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Mechanisms of intergranular cavity nucleation and growth during creep
Scripta METALLURGICA
Vol. 17, pp. 23-30, 1983 Printed in the U.S.A.
Pergamon
Press Ltd.
VIEWPOINT
SET No. S
MECHANISMS OF INTERGRANULARCAVITY NUCLEATIONAND GROWTHDURINGCREEP S. H. Goods Sandia National Laboratories Livermore, CA 94550 and T. G. Nieh Lockheed Palo Alto Research Laboratory Palo Alto, CA 94304 (Received
October
14, 1982)
Introduction I t is generally recognized that creep-rupture of metals and alloys is caused by the formation, growth and coalescence of grain boundary cavities. A large body of l i t e r a t u r e suggests that the nucleation of cavities rather than their growth controls the fracture of materials over a wide range of temperatures, especially at temperatures associated with creep deformation (1-4). Thus when cavities such as voids or gas bubbles exist in materials prior to the beginning of deformation a severe loss in creep d u c t i l i t y often results. Because cavities are such a potent source of embrittlement, a great many studies have attempted to i l l u s t r a t e both experimentally and theoretically the role of cavity growth in the creep fracture process. However, understanding the effect of grain boundary cavities on creep fracture is complicated by the i n a b i l i t y to accurately characterize the cavity structure prior to the onset and during creep deformation. For materials in which cavitation, the simultaneous nucleation and growth of cavities, occurs continuously during creep, the kinetics of cavity growth are often obscured by the kinetics of cavity nucleation. In order to understand the importance of e|t~mr cavity, nucleation or growth in creep fracture, the nucleation event must be clearly and unmbiguously separated from cavity gro~rth processes. In this a r t i c l e the c r i t i c a l factors which must be considered in the study of creep cavitation are discussed. We f i r s t examine a number of key observations regarding the processes responsible for cavity nucleation, then discuss the importance of nucleation in creep fracture and conclude with an analysis of important theoretical and experimental results found in the recent l i t e r a t u r e which i l l u s t r a t e the micromechanics of cavity growth. Cavity Nucleation Because of the d i f f i c u l t y in resolving small voids or determining the location of incipient cavity formation, there exist many contradictory findings regarding the location and characteristics of l i k e l y nucleation sites. I t is clear that while voids may nucleate within grains at second phase particles, the intergranular nature of creep fracture indicates that the most damaging cavities are those which exist at grain boundaries and for this reason our discussion is limited to grain boundary cavity formation. The role of grain boundaries in the development of cavities in deforming materials has been investigated extensively (5-8). From early observations of the creep of copper-based and magnesium alloys i t was proposed that l a t t i c e vacancies condensed under the action of an applied tensile stress to form voids on grain boundaries oriented normal to the tensile axis. While excess vacancies can be generated by dislocation cutting and non-conservative motion, i t is hard to visualize how they can agglomerate on grain boundaries (which act as sinks for point defects). I t has been shown theoretically that in the presence of high grain boundary stresses, a supersaturation of vacancies may produce voids but the vacancy concentrations (or local stress) required are
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higher than might r e a l i s t i c a l l y be expected (9). Therefore the idea that stress driven vacancy condensation alone can account for cavity nucleation seems unreasonable. The contribution of the deformation process in the formation of grain boundary cavities has been well established. Nucleation invariably occurs in regions where deformation is non-uniform. I t is precisely for this reason that cavity nucleation and creep fracture are localized in grain boundaries. Some of the basic processes involved in creep cavity nucleation are i l l u s t r a t e d in Figure 1. The intersection of a slip band and a grain boundary is a l i k e l y site for cavity nucleation. Dyson et al. (5) using TEM have shown that this occurs in Nimonic 80A. Nucleation occurs readily under conditions which favor a change in deformation mode from bulk deformation to grain boundary sliding. Triple points (10) or ledges (11) induce s u f f i c i e n t stress concentrations to nucleate cavities in grain boundaries by blocking sliding. The Fig. i . Intergranular cavity presence of non-deforming particle in the boundary w i l l nucleation processes. have much the same effect. Thus, while the localized deformation must be accommodated by bulk p l a s t i c i t y , in each case decohesion of an interface is caused by tensile stresses that develop where plastic flow is inhomogeneous. In qualitative terms, all of the above processes have been thoroughly discussed in the l i t e r a t u r e . However, there exist only a few theoretical treatments which can quantitatively describe the onset of cavity nucleation (12)*. Most of these models require that nucleation occurs as a stress or strain localization at particles and that the particles are located well away from grain boundaries. They result in predictions of threshold strains below which nucleation does not occur. In predicting the creep behavior of metals these models are therefore deficient in several important respects. F i r s t , they ignore the morphology of creep fracture ( i . e . that nucleation and fracture invariably occur at grain boundaries as described above) and require that nucleation occur only at r i g i d particles. Further, while there is some evidence that there may be a c r i t i c a l nucleation strain for engineering and particle strengthened alloys, for materials subject to creep deformation the evidence is to the contrary (13,14). More recently, Argon et. al. (15) have addressed many of the specific micromechanical aspects of creep cavity nucleation. While theirs is the most accurate description to date, in quantitative terms, the influence of nucleation on the creep behavior of materials remains poorly defined. Even though theoretical descriptions of creep cavity formation are inadequate, the role of nucleation in determining the creep l i f e and fracture characteristics of materials can s t i l l be assessed. For many metals and engineering alloys, the product of the steady state strain rate, ~s~and the time to rupture, t r, is found to be a constant. This is the familiar Monkman-Grant relationship and is expressed as: Ess t r : CMG
[l]
Further, in the power law regime the creep rate can be related to the applied stress, o, temperature, T, by the expression: Ess = A gnexp (-Qc/RT)
and
[2]
where A is a structure dependent constant, n is the stress exponent t y p i c a l l y equal to 5 (above the power law have regime n w i l l vary and have a higher value) and Qc is the activation energy, similar to that for l a t t i c e self diffusion. Equation 1 indicates that the processes responsible for creep-rupture are controlled by creep deformation. Since the inhomogeneous plastic flow responsible for cavity nucleation must be accommodated by the bulk deformation of the creeping material, the a p p l i c a b i l i t y of the Monkman-Grant relationship to a wide range of materials strongly suggests that in most cases, fracture *While not specifically intended to describe creep cavity nucleation, these models are cited as they attempt to formally describe deformation-induced cavity formation.
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is controlled by the nucleation of grain boundary creep cavities rather than by their growth. Experiments which report high stress exponents and high activation energies for creep fracture, similar to those for ~ss in Equation 2, are often erroneously interpreted to imply a fracture process controlled By power law cavity growth (see the following sections). These results are more correctly interpreted to mean that cavity nucleation is controlling creep rupture reflecting the deformation dependence of the nucleation process. Cavity Growth Many experiments have been performed which have attempted to determine the mechanisms involved in creep cavity growth. The results although often contradictory, indicate that two distinct processes may be involved. Greenwood et al. (8) found that cavities appeared along grain boundaries transverse to the tensile axis in copper, brass and magnesium. This suggested that cavity growth can occur by the absorption of stress induced excess vacancies. Hull and Rimmer (16), showed in copper that only tensile stresses resulted in void growth and that superpositon of a hydrostatic stress upon an applied tensile stress eliminated cavity growth, a result which could be explained in terms of a vacancy condensation mechanism. Alternatively, certain results have indicated that deformation is required in order to produce cavity growth. Davis and Dutton (17) suggested in their work on copper-aluminum that plastic deformation via grain boundary sliding controlled cavity growth. Chen and Machlin (6) found that under appropriate conditions, cavities in bi-crystals form only when a shear stress rather than a normal stress acts on a grain boundary. The reverse strain experiments of Davis and Williams (18) and Gittins (13) also indicated the importance of deformation rather than stress state on cavity growth. In each case i t was the creep behavior of the material, i t s plastic response to an applied stress, which governed the rate of cavity growth and ultimately fracture. Diffusion Controlled Models
tit'
A cavity growth model involving vacancy condensation is fundamentally a diffusion controlled process. The kinetics of cavity growth and therefore of fracture (fracture occurring as the result of cavity linkage) are limited by the diffusive path by which vacancies are transported to the cavity. Figure 2 shows schematically the diffusional paths vacancies may iiio follow. Because cavities l i e on grain boundaries, diffusion is assumed to occur along the boundary since Fig. 2. Vacancy diffusion paths the flux of l a t t i c e vacancies at creep temperatures is leading to cavity growth. usually orders of magnitude lower. The driving force for vacancy transport through the boundary are the gradients in the local tractions established during creep. Thus cavity growth occurs by stress driven vacancy diffusion to the cavity-grain boundary junction. Vacancies must then diffuse along the internal cavity surfaces to a point of minimum chemical potential. The driving force within the cavity then is governed by the varying curvature of the internal surface. In order for the cavity shape to remain invariant, surface transport along the internal surface must be fast compared with grain boundary transport otherwise the cavity profile w i l l evolve with time into an elongated crack-like defect. Early models of cavity growth assumed that grain boundary cavities were spherical (16,19) and remained so. By accounting for the difference in surface free energy and grain boundary energy other workers have constructed models for growth based on lenticular cavity profiles (20). Here too, the cavity shapes were assumed to remain constant. In these cases, where the vacancy flux across the internal cavity surface is ignored as a possible rate l i m i t i n g step for growth, the models predict cavity growth rates that have a linear stress dependence (n=1) and activation energies equal to that for grain boundary diffusion (Qgb)- The creep rupture time is then related to these same parameters as:
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t r l T ~ o-n
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(n:l)
and t r Io ~exp (Qgb/RT) Experiments (21,22) have shown and theory (23) predicts that grain boundary cavities change shape as they grow. Thus the assumption that cavities maintain an equilibrium shape during stress driven growth may not always be correct. Stated another way, the effect of surface diffusion on the redistribution of material along the internal cavity surface and its subsequent effect on cavity growth kinetics must be considered. Chuangand Rice (24) introduced a model which accounted for the effect of surface diffusion by coupling the grain boundary flux at the cavity-grain boundary triple point to the surface flux within the cavity at the triple point by requiring continuity of chemical potential and atomic flux. The rate controlling process depends on the relative magnitudes of the surface (Ds) and grain boundary (Dab) diffusivities. Since the two processes operate in series, the slower one is rate co~trolling. For the case where Ds<
trlT ~ ~-N and t r l o ~ exp (Qc/RT) where N is very nearly equal to n, the stress exponent for creep and Qc =Qz, the activation for lattice self diffusion. This is a satisfying prediction in that i t predicts the stress and temperature sensitivity for the rupture time commonly found for engineering alloys but as will be shown, fails to predict the appropriate stress and temperature dependencewhen cavity growth is clearly separated from nucleation.
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Coupled D i f f u s i o n - P l a s t i c i t y Models Recently a number of models have described cavity growth by coupling diffusional cavity growth to creep deformation of the material surrounding the cavity. Beere and Speight {29) were the f i r s t to quantitatively address this problem by assuming that each cavity was surrounded by a non-deforming zone of material within which diffusional cavity growth (Hull-Rimmer) occurs. These r i g i d regions are then imbedded in a power-law creep controlled matrix. Edward and Ashby (30) have proposed a similar model (with different boundary conditions from those of (29)) and their results y i e l d e x p l i c i t equations for t r and mr governed by either power-law creep or by a coupled diffusion-creep process. Because of the simplified constitutive laws used to describe creep deformation and diffusive growth, these treatments only approximate the physical processes occurring. Needleman and Rice (31) used a numerical technique to allow for creep deformation to occur in the material immediately surrounding a cavity. In this model surface diffusion is considered to be s u f f i c i e n t l y rapid to maintain a spherical quasi-equilibrium shape. The material diffused from the internal cavity surface is accommodated by creep deformation of the matrix adjoining the cavity. This shortens the effective diffusion path length and results in a faster rate of cavity growth (decreased t r ) than would occur by either diffusion or creep alone. In a similar manner Chen and Argon (32) found that cavity growth rates were sensitive to the ratio of D~/Omb. Thus when surface diffusion is the controlling parameter, cavities grow in non-equilibridm crack like shapes and growth rates are accelerated. The central result of these models is that cavity growth is controlled by diffusional processes when they are close together and by creep deformation when they are far apart. Thus the stress and temperature dependences of the rupture time may vary and are functions of the i n i t i a l cavity morphology. Comparison of Experiment and Theory A great many techniques have been used to study the kinetics of cavity growth. An important question to ask then is, which are the most appropriate experiments to perform in order to unambiguously determine the processes responsible for cavity growth? From the previous discussion of diffusion and creep controlled cavity growth models, i t is clear that measurements of the stress and temperature dependence of cavity growth or time to rupture may be used to infer the mechanisms responsible. To this end, measurements of density changes during creep deformation have been used to deduce the rate of cavity growth. However, this technique cannot differentiate between nucleation and growth processes (also, in complex alloys, strain and temperature induced precipitation may account for decreased density). Other experiments have been conducted on materials containing r i g i d , grain boundary particles. Cavity nucleation in these studies is assumed to occur by the separation of the particle matrix interface early in primary creep and thus over the major portion of the creep l i f e of the material only cavity growth is occurring. This assumption in generally not valid as studies have shown that in creep, nucleation occurs continuously (13, 14). Thus, once again, cavity growth kinetics are obscured by nucleation processes. There remain two alternatives: to measure the growth rate of grain boundary cavities after nucleation by direct microscopical observation or to determine the fracture kinetics of specimens in which a pre-existing and well defined cavity structure exists prior to the onset of creep deformation. In the l a t t e r case, there must be a one-to-one correspondence between the cavity spacing prior to the onset of creep deformation and the dimple spacing on the fracture surface. While a number of studies have q u a l i t a t i v e l y examined cavity growth via optical and electron microscopy, only a few studies have measured the stress dependence for cavity growth during creep by direct observation. Cane and Greenwood (22) have reported results for cavity growth in m-iron at 700°C. The cavity structure studied was subjected to continuous nucleation and they correctly point out that the measure of mean cavity size was therefore inappropriate. Their measure of maximum cavity size was more reliable as i t could be assumed that the largest cavities present nucleated at zero time. Their results, shown in Figure 3 indicate that the areal growth rate of cavities was proportional to ~+3. I f a fracture c r i t e r i o n is based on a c r i t i c a l grain boundary area fraction consumed in cavity growth, t r has a stress dependence given by Eq. 4a. Mancusoand Li (33) measured the cavity growth rate for pre-
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i
10"11
,
r
10"14~
o :c~-Fe 7O0°C Ni 500°C
• :
Ni 550°C
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I
C A V I T Y GROWTH IN c( - F e A N D Ni
• :
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existing CH4 bubbles in nickel. Their results, also shown in Figure 3, y i e l d a similar stress dependence. In both cases, the stress dependence is indicative of a surface diffusion controlled cavity growth process.
o
The kinetics of rupture have also been studied by implanting grain boundary cavities 1015~ ~ 1 0 -12 in several different materials prior to creep deO ~ g formation. Goods and Nix (3,34) applied this technique by implanting H20(v) bubbles in s i l v e r . In order to determine the stress dependence u and activation energy for t r , tests were conducted over a wide range of stress and temperature. Their 1013 I I I 10 16 i i i i ~ i results are shown in Figure 4 (note that the rupture 100 10 time has been temperature and diffusion compensated) STRESS, MPa along with those of Nieh and Nix (4) who reported similar measurements for OFHC copper. The data Fig. 3. Stress dependence of creep correlate with the surfac~ diffusion term DS, and cavity growth in ~-Fe (22) and i t can be seen that tr~ 0-5. This strongly inNi (33). , dicates that cavity growth and the rupture time kinetics , i ' I i were rate limited by mass transport along the internal "~rV5 APPLIED STRESS FOR Ag A N D cavity surface (Eqs. 4). Comparing the data to the Cu WITH H20(v ) BUBBLES 10"4 10 2 \ Chuang-Rice surface diffusion controlled model (24) - Ag ~ PREDICTIONS yields close agreement. We do not emphasize this point, % - ~ [7 . . . . Cu~' (REF. 24) 10 5 10 3 however,as measured values of Qs and Ds n vary greatly. \n, 9 10 "4 \ \ ,~ ,j 106 There are, nevertheless, two key observations. F i r s t , the non-linear stress dependence indicates that a Hull• '% _ Rimmer, grain boundary diffusion controlled process can 10 5 be discounted. Secondly, the stress dependence and activation energy for t r are well below those for ~ss 10.8 E 10-6 ~ m" \ "% • \\c, A indicating that cavity growth in these materials is-hot <~ a deformation controlled process. Additional work by o• ~ '~ 10.9 10"7 Nieh and Nix (35) further showed that creep deformation Ag Cu ~ k = ~.% cannot account for cavity growth. By strengthening s i l 10"8 550Oc • {3 - ~'~e - 1 0 10 ver with a fine dispersion of MgO particles they showed (Figure 5a) that the presence of the dispersoid resulted 10"9 in a dramatic decrease in ~ss relative to that for pure ~o~. o I",.. % . " 3o0°c • , n-01 ~ "<.--1012 s i l v e r . The stress exponent remained high having a value 10-10 approximately equal to 9. I f cavity growth was 2o0oc , [ ~. * \ controlled by bulk p l a s t i c i t y , the order of magnitude i i i i I i i L iA decrease in ~ss would have resulted in an increased 10 100 rupture l i f e . Figure 5b shows however that when H20(v) STRESS MPa Fig. 4. Stress dependence of temperbubbles were implanted along the grain boundaries of the strengthened alloy, t r was identical to that measured ature compensated rupture time, tr= for pure s i l v e r with a similar distribution of grain trD~/T. Axisymmetric predictions a~e-based on model for surface boundary bubbles. Thus cavity growth was found to be a process which occurs independently from creep deformadiffusion controlled cavity tion. The Monkman-Grant relationship did not describe growth (23,24). cavity growth and the stress and temperature dependence of t r strongly suggested that a diffusional process controlled the kinetics of fracture. c)
~
We realize that the findings presented in these last two paragraphs contrast the results reported by many other investigators. They are though among the few results which meet the important c r i t e r i a that cavity nucleation be clearly separated from cavity growth. In almost every instance when creep p l a s t i c i t y is reported as the controlling factor in cavity growth, we find that nucleation has been either ignored or erroneously discounted. Conclusions A l a r g e body of experimental observation provides a q u a l i t a t i v e understanding of the importance of c a v i t y n u c l e a t i o n in the creep f r a c t u r e process. Inhomogeneous
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deformation accommodated by bulk p l a s t i c i t y is required to nucleate grain boundary cavities. This, along with the interrelationship between ~ss and t r , as given by the Monkman-Grant relationship, strongly suggests that the creep l i f e of a material is limited by deformation induced cavity nucleation. Since i t is the formation of cavities which appears to control creep fracture i t would seem that future work should be directed towards a more quantitative understanding of those processes responsible for creep cavity nucleation.
5a) STEADY STATE STRAIN RATE vs. APPLIED STRESS Ag ~ MgO
} j
Aq om A
_z ~10 6 oo ,~10 7
!
I
/
! !
<10 8
!
10 9
300°C ....... IH
i
20 101
, 5bP
,
100 20 100 APPLIED STRESS, MPa ,
,
,
, , r, I
CREEP RUPTURE
,
20 r
100 ]
,
l
,
,
L I F E vs S T R E S S F O R
Ag AND Ag + MqO WITH H20(v I 8UBBLES 106
~ 104 I A(J
300°C 10 ~
L
MqO
• ~
,
, J,~l
,
10
.
.
.
.
.
29
,
10o
Creep cavity growth has been assumed to occur by either stress driven diffusion or creep deformation. Of these two processes, the one which is actually responsible for cavity growth can be ascertained by determining the stress and temperature dependence of either the cavity growth rate or rupture time. This can only be done when cavity nucleation is unambiguously separated from growth processes. We have shown that cavity growth predictions based on deformation controlled processes w i l l exhibit higher stress and temperature s e n s i t i v i t i e s than those predictions based solely on diffusive processes. While there exist a great many experiments which purport to measure cavity growth and fracture kinetics, we have reviewed only those which we feel have clearly avoided the obscuring effects of nucleation. The results of these experiments are clear:
APPLIED STRESS, MPa
Fig. 5. Stress dependence of ~ss and t r for Ag and Ag-Mg0, 5a. Dispersion strengthening r e s u l t s in dramatic decrease in Ess. 5b. For Ag and Ag-Mg0 with H20(y ) bubbles t r 3 i s nearly proportional to c~ .
i.
Studies of stress induced cavity growth report the continuous shape evolution of grain boundary cavities, supporting a surface diffusion controlled process.
2.
Whenever the nucleation step is circumvented, the Monkman-Grant relationship f a i l s to describe the stress and temperature dependence of creep fracture.
3.
The stress dependence for cavity growth or t r is less than that predicted for a deformation controlled process but greater than that predicted for a process controlled by grain boundary vacancy diffusion (Hull-Rimmer). This intermediate stress dependence suggests that the redistribution of material along the internal cavity surface ( i . e . , surface diffusion) is the rate l i m i t i n g process for cavity growth.
4.
In each case, where i t has been measured, we find that temperature dependence for t r is clearly much less than that for Css and again suggests a diffusion controlled process.
The study of creep cavity growth kinetics using materials with pre-existing grain boundary cavities is a powerful technique for uncoupling nucleation and growth processes. Additional experimental work is necessary to develop material systems in which both the cavity (or inert bubble) size and spacing can be widely varied. In this way, the v a l i d i t y of the more sophisticated coupled diffusion-plasticity models (which may be the most accurate descriptions of cavity growth) can be assessed. Continued study of the shape evolution of cavities during creep is needed to more clearly define the role of surface diffusion in cavity growth and creep fracture.
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Acknowledgments This work supported by U. S. Department of Energy, DOE, under Contract Number DE-ACO4-76DPO0789 (SHG) and by Lockheed Missles and Space Company, Inc. under Research Program Number IR 61-9313 (TGN). References 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
D. K. Matlock and W. D. Nix, J. Nucl. Mater. 56, 145 (1975). K. Erlich, H. Bohm, C. Wassilew, ASTM-STP-484, p. 495 (1970). S. H. Goods and W. D. Nix, Acta Metall. 26, 739 (1977). T. G. Nieh and W. D. Nix, Acta Metall. 2 ~ 557 (1980). B. F. Dyson, M. S. Loveday and M. S. Rodgers, Proc. R. Soc. (A)349, 245 (1976). C. W. Chen and K. S. Machlin, Trans. AIME 209, 929 (1957). R. T. Ratcliffe and G. W. Greenwood, P h i l . ~ g . 12, 59 (1965). J. M. Greenwood, D. R. Miller and J. W. Suiter, A~ta Metall. 2, 250 (1954). R. W. B a l l u f f i , and L. L. Seigle, Acta Metall. 5, 449 (1957). C. Zener, Fracturing of Metals, 17, (ASM, Metals Park, OH) (1948). R. W. Davies and B. Wilshire, St~ctural Process In Creep #70, P. 34 (1965). S. H. Goods and L. M. Brown, Acta Metall. 27, 1 (1979). A. Gittins, Metall. Sci. 1, 214 (1967). R. C. Boettner, and W. D. Robertson, Trans. AIME 221, 613 (1961). A. S. Argon, I. W. Chen and C. W. Lau, Three Dimensional Constitutive Relations and Ductile Fracture, (North Holland Pub. Co.), p. 23 (1981). D. Hull and D. E. Rimmer, Phil. Mag. 4~ 673 (1959). P. W. Davis and R. Dutton, Acta Metall. 14, 1138 (1966). R. W. Davis and K. R. Williams, Metall. ~ i . J. 1, 48 (1968). M. V. Speight and J. E. Harris, Metall. Sci. J. 1, 83 (1967). R. Raj and M. F. Ashby, Acta Metall. 23, 653 (1975). F. Dobes, Scripta Metall. L 1231 (19~). B. S. Cane and G. W. Greenwood, Metal. Sci. 9_, 55 (1975). T. Z. Chuang, Ph.D. Thesis, Brown Univ. (1974). T. Z. Chuang and J. R. Rice, Acta Metall. 21, 1625 (1973). G. M. Pharr and W. D. Nix, Acta Metall. 27,~615 (1979). H. E. Evans, Phil. Mag. 23, 195 (1971). - J. W. Hancock, Metal Sci-/-lO, 319 (1976). W. D. Nix, D. K. Matlock aN R. J. Dimelfi, Acta Metall. 2__55,495 (1977). W. Beere and M. V. Speight, Metal Sci. 12, 172 (1978). G. H. Edward and M. F. Ashby, Acta Metall. 27, 1505 (1979). A. Needleman andJ. R. Rice, Acta Metall. 2~-~,1315 (1980). I. W. Chen and A. S. Argon, Acta Metall. 2~, 1768 (1981). J. F. Mancuso and C. Y. Li, Metall. Trans--(A) 10, 1719 (1979). S. H. Goods and W. D. Nix, Acta Metall. 24, 1041--(1976). T. G. Nieh and W. D. Nix, Acta Metall. 2S1097 (1979).
No.
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Vol. 17, pp. 23-30, 1983 Printed in the U.S.A.
Pergamon
Press Ltd.
VIEWPOINT
SET No. S
MECHANISMS OF INTERGRANULARCAVITY NUCLEATIONAND GROWTHDURINGCREEP S. H. Goods Sandia National Laboratories Livermore, CA 94550 and T. G. Nieh Lockheed Palo Alto Research Laboratory Palo Alto, CA 94304 (Received
October
14, 1982)
Introduction I t is generally recognized that creep-rupture of metals and alloys is caused by the formation, growth and coalescence of grain boundary cavities. A large body of l i t e r a t u r e suggests that the nucleation of cavities rather than their growth controls the fracture of materials over a wide range of temperatures, especially at temperatures associated with creep deformation (1-4). Thus when cavities such as voids or gas bubbles exist in materials prior to the beginning of deformation a severe loss in creep d u c t i l i t y often results. Because cavities are such a potent source of embrittlement, a great many studies have attempted to i l l u s t r a t e both experimentally and theoretically the role of cavity growth in the creep fracture process. However, understanding the effect of grain boundary cavities on creep fracture is complicated by the i n a b i l i t y to accurately characterize the cavity structure prior to the onset and during creep deformation. For materials in which cavitation, the simultaneous nucleation and growth of cavities, occurs continuously during creep, the kinetics of cavity growth are often obscured by the kinetics of cavity nucleation. In order to understand the importance of e|t~mr cavity, nucleation or growth in creep fracture, the nucleation event must be clearly and unmbiguously separated from cavity gro~rth processes. In this a r t i c l e the c r i t i c a l factors which must be considered in the study of creep cavitation are discussed. We f i r s t examine a number of key observations regarding the processes responsible for cavity nucleation, then discuss the importance of nucleation in creep fracture and conclude with an analysis of important theoretical and experimental results found in the recent l i t e r a t u r e which i l l u s t r a t e the micromechanics of cavity growth. Cavity Nucleation Because of the d i f f i c u l t y in resolving small voids or determining the location of incipient cavity formation, there exist many contradictory findings regarding the location and characteristics of l i k e l y nucleation sites. I t is clear that while voids may nucleate within grains at second phase particles, the intergranular nature of creep fracture indicates that the most damaging cavities are those which exist at grain boundaries and for this reason our discussion is limited to grain boundary cavity formation. The role of grain boundaries in the development of cavities in deforming materials has been investigated extensively (5-8). From early observations of the creep of copper-based and magnesium alloys i t was proposed that l a t t i c e vacancies condensed under the action of an applied tensile stress to form voids on grain boundaries oriented normal to the tensile axis. While excess vacancies can be generated by dislocation cutting and non-conservative motion, i t is hard to visualize how they can agglomerate on grain boundaries (which act as sinks for point defects). I t has been shown theoretically that in the presence of high grain boundary stresses, a supersaturation of vacancies may produce voids but the vacancy concentrations (or local stress) required are
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higher than might r e a l i s t i c a l l y be expected (9). Therefore the idea that stress driven vacancy condensation alone can account for cavity nucleation seems unreasonable. The contribution of the deformation process in the formation of grain boundary cavities has been well established. Nucleation invariably occurs in regions where deformation is non-uniform. I t is precisely for this reason that cavity nucleation and creep fracture are localized in grain boundaries. Some of the basic processes involved in creep cavity nucleation are i l l u s t r a t e d in Figure 1. The intersection of a slip band and a grain boundary is a l i k e l y site for cavity nucleation. Dyson et al. (5) using TEM have shown that this occurs in Nimonic 80A. Nucleation occurs readily under conditions which favor a change in deformation mode from bulk deformation to grain boundary sliding. Triple points (10) or ledges (11) induce s u f f i c i e n t stress concentrations to nucleate cavities in grain boundaries by blocking sliding. The Fig. i . Intergranular cavity presence of non-deforming particle in the boundary w i l l nucleation processes. have much the same effect. Thus, while the localized deformation must be accommodated by bulk p l a s t i c i t y , in each case decohesion of an interface is caused by tensile stresses that develop where plastic flow is inhomogeneous. In qualitative terms, all of the above processes have been thoroughly discussed in the l i t e r a t u r e . However, there exist only a few theoretical treatments which can quantitatively describe the onset of cavity nucleation (12)*. Most of these models require that nucleation occurs as a stress or strain localization at particles and that the particles are located well away from grain boundaries. They result in predictions of threshold strains below which nucleation does not occur. In predicting the creep behavior of metals these models are therefore deficient in several important respects. F i r s t , they ignore the morphology of creep fracture ( i . e . that nucleation and fracture invariably occur at grain boundaries as described above) and require that nucleation occur only at r i g i d particles. Further, while there is some evidence that there may be a c r i t i c a l nucleation strain for engineering and particle strengthened alloys, for materials subject to creep deformation the evidence is to the contrary (13,14). More recently, Argon et. al. (15) have addressed many of the specific micromechanical aspects of creep cavity nucleation. While theirs is the most accurate description to date, in quantitative terms, the influence of nucleation on the creep behavior of materials remains poorly defined. Even though theoretical descriptions of creep cavity formation are inadequate, the role of nucleation in determining the creep l i f e and fracture characteristics of materials can s t i l l be assessed. For many metals and engineering alloys, the product of the steady state strain rate, ~s~and the time to rupture, t r, is found to be a constant. This is the familiar Monkman-Grant relationship and is expressed as: Ess t r : CMG
[l]
Further, in the power law regime the creep rate can be related to the applied stress, o, temperature, T, by the expression: Ess = A gnexp (-Qc/RT)
and
[2]
where A is a structure dependent constant, n is the stress exponent t y p i c a l l y equal to 5 (above the power law have regime n w i l l vary and have a higher value) and Qc is the activation energy, similar to that for l a t t i c e self diffusion. Equation 1 indicates that the processes responsible for creep-rupture are controlled by creep deformation. Since the inhomogeneous plastic flow responsible for cavity nucleation must be accommodated by the bulk deformation of the creeping material, the a p p l i c a b i l i t y of the Monkman-Grant relationship to a wide range of materials strongly suggests that in most cases, fracture *While not specifically intended to describe creep cavity nucleation, these models are cited as they attempt to formally describe deformation-induced cavity formation.
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CAVITY NUCLEATION
i
25
is controlled by the nucleation of grain boundary creep cavities rather than by their growth. Experiments which report high stress exponents and high activation energies for creep fracture, similar to those for ~ss in Equation 2, are often erroneously interpreted to imply a fracture process controlled By power law cavity growth (see the following sections). These results are more correctly interpreted to mean that cavity nucleation is controlling creep rupture reflecting the deformation dependence of the nucleation process. Cavity Growth Many experiments have been performed which have attempted to determine the mechanisms involved in creep cavity growth. The results although often contradictory, indicate that two distinct processes may be involved. Greenwood et al. (8) found that cavities appeared along grain boundaries transverse to the tensile axis in copper, brass and magnesium. This suggested that cavity growth can occur by the absorption of stress induced excess vacancies. Hull and Rimmer (16), showed in copper that only tensile stresses resulted in void growth and that superpositon of a hydrostatic stress upon an applied tensile stress eliminated cavity growth, a result which could be explained in terms of a vacancy condensation mechanism. Alternatively, certain results have indicated that deformation is required in order to produce cavity growth. Davis and Dutton (17) suggested in their work on copper-aluminum that plastic deformation via grain boundary sliding controlled cavity growth. Chen and Machlin (6) found that under appropriate conditions, cavities in bi-crystals form only when a shear stress rather than a normal stress acts on a grain boundary. The reverse strain experiments of Davis and Williams (18) and Gittins (13) also indicated the importance of deformation rather than stress state on cavity growth. In each case i t was the creep behavior of the material, i t s plastic response to an applied stress, which governed the rate of cavity growth and ultimately fracture. Diffusion Controlled Models
tit'
A cavity growth model involving vacancy condensation is fundamentally a diffusion controlled process. The kinetics of cavity growth and therefore of fracture (fracture occurring as the result of cavity linkage) are limited by the diffusive path by which vacancies are transported to the cavity. Figure 2 shows schematically the diffusional paths vacancies may iiio follow. Because cavities l i e on grain boundaries, diffusion is assumed to occur along the boundary since Fig. 2. Vacancy diffusion paths the flux of l a t t i c e vacancies at creep temperatures is leading to cavity growth. usually orders of magnitude lower. The driving force for vacancy transport through the boundary are the gradients in the local tractions established during creep. Thus cavity growth occurs by stress driven vacancy diffusion to the cavity-grain boundary junction. Vacancies must then diffuse along the internal cavity surfaces to a point of minimum chemical potential. The driving force within the cavity then is governed by the varying curvature of the internal surface. In order for the cavity shape to remain invariant, surface transport along the internal surface must be fast compared with grain boundary transport otherwise the cavity profile w i l l evolve with time into an elongated crack-like defect. Early models of cavity growth assumed that grain boundary cavities were spherical (16,19) and remained so. By accounting for the difference in surface free energy and grain boundary energy other workers have constructed models for growth based on lenticular cavity profiles (20). Here too, the cavity shapes were assumed to remain constant. In these cases, where the vacancy flux across the internal cavity surface is ignored as a possible rate l i m i t i n g step for growth, the models predict cavity growth rates that have a linear stress dependence (n=1) and activation energies equal to that for grain boundary diffusion (Qgb)- The creep rupture time is then related to these same parameters as:
26
CAVITY NUCLEATION
t r l T ~ o-n
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(n:l)
and t r Io ~exp (Qgb/RT) Experiments (21,22) have shown and theory (23) predicts that grain boundary cavities change shape as they grow. Thus the assumption that cavities maintain an equilibrium shape during stress driven growth may not always be correct. Stated another way, the effect of surface diffusion on the redistribution of material along the internal cavity surface and its subsequent effect on cavity growth kinetics must be considered. Chuangand Rice (24) introduced a model which accounted for the effect of surface diffusion by coupling the grain boundary flux at the cavity-grain boundary triple point to the surface flux within the cavity at the triple point by requiring continuity of chemical potential and atomic flux. The rate controlling process depends on the relative magnitudes of the surface (Ds) and grain boundary (Dab) diffusivities. Since the two processes operate in series, the slower one is rate co~trolling. For the case where Ds<
trlT ~ ~-N and t r l o ~ exp (Qc/RT) where N is very nearly equal to n, the stress exponent for creep and Qc =Qz, the activation for lattice self diffusion. This is a satisfying prediction in that i t predicts the stress and temperature sensitivity for the rupture time commonly found for engineering alloys but as will be shown, fails to predict the appropriate stress and temperature dependencewhen cavity growth is clearly separated from nucleation.
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CAVITY N U C L E A T I O N
27
Coupled D i f f u s i o n - P l a s t i c i t y Models Recently a number of models have described cavity growth by coupling diffusional cavity growth to creep deformation of the material surrounding the cavity. Beere and Speight {29) were the f i r s t to quantitatively address this problem by assuming that each cavity was surrounded by a non-deforming zone of material within which diffusional cavity growth (Hull-Rimmer) occurs. These r i g i d regions are then imbedded in a power-law creep controlled matrix. Edward and Ashby (30) have proposed a similar model (with different boundary conditions from those of (29)) and their results y i e l d e x p l i c i t equations for t r and mr governed by either power-law creep or by a coupled diffusion-creep process. Because of the simplified constitutive laws used to describe creep deformation and diffusive growth, these treatments only approximate the physical processes occurring. Needleman and Rice (31) used a numerical technique to allow for creep deformation to occur in the material immediately surrounding a cavity. In this model surface diffusion is considered to be s u f f i c i e n t l y rapid to maintain a spherical quasi-equilibrium shape. The material diffused from the internal cavity surface is accommodated by creep deformation of the matrix adjoining the cavity. This shortens the effective diffusion path length and results in a faster rate of cavity growth (decreased t r ) than would occur by either diffusion or creep alone. In a similar manner Chen and Argon (32) found that cavity growth rates were sensitive to the ratio of D~/Omb. Thus when surface diffusion is the controlling parameter, cavities grow in non-equilibridm crack like shapes and growth rates are accelerated. The central result of these models is that cavity growth is controlled by diffusional processes when they are close together and by creep deformation when they are far apart. Thus the stress and temperature dependences of the rupture time may vary and are functions of the i n i t i a l cavity morphology. Comparison of Experiment and Theory A great many techniques have been used to study the kinetics of cavity growth. An important question to ask then is, which are the most appropriate experiments to perform in order to unambiguously determine the processes responsible for cavity growth? From the previous discussion of diffusion and creep controlled cavity growth models, i t is clear that measurements of the stress and temperature dependence of cavity growth or time to rupture may be used to infer the mechanisms responsible. To this end, measurements of density changes during creep deformation have been used to deduce the rate of cavity growth. However, this technique cannot differentiate between nucleation and growth processes (also, in complex alloys, strain and temperature induced precipitation may account for decreased density). Other experiments have been conducted on materials containing r i g i d , grain boundary particles. Cavity nucleation in these studies is assumed to occur by the separation of the particle matrix interface early in primary creep and thus over the major portion of the creep l i f e of the material only cavity growth is occurring. This assumption in generally not valid as studies have shown that in creep, nucleation occurs continuously (13, 14). Thus, once again, cavity growth kinetics are obscured by nucleation processes. There remain two alternatives: to measure the growth rate of grain boundary cavities after nucleation by direct microscopical observation or to determine the fracture kinetics of specimens in which a pre-existing and well defined cavity structure exists prior to the onset of creep deformation. In the l a t t e r case, there must be a one-to-one correspondence between the cavity spacing prior to the onset of creep deformation and the dimple spacing on the fracture surface. While a number of studies have q u a l i t a t i v e l y examined cavity growth via optical and electron microscopy, only a few studies have measured the stress dependence for cavity growth during creep by direct observation. Cane and Greenwood (22) have reported results for cavity growth in m-iron at 700°C. The cavity structure studied was subjected to continuous nucleation and they correctly point out that the measure of mean cavity size was therefore inappropriate. Their measure of maximum cavity size was more reliable as i t could be assumed that the largest cavities present nucleated at zero time. Their results, shown in Figure 3 indicate that the areal growth rate of cavities was proportional to ~+3. I f a fracture c r i t e r i o n is based on a c r i t i c a l grain boundary area fraction consumed in cavity growth, t r has a stress dependence given by Eq. 4a. Mancusoand Li (33) measured the cavity growth rate for pre-
28
CAVITY NUCLEATION
i
10"11
,
r
10"14~
o :c~-Fe 7O0°C Ni 500°C
• :
Ni 550°C
17, No.
1
I
C A V I T Y GROWTH IN c( - F e A N D Ni
• :
Vol.
o
existing CH4 bubbles in nickel. Their results, also shown in Figure 3, y i e l d a similar stress dependence. In both cases, the stress dependence is indicative of a surface diffusion controlled cavity growth process.
o
The kinetics of rupture have also been studied by implanting grain boundary cavities 1015~ ~ 1 0 -12 in several different materials prior to creep deO ~ g formation. Goods and Nix (3,34) applied this technique by implanting H20(v) bubbles in s i l v e r . In order to determine the stress dependence u and activation energy for t r , tests were conducted over a wide range of stress and temperature. Their 1013 I I I 10 16 i i i i ~ i results are shown in Figure 4 (note that the rupture 100 10 time has been temperature and diffusion compensated) STRESS, MPa along with those of Nieh and Nix (4) who reported similar measurements for OFHC copper. The data Fig. 3. Stress dependence of creep correlate with the surfac~ diffusion term DS, and cavity growth in ~-Fe (22) and i t can be seen that tr~ 0-5. This strongly inNi (33). , dicates that cavity growth and the rupture time kinetics , i ' I i were rate limited by mass transport along the internal "~rV5 APPLIED STRESS FOR Ag A N D cavity surface (Eqs. 4). Comparing the data to the Cu WITH H20(v ) BUBBLES 10"4 10 2 \ Chuang-Rice surface diffusion controlled model (24) - Ag ~ PREDICTIONS yields close agreement. We do not emphasize this point, % - ~ [7 . . . . Cu~' (REF. 24) 10 5 10 3 however,as measured values of Qs and Ds n vary greatly. \n, 9 10 "4 \ \ ,~ ,j 106 There are, nevertheless, two key observations. F i r s t , the non-linear stress dependence indicates that a Hull• '% _ Rimmer, grain boundary diffusion controlled process can 10 5 be discounted. Secondly, the stress dependence and activation energy for t r are well below those for ~ss 10.8 E 10-6 ~ m" \ "% • \\c, A indicating that cavity growth in these materials is-hot <~ a deformation controlled process. Additional work by o• ~ '~ 10.9 10"7 Nieh and Nix (35) further showed that creep deformation Ag Cu ~ k = ~.% cannot account for cavity growth. By strengthening s i l 10"8 550Oc • {3 - ~'~e - 1 0 10 ver with a fine dispersion of MgO particles they showed (Figure 5a) that the presence of the dispersoid resulted 10"9 in a dramatic decrease in ~ss relative to that for pure ~o~. o I",.. % . " 3o0°c • , n-01 ~ "<.--1012 s i l v e r . The stress exponent remained high having a value 10-10 approximately equal to 9. I f cavity growth was 2o0oc , [ ~. * \ controlled by bulk p l a s t i c i t y , the order of magnitude i i i i I i i L iA decrease in ~ss would have resulted in an increased 10 100 rupture l i f e . Figure 5b shows however that when H20(v) STRESS MPa Fig. 4. Stress dependence of temperbubbles were implanted along the grain boundaries of the strengthened alloy, t r was identical to that measured ature compensated rupture time, tr= for pure s i l v e r with a similar distribution of grain trD~/T. Axisymmetric predictions a~e-based on model for surface boundary bubbles. Thus cavity growth was found to be a process which occurs independently from creep deformadiffusion controlled cavity tion. The Monkman-Grant relationship did not describe growth (23,24). cavity growth and the stress and temperature dependence of t r strongly suggested that a diffusional process controlled the kinetics of fracture. c)
~
We realize that the findings presented in these last two paragraphs contrast the results reported by many other investigators. They are though among the few results which meet the important c r i t e r i a that cavity nucleation be clearly separated from cavity growth. In almost every instance when creep p l a s t i c i t y is reported as the controlling factor in cavity growth, we find that nucleation has been either ignored or erroneously discounted. Conclusions A l a r g e body of experimental observation provides a q u a l i t a t i v e understanding of the importance of c a v i t y n u c l e a t i o n in the creep f r a c t u r e process. Inhomogeneous
Vol.
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10 4 L~ ~10 5 <
1
CAVITY N U C L E A T I O N
deformation accommodated by bulk p l a s t i c i t y is required to nucleate grain boundary cavities. This, along with the interrelationship between ~ss and t r , as given by the Monkman-Grant relationship, strongly suggests that the creep l i f e of a material is limited by deformation induced cavity nucleation. Since i t is the formation of cavities which appears to control creep fracture i t would seem that future work should be directed towards a more quantitative understanding of those processes responsible for creep cavity nucleation.
5a) STEADY STATE STRAIN RATE vs. APPLIED STRESS Ag ~ MgO
} j
Aq om A
_z ~10 6 oo ,~10 7
!
I
/
! !
<10 8
!
10 9
300°C ....... IH
i
20 101
, 5bP
,
100 20 100 APPLIED STRESS, MPa ,
,
,
, , r, I
CREEP RUPTURE
,
20 r
100 ]
,
l
,
,
L I F E vs S T R E S S F O R
Ag AND Ag + MqO WITH H20(v I 8UBBLES 106
~ 104 I A(J
300°C 10 ~
L
MqO
• ~
,
, J,~l
,
10
.
.
.
.
.
29
,
10o
Creep cavity growth has been assumed to occur by either stress driven diffusion or creep deformation. Of these two processes, the one which is actually responsible for cavity growth can be ascertained by determining the stress and temperature dependence of either the cavity growth rate or rupture time. This can only be done when cavity nucleation is unambiguously separated from growth processes. We have shown that cavity growth predictions based on deformation controlled processes w i l l exhibit higher stress and temperature s e n s i t i v i t i e s than those predictions based solely on diffusive processes. While there exist a great many experiments which purport to measure cavity growth and fracture kinetics, we have reviewed only those which we feel have clearly avoided the obscuring effects of nucleation. The results of these experiments are clear:
APPLIED STRESS, MPa
Fig. 5. Stress dependence of ~ss and t r for Ag and Ag-Mg0, 5a. Dispersion strengthening r e s u l t s in dramatic decrease in Ess. 5b. For Ag and Ag-Mg0 with H20(y ) bubbles t r 3 i s nearly proportional to c~ .
i.
Studies of stress induced cavity growth report the continuous shape evolution of grain boundary cavities, supporting a surface diffusion controlled process.
2.
Whenever the nucleation step is circumvented, the Monkman-Grant relationship f a i l s to describe the stress and temperature dependence of creep fracture.
3.
The stress dependence for cavity growth or t r is less than that predicted for a deformation controlled process but greater than that predicted for a process controlled by grain boundary vacancy diffusion (Hull-Rimmer). This intermediate stress dependence suggests that the redistribution of material along the internal cavity surface ( i . e . , surface diffusion) is the rate l i m i t i n g process for cavity growth.
4.
In each case, where i t has been measured, we find that temperature dependence for t r is clearly much less than that for Css and again suggests a diffusion controlled process.
The study of creep cavity growth kinetics using materials with pre-existing grain boundary cavities is a powerful technique for uncoupling nucleation and growth processes. Additional experimental work is necessary to develop material systems in which both the cavity (or inert bubble) size and spacing can be widely varied. In this way, the v a l i d i t y of the more sophisticated coupled diffusion-plasticity models (which may be the most accurate descriptions of cavity growth) can be assessed. Continued study of the shape evolution of cavities during creep is needed to more clearly define the role of surface diffusion in cavity growth and creep fracture.
30
CAVITY NUCLEATION
Vol.
17,
Acknowledgments This work supported by U. S. Department of Energy, DOE, under Contract Number DE-ACO4-76DPO0789 (SHG) and by Lockheed Missles and Space Company, Inc. under Research Program Number IR 61-9313 (TGN). References 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
D. K. Matlock and W. D. Nix, J. Nucl. Mater. 56, 145 (1975). K. Erlich, H. Bohm, C. Wassilew, ASTM-STP-484, p. 495 (1970). S. H. Goods and W. D. Nix, Acta Metall. 26, 739 (1977). T. G. Nieh and W. D. Nix, Acta Metall. 2 ~ 557 (1980). B. F. Dyson, M. S. Loveday and M. S. Rodgers, Proc. R. Soc. (A)349, 245 (1976). C. W. Chen and K. S. Machlin, Trans. AIME 209, 929 (1957). R. T. Ratcliffe and G. W. Greenwood, P h i l . ~ g . 12, 59 (1965). J. M. Greenwood, D. R. Miller and J. W. Suiter, A~ta Metall. 2, 250 (1954). R. W. B a l l u f f i , and L. L. Seigle, Acta Metall. 5, 449 (1957). C. Zener, Fracturing of Metals, 17, (ASM, Metals Park, OH) (1948). R. W. Davies and B. Wilshire, St~ctural Process In Creep #70, P. 34 (1965). S. H. Goods and L. M. Brown, Acta Metall. 27, 1 (1979). A. Gittins, Metall. Sci. 1, 214 (1967). R. C. Boettner, and W. D. Robertson, Trans. AIME 221, 613 (1961). A. S. Argon, I. W. Chen and C. W. Lau, Three Dimensional Constitutive Relations and Ductile Fracture, (North Holland Pub. Co.), p. 23 (1981). D. Hull and D. E. Rimmer, Phil. Mag. 4~ 673 (1959). P. W. Davis and R. Dutton, Acta Metall. 14, 1138 (1966). R. W. Davis and K. R. Williams, Metall. ~ i . J. 1, 48 (1968). M. V. Speight and J. E. Harris, Metall. Sci. J. 1, 83 (1967). R. Raj and M. F. Ashby, Acta Metall. 23, 653 (1975). F. Dobes, Scripta Metall. L 1231 (19~). B. S. Cane and G. W. Greenwood, Metal. Sci. 9_, 55 (1975). T. Z. Chuang, Ph.D. Thesis, Brown Univ. (1974). T. Z. Chuang and J. R. Rice, Acta Metall. 21, 1625 (1973). G. M. Pharr and W. D. Nix, Acta Metall. 27,~615 (1979). H. E. Evans, Phil. Mag. 23, 195 (1971). - J. W. Hancock, Metal Sci-/-lO, 319 (1976). W. D. Nix, D. K. Matlock aN R. J. Dimelfi, Acta Metall. 2__55,495 (1977). W. Beere and M. V. Speight, Metal Sci. 12, 172 (1978). G. H. Edward and M. F. Ashby, Acta Metall. 27, 1505 (1979). A. Needleman andJ. R. Rice, Acta Metall. 2~-~,1315 (1980). I. W. Chen and A. S. Argon, Acta Metall. 2~, 1768 (1981). J. F. Mancuso and C. Y. Li, Metall. Trans--(A) 10, 1719 (1979). S. H. Goods and W. D. Nix, Acta Metall. 24, 1041--(1976). T. G. Nieh and W. D. Nix, Acta Metall. 2S1097 (1979).
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