Ductile-to-brittle transition caused by dynamical work hardening at a crack tip

Scripta METALLURGICA

Vol. 23, pp. 383-388, 1989 Printed in the U.S.A.

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DUCTILE-TO-BRITTLE TRANSITION C A U S E D BY D Y N A M I C A L W O R K H A R D E N I N G A T A C R A C K TIP K. Maeda and S. Fujita Department of Applied Physics, Faculty of Engineering, Universityof Tokyo Hondo, Bunkyo-ku, Tokyo 1 13, Japan (Received December 30, 1988)

InIrto~l~c~ion Rice and T h o m s o n ( I ) p r o p o s e d that a material will be ductile or brittle according to w h e t h e r a d i s l o c a t i o n can o r can not be emitted from crack tips and thereby shield the concentrated stress field b e f o r e the stress intensity factor K reaches the ideal fracture toughness K c. Ohr and Chang (2) a p p l i e d the same argument to Mode I l l cracks and obtained a f o r m u l a for the critical stress intensity factor KIIIe required to generate a d i s l o c a t i o n at the crack tips:

Km*

~

+

where #, b, r c are the shear modulus, the Burgers vector of the dislocations to be emitted and their core radius, r e s p e c t i v e l y , a £ is the frictional stress w h i c h dislocations m u s t o v e r c o m e in o r d e r to glide and is u s u a l l y f i x e d c o n s t a n t e v e r y w h e r e in the crystal. This model asserts that the increase of oi, with d e c r e a s i n g temperature, w h i c h eventually satisfies the condition Ke > Kc f o r brittle fracture, brings about the d u c t i l e - t o - b r i t t l e transition (DBT). This model qu-alitati-vely explains also the decrease of Kic w i t h hardening of crystals by obstacles o b s e r v e d in many metallic alloys (3). In covalent c r y s t a l s , however, the concept of the frictional stress is not clear, because the flow s t r e s s , a usual measure of o £ , depends s e n s i t i v e l y on the strain-rate. St. J o h n (4), later Brede and H a a s e n (5,6), and more recently H i r s c h et al (7,8). p e r f o r m e d measurements o f fracture t o u g h n e s s in Si single crystals and s h o w e d that the DBT t e m p e r a t u r e T c increases w i t h increasing l o a d i n g rate d P / d t or d K / d t . They also s h o w e d that the relation between I o g ( d K / d t ) and I / T c f o l l o w s an Arrhenius type equation w i t h an activation energy close to that f o r d i s l o c a t i o n glides in Si c r y s t a l s . This fact s u g g e s t s that dynamics of d i s l o c a t i o n glides plays an i m p o r t a n t role in the DBT phenomenon. Although this effect might be explained in terms of o f if one assumes an increase of o f w i t h strain-rate, the functional dependence of o f on the e x t e r n a l p a r a m e t e r d P / d t is not at all ODVIOHS.

St. John (4) e x p l a i n e d the strain-rate dependence of T c in terms of crack-tip blunting by plastic d e f o r m a t i o n w h i c h competes w i t h increase of applied stress. Haasen (9) and Brede and H a a s e n (6) analyzed this blunting p r o b l e m at a more m i c r o s c o p i c level in terms of d i s l o c a t i o n e m i s s i o n from a crack tip. H i r s c h et al. (7) p r o p o s e d a d i s l o c a t i o n s h i e l d i n g model w h i c h considers screening of the crack tip stress field by dislocations m u l t i p l i e d at sources near the crack tip and f o r m a t i o n of the multiplication sources b y cross slip of p r e - e x i s t i n g dislocations that have traveled,

383 0036-9748/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press plc

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TRANSITION

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with a finite time, from far places I. In their model the strain-rate d e p e n d e n c e of T c arises from the finite time required for dislocations to travel from some places in the crystal to vulnerable parts of the crack tip, which ought to be in i n v e r s e p r o p o r t i o n to the d i s l o c a t i o n mobility.

~.- xj--.] _ _

s-s-s-s-s-s

(a)

F I G . 1 Two modes of fracture (a) Mode III; (b) Mode I

(b)

S_imu_La_tio_n_o__f__diil_ocation__emiss_ion_~t__a__cr~K~ In the usual argument of d i s l o c a t i o n shielding models, the value of K e n e c e s s a r y for d i s l o c a t i o n e m i s s i o n is a s s u m e d to be invariable all through the s u c c e s s i v e e m i s s i o n s . H o w e v e r , this a s s u m p t i o n is not quite correct. In w h a t f o l l o w s we treat, for the sake of simplicity, mainly Mode I l l cracks with screw dislocations on a s l i p plane coplaner with the cleavage plane (Pig. 1(a)). The force acting on a unit length of d i s l o c a t i o n located at a distance x from a crack tip in the presence of other dislocations at ~ is given by (10) 1{2

(2)

E./

=

,

where K is the stress intensity factor in the absence of d i s l o c a t i o n s . If one assumes that a d i s l o c a t i o n is emitted only when £d (re) becomes positive,

K. =

.b

.b E

÷

--0

(3)

The s e c o n d term on the right hand side o f eq. (3) is variable d e p e n d i n g on the p o s i t i o n s of other dislocations already emitted. Since this term is p o s i t i v e for emitted dislocations (b • 0 ), it represents a s o r t o f w o r k hardening at the crack tip in a m i c r o s c o p i c scale. This term has taken the place of the s e c o n d term of eq.(1) arising from of. The force acting on a crack is ( l 0) k=

(4)

Here the local stress intensity factor k is given by ( I 0 )

k =K - ~ .

pb

(5)

r e p r e s e n t i n g the d i s l o c a t i o n s h i e l d i n g effect. We assume that fracture takes place w h e n the G r i f f i t h condition (I l ) lln a more recent paper (8), they r e c o n s i d e r e d their p r e v i o u s l y d i s c a r d e d situation (7) that s h i e l d i n g of the m o s t vulnerable parts of the crack can be achieved only after p a s s a g e o f dislocations multiplied at sources located along the crack f r o n t but far from the concerned points.

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£,-2r

385

(6)

is s a t i s f i e d w h e r e I' is the surface energy per unit area. Hence, the condition for crack extension is, from eqs. (4) and (6), k > k. = 2 3 ~

(7)

and therefore the apparent fracture toughness is written as

pb

=

23/-~

(8)

+

X i

It is experimentally k n o w n in covalent crystals (I 2) that a dislocation driven by a force fd glide with a velocity

v , = C ( f ,)=exp(" - - ~ ) '

(9)

w h e r e the factor C , the stress e x p o n e n t m and the activation energy E are all constants i n d e p e n d e n t of £d and temperature T. C o m b i n i n g eqs. (2) and (9) and using the e m i s s i o n c o n d i t i o n e x p r e s s e d by eq. (3), one can calculate the time e v o l u t i o n of d i s l o c a t i o n p o s i t i o n s from a crack tip w i t h increase of K. F o r a general a r g u m e n t w e a d o p t the f o l l o w i n g units for r e s p e c t i v e quantities; b for distances, p ( b / 2 ~ r ) I/2 for stress intensity factors, pb / 2 x f o r forces on a d i s l o c a t i o n and (2 x / # b ) m (b / C ) e x p ( E / k T ) for times. Equations (2), (3), (5) and (9) are rewritten to d i m e n s i o n l e s s equations using these n o r m a l i z e d quantities i n d i c a t e d by < >. .

-

l

{K.> 2(x/(r)

+v

(J',>--

It>

. E (%>. l

~ -
(k) = (K) - E 1 I (3~1)

'

~

112

,

(io)

(11)

(12)

and d
(13)

On the basis of these equations w i t h - I , sequences of d i s l o c a t i o n e m i s s i o n under various values o f d/d w e r e simulated in a computer. F i g u r e 2 s h o w s the calculated variations of the local stress intensity factor < k > w i t h < t >. The horizontal axis in Pig. 2 is s c a l e d in i n v e r s e p r o p o r t i o n to d/d. One s h o u l d note that, since the external parameters that g o v e r n the time e v o l u t i o n o f the d i s l o c a t i o n d i s t r i b u t i o n are o n l y d K / dt and T and they are connected to each other

by

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04) the system of crack and dislocations follows the same trajectory in the (K, x i ) space as far as the value of d/d or the ratio of e x p ( - E / k T ) to d K / d t is the same. Therefore one can view Fig. 2 as k ( t ) curves plotted for various temperatures u n d e r a common value of d K / d t or otherwise as £ ( t ) curves plotted for various values of d K / d t at a common temperature. I n any case, w h e n < K > reaches < K e 0 > (the first term of eqs.(3) and (11)), the first dislocation is emitted and s i m u l t a n e o u s l y the local stress intensity factor < k > drops to a substantially low level ( becomes negative in this mode). As the emitted dislocation is expelled from the crack tip by the stress field, increases again and exceeds the previous level because < K > increases steadily in the meantime. When < K > becomes high ( < K > > < K e >) enough for the second dislocation to be emitted, drops again. If exceeds the critical value (eq. (7)) before this happens, the crystal will fracture; otherwise the above process will repeat until eventually reaches < k c >. As demonstrated in Fig. 2, the value of < K > at which fracture occurs becomes higher with decreasing d/d. This is due to the fact that the dislocation shielding works effectively only if dislocations once emitted have sufficient mobility to get remote from the crack tip so that the next dislocation can be emitted successively. In other words, through such a dynamical effect, the work hardening effect becomes greatly enhanced at large values of d/d< t >.

(K) , "

,"" 1.0

/~x /

//

/

I

i

/ ' / ' I ..":'./ "' / ,."i.."': A/~.'i" ," , - 2 10-1 . /z.'i ,, ~o- ....!/./... ! i "'~

VVV

20.5 v

-0.5

d (K)/d (t) =10 1 //

0



J 3 d(l~/d(t)

FIG. 2 Variations with dimensionless time < t > of the local stress intensity factor <• > for various values or dlcl. Note that the horizontal scale Is in inverse proportion to d / d < t >. The apparent fracture toughness or the critical value of < K > at which fracture takes place is plotted in Fig. 3 as a function of d/d using an approximation y , p ( l + v) / 10 (14) (.'. = 1.75). Here v is the P o i s s o n ratio which was taken to be 0.22 in this paper bearing Si in mind. As previously noted, whatever the respective value of T and dK / dt may be, the dislocation configuration developed under the same value of d/d is determined solely by K at each moment; the only difference is in the time at which the system of crack and dislocations reaches the same state. As eq.(14) holds, this means that the DBT temperature T c must satisfy E

dK

Cp

( Pb'~'cl(Ki

= constant,

(15)

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387

where d < K > / d < t >[c represents the value of d < K > / d < t > below which < K c > starts to increase sharply. The Arrhenius-type relation between d K / d t and T c experimentally observed (4-8) is thus explained.

9

T (K)

~o

= 175 z.,

Ii

#

3~ ,¢,-

i

I

I

10 I (2'n)3/2 e

102 /E~dK

FIG. 3 Dependence of apparent fracture toughness < K c > in Mode III on d < K > / d < t > calculated for < k c >= 1.75. Temperature and K c scales comparable with experiments in Si (4-8) are indicated by the top and right scales for d K ~ d r = 4.5 x 104 P a ' ~ - s - 1 . The scatters around the smooth curve arise from the discreteness of the k ( t ) curve in Fig. 2.

__Dis_cu_ss_ion The top and right scales in Fig. 3 indicate T and K c , respectively, for a fixed value of d K / d t which can be compared with the experiments in Si (b = 3.84 x l0 "10 m, p = 68 GPa, C = 8.6 x 104 p a - l s - l ( 6 ) , m = 1 (13), E = 1.66 eV (13), d K / d t = 4.5 x l04 Pa, m)m-s'l(6)). The c a l c u l a t e d D B T temperature is, however, in a range far below the experimental one (= 1000 K (6)). This discrepancy is considered to be due partly to the use of Mode lII in stead of Mode I, the actual mode of fracture in the experiments, and partly to the experimental geometry of the crack and the dislocation glide planes, which is not ideal for maximum shielding. For simplicity, we consider a Mode I crack with edge dislocations on a slip plane cutting the cleavage plane at the crack tip with an angle 0 (Fig. l (b)). The force acting on a dislocation at the crack tip is, in stead of eq. (2), written as

(I0)

f# (r)

Kb

2V/2#r

sin 0 cos 0

2

Wb~

" 4tr(1 - v ) r

~ "

# b2 4Jr(ll - v) g ( r ' r i ; O )

(16)

where r and r i are positions of dislocations from the crack tip. The function g (r, r i ;8 ) represents dislScation interaction which depends on O in a complicated manner. Similarly eq.(5) for the local k is replaced by (l 0) 0 ~d ~ / ~/Jb- ~ j • k = K - 2(1 3- v) sin 8c.o6 ~-

(17)

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Comparison of eqs. (16) and (17) with eqs.(2) and (5) indicates that if the factor s i n e c o s 8 / 2 is sufficiently small, K e 0. the value of K necessary for emission of the first dislocation, can be larger than that in Mode l l l w h i l e the effect of dislocation shielding, expressed by the second term of eq. (17), can be conversely smaller. Thus Mode I fracture can occur more readily and therefore the DBT temperature can be much higher than that calculated for Mode III. Calculations for Mode I cracks, which consume much computation time though, are now in progress.

Ackn~ent This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education and Culture.

I. 2. 3. 4. 5. 6. 7. 8. 9. I0. I I. 12. 13. 14.

F~.~eren_ces J.R. Rice and R. Thomson, Philos. Mag. 29, 73 (1974). S.M. Ohr and S. -J. Chang, J. Appl. Phys. 53, 5645 (1982). D. Broek, Elementary Engineering Fracture Mechanics, Noordhoff, Leyden (1974). C. St. John, Philos. Mag. 32, I193 (1975). M. Brede and P. Haasen, Proc. 7th Int. School on Defects in Crystals, p.529, World Scientific Press, Singapore (1985). M. Brede and P. Haasen, Acta metall. 36, 2003 (1988). P.B. Hirsch, S. G. Roberts and J. Samuels, Revue. Phys. Appl. 23,409 (1987). P.B. Hirsch, S. G. Roberts, J. Samuels and P. D. Warren, Proc. 8th Int. Conf. Strength of Metals and Alloys, p. I083, Pergamon Press, Oxford (1988). P. Haasen, Atomistics of Fracture, N A T O Conf. Ser. VI: Materials Sciences, p.707, Plenum Press, N e w York (1983). R. Thomson, Solid State Physics vol. 39, p.2, Academic Press, London (1986). A.A. Griffith, Philos. Trans. R. S oc. London Ser. A221, 163 (1920). For review, see H. Alexander, Dislocations in Solids vol. 7, p. 113, North-Holland, Netherlands ( 1986). M. Imai and K. Sumino, Philos. Mag. A 47, 599 (1983). J . J . Gilman, Strength of Ceramic Crystal, Amer. Cer. Soc., New York (1962).