coefficients a _ . , b_ n, c., dn for n = 1, 2, ... are real, with the a_,,, c,, given by (17). Expanding these coefficients in powers of c and retaining terms up to first-order provides
(l-r) a_2,,[1X . -
c . = - M ( K 2+ 1)
,(N~
-
2)A,,],
(20)
for n = 1, 2, ..., and where
M = G , b , , / { ~ ( K , + 1)}, No = 4 [ ( g 2 - 1)-F(K,- 1)] (1 - r ) , F(K 2 + I)(~, + I)
Nl
+
+ i)
i)] (i +
FK,).
(21)
Evaluation of the potential functions for Region 1 using the coefficients of (20) provide, after transferring to the z-plane ---"wm~ "0 - -'o real, cb,(z) = M [ l n ( z - Xo) + o N o
~,(z)=M(ln(z-xo)
V/z2 -- 4R2 1
(z-xO) ]"
Xo
(Z-Xo)
, [ x0¢z 2 - 4R2
+ cN 1
~/z 2 - 4R ~
(22)
(Z-Xo)
4R2(1 + 2¢ 2) Q
Of particular interest to the investigation of craze growth in glassy polymers is the nature of the stress field at the tip of the thin elliptical inclusion. To investigate this we now establish a polar coordinate system in the z-plane with origin at the fight-hand foci z = 2R. Accordingly, we set z = 2R + re iO, r << R, and introduce a scale factor ro equal to the minimum distance from the foci to the material interface. In terms of the parameter c, the minimum distance ro and the radius of curvature rc at the tip of the inclusion are
"-° = , , ( i + R
+ . . . ),
r~=2c2(1+½( 2+ "") R
(23)
and we note lhat for the thin ellipse, rc = 2ro. The stress field around the tip then becomes Oyy
(! ~ x 0) 1 + No
cos 0 / 2 , (24)
o Jy' ) - - o;~ ( ' ) + 2 i o,(, 'y)--' ( l _ 2x M o)
{( N , - N o )
~ r ° e-i#/2 + N o ~ r ° e-i3°/2 [2(-~) + i sin 0]},
where I --- 2R is the half craze length, and x 0 < I. This stress field has the same form as the stress field at the tip of a simple craze in a glassy polymer subjected to a uniform stress in the far field as obtained by Warren (1983). In particular, we note that since r >/r 0, the stress field remains bounded around the craze tip. Figs. 2 and 3 show typical contour lines of constant dimensionless hydrostatic stress S = ,tr(! - xo)S/Glby and constant dimensionless maximum shear stress :r = ,rr(l - Xo)'r/Glby around the craze tip, where S is the hydrostatic stress and ~- the maximum shear stress. In obtaining these results, we have taken G~/G2 = 4, v~ = -~, v2 = 0 which are material properties representative of crazes in glassy polymers.
W.E. Warren/ Edgedislocationinsideinclusion
325
The hydrostatic stress S in Region 1 for conditions of plane strain is given by S =-~(1 + lq)(o.:?~) + O.v(yl'),
(25)
while the m a x i m u m shear stress ~- in this region is given by
~.__½{(_(t) _(,,±T 2io(tv, _,t, - o.~.~ _,t, - 2:_(,,)} '/2. o>,), - "x.~ . )( "YV _,U.~y
(26)
These two quantities are of interest since they relate to numerous proposed criteria for craze growth in glassy polymers. Figs. 2 a n d 3 s h o w the highly localized nature of the stress e n h a n c e m e n t at the craze tip. The m a x i m u m values of S and T occur at the tip r = r0, 0 = 0, and for this example,
Smax = 1.53,
"rmax =
1.75.
Along the craze extension line, the shear stress • decays much more rapidly than S although it spreads out considerably in directions normal to the craze extension line. In certain polymers this condition may assist in the formation of shear bands which have been observed running at - 45 ° to the plane of the craze. In the absence of a craze, the stresses at the point corresponding to the craze tip x = ! for a dislocation located at x 0 < 1 are S = 0.55 and ~r= 0, so the stress enhancing effect of the craze is significant. The thin elliptical interface occurring at r = p(0) is defined by m
P( g ) = [c°s2 0/2] -'[ 1 - ~'2 ( 1 - c°s O)
70
-6 ¥
]
+
(27)
and suppressing terms of order c 2, the stress c o m b i n a t i o n s of (24) at the b o u n d a r y b e c o m e
[_(~) ± _(1)1 4M ox.~ T%.y l h - ( I - - x 0 ) [ 1
+ ½N0(1 + cos O)],
(28) _it) _(]) . (,) 2M [No( 1 + c o s 0 ) e _ i O + ½ ( N l _ N o ) ( 1 + e _ i a ) ] " 0,'.,' - Oxx + 2,O~y ] b - ( l _ xo)
CRAZE
r/~ 0
5
10
15
20
C
R
25
Fig. 2. Lines of constant dimensionless hydrostatic stress, g = [{ ~ ( I - Xo)}/(Glby)lS.
A O
~ 5
r/ro
10
15
Fig. 3. Lines of constant dimensionless maximum shear stress,
-- {~(i- Xo)}/(ciby),.
326
W.E. Warren / Edge dislocation inside inclusion
Fig. 4 shows the dimensionless stresses S and ~ around the boundary and we note here again that the shear stress ~"is more localized than the hydrostatic stress S at the craze tip. The potential functions for the inclusion Region 2 for this case are not uniformly convergent for small c and for all z in Region 2. These potential functions contain singular points in Region 1 which collapse down into Region 2 in the limit as c--* 0. For z sufficiently removed from x o, these potential functions provide
02(')- (~2M+ ]) [2 +(~,- ~)r] ln(z-xo) +O(t2),
M ( Xo / +o(C), ¢2(z)-(r2+1) [2~2-(K'-l)r]In(z-x°)-[2+(K'-l)r](z-xo)
(29)
which cannot be considered valid for all z in Region 2. In a neighborhood of the craze tip the stresses within the inclusion are effectively those due to a resultant force
4Glby X2 +iY2= - (K1 + 1)(K 2 + 1 ) [ ( K 2 - 1 ) - r ( , , -
])1
(30)
and a dislocation with complex Burger's vector lby, where
b. = by{l + 2 [(r2- r~,) + r2(1- r)] I -
r ( ~ , + I)(K 2 + i)
(31)
'
both acting at x 0 within the inclusion. The stresses near the craze tip are uniform and given by 4M
[2 + ( ~ , - 1 ) r l °~' + °Y(~)= ( l - xo ) (K2+ 1) (32)
and --
(2,
4M (i-xo)
_ ( 2 ) _ _ ( 2 ) 4. 2 1 0 x v =
%,y
~'x~
[(~2- 1)-(~,(~2 + 1)
1)r]
and satisfy the appropriate boundary conditions to first order in c around the craze tip. 2.5
I
I
I
I
lr(£-Xo) s
2.0
G I by
m Ill i-m
'a'(J~- Xo)
' - ""-1.5
m Ill _J Z _o 1.o
.
"
Z III
g Q
%
0.5
%
0 0
I 30
I 60
| 90 ANGLE
Fig.
4.
Boundary stress combinations S and ~.
I 120
150
327
W.E. Warren / Edge dislocation inside inclusion
4. Almost circular inclusion We consider in this section the case where the elliptical inclusion is almost circular corresponding to large values of a, and c close to 1. The results we are after are most readily obtained by mapping the ~'-plane into a t-plane with a simple scale change, ~"= at. With this transformation, the unit circle in the ~'-plane maps into a circle of radius 1 / a in the t-plane, and the interface boundary of radius a in the ~'-plane goes into the unit circle in the t-plane. The map from the z-plane to the t-plane then becomes =
=
Ro( +
(33)
where R 0 - R a , 0 <.%m = 1 / a < 1. The ratio of minor to major axis c is now given by c = (1 - m2)/(1 + m 2)
(34)
and m = 0 corresponds to the circular inclusion. We are interested in the solution for small m (large a), and * andc,,-~ c,,, * wherea*,,, for large a and n, the coefficients a ,, -~ a_,, _ c,* are given by (1¢2 + 1 )
1
a*-" = - K (r~, + 1) n A ' ' ¢1
a2"c * =
'"
P~
(35)
[ 1
_
*' In
- K ( K2 +'1"3
--A. +
(1
- a-
_
2)A.
(a2
a2
1)2
T) o)
"
of a _ . . c,, respectively in the series terms for n > N. where N is some integer, the Using a * . and c.* in vlace . potential functions for Region 1 become *l(t)=r
(K 2 + 1) {
(/'1¢ 1 + 1) l n [ ( t - t ° ) ( 1 - m 2 / ~ t 5 ° ) ]
(1 -
r)
+ F(K, + 1)
ln )
N
(36)
+ ~ . ( a_,, -- a*,, ) r e " t - " + ao + O ( m 2N ), n=l
(~2 + 1) (
r)
ln[(~-t°)(1-
( g 2 -- /'K1)
m2/tt°)]+
F-(~+ij
(1 - m2)(1 - t 2 ) lnt+t(t-to)(1-m2/tto )
+ (l-r) (1 + FK,) t 2 ( 1 - m Z / t 2) ( t - to)(1 - m21tto ) r ( + 1) (~ + r)
(1 + m 2 t 2 )
(K 2 + 1) N+l + (i-r) E [a2"c.-a2"c*]m"t
t(1 - m21t 2 )
-"
n=l
(1 + m2t z) +t2(1-m2/t
N
Y'. n[a_,,-a*_.]m"U" + bo + O(m2N), z) .=,
(37)
where we have made use of (19). With this representation, the function ~ l ( t ) and ~t,~(t) may be evaluated to any accuracy desired. Similar expressions may be obtained for ~2(t), ~2(t) of Region 2. It can be readily shown that the series terms of (36) and (37) have the order relations (38) for n = 1, 2, ..., N. We note that for the circular inclusion, m = 0, the n = 1 term of (38) 2 contributes a
W.E. Warren / Edge dislocation inside inclusion
328
non-zero term to the solution. For small eccentricities from a circular inclusion, rn << 1, and retaining terms to oider m 2, the potential functions of (36) and (37) become
K((1- B)ln(~-~0)-
(
[
~,, (~) = K (1 - A-) ln(~ - ~o)
-(H-A-)
(H- B)In ~-m2(1-
,o
(/j 5 ~o ) -
B)(1 + Q~02)~-~/, so~ /
1
Q~o/~
l n ~ + f~(f~-f~o) (B-A-) +(H-B)-~ 1 -m2
-m2(1- A-")['o('2'o)-
(39)
~0 1 A--(3'~ ~ 2 ) + Q(~-3 + ~o~)]
~o2~(~_ ~o) +
~
-~o~: -+
Q
1+ ~
'
where -
F-
A - ~F ~+
1
-
'
B=
F~ - K2 FK! + 1 '
~2 + 1 (1 - H)= r(,, + 1)
2(1 - F) Q = ( 2 F + K2- 1)
(40)
are the material coefficients defined by Dundurs and Sendeckyj (1965) in their investigation of the circular inclusion. For the circular inclusion of radius R o, m = O, ~ = z / R o , ~o = x o / R o and (39) provides , , ( z ) = K { (1 - H) In( z - Xo) - ( H - B) In z },
(
[
ff,(z)=K (l-A-)ln(z-xo) -(H-A-)Iag+(H-A-)R~
(Z_Xo)
~(Z-~o)
Qz
+(H-B)
1
- R!)
(41)
,
which agree with the results of Dundurs and Sendeckyj (1965) who obtain the solution in terms of the real Airy stress function. The effect of small eccentricities of the inclusion shape on the resultant force acting on an edge dislocation at some point x 0 on the major axis of the inclusion and directed perpendicular to this axis may be obtained directly from these results. The strain energy due to the dislocation is equal to the work done in forming the dislocation. The displacement along the dislocation is just the Burger's vector "t = bx + ib.v, and the total force on the dislocation line may be obtained directly from (4). The strain energy W is given by W = R e { ~ r' ~ ,"~ - i Y )} ,
(42)
where X - iF is the total force along the dislocation discontinuity running through Regions 1 and 2. Thus ~_.,,
,o'(~)
j~=, (43)
and from the boundary condition (8), the values of these terms cancel each other at ~ = 1. It remains only to evaluate the first bracketed term of (43) as ~--, N, a large constant, and the second term at the
W.E. Warren / Edge dislocation inside inclusion
329
dislocation core radius 8, or at ~--4o for terms which remain bounded at ~--40- Carrying out this evaluation, the strain energy due to the dislocation is G2bf ( 2 1 n ( N / 8 ) _ 2 H ln N + ( , 4 + B ) l n ( l _ /~2)(l _ m 4 / ~ 2 ) W - - 2,lr(1¢2 + 1) -- 2A-"Q(~o + m 2 / ~ o ) 2 + 9m2A2(~o + m 4 / ~ o ) 4
+ m212A- - 12A-'z + (1 + 3A-- 2 B ) Q ] ( 4o + m2/~o )2 + Coast.},
(44)
which is real since, for z o real, 4o is either real or on the circle of radius m. In t e ~ s of z o = x o, the strain energy W becomes . 2 C2b;
W = 2~r(1¢2 + 1) { 2 l n ( N / 8 ) - 2 H In N + Coast. + ( A-+ B ) In[ (1 + m 2 )2 _ x ~ / R 2 ] - 2 A Q x 2 o / R 2 + 9m2A'Zx4/R~ + m 2 1 2 A - 12A-2 +(1 + 3 ~ , 4 - 2 B ) Q ] x 2 / R 2 } .
(45)
The force F acting on the dislocation is defined as the negative gradient with respect to position of the dislocation of the elastic strain energy W. Thus, F = - a W / a x o,
(46)
and direct evaluation from (45) shows that
F=
G b;Xo
2(1,R0 .
.
.
[('.-1+.
(
. 2-
x /R o]
+ 2iQ-
-m212A---12A'2 + ( 1 + 3A---2B)Q] }.
A x o2/ R 2o 18 m 2,-2
(47)
The force on the dislocation becomes arbitrarily large as x o approaches the interface at the ellipse n, ajor axis R0(1 + m2). As in the circular case m = 0, the center of the ellipse x 0 = 0 is an equilibrium position and the stability of this equilibrium position for the circular case is discussed extensively by Dundurs and Sendeckyj (1965). For a given eccentricity m, the force expression of (47) indicates the possible presence of two other equilibrium positions in the interval 0 < x o < R0(1 + m2), one of which may occur in the circular case and has been found to be unstable if it exists. The effect of small eccentricities on the equilibrium positions for particular material combinations may be investigated from these results. Expressions for the force on a dislocation with the more general Burger's vector , / = bx + ib,, located at an arbitrary point z o within the inclusion may be obtained from the exact solution presented earlier, but such an investigation is beyond the scope of the present work.
References Bilby, B.A., A.H. Cottrell and K.H. Swinden (1963), "The spread of plastic yield from a notch", Proc. Roy. Soc. Ser. A272, 304. Dundurs, J. (1969), "Elastic interactions of dislocations with inclusions", in: T. Mura, ed., Mathematical Theory of Dislocations, ASME, 70.
Dundurs, J. and G.P. Sendeckyj (1965), "'Edge dislocation inside a circular inclusion", J. Mech. Phys. Solids 13, 141. Hirth, J.P. and J. Lothe (1968), Theory of Dislocations, McGraw-Hill, New York. Kambour, R.P. (1973), "A review of crazing and fracture in thermoplastics", J. Polymer Sci.: Macromolecular Rev. 7, !. Kramer, E.J. (1979), "Environmental cracking of polymers", in E.H. Andrews, ed., : Developments in Polymer Fracture 1, Applied Science Pub., London, .~.
330
W.E. Warren / Edge dislocation inside inclusion
Mura, T. (1969), "Method of continuously distributed dislocations", in: T. Mura, ed., Mathematical Theory of Dislocations, ASME, 25. Muskhelishvili, N.I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, 3rd ed., Noordhoff, Groningen.
Nabarro, F.R.N. (1967), Theory of Crystal Dislocations, Oxford University Press, London. Warren, W.E. (1983), "The stress and displacement fields at the tip of crazes in glassy polymers", Polymer., to appear.