Inclined pileup of screw dislocations at the crack tip with a dislocation-free zone

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INCLINED PILEUP OF SCREW DlSLOCATIONS AT THE CRACK TTP WITH A DISLOCATION-FREE ZONE* S.-J. CHANG’

and T. MURA’

‘Engineering Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. ‘Department of CIVIL Engineering, Northwestern University, Evanston, IL60201, U.S.A. Abstract ~A single pileup of screw dislocations extending from the crack tip along an inclined direction has been observed in experiments. it is often associated with dislocation emission mechanisms at the crack tip. This linear pileup is a microplastic slipline emanating from the crack tip. A region near the crack tip is often free from dislocations because of a finite resistance value for the crack tip to emit dislocations. The mathematical problem is solved in this paper hy applying the extended Wiener--Hopf method. The condition of finite stress at the end of the plastic zone. the crack opening displacement, and the stress distribution along the slipline are obtalned in analytical expressions. Numerical values are calculated and the results can be used to discuss brittle versus ductile fracture for metals as treated in previous studies, A method to approximately calculate the corresponding results for edge dislocations is suggested.

1. INTRODUCTION

The problem of a linear pileup of screw dislocations extending from a crack tip is of interest because of the effect of plastic slip on metal fracture. A single pileup of dislocations along an inclined direction from the crack tip has been observed in experiments and often the pileup is associated with the dislocation emission mechanism at the crack tip. Between the pileup and the crack tip, the region is frequently free from dislocations [l-3]. This model and the significance of the dislocation-free region have been discussed by several authors [4-s]. The physical implication of this model is a subject of current interest, and a review was written by Thomson [9]. The elastic problem of the inclined pileup of screw dislocations at a crack tip with a dislocation free region is solved in the present paper. The dislocation configuration of the present problem is not symmetric with respect to the crack plane. The solution is an extension of an earlier solution [lo] in which the dislocations were also piled up from the crack tip along an inclined direction, but there was no dislocation-free region (the BCS case [12]). The existence of the dislocation-free region leads to a finite stress intensity factor in the present solution. A quantitative determination of the stress intensity factor is therefore possible from this solution for a discussion of ductile versus brittle fracture of the material. The physical discussion has been described in the previous studies [7,10,1 I] and will not be repeated in this work. The present solution may aiso be relevant to other fracture problems. It is clear that if we assume the friction stress, dlt to be zero, then the dislocation pileup region will be reduced to a slit in which the two surfaces arc free from sliding without normal separation. This problem can be viewed as a void located in front of a crack or a model of crack branching. Another relevant problem may be the zigzag crack propagation due to cyclic loading. The inclined plastic zone around the crack tip represents a realistic crack tip behavior under mode I load. The numerical treatment for two inclined plastic zones symmetric to the crack plane was studied by Bilby and Swinden [ 131, Vitek [ 141 and Riedel [IS]. The analytical solution for the double pileup problem was examined recently by Cheropanov (1161. He obtained results for the BCS (Bilby, Cottrell and Swinden [IZ]) condition and the total number of dislocations, but the results were obtained partly by numerical integration. As mentioned earlier, the problem of a single inclined pileup of screw dislocations was solved by Chang and Ohr [lo] recently by applying the WienerHopf technique. The distribution function of dislocations and the stress distribution were *Research sponsored jointly by AR0 DAAG 29-85-K-0134, NSF-MRL DMR-8216972, Northwestern University; the Division of Materials Science and the Division of Mathematical Science. U.S. Department of Energy. under contract DE-AC@%840R21400 with Martin Marietta Energy Systems, Inc.

S.-J. (‘HANG

S62

and T MURA

obtained in simple series forms in which each coefficient is expressed in terms of a beta function. The BCS condition and the total number of dislocations were obtained in terms of simple algebraic functions. The Wiener-Hopf method is usually applied for solving wave diffraction problems [ 171 and was first applied to solve the dislocation pileup problem by Kuang and Mura [I81 and later by Tucker [ 191. In the present problem, the existence of an elastic region between the crack tip and the pileup leads to the extended Wiener-Hopf equation, The extended equation is then reduced to a system of coupled integral equations from which the solution is obtained as an analytical expression. It is fortunate that the stress distribution, the BCS condition, and the total number of dislocations have been obtained completely without approximation. Usually the solutions for the extended Wiener-- Hopf equation can be obtained only approximately as can be seen from a standard treatment of the method [20]. Because the method of solution to the coupled integral equations represents a successive interaction of two standard WienerHopf solutions, the solution is often approximate. The result obtained here has been compared with that by using a direct numerical method and shows an excellent agreement.

2.

PILEUP

INTEGRAL

EQUATION

Antiplane shear stress is applied to an infinite elastic body which contains a semi-infinite crack as shown in Fig. 1. The crack is chosen to coincide with the negative x-axis of the complex z-plane (z = x + iy). A linear array of screw dislocations is piled up aiong a

Fig. 1. Inclined

pile-up of screw dislocations.

straight line, starting at a distance e from the crack tip pileup is inclined at an angle 4 = a~ to the crack plane. z-plane to e is the dislocation-free zone (DFZ) and that By considering the statical equilibrium of forces acting due to the external force with the stress intensity factor forces due to other dislocations in the pileup region, integral equation:

up to a distance 1. The line of the The region from the origin of the from e to 1 is the plastic zone. on each dislocation at z (z = re’“) K,, the friction force or and the we obtain the dislocation pileup

(2.1)

inclined pileup of screw dislocations at the crack tip with a dislocation-free zone

563

where j(r) is the distribution function of the dislocations, p is the shear modulus, and h is the Burgers vector. Similar to the previous paper [IO], the above integral equation is reduced-to a non-dimensional form by introducing a new variable< = Jr/l with t, = &IL

s0 1

A--t=

k ;

F(t)?

(t, < t < 1)

(2.2)

2,

k(x) =

_J”.._

1 - x + 2(x

1 +

26)

+

1

(2.3)

2(X + Kid) (2.4)

and

(2.5) Equation (2.2) wiil be solved in the subsequent sections. For I, = 0, it reduces to the governing integral equation for the BCS case.

3. EXTENDED

WIENER-HOPF

EQUATION

The integral equation (2.2) is defined for t, < t < 1. The range of t for the integral on the right hand side of equation (2.2) is extended to the whole of positive t and two functions are defined:

h(t)

(3-f)

=

and

s(i 1

fdt) =

$ F(t$

k

o
(3.2)

fe

Physically, they represent the stress distribution due to dislocations in the pileup region with the property that the total stress r~is A ~~

(T

-=;

h(t) t

01

l-ct-cca

(3.3)

and (3.4)

The functions g(t) and h(t) have been referred to as the effects of dislocation shielding [Zl]. Taking MeIlin transform of equations (2.2), (3.1) and (3.2), we obtain 1,

I

0

g(t)t"-'dt

I-

'(A - t)t”-‘dt

sti?

+

m h(r)t”-‘dt=S:t’-‘[~~k(~)i(f’)~]dt

s1

(3.5)

S.-J. C-HANG and T. MURA

544

or the extended

Wiener-~opf

equation t”,G+(s) + T(s) + U_(s) = B(s)K(s)

(3.6)

s

F(t)t”-- * dt

(3.7)

k(t)t”- “dt

(3.X}

where 1

B(s)

=

t..

<*I

K(s) =

I

0

T(s) =

(A - t)t”-

I dt = T,(s) + t;T2(s)

(3.9)

(3.10)

(3.11)

,*r

H_(s) =

h(t)t’- I dt

(3.12)

and

s 1,

t';G+(s)

=

0

K(s) in equation

s 1

g(t)t”_ ’ dt = t”,

(3.8) can be reduced

g(t,t)t”-‘dt

(3.13)

0

to

(3.14)

where 0: = 2/( 1 + a) and p = 2/( 1 - u). S, for &T(S) > 0 and S- for Re(s) -C 1. K(s) is the analytic and nonzero in the K(s) has been factored to K+(S) and

W e may define two regions in the complex s-plane, G+(s) is analytic in S, and H_(s) is analytic in S_, intersection of the two regions S, and S- . K_(s) in [lo] as

(3.15)

K_(s) =

Jzx[x_(l)]”

f-(1 - s) I(&+(I,-)

where x-(l)

= ,$,$J$.

(3.16)

inclined

pihup

of screw dislocations

at the crack

tip with a dislo~ti~n-rree

zone

565

In order to solve the extended Wiener-Hopf equation, (3.6), we need to esti_mate the order of magnitude for the functions in (3.6). We have shown that K +(sj --+l/Js and K_(s)-+&for s-+co. Ifg(t) -+ 1 for t -+ I, then the function G+(s) -+ l/s. If h(t) -+ 1 for t --* 1 then H_(s) -+ l/s. This condition has to be satisfied in order to be consistent with the condition of finite stress at t = 1. With respect to these conditions, the usual argument of analytic continuation and Liouville’s theorem implies that the extended Weiner-~opf equation, (3.Q can be decomposed to a system of coupled integral equations in the following form:

s

c+im

G+(4 -

1 K+(S)& K+@)(” c_im

s d+im

K-(s)&

1

d-im E(-(“Hu - ‘f

‘1

+ T(u))] du = 0

[r;“(H_(u)

[t:(G+(u) + T(u)] dzr + II_(s) = 0

(3.17)

(3.18)

These equations can further be reduced to

s

c+im t-“[H’fu) .e._-

+ T,(u)] du = o K +(u)(u - s)

c-ice

s

d+im t:[G+(u).-~+ T’(u)]du K -(U)(U- s)

d-im

+ HZ(s) = 0

where T;(s) and 7;(s) have been defined in equations decomposed as H_(s) = H’(s) f

(3.19)

(3.20)

(3.10) and (3.11) and N_(s) is

U?(s)

(3.21)

with

(3.22)

In the previous paper [lo], ZE(s) and T,(s) have been shown as the transformed dislocation stress and the applied stress, respectively, for the problem which does not contain a dislocation-free zone. The reduction of equations (3.17) and (3.18) to (3.19) and (3.20) has made use of this result. The function H?(s), therefore, can be regarded as the correction term owing to the presence of the dislocation-free zone. Two results can be observed immediately. From equation (3.19),

G+(s)+

-

7;(s)+; asE,-+O

(3.23)

After inverting G+(s),

&)-+A

ast,-+O

(3.24)

S.-J. CHANG and ‘I‘. MURA

566

Hence, from equation (3.4), we obtain the result that for t,, --t 0 the stress intensity i- --K = lii/2nlcqA

13.31

- g(t)] = 0

which implies the BCS type of stress distribution equation (3.19), we observe that

G+(s) + T;(s) -+

ktctor

near the crack tip, From the same

---!rass ---t ixi

(3.26)

J’s3

By applying the asymptotic property of the Mellin transform, the inversion of the above equation will lead to g(t) -+ A

‘-

t

as t 3 t,

(3.27)

which implies that

Therefore, from these two results of the function g(t), the stress or (T-+LT/ as t-t,. distribution CTwithin the DFZ has the property that if t, 3 0 then the stress intensity factor K -+ 0 and if t, # 0 then at t = t, the stress c is continuous with the value of CJ~.

4. SOLUTION

TO THE

COUPLED

INTEGRAL

EQUATiONS

The coupled integral equations, equations (3.19) and (3.20), will be solved after reducing them into a system of algebraic equations in the following manner. The contour of integration in equation (3.19) is completed by a large semicircle in the left half plane. The integrals are calculated by evaIuating their residues and equation (3.19) is reduced to

G+(s)+ 722) -

K.,(s~~~~ ;r,_s t,’ 1

_

-‘@“W:(1 - an)

+ H’(1 - cxn)+ 7;(1 - an) +

h

t-l+pn

1 -fin-s

(H4(1 - pn) + Hi(l

p

- j%z)+ 7-,(I - @)f] = 0

(4.1)

where, in terms of beta function B&y),

(4.2)

(4.3)

H'(s)

+

T,(s)

=

X_(s)

1 A K-(O) s

--

1 -

_____~

K-(-1)s

1

+ 1

(4.41

tnciined

pileup of screw dislocations

at the crack

tip with a disio~tion-free

Similarly, the contour of the integration in equation (3.20) is completed the right half plant and equation (3.20) is reduced to

zone

567

by a semicircle

in

(G+(i+ pn) + T,(l + fin))] + H?(s) = 0

(4.5)

J&l),

(4.6)

where

p=

(4.7)

(4.8)

In equations (4.1) and (4.5) the functions G+(s) and H?(s) are expressed in terms of their values at 1 _t WI (n = 1,2,. . .) and 1 + /hz (n = I, 2,. . .). The functions G+(s) and H’..(s) can be obtained explicitly from equations (4.1) and (4.5) provided that their values at 1 + an (n = I,2 ,...I and 1 +- /I% (n = I,2 ,...) are known. The values of G +( 1 + cm), G +(l $- In), H2.( 1 --- cm)und H?( I - [in) can be solved from equation (4.1) and (4.5) by setting s = I + tm and 1 + n/j (n = 0,1,2,. . .) in equation (4.1) and s = 1 - m and 1 - n/3 (n = 1,2,. . .) in equation (4.5). The calculation is shown in Appendix A.

5. STRESS

DISTRIBUTION

WITHIN

DFZ

It has been shown

that the stress distribution within the DFZ is cr/aj = (A - g(t))/t. The function g(t) has been referred to as the effect of dislocation shielding 1211. The Mellin transform of g(r) is G+(s). To solve g(r) from equation (4.1) we require the inversion of G+(s). From equation (3.13).

s r+ix

G+(s)-“ds

forO
1

c-ia

Substituting

G+(s) from equation

(4.1) into the above equation,

(H-(1 -- (In’) + T*(l .- pl’))]

forO
1

(5.1)

we obtain

(5.2)

56X

Defining replacing

S-J. CHANG

and T. MURA

t,

= t,t, the function g(ll) for 0 < t, < t, can be obtained from equation (5.2) by t with t,/t,. The stress intensity factor K is derived from the above equation as

K

zz

-q&;igo

-an)

b

--“tC’+@“H_(I

by the following

(5.3)

limit:

K =?iqj2nt,t

The numerical

-

A - d&t) f t e

(5.4)

values of a/~~ are shown in Fig. 2 and values of K/K,

1.0 0.00

1

1 0.04

’

an))

- Bn) + T,(l -/In))]

+1-/W

where K is defined

+ T,(l

1 \ 0.06

1

1

’

0.12

0.16

1

h.

1

0.20

’

’

0.24

are plotted in Fig. 4.

1-b 1

0.26

r/t

Fig. 2. Stress distribution

within the dislocation

free zone.

Equation (5.2) can be proved by using the following results. inversion formula for G+(s) is K +(s)t-“/(l - an’ - s). The function ats=O, -1, -2,...and

(-1) T(s)l-(1 - s) + s+n

A typical

term

K+(s) has simple

ass -+ - n (n = 1,2,. . .)

in the poles

(5.5)

The residue due to the pole of K+(s) is

J2n( [x-(l),-“l-(l

1)"P

+q(1-+)r(l = gJ-x-(l)t]”

_!$2)1-ak~+n

l 1 - an’ + n

(5.6)

Inclined

pileup of screw dislocations

at the crack tip with a dislocation-free

and the residue due to l/( 1 - an’ + s) is zero because coefficients g, are

x,=&sin

6. STRESS

569

zone

K +( 1 - zn’) = 0 for n’ = 1,2,. . The

71 0x

DlSTRIBUTION

OUTSIDE

OF

THE

PLASTIC

ZONE

The stress distribution outside of the plastic zone, or 1 < t < cm, has been shown as o/a, = (A - h(t))/t. The Mcllin transform of h(t) has been represented as H-(s). As mentioned before, H_(s) is decomposed into two parts Hi(s) and H?(s). H’(s) is the part for the corresponding problem which does not contain a dislocation-free zone and the remaining part H?(s) can be regarded as a perturbation owing to the existence of the dislocation-free zone. Hy as well as its inversion k’(t) have been obtained previously [lo]. The remaining part k2(t) will be computed in the following. The function k’(t) is the inversion of H?(s),

k’(t) = &

s

d+iaa H?(s)t-‘ds

(6.1)

1
d-ix

Substituting calculating

the function H?(2), shown the residues, we obtain

+

in equation

c

n,&, +:‘;,_ n tr+a”’(G+(l

(4.5) into the above

equation

and

+ rn’) + T,(l + ctn’))

+ _n’_ d t: +pn’(G +( 1 + pn’) + T,( 1 + fin’))] 1 + fin’ - n I

l
(6.2)

where

k,

=o

(6.4)

Numerical values of o/a, are shown in Fig. 3. We shall prove equation (6.2) by using the following results. A typical term in the inversion formula for H?(s) is -K -(s)t-‘/(l + cm’ ~ s) or - K(s)t--“/( 1 - s). The function K-(s) has simple poles at s = 1, 2,. . But, at s = 1, K _(l) = 0 and it is removable. Using the relation that

(- 1)

I-(1 - s)I-(s) + __ S--ll

as .s+n(n

= 1,2,...)

(6.5)

S.-J. CHANG

and

T. MURA

08

0.6

e/(=02-

b‘ \

:

0.4 b 02

1

0 = 000

2. 0 =02.5 3

0 = 0.50

t

c

4 I

0.0

I,

I

04

08

I

I

I

I

12

16

I

I

I

2.0

I

I

I

24

28

r/l

Fig. 3. Stress distribution outside of the plastiz zone.

the residue

due to K-(s)

is

and the residues due to l/(1 + an’ - s) and l/(1 - s) are zero because 1, 2,... and K(1) = 0. The previous paper [lo],

K-(1

+ cm’) = 0 for

n’ =

(6.7)

The resulting

dislocation

stress is the sum of h’(t) and h’(t).

7. DISLOCATION-FREE

ZONE

(DFZ)

CONDITION

The inversion of H?(s) and If?(s) has been denoted as h’(t) and h’(t) respectively, so that h’(t) + h’(t) = h(t). The function h(t) is the dislocation stress for t > 1. From equations (3.22) and (4.5) both

fors-+

HL(s)andHZ_(s)-+I

=c

(7.1)

fort + 1

(7.2)

Js which, from the Abeiian

theorem,

implies that both

h’(t) and h’(t) -+ ~

Hence the stress at t = 1 is unbounded. Physical condition requires that a relation is imposed in order to reduce the order of magnitude of H_(s) as s -+ cc so that its inversion h(t) will be bounded at t = 1. This condition will be referred to as the DFZ condition.

Inclined

pileup of screw dislocations

at the crack

tip with a dislocation-free

zone

From an algebraic consideration of the coefficients in H_(s), we obtain condition for these coefficients, the DFZ condition, as -.-

A

K..(O)

_ ---

+ pt,(G+(l)

K-(-l)

+ T,(l)) + f n=l

an additional

[c,t:+""(G+(l + an) + T,(l + an)) /In) + T2(1 + @I))] = 0

-I- d,t: +pn(G+( 1 +

The above equation

571

(7.3)

implies

(7.4)

and, therefore,

From

Appendix

A, an integral

h(t) --+Jt-1

fort + 1

form of equation

(7.3) can be obtained

A

(7.5)

+ T,(u)]du

t:[G+fu)

as 1= o (7.6)

K-(u)

K-(O) or

+ T(u)]du

[tfG+(u)

= o (7.7)

K-(u)

Obviously,

for t, + 0, it reduces to the BCS condition

obtained

previously,

i.e.

UO)

A=K-(-l) Numerical

values of equation 2&%

(7.3)

(7.3) or (7.6) are shown in the form

=g

xK3

sin(un) -&-[I

‘i,iI (

+ a]W[l

- a]+

I

(7.9)

where the factor g(e/l, a) is essentially independent of U~Land is equal to one for e/l = 0. For g = 1, equation (7.9) reduces to the BCS case, equation (7.8). The function g is related to a factor Fiq by g = A,. The factor F,,, as plotted in Fig. 4 is a ratio of the pileup length I for the BCS case to that for the DFZ case if the applied rorce G = Fc$/2~ is kept constant. The ratio l/G for the BCS case is plotted in Fig. 5.

8. TOTAL

NUMBER

OF

DISLOCATIONS

The total number of dislocations N in the plastic zone can be calculated directly the transformed distribution function B(s) by applying the following relation:

s 1

N

=

f(r)dr

P

47r10f l = ~ F(t)dt = I*b sIp

from

(8.1)

S.-J, CHANCi

and T. MUKA

1 .c“F;

0.s I -

OE I-

k> g

07

4: 0.t

O! j-

0.L I._ O(

I

0.1

0.3

0.2 e/l

Fig. 4. Stress intensity factor K and reduction factors F,, and F,@for I/N and l/G, respectively, in the BCS conditions as functions of the ratio e/l. F,, and E,, are conversion factors which reduce l/N and 1/G, respectively, from the BCS case to the DFZ case.

0.0

Fig. 5. BCS condition

0.2

0.4

0.6

0.8

as expressed in terms of the pile-up length I, the applied number of dislocations N in polar coordinates.

The value of the function B(s) for s + 1 will be calculated equation, equation (3.6), in a modified form,

m)K +(s)=

1.0

stress G, and the total

from the extended

Wiener-Hopf

&W+is) +G(s) + H%)l J--CT,(s) 4 Hi(s)]

+ K-(s)

(X.2)

Inclined

Observe

pileup of screw dislocations

that, from equations

at the crack

tip with a dislocation-free

zone

513

(3.15), (3.16), (4.4) and (4.5), for s -+ 1 fin K+(s) +--x-(l)

K_(s) --f -

t”,[G+(s) + &(s)] + HT. -+ -Km(s)

f n=l

J2w(l

_ s) _ 1 - s

(8.4)

P

aB

+ an)

c(G+(l an d

+

(8.3)

t’+h

T,(l + an)) + ne Bn

(G +(I + Bn) + T,(l +

LWI

(8.5)

and

7’,(s) + Ht(s) + K_(s)

From

equation

A 1 ~ K_(O) 2K_(-1)

(8.6)

(8.2). we obtain

+ an) + I”,(1 + an))

___

1 +/In

d t,

L-(G+(l +

Substituting

+ fin) + T,(l + on))]

(8.7)

fin

equation

(8.2) into (8.1), the total number

+(G+(l + d,t: +(ln pn(G+(l

of dislocations

N is

+ an) + T’(1 + an))

+ Bn) + T,(l +

‘1 IW)lj

(8.8)

which may be considered as a modification of our previous result. If the dislocation-free zone is vanishingly small, t, + 0, and the BCS condition is satisfied, then equation (8.8) reduces to our previous result

(8.9)

To calculate N from equation (8.8), the DFZ implemented. Equation (8.7) can be represented by an integral

condition,

equation

(7.3), has

to be

form as

x_(l) 1 d+im - [t:G+(u) + T(u)]du li_T B(s) = __ K-(u)(u - 1) J% 2xi s d-is,

(8.10)

574

S.-J. (‘HANG

where the following

relation

and

I. MUKA

has been used: d~kir

A I T,(U)dU __ _ ____._~~~ - 1 K_(O) 2K--1) 27ri d+il K-(u)(u 1)

-.-J’

Numerical

values of N are shown according

zL(!+

to a modified

(1

sin(a7t) ;,a ~ arc

form of equation

(X.1 Ii

(X.8).

(8.12)

where G = Kz/2~ and h(e/l,a) is F,$F,,. Both F,, and Fl,, are plotted in Fig. 4. The factor F,, has been defined earlier and the factor F,, is a ratio of l/N for the BCS case to l/N for the DFZ case provided that the applied force G is kept constant. The function h is found to be independent of arc and is equal to one as e/l = 0. The ratio N/G for the BCS case is plotted in Fig. 5.

9. CONCLUSIONS The inclined pileup of screw dislocations at a crack tip with a region free from dislocation is solved by the extended method of WienerHopf. The stress distribution along the line of pileup, the condition of finite stress at the end of plastic zone, and the total number of dislocations (the crack opening displacement) are obtained in analytical expressions. These quantities can be used to discuss the dislocation generation and crack tip shielding due to dislocations. For a specific metal, a conclusion on ductile versus brittle fracture can be reached by the usual argument. As mentioned earlier, several other fracture problems may be discussed by applying the present solution. Only screw dislocations have been discussed in the text. More often edge dislocations are piled up along an inclined direction to the crack for the specimen under mode I loading. The elastic interaction force between two edge dislocations has a more complex form as is expected but the first order interaction term assumes the same form as that for the screw dislocations. Therefore the integral equations of pileups for screw dislocations versus edge dislocations are only different for externally applied loads, that is, the different stress intensity factors and their angular dependence on the applied stress. Since the method of solution is only concerned with the solution of the integral, an estimate of the mode I or mode II solution can be approximated immediately by using the mode III solution with the stress intensity factor of the other mode. Based on this approximation, the plastic zones for mode II case and mode III case are plotted in Fig. 6. Acknowledgement~The research for the first author was supported by the Division of Mathematical Science and the Division of Materials Science, U.S. Department of Energy under contract DE-AC05840R21400 with Martin Marietta Energy Systems, Inc. The research for the second author was supported partially by AR0 DAAG 2985-K-0134, NSF-MRL DMR-8216972.

REFERENCES [I] S. M. OHR and J. NARAYAN, Phil. Mag. A41, 81-92 (1980). [2] S. KOBAYASHI and S. M. OHR, Phil. Mag. A42, 763-772 (1980). [3] S. M. OHR, J. A. HORTON and S-J. CHANG, In Defects, Fracture and Fatigue (Edited by G. C. SIH and J. W. PROVAN), pp. 3-15. Martinus Nijhoff Publishers (1983). [4] R. THOMSON, J. Materiais Sci. 13, 1288142 (1978). [S] J. WEERTMAN, Acta metall. 26, 1731-1738 (1978). [6] S.-J. CHANG and S. M. OHR, In Dislocation Mode&g qf PhysicalSystems (Edited by M. F. ASHBY). pp. 23-27. Pergamon Press, Oxford (1981). ]7] S.-J. CHANG and S. M. OHR, J. appl. Phys. 52, 7174-7181 (1981). [S] S. M. OHR and S.-J. CHANG, J. appl. Phys. 53, 564555651 (1982). [9] R. THOMSON, In Solid State Physics (Edited by F. SEITZ and D. TURNBULL). In press. [lo] S.-J. CHANG and S. M. OHR, J. appl. Phys. 55, 350553513 (1984). Ill] J. R. RICE and R. THOMSON, Phil. Msg. A29, 73-97 (1974).

Inclined

pileup of screw dislocations

00

Fig 6. A comparison

[I?] 1131 1141 [IS] [I61 L17] LlX] 1191 [ZO] 121 J

at the crack

02

tip wth

04

06

a dislocation-free

08

rone

575

10

of mode II versus mode III plaatlc zones.

B. A. BILBY, A. H. COTTRELL and K. H. SWINDEN, Pro<. R. SM. Lord. A272, 304 312 (1963) B. A. BILBY and K. H. SWINDEN. P WC. R. Sot. Lord A285, 22-23 (1965). V. VITFK, J. Mech. Phys. Solrds 24, 263-276 (1976). H. RIEDEL. J. Mrch. Phrs. Solids 24, 277 -289 (1976). G. P. CHEREPANOV. J. uppl. Murh. Mech. 40, 720-728 (1976). S.-J. CHANG. Q. J. .tfcc,h. appl. Math. 24, 423~m443 (1971). J G. KUANG and T. MURA. J. uppl. Phrs. 40, 5017-SO21 11969). M. 0. ‘TUCKER, J. Mech. Phys. Solids 21, 41 l-426 (1973). B. NOBLE. The Wiener-Hopf Technique. Pergamon Press. London (1958). R. THOMSON and J. E. SINCLAIR, Actu merall.30, 13251334 (1982).

APPENDIX

A

To obtain values of G+(l + xn). G+(l + /hl), H?(l ~ zn) and H’(I ~ [In) (n = I. Z....) from cquatlons (4.1) and (4.5). we substitute s = I + x,x and I + /jn in equation (4.1) and s = I - an and I - /jn in equation (4.5). A system of algebraic equations is obtained that is represented in matrix notation as:

I.4 I I

lwherc

the vectors

G and H have the form

G’ = ICC;, + T,HI).(G+ + r,xl (G,

+ TzHl + Z/%...(G+

+ Zl.(G+ + T,HI + /&(G+ + r,)(I + r,)(1 + mrl.(G+

+ ‘z).

+ 7;HI + m/j)]

(A71

H’ = [H’(l -zJ,H<(l -/j).H?(l -&).H?(I -2/j),....HL(I -mx),H?(l -m/j)] The matrix c‘ IS of order 7m + I x 2m and the matrix D is of order 2m x 2m + I with the followng

(A?I expressIons:

Zm

K+(lJU,f~“’ 1-z-I

K+(l)h,r~‘+P,,,~K+(I)u,r ‘-

I-b-1

Itrnz l-ma-l

K+(l)h,t~“” ‘_1--mfi-

I

I

S-J. CHANG

K_(l

-a)pt

K-(1

I - (1 -z)’ K_(l

-&t

1 -(1

.

D=

/

K-(1

.

-%)c,r’+’

1 + 1 -(I K-(1

-fit’1

-,/l)c,t””

+I-(1

.

.

- ma)pr K_tl

K-(1

“- J,’

-z)d,t””

K-(1

.

- mak,f’+a

K..(l

I + p ~ (I -- a) %

-.jii’

.

and 7‘. MlJRA

-/l)d,r’+P

I +p-Cl

. K_(l

.

K(l

-fir”“‘--

.

+ nxa K..(l

.

.

.

!-_(I

-m~)c_t’+~’

K-(1

-mi)‘

-m&x

1 +a--(1 K(1

.-ma)' 1 -4-p-(1

- m@)c,r’*’

K-(1

-ma)“‘.

- m,8)d,t’+P

1 i-ma-f1 I(..(1

1-(l-~-m~)‘1+z-(1-m~)‘1+~-(1-m~)”“

-ma)‘!

-m&mt’*m’

1 +mcc-(1

.

.

K_tI

I.~

1 -(l

--- (I - 1)

K.(I -- /j)d,r’+*” ~. +mp-(I -p,

-[f)‘I

.

- x)d,r”““f

xl-.TT-iB

- /J’)c,J”“’

1 +ml-(l

.

- maBd,PiS

- n)c,r’

I + “,OL- (1 -

-m/?)‘-1

.

.

-mrkf_t”m8 “.

+mfi--(1

;

-mmzf

I(_(1 - m/?)d,t”“~ +mp-(1 -m/I)

L The vector E’ has the form

where

The vector F’ is zero. In the calculation for the vectors G and H, we have made use of the DFZ condition as described by equation (7.3). From this condition, the constant A is solved by incorporating the additional equation, (7.31,

+ to the system of equations,

d,t: “B”(G+(l

(Al). (Received

ii

March

19%)

+ Bn) + &(I

+ ,Gi))] = 0

(A@