Analytic solution for linear dislocation arrays in an arbitrary stress field

77 Materials Science and Engineering, 21 (1975) 77--91 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

Analytic Solution for Linear Dislocation Arrays in an Arbitrary Stress Field

D. PREININGER Institut fiir Material- und Festk6rperforschung, Kernforschungszentrum Karlsruhe (Germany) (Received February 10, 1975; in revised form June 20, 1975)

SUMMARY Applying the c o n t i n u u m approximation, the behaviour of a dislocation pile-up in an arbitrary periodic stress field o(x) is examined. The distribution function, as well as the length of pile-up~ the individual dislocation positions and stresses outside the pile-up that results from it, can be derived in an analytical way. The main result of this work shows that the dislocation distribution depends not only on the kind of o(x) function but also additionally on the position of o(x) relative to the pile-up. Therefore, in the case where o(x) is given by a single arbitrary periodic stress field, the phase shift ~x of the stress amplitude of o(x) with respect to the center of pile-up appears as a parameter only. Further, the conditions for splitting of pile-ups depends also on the wavelength and stress amplitude of the stress field additionally from ~x. The consequences of these results are discussed and compared with earlier results of an asymmetric periodic as well as uniform stress field.

1. INTRODUCTION In real crystals, owing to the presence of defects, there are internal stresses [ 1]. Besides other factors these define mainly their mechanical behaviour. This depends on the mean internal stress ~i values as well as the variation of stresses along the glide planes (e.g. the average dislocation velocity for dislocation glide and from this, the resulting activation parameters [2]). Measurements of dislocation bowing and from the calculated internal stresses show that the dependence oi(x) is generally of a periodic nature [3]. Pile-ups have been considered frequently to explain the observed mechanical as well as physical properties. It is generally accepted that microcracks are generated in crystals because of an increase in local stresses in front of the pile-up [4]. This occurs particularly in the case of cooperative effects of several slip lines [5]. Further, in stage II of workhardening of face-centered cubic crystals the theory of Hirsch and Mitchell [6] postulates the existence of pile-ups, stabilized on secondary dislocations. This theory results in a linear workhardening (o~ ~ a) [7] in basic agreement with experimental observations [8]. Moreover pile-ups can initiate yielding in polycrystals [9,10]. In this case the formation of pile-ups at grain boundaries is probable. Then the glide of dislocations from one to the adjacent grain is the result of local stress concentrations caused by those pile-ups. Pile-ups are also considered to be the cause for anelastic relaxation [11] in special cases. All these considerations start from dislocation pile-ups in the presence of uniform stresses. They do not consider the in~ernal stresses oi(x). Very little work is reported in the literature on the evaluation of dislocation distribution of pile-ups in static equilibrium [12,16,25]. Chou and Louat [12] applied the continuum approximation and found for the function a ~ e ~, o ~ sin(cx) (c is a constant) only solutions in terms of infinite series. Alekseev and Strunin [25] applied the same method for a double-ended pile-up in a a(x) which originates from an irregular arrangement of dislocations. Vladimirow and Khannanov [13] examined a dislocation distribution for edge

78 dislocations in tilt walls and could give only an approximate solution. Kronmtiller and Marik [14] carried out calculations for a pile-up in special cases of periodic stresses. The aim of the present study is to examine in detail the dislocation distribution and the properties in an " a r b i t r a r y " periodic stress field. As described before, this special case of the periodic function a(x) was chosen because of the physical relevance. This work is an extension of a previous one [16] that considered a pile-up in a single periodic stress field which lay symmetrical to the center of pile-up. In comparison with this the influence of the asymmetry of a single periodic stress will be investigated. This has special significance for the study of anelastic relaxation in internal stress fields. Furthermore for the examination of by-passing stresses of coplanar dislocation groups in the presence of oi(x) [17,20]. This problem is also of particular interest for examining a static dislocation distribution in the interval --~ < x < + ~ for the study of dislocation motion on slip or climb planes in the presence of periodic stresses in the subsonic velocity range, (v < c (shear wave velocity)), transonic (v ~- c) and supersonic (v > c) [18].

2. F O R M U L A T I O N O F T H E P R O B L E M

The calculations that follow are based principally on the same model as described earlier [16]. Assume n straight and infinite dislocations lying parallel to the z-axis between --a < x < + a in the glide plane y = 0 with the same Burgers vectors b > 0. In the considered isotropic and homogeneous continuum at x = --a these dislocations are held up owing to the action of a stress o ( x ) . The stress o ( x ) is given as a superposition of an applied stress u a with an internal stress oi(x). The resulting effective stress can now be described generally as

(={

o

O(X) =

Ua -

~

O? COS

X --

(1)

i=1,2

where hi is the wavelength, o o the stress amplitude and ~ the phaseshift from x = 0 -~ ~0~ = 0 of the stress field c o m p o n e n t j. Applying the concept of continuous dislocation distribution [19] the individual dislocation positions at static equilibrium, can be calculated. Let 7(}) = + ( 1 / b ) ( 6 b / 8 } ) be the dislocation density function such that the number of dislocations between } and ~ + d} is 7(}) d} with a total Burgers vector b 7(}) d}. Negative values of 7(}) signify dislocations with negative Burgers vectors. Under this condition and owing to eqn. (1) at equilibrium, the total stress at any point } is zero and for the u n k n o w n 7(}) we get:

(2)

j=l

--a

with z i = a°/aa and k i = 2arc/Xi. For a screw dislocation: A = pb/2rr and for an edge dislocation: A = pb/(2r~(1 -- v)) p and v have their usual meanings.

3. S O L U T I O N O F T H E I N T E G R A L E Q U A T I O N

With the aid of eqn. (A1), eqn. (A3) and eqn. (A9) (see Appendix) we get as the solution of eqn. (2) ~(}) = (EV~__

}2)--1 t~ - - ~

Oa I1 - - j= q1,2 z l ( J o ( k j )

cos ~flS--Jl(kj)sin~X ( a - - V r ~ - - }2))1 }

q

oa

.

2~"

(3)

79

cos SOs ?~i , = --SO72rra for t h e b o u n d a r y c o n d i t i o n of/2 dislocations lying b e t w e e n a prescribed interval o f --a ~< ~ ~< + a. JN(k~) implies the Bessel f u n c t i o n s o f o r d e r N w i t h the a r g u m e n t s k i. F o r t h e o t h e r b o u n d a r y cond i t i o n o f static equilibrium at the pile-up end o f 7(~) = 0 at ~ = + a, t h e r e is the f u r t h e r result f r o m eqn. (2) and eqn. (3): 7(~) = ~

~ z i cos soSJo(ki) j=l

--

( a - - ~) + ~ z i sm . _-2rr ~ + SO~ s + J l ( k i ) sin SOs i x/ra2 -- ~ 2 ]=1 ]kj

(4)

with the t o t a l n u m b e r o f dislocations n o w c o n t a i n e d in the pile-up o f

oo(

}

/2 = . 4 a 1 + ~ z i [ s i n so)SJl(kj) -- cos sosj0(kj-)] .

(A12)

1=1

Moreover, f r o m eqn. (3) an a s y m p t o t i c s o l u t i o n for Xj = ~ , kj = 0 for 1 ~ j ~ q results:

(

Oa

"y(~)= / 2 - - ~

E

1---~zjcossc s

]1

1

•

(5a)

for n dislocation lying b e t w e e n --a and +a. F o r the c o n d i t i o n o f "),(~ = 0 at ~ = +a we have f r o m eqn. (A4) for this

v(~) = 1 -~=l~Zj cos so~

~2 __ ~2

~

(5b)

with:

n=~a

-~z~cos

(5c)

]=1

E q u a t i o n s 5a - 5c agree with the results given b y L e i b f r i e d [19] for a pile-up in a stress o ~ x o f o = aa -- ET= lO°COS SOs.T h e same f o r m o f eqn. ( 5 a ) - eqn. (5c) follows also f r o m eqn. (3) and eqn. (4) for o ° = 0 at 1 ~ j ~< q and o = o~.

4. SOLUTIONS OF EQN. (2) FOR SPECIAL CASES OF o(x) F o r special cases given b e l o w it follows f r o m eqn. (2): (a) F o r o ° = 0 a t j > 2

010 : O0,

~1 ::~ SOS,

~1 ~: ~'2

we have f r o m eqn. (1): (6a)

ASO,;';, =

sot

-

so~,

so~

= 1~(SOl x +

SO~),

kl + ]~2

Ak =kl--k2,

km

.....

2 ....... .

Thus in this case 7(~) is given b y

,y(~)

=

--(a

(Trv/-a2 _ ~2)--1 /2 + ~ N//a~

1 +z

Jo(]zl) cos

~2)(j1(~1)sinsoS + j1(~2) sinsoS

SOS1+ Jo(k2) + ~2z

cos

SOS)

sin(2~ ~

SO~)

+

(6b)

80

(a)

k=3,14

1 8

k=6,24

(b}

21

~.;

.

--

•~ \ , ' k . \ \

o~a \ /V / I ?'~V/ -'.w;,,Q/\ / / 1\ ~i~\

T

"\./'\

/\

\! / / /

! A,~\\

/

\.

\ '\ %-'

i 0

'°F L

/-

.~

\,

•

\ .~ /

/,

ir 8-

i

+

z=O,4.

\

Z=O,4

I

*----

Z,i

~0/4

0

E

I

J

E

i

Z=12

i

't \V

T

!

I I

/

0

,o

"\.

i

"<.,,~_¢/

't)

8r~~ 0/4

Z=2

Z=2

- \. \.~.~

¢'~,

T\\

•, \ \ / "

i i~/\~,V ,I A \ \A

t

t F i g . 1.

'~

i

/'-'~"-~ I

81 (C )

k = 9,/~

'j',, t/l/~ o

--/\.U,.,~ I

Iii~ ~. "~t

!

t

ei .~~

8- ~

Z,

10~ ~

t

6~ o"

~/~

i

z=o,4

~. ~.__..

--

Z=12

~\ '~'~'--~

rCl2 11;/4

1°~T~ Z=2

i

Y_~ \',t+" D.' "

\ t \\.~ 1/~ '..7\/I '

o~ ...............\... ]...~......

7,XI;' J"~\7."?~ W:"

Fig. l a - l c . Effect of the phase shift ~ o n the dislocation distribution f u n c t i o n g ( t ) = A 7 ¢ / 0 a • 7(~ = --a cos (t)) for different periodic stress field parameters k, z (the numbers d e n o t e the values o f ~0S ). The transformed stress function~ are s h o w n in the top o f each Figure.

82

for n dislocations lying b e t w e e n --a and +a, and for ~/(~) = 0 at ~ = +a we have

°r[r I --z(g°(kx)

= rA|L

7(})

+ 2 sin(2~

]

cos ~1s + g0(k2) cos ~os2) w (a--G) / ~ 2 _ ~2 + z [ ( J l ( k l ) s i n ~ S + j l ( k 2 ) s i n ~ ) S )

+ ~s) cos(-~ + ~)1}

(6c)

with Or n = ~ a { 1 + z[sin ~ J l ( k l )

-- cos ~oSJo(kl) + sin ~0sj~(k2) -- cos .~s Jo(k2)] }.

(b) For o°= 0 a t / > 1 eqn. (1) reduces to o = or -- o ° cos (kx/a _ ~ s ) and for the solution of eqn. (2) t h e results for the b o u n d a r y c o n d i t i o n 3'(~ = +a) = 0 are: 7 ( } ) = ~Or

]/-(a=~)2+z(sin(~+~s)+j1(k)sin~S)}

{ (1 - z c o s ~SJo@)) V a 2 - ~2

(7a)

with Oa n = ~ a { 1 + z[sin ~ s J l ( k ) -- cos ~osJo(k)] }.

(75)

For the case of a single periodic stress o(x), Fig. l a - l c shows for specific values of a/X the influence of the phase shift ~os = arc cos(--~X/27ra) and o°/o, on t h e distribution f u n c t i o n g(t) = ATr o~- 1~, (} = --a cos(t)) (see A p p e n d i x 1). Accordingly for all values of ~os and for o°/Or > 0 the pile-up splits into dislocation subgroups. The position of these subgroups (extreme values of g(t)) are in a d d i t i o n to a/3. [16] and also d e p e n d e n t on ~oS. These however are in first a p p r o x i m a t i o n independ e n t of o°/Or . The n u m b e r of dislocations n~i containing these particular subgroups (magnitude of e x t r e m e values in g(t)) depends m a i n l y on o°/o a (see Fig. l a - l c ) . Moreover for a given or, nsi increases u n i f o r m l y with the stress a m p l i t u d e . The n u m b e r of subgroups f o r m e d decreases w i t h a/X and is o n l y slightly d e p e n d e n t on ~s at low values of a/X < 0.5 (k < 3). 5. RANGE OF VALIDITY OF SOLUTIONS OF THE INTEGRAL EQUATION For special values of k, ~s, ~/(~) can be negative inside the pile-up at o°/or > 0. A negative -),(~) refers to dislocations with negative Burgers vectors. In these cases of occurrences of ~,(~) X 0 static equilibrium can o n l y be attained if the pile up contains additional negative dislocations, respectively subgroups. These c o n d i t i o n s for c o n s t r u c t i o n of negative (b < 0) dislocation subgroups at ~(o) are given by z/> B: 1 - z c o s ~SJo(k )

z~>

( ~ ) sin

~(0) --

~s __J1(k) sin ~s

(a - ~(0)) 2

~-

=

B.

(8a,

~0)

The behaviour of negative s u b g r o u p - f o r m a t i o n is analogous to t h a t of positive ones (Fig. l a l c ) . Thus, n~ increases strongly with o°/Oa and its d e p e n d e n c e on ~s. The d e p e n d e n c e of the b o u n d a r y range b > 0, b X 0 of solutions of eqn. (2) f r o m ~s is s h o w n in Fig. 2a - 2b. Figure 2a - 2b shows the phase diagram z = f ( L / X ) a t n - / n + = const., derived f r o m eqn. (8a) for o°= 0, j ~ 2 at special values of ~s = 0, r / 4 , 7r/2, r . Below the lowest curve in Fig. 2a solutions of eqn. (2) exist for o n l y b > 0 (z < B). Above this curve, solutions for n - / n + > 0 respectively b X 0 exist. The d e p e n d e n c e n - / n + = f ( z ) s k f r o m the parameters ~ s k, are also d e m o n s t r a t e d in Fig. 2a - 2b. Accordingly for all values of ~ ~nd ~s, n-/n+ increases with z. Beside this, the d e p e n d e n c e n - / n = f(~s)k, z is of a periodic nature. In all cases of validity of eqn. (8a), in eqns. (3), (A12), (5a), (5c), (6c) and (7b) n has to be replaced by n = n + -- n - . T h e n the t o t a l n u m b e r ng = n - + n + of dislocation with sign b ~ 0 contained

83 '- L] 05

Off

0,8 iI

o,5" LOS

i

E, > i 0,8

@,3.

_ _.

0,2

0,5

I I

o

i Jrgers vectors

3

1

-

i.O,3 ]5_

2

01

iI

l

2

3

~

5

.....

6

7

0

2

L

3

4

i

5

L

X Fig. 2a. T h e z ( L / ~ ) - - p h a s e d i a g r a m for a single periodic stress field at ~z = 0 ( t h e n u m b e r s d e n o t e t h e r~tio n In+). Fig. 2b. E f f e c t o f t h e p h a s e s h i f t ~ s o n t h e z(L/~.)-phase diagram for a single p e r i o d i c stress field. ( R a n g e b e l o w t h e l o w e s t curves ( t h i c k lines) d e n o t e s t h e field o f b ~ 0.) T h e t h i n lines signify the f u n c t i o n z ( L / ~ ) for n--/n + = 1.

in the pile-up gwen by

ng=~a

zjsin~Sgl(kj)--cos~go(k]

1+

+2

~

ni[~2j,~(~.__l) ]

(85)

j=l

with ~a

ni(~,u )

j= 1

c cos

2 .

1

-- arc cos

+

si L ,

or m(lej,~jS )

ng=

~

j=l

m(kj, !

ni[~+l),~]

v

+

~

~o~)

j=l

r

p

~i{~,~(~-1)] ,J

r

+ ~[~l,-a]

.

This e q u a t i o n in c o n n e c t i o n with eqn. (4) is the m o s t general form of s o l u t i o n of eqn. (2), valid for

84

~/(~ = +a) = 0 as well as b <> 0. In eqn. (8b) m = f(k,~S), ~ = f ( k , ~ s ) signify respectively t h e n u m b e r o f groups o f positive and negative dislocations. ~ is the i-th zero o f ~,(~) b e t w e e n --a < ~ < +a c o u n t e d in the positive d i r e c t i o n o f } axis. F o r 7(~ = --a) = + ~ it holds f o r (57/5~)~=5 > 0 n~'i

m-

n~' i

2

m=-2 -+1

and in t h e o t h e r case o f (d3'/d})~=+a < 0 it follows t h a t m = n ~ / 2 + 1, ~ = n ~ / 2 , w h e r e n~.~d e n o t e s t h e n u m b e r o f zeros t h a t are given b y z = B.

(8c)

6. D O U B L E E N D E D PILE-UP

In this case an equal n u m b e r o f positive (b > 0) as well as negative (b < 0) dislocations generates f r o m a c e n t e r scource at ~ = 0 and is b l o c k e d at t w o ends ~ = --a, ~ = +a. T h e interval --a ~< ~ ~< 0 c o n t a i n s o n l y dislocations w i t h b > 0, t h e interval +a/> ~ ~> 0 o n l y t h a t o f b < 0. Because n - / n + = 1 it follows f r o m eqn. (4)

7(~)=([1--~zsk(ks)cos~]--Jl(ks)sin~(a--xira2--~2)]l ~ t2 s: 1 t _1J + =~lZj sin i=

~ +~]

(9)

A~"

7. T H E L E N G T H O F PILE-UP

This is o b t a i n e d b y using t h e r e l a t i o n +5

where 7 ( { ) refers to that for ~/({) = 0 at { = a, From this and f r o m eqn. (4) it follows that 2An

L =

o5 1 + ~ z j ( sin ~ J l ( k s )

- - cos ~oSJo(kj)

j=l

respectively for eqn. (6a) as a special case (o ° = 0 a t j > 2) o f eqn. (.1) and we have L = o5{1 + z [ s i n ~1 J l ( k l ) + sin ~

2An J l ( k 2 ) -- cos ~s J o ( k l ) _ cos ~s Jo(k2)] }"

A n a l o g o u s t o t h a t , f o r a single periodic stress field (o ° = 0 a t j > 1) L is given b y L = 2An {1 + z [ s i n ~pS J l ( k ) - - cos ~ S J o ( k ) ] }-1. ga

F r o m this r e l a t i o n it follows t h a t the m a x i m u m change o f t h e pile-up length as a c o n s e q u e n c e o f s u p e r p o s i t i o n o f oa ¢ f(x) with a stress field oi = f(x) occurs at t h e phase shifts ~0~jmax q

[cos(~Smax)Jl(ks) + sin@Smax)J2(kj)] = 0. 1=1

85 Respectively for o ° = 0 at j >1 1 we have ~max

jo(k)].

arc tg

=

This relation shows that ~OSax is independent of o0/%. Thus s0Smaxis only determined by a/~. Moreover the function ~0maxS= f(a/~) is of a periodic nature.

8. THE EQUILIBRIUM POSITION OF DISLOCATIONS The number n of dislocations lying between --a and --a +~' is determined by -a+~

,/

~

a

From this we get n, =

x

c cos

+ x%~ a

i,

~_ _

~,2

+

.=

do(kj) cos Cs + J1 (ks) sin ¢~

z~

arcc°s(--~'/a)--~'/a--1))--~ z j ~ " [27r~' ¢~) (sink/ kj~)l } 1/,~ ~v"~-- ,~'~ ;= lk, Ls'n [ ~ -~ + + cos

(lO)

For the case of infinite wavelength h i = oo, kj. = 0 j >~ 1 eqn. (10) reduces to q

n -~(t

+sint')

~--~

o~cos

.i=1

in agreement with the solution of Leibfried [19] for a pile-up in a uniform stress o = oa - Z~= ~ o° cos ¢~. For approximately t' ~< ~r/2, eqn. (10) reduces to a simple form: q

n' =~-A1rLI % - i__~1oocos ~0sI I7r - - a r c cos(~)l .

9. THE STRAIN Let us now consider an exhausted one-dimensional Frank - Read source in the sense of Leibfried [19]. The dislocation distribution is given by eqn. (9). The shear associated with this follows +a

--a

with/3 a model constant and R the length of the glide plane. Using eqn. (9) we get:

re%a2 5 = - - R4A-

~°z [1 -- j=l

i cos ~s(J0(kj ) + J2(kj)

.

For the boundary condition }~y-+ oo, kj = 0 and j t> 1 there follows:

o- o cos q

R4A

j= 1

86 in agreement with that of Leibfried [19]. The maximum change of the shears Ae = f(oi(x))Oa as the result of the presence of oi(x) occurs at the phase shifts ~O)m ¢ a x •• q

sin(~]m~x)[Jo(k i) + J2(kj)] = O. j=l

For a single periodic stress field oi(x) we have further that sin

C x - - 1~ a r C qOma

t j ~ )

•

This relation shows that ~max depends only on a/h. It is independent of o°/oa. This result is similar that for the influence of oi(x) on the pile-up length (~Sax } as postulated in section 7. As postulated in the preceding work [16], at ~s = 0 and o ° = 0 a t ] > 1, it appears in these cases (k) for the maxim u m changes of the pile-up length which produces, from a superposition of oi(x) with Oa ¢ x, no changes in the shears. This is valid for all values of ~s as demonstrated in the above relations.

10. THE STRESSES According to the results in section 1 it is particularly interesting to know the stress concentration in front of pile-up (important for microcrack formation and glide in polycrystals). Besides this it is also of importance for workhardening theories to know the stresses outside the pile-up that arise from it.

10.1. Stress on the head of pile-up For a given pile-up in a uniform stress o ¢ x, the stress o 1 tending to move pile-ups is given exactly by Ol = no [21]. From this it follows for ol in a non-uniform stress o(x) [19] -I- a

ol = f

7(})

--a

Substituting this in eqn. (4) we obtain

° ° I1 -- j~ZjJo(kj)cos ~s; ~ - ~ z~ [(J1 (kj)sin ~ s ) 2 _ Jo(kj)jl(kj)sin ~ps O 1 = (/an

---

-q

.

.i=1 . .

.

.

1 + ~ z j [sin ~iJl(k./) --

(11)

.

cos

./=1

¢flSJo(kj) ]

From these further results for ~i = oo, k./= 0 j >~ 1 q.

o1

(°a

in agreement with the solution al = a n for a uniform stress a = o, -- E~= 1 o° cos ~s. Now for the special case of o(x) described in subsection 4(a) we get for eqn. (6a) -

O 1 = Oan

-

2ZJo(k) cos ~ s cos

--4Z2Jl(k)

l ( k ) c°s2~S~ c ° s 2 2

4

cos

1 + 2z cos(a~os/2)[gi(h) sin ~osm--Jo(k) cos ~ s ]

with ~ s = 1 / 2 ( ~ + ~s) and ~ s = ~s .... gr~2. Equation (11) shows that Ol depends generally on parameters a/h and o°/oa. Furthermore the point ~s of the strongest influence of oi (x) upon ol is determined from b o t h parameters o°/oa,

a/h.

87 10.2. S t r e s s e s o u t s i d e the p i l e - u p (a) In the glide p l a n e

For x > +a and y = 0, o~,y is given by o~,,=A f

(12)

7(~)5~ .....1

a where ~ = - ( a + x). F r o m eqn. (12) we get, according to the p e r f o r m a n c e in A p p e n d i x 2, finally

(

q

1 0 ~ ___

1

[

0 .....

~zj

j= 1 × (101 - - ~ - -

1)2N--1 + sin ~ s

N=1,2

i::°i 2" cos~ s

~ [J2N- 2(kj)+J2N(kj)][t0i ] N = 1,2

[(2N-- l(~j) +J2N + l(~j)] (--1) ~':

10{

(--1) ~v+l (12a)

(i 01 ---~-02 ~ 1 ) 2N

because sinh (N~) -- cosh ( N ~ ) = _ [10l --X/r0-2 -- 112 = ZN(O). sinh (~) x / ~ -- 1 0 = --rl/a and K~ = 1 for 7(~ = +a) = 0 and K~ = n A / a o , containing n dislocations in 2a. For large values of ~/also 0 >> 1, ZN(O) is a p p r o x i m a t e l y z~(o)

-

1

2NI0 IN+I"

oo{ (

F u r t h e r m o r e for t h e c o n d i t i o n hj = ~ , kj = 0 j >~ 1 f r o m eqn. (12a) we get

o~,.v=-~

~ + 1-~z

o

icos~s~

)( 1

02

in a g r e e m e n t with t h a t of reference 24. (b) O u t s i d e the glide p l a n e In this case y ¢ 0. Let us n o w consider a pile-up of screw dislocations. The stress ox, y at y ¢ 0 originated f r o m the pile-up is given by

ox, y : A

/

a 7(~) (x -(x ~)2 - ~)+ y2@" a

Substituting in this x 2=-~

y2=7~

( l + b '2)+

(1+c 2

1+C 2 1+

we get finally

Oa

~,y = ~-a R(c)

__

XC

a(c - - b')(1 -- c b ' )

Zj COS~ ~

N= I

( - - 1 ) N - I ( 2 N - - 1 ) !2N--I(kJ) L' kj - "

~'

q

+ sin ~s ~ ( _ I )N + I( 2N) - ~ J - N:I

L2N(e)

--

(C-- b')(1 --cb')

(b')--j=l~Zj(COS@@\j

oo

X N=I ~ (--1)N-- 1(2N)

'~j

L ' 2 N - l ( b ' ) + s i n ~ o s ~ N = l (--1)

(2N) - ~ i j - L2N(t,')5 .

1(C)

88 where for Ipl > 1 r(1 + p)(Kz + p 2 - 1) p(p2 _ 1)

R(p) =

and L~v(p) - 7r(1 - - p )

pN+ 1

For IPl < 1 we obtain:

gz R(p) = ~r(1 + P ) ( 1 _p2 + P ) and L~v(p) = 7rpN(1 + p). As an asymptotic solution from eqn. (12a) for h~ = ~ , kj = 0 and j/> 1 we obtain q

q

(c -- b')(1 -- cb')

11. CONCLUSIONS In addition to the rate LI/Xj and the amplitudes a ° of the j-th stress c o m p o n e n t of an arbitrary stress o(x) defined in eqn. (1), the dislocation distribution within a pile-up that is exposed to a(x) is also mainly determined b y the position of a(x) relative to the pile-up length. For a single arbitrary periodic stress, then, for this geometric parameter it appears only the phase shift ~x of o ° from ai(x) counted at the center of the pile-up. The influence of the parameter ~ on the dislocation distribution function ~,(}), stresses on the head as well as outside the pile-up is in general not the same. Besides this, ~d' determined additionally to L/X the splitting behaviour of pile-ups in dislocation subgroups.

APPENDIX 1 Substituting X =--a

COSS

~ =-a

a a -

~[(~) = ~-~g(t),

COS t

y(t) = g(t) sin t,

k

2aTr

(A1)

X

into eqn. (2) and multiplying this form with sin s it follows (for details see reference 16) q

1/

sY(t) S t = f ( s ) = s i n s [ l _ ~ z j c o s ( k i c o s s _ ~ i

sins

fl"

COS t --

]= 1

COS

~

(A2)

0

with ~s = arc cos

~

.

Performing a Fourier transform on f(s) the solution of eqn. (A2) is [22,23]

g(t) = ~

+

~ as cos((~t O~=1

(A3)

89 where the Fourier coefficients are given by as =

(2)1/2 ~ /

[1 -- zi cos(kj cos s -- s ) ]

sin as sin s ds.

(A4)

J=l 0 The factor (A/aoa)n = (A/aoa) f+--~a~'(~) d~ is the boundary condition of eqn. (A2); those values can be chosen arbitrarily, n denotes the number of dislocations lying between --a and +a. Transforming eqn. (A4) gives

a,~

\~]

[_2--j:

zj(a~sj c o s ~ + a,~Dysin,pS)

7r a~Bj = f cos(kj cos s) sin as sin s ds

(A5)

0 ?r aaDj = f

sin(kj cos s) sin s sin as ds.

o Now for a~Bj it holds [16]

a~Bj=--~r

'

w

N=W ~ i2NJN+l(kj)

N = 0,1 W= 1,2, "'"

OPN'W

(A6)

kJV+ 1

1

]

+

\

3

1)(N_I)

+

\ 2 N + l W1)(N

'::)1 .

where JN(ky) signifies the Bessel functions of the order N with argument k i. Because JY - l(kj) + JN ÷1 (lej) = 2N Jy(tej)/kj eqn. (A6) can be rewritten in the form

aaBj = 2(__1)N-- 1 [J2N 2(ki) + J2N(ki)] = a ( 2 N - -

1)BIN = 1,

2, 3 ...

(A6a)

Applying the same method as described in reference 16 we get for aaDi a~D i = a~z~ ) .~ =

2(--1)N ÷ 1 [ J 2 N - -

l ( k j ) + J 2 N ÷ ~(k~.)].

(A6b)

Substituting eqn. (A6a), eqn. (A6b) and eqn. (A6a) into eqn. (A3) gives further

g(t)=~:ln-t[~+

os t-- z~j 1j = COS~jN= ~1,2 (--1)N--I[J2N--2(kJ)+J2N(kj)]

× c o s [ ( 2 N - - 1 ) t ] +sin~0 sN=l,2~ (--1)N+l[J2N_l(kj)+J2N+l(kj)]cos2Nt)l } .

(A7)

On the other hand for the boundary condition g(t) = 0 at t = 7r (3'(}) = 0 at } = +a) it follows from eqn. (A7) and eqn. (A2)

g(t) = ctg (t/2)

sin

t

"~-COS~J IJ0(kj)"~" N~

~"=z i sin ~/

1(ki) + Y~~- 1 (--1)N+I(J2N--I(k]) + J2N+l(kj)) cos 2N

(--1)N-I(J~-2(kj) "~J2N(I'~j))COS(2N--1)tl}

(A8)

90 and

~]Jl(ki) -- cos ~sJ0(k/)] .

n = -~a 1 + ~ z / [ s i n

(A8a)

j=l

Furthermore with q

oo

q

~Jo(kj )

j=l,2 &=l,2

j=l

and q

oo

q

j=l,2

c~= 1,2

zr j ~= l jl(kj)"

Finally after some manipulation the solution of the integral equation becomes

g(t)=(sint) -1 InA + c o s t II--j~=l zj ( Jo(kj) c o s ~ S + j t ( k i ) s i n ~ s l c o- s- ts i n t ) l ( ClOa

"

-- ~ zj sin(kj cos t _ ~s/ )

}

(A9)

j=l

for n dislocations lying between --a and

g(t) = 1 - ~ z / cos ~sJ0(kj .) c t g t - - j j=l

+a, and z/[sin(k/cos t -- ~/) -- Jl(kj) sin ~s ]

(A10)

"=

for the case of g(t) = 0 at t = ~r.

APPENDIX

2

Substituting eqn. (A1) into eqn. (12a) we get

oaf y(t) 6t. o(t,~) = ~ --~?la + cos t

(All)

o

For this solution arise a sum of definite integrals of the form

IN(O)

=

~

cos

(Yt) 5t with O=--~/a

0 + cos t

10]>1

(A12)

o

w i t h N = 0, 1, 2 .... For the case o f N = 0, 1 0

Io(0 ) -

r01

I1(0)

= 1

101

where the sign of the r o o t is only positive. For general values of N it follows

(--0~ N+I s i n h ( N ~ ) - - c o s h ( N ~ ) (0)g+, IN(O) = \ ~ [ ] sinh (~) - - = -- __ ~ _

e--N~ sinh (~)

(A13)

with JOl = cosh (~) and ~ > 0. Finally there follows from eqn. (A12) because of eqn. (A13), eqn. (A10) and eqn. (A9)

91

+ I O , Vo~ 2 ~

Eq I - - 0,oL - ]2 j=l z1 C O S ~ S N~= I (--1)N--I(J2N--2(kl)+J2N(I~J))

e -2N ¢ + sin ~ s XE= 1 This yields ~=+a.

Kz = nA/aea

(--1)N+I(J2N_I(]~j) +J2N+l(ki))~[]

f o r n d i s l o c a t i o n s i n t h e i n t e r v a l --a ~< x ~< +a a n d

e--(2X+l)~ sin Kz =

1 f o r ~(~) = 0 a t

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