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The distribution of dislocations in two pile-ups, arranged one above the other
ht.
J. Engng
THE
Sci.
Vol.4,pp. 451-461. Pergamon Press 1966. Printed in Great Britain
DISTRIBUTION OF DISLOCATIONS IN TWO PILE-UPS, ARRANGED ONE ABOVE THE OTHER E. SMITH Central Electricity Research Laboratories, (Communicated
Leatherhead,
Surrey
by B. A. BILBY)
Abstract-The theory of continuous distributions of dislocations is used to consider the interaction between two identical non-coplanar dislocation pile-ups. The distribution function is determined for widely spaced pile-ups using a series solution approach, and it is shown that interaction becomes important when the spacing between them is approximately equal to the total length of each pile-up. Results are significantly different for screw and edge dislocations; thus whereas the number of screw dislocations in each half of a pile-up decreases as the pile-ups become closer, the n~ber of edges increases. The stresses in front of the pile-ups are determined, and these results are used in a brief discussion of the validity of the isolated slip band model for yiefd propa~tion in polyc~s~~ine material. 1. INTPODIJCTION
PROBLEMS involving distributions
of dislocations in linear arrays have frequently been solved, using an approach whereby the discrete dislocations in a plane are smeared out into a continuous distribution of infinitesimal dislocations, in a manner o~~nally outlined by Eshelby [l] and Leibfried [2]; each problem thus reduces to the solution of a singular integral equation which expresses the requirement that each dislocation in the distribution is in equilibrium. If the resistance to motion of the dislocations is zero, the smeared-out arrays can represent cracks, and problems involving these have also been discussed using other methods such as, for example, dual integral equation techniques. Considerable attention has been given to problems where the dislocations all he in one plane, and thus coplanar crack problems have been studied, but it is only quite recently that successful attempts have been made to solve problems where the dislocations are not coplanar. Unfortunately, unlike the coplanar case, it is generally not possible to obtain an exact solution for the dislocation distribution function when the dislocations are noncoplanar, and approximate methods must be used. The simplest situation to study 13-61, which involves dist~butions of non-coplanar dislocations, is that of an infinite body, subject to an externally applied shear stressp2s =o, at infinity, containing identical pile-ups of screw dislocations on an infinite sequence of planes with a constant distance of separation h; positive and negative screw dislocations are parallel to the xa axis and lie respectively in the intervals 0~ x1 6 c, x2 = +nh (n = 0, 1,2,etc.)and -c
4.52
E. SMITH
parallel to the x2 axis and the applied stress system is a tensile stress pzz =ar: the model then corresponds to a series of cracks under tension. The results for the preceding shear-type models are appreciably different for the screw and edge cases; thus whereas the number of screw dislocations in each half of any one of the pile-ups decreases with the spacing h of the pile-ups, the corresponding number of edge dislocations increases. The shear-type models involve an infinite sequence of pile-ups constructed from infinite walls of either edge or screw dislocations; these walis have wellknown special characteristics and because of this it would appear desirable also to study the behaviour of a finite number of non-coplanar pile-ups, as such a model is probably more relevant in discussing some physical applications. Thus the present paper considers the behaviour of two identical non-coplanar dislocation pile-ups directly above each other and at a distance h apart. Both screw and edge shear-type models are discussed; the correspon~n~ tensile crack problem has been dealt with elsewhere [9]. The continuous distribution approach is used for both problems and the appropriate singular integral equations are solved for widely spaced pile-ups using the series solution approach outlined earlier [6]; this method is particularly useful as the primary concern is in assessing when interaction effects become important. The associated dislocation distribution functions are determined; these give the number of dislocations in each half of one of the pile-ups and also the shear stresses in front of the pile-ups. These latter results are used in a brief discussion of the problem of yield propagation in polycrystalline materials, when the conditions are defined under which the isolated slip band model for yielding is valid. 2. THE
SCREW
DISLOCATION
PROBLEM
Consider an infinite body, subject to an externally applied shear stress pz3 =cA at infinity, and containing pile-ups of screw dislocations on two parallel planes a distance h apart (Fig. l), such that the body deforms in an anti-plane strain mode, the displacement us (x,, x2) being parallel to the x3 axis which is the outward normal to the plane of the figure. The positive and negative screw dislocations, using the RH/FS convention [lo], are Pzs=Q,,AT
00
x2 s 7
I P1,=U,, AT FIG.
co
1. Two screw dislocation pile-ups. @ positive screw dislocations: @ negative screw disl~tions.
The distribution
of dislocations in two pile-ups, arranged one above the other
4.53
parallel to the xa axis and lie respectively in the intervals 0~ x1 < c, x2=&h/2, and - c < x1 < 0, x2 = f h/2, and it is supposed that the resistance to motion of these dislocations is due to a friction stress rrl < cA. The dislocations in each array are now smeared out into a continuous surface distribution of infinitesimal dislocations, in a manner originally outlined by Eshelby [l] and Leibfried [2]; thus since the distribution of dislocations will be the same in each of the planes .Y~= Jr h/2, it is supposed that there arefA(X1)bX1 dislocations each of Burgers vector b>O in any interval 6x, so that with the RH/FS convention fA(xl) is positive for right-hand screw dislocations. To determine~~(~~) it is only necessary to consider the equi~brium of the dislocations in one of the arrays, and since the shear stresspsa due to an isolated positive screw dislocation situated at (A,, A,) and with a Burgers vector b is [l 11 Gb
p23=G*
[(x1-a$;2:~-n2)2]
at a point (xi, x2) where G is the shear modulus, the integral equation which expresses the requirement that the resultant shear stress on any dislocation in the distribution is zero, can be obtained by summing the effects due to all the other dislocations, and is
(x1-4) [(x,-n~)2+h21
(2)
for feel& c, where it is understood that the Cauchy principal value of the singular integral is used. Introducing the dimensionless variables x=x1/c, the form A(& k) where k = c/h, then (2) becomes
+ s
1 =A&, and noting that fA(A,) is of
’ A(& k)dA + + 1 k2(x - A))A(k,k)dd
-1
(X-A)
f _ 1 [k2(X -
ay+1]
271 +Fb(Q-O’)=O
for 1x1~ 1. For two widely spaced arrays, k is small and A(,%,k) can be expressed as a power series in k, i.e. (4) where the A,(A) are functions of A. Moreover since k2(x -A) =k2(x-jZ)-k4(x-;1)3+O(k6) [kZ(x--12)2+ 1-j substitution in equation (3) and equating coe~~ents
o=
s
+ ’ Ao(A)dA 2n -+$---fa,-~1) -I (X-A) Gb
of k”-k4 gives
(5)
454
E. SMITH
s
o= +‘Ar(l)dk -1 (x--I*)
Integral equations of the type
+ s
l4(;l>dn
_l(x+*(x)=O
(7)
for 1x1< 1, have been discussed extensively by Muskhelishvili [ 121and Head and Louat [ 131. For our particular problem, the solution is $(A)=-
s
+‘(l-x’)*$(x)dx 7r2(1- n2)* _ 1 (x-4
l
where K is a constant that has to be determined. noting thatf,(l,) is an odd function of rZI, then A,(l)=
W,4--o1) Gb *
A2(4=
39
K +(l --P)+
Applying this result to equations (6) and
J.
(1 -n2y
*&
A1(I)=O
A,(A)=0
(9) whereupon the dislocation distribution function is given by
ml)=
%,-ad Gb
,
Xl
l-
(C2_x2)*
[
4 *-C4 $ - $ 1 +3(o;;c1) - c2(c2x:p h4
The number of dislocations in each half of one of the planes x2 = + h/2 is nA= $x,)dx, s
(11)
The distribution of dislocations in two pile-ups, arranged one above the other
whereupon use of reiation (10) gives
This relation is shown in Fig. 2 (full curve) for values of h/c>2, and since nA must equal as h/c+O, the inferred behaviour for values of h/c<2 is shown by the dotted portion in Fig. 2. The stress pz3 at a point (_Y~,h/2) ahead of one of the pife-ups and along
((z~-cT&/GB
FIG. 2. The number rt~ of screw dislocations in each half of one of the pile-ups for different values of h/c. The full curve is obtained from equation (12) while the dotted curve shows the inferred behaviour for h/c < 2.
its plane is given by pz3=
s l
cGb -*-
-,2x
f(x,)dx, (13)
(x0--x1)
which, by virtue of equation (10) becomes
at a point a distance r ahead of the pile-up. 3. THE EDGE
DISLOCATION
PROBLEM
If the screw dislocations of the model in the previous section are replaced by edge dislocations with their Burgers vectors parallel to the xl axis and the external stress is altered to pr2 = bp, the new model is that of an infinite body containing two edge dislocation pile-ups, and the body deforms under plane strain conditions (Fig. 3). Positive and negative edge dislocations are parallel to the x3 axis and lie respectively in the intervals 0 < x1 < c, x2 = &h/2, and -c< x1 ,< 0, x2 = rt h/2; with the RH/FS invention, positive edge dislocations have their associated extra half planes in the positive x2 direction. It is again supposed that the resistance to motion of the dislocations is due to a friction stress c1 CCF~. As in section 2, the dislocations can be smeared out into two surface distributions of infinitesimal dislocations, and since the distribution will be the same in each of the planes x2 = + h/2, it is supposed that there are fp(xl)Sxl dislocations each of Burgers vector b > 0 in any interval &xi, so that with the RH/FS convention~~(~~) is positive for positive edge
456
E. SMITH
P,*= U,, AT 00
FIG. 3. The edge dislocation pile-ups. _I_positive edge dislocations; T negative edge dislocations.
dislocations. Since the shear stress plz due to an isolated positive edge dislocation situated at (A,, A,) and with a Burgers vector b is [l l]
-
Gb
. (x~-~~)c(x~-~~)2-(x2-~2)21
P==27r(l -V)
(15)
Kxl-~l)2+t~2--2)232
at a point (x1, x2) where v is Poisson’s ratio, the integral equation which determines the equilibrium distribution of the dislocations is
for Ix, 16 c, and it is understood that the Cauchy principal value of the singular integral is used. Introducing, as before, the dimensionless variables x=x1/c, A=2,/c, and noting that f&) is of the form P(1, k) where k =c/h, then equation (16) becomes
+ s
1P(1, k)dl
-1
(X-A)
+ ’ k’(x - Iz)[k2(x - 2,)’ - l]P(A, k)dl+ 27r(l- v) -(% [k2(x-A)2+1]2 Gb + s -1
- CT,)=0
(17)
for lx11< 1. When the arrays are widely spaced, k is small and P(I, k) can be expressed as a power series in k, i.e.
where the P,,(A) are functions of A. Moreover since k2(x - A)[k*(x - A)2 - l] [k*(x-A)*
+ l-J2
= - k2(x -A) + 3k4(x - A)’ + 0(k6)
The distribution
of dislocationsin two pile-ups,arranged one above
the
other
457
substitution in equation (17) and equating coefbcients of k”-k4 gives
l P&da -1 (x-2) J +
I&
o=
+ 2x(1 - v) -
Gb
((Tp-od
l P,(a)dn
+
____1
J
-1
b-4
for IX/< 1. Following the lines of the previous section, and noting that function of AI, then P*(A)=
2(1 -Y)(+-(Tr) Gh
P4(4=
ll(l -v)((Tp-o~) 4Gh
.-
a (1 --nz)+’
a. q (1 -- -A2)*
f,&) is an odd
P,(A)=0
9(1-$0(,-a,) Gb
A3 l
(f
whereupon the dislocation distributia~ function is given by
The number of dislocations in each half of one of the planes x2 = &h/2 is np= whereupon use of relation (22) gives n
J
(23)
3--vXap-Cl) P
Gb
4
(24)
This relation is shown in Fig* 3 (full curve) for vahres of ftjo3; since nP must equal o,)c/Gb as +-PO, the behaviour for values of h/c<3 should follow a similar 0 -v)(cpform to that indicated by the dotted portion in Fig. 3. The stress plz at a point a distance r ahead of one of the pile-ups and along its plane is
458
E.
SMITH
(25)
4. DISCUSSION III the preceding sections we have been concerned with the interaction between two dislocation pile-ups, and have given particular consideration to the case, when the pile-ups are widely spaced. The problems have been treated in terms of the theory of continuous distributions of dislocations and solved using a series solution approach that has been applied previously to other models. Thus expressions for the dislocation distribution function, the number of dislocations in each half of a pile-up, and the stress ahead of a pile-up have been obtained for h/c32 or 3, where 2c is the total length of each pile-up and h is the spacing between them. The series solution approach is ideal for such h/c values but breaks down for lower values, although it is then possible to infer the magnitude of the number of dislocations in each half of a pile-up (Figs. 2 and 4), using the calculated behaviour for h/c>,2 and the obvious value for h/c-+0. For the screw problem it is in fact
FIG. 4. The number np of edge dislocations in each half of one of the pile-ups for different values of h/c. The full curve is obtained from equation (24) while the dotted curve represents the anticipated form for h/c < 3.
possible to obtain a solution for all h/c by treating the problem as one in potential theory; since for antiplane strain deformation the displacement u3 satisfies Laplace’s equation V’us =O, the region external to the H-shaped figure, formed by the two pile-ups and the portion of the x2 axis joining them, can be transformed into the region external to a rectangle [14]. Thus it is possible to derive equations from which u1 can be obtained for all h/c; however, these are complex and since we are primarily interested in the behaviour for large h/c, the series solution approach is preferred in the present paper, particularly as the same method has been applied to the corresponding edge dislocation problem, which cannot of course be treated from the potential theory view-point. The general conclusion that emerges from the present analysis is that the pile-ups
The distribution
of dislocations
in two pile-ups, arranged one above the other
459
interfere with each other when their spacing is about equal to their length; then for exampIe there is approximately a 10 per cent change in the number of dislocations in each half of a pile-up. When the pile-up spacing becomes smaller, these changes become more marked and interaction between the pile-ups becomes appreciabIe. There is a significant difference between the behaviour of screw and edge dislocations, thus whereas the number of screws in each half of a pile-up decreases as the pile-ups become closer, the number of edges at first increases. This behaviour is different to that of coplanar pile-ups where edges and screws act in a similar manner, but is in accord with that of an infinite array of non-coplanar pile-ups [6]. For this latter model the relations corresponding to (10) and (22) are
(26) and f(
P Xl
)=w-VMP--01) *
(27)
Gb
where in these cases h is the spacing between neighbouring pile-ups in the infinite sequence. For large c/Ii, therefore, the characteristics of the two pile-ups and infinite sequence models are essentially similar. This is not the case, however, as h-+0; thus for example the number of screw dislocations in each half of a pile-up tends to the finite limit (0;~- o,)c/Gb when there are two pile-ups and to oh/i% for an infinite sequence [.5, 61. With the two pile-up model there is also a difference in the behaviour of screw and edge dislocations with respect to the shear stress in front of a pile-up; thus the appropriate stress decreases for screws (equation (14)) whereas it increases for edges (equation (25)). Now isoiated pile-up modefs have been used to explain the grain size dependency of the lower yield stress of metals and alloys. At this stress a Luders band propagates down a specimen, and experimental results have shown that its value follows a relationship of the form : z=r,+kZ-+
(28)
where 21 is the grain diameter and z. and k are constants for a given system of testing conditions. The form of equation (28) is consistent with the spread of plasticity occurring by the concentrated shear stress ahead of a slip band in one grain initiating plastic flow in a neighbouring grain. Theoretical considerations [15-17J of this problem have used a model of an isolated slip band; if z is the applied shear stress acting on the band, the shear stress at a distance r directly ahead of it is (see equations (14) and (25)) (z--z1
0 ;
*
(2%
where z1 is the lattice friction stress opposing the motion of the dislocations in the band. Thus, if a dislocation source is situated at the point I and a critical shear stress z, is required to operate it, then the criterion for yield propagation is (z--zl) or
1 *
0 2r
=z,
(301
2==ZI+(2r)*z,l-”
(31)
440
E. SMITH
and hence in the interpretation of experimental results, rO and k in relation (28) are associated respectively with tr and (2r)%,. The analysis of the present paper suggests that the results obtained from the isolated slip band model are probably quite adequate provided that the yield stress is not controlled by a deformation process involving the operation of more than, say, one slip band per grain. Thus, if the region at the front of a Luders band consists of a few grains with the occasional slip band in each, it is satisfactory to use the isolated slip band model; however, if slip occurs on more than one slip plane then this fact should be taken into account. The edge dislocation model is probably more suitable for discussing such a situation and equation (25) suggests that ex~rimentaliy measured k values will be lower than when the slip bands act in isolation. However, there is little ex~rime~tal evidence to guide us at this stage, as very few observations have been recorded of the detailed processes operating at a Luders band front. Finally, it is worth mentioning that although pzz is zero along each pile-up for the screw model and so the present analysis (with cur-0) can be used for two cracks subject to an applied stress pZ3, the same is not true for the edge model; pzz is non-zero in this latter situation and the cracks will open up. To solve this problem it is necessary to insert an additional distributioI1 of edge dislocations with their Burgers vectors parallel to the x2 axis, and then formulate equations which give zero p12 and p2. stresses across each crack face. ~ck~ow~e~~~a~-This published by ~rmission
work was conducted at the Central Eiectricity of the Central Electricity Generating Board.
Research
Laboratories,
and is
REFERENCES [I] [2] [3] [4] [SJ [6] [7l
J. D. ES3ELBY, Phil. Afag, 40,903 (1949). G. LEIBFRIED, Z. Phrys. 130,214 (1951). G. I. BARENBLATFand G. P. ~E~PANOV, Appl. auto. Mech, 25, 1654 (1961). N. LOUAT,PM Msg. 8, 1219 (1963). E. SMITH,Phi% Mag. 9, 879 (1964). E. SM~H, Proc. R. foe. A 282,422 (1964). Philadelphia, Society of Industrial and Applied W. T. KOITER, Problems qf Cantinwm Me&u&s. Mathematics, 246 (1961). [g] A. II. ENGLAND and A. E. GREEN, Proc. Cumb. Phil. Sot. 59,489 (1963). [9] E. SMlTH,Int. J. Ei?g??gSci. 4, 41 (1966). [lo] B. A. BILBY,R. BULLOUGH and E. SMITH,Proc. R. Sot. A 231,263 (1955). 111J A. H. CO~ELL, ~jslocat~~ns and Plastic Flow in Crystals. Oxford University Press (1953). 114 N. I. ~USKHELI~H~~I, Singa~ar Integral ~~at~oas (translated by J. M. Radok). P. Noor~off, N. V. (1953)‘ [13] A. K. HEAD and N. LOUT, Amt. J. Pkys. 8, 1 (1955). [14] C. DARWIN, Phil. Msg. 41, 1 (1950). [lS] N. J. FEKX, J. Iron Steel Inst. 174, 25 (1953). 1161 A. H. COTTRELL,Trans. Am. Inst. i34in. ~etall~ Ergrs 212, 192 (19%). [17] 1. Conr,and N. J. PETCH,Phi/. &fug.5,30 (196% (Received 8 November 1965f R&n&--L’auteur utilise la theorie de Ia distribution continue des dislocations pour Ctudier I’interaction entre deux empilages de dislocations, identiques, mais ne se trouvent pas dans un m&ne plan. La fonction de distribution est d&em&% pour des empilages tr&s espac& en utilisant nne solution d’approache par series, et ii est montr6 que l’interaction devient important lorsque I’intervalle, qui les &pare, devient ~~iblement Cgal B la fongueur totale de chaque empilage. Les r&ultats sont tr& dif%rents en ce qui conceme les dislocations par rotation et les dislocations sur les bords. C’est ainsi que le nombre des dislocations par rotation dans chaque moitiit d’un empiIage, ditcroit lorsque l’empilage se ressert tandis que ie nombre des dislocations sur les bords augmente.
The Les contraintes
distribution
of dislocations
sur le front
Ctudier la validitd du modele materiau poly-cristallin.
des empilages
d’un
plan
in two pile-ups, sont
de glissement
arranged
determinees isole
dans
one above
par l’auteur
et il utilise
la propagation
461
the other ces n+sultats
de l’t?coulement
pour
dans
un
Zusammenfassung-Die Theorie der kontinuierlichen Verteilung von Versetzungen wird henutzt, urn die Wechselwirkung zwischen zwei identischen nicht-coplanaren Versetzungsschichtungen zu betrachten. Die Verteilungsfunktion wird ftir weitrlumige Schichtungen unter Verwendung einer Seriennaherungsmethode hestimmt, und es wird gezeigt, dass die Wechselwirkung an EinUuss gewinnt, wenn der Abstand zwischen ihnen ungefahr gleich der Gesamtlange jeder der beiden Schichtungen ist. Die Ergebnisse sind deutlich verschieden fiir Schrauhenund fur Stufenversetzungen; wahrend die Zahl der Schraubenversetzungen in jeder Hllfte einer Schichtung abnimmt, wenn die Schichtungen enger werden, nimmt die Zahl der Stufenversetzungen zu. Die Spannungen vor den Schichtungen werden bestimmt, und diese Ergebnisse werden in einer kurzen Besprechung der Gtiltigkeit des isolierten Gleitbandmodells fur die Fliessausbreitung in polykristallinem Material verwendet.
Sonunario-Si fa ricorso alla teoria della distribuzione continua delle dislocazioni per studiare l’azione reciproca fra due scontri non coplanari identici di dislocazione. La funzione della distribuzione e determinata per accatastamenti a notevole distanza impiegando un’impostazione di soluzione di serie ed i: dimostrato the l’azione reciproca assume un carattere importante quando la distanza fra di loro I: approssimativamente identica alla lunghezza totale di ciascun accatastamento. I risultati sono significativamente dalle dislocazioni di vite e bordo; cosi, mentre il numero di dislocazioni di vite in ciascuna meta
differenti
di accatastamento diminuisce con I’awicinarsi dell’accatastamento, il numero di bordi aumenta. Si determinano le sollecitazioni di fronte agli accatastamenti e i risultati vengono impiegati in una breve discussione sulla validita de1 modello a banda di slittamento per la propagazione de1 carico di snervamento in materiali policristallini. TeOpBs AIICIIOKaIIEU npBMeHaeTCs KpaCCMOTpeHUFJ B3aBMOBetiCTBIBI MeXJIy ABYMB OHEIAaKoBbIbIB IIeKOMIIJIaHapHblMB CKOIIJIeBBsMB ,mIcnoKaIIBB. B CnyYae 6OBbmero pacCTOKHIUI CKOMeHII& &HKIIBR paCIIpeAeJIeIIBs OIIpeAeJIReTCff C nOMOIIIbIO peIIIeHIUI B PsAaX; BOKa3bIBaeTCII 9YO B3aBMOAe&CTBEe BhIeeT 3Ha¶eHBe KOrAa paCCTOIIHBe MeXCAy CKOIIJIeIIIUIMII 6yAeT npn6nnmeBBo paBHo o6meti ,IumIIe Kamoro a3 HBX. Pe3yJIbTaTbI pa3HRTbC.K 3Ha’IBTeJIbHO AJI5I BBIITOBOt II KpaeBOt$ AHCJIOKilIWW. %CJIO BBHTOBbIX ,IIIICBOKauti B KaXCBOn nOJIOBBHe CKOMeHB5I yMeHbmaeTCff C npB6BmKeHBeM CKOnHeHti, a SIICJIO KpaeBbIX AIICJIOKaII& BO3paCTaeT. GITpeAeJIeHbI HaIIpSxeHIBI Ha @pOBTe CKOIIJIeHIUI; IIOJIy’IeHIIbIe pe3yJIbTaTbI IICnOJIb30BaHbI B KpaTKOit ABCKyCCBB 0 npBMeHseMOCTII MOAeIIB 830,-IBpoBaHIIOti CKoJIbbsIImefi IIOJIOCbl K paCIIpOCTpaHeIIIIKJ IIJIaCTIFIecKOti 30HbI B IIO,IBKpBCTa,IBIBIeCKIIX
~~LUUC’r--Konrnuyanbuarr
h4arepuaaJrax.
J. Engng
THE
Sci.
Vol.4,pp. 451-461. Pergamon Press 1966. Printed in Great Britain
DISTRIBUTION OF DISLOCATIONS IN TWO PILE-UPS, ARRANGED ONE ABOVE THE OTHER E. SMITH Central Electricity Research Laboratories, (Communicated
Leatherhead,
Surrey
by B. A. BILBY)
Abstract-The theory of continuous distributions of dislocations is used to consider the interaction between two identical non-coplanar dislocation pile-ups. The distribution function is determined for widely spaced pile-ups using a series solution approach, and it is shown that interaction becomes important when the spacing between them is approximately equal to the total length of each pile-up. Results are significantly different for screw and edge dislocations; thus whereas the number of screw dislocations in each half of a pile-up decreases as the pile-ups become closer, the n~ber of edges increases. The stresses in front of the pile-ups are determined, and these results are used in a brief discussion of the validity of the isolated slip band model for yiefd propa~tion in polyc~s~~ine material. 1. INTPODIJCTION
PROBLEMS involving distributions
of dislocations in linear arrays have frequently been solved, using an approach whereby the discrete dislocations in a plane are smeared out into a continuous distribution of infinitesimal dislocations, in a manner o~~nally outlined by Eshelby [l] and Leibfried [2]; each problem thus reduces to the solution of a singular integral equation which expresses the requirement that each dislocation in the distribution is in equilibrium. If the resistance to motion of the dislocations is zero, the smeared-out arrays can represent cracks, and problems involving these have also been discussed using other methods such as, for example, dual integral equation techniques. Considerable attention has been given to problems where the dislocations all he in one plane, and thus coplanar crack problems have been studied, but it is only quite recently that successful attempts have been made to solve problems where the dislocations are not coplanar. Unfortunately, unlike the coplanar case, it is generally not possible to obtain an exact solution for the dislocation distribution function when the dislocations are noncoplanar, and approximate methods must be used. The simplest situation to study 13-61, which involves dist~butions of non-coplanar dislocations, is that of an infinite body, subject to an externally applied shear stressp2s =o, at infinity, containing identical pile-ups of screw dislocations on an infinite sequence of planes with a constant distance of separation h; positive and negative screw dislocations are parallel to the xa axis and lie respectively in the intervals 0~ x1 6 c, x2 = +nh (n = 0, 1,2,etc.)and -c
4.52
E. SMITH
parallel to the x2 axis and the applied stress system is a tensile stress pzz =ar: the model then corresponds to a series of cracks under tension. The results for the preceding shear-type models are appreciably different for the screw and edge cases; thus whereas the number of screw dislocations in each half of any one of the pile-ups decreases with the spacing h of the pile-ups, the corresponding number of edge dislocations increases. The shear-type models involve an infinite sequence of pile-ups constructed from infinite walls of either edge or screw dislocations; these walis have wellknown special characteristics and because of this it would appear desirable also to study the behaviour of a finite number of non-coplanar pile-ups, as such a model is probably more relevant in discussing some physical applications. Thus the present paper considers the behaviour of two identical non-coplanar dislocation pile-ups directly above each other and at a distance h apart. Both screw and edge shear-type models are discussed; the correspon~n~ tensile crack problem has been dealt with elsewhere [9]. The continuous distribution approach is used for both problems and the appropriate singular integral equations are solved for widely spaced pile-ups using the series solution approach outlined earlier [6]; this method is particularly useful as the primary concern is in assessing when interaction effects become important. The associated dislocation distribution functions are determined; these give the number of dislocations in each half of one of the pile-ups and also the shear stresses in front of the pile-ups. These latter results are used in a brief discussion of the problem of yield propagation in polycrystalline materials, when the conditions are defined under which the isolated slip band model for yielding is valid. 2. THE
SCREW
DISLOCATION
PROBLEM
Consider an infinite body, subject to an externally applied shear stress pz3 =cA at infinity, and containing pile-ups of screw dislocations on two parallel planes a distance h apart (Fig. l), such that the body deforms in an anti-plane strain mode, the displacement us (x,, x2) being parallel to the x3 axis which is the outward normal to the plane of the figure. The positive and negative screw dislocations, using the RH/FS convention [lo], are Pzs=Q,,AT
00
x2 s 7
I P1,=U,, AT FIG.
co
1. Two screw dislocation pile-ups. @ positive screw dislocations: @ negative screw disl~tions.
The distribution
of dislocations in two pile-ups, arranged one above the other
4.53
parallel to the xa axis and lie respectively in the intervals 0~ x1 < c, x2=&h/2, and - c < x1 < 0, x2 = f h/2, and it is supposed that the resistance to motion of these dislocations is due to a friction stress rrl < cA. The dislocations in each array are now smeared out into a continuous surface distribution of infinitesimal dislocations, in a manner originally outlined by Eshelby [l] and Leibfried [2]; thus since the distribution of dislocations will be the same in each of the planes .Y~= Jr h/2, it is supposed that there arefA(X1)bX1 dislocations each of Burgers vector b>O in any interval 6x, so that with the RH/FS convention fA(xl) is positive for right-hand screw dislocations. To determine~~(~~) it is only necessary to consider the equi~brium of the dislocations in one of the arrays, and since the shear stresspsa due to an isolated positive screw dislocation situated at (A,, A,) and with a Burgers vector b is [l 11 Gb
p23=G*
[(x1-a$;2:~-n2)2]
at a point (xi, x2) where G is the shear modulus, the integral equation which expresses the requirement that the resultant shear stress on any dislocation in the distribution is zero, can be obtained by summing the effects due to all the other dislocations, and is
(x1-4) [(x,-n~)2+h21
(2)
for feel& c, where it is understood that the Cauchy principal value of the singular integral is used. Introducing the dimensionless variables x=x1/c, the form A(& k) where k = c/h, then (2) becomes
+ s
1 =A&, and noting that fA(A,) is of
’ A(& k)dA + + 1 k2(x - A))A(k,k)dd
-1
(X-A)
f _ 1 [k2(X -
ay+1]
271 +Fb(Q-O’)=O
for 1x1~ 1. For two widely spaced arrays, k is small and A(,%,k) can be expressed as a power series in k, i.e. (4) where the A,(A) are functions of A. Moreover since k2(x -A) =k2(x-jZ)-k4(x-;1)3+O(k6) [kZ(x--12)2+ 1-j substitution in equation (3) and equating coe~~ents
o=
s
+ ’ Ao(A)dA 2n -+$---fa,-~1) -I (X-A) Gb
of k”-k4 gives
(5)
454
E. SMITH
s
o= +‘Ar(l)dk -1 (x--I*)
Integral equations of the type
+ s
l4(;l>dn
_l(x+*(x)=O
(7)
for 1x1< 1, have been discussed extensively by Muskhelishvili [ 121and Head and Louat [ 131. For our particular problem, the solution is $(A)=-
s
+‘(l-x’)*$(x)dx 7r2(1- n2)* _ 1 (x-4
l
where K is a constant that has to be determined. noting thatf,(l,) is an odd function of rZI, then A,(l)=
W,4--o1) Gb *
A2(4=
39
K +(l --P)+
Applying this result to equations (6) and
J.
(1 -n2y
*&
A1(I)=O
A,(A)=0
(9) whereupon the dislocation distribution function is given by
ml)=
%,-ad Gb
,
Xl
l-
(C2_x2)*
[
4 *-C4 $ - $ 1 +3(o;;c1) - c2(c2x:p h4
The number of dislocations in each half of one of the planes x2 = + h/2 is nA= $x,)dx, s
(11)
The distribution of dislocations in two pile-ups, arranged one above the other
whereupon use of reiation (10) gives
This relation is shown in Fig. 2 (full curve) for values of h/c>2, and since nA must equal as h/c+O, the inferred behaviour for values of h/c<2 is shown by the dotted portion in Fig. 2. The stress pz3 at a point (_Y~,h/2) ahead of one of the pife-ups and along
((z~-cT&/GB
FIG. 2. The number rt~ of screw dislocations in each half of one of the pile-ups for different values of h/c. The full curve is obtained from equation (12) while the dotted curve shows the inferred behaviour for h/c < 2.
its plane is given by pz3=
s l
cGb -*-
-,2x
f(x,)dx, (13)
(x0--x1)
which, by virtue of equation (10) becomes
at a point a distance r ahead of the pile-up. 3. THE EDGE
DISLOCATION
PROBLEM
If the screw dislocations of the model in the previous section are replaced by edge dislocations with their Burgers vectors parallel to the xl axis and the external stress is altered to pr2 = bp, the new model is that of an infinite body containing two edge dislocation pile-ups, and the body deforms under plane strain conditions (Fig. 3). Positive and negative edge dislocations are parallel to the x3 axis and lie respectively in the intervals 0 < x1 < c, x2 = &h/2, and -c< x1 ,< 0, x2 = rt h/2; with the RH/FS invention, positive edge dislocations have their associated extra half planes in the positive x2 direction. It is again supposed that the resistance to motion of the dislocations is due to a friction stress c1 CCF~. As in section 2, the dislocations can be smeared out into two surface distributions of infinitesimal dislocations, and since the distribution will be the same in each of the planes x2 = + h/2, it is supposed that there are fp(xl)Sxl dislocations each of Burgers vector b > 0 in any interval &xi, so that with the RH/FS convention~~(~~) is positive for positive edge
456
E. SMITH
P,*= U,, AT 00
FIG. 3. The edge dislocation pile-ups. _I_positive edge dislocations; T negative edge dislocations.
dislocations. Since the shear stress plz due to an isolated positive edge dislocation situated at (A,, A,) and with a Burgers vector b is [l l]
-
Gb
. (x~-~~)c(x~-~~)2-(x2-~2)21
P==27r(l -V)
(15)
Kxl-~l)2+t~2--2)232
at a point (x1, x2) where v is Poisson’s ratio, the integral equation which determines the equilibrium distribution of the dislocations is
for Ix, 16 c, and it is understood that the Cauchy principal value of the singular integral is used. Introducing, as before, the dimensionless variables x=x1/c, A=2,/c, and noting that f&) is of the form P(1, k) where k =c/h, then equation (16) becomes
+ s
1P(1, k)dl
-1
(X-A)
+ ’ k’(x - Iz)[k2(x - 2,)’ - l]P(A, k)dl+ 27r(l- v) -(% [k2(x-A)2+1]2 Gb + s -1
- CT,)=0
(17)
for lx11< 1. When the arrays are widely spaced, k is small and P(I, k) can be expressed as a power series in k, i.e.
where the P,,(A) are functions of A. Moreover since k2(x - A)[k*(x - A)2 - l] [k*(x-A)*
+ l-J2
= - k2(x -A) + 3k4(x - A)’ + 0(k6)
The distribution
of dislocationsin two pile-ups,arranged one above
the
other
457
substitution in equation (17) and equating coefbcients of k”-k4 gives
l P&da -1 (x-2) J +
I&
o=
+ 2x(1 - v) -
Gb
((Tp-od
l P,(a)dn
+
____1
J
-1
b-4
for IX/< 1. Following the lines of the previous section, and noting that function of AI, then P*(A)=
2(1 -Y)(+-(Tr) Gh
P4(4=
ll(l -v)((Tp-o~) 4Gh
.-
a (1 --nz)+’
a. q (1 -- -A2)*
f,&) is an odd
P,(A)=0
9(1-$0(,-a,) Gb
A3 l
(f
whereupon the dislocation distributia~ function is given by
The number of dislocations in each half of one of the planes x2 = &h/2 is np= whereupon use of relation (22) gives n
J
(23)
3--vXap-Cl) P
Gb
4
(24)
This relation is shown in Fig* 3 (full curve) for vahres of ftjo3; since nP must equal o,)c/Gb as +-PO, the behaviour for values of h/c<3 should follow a similar 0 -v)(cpform to that indicated by the dotted portion in Fig. 3. The stress plz at a point a distance r ahead of one of the pile-ups and along its plane is
458
E.
SMITH
(25)
4. DISCUSSION III the preceding sections we have been concerned with the interaction between two dislocation pile-ups, and have given particular consideration to the case, when the pile-ups are widely spaced. The problems have been treated in terms of the theory of continuous distributions of dislocations and solved using a series solution approach that has been applied previously to other models. Thus expressions for the dislocation distribution function, the number of dislocations in each half of a pile-up, and the stress ahead of a pile-up have been obtained for h/c32 or 3, where 2c is the total length of each pile-up and h is the spacing between them. The series solution approach is ideal for such h/c values but breaks down for lower values, although it is then possible to infer the magnitude of the number of dislocations in each half of a pile-up (Figs. 2 and 4), using the calculated behaviour for h/c>,2 and the obvious value for h/c-+0. For the screw problem it is in fact
FIG. 4. The number np of edge dislocations in each half of one of the pile-ups for different values of h/c. The full curve is obtained from equation (24) while the dotted curve represents the anticipated form for h/c < 3.
possible to obtain a solution for all h/c by treating the problem as one in potential theory; since for antiplane strain deformation the displacement u3 satisfies Laplace’s equation V’us =O, the region external to the H-shaped figure, formed by the two pile-ups and the portion of the x2 axis joining them, can be transformed into the region external to a rectangle [14]. Thus it is possible to derive equations from which u1 can be obtained for all h/c; however, these are complex and since we are primarily interested in the behaviour for large h/c, the series solution approach is preferred in the present paper, particularly as the same method has been applied to the corresponding edge dislocation problem, which cannot of course be treated from the potential theory view-point. The general conclusion that emerges from the present analysis is that the pile-ups
The distribution
of dislocations
in two pile-ups, arranged one above the other
459
interfere with each other when their spacing is about equal to their length; then for exampIe there is approximately a 10 per cent change in the number of dislocations in each half of a pile-up. When the pile-up spacing becomes smaller, these changes become more marked and interaction between the pile-ups becomes appreciabIe. There is a significant difference between the behaviour of screw and edge dislocations, thus whereas the number of screws in each half of a pile-up decreases as the pile-ups become closer, the number of edges at first increases. This behaviour is different to that of coplanar pile-ups where edges and screws act in a similar manner, but is in accord with that of an infinite array of non-coplanar pile-ups [6]. For this latter model the relations corresponding to (10) and (22) are
(26) and f(
P Xl
)=w-VMP--01) *
(27)
Gb
where in these cases h is the spacing between neighbouring pile-ups in the infinite sequence. For large c/Ii, therefore, the characteristics of the two pile-ups and infinite sequence models are essentially similar. This is not the case, however, as h-+0; thus for example the number of screw dislocations in each half of a pile-up tends to the finite limit (0;~- o,)c/Gb when there are two pile-ups and to oh/i% for an infinite sequence [.5, 61. With the two pile-up model there is also a difference in the behaviour of screw and edge dislocations with respect to the shear stress in front of a pile-up; thus the appropriate stress decreases for screws (equation (14)) whereas it increases for edges (equation (25)). Now isoiated pile-up modefs have been used to explain the grain size dependency of the lower yield stress of metals and alloys. At this stress a Luders band propagates down a specimen, and experimental results have shown that its value follows a relationship of the form : z=r,+kZ-+
(28)
where 21 is the grain diameter and z. and k are constants for a given system of testing conditions. The form of equation (28) is consistent with the spread of plasticity occurring by the concentrated shear stress ahead of a slip band in one grain initiating plastic flow in a neighbouring grain. Theoretical considerations [15-17J of this problem have used a model of an isolated slip band; if z is the applied shear stress acting on the band, the shear stress at a distance r directly ahead of it is (see equations (14) and (25)) (z--z1
0 ;
*
(2%
where z1 is the lattice friction stress opposing the motion of the dislocations in the band. Thus, if a dislocation source is situated at the point I and a critical shear stress z, is required to operate it, then the criterion for yield propagation is (z--zl) or
1 *
0 2r
=z,
(301
2==ZI+(2r)*z,l-”
(31)
440
E. SMITH
and hence in the interpretation of experimental results, rO and k in relation (28) are associated respectively with tr and (2r)%,. The analysis of the present paper suggests that the results obtained from the isolated slip band model are probably quite adequate provided that the yield stress is not controlled by a deformation process involving the operation of more than, say, one slip band per grain. Thus, if the region at the front of a Luders band consists of a few grains with the occasional slip band in each, it is satisfactory to use the isolated slip band model; however, if slip occurs on more than one slip plane then this fact should be taken into account. The edge dislocation model is probably more suitable for discussing such a situation and equation (25) suggests that ex~rimentaliy measured k values will be lower than when the slip bands act in isolation. However, there is little ex~rime~tal evidence to guide us at this stage, as very few observations have been recorded of the detailed processes operating at a Luders band front. Finally, it is worth mentioning that although pzz is zero along each pile-up for the screw model and so the present analysis (with cur-0) can be used for two cracks subject to an applied stress pZ3, the same is not true for the edge model; pzz is non-zero in this latter situation and the cracks will open up. To solve this problem it is necessary to insert an additional distributioI1 of edge dislocations with their Burgers vectors parallel to the x2 axis, and then formulate equations which give zero p12 and p2. stresses across each crack face. ~ck~ow~e~~~a~-This published by ~rmission
work was conducted at the Central Eiectricity of the Central Electricity Generating Board.
Research
Laboratories,
and is
REFERENCES [I] [2] [3] [4] [SJ [6] [7l
J. D. ES3ELBY, Phil. Afag, 40,903 (1949). G. LEIBFRIED, Z. Phrys. 130,214 (1951). G. I. BARENBLATFand G. P. ~E~PANOV, Appl. auto. Mech, 25, 1654 (1961). N. LOUAT,PM Msg. 8, 1219 (1963). E. SMITH,Phi% Mag. 9, 879 (1964). E. SM~H, Proc. R. foe. A 282,422 (1964). Philadelphia, Society of Industrial and Applied W. T. KOITER, Problems qf Cantinwm Me&u&s. Mathematics, 246 (1961). [g] A. II. ENGLAND and A. E. GREEN, Proc. Cumb. Phil. Sot. 59,489 (1963). [9] E. SMlTH,Int. J. Ei?g??gSci. 4, 41 (1966). [lo] B. A. BILBY,R. BULLOUGH and E. SMITH,Proc. R. Sot. A 231,263 (1955). 111J A. H. CO~ELL, ~jslocat~~ns and Plastic Flow in Crystals. Oxford University Press (1953). 114 N. I. ~USKHELI~H~~I, Singa~ar Integral ~~at~oas (translated by J. M. Radok). P. Noor~off, N. V. (1953)‘ [13] A. K. HEAD and N. LOUT, Amt. J. Pkys. 8, 1 (1955). [14] C. DARWIN, Phil. Msg. 41, 1 (1950). [lS] N. J. FEKX, J. Iron Steel Inst. 174, 25 (1953). 1161 A. H. COTTRELL,Trans. Am. Inst. i34in. ~etall~ Ergrs 212, 192 (19%). [17] 1. Conr,and N. J. PETCH,Phi/. &fug.5,30 (196% (Received 8 November 1965f R&n&--L’auteur utilise la theorie de Ia distribution continue des dislocations pour Ctudier I’interaction entre deux empilages de dislocations, identiques, mais ne se trouvent pas dans un m&ne plan. La fonction de distribution est d&em&% pour des empilages tr&s espac& en utilisant nne solution d’approache par series, et ii est montr6 que l’interaction devient important lorsque I’intervalle, qui les &pare, devient ~~iblement Cgal B la fongueur totale de chaque empilage. Les r&ultats sont tr& dif%rents en ce qui conceme les dislocations par rotation et les dislocations sur les bords. C’est ainsi que le nombre des dislocations par rotation dans chaque moitiit d’un empiIage, ditcroit lorsque l’empilage se ressert tandis que ie nombre des dislocations sur les bords augmente.
The Les contraintes
distribution
of dislocations
sur le front
Ctudier la validitd du modele materiau poly-cristallin.
des empilages
d’un
plan
in two pile-ups, sont
de glissement
arranged
determinees isole
dans
one above
par l’auteur
et il utilise
la propagation
461
the other ces n+sultats
de l’t?coulement
pour
dans
un
Zusammenfassung-Die Theorie der kontinuierlichen Verteilung von Versetzungen wird henutzt, urn die Wechselwirkung zwischen zwei identischen nicht-coplanaren Versetzungsschichtungen zu betrachten. Die Verteilungsfunktion wird ftir weitrlumige Schichtungen unter Verwendung einer Seriennaherungsmethode hestimmt, und es wird gezeigt, dass die Wechselwirkung an EinUuss gewinnt, wenn der Abstand zwischen ihnen ungefahr gleich der Gesamtlange jeder der beiden Schichtungen ist. Die Ergebnisse sind deutlich verschieden fiir Schrauhenund fur Stufenversetzungen; wahrend die Zahl der Schraubenversetzungen in jeder Hllfte einer Schichtung abnimmt, wenn die Schichtungen enger werden, nimmt die Zahl der Stufenversetzungen zu. Die Spannungen vor den Schichtungen werden bestimmt, und diese Ergebnisse werden in einer kurzen Besprechung der Gtiltigkeit des isolierten Gleitbandmodells fur die Fliessausbreitung in polykristallinem Material verwendet.
Sonunario-Si fa ricorso alla teoria della distribuzione continua delle dislocazioni per studiare l’azione reciproca fra due scontri non coplanari identici di dislocazione. La funzione della distribuzione e determinata per accatastamenti a notevole distanza impiegando un’impostazione di soluzione di serie ed i: dimostrato the l’azione reciproca assume un carattere importante quando la distanza fra di loro I: approssimativamente identica alla lunghezza totale di ciascun accatastamento. I risultati sono significativamente dalle dislocazioni di vite e bordo; cosi, mentre il numero di dislocazioni di vite in ciascuna meta
differenti
di accatastamento diminuisce con I’awicinarsi dell’accatastamento, il numero di bordi aumenta. Si determinano le sollecitazioni di fronte agli accatastamenti e i risultati vengono impiegati in una breve discussione sulla validita de1 modello a banda di slittamento per la propagazione de1 carico di snervamento in materiali policristallini. TeOpBs AIICIIOKaIIEU npBMeHaeTCs KpaCCMOTpeHUFJ B3aBMOBetiCTBIBI MeXJIy ABYMB OHEIAaKoBbIbIB IIeKOMIIJIaHapHblMB CKOIIJIeBBsMB ,mIcnoKaIIBB. B CnyYae 6OBbmero pacCTOKHIUI CKOMeHII& &HKIIBR paCIIpeAeJIeIIBs OIIpeAeJIReTCff C nOMOIIIbIO peIIIeHIUI B PsAaX; BOKa3bIBaeTCII 9YO B3aBMOAe&CTBEe BhIeeT 3Ha¶eHBe KOrAa paCCTOIIHBe MeXCAy CKOIIJIeIIIUIMII 6yAeT npn6nnmeBBo paBHo o6meti ,IumIIe Kamoro a3 HBX. Pe3yJIbTaTbI pa3HRTbC.K 3Ha’IBTeJIbHO AJI5I BBIITOBOt II KpaeBOt$ AHCJIOKilIWW. %CJIO BBHTOBbIX ,IIIICBOKauti B KaXCBOn nOJIOBBHe CKOMeHB5I yMeHbmaeTCff C npB6BmKeHBeM CKOnHeHti, a SIICJIO KpaeBbIX AIICJIOKaII& BO3paCTaeT. GITpeAeJIeHbI HaIIpSxeHIBI Ha @pOBTe CKOIIJIeHIUI; IIOJIy’IeHIIbIe pe3yJIbTaTbI IICnOJIb30BaHbI B KpaTKOit ABCKyCCBB 0 npBMeHseMOCTII MOAeIIB 830,-IBpoBaHIIOti CKoJIbbsIImefi IIOJIOCbl K paCIIpOCTpaHeIIIIKJ IIJIaCTIFIecKOti 30HbI B IIO,IBKpBCTa,IBIBIeCKIIX
~~LUUC’r--Konrnuyanbuarr
h4arepuaaJrax.