Toughening by stress induced phase transformation

Scripta METALLURGICA

Vol. 19, pp. 935-939, 1985 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

TOUGHENING BY STRESS INDUCED PHASE TRANSFORMATION J. C. M. L i * and S. C. Sanday D i v i s i o n of Material Science and Technology Naval Research Laboratory Washington, D.C. 20375-5000

(Received April 15, 1985) ~Revised May 30, 1985) Introduction Materials can be made to r e s i s t f r a c t u r e by introducing mechanisms which can i n h i b i t crack propagation. Without these mechanisms the material is b r i t t l e such as is the case for glasses. One mechanism is p l a s t i c flow and t h i s is why metals are d u c t i l e . Another mechanism which has increased the toughness of ceramics by a f a c t o r of ten in the recent years is the stress induced phase transformation. Like p l a s t i c flow, a phase transformation could blunt the crack, reduce the stress concentration and increase the energy required f o r f r a c t u r e . However, McMeeking and Evans(1) and Budiansky, et a l ( 2 ) showed that a f t e r the transformation at the crack t i p caused by a d i l a t i o n a l m i s f i t the stress i n t e n s i t y f a c t o r is not reduced. In other words, the stress i n t e n s i t y f a c t o r in Mode I is the same before and a f t e r the transformation which takes place everywhere w i t h i n a region in which the hydrostatic stress exceeds a c r i t i c a l value needed for the transformation. This r e s u l t is puzzling. While the crack t i p stress f i e l d is the cause f o r the transformation, somehow the transformation does not modify the stress i n t e n s i t y factor. It is the purpose of t h i s note to attempt to c l a r i f y the u n s a t i s f a c t o r y state of a f f a i r s and to propose a proper way of c a l c u l a t i n g the stress i n t e n s i t y f a c t o r due to stress induced t r a n s f o r mation, A Simple Dislocation Model Let us consider a Mode I crack which covers the -x h a l f of the xz plane. Under an applied t e n s i l e stress oo , a stress i n t e n s i t y f a c t o r Ka = O~y / ~ a p ~ r s at the t i p with L being the length of the c r ~ k . Furthermore, a t e n s i l e stress Oyy = Ka/W2~x is created along the xz plane in f r o n t of the crack. Because o f t h i s t e n s i l e stresS, which can be much l a r g e r than the applied stress, a stress induced phase transformation could take place which can be modeled by a negative edge d i s l o c a t i o n at x and a p o s i t i v e edge d i s l o c a t i o n at x+a. The d i s l o c a t i o n dipole represents a new ~hase of thickness h which is small compared to "a" and has a normal s t r a i n of transformat i o n eyy.L These q u a n t i t i e s are related to the Burgers vector by of the dislocations as f o l l o w s : by = heTy

(I)

For a Mode I I crack the applied stress is O°v and the stress i n t e n s i t y f a c t o r is Ka = ax°v /~. A shear stress concentration of Oxy = K a ~ appears along the xz-plane in f r o n t of the crack. Because of t h i s stress concentration, a shear transformation can be induced to take place which can be modeled again by a d i s l o c a t i o n dipole which represents a new phase of t h i c k ness h and a shear s t r a i n o f transformation exTy. The Burgers vector of the d i s l o c a t i o n is again given by bx = heTy

(2)

S i m i l a r l ~ f o r a Mode I I I crack, the applied stress is o°wz ~ and the stress i n t e n s i t y f a c t o r is Ka = O~z~2~L. A shear stress concentration of Oyz = K a ~ appears along the xz plane in

*On leave from the Department o f Mechanical Engineering, U n i v e r s i t y of Rochester, Rochester, N.Y. 14627.

935 0036 9748/85 $3.00 + .00 Copyright (c) 1985 Pergamon Press Ltd.

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PHASE TRANSFORMATION

TOUGHENING

front of the crack which can induce a shear transformation to take place. dipole can model this transformation with a Burgers vector given by

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A screw dislocation

bz = heyz T

(3)

These three modes of crack extension are so similar that they are treated together. More realistic but more involved calculations have been reported by Porter, et al(3) but their calculations are too complicated to reveal the relationship between the work of transformation and the dynamic or the steady state stress intensity factor. Other dislocation models have been used by Seyler, et al(4) although no attempt was made to resolve the d i f f i c u l t y that the hydrostatic stress induced phase transformation has no effect on the stress intensity factor. The Work of Transformation The work of transformation can be calculated by allowing the crack to emit a positive dislocation to x+a and then a negative dislocation to x. The work per unit length (in the z direction) of the new phase is

~ ( ~ w = Kab q T , q a

+ l -

3

)+ Ab2 Zn

(~+ ~)2 ~ )

ro

(4)

a

where b is given by either of Eqs. (I-3), r o is the core radius of the dislocations and the quantity A is p/2~(l-~) for Mode I and I I and p/2~ for Mode I l l with p being the shear modulus and ~ the Poisson ratio. The f i r s t term which contains the length of the crack is more important for long cracks than the second term and i t is plotted in Fig. I . I t is seen that the work is always positive and is the largest when x=O. The Stress Intensity Factor However, the stress intensity factor K at the crack tip after the transformation is K = Ka - Ks where Ks i s the s h i e l d i n g e f f e c t ( 5 - 1 1 )

o f the d i s l o c a t i o n d i p o l e and can be c a l c u l a t e d ( l l )

Ks = ' A b J ~ ( 3

- ~ )

(5) t o be

(6)

which is also plotted in Fig. I . Since the crack is single-ended and a dislocation dipole is emitted, the effect pointed out by Lin and Hirth(12) does not apply here. I t is seen that Ks is negative for all positive x. In other words, the stress intensitY factor K is larger than Ka after the transformation. Instead of blocking the crack advance, the transformation appears to make the crack extension easier than before, provided of course that the intrinsic resistance for crack extension is the same as before such as twice the free surface energy. Because of the work done associated with the transformation, the energy of the system is lowered. In other words i f the applied stress (O~y, o~y or O~z) is maintained constant, i t s compliance w i l l increase during the transformatioB% Alternatively, i f the compliance is maintained constant, the applied stress will decrease instead. In any case, the energy of the system is reduced by the amount shown in Eq. (4). The ener~v of the system w i l l decrease further when the crack advances because w increases with decreasing x. The extra force exerted on the crack by the transformed phase (the dislocation dipole) can be calculated by

gd = - dx

/~

x

-

)+ 2a Wx -

Together with the force exerted by the external stress oyy, ° o or oo Oxy yz".

(7)

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PHASE TRANSFOR/Vaa.TION TOUGHENING

937

gO = K~I4~A the total force is g = gO + gd (9) = (Ka - Ks)2/4~A = K2/4~A confirming the relation between the strain energy release rate g and the local stress intensity factor at the crack t i p . Hence the crack extension force is actually increased by the transformation despite the fact that the transformation is caused by the stress amplification of the crack. While such analysis is correct, i t s t i l l is not satisfactory because i t does not reveal the extent of toughening by stress induced transformation. Maximum Toughenin9 Without Reverse Transformation When the crack extends further into the transformation zone, i t s stress intensity factor decreases due to a positive Ks and the crack should stop at K=O or at 2~AZ/K~ from the positive dislocation. Here we f i n a l l y see the toughening effect of the transformation. Further extension into the transformation zone the K becomes negative but the crack extension force is s t i l l positive for both Mode I I and Mode I I I cracks and the crack continues to advance in these cases. Another way of looking at the situation is that a negative K can cause reverse transformation so that the new phase may disappear and then the K returns to Ka. In the absence of reverse transformation, the average crack extension force while the crack advances through the transformation zone is

1 (a 1 la = ~ J fdx - 4~Aa (Ka - Ab 0

r0

K2 _

a 4~A

)2 dx

_

F-GbKa /~~a

+

(lo)

a Ab2 ~ i n ro

in which the last two terms turn out to be -w/a of Eq. (4) for x=O or r o. Since the last two terms must end up negative so as to have a toughening effect, the size "a" must be such that Ab ~r~-, a Ja> 2Ta~]~ ,n r-o I t turns out that the maximum effect comes about when ~ is twice that value. K2 a gmin = 4~A

(ll)

Then

Kab ~

which gives the maximum toughening effect for transformation zone the K value returns to Ka and the process repeats i t s e l f . As a result, force for repeated transformation in front of

(12) t h i s simple case. Whenthe crack comes out of the so that another transformation event takes place Eq. (12) gives the minimum average crack extension the crack.

Thus the foregoing consideration shows that, despite the variety of K values which can arise because of the transformation, the toughening effect comes from the energy or work of transformation. This work of transformation can be considered as part of resistance for the crack advance i f the driving force is s t i l l taken as K~/4~A or as a reduction of driving force as seen in Eqo (I0). Physically i f the crack advance is nothing but a bond breaking process, i t seems preferable to consider 2Y (where y is the surface energy) as the only resistance and the work of transformation a decrease of the driving force.

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PHASE T R A N S F O R M A T I O N

TOUGHENING

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Maximum Toughening with Reverse Transformation I f the transformation is reversible, the new phase w i l l disappear when the crack extends into the transformation zone so that K is zero or negative. However, as soon as the new phase disappears, the K returns to Ka and hence the new phase appears again. As a result the steady state situation would be a constant transformation zone of length a propagating with the crack. The crack t i p K is smaller than Ka but sufficient to maintain the transformation zone. Since the crack t i p condition remains the same a l l the time, the strain energy release rate is s t i l l K~/4~A (not Kz/4~A) per unit crack extension. However, part of the strain energy is used to do t~e work of transformation which is w/a with w given by'Eq. (4) for x=O or r o. The net driving force for the steady state crack extension with the transformation zone is the d i f f e r ence between the two which turns out to be the same as Eq. (10). Hence the maximumtoughening is s t i l l given by Eq. (12). The Proper K Value at the Crack Tip The agreement between the toughening effects of reversible and irreversible transformation suggests that a proper way of obtaining the crack t i p K for transformation toughening is to use Eq. (10): K:

J

K - 4~A~

(13)

which is seen to be always smaller than Ka and the extent of toughening depends on the work of transformation per unit crack extension. Eq. (13) can be used to determine the steady state shape of the transformation zone as shown in Fig. 2 for a Mode I crack resisted by a phase transformation of dilatational strain of m i s f i t . The stress intensity factor at the crack t i p is smaller than that supplied by the applied stress but sufficient to effect a continuous transformation in front of the crack t i p . Behind the crack t i p , the reverse transformation may take place. The steady state shape w i l l depend on the kinetics of both the stress dependent forward and reverse transformations. I t is noted that Eq. (13) cannot be used for a system dispersed with transformed inclusions and in which there is no more transformation taking place at the crack t i p . In other words the toughening effect of existing inclusions is different from that of stress induced phase transformation and w i l l be discussed in a l a t e r communication. References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12.

R.M. McMeeking and A.G. Evans, J. Am. Ceramic Soc., 65, 242-246 (1982). B. Budiansky, J.W. Hutchinson and J.C. Lambropoulos, Int. J. Solids Structure 19, 337-355 (1983). D . L . Porter, A.G. Evans and A.H. Heuer, Acta Met. 27, 1649-54 (1979). R.J. Seyler, S. Lee and S.J. Burns, Advances in Ce~mics, Vol. 12,213-22.4 (1984). N.P. Louat, Prec. First Int. Conf. on Fracture (Sendai, Japan, 1965) pp. I17-32. E. Smith, Acta. Met. 14, 556-7 (1966). J.R. Rice and R. Thom~n, Phil. Mag. 29, 73-97 (1973). E.W. Hart, Int. J, Solids Structures ~ , 807-823 (1980). B . S . Majumdar and S.J. Burns, Acta. Me-i-. 29, 579-588 (1981). J.C.M. Li, "Dislocation Modelling of Physical Systems," tEd. by M.F. Ashby, et a l , Pergamon, 1981) pp. 498-518. Shu-Ho Dai and J.C.M. Li, Scripta Met. 16, 183-8 (1982). I.-H. Lin and J.P. Hirth, Phil. Mag. A5_~, L43-L46 (1984).

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PHASE TRANSFORMATION

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939

Y OF TRANSFORMATION, ON Kab-,~=Z67~, b= he=Ty

O

x /a

I~3

KsinAb2 - [ ~ ' ~ ~

y

....

-21 Fig. I .

The Energy of Transformation of a New Phase

/ \

%

f •

Fig. 2.

d /

Steady State Shape of a Transformation Zone for a Mode I Crack