Elastic interaction between screw dislocations and collinear edge and internal cracks

Engineering Fracture Mechanics Vol. 39, No. 3, pp. 59142,

0013.7944/91 $3.00 + 0.00 Q 1991 Pergamon Press pk.

1991

Printed in Great Bntam.

ELASTIC INTERACTION BETWEEN SCREW DISLOCATIONS AND COLLINEAR AND INTERNAL CRACKS tTelecommunication fDepartment

EDGE

SHAM-TSONG SHIUEt and SANBOH LEE# Laboratories, Ministry of Communications, P.O. Box 71, Chung-Li, Taiwan, R.O.C.

of Materials

Science

and Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C.

Abstract-The elastic interaction between screw dislocation and collinear internal and surface cracks has been investigated by the dislocation modelling method. The dislocation distribution function to simulate the cracks, the stress field and the stress intensity factor at the crack tip are obtained. From the stress field, we have derived the force on the dislocation, the zero-force position and the critical stress intensity factor for dislocation emitted from the crack tip. The critical stress intensity factor for dislocation emitted from the crack tip is affected by the other crack. The dislocations inside the crack also play an important role in fracture. Finally, four special cases can be obtained from our results: (1) the interaction between a dislocation and a surface crack; (2) the interaction between a dislocation and an internal crack, (3) the interaction between two dislocations with a surface crack; (4) the interaction between a dislocation and an internal crack near a free surface.

1. INTRODUCTION IT IS KNOWN that the dislocations

in the neighbourhood of crack tip play an important role in fracture mechanics. Thus the elastic interaction between dislocations and crack has been studied by many. Bilby et aZ.[l] used the continuous dislocation distribution to simulate both the internal crack and plastic zone and found that the size of plastic zone agrees excellently with the result of Hult and McClintock[2] using the classical plastic theory. Rice and Thomson[3] studied the ductile versus brittle behaviour of material and found that the ratio of theoretical shear strength to tensile strength to determine the toughness of material proposed by Kelly[4] was not a good model. They proposed that the criterion of dislocation emission from the crack tip was also an important parameter to determine the toughness of material. Shiue and Lee]51 studied the slip and climb components of image force on the mixed dislocation in the neighbourhood of the semi-infinite crack tip. Louat[6] used the screw dislocation distribution to simulate the internal crack and derived the stress field arising from the lattice dislocation and the image force on the dislocation. Lee[7] pointed out that the dislocation inside the finite crack plays a very important role in fracture and corrected the image force on the screw dislocation due to elliptic hole derived by Lung and Wang[S]. Chang and Ohr[9] found the dislocation-free zone between the internal crack tip and plastic zone which was not obtained by Bilby et a1.[1]. Chu[lO] analysed the elastic interaction between the surface crack and screw dislocation using the conformal mapping technique. Lee]1 l] and Juang and Lee[ 121 investigated the elastic interaction between dislocation and surface crack and proved that Newton’s third law is satisfied. Lin et a/.[ 131compare the elastic behaviour of surface crack and internal crack interacting with the screw dislocation. Shiue and Lee[14-151 studied the elastic interaction between screw dislocations and cracks emanating from an elliptic hole and proposed the effect of crack blunting on fracture. Shiue et aZ.[16-171 studied the elastic interaction between screw dislocations and a welded surface crack in composite materials. Those investigations stated above were applied to a single crack, i.e. semi-infinite crack, surface crack, internal crack and crack emanating from the elliptic hole. It thus stimulated us to study the interaction between screw dislocations and collinear cracks. In this paper, the crack surface is simulated by the pile-up of dislocations. Then the stress field and stress intensity factor at the crack tip are obtained. From the stress field, the force on the dislocation, the zero-force points and the critical stress intensity factor for dislocation emission are derived. The dislocation source is also discussed. 591

SHAM-TSONG SHIUE and SANBOH LEE

592

Y

b

b

0

0

lb)

(4

Fig. 1. Schematic diagram of a screw dislocation (a) and collinear internal and surface cracks; (b) and its image near collinear three internal cracks in an infinite medium.

2. DISLOCATION DISTRIBUTION FUNCTION The problem considered is as follows. An internal crack of length (a - b) and a surface crack of length c are coplanar on the xz plane as shown in Fig. l(a). A screw dislocation of positive Burgers vector b, (RHFS) is situated at zO(= x,, + iyO). This problem is equivalent to the mirror image with respect to the free surface (_yzplane) so that the stress component ~~~on the free surface is zero, i.e. the problem can be reconsidered as three internal cracks interacting with a pair of antisymmetric screw dislocations in an infinite medium as shown in Fig. l(b). Using the dislocation modelling method, the cracks can be simulated by a distribution of continuous dislocations and the method is the same as that derived by Juang and Lee[lZ]. The distributionf(x) inside the crack arising from a lattice dislocation at z,, under the action of mode III applied stress cr is x2 - c2 ~z;-x2

+&-=&/m x2 -c*

(b2 - c’)l-I(&

--

&p _m71

w

K(k)

(2;

+X2

-

b2)K(k)

+ 2ax[x2 - (a2 - c2)E(k)/K(k) pJizy/~Jn

k)

(b2 - c’)l-I(u;, k)

(z; - b2)K(k)

1

1 - c’] ’

(1)

where

u;= (z; -

c2)(a2- b2)/(a2 - c’)(z; - b2),

k2 = (a’ - b2)/(u2 - c2),

(24 (2b)

and K(k), E(k) and lI(at, k) are the first, second and third kinds of complete elliptic integrals as defined in ref. [18], respectively. 4 and $ are the complex conjugates of z,, and a,, respectively. The value m in eq. (1) represents the number of dislocations inside the internal crack which is satisfied with the following condition, :f(x) dx = mb,. (3) s Assume no dislocation is inside the internal crack before a lattice dislocation is created. Based on the Burgers vector conservation, when the lattice dislocation is emitted from the internal crack, m is equal to - 1. On the other hand, if the lattice dislocation is come from infinity or the surface crack, then m is zero.

593

Elastic interaction between dislocations and cracks

3. STRESS FIELD

The complex stress field S(z) is arising from the dislocation distribution inside the crack, the lattice dislocation and its image, and the applied stress. Using the method proposed by Juang and Lee[l2], we obtain the stress field S(z) as S(z) = z,, + irx; =-

& 2lt X

z:-c:

+dm,/m

I (b2-c2)II(a;,k) (z; - b’)K(k)

z -zo

+(2; (b2 - c2)II(Ef

k)

22-5;

JW

+ m x,/m

b*)K(k)

zz-c2

K(k)

+az[z2-(a2c*)E(k)/K( c'] II Jgyi&r-gJ~

.

@I

It is noted that the symbol and subscript of z represent the complex variable and Cartesian coordinate, respectively. The stress components have been checked by numerical calculation and the boundary conditions are satisfied. The boundary conditions are rXZ= 0 at x = 0; rYZ= 0 in the range 0 d x c c, b < x < a and y = 0; ryr = u, at y = f co. There are four special cases worthy of mention. Firstly, when b is equal to c, the problem becomes that of a surface crack of length a[12]. Secondly, when all z, zO, a and b approach infinity and their differences remain finite, then the problem is reduced to that of an internal crack[6, 131. Thirdly, when a = b, it reduces to a dislocation of Burgers vector mb, located at a (or b) without the internal crack. Finally, when c is equal to zero, the problem becomes that of an internal crack near a free surface[l9]. 4. STRESS INTENSITY FACTORS The stress intensity factors at the right-hand obtained as

and left-hand

side internal crack tips are

K”= lim [J271(~-u))r,~,,=ol=K”s+K~+KP, x-u+

]

-m&}

W

and

+ia$Z

b2 - c2

(a2 - c2)E(k)

a2

(b2-

J-7

c2)K(k)

1

_ 1

’

(5b)

594

SHAM-TSONG SHIUE

and SANBOHLEE

respectively. The stress intensity factor at the surface crack tip is obtained as K’ = lim [v/20rYz,,0] x-e+

= Kg + Kf, + Kf4

All the K”, Kb and K’ include three parts, and the subscripts S, m and A represent the stress intensity factors induced by lattice dislocation (m = a = 0), m didocations inside the crack and apphed stress a, respectively. It is noted that the stress intensity factors induced by applied stress K;, Ki and KEAare in good agreement with the results of Sih[20], if the constant is properly selected. The contours of position of dislocation to generate the same stress intensity factor K”, Kb and K’ are shown in Fig. 2(a), (b) and (c), respectively, where a = 0, m = - I and a/3 = b/2 = c. It can be seen that the shape of the contour shown in Fig. 2(a) and (b) looks like a spider whose head, i.e. the region between the surface and internal cracks, was indented symmetrically with respect to the x-axis. On the other hand, the contour shape of Fig. 2(c) looks like an octopus. Note that Ku and Kc have always different signs from Ki and Ki, i.e. the disl~ation plays the role of preventing crack propagation; but Kb have the same sign as Kf,, so it will enhance the crack movement. The shielding (or enhancing) effect of dislocation on the stress intensity factor is pronounced when the dislocation is close to the tip. When the dislocation is near the free surface, the contour is flattened. When the lattice dislocation is moved into the internal crack, then no dislocation exists inside the internal crack. That is, no dislocation induces the stress intensity factor. On the other hand, when the dislocation moves into one of the free surface, surface crack and infinity, the dislocation of Burgers vector (-b,) is inside the crack, so the stress intensity factors K”, Kb and K’ are -0.615&/d-, 0.82~~b~]J~, and -0.46pb,/,,& respectively.

x/c Fig. 2(a).

Elastic interaction between dislocations and cracks

x/c Fig. 2(b).

0.5

<

1

0

-05

-1

-1.5

x/c Fig. 2(c). Fig. 2. The contours of dislocation to generate the same stress intensity factor (a) at the right-hand internal crack tip (KY), (b) at the left-hand internal crack tip (K*), and (c) at the surface a/3 = b/2 = c, a = 0, m = - I. The units of K”, Kb and Kc are -fi,/,/m, -~b,/&c, respectively.

595

SHAM-TSONG SHIUE and SANBOH LEE

596

-,&.n

0

0.125 I

Q25

a375

I 0.5

1 0.625

I a75

I

0.875

I

1

c/b Fig. 3(a).

.25 /

K,b, Q -135’

-025 -

-0.5 -

475

Kb m-l mL____

- __ _-_____

-1 0 b

I

a125

I

a25

I

a375

0.5

0625

0.75

0875

1

c/b

Fig. 3(b) Fig. 3. The stress intensity factor as a function of c/b, (a) at right-hand internal crack tip (K”): (b) at left-hand internal crack tip (K”). Here a/b = 1.5.

597

Elastic interaction between dislocations and cracks

In order to understand the effect of a surface crack on the stress intensity factor, the stress intensity factor K” as a function of c/b is plotted in Fig. 3(a), where a/b = 1.5. Note that the distance between the lattice dislocation and the right-hand side tip of the internal crack is (a - b)/2 and Y is the angle between the line from the right-hand side tip of internal crack to dislocation and the positive x-axis. It is found that (1) K”, and K; decrease with increasing the ratio of c to b, but K;; increases with increasing the ratio of c to 6; (2) when c is equal to b, the result is reduced to that of the surface crack of length a[12]; and (3) when c is equal to zero, the result is reduced to that of an internal crack of length (a - b) near a free surface[l9]. Alternatively, the stress intensity factor at the left-hand side tip of internal crack as a function of c/b is plotted in Fig. 3(b), where the distance between the lattice dislocation and the left-hand side tip of internal crack is (a - b)/2 and Y is the angle between the line from the left-hand side tip of internal crack to dislocation and the negative x-axis. It is found that (1) Ki and luk increase with decreasing the ratio of c to b, but Ki increases with increasing c/b; (2) when c is equal to 6, the left-hand crack tip disappears. It means that the stress intensity factor at the left-hand tip of internal crack is zero; (3) when c is equal to zero, the result is reduced to that of an internal crack of length (a - b) near a free surface[l9]; (4) when c approaches b (but not equal to b), Kk and Ki will approach infinity. Therefore, an internal crack which is very close to a surface crack will be easily extended to a surface crack. On the other hand, in order to understand the effect of internal crack on the stress intensity factor at the surface crack tip, the stress intensity factor at the surface crack tip as a function of (b - c)/c is plotted in Fig. 4, where (a -b) = c. The distance between the lattice dislocation and the surface crack tip is c/2 and II/ is the angle between the line from the surface crack tip to dislocation and the positive x-axis, It is found that (1) when angle I,? is larger than zero, the magnitude of Kg decreases abruptly with increasing (b - c)/c at the beginning and then increases slowly. When angle $ is zero, the magnitude of Ki will increase with (b - c)/c; (2) when b is equal to c, then the surface crack tip disappears. It means that Kc is zero; (3) when (b - c) approaches infinity, then the problem is reduced to that of a surface crack of length c[12]; (4) both magnitudes of Kf, and Kf, decrease with increasing (b - c)/c. When (b - c) approaches zero (but not equal

P

a

1

-.-

-

(b-cl/c Fig. 4. The stress intensity factor at the surface crack tip (Kc) as a function of (b -c)/c, (a - b)/c = 1.

where

598

SHAM-TSONG SHIUE and SANBOH LEE

to zero), Kf, and KA approach infinity. Therefore, a surface crack which is very close to an internal crack will also be easily extended to the internal crack.

5. FORCE ON THE DISLOCATION The force components formula[21] as pbj F, + iFy = X

on dislocation

per unit length can be obtained by Peach-Koehler

(c2 - a2 - b’)zi + 2db2z~ - a2b2c2

22,(.2~- a’)(zg - b’)(zfj - c2)

(b2-c*)z~~(ff~,k~

ZO

+ (z; - b2) (2; - c2)K(k) - 2zo - zo

It is found that Fz is equal to the negative value of summation of crack extension force at each tip, where the crack extension forces at right-hand tip of internal crack and at the surface are positive (cracks propagate along the positive x-direction) and the crack extension force at the left-hand tip of internal crack is negative (crack propagates along the negative x-axis). The implication is that the Newton’s third law is valid in this system. If the dislocation is situated on the crack coplanar surface (z. = x0), the eq. (6) is reduced to

F=& x

2lt

-xi

+ (3c2 - a2 - b2)xi + (3a2b2 - bZc2 - a2c2)xi - a2b’c2 2x0(x; - a2)(xi - b2)(xfj - c2)

+ 2(b2 - c2)xJI[kz(x; (x; - b;)(x;

m7E7 a -cx,

- c”),(x; - b’), k] + - c2)K(k)

J-,/m,/-

K(k)

(7)

and I;, is equal to zero. If the force on the dislocation is zero, then the dislocation can stay in the medium. Although the interaction of new dislocation disturbs the stress field of existing dislocation in the system, it is possible to form a plastic zone in the region containing the zero-force point by collecting other dislocations. The size of plastic zone determines the ductility of material. Since the force component Fy is always zero along the x-axis, eq. (7) can be adopted to obtain the positions of zero-force. Figure 5 shows the curves of applied stress versus the dislocation position of zero resultant force with the parameter m, where a/3 = b/2 = c. It is interesting to point out that even no applied stress acts on the system, the zero-force point exists in the region between the surface crack and internal crack. It is because both surface crack and internal crack attract the lattice dislocation, respectively. When m is negative, the number of zero-force point increases from one to two with increasing applied stress. When m is more negative, the zero-force points are farther away from the internal crack tip but closer to the surface crack tip. When m is equal to zero, both zero-force points are closer to the surface crack with increasing applied stress. When m is positive, the number of zero-force point varies from two to four depending on the applied stress. The m is more positive, the zero-force point positions are closer to the internal crack tip but farther away from the surface crack tip.

599

Elastic interaction between dislocations and cracks 161

I

I

I

I

. *I

I

I

I

R

o/3=b/2:c

-8 -

-12 -

x/to-bl Fig. 5. The positions of zero-force on the dislocation along the x-axis, where a/3 = b/2 = c.

6. DISLOCATION

EMISSION

CRITERION

According to Rice and Thomson[3], when a zero-force position of a dislocation is less than its core radius a,, the dislocation is spontaneously emitted from the crack tip. We extend their approach to our case. From eq. (7) we obtain the critical applied stresses trn for the dislocation emitted from the right-hand (a&), left-hand (cr;) internal crack tips and surface crack tip (ab) along the x-axis as -pbs,/m&j?,/m nD = 27rx, [x: - (a* - c2)E(k)/K(k) X

- c’]

-xy + (3c2 - a2 - b2)x’: + (3a2b2 - b2c2 - u2c2)xf - a2b2c2 2x,(x: - a’)@: - 6*)(x: - c’) 1

+ 2(b2 - c2)x, l-I[k’(x; - c*)/(x: - b*), k] + (x;

-

b*)(x: - c2)K(k)

Wl?CJ2TX,

dp/,2-,/^2Y(k)

I ’

(8)

where x1=a+a,,

for

o&,

(9a)

x, = b -a,,

for

ah,

(9b)

x1 =c +a,,

for

ab.

(9c)

and

Here we assume that m dislocations are inside the crack before dislocation emission. Combining eqs (5), (8) and (9), the critical stress intensity factors KD for the dislocation emission due to applied load from the right-hand and left-hand internal crack tips and surface crack tip are

SHAM-TSONG

600

SHIUE and SANBOH LEE

$_[

K+&fG

b2

c2 (a2- c2)E(k) _ 1

a*-

b2 (b*--c2)K(k)

~

(W

1’

respectively. The species Kg and Kb, as functions of c/b are shown in Fig. 6(a) by solid and dashed lines, respectively, where (a - b) = b = lOOa,. It is shown that K& decreases with increasing m. When m is large enough, a dislocation will be automatically emitted from the right-hand internal crack tip without any applied stress. The term K”, increases with increasing the ratio of c to b for large positive value of m, but decreases with increasing the ratio of c to b for large negative value of m. Thus, the influence of surface crack on the dislocation emitted from the right-hand internal crack tip is strongly dependent on the number of dislocations inside the internal crack. On the other hand, for the positive value of m, KL increases rapidly with increasing c/b until it reaches maximum, then decreases. When m is equal to zero, Kb, decreases slowly with increasing c/b in the range 0 SGc/b d 0.6 and then drops rapidly. It implies that when the surface crack is close to the internal crack, a dislocation is more easily emitted from the left-hand internal crack tip. It is noted that under an action of mode III applied stress, the Burgers vector of dislocation emitted from the left-hand internal crack tip is - b, not 6,. Hence when m is negative, the dislocations inside the crack will enhance the emission of dislocation from the left-hand internal crack tip. Alternatively, the critical stress intensity factor for dislocation emitted from the surface crack tip KS as a function of (b - c)/c is shown in Fig. 6(b), where (a - 6) = c = lOOa,. It is found that K’, increases with m. When the negative value of m is large enough, a dislocation will be automatically emitted from the surface crack tip without any applied stress. For the positive value of m, Kf, increases rapidly with increasing (b - c)/c until reaches maximum, then decreases, For semi-negative value of m, K;j increases with the ratio of (b - c) to c. Thus, the influence of internal crack on the dislocation emitted from the surface crack tip is also dependent on the number of dislocations inside the internal crack.

LO. 0

’

I

I

I

I

I

I

1

I

rbO I

1 I

/ ,fi 35-

30-

12,’

_c_

m--20

25z________-_-----

c/b Fig. 6(a).

$35

Elastic infraction

0

a2

Q4

0.6

between disi~ations

‘38

1

1.2

601

and cracks

1.4

1.6

1.8

2

(b-cl/c

Fig. 6(b). Fig. 6. (a) The stress intensity factors for dislocation emitted from the right-hand internal crack tip (K$) and left-hand crack tip (Xi) versus c/b are plotted as solid and dashed lines, respectively, where (a -b) = b = 100%; and (b) the stress intensity factor for dislocation emitted from the surface crack tip (KC,) versus (6 - c)/c, where (a - b) = c = lOOa,, and a, is the core radius of dislocation.

7. SUMMARY AND CONCLUSIONS The elastic interaction between a screw dislocation and collinear internal and surface cracks is investigated. Some important results are stated in follows: (1) The contours near the surface for all Kg, Kb, and lu: are almost parallel to the y-axis due to the effect of free surface. (2) Both Kg and Ki increase with decreasing the ratio of c to 6, but K$ and Ki increase with increasing the ratio of c to b. If m is positive, both K; and Kk increase with decreasing the ratio of c/b. (3) When angle J, is larger than zero, the magnitude of ius decreases rapidly with increasing (b - c)/c and then increases. When angle I&is zero, the magnitude of ICEwill increase with increasing (b - c)/c. If m is positive, the magnitude of KA and K; will decrease with increasing (b - c)/b. (4) The number of zero-force points along the x-axis varies from one to four, it depends on the applied stress and the m dislocations inside the internal crack. (5) Both Kc and Kh decrease with increasing m, but hrb increases with increasing m. For positive value of m, Kf; increases with increasing c/b; Kf, increases with increasing c/b until reaches maximum, then decreases rapidly; and K’, increases rapidly with increasing (b - c)/c until it reaches maximum, then decreases. For the negative value of m, K”, and Kb, decreases with increasing c/b; and Kf, increases with increasing (b - c)/c. When m is equal to zero, K$ increases with increasing c/b; K6, decreases with increasing c/b; and K’, increases with increasing (b - c)/c. (6) Our results can be reduced to the following four special cases. First, when b is equal to c, it can be treated as a surface crack[ 121.Secondly, when all z, zO,a and b approach infinity and their differences remain finite, it can be dealt with the internal crack[6, 131. Thirdly, when a is equal to b, this problem is reduced to the interaction of surface crack and two

SHAM-TSONG SHIUE and SANBOH LEE

602

dislocations and their Burgers vector are b, and mb,, respectively. Finally, when c is equal to zero, it becomes that of an internal crack near a free surface[l9]. Acknowledgement-This

work was partially supported by the National Science Council of the Republic of China.

REFERENCES Proc. R. Sot. A272, 304 (1963). J. A. H. Hult and F. A. McClintock, 9th Int. Cong. appl. Mech. 8, 51 (1957). J. R. Rice and R. Thomson, Phil. Msg. 29, 73 (1974). A. Kelly, Strong Solids, 2nd edn. Clarendon Press, Oxford (1973). S. T. Shiue and S. Lee, Engng Fracture Mech. 22, 1105 (1985). N. P. Louat, in Proc. First Znt. Conf Fracture, Sendai, Japan (1965), p. 117. Japan Society for of Materials (1966). [7] S. Lee, Engng Fracture hfech. 27, 539 (1987). [8] C. W. Lung and L. Wang, Phil. Msg. A50, L19 (1984). [9] S. J. Chang and S. M. Ohr, J. appl. Phys. 52, 7174 (1981). [lo] S. N. G. Chu, J. appl. Phys. 53, 8678 (1982). [ll] S. Lee, Engng Fracture Mech. 22, 429 (1985). [12] R. R. Juang and S. Lee, J. appl. Phys. 59, 3421 (1986). [13] K. M. Lin, C. T. Hu and S. Lee, Mater. Sci. Engng 95, 167 (1987). [14] S. T. Shiue and S. Lee, .I. appl. Phys. 64, 129 (1988). [15] S. T. Shiue and S. Lee, Phys. Stat. Sol. (a) 113, 365 (1989). [16] S. T. Shiue, C. T. Hu and S. Lee, Engng Fracture Afech. 33, 697 (1989). [17] S. T. Shiue, C. T. Hu and S. Lee, Mater. Sci. Engng A112, 59 (1989). [18] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integralsfor Engineers and Physicists, (1954). [19] S. T. Shiue and S. Lee, ht. J. Fracture (in press). (201 G. C. Sih, Handbook of Stress Intensity Factors, p. 2.2.5-1, Institute of Fracture and University, Bethlehem, Pennsylvania (1973). [21] J. P. Hirth and J. Lothe, Theory of Di.siocations 2nd edn, p. 109. John Wiley, New York

[l] B. A. Bilby, A. H. Cottrell and K. H. Swinden,

[2] [3] [4] [5] [6]

(Received 11 June 1990)

the Strength and Fracture

pp. 9-10. Springer, Berlin Solid Mechanics, Lehigh (1982).